Lecture 1 | Modern Physics: Special Relativity (Stanford)

Views:653279|Rating:4.86|View Time:1:49:24Minutes|Likes:2611|Dislikes:73
Lecture 1 of Leonard Susskind’s Modern Physics course concentrating on Special Relativity. Recorded April 14, 2008 at Stanford University.

This Stanford Continuing Studies course is the third of a six-quarter sequence of classes exploring the essential theoretical foundations of modern physics. The topics covered in this course focus on classical mechanics. Leonard Susskind is the Felix Bloch Professor of Physics at Stanford University.

Complete Playlist for the Course:

Stanford Continuing Studies:

About Leonard Susskind:

Stanford University Channel on YouTube:

this program is brought to you by Stanford University please visit us at stanford.edu this quarter we're going to learn about field theory classical field theory fields such as the electromagnetic field gravitational field other fields in nature which I won't name right now propagate which means they change according to rules which give them a wave-like character moving through space and one of the fundamental principles of field theory in fact more broadly nature in general is the principle of relativity the principle the special printless the the principle of special relativity in this particular case the principle of special relativity well let's just call it the principle of relativity goes way back there was not an invention of Einstein's I'm not absolutely sure when it was first announced or articulated in the form which I'll spell it out I don't know whether it was Galileo or Newton or those who came after them but those early pioneers certainly had the right idea it begins with the idea of an inertial reference frame now inertia reference frame this is something a bit tautological about an inertial reference frame Newton's equations F equals MA are satisfied in an inertial reference frame what is an inertial reference frame it's a frame of reference in which Newton's equations are satisfied I'm not going to explain any further what an inertial reference frame is except to say that the idea of an inertial reference frame is by no means unique a reference frame first of all was a reference frame in tale of a reference frame first of all entails a set of coordinate axes in ordinary space X Y & Z and you know how to think about those but it also entails the idea that the coordinate system may be moving or not moving relative to whom relative to whomever we sitting here or you sitting here in this classroom here define a frame of reference we can pick the vertical direction to be the z axis the horizontal direction along my arms here to be the x axis X plus that way X my X is minus in that direction and which one have I left out I've left out the y axis which points toward you from me so there are some coordinate axes for space XY and Z and I didn't this in addition to specify a frame of reference one also imagines that this entire coordinate system is moving in some way relative to you sitting there presumably with a uniform velocity in a definite direction if your frame of reference is an inertial frame of reference in other words if when you throw balls around or juggle or do whatever is supposed to do in an inertial frame of reference if you find yourself in an inertial frame of reference then every other frame of reference that's moving with uniform velocity relative to you now remember what uniform velocity means it doesn't just mean with uniform speed it means with uniform speed in an unchanging direction such a frame of reference is also inertial if it's accelerated or if it starts standing still and then suddenly picks up some speed then it's not an inertial frame of reference all inertial frames of reference according to Newton and also I think also Galileo Galileo was often credited with the idea but I never read enough of Galileo to know whether he actually had it or not neither did I read enough of Newtons they both wrote in languages that I don't understand what was I saying oh yes right according to both Newton and anybody else who thought about it very hard the laws of physics are the same in all inertial reference frames laws of physics meaning F equals MA the forces between objects all the things that we would normally call laws of nature or laws of physics don't distinguish between one frame of reference of and another if you want a kind of pictorial example that I like to use a lot when I'm explaining this to the children or to grownups I like to think about the laws of juggling there are very definite procedures that you train your body to do uh in order to be able to juggle balls correctly now you can imagine yourself being in a railroad car moving with perfectly uniform velocity down the x axis and trying to juggle do you have to compensate for the fact that the train is moving and for particular when you throw a ball up into the air that you have to reach over to the right to compensate for the fact that the train is moving to the left my left your right the answer is no you don't the laws of juggling are the same in every reference frame and every inertial reference frame whatever you do in one reference frame you do exactly the same thing and you'll succeed or fail depending on whether you're a good juggler or not but it will not depend on whether you're moving with uniform velocity so the laws of juggling are the same in every inertial reference frame the laws of mechanics are the same in every inertial reference frame the laws Newtonian laws of gravity are the same in every inertial frame according to Newton what about the laws of electrical phenomena well there there was a clash the clash had to do with Maxwell's equations Maxwell's equations were the field equations the field theory that governed the electromagnetic field and the way that it propagated and sent waves electromagnetic waves that we ordinarily call light or radio waves or so forth and the fundamental dilemma as you all know I'm sure you all know the fundamental dilemma was both according to well here was the dilemma Maxwell's equations said light moves with a certain velocity if you take the various constants that appear in Maxwell's equations and put them together in the right way you get the velocity of waves moving down an axis and that velocity comes out to be a certain number out of Maxwell's equations you have two choices one is to believe that Maxwell's equations are true laws of nature as good as any other laws of nature in which case the principle of relativity says they should be the same in every reference frame but if it follows from Maxwell's equations that the speed of light is three times ten to the eighth meters per second which is about what it is if it follows from Maxwell's equations that light moves that fast and if Maxwell's equations are laws of physics fundamental laws of physics and if the laws of physics are the same in every reference frame then the speed of light must be the same in every reference frame but that was very hard to swallow because if a light beam is going down that axis and you chase it and run along with it that lets say three-quarters of the speed of light then you want to see that light ray moving much more slowly than three times ten to the eighth meters per second relative to you on the other hand the light ray going in the other direction since you're sort of running into it you should see going even faster so all these possibilities could not simultaneously be correct that the laws of nature are the same in every reference frame and that Maxwell's equations are laws of physics in the same sense that Newton's laws of physics namely the same in every reference frame something had to give well the point was of course that they were good laws of nature and that they were the same in every reference frame the thing that had to give is our concepts of velocity space and time and how we measure velocity especially velocities were up which are up near the speed of light now I'm not going to spend the full amount of time that I did previously on the special theory of relativity that can be found on lectures from how long ago and there on the Internet I believe relativity and electromagnetism I think that was maybe about three quarters ago I've lost track yeah they're up there they're on the net and they're the lectures on relativity special relativity and electromagnetic theory we're just going to cut through it real fast we're going to cut through the basic ideas of relativity a little more mathematically than I would do if I were teaching it for the first time I teach it the first time I tend to teach it the way Einstein first conceived of it how do you measure distances how do you measure velocities how do how does the propagation of light influence these things instead I'm going to take a more mathematical view of it and think about the properties of various kinds of coordinate transformations coordinates now consists not only of XY and Z but also time T so imagine every event in the world is characterized by just like every particle would be characterized by a position x y&z every event taking place in space-time is characterized by four coordinates X Y Z and T let's suppress for the moment y&z let's just forget I forget them for the moment and concentrate on X and T that would be appropriate if we were mainly interested in motion along one axis let's focus on that motion along the x axis let's suppose there is no motion along y&z then we can forget y&z for the moment momentarily we'll come back to them and think of motion along X and T and the various reference frames that might be moving along the x axis alright here's here's time vertically is space horizontally physicists always draw space horizontally and time vertically I found out that mathematicians are at least certain computer scientists always draw time going horizontally I didn't know that and I got into an enormous argument with a quantum computer scientist which was ultimately resolved by the fact that he had time going horizontally and I had it going vertically these are traditions I guess traditions grow up around subjects but time is north and X is east I guess or at least time is upward yeah yeah yeah that's what that that that's the point that is the point yes they're thinking of time is the independent variable and everybody knows that it's a law of nature that the independent variable should be horizontal ok all right now let's in let's imagine a moving observer moving down the x axis with a velocity V let's take his origin of spatial coordinates his origin of spatial coordinates at time T equals zero is just the same let's assume that my I'll be the moving observer I move down the x-axis I am my own origin there's nobody who was your origin that seat is vacant over there so that absent a human over there is the center of the x-coordinates in your frame I'm the X prime coordinates and of course I being very egocentric will take my x-acto is origin to be where I am there I do I move down the x-axis we pass each other our origins pass each other at t equals 0 so that means at T equals 0 your axis and my axes are the same or your origin in my origin is the same but then as I move down the x axis my core my coordinate center moves to the right most of the right that's supposed to be a straight line that's as good as I can do under the circumstances that's a straight line and it's moving with velocity V which means it's X prime equals SR it means x equals VT but it's also that's the way you describe it in terms of your coordinates my centre you described by saying x equals VT how do I describe it I just say X prime my coordinate X prime is 0 X prime equals 0 is the same as x equals VT all right what's the relationship between X Prime and X and T well it's easy to work out if you believe this picture the X prime coordinate is the distance from my origin the x coordinate is the distance from your origin so one of these is X the other is X prime the upper one here is X prime the low and here is X and the relationship between them is that they differ by an amount VT in particular X is equal to X prime minus VT or X prime is equal to X plus VT will have it wrong yes I do X prime is X minus BT and X is X prime plus VT yeah I think I have that's correct now all right what about time itself well according to Newton and according to Galileo and according to everybody who came afterward up until Einstein time is just time is just time is just time there was no notion that time might be different in different reference frames Newton had the idea of a universal time sort of God's time God upon his cloud ticking off with his with his super accurate watch and that time was universal for everybody no matter how they were moving and so everybody would agree on what on the time of any given event in this map of space and time here and so the other equation that went with this is that T prime is equal to T let's forget the top equation here let's just forget it one might say that this was the Newtonian or the Galilean transformation properties between X and T your coordinates and the coordinates that I ascribe to a point in space-time now let's examine a light ray moving down the plus x axis if it starts at the origin here then it moves along a trajectory which is x equals CT C being the speed of light now shortly I'm going to set C equal to 1 we're going to work in units in which C is equal to 1 but not quite yet incidentally once you understand a bit of relativity working in coordinates in which C is not equal to 1 is about as stupid as using different units for x and y are if we used yards for x and feet for y then we will have all kinds of funny factors in our equations which would be conversion factors from X which is measured in feet to Y which is measured in our yards the cycle has its uses log scale has its uses no long skilling long scale well let common interest yep I'm not sure we good but okay I'm just saying it is quite often in practical circumstances that one uses different scales yeah you sometimes you might there might be a good reason I mean um it wouldn't be totally unreasonable for a sailor to use different units for horizontal direction and vertical direction hmm I mean he's used to moving around horizontally he might use what miles miles versus fathoms or something nautical miles versus paddles yeah Persian is relative but um when you talk about a frame of reference you need to specify a period of time because obviously goes that 15 billion years there is no yeah we're ignoring now the fact that the universe began at some time and we're imagining now as Newton did and as the early Einstein did that the universe has just been here forever and ever and ever unchanging totally static and space and time have properties which don't change with time now of course that's incorrect in the real world and at some point we will take up the subject of cosmology and find that's not right but as long as we're interested in time intervals which are not I suspect this is what you're getting at as long as we're interested in time intervals which are not too long in particular time intervals over which the universe doesn't expand very much and so forth we can mainly say the properties of space don't change over a period of time and so everything just stays the same as always was is that what you're asking it seems that that this assumption if it is made it needs to what you're describing so well so the question is without imagining to some point as it doesn't lead it doesn't lead to what I'm describing where is this this room for different formulas here this is a formula which is based on an assumption the assumption being that time is universal that's what Einstein found was wrong basically what he found is that when you're in a moving frame of reference to different the observers will not agree about what time a particular event takes place this is the culprit here this one and some modifications to this one but in any case to see what's wrong let's go to Maxwell's equations Maxwell's equations say that light always moves with this velocity C being some numbers in meters per second okay 3 times 10 to the 8th meters per second we will later as I said say C equals 1 let's imagine a light beam moving down the x axis let's describe how X prime sees it in other words you see the light move this way to the right how do I see the light well let's see what I see let's just work it out X prime will be X which is CT for that light ray minus VT which is the same as C minus VT all this says is that I see the light moving with a diminished velocity a velocity C minus V why is that because I'm moving along with the light so naturally I see it move slowly the slow compared to what you see it what about the light going in the other direction supposing it was a light beam going in the other direction then how would you describe it you would describe it as x equals minus CT and if I do exactly the same thing I will find that X prime is equal to X that's minus CT – VT which is the same as minus C plus V times T so what this says is that I will see the light moving also in the negative direction that's the minus sign but I'll see it moving with an enhanced velocity C plus V if this were the right story and if these were the right transformation laws for space and time then it could not be the case that Maxwell's equations are laws of physics or laws of nature in the sense that they were true in every reference frame they would have to be corrected in moving frames just like the juggler who had to reach to the right who didn't actually but who thought he had to reach to the right to collect the ball when train is moving the physicist interested in light beams would have to correct things for the motion of his reference frame now it's an experimental fact that this is not the case that you don't have to correct for motion was the famous Michelson Morley experiment Einstein he just rejected he just felt this can't be right Maxwell's equations were much too beautiful to be relegated to the approximate or to the contingent on which reference frame and so he said about to find a framework in which the speed of light would be the same in every reference frame and he basically focused on these equations and after various very very beautiful Gedanken experiments thought experiments about light and about measuring and so forth he came to a set of formulas called the Lorentz transformations I'm going to explain them the Lorentz transformations in a more mathematical way not fancy mathematics but just get we want to get right to the heart of it and not spend the three weeks doing it the best way is to a mathematical problem but before I do let me set up a different mathematical problem which is for most of you you've seen me do this before but nonetheless let's go through it again the problem of rotation of coordinates we're going to do this quickly let's just take spatial coordinates now for the moment two dimensional spatial coordinates let's forget X and T and just concentrate on X&Y two coordinates in space instead of events in space-time concentrate on a point in space a point in space has coordinates and we can determine those coordinates the x and y coordinates just by dropping perpendicular to the x axis in the y axis and we would describe this point as the point at position let's just call it X Y now there's nothing sacred about horizontal and vertical so somebody else may come along some crazy mathematician a really nutty one who wants to use coordinates which are at an angle relative to the vertical maybe a couple of beers and you don't know the difference between vertical and worth worth worth we should give this direction a name oblique yeah all right the oblique observer the blue observer can blue be seen everybody can see blue okay good ah the blue observer also characterizes points by coordinates which he calls X Prime and Y Prime the X Prime and the Y prime coordinates are found by dropping perpendicular to the X Prime and the Y prime axis so here's X prime is y prime and given a point X Y there's a role it must be a role if you know the value of x and y you should be able to deduce the value of X I'm in y-prime if you know the angle between the two coordinates between the x coordinate and the X prime coordinate and the formulas simple we've used it least in these classes many times I'll just remind you what it is that's X prime is equal to x times cosine of the angle between the two frames between the two coordinate systems minus y times sine of the angle and Y prime is equal to minus plus I think X sine of theta plus y cosine theta I just want to remind you about a little bit of trigonometry all of trigonometry is encoded in two very simple formulas I've used them this signs on these signs of are on the right let's Ella and X prime is bigger than X for small theta since ours here are all so it's Auto Expo Rhine is bigger than it is is it yeah let's see if you rotate it to the next so that y is y prime is zero it's further out X prime rook will have it backward yeah what's your gift I'm not gonna fit nobody so let's say just make sure the links take survive is the little perpendicular there no my life primary so that's y prime y prime is this is why I'm here right right that's why I'm in X prime is bigger than X so there has to be a plus sign on the second you know its prime is bigger than X let's see um yeah X prime is bigger than X yeah X prime is bigger than X looks like that's probably right probably sign but then this one must be man negative yeah okay there's an easy way to correct for it another way to correct for it just call this angle minus theta that would also do the trick because cosine of minus theta is the same as cosine of theta and sine changes sign when you change theta 2 minus theta so if instead of calling this angle theta I called it minus theta then my previous formulas would be right it's true true but the it's an excuse all right what do we know about sine and cosine it's important to understand sine and cosine everything you ever learned about trigonometry can be codified in two very simple formulas if you know about complex numbers the two very simple formulas are that cosine of theta is e to the I theta plus e to the minus I theta over 2 and sine of theta is e to the I theta minus e to the minus I theta over 2i those two formulas contain everything about trigonometry you don't have to know any other formulas other than these for example I will assign you the homework problem of using these two formulas to find cosine of the sum of two angles but the way you would do it is just write the sum of two angles in here and then reexpress the Exponential's in terms of cosine and sine that's easy to do e to the I theta is equal to cosine of theta plus I sine theta and e to the minus I theta is cosine of theta minus I sine theta so work through these formulas get familiar with them they're extremely useful formulas once you know them you will never have to remember any trigonometric formulas again the other thing to know is that e to the I theta times e to the minus I theta is 1 all right e to the anything times e to the minus the same thing is one those things characterize all trigonometric formulas in particular as was explained to me by Michael a number of times if we multiply e to the I theta times e to the minus I theta we will get one on this side but on this side we will get cosine squared of theta plus sine squared of theta naught minus sine squared but plus sine squared cosine squared and then ice minus I squared sine squared that gives us cosine squared plus sine squared cosine squared theta plus sine squared theta so that's equivalent to the fact that e to the I theta times e to the minus I theta is 1 all right now the most important fact that again follows from the simple trigonometry is that when you make the change of coordinates from XY to X prime Y prime something is left unchanged namely the distance from the origin to the point XY that's something which is you know you count the number of the molecules along the blackboard from here to here and that doesn't change when I change coordinates so the distance from the origin to the point XY has to be the same independent of which coordinate axes we use well let's take the square of that distance the square of that distance we know what it is let's call it s squared I'm not sure why I use s but s for distance s s for distance s for space I think it must be for space that I'm using it for the spaces for the spatial distance from the origin to the point XY we know what that is it's Pythagoras theorem x squared plus y squared but as I said there's nothing special about the XY axes we also ought to be able to calculate it as X prime squared plus y prime squared well it's not too hard to work out that X prime squared plus y prime squared is x squared plus y squared it's easy to use do X prime squared plus y prime squared will have x squared cosine squared theta it will also have x squared sine squared theta when you add them you'll get x squared plus y squared you know you know the rigmarole so it follows from cosine squared plus sine squared equals 1 that X prime squared plus y prime squared equals also equal is equal to x squared plus y squared work that out make sure that you have this on the control that you understand why from the trigonometry not from the the basic physics of it or the basic geometry of it is clear make sure that you understand that you can see that from the trigonometry okay one last thing about sines and cosines if I plot on the blackboard for every angle if I plot sine or cosine along the horizontal axis supposing I plot cosine of theta along the horizontal axis and sine of theta along the vertical axis then if I plot all possible angles they will correspond to a bunch of points that lie on a unit circle Y on a unit circle because sine squared plus cosine squared equals 1 so one might call the properties of sine and cosine the properties of circular functions circular in that they're convenient for rotating they're convenient for describing unit circles points on unit circles are described in terms of coordinates which are cosines and sines of angles and so forth it's natural to call them circular functions these are these are not the functions that come in to the transformation the new transformation properties first of all these are wrong and I don't want to use X what's X ya ya now just wrong Newton had it wrong Newton or Galileo however it was postulated who postulated it Einstein modified it now we're going to have to make sure that Einstein's modification doesn't change things in situations where Newton knew where Newton's equations were good approximations the situations where I'm Stan's modifications are important is when we're talking about frames of reference moving very rapidly up near the speed of light before the 20th century nobody or nothing had ever moved faster than a hundred miles an hour probably well of course some things did light did but for all practical purposes light didn't travel at all it's just when you turned on the switch the light just went on so light didn't travel nothing and anybody's experienced direct experience traveled faster than 100 or 200 miles an hour and well I should say nothing travels faster than 100 miles an hour and then live to tell about it so all of experience was about very slow velocities on the scale of the speed of light on the scale of such velocities newton's formulas must be correct they work they're they're very useful they work Nutan got away with it so there must be good approximations it better be that whatever einstein did to the equations in particular to these two equations here had been a reduced to newton's equations in the appropriate limit okay let's come back now to light light according to the Newton formulas doesn't always move with the speed of light but let's let's try to figure out what it would mean of a better formula of a replacement for this but light always moves with the speed of light first of all let's set the speed of light equal to one that's a choice of units in particular it's a choice of the relation between space units and time units if we work in our light years for spent for a distance and years for time then light moves one light year per year the speed of light is one if we use seconds and light seconds it's also one whatever whatever scale we use for space if we use for time the time that it takes light to go that distance one unit of space if we use that for time units then the speed of light is equal to one now from the ordinary point of view of very slowly moving things those are odd units but if we were electrons with neutrinos and whizzing around like photons they would be the natural units for us speed of light equals one so let's set the speed of light equal to one as I said it's just the choice of units and then a light ray moving to the right just moves along a trajectory x equals T C is just equal to one a light ray moving to the left is x equals minus T how can we take both of these equations and put them together sorry x equals minus T can I write a single equation which if it's satisfied is a light ray either moving to the left or to the right yes here's an equation x squared equals T squared it has two solutions x equals T and X equals minus T the two square roots or x squared equals T squared is equivalent to either x equals T or x equals minus T in other words this equation here has the necessary and sufficient condition for describing the motion of a light ray either to the right or to the left supposing we found a replacement for this equation which had the following interesting property that whenever let's let's write it this way X square minus T squared equals 0 this is even better for our purposes x squared minus T squared equals 0 that's the necessary and sufficient condition to describe the motion of a light ray supposing we found a new set of rules a new set of transformation properties which which um had the property that if x squared minus T squared is equal to 0 then we will find that X prime squared minus T prime squared is equal to 0 in other words supposing this implied this and vice-versa then it would follow that what the unprimed observer you and your seats see is a light ray the primed observer me moving along also see as a light ray both of us agreeing that light rays move with unit velocity now this doesn't work for Newton's formula here it just doesn't work if X is equal to T it does not follow that X prime is equal to the T prime in fact it says something quite different okay so the form of these equations must be wrong let's look for some better equations now at this point let's in fact let's even be a little bit more ambitious it turns out being a little bit more ambitious actually simplifies things let's not only say that when X square minus T squared is equal to zero then X prime squared minus T prime squared is equal to zero let's say something even bolder let's say the relation between XT and X prime T prime is such that x squared minus T squared is equal to X prime squared minus T prime squared in other words pick any X and any T and calculate X square minus T squared then take the same point except reckoned in the primed coordinates in other words we take a certain event a light bulb goes off someplace you say that corresponds to X and T I say it corresponds to X Prime and T Prime but let's require just to try it out see if we can do it let's look for transformations so that X square minus T squared will always be equal to X prime squared minus T's prime squared that would be enough to ensure that everybody will agree about the speed of light why if x squared minus T squared equals X prime minus T prime squared for all X and T and so forth then when X square minus T squared equals zero X prime minus T prime squared will be zero and then if this is a light ray so is this a light ready everybody get the logic ok good so let's assume now that let's ask can we find transformations which have this particular property now it's not so different from looking for transformations which preserve x squared plus y squared equals x prime squared plus y prime squared it's just a little minus sign other than a minus sign here X square minus T squared look of these two is very similar and the mathematics is quite similar here are the transformations which preserve x squared plus y squared what are the transformations which preserve x squared minus T squared well they are the Lorentz transformations they are the fundamental transformations of the special theory of relativity they're not this but they're closely related or perhaps one should say closely analogous to these equations here but we have to substitute for circular trigonometry hyperbolic trigonometry so let's go back and remember a little bit about hyperbolic functions instead of circular functions well I didn't want to erase that all right these are the basic rules governing circular functions cosine theta this sine theta is equal to this and the e to the I theta in terms of cosine and sine all right let's see if we have a yeah we do have a blank blackboard here let me write whoops what did I do here I erased something I didn't mean to erase incidentally does everybody see how I got this side from the side you just add and subtract the equations appropriately and you isolate it to the I theta e to the minus R theta that's elementary exercise alright hyperbolic functions what are hyperbolic functions alright those are functions of the form hyperbolic cosine cosh hyperbolic cosine first of all the angle theta is replaced by a variable called Omega which I will call Omega Omega is called a hyperbolic angle it doesn't go from zero to two pi and then wind around on a circle it goes from minus infinity to infinity goes from minus infinity to infinity so it's a variable that just extends over the entire real axis but it's defined in a manner fairly similar to cosine and sine cosh Omega is by definition you're not allowed to ask why this is definition e to the Omega plus e to the minus Omega over 2 all we do is substitute for theta or for Omega theta I theta substitute Omega and that gives you hyperbolic functions likewise or similarly there's the hyperbolic sine and that's given by e to the Omega minus e to the minus Omega over 2 essentially you throw away all eyes out of that formula out of the top formulas just throw away all Sun all eyes the equations on the right-hand side become e to the Omega equals hyperbolic cosh Omega plus sin Chi Omega and e to the minus Omega equals cosh so mega- cinch Omega I think that's right is it right gosh – cinch it is yeah it is right okay now what about the analog of cosine squared plus sine squared equals one that simply came by multiplying this one by this one so let's do the same operation multiplying e to the Omega by each by e to the minus Omega gives one and now that gives cosh squared minus cinch squared you see we're getting a minus what we want we want that minus the minus is important we want the well somewhere is under here was a formula with a minus sign yeah we want to get that – into play here that's cos Omega squared knockouts Prakash squared Omega minus sin squared Omega so it's very similar everything you want to know about hyperbolic trigonometry and the theory of these functions is called hyperbolic trigonometry everything you ever want to know is codified in these simple formulas these in these and they're more or less definitions but there are the useful definitions now yeah go ahead yeah not only is it worth mentioning I was just about to mention it so I squared minus y squared is what hyperbola yeah right exactly so if I were to play the same game that I did here namely plot on the horizontal and vertical axis the values not of cosine of theta and sine of theta but cosine cosine cosh of that of Omega and since Omega what's in other words on the x-axis now we're going to plot cos Omega and on the y-axis cinch Omega then this is a hyperbola not a circle but a hyperbola and it's a hyperbola with asymptotes that are at 45 degrees you can see let me show you why why the asymptotes are at 45 degrees when Omega is very large when Omega is very large then e to the minus Omega is very small right when Omega is very large e to the minus Omega is very small and that means both cosh and cinch are both essentially equal to e to the plus Omega in other words when Omega gets very big cosh and cinch become equal to each other and that's this line here cash equals cinch along this line here so when Omega gets very large the curve asymptotes to to a curve which is a 45 degrees it's not hard to see that in the other direction when Omega is very negative that that it asymptotes to the other asymptotic line here so that's why it's called hyperbolic geometry it the hyperbolic angle the hyperbolic angles the caches the cinches play the same role relative to hyperbolas as sines and cosines do two circles any questions No so cosh Omega equals zero how would you plot that hi purple okay show me hmm Oh cos squared minus sin squared equals zero no that's no no cos squared minus sin squared equals one in the same sense that sine squared plus cosine square it never equals zero I think what I think you want to ask a different question I think oh well since Omega equals zero is the horizontal axis the costume a equals zero is the vertical eyebrows right okay well this is the x-intercept yeah it's it's the vertex I just think here's one point on a minute oh man the x-intercept there is one yeah because Kostroma cost of zero is one to see that just plug one r 0 in here 1 plus 1 divided by 2 is 1 at least it was yesterday yeah stores okay so now we we're sort of starting to cook a little bit we're starting to see something that has that nice minus sign in it but what's it got to do with X and T and X Prime and T prime we're now set up to make let's call it a guess but it's a guess which is based on the extreme similarity between hyperbolas and circles cautions and cosines and so forth he is the guess I'm going to make and then we'll check it we'll see if it does the thing we wanted to do my formula instead of being this has gotten with and we're now going to have instead of x and y we're going to have x and t time and x later on we'll put back y&z we're going to have to put back y&z but they're very easy okay so let's start with X prime X prime is the coordinate given to a point of space-time by the moving observer namely me and I'm going to guess that it's some combination of X and T not too different but not the same as where is it X prime equals X minus VT I'm going to try cosh Omega X let's write X cos Omega minus T sin Omega sort of in parallel with this I could put a plus sign here but you can go back and forth between the plus and the minus by changing the sign of Omega just as you did here so this let's do it this way X cos Omega minus T sin Omega and T prime going to look similar but without the extra minus sign here this you know the relation between sines cosines and cautious and cinches is one of just leaving out an eye you go from sines and cosines the clashes and cinches by leaving out the I well if you track it through carefully you'll find that this minus sign was really an I squared it's not going to matter much I will just tell you it was really came from some I squared and if you leave out I I squared just becomes one squared is no minus sign so here's the guess for the formula connecting X prime T Prime with X and T it equals let's say X since Omega – no – plus T cos Omega in this case there are two minus signs in this case there was only one minus sign okay but but let's check what do we want to check we want to check that X prime squared minus T prime squared is equal to x squared minus T squared your ask you're probably asking yourself what is this Omega what does it have to do with moving reference frames I'll tell you right now what Omega is it's a stand-in for the velocity between the frames we're going to find the relationship between Omega and the relative velocity of the reference frames in a moment there has to be a parameter in the lower end these are the lines in these are the Lorentz transformations connecting two frames of reference in the Lorentz transformations as a parameter it's the velocity the relative velocity that parameter has been replaced by Omega it's a kind of angle relating the two frames a hyperbolic angle but we'll we'll come back to that for the moment let's prove that with this transformation law here that X prime squared minus T prime squared is equal to zero ah is equal to X square minus T squared I'm getting to that point in the evening where I'm going to make mistakes all right this is easy you just work it out you use all you have to use is that cosine squared minus sine squared is 1 you can work that out by yourself but we can just see little pieces of it here X prime squared will have x squared cos squared Omega t prime squared will have x squared sin squared Omega if I take the difference between them I'll get a term with an x squared times cos squared minus sin squared but cos squared minus sin squared is one fine so we'll find the term with an x squared when we square take the square of the difference between the squares of this and this and likewise will also find the T squared the cross term when you square X Prime you'll have XT cost cinch when you square T Prime you'll have XT costs inch when you subtract them it'll cancel and it's easy to check that's our basically one liner to show that with this transformation here x prime squared minus T's prime squared is x squared minus T squared which is exactly what we're looking for let me remind you why are we looking for it if we find the transformation for which the left-hand side and the right-hand side are equal then if x squared equals T squared in other words if the right-hand side is 0 the left-hand side will also be 0 but x squared but x equals T that's the same as something moving with the speed of light in the X frame of reference if this being 0 is equivalent to the left hand side being 0 it says that in both frames of reference the light rays move with the same velocity so that's the basic that's the basic tool that we're using here X prime squared minus T prime squared is equal to x squared minus T squared all right that does follow by a couple of lines using cos squared minus N squared equals 1 but what I want to do let's take another couple of minutes now let's take a break for five minutes and then come back and connect these variables Omega with the velocity of the moving frame of reference somebody asked me a question about the ether and what it was that people were thinking somehow Einstein never got trapped into this mode of thinking um well what were they thinking about when they were thinking about the ether what exactly was the michelson-morley experiment well I'll just spend the minute or two mentioning it certainly Maxwell understood that his equations were not consistent with with Newtonian relativity he understood that but his image of what was going on is that the propagation of light was very similar to the propagation of sound in a material or water waves propagating on water and of course it is true that if you move relative to the atmosphere or move relative to the substance that sound is propagating in you'll see sound move with different velocities depending on your motion if you're at rest in a gas of material isn't there's a natural sense in which is a particular rest frame the rest frame is the frame in which on the average the molecules have zero velocity if you're in that reference frame then first of all light has the same velocity that way as that way number one and it has a velocity that's determined by the properties of the fluid that the sound is moving in okay Maxwell more or less thought that light was the same kind of thing that there was a material and the material had a rest frame and that particular rest frame was the frame in which light would move with the same velocity to the left as to the right and he thought that he was working out the mechanics or the behavior of this particular material and that we were pretty much at rest relative to this material and that's why we saw light moving the same way to the left of the right one would have to say then that Maxwell did not believe that his equations were a universal set of laws of physics but that they would change when you moved from frame to frame just happened by some luck we happen to be more or less at rest relative to the ether to this strange material um of course you could do an experiment with sound if you're moving through the sound you can check that the velocity in different directions is different you do let's not worry exactly how you do that that's what the Michelson Morley experiment was Michelson and Morley I suppose said look the earth is going around in an orbit maybe at one season of the year we just happen to be at rest relative to the ether by accident and some other season six months later we're going to be moving in the opposite direction and we won't well we won't be at rest only at one point in the orbit could we be at rest relative the–this or at any other point in the orbit we wouldn't be so if we measure in November that light moves the same than all possible directions then in what's what's the opposite of November May then in May we should find that light is moving with great with the different velocities in different directions and he tried it and a very fancy and sophisticated way of measuring the relative velocity in different directions but he found that there was no discrepancy that the light traveled the same velocity in every direction at every time of year there were all sorts of ways to try to rescue the ether but none of them worked none of them work and the result was one had to somehow get into the heart of space and time and velocity and mid distance and all those things in a much deeper way in a way that didn't involve the idea of a material at rest in some frame of reference that that propagated the light okay oh where are we I forgotten where we were when we stopped somebody remind me whoo-hah Omega yeah what is Omega forgotten Omega Oh how Omega is really metal speed of light but to the velocity of the moving reference frame here we have two reference frames X T and X Prime and T prime what's the relationship between them well the whole goal here was to understand the relationship between frames of reference moving with relative velocity between them Omega is not exactly the relative velocity but it is closely related to it okay let's say let's see if we can work out the relationship we know enough to do it let's see if we can work out the relationship between Omega and the velocity of the moving frame all right again let's go back to this picture there's a frame of reference moving let's redraw it here's my origin moving along okay what does it mean to say that from your perspective my frame of reference so my origin is moving with velocity V well by definition this is not a law now this is a definition and says that this line here has the equation x equals VT that's the definition of this V here my origin moves relative to your origin it moves with a uniform constant velocity that's an assumption that we're talking about two inertial frames of reference and you in your frame of reference will write x equals VT that's the definition of V if you like what will I call it I will call it X prime equals zero all along there I will say X prime is equal to zero it's my origin of coordinates okay now let's come to this transformation law here and see if we can spot how to identify V well X prime equals zero that's this trajectory moving at an angle with a velocity V X prime equals zero is the same as saying X cos Omega equals T sin Omega X prime equals zero set this side equal to zero and that says that X cos Omega equals T sin Omega all right so whatever the connection between velocity and Omega is it must be such that when X prime is equal to zero X cos Omega equals T sin Omega well let's look at that equation it also says that X is equal to sin CH Omega over cos Omega times T well that's interesting because it's also supposed to be equivalent to x equals VT now I know exactly how to identify what the velocity is as a function of Omega the velocity of the moving transformation the moving coordinate system must just be sin Chi Omega over cos Omega that's the only way these two equations can be the same x equals VT x equals sin Chi Omega over cos Omega times T so now we know it we know what the relationship between velocity and Omega is write it down the velocity of the moving frame now this is not the velocity of light it's just the velocity of the moving frame must just be cinch Omega over cos omega well actually i want to invert this relationship i want to find sin and cos omega in terms of the velocity i want to rewrite these Lorentz transformations where are they i want to rewrite these Lorentz transformations in terms of the velocity that's the familiar form in which you learn about it in in elementary relativity books X prime is equal to something with velocities in it to exhibit that all we have to do is to find Cinch and cosh Omega in terms of the velocity that's not very hard let's let's work it out the first step is to square it and to write V squared is equal to cinch Omega squared over cosh Omega squared that was easy next I'm going to get rid of since Omega squared and substitute where is it I lost it one is equal to cos Omega squared minus cinch Omega squared alright so wherever I see cinch Omega squared I can substitute from here namely cosh squared Omega minus one is equal to sine squared Omega so here we are this is just equal to hash of Omega squared minus one divided by cost of Omega squared or let's multiply by what I want to do is solve for cost Omega in terms of velocity I want to get rid of all these cautions and cinches of Omega and rewrite it in terms of velocity so first x cost Omega squared we have cosh squared Omega times V squared equals cosh squared Omega minus one or it looks to me like this is cosh squared Omega times one minus V squared equals one what I've done is transpose yeah cos squared times V squared minus cos squared itself that gives you cos squared 1 minus V squared equals 1 change the sign can everybody see that the second line follows from the first I'll give you a second yeah yeah yeah it's clear ok finally we get that cos Omega is equal to 1 divided by 1 minus V squared but now I have to take the square root cos Omega / one minus V squared and then take the square root and that gives you cos Omega now we've all seen these square roots of 1 minus V squared in relativity formulas here's where it begins the kayne we begin to see it materializing what about sin Chi Omega let's also write down sin Chi Omega well from here we see that sin Chi Omega is just equal to V times cos Omega this is easy since Omega equals V times cos Omega sorrow sin Chi Omega is V divided by square root of 1 minus V squared let's go back to these Lorentz transformations over here and write them getting rid of the trigonometric functions the hyperbolic trigonometric functions and substituting good old familiar velocities let's get rid of this and substitute the good old ordinary velocities ok so we have here X prime equals x times cos Omega and that's divided by square root of 1 minus V squared then this minus T times sin Omega which is V over the square root of 1 minus V squared or if I put the two of them together and combine them over the same denominator it's just X minus VT divided by square root of 1 minus V squared I think most of you have probably seen that before maybe slightly different let's let's clean it up a little bit X prime equals X minus VT divided by the square root of 1 minus V squared what about T prime T Prime is equal to t minus V X over square root of 1 minus V squared T prime is equal to T times cos cost is just 1 over square root and then x times sin CH that gives us the extra V in other words the formulas are more or less symmetrical and those are all good old Lorentz transformations now what's missing is the speed of light let's put back the speed of light the put back the speed of light is an exercise in dimensional analysis there's only one possible way the speed of light can fit into these equations they have to be modified so that they're dimensionally correct first of all one is dimensionless has no dimensions it's just one velocity is not dimensionless unless of course we use dimensionless notation for it but if velocity is measured in meters per second then it's not dimensionless how do we make V squared dimensionless we divide it by the square of the speed of light in other words this V squared which is here which has been defined in units in which the speed of light is 1 has to be replaced by V squared over C squared likewise over here V squared over C squared now velocity times time does have notice first of all the left hand side has units of length the right hand side this is dimensionless X has units of length but so does velocity times time so this is okay this is dimensionally consistent as it is but over here it's not the left hand side has dimensions of time that's all right 1 minus V squared over C square that's dimensionless this has units of time but what about velocity times X velocity times X does not have units of time in order the given units of time you have to divide it by C square okay let's check that velocity is length all the time times length divided by C squared that's length square R which gets correct but it's correct all right this is probably familiar to most of you who've seen relativity once or twice before these are the equations relating to different moving coordinate systems moving relative to the x axis but you see the deep mathematics or the mathematical structure of it in many ways is best reflected by this kind of hyperbolic geometry here and you know most physicists by now never write down the Lorentz transformations in this form much more likely to write them in this form easier to manipulate easier to use trigonometry or or hyperbolic trigonometry it's a little exercise it's a nice little exercise to use this the hyperbolic trigonometry to compute their to compute the compounding of two Lorentz transformations if frame two is moving relative to frame one with velocity V and frame three Israel moving relative to two with velocity V Prime how is three moving relative to one the answer is very simple in terms of hyperbolic angles you add the hyperbolic angles not the velocities but the hyperbolic angles the hyperbolic angle of three moving relative to one is the hyperbolic angle of three moving relative to two plus two moving relative to one and then you use a bit of trigonometry or hyperbolic trigonometry to figure out how you do the inches and kosh's of the sum of 2 hyperbolic angles very straightforward and I'll leave it as an exercise to see if you can work that out much easier than anything else ok so there there we have the Lorentz transformations yeah oh oh absolutely yes that's that's that's a good point yeah when we that's right if we have frame 1 let's call this x1 and y1 x2 and y2 and finally x3 and y3 well then the angle of – let's call F of 3 relative to 1 let's call it theta 1 3 is just equal to theta 1 2 plus theta 2 3 the angle connecting frame one with frame 3 is just the sum of the angle theta 1 2 plus theta 2 3 so in that respect the Lorentz transformations are much simpler in terms of the Omegas it's the Omegas which combined together to add when you add velocities now how different is omega from the velocity let's work in units in which the speed of light is equal to 1 where is our formula for velocity all right let's take this formula over here what a cinch Omega 4 small Omega let's put the C squared there a let's not put the C square there or not put the C square there since Omega is essentially Omega when Omega is small just like sine is omega where is theta when theta is small the cinch function the cost function looks like like this the cinch function looks like this but it but it crosses the axis with a slope of 1 for small Omega cinch Omega is proportional to Omega for small velocity one minus V squared is very close to 1 if the velocity is a hundredth of the speed of light then this to within one ten-thousandth is just 1 if we're talking about velocities a millionth of the speed of light then this is very close to 1 and so since Omega and velocity are very close to each other it's what's going on here Thanks okay so for small velocities Omega and velocity are the same the actual correct statement is that V over C is like Omega the dimensionless velocity over the speed of light is like Omega for small Omega and small velocity so for small velocity adding velocities and adding omegas are the same things but when the velocities get large the right way to combine them to find relationships between different frames is by adding Omega and not adding velocities when you add Omega like compounding velocities as you've got it there I guess you won't go greater than 45 degrees that guess because that would be faster than light no but Omega no more you see this bit the speed of light is V equals one that corresponds to Omega equals infinity yeah yeah so Omega Omega runs over the whole range from minus infinity to infinity but when it does V goes from minus the speed of light to the speed of light so you can add any omegas and still add any omegas Omega that's right there's no there's no speed limit on Omega is this like we just go on that diagram it looks like it's greater than 45 degrees if here where where I make a and I guess they use the definition of state along the hyperbola yeah that's right sorry where are we right there today I guess that's theta though isn't it this is Theta that's a good oh god yeah right right yeah Omega is the distance along hyperbola that's right distances that's right Omega is a kind of distance along the hyperbola all right now let's let's talk about that a little bit all right now that we've established the basic mathematics structure of the transformations I think we should go back and talk about some simple relativity phenomena and derive them oh one thing which is important which I yeah well let's see we're here are my Lorentz transformations over here I said we should we ought to at the end make sure that our transformations are not too dissimilar from Newton's in particular when the velocities are small they should reduce to Newton that's all we really know that's or at least that's all that Newton really had a right to assume that when the velocities are smaller than something or other that his equations should be good approximations isn't adding velocity good enough isn't velocities adding good enough in fact you're right in fact you're right but let's just look at the transformations themselves all right as long as the velocity is a small percentage of the speed of light an ordinary velocities are what a hundred miles an hour versus 186,000 miles an hour what is that it's small right and it's doubly small when you square it so for typical ordinary velocities even the velocities of the earth around the Sun and so forth fairly large velocities what 60 kilometers per second or something like that 60 kilometers per second is pretty fast that's the that's the orbital earth around the Sun it's pretty fast but it's nowhere near 300,000 kilometers per No yeah looks here on a thousand meters per second we're I'm sorry three times ten to the eighth no three times three hundred thousand kilometers per second right 60 kilometers per second three hundred thousand kilometers per second small fraction and then square it so for ordinary motions this is so close to one that the deviation from one is negligible so let's start with the top equation for the top equation this is negligible and it's just x prime equals X minus VT the bottom equation here you have a C squared in the denominator whenever you have a C squared in the denominator that's a very very large thing in the denominator this is negligible compared to T so here the speed of light is also in the denominator just forget this and it's just T but it's just T prime equals T it's just D prime equals T so in fact Newton's formulas are essentially correct for slow velocities no no significant departure from Newton until the velocities get up to be some some appreciable fraction of the speed of light okay let's talk about proper time proper time and then let's do a couple of relativity examples yeah question the bottom equation when X is very large yes that's right when X is exceedingly large you get a correction but that correction that X has to be very large look let's let's discuss before we do anything else let's let's let's talk about that a little bit X minus VT one minus V squared over C squared yeah let's alright in my drawings I'm going to sitt C equal to one but in the equations you can leave the C there okay this equation we understand apart from this one minus V squared over C squared in the denominator it's just this x equals V T or X minus V X minus X minus VT that's Newton let's look at this one over here okay let's look at the surface T prime equals zero T prime equals zero is the set of points that I in my moving reference frame call T call time equals zero it's what I call the set of points which are all simultaneous with the origin T prime equals zero is just everyplace in space-time which has exactly the same time according to my frame of reference and I will therefore call all those points synchronous at the same time what do you say about them if T prime is equal to zero that says that T is equal to V over C squared X now let's set C equal to one for the purpose of drawing just for the purpose of drawing I don't want this huge number C squared to distort my drawings too much it says the T equals V X what does the surface T equals V X look like it looks like this T equals V X which is also X is equal to 1 over V T so it's just a uniform line like that all of these points are at different times from your reckoning this ones later this ones later this ones later and so forth according to my reckoning all these points are at the same time so we disagree about what's simultaneous this was this was the hang-up incidentally this was the basic hang-up that took so long to overcome that took Einstein to overcome it the idea that simultaneity was the same in every reference frame nobody in fact it was so obvious that nobody even thought to ask a question is simultaneous does it mean the same thing in every reference frame no it doesn't in more in your reference frame the horizontal points are all simultaneous with respect to each other in my reference frame what I call horizontal what I call simultaneous you do not okay so simultaneity had to go let me point out one more thing about these equations I'm not going to solve them for you but I will tell you the solution anyway how do you solve for X and T in terms of X Prime and T Prime well think about it in the case of angles supposing I have a relationship like X prime is equal to X cosine theta what is it plus plus y sine theta and y prime is equal to X minus X sine theta plus or Y cosine theta and supposing I want to solve for x and y in terms of X Prime and Y Prime you know what the solution is just change theta 2 minus theta and write that X is equal to X prime cosine of minus theta but what's cosine of minus theta right cosine theta plus y sine of minus theta what's sine of minus theta minus sine theta times y and likewise for y prime Y prime is equal to minus x times sine of minus theta so that becomes plus X sine theta plus y cosine of minus theta which is cosine theta you don't have to go through the business of solving the equations you know that if one set of axes is related to the other by rotation by angle theta the second one is related to the first one or vice versa the first one is related to the second one by the negative of the angle if to go from one frame to another you rotate by angle theta and to go from the second frame back to the first you rotate by angle minus theta so you just write down exactly the same equations interchange Prime and unprimed and substitute for theta minus theta same thing for the Lorentz transformations exactly the same thing if you want to solve these for X and T write down the same equations replace primed by unprimed and change the sign of omegas to minus the sines of omegas change sinus rgn of all the sign all the cinches okay in other words just send Omega 2 minus Omega and that will solve the equations in the other direction yeah yes it's also the same as changing V 2 minus V yes the way to see that is to go right what was it what do we have cosh Omega yep yeah that's right via sign yes that was correct yeah you just well you change Omega 2 minus Omega it has the action of changing V 2 minus V you can just check that from the equations good alright let's let's talk about proper time a little bit proper time if you're doing ordinary geometry you can measure the length along a curve for example and the way you do it is you take a tape measure and you you know sort of take off you take off equal intervals equal equal little separations you can think of these separations as differential distances DS squared small little differential distances and that differential distance is d x squared plus dy squared with the x squared and the y squared are just the differential increments in x and y DX and dy this is d s alright so that's the way and you add them up you add them up that's the way you compute distances along curves it's quite obvious that if you take two points the distance between those two points depends on what curve not the same for every curve so I'll measure the longer curve you have to know not only the two points but you have to know the curve in order to say what the distance between those points are of course the distance between its longer straight line that's that's well-defined but the distance along a curve depends on the curve in any case D s squared equals the x squared plus dy squared is the basic defining notion of distance between two neighboring points if you know the distance between any two neighboring points in a geometry you basically know that geometry almost essentially completely so given this formula for the distance between two points you can compute if you like the distance along a curve because you've got to take the square root of this and then add them up don't anhedonia the squares add the differential distances all right the important thing is here that square root of DX squared plus dy squared which is the distance between neighboring points doesn't depend on your choice of axes I could choose X Y axes I could choose X prime y prime axes if I take a little differential displacement the X and the y or I just take two points two neighboring points don't even give them labels and measure the distance between them the distance between them should not depend on conventions such as which axes are used and so when I make rotational transformations the X square plus dy squared doesn't change the X and the y may change but the x squared plus dy squared does not change the same thing is true in relativity or the analogous thing we don't measure distances along the paths of particles let's say now that this curve here is the path of a particle moving through space-time there's a particle moving through space-time and we want some notion of the distance along it the notion of distance along it another example would just be a particle standing still as a particle standing still particle standing still is still in some sense moving in time I wouldn't want to say that the distance between these two points and space-time is zero they're not the same point I wouldn't like to say it's zero I would like to say there's some kind of notion of distance between them but it's quite clear that that distance is not measured with a tape measure this point and this point are the same point of space boom here at this point of space and that at a later time boom again at the same point of space two events at the same point of space how do I characterize and some nice way the distance between those two events that occurred in the same place you don't do it with a tape measure all right what do you do with a clock a clock you take a clock and you start it at this point tic tic tic tic tic tic tic a stopwatch you press it at this point tic tic tic tic tic it picks off intervals and then you stop it at that point and you see how much time has evolved that's a notion of distance along a particle trajectory it's not the distance the particle moves in space it's a kind of distance that it's moved through space-time and it's not zero even if the particle is moving standing perfectly still in fact what it is is it's the time along the trajectory what about a moving particle well you can imagine that a moving particle carries a clock with it of course not all particles carry clocks but we can imagine they carry clocks with them as they move and we can start the clock over here and then the clock over here what is the time read off by this moving clock the time read off by a moving clock is much like the distance along a curve measured by a tape measure in particular it should not depend on the choice of coordinates why not this is a question that has nothing to do with coordinates I have a clock made in the standard clock Factory the standard clock Factory and I don't know we're in Switzerland someplace makes a certain kind of clock that clock gets carried along with a particle and we ask how much time evolves or how much time elapses or how much the clock changes between here and here that should not depend on a choice of coordinates it shouldn't depend on a choice of coordinates because it's a physical question that only involves looking at the hands of the clock in fact we can ask it for little intervals along along the trajectory we could ask how much time elapses according to the clock between here and here well the answer again should not depend on what coordinates you use which Lorentz frame you use and there's only one invariant quantity that you can make out of the D X's and DTS describing this point describing these two points there's a little interval DT and there's a little interval DX now we're in space and time not ordinary not ordinary space and the quantity which is invariant there's really only one invariant quantity that you can make out of it it is DT squared minus DX squared it's the same quantity x squared minus T squared for a whole you know for a whole interval the T squared minus DX squared that's the quantity which is invariant it's minus D it's the negative of what I wrote over here x squared minus T squared okay this quantity is equal to the X prime squared minus DT power sorry DT prime squared minus the X prime squared the same algebra goes into this as goes into showing that X prime squared minus T prime squared equals x squared minus T squared incidentally this is the same as saying T prime squared minus X prime squared equals T squared minus x squared doesn't matter which way you write it all right so that suggests that suggests that the time read off the invariant time read off along a trajectory between two points separated by DX and DT is just the square root of DT squared minus DX squared why the square-root incidentally okay you're going to integrate in detail I can integrate DT yeah well alright why not just DT square minus the x squared for the time between here and here is it here's an answer supposing we go to you two intervals exactly the same as the first one we go an interval over here DX and DT and then we go another DX in DT what happens when we double the interval to DT squared minus DX squared it gets multiplied by four because everything is squared well I wouldn't expect a clock when it goes along you know when it goes along a trajectory for twice the the interval here to measure four times the the time I expected to measure twice the time so for that reason the square root is the appropriate thing here okay that's called D tau squared the tau squared the proper time along the trajectory of an object you're right that's just the towel or D tau squared being the x squared minus DT squared the Tau is called the proper time let's go I think we'll let's see the towel is called the proper time and it is the time read by a clock moving along a trajectory it's not just DT that's the important thing it's not just DT the T squared minus the x squared let's do one last thing let's just do the twin paradox in this language I think I think I've had it I'm going to finish you can do the twin paradox in this language all you have to do is to compute the proper time along two trajectories one that goes out with a uniform velocity turns around and comes back with the same uniform velocity versa a trajectory which just goes from one point to the st. the another point along a straight line and it's no more weird it's no weirder really from this perspective than saying the distance from one point to another along two different curves do not have to agree the proper time along two different curves in general will not agree what is a little bit weird is that because of this minus sign the proper time this way is less than the proper time this way that's the consequence of this minus sign here moving with some DX decreases the proper time all right we'll do a little bit more next time but then I want to get to the principles of field theory and and connect some of this with field equations for interesting wave fields the preceding program is copyrighted by Stanford University please visit us at stanford.edu

Rocket Science: How Rockets Work – A Short and Basic Explanation

Views:44786|Rating:4.74|View Time:6:26Minutes|Likes:445|Dislikes:24
How do rockets work? What is the science behind a rocket launch? How does a rocket go into space? In this short and simple video, we discuss the science of how rockets work. It is a short animated video for kids and laymen to understand the basics of a rocket launch.

Rocket science is considered a highly technical, or even scary subject by many. But in reality, it’s not that complicated. Rocket launches are elaborate processes, which consist of a number of small, basic steps that need to be accomplished properly in order to reach the desired end result – the successful launch of the rocket.

A rocket is, in simple terms, a vehicle that is powered by rocket engines. NASA and other space agencies all over the world, including ISRO (Indian Space Research Organization), JAXA (Japanese Space Agency), Roscosmos (Russian Space Agency), ESA (European Space Agency) and CNSA have been involved in sending rockets into space for decades now. A rocket consists of three main parts – the guidance system, payload system and propulsion system. We discuss these systems in more detail in this short, simple video designed for kids.

on December 18 1958 America launched a rocket that broadcasts at a Christmas message from space the message was recorded by Dwight D Eisenhower who is the president of the United States at the time the mission was considered a great success as it launched the first ever communication satellite into orbit and laid the foundation for what is now an essential multibillion-dollar industry today the communication satellite used to broadcast the Christmas message came to be known as the talking Atlas as it was launched aboard an Atlas rocket a rocket is exactly what you think it is a long thin metallic cylinder with a pointed nose that shoots up from the ground leaving a gigantic cloud of smoke in its wake however there is more to it than that there are a number of other things that make rocket both functional and useful the word rocket can mean different things in different contexts simply put a rocket is a spacecraft missile aircraft or other vehicle that obtains thrust from a rocket engine from the outside the frame of rocket is very similar to that of an airplane it's made of various light but very strong materials like aluminum and titanium the skin of the rocket is covered with a thermal protection system that protects the rocket from extreme heat caused by air friction and helps maintain cold temperatures that are needed for certain fuels and oxidizers within the rocket the body of rocket is composed of different sections all of which are housed within the frame of the rocket the first component is the payload system of the rocket for the uninitiated the payload is the Rockets carrying capacity the payload depends on the type of mission the rocket is being used for it can consist of cargo a satellite a space probe and even a spacecraft carrying humans so if you want to send humans to space the payload of your rocket will contain a spacecraft whereas if you're using the rocket as a weapon then the payload would consist of a missile next is the guidance system the system which ensures that the rocket stays on its intended trajectory and goes where it's supposed to go the guidance system consists of onboard computers and sophisticated sensors as well as radar and communication systems to maneuver the rocket while in flight last is the propulsion system a majority of the entire length of a modern rocket is actually made up of the propulsion system as the name suggests the propulsion system consists of the components that help launch the rocket off the ground and subsequently propel the rocket in a given direction so how is this huge enormous ly heavy cylindrical metallic tube shot into space in order to get into space the rocket must first cross the thick layers of atmosphere that envelop our planet since the atmosphere is thickest near the ground the rocket has to go extremely fast in order to get past this part of the atmosphere so how does it climb so fast in the air the answer to this question lies in one of the most popular physical laws of the universe Newton's third law of motion according to the third law every action has an equal and opposite reaction in our case we have a rocket that we want to launch into space how does the third law help us this law tells us that if we can get the rocket to push against the ground with an enormous amount of force then the ground will respond by pushing the rocket upwards with an equivalent amount of force that's where the rocket engine comes into play the rocket engine works by burning either a liquid or solid fuel in the presence of an oxidizer when the combustion reaction occurs it throws out a great deal of mass as a byproduct of the reaction these byproducts are released at great speed through the bell shaped nozzles that you see at the bottom of rockets since the rocket pushes the exhaust down the exhaust responds by pushing the rocket up at great speeds which lifts the rocket off the launching pad and helsing upwards into space in a way you could say that a rocket shoots upward by throwing hot gases from its exhaust houses below if you have ever seen a rocket launch in person or even seen a rocket launch video on the internet beyond the liftoff phase you may have noticed that a rocket doesn't maintain a straight trajectory all the way up it lifts off perfectly vertically but at around the 1-minute mark of the flight it starts turning and going laterally that is a flight maneuver known as the gravity turn it's a trajectory optimization technique that's always employed while launching rockets because it offers two benefits first it uses gravity to steer the rocket onto its desired trajectory which helps to save rocket fuel second it helps to minimize aerodynamic stress on the launch vehicle if a rocket continued going up without tilting at all it would reach a point where it would run out of fuel that's why it tilts slightly after lifting off and straight up thanks to the exhaust analysis of the rocket which means swiveled from side to side in order to alter the direction of the thrust once a rocket lifts off parts of it are sequentially separated or jettisoned and predefined intervals for instance if a spacecraft is being launched with a rocket then as rocket boosters are separated first followed by the external tank these separated parts blast off from the spacecraft and splashdown in the Atlantic Ocean where they can be retrieved the spacecraft then maneuvers on its own using its main engines to reach the designer in orbit similarly if an unmanned satellite is launched on a rocket the sole purpose of the rocket is to get the satellite into its intended orbit once their satellite stays in the orbit and does a small amount of maneuvering using its own engines all in all rockets are used only to get stuff into space period once a rocket has done its job it's separated in parts from the stuff that it carries as it's no longer considered an operation requirement of the mission space agencies all over the world been sending men and materiel into space for decades now as such it's only fair to say that we wouldn't have been able to understand and explore space nearly as much as we have if not for those tall cylindrical metallic tubes that shoot up from the ground in a bid to expand man's reach beyond this planet

Carl Sagan, Stephen Hawking and Arthur C. Clarke – God, The Universe and Everything Else (1988)

Views:2444692|Rating:4.91|View Time:52:11Minutes|Likes:30589|Dislikes:549
Join me on facebook

Stephen Hawking, Arthur C. Clarke and Carl Sagan (via satellite) discuss the Big Bang theory, God, our existence as well as the possibility of extraterrestrial life.

General Relativity & Curved Spacetime Explained! | Space Time | PBS Digital Studios

Views:1151479|Rating:4.72|View Time:8:53Minutes|Likes:9983|Dislikes:591
The Final Installment of our General Relativity Series!!!

Tweet at us! @pbsspacetime
Facebook: facebook.com/pbsspacetime
Email us! pbsspacetime [at] gmail [dot] com
Comment on Reddit:
Support us on Patreon!

Help translate our videos!

We’ve been through the first few episodes of our crash course on general relativity, and came out alive! But it’s officially “time” for CURVED spacetime. Join Gabe on this week’s episode of PBS Space Time as he discusses Newton and Einstein’s dispute over inertial frames of reference. Is Einstein’s theory inconsistent? Is gravity even a force??? Check out the episode to find out!

Previous Installments of the General Relativity Series:

“Are Space And Time An Illusion?”:

“Is Gravity An Illusion?”

“Can A Circle Be A Straight Line?”

“Can You Trust Your Eyes In Spacetime?”:

Let us know what topics you want to learn more about:

well we're finally here a synopsis of general relativity that builds on these previous four episodes if you haven't seen them then pause me now go watch them in order and meet me back here after the music to hear about curved space-time mutants and Einstein's dispute over gravity comes down to competing notions of what constitutes an inertial frame of reference Newton says that a frame on Earth's surface is inertial and relative to that frame a freely falling Apple accelerates down because it's pulled by a gravitational force but Einstein says nah it's the apples frame that behaves like a frame in deep space so the apples frame is inertial and the earth frame is actually accelerating upward you just get a false impression of a gravitational force downward for the same reason that a train car accelerating forward gives you a false impression that there's a backward force so who's right well between our gravity illusion episode and your comments we've seen that Einstein's position seems internally inconsistent remember that inertial frame in Zeke's face well the Apple accelerates relative to it so even Urschel frames define the standard of non acceleration how can both of those frames be inertial today we're finally going to show how curved space-time makes einstein's model of the world just a self-consistent as newton's step one is to express both Newtons and einstein's view points in geometric space-time terms since that's the only way to compare them in a reliably objective way remember humans experience the world and talk about the world dynamically as things moving through space over time but even in a world without gravity we already know that clocks rulers and our eyes can all mislead us so to be sure we're talking about real things as opposed to just artifacts of our perspective we have to translate dynamical statements into tense lists statements about static geometric objects in 4d space-time let's start with Newton he says that space-time is flat just think about it on the flat space-time diagrams of inertial observers the world lines of other inertial observers are straight indicating constant spatial velocity this captures Newton's idea that inertial observers shouldn't accelerate relative to other inertial observers Newtonian gravity would just be an additional force we introduced like any other that would cause some world lines to become curved II's facially accelerated this is a bit oversimplified but for today it'll do now for Einstein's position this is actually more subtle and it'll be easier to explain if I first set up an analogy using our old friend the 2-dimensional ant on the surface of the sphere a tiny patch at the equator looks like a plane and within that patch two great circles both look straight but suppose the ant believes that he lives on an actual plane and decides to draw an XY grid on a large patch of the sphere with its x-axis along the equator and the y axis along a longitude line relative to this grid the second grade circle looks bent so the ant concludes that it's not a geodesic but you see the ants mistake right his grid is distorted you can't put a big rectangular grid on a sphere without bunching it up try it with some graph paper and a basketball it doesn't work stated another way a sphere can accommodate local Euclidean grids and tiny patches but not global ones so the ant can use his axes as rulers and protractors within a patch but not between patches flat space definitions of straightness apply over small areas but not big ones okay Einstein's position is that Newton is making the same mistake as the ant inertial frames that means axes plus clocks are the spacetime equivalent of ants XY grid if space-time is curved then those frames are only valid over tiny space-time patches so when an observer in deep space says that the falling Apple is accelerating he's pushing his frames past the point of reliability just like the ant did in other words global inertial frames don't exist in space-time however global inertial observers do their observers that have no forces on them their world lines will be geodesics and their axes and clocks can serve as local inertial frames provided that we think of them as being reset in each successive space-time patch and by the way pictures like this are not intended to make literal visual sense on the contrary they're designed to break your excessive reliance on your eyes so that your brain becomes more free to accept what reality isn't remember no one can really see or draw space-time there is no spoon now the world line of a falling Apple turns out to be a geodesic it has no forces on it so there's no need to invent gravity okay but what about two apples in a falling box like at the end of our gravity illusion episode remember they get closer as the box fault now according to mutant that happens because the apples fall radially instead of down but according to Einstein it happens because the apples are on initially parallel geodesics that since space-time is curved can and do cross just like on the sphere in contrast the world line of a point on Earth's surface is not a geodesic it has a net force on it and it's really accelerated so does that mean that Earth's surface has to be expanding radially well be careful in order to compare distant parts of Earth you'd need a single frame that extends across space-time patches but that frame can't be inertial so any conclusions you base on it have to be interpreted with a heavy grain of salt okay so Einstein's gravity free curved space-time sounds like it's self consistent but then again so does Newton's flat space-time picture that has gravity injected as a kicker so once again which of them is right the answer is whoever agrees better with experiments and there's over a century of experiments to refer to now we haven't probably fleshed out all of general relativity yet but there's one experimental fact that I can use to show you that space-time must be curved just based on what we've seen in this series of episodes so far it's a cool argument originally presented over 50 years ago by physicist Alfred shild and it goes like this fire a laser pulse from the ground floor of a building up to a photon detector on the roof now wait five seconds and then do it again on a flat space-time diagram the world lines of those photons should be parallel and congruent without making any assumptions about how gravity affects light that would be true even if it turned out that gravity slowed photons down and bent their world lines since both photons would be affected identically now space-time is flat then clocks on the ground and on the roof should run at the same rate they're both stationary thus the vertical lines at the ends of the photon world lines should also be parallel and congruent but if you actually do this experiment you find that photons arrive on the roof slightly more than five seconds apart the excess time is less than a nanosecond but any discrepancy means that clocks are running at different rates in which case the opposite sides of this parallelogram aren't congruent and that's geometrically impossible if space-time is flat thus the very existence of gravitational time dilation regardless of its degree requires that space-time be curved and that means game over for Newton in fact to the extent that we can speak about space sometimes separately at all most of the everyday effects on earth the Newton would attribute to gravity are due to curvature in time the 3d space around earth is almost exactly Euclidean those pictures that you see of Earth deforming a grid the way a bowling ball deforms a rubber sheet or even the pictures we sometimes use on this show they all suggest spatial curvature only so they're somewhat misleading remember a frame consists of axes and clocks and around Earth space-time curvature manifests itself seam clocks much more than in rulers so even though it's hard to visualize it's curved time that makes the freefall orbits of satellites look spatially circular in frames of reference that cover too big a space time patch so why is space-time curved in the first place unfortunately the math gets heavier here and good analogies are harder to come by but here's the flow chart level answer consider a region of space-time and remember that means a collection of events not just locations its curvature and geodesics are determined by how much energy is present at those events via a set of rules called no surprise the Einstein equations so for example say you stick the energy distribution of the Sun into the Einstein equations and turn a crank what comes out is a map of the geodesics in the sun's space-time neighborhood now when you translate those geodesics into 3d spatial and temporal terms what you find is planetary orbits or spatially straight radially inward trajectories along which you would see spatial speed increase or pretty much anything else that you would otherwise attribute to a gravitational force it's pretty amazing I want to conclude with a question once asked by one of our viewers Evan Hughes if there's no gravity and gravity is not a force and why do we keep using that word well physicists are still human as far as I know most of us have no special ability to visualize or directly experience 40 space-time so we often think in Newtonian gravitational terms because it's easier and because the resulting errors are usually small we just remind ourselves that it's just a crutch that we have to use with caution but even when people are referring to relativity or string theory or whatever it's just a lot easier to say the word gravity than to say curvature or four-dimensional space-time you

Einstein's Theory Of Relativity Made Easy

Views:3077145|Rating:4.16|View Time:8:30Minutes|Likes:12186|Dislikes:2452
… Albert Einstein’s Theory of Relativity (Chapter 1): Introduction.

The theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word “relativity” is sometimes used in reference to Galilean invariance.

The term “theory of relativity” was coined by Max Planck in 1908 to emphasize how special relativity (and later, general relativity) uses the principle of relativity.

Please subscribe to Science & Reason:


Special relativity is a theory of the structure of spacetime. It was introduced in Albert Einstein’s 1905 paper “On the Electrodynamics of Moving Bodies” (for the contributions of many other physicists see History of special relativity). Special relativity is based on two postulates which are contradictory in classical mechanics:

1. The laws of physics are the same for all observers in uniform motion relative to one another (principle of relativity),
2. The speed of light in a vacuum is the same for all observers, regardless of their relative motion or of the motion of the source of the light.

The resultant theory agrees with experiment better than classical mechanics, e.g. in the Michelson-Morley experiment that supports postulate 2, but also has many surprising consequences. Some of these are:

• Relativity of simultaneity: Two events, simultaneous for one observer, may not be simultaneous for another observer if the observers are in relative motion.
• Time dilation: Moving clocks are measured to tick more slowly than an observer’s “stationary” clock.
• Length contraction: Objects are measured to be shortened in the direction that they are moving with respect to the observer.
• Mass-energy equivalence: E = mc2, energy and mass are equivalent and transmutable.
• Maximum speed is finite: No physical object or message or field line can travel faster than light.

The defining feature of special relativity is the replacement of the Galilean transformations of classical mechanics by the Lorentz transformations. (See Maxwell’s equations of electromagnetism and introduction to special relativity).


General relativity is a theory of gravitation developed by Einstein in the years 1907–1915. The development of general relativity began with the equivalence principle, under which the states of accelerated motion and being at rest in a gravitational field (for example when standing on the surface of the Earth) are physically identical. The upshot of this is that free fall is inertial motion; an object in free fall is falling because that is how objects move when there is no force being exerted on them, instead of this being due to the force of gravity as is the case in classical mechanics.

This is incompatible with classical mechanics and special relativity because in those theories inertially moving objects cannot accelerate with respect to each other, but objects in free fall do so. To resolve this difficulty Einstein first proposed that spacetime is curved. In 1915, he devised the Einstein field equations which relate the curvature of spacetime with the mass, energy, and momentum within it.

Some of the consequences of general relativity are:

• Time goes slower in higher gravitational fields. This is called gravitational time dilation.
• Orbits precess in a way unexpected in Newton’s theory of gravity. (This has been observed in the orbit of Mercury and in binary pulsars).
• Rays of light bend in the presence of a gravitational field.
• Frame-dragging, in which a rotating mass “drags along” the space time around it.
• The Universe is expanding, and the far parts of it are moving away from us faster than the speed of light.

Technically, general relativity is a metric theory of gravitation whose defining feature is its use of the Einstein field equations. The solutions of the field equations are metric tensors which define the topology of the spacetime and how objects move inertially.

The Cassiopeia Project – making science simple!

The Cassiopeia Project is an effort to make high quality science videos available to everyone. If you can visualize it, then understanding is not far behind.


relativity relativity is just a method for two people to agree on what they see if one of them is moving and since we all move about pretty regularly we can find many examples of how useful relativity is in everyday life even if we don't call it by name one miracle of modern life is the global positioning system or GPS it is pretty amazing that the GPS can pinpoint your location anywhere on earth to within a few yards and this magic depends entirely on the existence of the two dozen satellites 12,000 miles above the earth and a little relativity briefly here's how it works the GPS receiver gives a timing signal from several different high flying satellites and using Einstein's theory of relativity it calculates the distance from each satellite throw in a little triangulation and I'll come to your location simple and concept but to do this successfully the timing signals must be accurate to a few billions of a second so that the distance calculations can be accurate to a few yards but with all this motion going on time and distance must be reconciled carefully without an Stein's version of relativity the accuracy of the global positioning system would drift more than seven miles every day but of course relativity was not a new concept with Einstein the problem of how two people reconcile their observations about the world if one of them is moving has been addressed for centuries let's easier way into relativity with some common experiences if you are travelling in a car on a smooth straight stretch of highway there's no sensation of motion at all you mean I could read a book or a drink flip a coin and everything looks and feels the same as if the car we're sitting still that's because relative to the car view of the book the drink and the coin are not moving notice that this works only if the car is not changing direction or speed so if the car accelerates or turns pouring that drink becomes a real problem but constant motion feels just like sitting still and if you want to know what it feels like to move at a thousand miles per hour just look around because of the Earth's spin we zip along our time zone at a speedy 1,000 miles per hour and because of its motion around the Sun the earth carries us through space about 67,000 miles per hour and because of the motion of our solar system about the center of our galaxy we are moving at more than half a million miles an hour but it's not enough to ask how fast am i moving we must ask how fast am i moving relative to some other thing let's make up a simple rule that allows two observers to agree on how fast something is moving we begin at a moving walkway at the airport the walkway is moving at a brisk 3 miles per hour so if Susan simply stands on the walkway she is moving at 3 miles per hour relative to Sara who is standing still but not on the walkway if Susan walks on the walkway at 3 miles per hour she can accurately say she is walking at 3 miles per hour but Sara sees her moving at 6 miles per hour and if Susan walks against the walkway at 3 miles per hour Susan can still say she's walking at 3 miles per hour but now Sara sees her as standing still zero miles per hour so our first conclusion is that two observers can simply add or subtract their speed with respect to each other to any measurement of velocity they make this idea is the basis of classical relativity here's another scenario suppose there's a truck moving down the road at a constant speed of 50 miles per hour on the back or a baseball pitcher a catcher and their pitching coach armed with the speed gun as long as the truck doesn't speed up or slow down or hit any large bumps they can conduct pitching practice just the same as they would on the baseball field and when the pitcher throws a 100 mile-per-hour fastball the coaches speed gun will read 100 miles per hour the ball is indeed moving 100 miles per hour relative to the pitcher the catcher the coach and the truck but suppose an observer standing by the side of the road plucks the speed of that same baseball what speed would this observer measure for the ball well the ball would already be moving at 50 miles per hour when the pitcher was just holding it so this observer would measure a speed of a hundred and fifty miles per hour for the pitch the speed of the ball relative to the truck plus the speed of the truck relative to the observer the example of adding velocities in the bullet and plane example is classical relativity at its finest this classical version of relativity simply add in the velocities worked perfectly well for centuries for describing horse carts and ships or baseballs and trucks even airplanes and rockets and bullets but the relativity of classical physics is merely a very close approximation to reality at very very fast speeds classical relativity breaks down but this wouldn't be clear until scientists began flying Sopwith camels and examining the nature of the fastest known thing light

Scientific Method in Physics | Measurement | Phyacademy

Views:1511|Rating:4.09|View Time:2:21Minutes|Likes:9|Dislikes:2
Support Phyacademy on Patreon!

Phyacademy is the site for those one who think Physics is some kind of rocket science. Here we are trying our best to provide you all kind of knowledge and information about every topic in very simple way. But we need your support and collaboration. So Please subscribe our channel, like, share and comment so we can work more and more for you.

Subscribe to our channel:
Twitter –
And Google+:

Which Way Is Down?

Views:10819093|Rating:4.91|View Time:26:11Minutes|Likes:353304|Dislikes:6257
Links to sources and to learn more below!

my twitter
my instagram

Thanks to Eric Langlay ( for producing, editing, and animating this episode with me. Thanks also to Henry Reich ( for his advice and guidance.

Universe Sandbox² :

Mass vs weight:

Great Veritasium video:
two other great videos: and

baseballs coming together under gravitational attraction can be simulated in Universe Sandbox 2. More math behind it can be found here:

Weight to mass (on surface of Earth) convertor:

pencil and Earth falling numbers:

NASA HD footage:


Earth’s spin and its effect on ‘down’:

The measurement of Earth and its gravity:

Movement of Earth’s center of mass:

you get heavier before you get lighter as you descend into Earth:

vertical deflection:

practical uses of measuring gravity:


Interactive Earth geoid:

your weight when moon is overhead:

Hammer and feather drop on moon:

Why things fall at the same rate:

Wolfram Alpha cone geodesic tool:

General Relativity:

simple animation showing geodesic on cone and how it causes motion DOWN in space:
GREAT pbs spacetime video (watch the whole channel):

time and gravity:

tests of general relativity:


great introductory texts:

“Relativity Visualized” by Lewis Carroll Epstein

There’s also this PDF that takes Epstein’s diagrams into more detail:

“Relativity Simply Explained” by Martin Gardener

great intro to the math of general relativity:

“A Most Incomprehensible Thing: Notes Towards a Very Gentle Introduction to the Mathematics of Relativity” by Peter Collier

Requires some background in relevant math topics (see above) but very very good:

“Spacetime And Geometry: An Introduction To General Relativity” by Sean Carrol

مرحباً مشاهدي Vsauce مايكل هنا بالأسفل هنا ولكن إلى أي اتجاه يكون الأسفل؟ وكم وزن أخف شيء ينزل للأسفل؟ حسناً, الريشة تزن 0.01 جرام لكل سنتيميتر مكعب خفيفة جداً مما يسمح للطيور بالطفو على سطح الماء ولكن مجرّد أن تتركها .. فإنها تسقط للأسفل إذاً هذا هو اتجاه الأسفل إنه الاتجاه الذي تسحبنا إليه الجاذبية. حسناً, بالنسبة لشخص في الجهة الأخرى من الأرض,
فإن اتجاه الأسفل عندي هو اتجاه الأعلى عنده ولكن أين تذهب الأجسام التي تسقط؟ ولماذا تسقط؟ هل تٌسحب أم تُدفع أو أنها تسقط بفعل السفر عبر الزمن؟ لنبدأ بالأولويات .. دعونا نحوّل الشمس إلى ثقب أسود يمكننا فعل ذلك بواسطة لعبة Universe Sandbox 2 هذا المُحاكي سيُدهشك .. يروقني جداً أحبه لدرجة وضعت كود لتحميل اللعبة بشكل مجّاني في Curiosity Box إن لم تشترك معنا في Curiosity Box فقد فاتك الكثير. حسناً, بناءً على الهدف من هذا الفيديو نريد المجموعة الشمسية ..
وها هي .. لاحظ أن كل شئ يتحرك بسرعة حول الشمس والسبب أننا ضبطنا إعدادات اللعبة بحيث كل ثانية تمرّ بالنسبة لنا عبارة عن 14 يوم تقريباً داخل اللعبة إن قمت بتغيير هذه إلى ثانية واحدة, سوف ننظر للمجموعة الشمسية بنفس سرعة وقتنا قد لاحظتم أن الكواكب بالكاد تتحرّك رغم أن الأرض تدور حول الشمس بسرعة 30 كم/ثانية, إلا أنها تبدو بالكاد تتحرّك .. وهذا يعطينا انطباع عن أن الكون شاسع أحببت هذه الحركة .. الآن انظروا للشمس حتى الآن لم نجعلها كثقب أسود, لكن يمكننا تغيير ذلك.
ما نحتاج فعله هو أن نقوم بضغط الشمس. إذاً دعونا نقفل الكتلة, لتبقى ثابتة حتى عندما نقوم بجعل قطرها أقصر. دعونا نقوم بتصغير قطرها لأصغر طول ممكن ثم, أوه, أين اختفت؟
حسناً هي لازالت موجودة .. لكنها أصبحت ثقب أسود هذا مرعب قليلاً,
لكن الآن دعونا ننظر لبقيّة عناصر المجموعة الشمسية لم يتغيّر شيء, أعني تغيّرت حاجات صغيرة. أصبح بارد ومظلم, لكن الكواكب لا تطفو بحرّية في الفضاء
ولم يمتصّها الثقب الأسود. مثل ما شاهدتم, عندما قلّصنا حجم الشمس,
لم يتغيّر اتجاه الأسفل بالنسبة للكواكب. يتم سحبهم بشكل دائم بواسطة الجاذبيّة باتجاه الوسط ..
وتقليصنا لحجم الشمس لم يحرّك الشمس. أيضاً, قوّة الجذب التي كان تسحبهم لاتجاه الشمس بقيت كما هي هذا يعطينا دليل حول اتجاه الأسفل. الدليل هو الشيء الآخر الذي لم نقم بتغييره: الكتلة. الكتلة هي مقياس لكم من الصعوبة يمكنك جعل جسمٍ ما يتسارع; تغيّر سرعته. الآن, هذه الكرتين ليس عليهم أي تأثير منّي بخصوص الحركة. لكن عندما أدفع هذه الكرة البلاستيكية المجوّفة بشكل متكرّر عمليّة سهلة جداً, لكن عندما أفعل نفس الشيء مع الكرة الفولاذية الصلبة العمليّة أصعب بشكل كبير. الجاذبيّة والوزن ليس لهما أي علاقة في هذا. الجاذبيّة تسحب للأسفل, وليس عكس اتجاه الدفع الذي قمت به بالطبع, الجاذبيّة تؤثر على الاحتكاك,
لكن الاحتكاك يؤثر عكس الاتجاه عندما أبدأ بتحريك الكرة, لكنه يؤثر معي بنفس الاتجاه عندما أقوم بإيقاف الكرة. والكرة الفولاذية أصعب في إيقافها من البلاستيكية الاختلاف هو الكتلة.
الكرة الفولاذية أكثر كتلة أكثر مقاومة لتغيير حركتها. الكتلة هي خاصية باقية على طبيعتها;
لا تعتمد على ما حولها ولا تتغيّر من مكان لآخر. يمكن تعريفها أحياناً أنها مقدار المادة الموجودة في جسمٍ ما كتلك ثابتة بغض النظر عن مكانك,
على القمر .. الأرض .. أو حتى حُراً في الفضاء ولكن يُقال أن الكتلة تهتمّ بما حولها على ما يبدو. الكتلة تحب الصحبة. الأجسام التي لديها كتلة (أو طاقة)
تتجاذب فيما بينها بواسطة قوّة نسميّها الجاذبيّة. شعور الجاذبيّة هو فقط عبارة عن
نت والأرض تتجاذبون لبعضكم كل نقطة في أي جسم لديه كتلة,
تنجذب إلى نقطة أخرى في جسم آخر. متوسط عمليات السحب هذه
هي جاذبيّة بين مراكز الكتلة لهذه الأجسام. الأشياء الضخمة مثل الأرض تبذل قوة سحب واضحة,
لكن كل شيء يقوم بذلك فعلاً. حتى كرة البيسبول. هذه الكرات تسحب بعضها البعض بواسطة جاذبيّة كلٍّ منهما. باستثناء أن كتلة كلٍّ منهم صغيرة جداً, لذلك القوّة ضئيلة,
ولا يمكنها التغلّب على قوّة الاحتكاك أو دفع الهواء عن طريقها. لن تنجذب هذه الكرتين لبعض لكن .. إن وضعت كرتي بيسبول لمسافة متر واحد بينهما
في وسط الفضاء ..حيث لا يوجد قوّة تؤثر عليهما سوف تسقط كلٍّ منها للثانية وتتصادم حرفياً. سوف يستغرق ثلاثة أيام, لكنه سيحدث. إسحاق نيوتن وجد أن كميّة القوة التي تجذب جسمين لبعض تساوي:
حاصل ضرب كتلتيهما مقسوماً على المسافة بين مركزي الكتلة للجسمين مربعاً,
مضروباً في ثابت الجاذبية الأرضية إن قمت بجعل أحد الجسمين أو كلاهما أكثر كتلة أو قرّبتهما لبعض,
كميّة القوة سوف تصبح أكبر قوّة الجذب هذه .. هي ما نسميّها: الوزن إذاً, الكتلة تبقى على طبيعتها بينما الوزن يعتمد على ما حوله شيء غريب يحصل عندما تقيس وزنك على أغلب مقاييس الوزن الوزن عبارة عن قوّة
لكن مقاييس الوزن تُظهِر باوند وكليو جرام مما هي وحدات للكتلة,
ماذا يحدث؟ أن الميزان يعمل بواسطة القوّة أي قوّة ليس شرطاً أن يكون سببها الجاذبيّة. الميزان يعرض كميّة الكتلة حول سطح الأرض التي تنجذب للأرض
بواسطة القوّة التي يقيسها. حسناً, بما أن مقاييس الوزن يُعرف أنها تستخدم على سطح الأرض بواسطة أشخاص لا تؤثّر عليهم قوّة غير الجاذبيّة الأرضيّة فإنها لا تخطئ في الغالب,
ولكن يمكن خداعها بسهولة لأنها لا تفرّق بين الكتلة والوزن لاحظ أن الوزن خاصيّة متبادلة. تقوم الأرض بجذبك للأسفل
بنفس القوّة التي تقوم أنت فيها بجذب الأرض للأعلى. بناءً على الميزان أنا أزن 180 رطلاً على الأرض والأرض تزن 180 رطلاً عليّ ولكن لأن كتلة الأرض أكبر من كتلتي أو لأن كلما كان الجسم أكثر كتلة,
كلما كان أقوى في مقاومة أن يتحرك قوّة الجذب المتساوية بيننا
تقوم بجعلي أتسارع أكثر بكثير من الأرض إن قمت بإسقاط قلم رصاص من ارتفاع 6 أقدام,
فإن القلم لا يسقط على الأرض وحسب بشكل أدقّ: كليهما ينجذبان لبعض ينجذبان لبعض بواسطة قوى متساوية لكن القوّة نفسها تحرّك القلم أكثر بكثير من الأرض عندما تترك القلم,
فإن الأرض حرفياً تنجذب للأعلى باتجاه القلم بواسطة جاذبيّة القلم لمسافة تساوي تقريباً
جزء من 9 تريليون من عرض البروتون. هذه القوّة نفسها تحرّك القلم للمسافة المتبقيّة
والتي هي بالطبع 6 أقدام لو كنت على مدار محطة الفضاء الدولية,
فإنك والأرض تنجذبون لبعضكما بنسبة 10% أقل من لو كنت على سطح الأرض,
سوف تصبح كتلتك أقل بـ8.8 مرات, لكن ليس صفر لهذا السبب .. فإن روّاد الفضاء عند انعدام الجاذبيّة,
هم ليسوا بلا وزن وليسوا في منطقة انعدام الجاذبيّة قوّة أوزانهم لا تنجح في انجذابهم للأرض لأنهم يتحرّكون أفقياً بشكل سريع حيث أنهم يسقطون بنفس سرعة انحناء سطح الأرض عنهم رغم أنهم يشعرون بـ90% من الجاذبيّة التي نشعر بها نحن الآن ولهذا هم لا يحلّقون بعيداً لا يوجد قوى جاذبية تقاوم أوزانهم, حيث كل شيء حولهم يسقط أيضاً قوة المقاومة التي تشعر بها بسبب وزنك ما لا نراه على روّاد الفضاء هو الوزن الظاهري كذلك أيضاً بالون الهيليوم له وزن أعني أنه مصنوع من مادة,
من الواضح أن له كتلة .. لذا البالون ينجذب للأرض دعونا نقيس قوة وزنها حسناً, لديه وزن ظاهري سلبي وهذا لأن انجذابه للأرض
أضعف من قوّة قابليّة الطفو من الهواء حوله الذي يدفعه للأعلى بينما يتحرك للأعلى يضغط على جزيئات الهواء للأسفل, لكنها تنقل هذه القوّة بشكل واسع.
ليس مباشرة باتجاه الأسفل على الميزان سبب قوى قابليّة الطفو
هو حقيقة أنه عندما تكون مغموراً بمائع كالماء أو الهواء الجزئيات السفلى تتعرّض لضغط أكبر حيث أنه يتم ضغطها بواسطة أوزان الجزيئات التي فوقهم جميعاً
والأقرب للأرض لذا يتم سحبها بواسطة قوّة أكبر حسناً كون لديها ضغط أكبر يعني أنه يمكنها تكوين قوّة تصادم أكبر. لذا أفقياً, هذه التصادمات تلتغي أما عمودياً, قوّة التصادم الكبرى اختفت لتقدّم قوّة رفع تمكّن البالون على الطوفان هذا يحدث أيضاً مع جسمك, من خلال السطح الخارجي للجسم
الهواء يرفعك للأعلى بقوّة مقدارها 1 نيوتن مما يساوي قوة وزن تفاحة حسناً, إذا وزنت نفسك في الفراغ بلا هواء سوف يكون وزنك أكثر من وزنك الحالي بمقدار وزن تفاحة لكن هذا ليس كل شيء, عندما تدور الأرض حول نفسها فإنها تتسبب في انتفاخ عند خط الاستواء لذا, كل ما كنت أقرب لخط الاستواء ستكون أبعد عن مركز كتلة الأرض وسيكون وزنك أقل اتجاه الأسفل يتغيّر دائماً أعني أين هو مركز كتلة الأرض؟ سيكون نفس مركز الشكل الهندسي للأرض
في حالة كان شكل الأرض منتظماً لكن الأرض عليها كمّيات من الصخور الضخمة بارتفاعات مختلفة ..
وماء وجبال وبداخلها أشياء تتحرّك باستمرار,
والهواء, والثلوج الموسمية وبرغم أن هذه العناصر بعيدة, إلا أنه الجاذبية تستمر للأبد لذا .. فإن القمر, والشمس, والنباتات .. جميعها تجذبك بمقدار معيّن لا يُذكر لكن حقاً, سوف يكون وزنك أقل بمليون مرّة من وزنك الحالي
لو كان القمر فوقك مباشرة هذا التغيير المستمرّ في التوازن للمواد على الأرض وفي الكون بشكل عام يعني أن اتجاه الأسفل يتغيّر باستمرار بالإضافة لهذا, دوران الأرض حول نفسها
يسبب انحراف ما نسميّه اتجاه الأسفل عن مركز كتلة الأرض لأنه كما يبدو أن الدفع الذي تتعرض له بسبب دوران الأرض
يتسبب في ارتفاعك قليلاً وتقليل وزنك الظاهري ويحني الأرض باتجاه خط الاستواء النتيجة هي وزن ظاهري أقل بحوالي نصف بالمئة
على خط الاستواء إن كان وزنك يظهر على الميزان 200 رطل على القطبين فإنه عند خط الاستواء سيظهر 199 باوند الرقم 9.8 الذي يستخدم في الفيزياء كقوّة جاذبيّة الأرض
تم حسابه عن طريق تأثير هذه العوامل على شخصٍ ما على خط عرض 45 درجة كل هذه التأثيرات على اتجاه الأسفل
يصبح نتيجتها انحراف عمودي هذا أكثر مقدار ثوان قوسية على الأرض "ثانية قوسية يساوي حجم قطعة نقود معدنية من على بُعد
4 كيلومترات (2.5 ميل) ضئيل جداً لدرجة لا يمكن الشعور به,
لكن تغيير الاتجاه والقوة يمكن استخدامه لدراسة شكل قاع البحر تحديد ما يكون تحتك عندما تكون على سطح البحر
أو حتى مساعدتك لاكتشاف أشياء مدفونة قديمة المقصود هو أنه ليس كل اتجاهات الأسفل لدينا
هي خطوط مستقيمة ومتوازية باتجاه مركز الأرض اتجاهات الأسفل هي خطوط متعرجة بما أن الأجسام الصلبة لا تتحرك, فإنها لا تتأثر بهذا الموضوع
بينما الماء يتأثر لذا, مع تجاهل عوامل مؤثرة مثل الرياح والمد والجز,
فإن أسطح المحيطات والبحيرات عموديّة على اتجاه الأسفل لو كان الماء يجري خلال اليابسة,
أو لو كان سطح الأرض مغمور بالمياه سوف تكون الجاذبيّة نفسها على
سطحها الوعر سطح مثل هذا يُطلق عليه اسم
(مجسم أرضي) يمكن رسمه على أي خط عمودي لو أردت صناعة طاولة تغطي كل سطح الأرض
سوف يكون سطحها مموّجاً بعمق 100 متر تقريبأ في بعض المناطق
من أجل أن تكون مستوية بحيث لو وضعنا كرة في أي مكان على الطاولة
لا تتدحرج هذا مجسم أرضي مضاعف إلى 1000 مرة سوف تزن حوالي 100% أقل من وزنك لو كنت هنا عن لو كنت هنا,
حيث الجاذبيّة أقوى المقصود هو أن قوّة واتجاه الأسفل يتغيّران
باختلاف المكان والتغييرات التي تحصل عبر الزمن إذاً, الأسفل هو قوّة موجهة متغيّرة.
هذا سهل جداً لكن لماذا يجب على مادةٍ ما
أن تجذب مادة أخرى من الأساس؟ كان (إسحاق نيوتن) قادراً على وصف الجاذبيّة,
لكنه لم يفسّرها اقتربت البشرية من تفسيرها عندما قدّم (ألبرت أينشتاين)
نظريّة النسبيّة العامّة فكّر أينشتاين كثيراً حول حقيقة سقوط الأشياء إلى الأرض بنفس المعدّل مهما كانت كتلة الجسم كبيرة, فإنه إن تم إسقاطه
سوف يتسارع باتجاه الأرض بمقدار 9.8 متر بالثانية لكل ثانية أثناء سقوطه هذا يعني أن مطرقة ذات كتلة كبيرة
وريشة ذات كتلة ضئيلة عندما يتم إسقاطهما معاً من نفس الارتفاع
سوف يرتطمون بالأرض بنفس الوقت حسناً, ما حدث للتو.. كان خطأ بسبب الهواء لكي تسقط الأشياء من خلال الهواء
فإنها يجب أن تدفع الهواء عن طريقها لكن إن كان لديها سطح واسع وقوّة وزن خفيفة
فإنه سيكون لديها الكثير من الهواء لدفعه عن الطريق لكنها لن تستطيع دفع الهواء بشكل سريع في الفراغ, تسقط الأشياء بنفس معدل التسارع
بغض النظر عن الكتلة تم وصف هذه الحالة على القمر بواسطة قائد رحلة أبولو 15 ديفيد سكوت ريشة مطرقة هذا عجيب أليس كذلك؟
أعني إن كان جسم ذو كتلة كبيرة يتم سحبه بقوّة أكبر ألا يجب أن يسقط بشكل أسرع؟
حسناً, تفسير نيوتن كان بسيطاً: تنجذب الأجسام ذات الكتل الكبيرة بواسطة قوى كبيرة أيضاً سوف تتطلّب قوّة أكبر ليتم تحريكها
مقارنةً مع الأجسام ذات الكتل الأقل جسم أكبر في الكتلة من الثاني بمئة مرة
سوف يتطلّب قوّة أكبر بمئة مرة ليتم تحريكه نفس الحركة وسيتم سحبه بواسطة الجاذبيّة أكثر بمئة مرّة أيضاً إذاً, كل شيء يسقط على الأرض بنفس المعدل يالها من صدفة ممتعة, أليس كذلك؟ ربما لا أدرك أينشتاين أنه يوجد طريقة أخرى لإظهار
الأجسام تسقط بنفس السرعة بغض النظر عن كتلها تخيّل ريشة ومطرقة في غرفة في الفضاء
إن تحرّكت الغرفة للأعلى بشكل مفاجئ بسرعة 9.8 متر بالثانية سوف تصدم المطرقة والرشية بالأرض في نفس الوقت سواءً كانت الغرفة ارتفعت لتصدم بهما,
أو أنه تم تشغيل الجاذبيّة بشكل مفاجئ لن يشعر أي جسم بأي قوّة تدفعه لا يوجد طريقة لمعرفة أيّ من هذه حدث هذا هو مبدأ التكافؤ الذي طوّره أينشتاين اعترف مرّة أن أحد أعظم أفكاره هي عن رجل يسقط من السطح أثناء سقوطه, لن يشعر بأي قوّة تؤثر عليه
برغم أنه يتسارع السقوط الحرّ لا يمكن تمييزه عن العوم الحر في الفضاء
من جانب أنك لا تشعر بأي قوّة مؤثرة وأنه لا يتم تحريكك ماذا لو لم تكن الجاذبيّة قوّة على الإطلاق؟
ماذا لو لم تكن الأشياء تسقط بسبب أنه يتم دفعها أو سحبها؟ لكن بسبب أنه لا يتم دفعها ولا سحبها. لنرى كيف يمكن هذا, نحتاج للحديث عن الخطوط المستقيمة لديّ هنا حامل بطاقة مع خيط قابل للسحب هذه طريقة رائعة لاختبار المسارات المستقيمة لأنه يتم الحفاظ على الخيط مشدود دائماً البطاقة مرسوم عليها خطّين مستقيمين وإن أبقيت الخيط مستقيماً أثناء سحبه للخارج
فإنه سيبقى بين الخطّين سأعرف أنني لم أثنيه أثناء سحبه لأنه أي ثني يتم تعريفه على البطاقة بشكل زاوية مختلفة بين الخطّين والحبل حسناً, إن وضعت اثنان من هذه على طاولة مسطحة
وقمت بسحبهما, أستطيع ضمان أنهما يسيران بشكل مستقيم لن يتقاطعا أبداً, وسيكونان متوازيين للأبد. لكن الآن دعونا نضعها على جسم كروي مرة أخرى, أقوم بسحب الخيطين معاً للأمام,
مع التأكد أنني أسحبهما بشكل مستقيم, بلا ثني انتظر .. لقد تقاطعا حسناً, إنهما لم ينثنيا .. انظر ربما هناك قوّة من نوع غريب كانت تسحب يديّ
مثل الجاذبيّة, لم أشعر بها, لكنها حدثت لا ما حدث لم يكن نتيجة قوّة,
بل كان مجرّد نتيجة طبيعية لـ؟ الانحناء قد تقول انتظر, هل هذه حقاً خطوط مستقيمة؟
انظر, لا يبدوان مستقيمين بالنسبة لي. أيضاً, ماذا لو نقلتهما إلى خطوط العرض؟
هكذا لن تتقاطع أبداً, وتبدو مستقيمة أيضاً لكنهما ليسوا كذلك!
الخط المستقيم لا ينثني
وبرغم أن خطوط العرض تبدو مستقيمة للوهلة الأولى عندما نريد تتبّع أحدها
يتطلّب منا الثني للعثور على مسارات خط مستقيم على أسطح
سواءً كانت مسطحة مثل هذا أو منحنية أنا أفضّل اختبار الشريط
يمكنك استخدام شريط من القماش لكني وجدت شريط ورقي, يعمل بشكل أفضل,
دعونا نلقي نظرة على هذا المسار هنا مستقيم في البداية, لكنه بعد ذلك ينحني,
الآن, لو لدينا شخصين يمشون على هذا المنحنى ويريدون البقاء معاً, الشخص الذي بالداخل سوف يقطع مسافة أقصر
من الشخص الذي بالخارج بما أن كلا الجانبين من هذا الشريط الورقي لا يمكن تغيير أطوالها سوف يساعدنا هذا في العثور على مسار مستقيم
إن كان يمكننا وضع الشريط الورقي بشكل مسطح سنعرف أننا قد وجدنا خط مستقيم,
وكما تشاهدون يمكننا وضع الشريط بشكل مسطح على الجزء المستقيم من هذا المسار لكن عندما نصل إلى المنحنى, من أجل تتبع المسار الآن وبما أن الشريط لديه زيادة في الجزء الداخلي من المسار, وهذا الجزء ينثني ويرتفع عن السطح,
لذا نعلم أن هذا الجزء من المسار ليس مستقيماً دعونا نستخدم اختبار الشريط للعثور على خطوط مستقيمة
على سطح مخروط حسناً, بمحاذاة مباشرة من القاعدة إلى القمة,
يبدو أن الاختبار يُظهر لنا خطوط مستقيمة على سطح المخروط نعم, الشريط ينبسط بشكل مسطح على هذا المسار,
لكن ماذا عن حلقة حول المخروط؟ لا, لا تعمل
المسافات القصيرة حول قمة المخروط
تعني أنه هناك زيادة في الشريط في الأعلى لذا لا تنبسط بشكل مسطح دعونا نرى ماذا أيضاً هناك
سوف أبدأ هنا, ثم أسمح للشريط بأن ينبسط حسناً, حصلت على شكل يبدو متعرّج قليلاً أقول: يبدو متعرّج, لأنه بالنسبة لشخص في قاعدة هذا المسار
قد يبدو أن المسار يرتفع قليلاً يتباطأ قليلاً, ثم يغيّر الاتجاه ويسقط بشكل أسرع وأسرع
بما أن الشريط على سطح مستوي مثل هذا في الحقيقة أنه بالنسبة للأشخاص الذين على سطح المخروط مستقيم تماماً,
لو تتبّعنا مسار الشريط على المخروط يمكننا أن نرى بوضوح أن المخروط يمكنه تسوية خط مستقيم على سطح منحني
وهذا ما يسمّى بالجيوديسية ( تعميم للخط المستقيم على الأسطح المنحنية) هنا جيوديسي على شكل كرة
خط الاستواء هو واحد هنا خط آخر,
خطوط العرض ليست جيوديسية ليس خط مستقيم,
لنرى لماذا, دعونا نحاول تتبع الخط بواسطة الشريط أتعلم؟ يجب أن أبقى أرفعه نعم, انظر
المسافات حول الكرة تصبح أقصر كلما ارتفعنا للأعلى لذا, يوجد زيادة في الجزء العلوي من الشريط وهي ترتفع عن السطح,
هذا المسار يحتوي منحنيات ومن أجل أن نثنيه,
يجب أن يكون هناك قوّة تؤثر عليه. إن لم يكن هناك أي قوّة مؤثرة,
هذا هو المسار الذي سوف يتخذه لاحظ أن الشريط يبدأ بالتحرّك من الشرق,
لكنه بعد ذلك يسقط باتجاه الجنوب يسقط أدرك أينشتاين أن الانحناء يمكن أن يتسبب في جذب الأشياء إلى بعضها البعض بدون الحاجة لاختراع وجود قوى مثل الجاذبيّة لكن التجاذب يحدث فقط إذا تحرّكت الأشياء على سطحٍ ما, لكن إن بقيت ثابتة فإنها لا تتجاذب لبعضها حسناً بالنسبة لشيءٍ ما ساكن
كيف يبدأ السقوط؟ أعني أن الأشياء يجب أن تتحرّك إلى هذا الاتجاه,
لكنها الآن في حالة سكون, صحيح؟ نعم لكنها فقط في حالة سكون في الفضاء,
وهذه ليست القصة بأكملها أعلى .. أسفل .. أمام .. خلف .. يسار .. يمين هي كل ما تحتاجه لوصف مكان وقوع حدثٍ ما, ولكن لوصفٍ كامل تحتاج أيضاً إلى وصف:
متى هذه الأبعاد الأربعة معاً
تشكّل الإعداد الذي يجعل كل شيء يحدث في عالمنا الزمكان وبما أننا استطعنا الحديث عن سقوط القلم باستخدام بُعد مكاني واحد:
فوق وأسفل يمكننا استخدام ورقة لاستعراض الزمكان لسقوط القلم حسناً, لدينا أعلى وأسفل,
لكن يجب أن نضيف اتجاه آخر يتحرّك فيه القلم الزمن الآن إن لم تؤثر أي قوّة على القلم
فإنه لن يتحرّك عبر المكان سوف يمر الوقت وهو ساكن,
وكما تشاهدون, إن بقي ساكناً مع مرور الزمن فإنه لن يسقط إن كان الزمكان مسطحاً,
فإني عندما أترك القلم فإنه لن يتحرّك لكن دعونا الآن نسمح للأرض والتي هي بطبيعتها ضخمة
للتلاعب بالزمكان لكن لنفرض أنها مخروط الآن مع عدم وجود أي قوّة مؤثرة على القلم,
فإن كل جزء من القلم يتبع خط مستقيم ولكن على مخروط, كما شاهدنا سابقاً
مثل هذا المسار سوف يُظهر سقوط القلم وذلك لأن المسافات تكون أقصر حول المخروط كلما ارتفعت للأعلى الوقت يمضي بشكل أسرع ولكن ليتحرّك القلم بشكل مستقيم وبدون انحناء,
يجب أن يقطع كل جزء من القلم مسافة متساوية في الزمكان هكذا فقط عندما يصطدم القلم بالأرض, فإن التنافر بين الألكترونات المتبادلة يسبب قوّة تدفع القلم من الخط الجيوديسي بالنسبة للأرض, فإن الزمن هو عبارة عن سلسلة شرائح من هذا التطوّر, جيوديسية القلم الحرّة من أي قوّة مؤثرة
هي السبب في سقوطه ليس بسبب قوّة دفع أو سحب,
فقط نزعة طبيعية للمتابعة على خط مستقيم حتى يؤثر عليه شيء آخر, حسناً, نحن لم نستخدم إلا بُعداً واحداً فقط من المكان,
وواحداً فقط من الزمان لأنه عمليّة تصوّر الأكوان, ثلاثة أبعاد من المكان, وواحداً من الزمان
سيكون خارج حدود ما يمكننا عرضه على الورق والشاشات ولكن الرياضيات يمكن أن تأخذنا هناك النسبيّة العامة تتيح لنا حساب مقدار الكتلة والطاقة
التي تسببت في انحناء الزمكان وقد كانت تستخدم لشرح أشياء مثل نظرية نيوتن القديمة للسقوط
كنتيجة أن القوى التي لا تؤثر مثل الشذوذ في مدار كوكب عطارد الذي يدور قريباً من الشمس,
وبالتالي فهو الأكثر تأثراً من الشمس على الزمكان وقد أكدت العديد من التجارب الأخرى صورة نظرية النسبيّة العامة للكون مما يوصلنا إلى استنتاج أنه لا يوجد جاذبيّة! هناك فقط .. الزمكان كونه منحني, وكوننا نعيش فيه مثل المقولة الشهيرة لـ(جون ويلر):
الزمكان يُملي على الكتلة كيف تتحرّك, والكلتة تُملي على الزمكان كيف ينحني بالنسبة للأرض .. نحن لا نتحرّك بسرعة كبيرة, حتى الطائرات النفّاثة
تتحرّك بشكل لا يُصدّق بسرعة قريبة من سرعة الضوء لذا, بالنسبة إلى الأرض, نحن نتحرّك تقريباً
خلال الزمن فقط على هذا النحو, نحن أكثر تأثراً بالطريقة التي ينثني بها الزمان بواسطة الكتلة أكثر من مقدار انحناء المكان. وقد دفع هذا الكثير إلى الادّعاء بأنك في معظم الأحيان
تشعر وكأنك تُدفع إلى الأرض ليس بسبب قوّة تسمّى الجاذبيّة ولكن لأن الوقت يتحرّك بشكل أسرع بالنسبة لرأسك عن قدميك اتجاه الأسفل نسبيّ ودائم التغيّر,
ولكنه موجود بسبب وباتجاه الزمن الأبطأ (برتراند راسل) سمّى هذا بـ(قانون الكسل الكوني) كل شيء موجّه إلى حيث يكون الوقت أبطأ,
هذا نسمّيه الذهاب للأسفل لذلك لا يجب عليك التحفّظ على شيء الوقت سوف يتولّى هذا الأمر وكالعادة, شكراً لمشاهدتكم ترجمة: عبدالرحمن التميمي
[email protected] تذكّر أنه يمكنك دعم Vsauce والبحوث الخاصة بمرض ألزهايمر عن طريق الاشتراك في (Curiosity Box) الاشتراك الحالي يأتي مع كود نسخة مجانية للعبة Universe Sanbox 2 وهذا مدهش ومجموعة كاملة من الألعاب العلميّة الأخرى
والأدوات التي اخترتها أنا و(جيك) و (كيفين) أحبكم, وأيضاً آمل أن أراكم في عروض (Brain Candy) المباشرة.
نحن قادمون إلى العديد من المدن قريباً نأمل أن يكون أحد هذه العروض بالقرب منكم,
بالذهاب للعرض سوف تروني أنا و(آدم) نفعل أشياء قد لا تكونوا رأيتمونا نفعلها من قبل,
نستكشف العلم والمفاهيم الخاطئة الشائعة وراء كل شي ربما قلت كثيراً, ربما لا.
آمل أن أراكم هناك
وكالعادة .. شكراً لمشاهدتكم

Gravity Equations in International Economics

Views:16497|Rating:4.72|View Time:9:34Minutes|Likes:137|Dislikes:8
Gravity equations.

in 1687 Sir Isaac Newton discovered the law of universal gravitation where f the attractive force between two objects is equal to a gravitational constant multiply it by the product of the masses of the two objects divided by the distance between them squared now that's a bit of a mouthful but it led to the development of the basic gravity equation by the famous economist Ian Tim Bergen in 1962 he slightly adapted Newton's equation to model trade flows between any two countries now trade flows can be any bilateral relationship between the two countries be it migration FDI or imports and exports in Tim bergens model X represents the volume of trade from I to J mi in MJ represent the respective forces of supply and demand in the country and D which proxies for transport costs represents the distance between countries I and J we'd expect that the further apart two countries are for higher of the transport costs of trade to obtain a linear relationship we take the locus of both sides to form the basic gravity equation in order to analyze and explain bilateral trade flows the basic gravity equation is often augmented for example you can add variables for the respective incomes of the exporter and for the importer we expect that the higher the incomes of either country the more trade occurs between them Tetris paribus for example we know that the UK trades with Africa but we expect they would trade more with North America where income per capita is higher than in Africa another interesting adjustment is adding a dummy variable for adjacency for example city B in Poland and city C in Sweden or equidistant from city a in Germany however since Germany and Poland are adjacent countries we would expect the volume of trade between a and B to be greater than that between a and C even though the relative distances are the same we also see an effect when we add a variable which measures the level of remoteness from alternative trading partners for example the distance between city a in France and city B in Switzerland is the same as the distance between city C in Hungary and city D in Romania however since it is a and B are closer to a mutual market such as Germany we expect that they would trade less with each other and cities C and D who are more remote from an outside market and so must trade more between themselves in case you hadn't realized the pattern here the general case of gravity equations is just a linear model proposed by Tim bogan with the addition of a dummy variable which represents a variable of interest these dummies can be almost anything from whether there is a colonial link between countries I and J to whether or not they're in the same trade area one application of the gravity equation was explored by John McCallum 1995 when he examined the impact of national borders on trade flows between the u.s. and Canada where we would expect that the border should have a negligible effect on trade McConnell's findings were later identified as one of the six major puzzles in international economics bios failed and robots so what exactly did he find well in examining the trade between ten Canadian provinces which yielded 90 observations and the trade between the same temper winces and 30 US states yielding a further 600 observations he discovered the existence of a substantial border effect this is unexpected because the US and Canada are so similar in terms of culture language and institutions he estimated the following gravity equation where the dummy variable home is equal to 1 for interprovincial trade and 0 for province to state trade and the exponential of the coefficient D gives us the border effect which basically symbolizes how much more the Canadian provinces trade with with themselves rather than with the US states McCarran calculated this to be about 22 times there has since been further evidence of the existence of border effects such as by way in 1996 who focus on oacd countries and found a border effect of 2.5 also Hendon Mayer use data on 98 industries in 11 new countries to find a border effect of between 10 and 22 more recently still Chen news data on 78 industries from 70 u countries to find a border effect of around 3.7 as you can see these border effects vary a magnitude but they are always highly significant in a world where perfect competition existed border effects would be zero so these positive border effects imply less than per market integration so why do these border effects exist theoretically speaking the border effect is composed of two factors Sigma the ERISA T of substitution between domestic and foreign goods and T the ad valorem tariffs equivalent of the border barrier in the year 2000 evidence estimated the follow model across osed countries where Z is one of the share of intra industry trade an index of product differentiation the level of R&D spending or the level of advertising intensity all of these proxy for the level of product differentiation she finds that the coefficient e is negative implying that more differentiated products ie those with a small elasticity of substitution have smaller border effects in 2004 Chen recalls ten bergens model with the addition of a home trade done she then defines the volume of trade between countries I MJ in terms of each country's income per capita and the value of the advil and tariffs equivalent of the border between them she further defines this tariff equivalent in terms of distance by taking the natural logarithm of X IJ now we end up with the following where one minus Sigma B 1 is a distance coefficient and one minus Sigma B 2 is the border coefficient Chen further computes the way to value ratio of exports where Q is the weight of lateral exports a large WV indicates high transport costs and therefore a low level of transport ability including WV weight to value in her model reduces the size of the border effect in a similar model Hillberry discovered that the following trade barriers were not significant between Canada and the US and so trade patterns are explicable through the Bohr effect an interesting paper by row in 1999 investigates the idea that when matching international buyers and sellers proximity and a common language are more important for differentiated products and for homogeneous products and that search perilous to Trainor therefore higher for differentiated products he classifies different groups according to their own information costs will Group one he puts those codes which are traded on an organized exchange which has gold and oil where price information is centralized in group two he puts differentiated products such as Footwear and clothes where there is no organized exchange market and in the final group Group three he puts reference prices such as chemical products whose prices are in weekly in specialized magazines Clisson group three are a somewhat intermediate class of product because they're homogeneous while group one goods but two not traded on an organized exchange which is the same as group to its his methodology includes estimating a gravity equation for each of the types of goods with GDPs and distance but also dummies for adjacency colonia links language and for the european economic community using 63 countries data spending 20 years our disaggregated trade data at the industry level the coefficients on the distance variable were found to be different for each model implying that product specific information matters the distance coefficient on the group one products was less an absolute value that of group two boroughs which itself was lower than that of the model with group three products this implies that distance inhibits international trade more when goods are differentiated an alternative explanation to the existence of god border effects was proposed by wolf in 1997 who discovered these so-called border effects existed between US states which are surprising since they share the same constitution currency an exchange rate and culture therefore wolf proposes that these apparent border effects are as a result of the spatial elimination of firms anderson and van window coupe propose to miss specification of the usual gravity equation g2 emitted relevant variable bias they therefore propose a new model where goods are differentiated by place for engine and each region is specialized in only one good consumers in their movie maximize utility which depends on the consumption of goods produced in different regions subject to a budget constraint preferences are identical across consumers they derive the following model where YW is world income which is the sum of the incomes of each country and sigma is the es sisty of substitution this is maximize subject to the following constraints we now define the trade cost function t as a local linear function of bilateral distance d and whether or not there exists a border b where rho is the elasticity of trade costs with respect to distance this trade cost function is then substituted into the gravity equation which yields this scary-lookin equation where pi and pj represent multilateral trade resistance and a depending on the trade cost function which itself depends on distance and the existence of a border once we estimate this equation we find that P I and P J are correlated with both distance and the border dummy which points to the existence of a limited relevant variable therefore they conclude that trade between regions is not only determined by bilateral trade barriers but also by relative trade barriers with all ports as a result they argue that McCallum's border effect of 22 is upwardly biased due to the P's being committed variables in his specification and also due to the size of the Canadian economy the revised border effect is about half of that McCollum's at ten point seven thank you for watching and for more videos on economics and a range of other topics please see our child you

Interpretation & Understanding: Language & Beyond (Noam Chomsky)

Views:116930|Rating:4.87|View Time:1:1:42Minutes|Likes:1344|Dislikes:36
Noam Chomsky discusses some philosophical issues involving the limits of language and mind. In particular, he considers three intellectual problems: what he calls Plato’s problem, Orwell’s problem, and Descartes’s problem. Plato’s problem has to do with how human beings, whose contacts with the world are limited and brief, are nevertheless able to know so much (i.e. issues involving innate knowledge & the poverty of the stimulus). Orwell’s problem is the opposite: how human beings, with amble and reliable information, nonetheless know and understand so little. And finally Descartes’ problem, which has to do with issues of freedom of choice and action. This talk is from the College de France.

The Speed of Light is NOT About Light

Views:|Rating:|View Time:Minutes|Likes:[vid_likes]|Dislikes:[vid_dislikes]
The Speed of Causality Tweet at us! @pbsspacetime Facebook: facebook.com/pbsspacetime Email us! pbsspacetime [at] gmail [dot] com Comment on Reddit: …

Newton's Laws: Crash Course Physics #5

Views:|Rating:|View Time:Minutes|Likes:[vid_likes]|Dislikes:[vid_dislikes]
I’m sure you’ve heard of Isaac Newton and maybe of some of his laws. Like, that thing about “equal and opposite reactions” and such. But what do his laws …

Scientific Method vs. Theology – Dr. Peter Atkins

Views:|Rating:|View Time:Minutes|Likes:[vid_likes]|Dislikes:[vid_dislikes]
Peter Atkins is Professor of Chemistry and Fellow of Lincoln College at Oxford University. He is the author of several world-famous chemistry textbooks.

Interface Lectures – Re-enchanting the Universe: Evangelicals and the Rise of Science

Views:457|Rating:0.00|View Time:1:45:24Minutes|Likes:|Dislikes:
Continue exploring the intersection between science and theology at regentinterface.com, or download the audio of this lecture at regentaudio.com.

The popular idea that Christianity and Science have always been fundamentally in conflict dissolves upon closer historical examination. This is true even for popular Protestant spirituality. The significant evangelical spiritual awakening in the North Atlantic that appeared in the eighteenth century took place among those who were the first generation to accept the basic postulates of Isaac Newton and to embrace the new science. The world of nature was now neither possessed of a transcendent spiritual form (Plato) nor an immanent spiritual form (Aristotle), so how was one to understand the relation of things spiritual and things material? A number of the early evangelicals engaged with this question in a sophisticated way. Jonathan Edwards was a young undergraduate at Yale when Newton’s Principia and Opticks were first taken out of their wooden crates and added to the college library collection, and he studied these works exhaustively. So also John Wesley produced one of the most comprehensive compendia of the period of the latest findings of science. To these can be added a number of other figures over the course of the century—devout poets, artists, practicing scientists, and theologians who responded to the rise of science with “wonder, love, and praise.”

Bruce Hindmarsh took his DPhil degree at Oxford University in 1993. From 1995 to 1997, he was also a research fellow at Christ Church, Oxford. He has since published and spoken widely to international audiences on the history of early British evangelicalism. His articles have appeared in respected academic journals, and he is the author of three major books: John Newton and the English Evangelical Tradition (Oxford University Press, 1996), The Evangelical Conversion Narrative (Oxford University Press, 2005, and The Spirit of Early Evangelicalism (Oxford University Press, 2018). Bruce has been the recipient of numerous teaching awards, research grants and fellowships. He has also been a Mayers Research Fellow at the Huntington Library and a holder of the Henry Luce III Theological Fellowship. A fellow of the Royal Historical Society, he is a past-president of the American Society of Church History. He is also an active member of the Anglican Church.

This lecture series is supported by a generous grant from the John Templeton Foundation.

while I'd like to add to Brianna's welcome to region college my welcome to this third installment in interface our autumn lecture series in theology and science the purpose of these lectures is to explore the many and often surprising ways that theology and science can enrich one another to do that we therefore seek to challenge the myth the largely 19th century myth that there must be a conflict perhaps a kind of cold war between these fields of study now one of the best ways to counter mythology is through better historical understanding after all theology and science have not always existed in the way we think of them today as we will here in tonight's lecture the empirical study of nature often went by the name natural philosophy as recently as the 17th century it was a discipline that saw itself as vitally connected to the study of God what's more people of faith have shown a variety of postures towards advance in scientific understanding often exhibiting curiosity humility and even wonder rather than offensiveness as such they remain an inspiration for what we do here now our interface series is made possible by a generous grant from the John Templeton Foundation and we are grateful to the foundation for providing the means to bring you tonight's distinguished speaker Bruce hindmarsh is the James M who stand professor of spiritual theology here at Regent he is a fellow of the Royal Historical Society as well as a past president of the American Society of church history professor hindmarsh holds a DPhil in theology from the University of Oxford and is the author of three books including most recently the spirit of early Evangelic ilysm true religion in a modern world I would especially recommend this book for those who like me are looking for clarity and hope about whatever angelical ism means today as James Houston has said contemporary confusion is often the result of a historicism which is to say we often don't understand our present become because we haven't properly taken our bearings from the past it's therefore fitting that as the holder of the Houston chair Bruce hindmarsh counters this trend by telling our history with scholarly rigor and stylistic grace tonight were delighted then to have him deliver a lecture entitled rien chanting the universe have angelical x' and the rise of science please join me in welcoming in thank you David for that thoughtful introduction and thank you for coming out on this Tuesday evening if you've been taking in this whole series this autumn at Regent College on the interface of faith and science you may recall that Alister McGrath began by challenging the idea that religion and science are necessarily in conflict he argued instead that the discourses of religion and science may be seen as layered and complementary religion addresses questions that really can't be adequately addressed within a naturalistic scientific framework questions that science takes as its purview in particular questions of moral judgment what ought we to do questions of meaning and purpose why are we here you invite a religious perspective Sara Coakley then spoke on the possibility of a kind of natural theology thinking about God from what nature tells us a kind of natural theology that reflects on the layered phenomenon of cooperation as observed in the mathematical modeling of evolution a process to be reckoned with in addition to the standard account a mutation and selection she further invited us to consider the possibility that the ancient teaching about the cultivated practice of spiritual senses might be drawn upon to speak today in illness terms of the presence of God in the world and to offer challenges metaphysical challenges to the naturalism that can creep into the discussion of science in my lecture this evening I would like to offer an exploration in the history of science and faith that illustrates both McGrath's model of complementarity as well as the significance of Coakley's attention to the spiritual census I'll offer a series of historical footnotes on these things I'll be focusing on the 18th century as the moment when Isaac when the theories of Isaac Newton were being debated vigorously in society and received into mainstream culture in the english-speaking world the reception of Newtonian ism in the middle third of the century coincided with the rise of a significant broad-based devotional movement among Protestants that historians refer to as a transatlantic evangelical Awakening it was the very same generation that witnessed this transatlantic revival that also came to accept the postulates of Newtonian science we may be accustomed today for various reasons to think of popular evangelicalism as inimical to science but with only a few exceptions it was not so in the beginning my argument tonight is that English speaking of angelical x' in the 18th century largely embraced the new science but they nested the insights of materialist science within a larger vision of the universe in which God was a creating animating guiding and redeeming presence this I think illustrates something of McGrath's thesis the key note of their response to the natural world as revealed by science was wonder love and praise their devotion acted to Rhian that the new science had seemingly drained of transcendence and they often did so precisely in the language of the spiritual senses this illustrates something I think of Coakley's thesis or her concerns I'll proceed this evening by discussing first newtonian ISM and its cultural reception and then secondly by examining various case studies in which we can observe evangelical is interacting with modern scientific methods findings and practices pardon me while we deal with the technology all right still not up okay I have it here okay okay we're back in business there we go okay at the beginning of the 18th century when Isaac Newton's Principia published in 1687 was only making its way into mainstream culture it was published in Latin and so it took a while for it to make its way into mainstream culture natural philosophy was still properly a branch of philosophy this is David was saying was the word that was used for science natural philosophy and it was a branch of philosophy closely related to metaphysics logic and moral philosophy it was a part of a comprehensive vision of the unity of all knowledge in the general school ium of the second edition of the Principia inset in 1729 I think that's when it appeared in English Newton himself engaged in theological reflection and he added this he said and thus much concerning God to discourse of whom from the appearance of things from nature certainly does belong to natural philosophy to discuss God and his attributes theology was understood as a natural conversation part partner with physics in the philosophical Enterprise over the course of the century however philosophy came to stand apart natural philosophy came to stand apart from other fields with its distinctive emphasis on observation experiment and mathematics those three things observation experiment and mathematics these were key to Newtonian ISM so this diagram is from a schematic of a popular book astronomy explained on Sir Isaac Newton's principles and made easy to those who have not studied mathematics and that a great title Isaac Newton for dummies Newtonian Newtonian ISM then is just a couple other comments about that the empirical methodology of the optics so his book on the optics the the Principia included his laws of motion it included his inverse law of gravitation included a lot of math it included United's celestial and terrestrial mechanics it doesn't work differently up there than it does down here there's a lot this is one of the most important science texts that was ever written but the optics was much more experimental and the empirical methodology of the optics was what counted foremost in terms of the take off of Newtonian ism in the 18th century over time physics became became less a synonym for natural philosophy than a field concerned with the study of inanimate nature by experimental procedure and observation as it is today Newtonian ISM then is a term when it which in its narrowest sense refers to scientific theories and scientific endeavors that drew upon the precedent set in the Principia and the optics but in its broadest sense has been extended just about any idea or any activity or any group that identified itself with a growing prestige of Isaac Newton and his prestige just grew and grew in the 18th century and there were many that identified with Newton Newtonian ism was a house with many mansions modeling nature with mathematical laws adopting an empirical approach to nature having recourse to experimental demonstration all of these were Newtonian but within this there was room for a range of contestable theories contestable scientific ideas within this legacy tremendous pluralism in the social contexts where Newtonian ism was received and where his ideas were legitimated one of the most important grounds of debate related to the religious implications of all this science the Boyle lectures were established in 1690 one quote for proving the Christian religion against notorious infidels and these became a key Aurum for the development of natural theology that drew on Isaac Newton's philosophy to make a posteriori theological arguments that is arguments from the evidence from the design of the universe this is a book by William Weston one of these physical theologians trying to unite Newtonian ISM and his reading of the Bible then mode of the new natural theology would be typically called physical theology uniting physics and theology but ideas about what constituted natural theology or natural religion is reading it off of nature were diverse and could be enlisted in support of Christianity or against it several of the early theological supporters of Newtonian ism favored an unorthodox Christology many of them were Aryans such as William wisdom and when others such as Matthew Toland used the idea of natural religion the religion of nature to dispense with Christianity altogether religious anxieties rose accordingly and the Newtonian system did leave a number of important questions open for discussion it opened new questions that would have serious metaphysical and religious implications the status of matter as inert particles these inertia nothing moves until it's move matter is inert no form principle no no inner animation the determinism of mechanics everything caught in the system would cause an effect the origin of motion if everything is inertial until it receives motion where does motion originate the non mechanical nature of gravity this freaked everybody out non-contact action it sounds like magic how does that work what is gravity what is action at a distance the ontological status of void space that space is empty all of these things invited reflection religious reflection philosophical reflection on on the meaning of these things how did the knowledge of this new science make its way into the general culture as historians have pointed out it wasn't a fried egg model of diffusion with the new knowledge of the scientist at the center like an egg yolk and then an ever thinning egg white that symbolizes the dissemination of the knowledge into the culture there's the scientist and then the knowledge just flows out into the culture it's a lot messier than this it is there's negotiation there's contestation bits are picked up things that are practical or seized on Newton's ideas were vigorously debated and explored in a number of social contexts in these Boyle lectures in a number of political contexts it was seen to be the model of the English government as opposed to French absolutism and it was picked up in popular science demonstrations in polite books for ladies in storybooks for children even in new explanations for folk remedies in the practical application of scientific ideas in agriculture in medicine and botany natural knowledge didn't just flow smoothly from theory to practice but in every case there's social negotiation practical considerations local appropriations and when you when you when you realize that this is what's going on it becomes clear that the early evangelical communities that emerged in the middle third of the 18th century were one of these important local contexts for negotiating the meaning and the application of natural knowledge in the 18th century I mean we could do a whole lecture just John Wesley on Medicine John Wesley's fascination with electricity and he's mediating this to very ordinary people he attended the early electrical demonstrations he wrote a whole book on electricity he wondered if this could be a universal cure he had a little elector electrical machine he'd like to electrify everything see if it was speed healing even baldness could be cured he thought with electricity everybody was fascinated with him well so this is a context for absorbing the new science although evangelicals engaged in some detail with the findings of the new science there were some that were involved in the production of scientific knowledge their approach was largely not that of these earlier Boyle lecturers and the early apologists who work to construct a physical theology they would stand up to the challenge of deism and natural religion in one sense that kind of work had been done the Orthodox apologists had kind of done their work instead again what was the average elavil concern in these communities it was to nest the insights that are being received from science in a larger vision of the universe in which God is still emphatically present they would accept if you like methodological naturalism but not metaphysical naturalism God was still present in this sense their devotion even their ordinary popular devotion did a kind of metaphysical work above its pay grade some of you may know Charles Wesley's famous hymn love divine all loves excelling still sometimes sung at weddings by the well-meaning sister of the bride it is a celebration of a celebration of love come down God's love come down to us in the coming of Jesus Christ into the world but the hymn ends in this great parabolic motion Christ descends and then we ascend we cast our crowns before thee changed from glory into glory we cast our crowns before the lost in wonder love and praise this moment of ecstasy of Apple fascist being speechless in wonder love and praise lay Methodists all over England and Scotland and elsewhere would have been singing this hymn at the time that John's the Charles brother John John Wesley published his multi volume compendium a survey of the wisdom of God in creation here John presented the latest findings of 18th century science in the post-newtonian world it was a multi-volume compendium for editions he kept adding to it until it reached five volumes a survey of the wisdom of God in creation of all the scientific findings he could read about he revised the work he added to the work as I say until it reached five volumes the conclusion of his preface in all the volumes and all the editions though takes us full circle back to Charles as him really interesting what did he hoped this account would do for people just educating them about science he said he hoped that this would display the amazing power wisdom and goodness of the Great Creator and that it would serve to warm our hearts and to fill our mouths with what with wonder love and praise it's the very same language as Charles Wesley's him the work of God in creation the work of God and redemption invited and stirred the same religious emotions the same responsive doxology it even called forth exactly the same phrase evangelical devotion then proved enormously significant for cultivating a worshipful response to the natural world and a readiness to perceive God's continuous presence in the world revealed by science it's this kind of cultivation of certain kinds of sensibility that I think Sarah Coakley was talking about this devote reception of science reached ordinary people Wesley was confident that trades men and trades women would easily understand it leads least nine-tenths of the content of his book is among these sort that he sent his co-workers hustling for subscriptions he suggests that his followers might save up six pence or shilling a week two or three people could go together and pre-purchase a set how though did one perceive God's presence in this natural world that was described this way by Wesley and others Wesley used the language of the spiritual senses the invisible world was he said above beneath on every side only the natural man discern if it not partly because he said he hath no spiritual senses however the human person redeemed and renewed by God's Spirit could he felt be keenly conscious of God's presence everywhere in the material world they see him in the height above and in the depth beneath they see him feeling all in all the pure in heart see all things full of God they see him in the firmament of heaven in the moon walking in brightness in the Sun when he rejoices as a giant to run his course they see God in the wind and his rain and in the fields this is to directly counter metaphysical naturalism we are to see the Creator he said in the glass that is in the mirror of every creature look upon nothing as separate from God he said which is indeed a kind of practical atheism I didn't know that term existed in the 18th century a kind of practical atheism he even drew upon the language of an earlier age saying that God upholds all things as the anima Mundi the soul of the universe he pervades and actuates the hole created frame and isn't a truce as he said the soul of the universe but we find a similar attitude on the other side of the Atlantic in the evangelical awakening with the young Jonathan Edwards who came across the new science during his years as a student at Yale College in New Haven Isaac Newton himself had contributed a copy of his Principia and optics to the collection of volumes given to establish the library at the infant college the collection is unfortunately called the Dumber collection and that was the donor but this collection given to establish the infant college these books were taken out of their crates in 1718 in the middle of Edwards undergraduate education and it produced an awakening for him the new physics literally arrived on his doorstep as a student he was soon keeping his own science notebooks including a notebook called my daughter loves this title she said she's got to keep a notebook like this now things to be considered and fully written about which was one of his main science notebooks evidently there were a lot of things to be considered and fully written about why Thunder a great way off sound Grum but nearer sounds very sharp how arterial blood descends and vino blood ascends according to the laws of hydrostatics why the fixed stars twinkle but not the planets why we need not think that the soul is in the fingers just because we have feeling in her fingers why the sea is salty and the list goes on and on and on there seemed no end to Edwards intellectual curiosity about the natural world and he also explored a number of specifically Newtonian subjects refraction of light and the inverse law of gravitation and a number of these subjects as solidity he was very interested in the concept of solidity what makes something solid which was in query 32 of Newton's optics at the same time as he was experiencing a genuine scientific awakening there was a parallel spiritual awakening going on for Edwards as a student at Yale he wrote of the new sense quite different from anything I ever experienced before here's the language of the senses nature was transfigured every trees in for him to be a burning bush every cloud a pillar of fire the appearance of everything was altered he said there seemed to be as it were a calm sweet cast or appearance of divine glory in almost everything the divine glory notice was not just deducible from nature he was perceived in nature it wasn't just the QED of a physical theater logical argument it was a new kind of perception all the objects of inquiry in his natural philosophy notebooks now seemed to shine with a borrowed light from beyond the walls of the world God's excellency he said his wisdom his purity and love seemed to appear in everything in the Sun moon and stars in the clouds and bluegrass blue sky in the grass the flowers and trees in the water and in all nature he grasped all of these things with a kind of direct devotional intelligence that concentrated his powers these natural phenomena viewed this way he said used to greatly fix my mind I used to often sit and view the moon for a long time and so in the daytime spent much time viewing the clouds and the sky to behold the sweet glory of God in these things there's a new sensibility a new layer if you like he offers another example of the spiritual senses in the age of materialist science we don't have time to talk about it this evening but in time Edwards works all this out in a impressive Magisterial comprehensive intellectual program uniting science ontology ethics psychology history and theology and he genuinely tries to respond at a very deep level to nest Newtonian science within a world in which God is intimately present and he works this out intellectually but where else might we witness the sensibility I thought I'd take us somewhere different this evening to the Arts TS Eliot admired the way earlier 17th century poets could bring diverse the metaphysical poet's John Donne and George Herbert and these sorts of people could bring diverse materials together into harmony poets for whom thought and feeling were combined in a unified sensibility in their responses to the world this is before kind of fact value distinctions and all that kind of language however TS Eliot famously described a kind of dissociated sensibility that's his word dissociated sensibility that said in in the 18th century with the advent a mechanical philosophy and materialist science the new science made it more difficult to hold together the spiritual with the material after all what is the response of a unified Sensibility thought feeling religious devotion to particulate matter in void space obeying abstract mathematical laws how do you respond to that well his focus on the arts gives us a clue where to look in this period where Newtonian ism is being received and evangelicals are responding here's where we might return now to examine some of the lesser-known evangelical poets and literary figures from this period to see how they responded to the world revealed by science in their writings we see science embraced not principally with apologetic intentions not defensively or aiming to construct a rational physical theology but with devotional aspirations searching out more profound occasions for praise and we'll look at this in the work of the Anglican writer James Harvey the Baptist poet and Steele and the african-american Congregationalist poet Phyllis Wheatley those are our three remaining case studies James Harvey had been part of John Wesley's holy club at Oxford later as a hard-working parish priest often in poor health he still found time to write and published a series of what can only be called Pro prose poems on nature he wrote in a rhapsodic style that was suited to the taste of his contemporaries it's actually very difficult to read today it's um yeah the first time I picked it up I found it almost unreadable many of us would find it too flowery but it was exactly the taste of his period his mid century collection meditations and contemplations a kind of compendium of his writings of this sort was immensely popular I want to emphasize this for a moment by the mid in 1748 when he published it it immediately went into four editions there were over a hundred editions by the end of the century including imprints in London Edinburgh Glasgow Dublin Belfast New York Philadelphia and Boston is translated into sales are huge additions typically running in the range of 5,000 to 6,000 in each Edition app a friend who just published her magnum opus with Oxford and the print run was 200 books you know hopefully they'll go into reprinting soon but Harvey was widely reprinted and widely read very popular he William Blake was a great admirer he would inspire later Gothic literature and so on it's worth recalling all of this popularity the influence of Harvey since he remains less well-known today and because of the way his style went out of fashion after the first wave of 19th century Romanticism he was self-consciously imitating the sort of style one finds in the third Earl of Shaftesbury still Asafa collapse ADIZ and in part this was an evangelical tactic it was he said quote to bait the gospel hook agreeably to the prevailing tastes but his response to the natural world was no less genuine for this and he clearly found an eager and receptive audience meditations and contemplations this book is probably the most representative example of evangelical devotion as it engaged with Newtonian ideas in Wesley's survey that I mentioned earlier he repeatedly turns to Harvey for exemplary sentiments a praise in response to the terraqueous globe the meteors the heavenly bodies and other phenomenon when Jonathan Edwards we just talked about was introduced to the book by a Scottish correspondent as quote the most polite of any that has been written in an ethnic Jellicle strain in this century he quickly got himself a copy quoted it loaned it out to others and mined it for other sources to read Harvey circulates very early he sent out his program we should always view the visible system he said with an evangelical telescope and with an evangelical microscope regarding Christ Jesus is the great projector and architect that planned and executed the amazing scheme this is the Christians natural philosophy the word evangelical has all sorts of associations but here he's really talking about the gospel he's using the word Ave angelical like we talked about the four Gospels it's that we don't we don't leave Jesus Christ out of the equation he is saying and this is his sort of phrase for that his motivation is equally clear did we carefully attend to this leading principle he said in all our examinations of nature it would doubtless be the most powerful means of in kindling our love and strengthening our faith was the cultivation of these virtues when one looks around and sees all the millions of substances as an important term he said one should carry a single transporting reflection namely the maker of them all died on the cross for me how can I remain any longer indifferent anyways here Harvey developed a set of theologically robust principles for contemplating what he called the teeming earth and the smiling skies and all were Christ centered all were christological first he articulates the patristic karati o ex nihilo doctrine that God creates out of nothing Christ made them when they were not he fetched them up from utter darkness and gave them both their being and their beauty secondly he affirms the cosmic scope of Christ's Redemption Christ recovered them this is like the language of Romans 8 that all of creation is caught up in the work of redemption Christ recovered them when they were forfeited the cosmic scope of Christ's Redemption thirdly he points to the contingency of the created order Christ upholds them which would otherwise tumble into ruin or more poetically his finger rolls the seasons round and presides over all the celestial revolutions and finally he extends this contingency we'll talk about that word more later to life itself fusing the ancient language of vitalism were emanation with a newtonian idea of inert matter Christ actuates them which would otherwise be lifeless and insignificant he only has life and whatever operates operates by an emanation from his all sufficiency as he continues his metaphor for this contingency the contingency means it might it might not have been that's what contingency means it might not have been it could have been otherwise it's not necessary that the act of creation was that God wanted beings to be blessed with his blessedness it might not have been his metaphor for this contingency is economic but he merges this with the PAS line notion of play Roma or fullness in Christ pensioners they are this whole world that we observe this whole universe pensioners constant pensioners on his bounty and they bore oh they're all from his fullness it's almost a doctrine of analogous antis Harvey who had often turned thus to text from st. Paul's letter to the Colossians for this cosmic Christological doctrine he defended what he was doing as more than lawless flights of fancy and there are lots of people who engaged nature that wage is merely emblematic Glee with arbitrary pretty illustrations of doctrinal truths and he was saying this is more than fancy precisely because his observations were grounded in what he called this grand doctrine the hinge upon which they all turn and then he quotes the Christ him in Colossians that all things were created by him and for him and that in him all things consist so all told his contemplation of nature was deeply considered theologically in Harvey's final contemplation of the starry heavens and innocence he sort of begins with the tombs and the earth and he works his way up into the heavens in his contemplation of the starry heavens it's clear how wholeheartedly he embraced Newtonian ISM and observational astronomy . in fact he added a preparatory note to the reader who might have doubts about astronomy he said it may seem unaccountable to an unlearned reader that astronomers should speak such amazing things and speak them with such an air of assurance concerning the distances and the magnitude and the motions and the relations of the heavenly bodies harvey reassures this uneducated reader he said consider the cases of eclipses and with what exactness they are calculated they are not only foretold but the very instant of their beginning is determined this can be done he said down to the nicety of a moment and calculated hundreds of years into the future all this is he says a matter of fact absolutely indisputable the principles of science on which these calculations are based he said are not mere conjecture or precarious supposition they have a real certain foundation in the nature and constitution of things very robust statement still after all this has been said in his preface he said what are these things though above all there are objects of admiration rather than science science is going to be taken up into his religious devotion these sentiments were Illustrated for the reader in a frontispiece that harvey added to the second volume of his third edition he added a two-page explanation in the picture there is observational astronomy at work on the left with a telescope pointed toward the Stars can you see that there is also a natural philosopher on the right instructing students quote according to the Newtonian hypothesis there's a book sticking out from under the schematic of the planetary orbits there on the right that prominently displays Isaac Newton's name right next to Newton's name we can read the Latin words in our aunt Gloria Dei on the map words from the opening of Psalm 19 explaining that the heavens do what they declare the glory of God another student head bowed in an attitude of humility is unfurling on the far left unfurling a diagram illustrating the centrifugal and centripetal forces that combine in perfect balance to keep the planets in their orbits here the Latin motto is SiC gratia Dei or such as the operation of divine grace amidst all this scientific activity the key figure in the picture is the one most brightly illuminated just left of center on his knees he is a youth in the middle overwhelmed with veneration for the Almighty maker in a transport of gratitude and a posture profound adoration he seems to pour out his soul in those emphatic words when I consider the heavens the work of thy fingers the moon and the stars which thou hast ordained what is man that thou art mindful of him and so on he's positioned in the middle between observational astronomy and philosophy and he provides the exemplary response of wonder love and praise as I say he commissioned this picture there's a two-page description in Harvey then we see an emblematic imagination or fancy passing over into something more transcendental whatever is pleasurable or charming below he says let it raise my desire to those sublime delights which are above which yield not partial the perfect Felicity here for a moment virgil leads to Beatrice earthly loves are caught up in an Augustinian rush of desire turning again to Pauline Christology Harvey continues yes my soul let these beauties in miniature this is in the early section of the book looking at the beauties of creation of the terrestrial world let these beauties in miniature always remind me of that glorious person in whom dwells all the fullness of the Godhead bodily adding let these little emanations teach thee to thirst after the eternal fountain from the domestic the miniature and the picturesque Harvey rises to contemplate the whole compass of finite perfection and says that it is all only a faint ray shot from that immense source it is only a drop derived from that inexhaustible ocean of all good somehow Harvey has managed to graft onto his Newtonian ISM the sort of transcendental language the transfiguring language some of the very same metaphors that one finds in the previous century and platanus like Henry Moore or Ralph Cudworth and a much earlier age right back to people like a Basu J in one sense then Harvey's project was to develop a new religious aesthetic for his generation of Newtonian readers to appreciate his response to the natural world that's described by science three features of the Newtonian system may be picked out and then we can look at how he treats them in these meditations first of all the vast dimensions of the universe that's one of the things that this Newtonian picture introduces secondly the nature of void space and thirdly the fundamental forces of attraction especially gravity as for many others Harvey was led often from his contemplation of the sheer magnitude of the Newtonian universe to an experience of astonishment of utter awe he takes the reader in his contemplation of the starry heavens upward more and more farther and farther soaring beyond the moon passing through the planets to wing your way he says to the highest apparent star what would that be I don't know serious he says what would you see there only others sky is expanded and other sons and other star systems in unknown profusion through all the boundless dimensions of space and what about after this keep going he says one remains only in the suburbs of creation clearly he has a sense for the infinite and the infinite is a big deal in the 18th century in calculus and in all sorts of ways if fascinated with the infinite he has a sense for this the aesthetics of the infinite and he is aware that of the as yet undetectable stellar parallax that this implies unfathomable distances he tries to recreate the sense for the reader in his prose but then suddenly he turns all this on its head he pivots saying that all of this vastness this prodigious extent of space is comparatively a mathematical point in God's presence he's a vividly responsive to the mental vertigo produced by contemplating Newtonian space he says my very thoughts are lost in the abyss of space now at the same time for the French thinker fontanelle this sort of contemplation led to a profound sense of alienation and insignificance and being alone in the universe but Harvey provides a different response he says how I am overwhelmed with awe and sunk to the lowest prostration of mind behold this immense expanse and admire the condescension of your God that God condescended so much response of Harvey and even fontanel to the vastness of space in Newtonian cosmology may be compared to the reactions people had in our lifetimes to the famous pale blue dot photograph remember this taken by the Voyager 1 spacecraft on February 14th 1990 from a distance of more than six billion kilometers as the probe was leaving the solar system that turned around to take a picture of us in the in this image the earth is a mere point of light do you see it it's a crescent only point 1 2 pixels in size on voyagers camera Carl Sagan famously captured the feelings of a generation and when he wrote about this look again at that dot he says that's here that's home that's us on it everyone you love everyone you know everyone you ever heard of every human being who ever was lived out their lives the aggregate of our joy and suffering every saint and sinner in the history of our species lived there on a mote of dust suspended in a sunbeam for Harvey the awareness of how vast the universe is led him to an intensified appreciation of how far God condescended to save the human race it measured the extent to which Christ humbled himself to save us space wasn't as vast in the Newtonian system but secondly it was also empty and for Harvey this pointed to the sheer contingency of material worlds there's that word again his favorite metaphor for this was to describe the earth as a pendant like hanging a picture Angeles hanging in Newton's speculative rarefied ether this is not the old idea of ether of the quintessence and of a fifth element and Hellenistic philosophy above above the moon and so on but this was speculating about what it is that it operates in empty space he says what holds up the planetary and stellar orbs in space are they hung in golden or adamant I'm Shane's rest they their enormous load on rocks of marble or columns of brass no he says they are pendulous in fluid ether so what makes them amove ibly fixed in their places he says it's the work of God alone he hangs the earth and all the ethereal Globes upon nothing he rounded in his palm those dreadfully large globes which are pendulous in the vault of heaven he likes that word and by him they are suspended in a fluid ether and cannot be shaken the metaphysical correlative empty space was not thus atheism but the sense that everything is held in existence its pendant it's like it's held by God's hand a God of love and faithfulness the repeated refrain of the psalmist that the divine love and faith one has reached to the skies it's like it's taken on a vast Newtonian dimension for Harvey when you realize how big the skies are thirdly in dealing with the attractive forces of gravitation and cohesion Harvey Sabo with a beautiful simplicity and the design of the universe and quote a prodigious ballast which composes the equilibrium and constitutes the stability of things attraction is the great chain he said which forms the connections of universal nature and it raised the question he knew everybody was asking and literally everybody was asking like what is gravity what is it what is this attraction harvey asks is it a quality in its existence inseparable from matter and in its acting independent of the deity by this point you'll be no surprise to the reader that he answers quite the reverse it's the very finger of God the cost and impression of divine power a principle neither innate in matter so he rejects the Aristotelian form principle this is not innate in matter nor intelligible my mortals he does not quite equate gravity with the holy spirit directly in a vitalist way but he is willing to say that gravity does bear a considerable analogy to the agency of the Holy Ghost but in the end he concludes that even if he could see as far into things as Isaac Newton could his vision would still be as dim he says as what a mola sees emerging from a hole in the ground what really mattered was devotion so he says quote having now walked and worshiped in this universal temple having cast an eye of reason and devotion over at all having discovered an infinitude of worlds having met the deity in every view it remained only to close his meditations with a hymn of praise and that's what he does harvey was hugely successful in cultivating a devotional sensibility remember how much how wide he was read among a large following one of the many who wrote their imitations of Harvey was the Baptist poet and steel or Theodosia as she styled herself in publication living in rural Hampshire this is my next case study as a single woman with her family she published her poems on subjects chiefly devotional at 43 years of age in 1760 and her poems in her hymns soon circulated among nonconformists becoming widely known on both sides of the Atlantic in the 19th century her prosperous family were the mains day of the particular Baptist Church in the village in West Hampshire where she lived and they were as Cindy alder says attempting to negotiate a new world a provincial politeness the family was literate and literary and it seemed natural for Anne steel to compose poems and hymns from a young age she is often described as the first British Hemme writer while she was fond of recording walks in the garden near her family home and she wrote often about nature among her compositions there is a poem called this is the title on reading mr. Harvey's meditations it not only pays tribute a homage to Harvey but also raging over four pages traces his contemplations likewise from the tombs to the flower garden in the evening and an evening walk up into the night sky the poem concludes with the shining laps of heaven the stars advancing in the darkening skies these stars and planets seem so far away to the naked eye like lamps they sparkle on the unaided sight but nearer viewed in philosophic light prodigious orbs a numbered worlds arise new scenes of wonder meet our gazing eyes it's the findings of the telescope nearer viewed nearer viewed it's the findings of the telescope and the reasoning of natural philosophy philosophic light she refers to that have cultivated for an steal a sensibility a wonder in response to the vastness of the universe and the newly appreciated magnitudes of observational astronomy her instinctive responses to do what with us is to turn it into a personal hymn of praise Jesus thy glory beaming from afar great source of light illumines every star thy word informed the planets where to roll and stationed every orb that gilds the pole to the midst all the glories of the skies to the alone I raise my longing eyes the language of infor the planets seems to mean here not only to instruct the planets and form them you go here you go there but also to animate a deeper forming principle implied by creation by the word crotchy of par barbam in another poem that steele wrote an evening meditation she similarly surveys the scene just after sunset looks over the landscape and concludes again by looking up at the night sky the aim of the poem is devotional it's prayerful she says her aim is to trace thy form again and in thy works to meditate thy praise and so she addresses God as thou both a nature's author and her Lord in a nice coda at the end she encloses these sentiments of praise in the double ascent of the moon and the soil together in the dark ming sky serenely ascends the silver moon she says attended by its train of innumerable stars and the vast expanse of space but then about ten lions later instructed by nature's author and our Lord in the final terrset you can see it is the soul that rises to conclude the poem led by these my raptured soul ascends on Heaven's heavenly contemplations soaring wing to thee the sacred source of all perfection psyche as the moon and the stars rise in the night sky her soul Rises with them again this transcendental impulse and steel like James Harvey like Jonathan Edwards like John and Charles Wesley responds to the 18th century ideas of nature above all with devotional intent and spiritual aspiration this same devotional attitude that we've been seeing appears in the first published book by an african-american the devote evangelical poet phillis wheatley taken forcibly from near Senegal West Africa at about seven years of age and transported to America the young girl who would be named Phyllis Wheatley was purchased as a slave in August 1761 by the evangelical Boston Taylor John Wheatley as a gift for his wife Susanna Phyllis was one of the millions of enslaved Africans who had arrived in the Americas in the 18th century unlike most of those slaves though she was given unusual freedoms opportunities and education and treated like a family member in the Wheatley home precocious intellectually gifted she developed her talents quickly and was soon sufficiently literate to be corresponding with educated clergy and composing classically styled verse her writings show an extensive knowledge of the classics probably in translation along with an education in geography history politics literature and as we shall see astronomy she was baptised at the Old South Congregational Church in Boston in 1771 where George Whitfield and sometimes preached when in Boston it's not clear that she ever heard him however it is clear that she held him in the highest regard the most famous preacher of the 18th century George Whitfield it was her elegiac poem on the death of that celebrated divine George Whitfield 1770 that brought her international renown she was soon drawn into the circle of many influential of angelicus her poems on various subjects religious and moral 1773 was published when she was scarcely 20 and it was dedicated the evangelical countess of Huntington her writing her Travel deepened her evangelical connections and introduced her to abolitionist circles gaining her freedom in 1774 she issued a strong condemnation of slavery in a letter to Samson akhom she was married in 1778 but little is known of her life after that until her death in 1784 so she's a very significant figure in her own right her life stands uniquely at the crossroads of religion race gender literature as recent scholarship has been keen to explore it's also clear that she appropriated 18th century literary culture creatively and intelligently and in a remarkable short period of time and did so in a way that reflected her own concerns her work showed as one critic comments that culture was and could be the equal possession of all humanity Jennifer Billingsley argues that Wheatley established a new poetics out of the classical tradition by placing this is important an attitude a wonder at the center of her work but instead of simply being an object of wonder as an enslaved female poet that she's kind of like this exotic specimen of wonder and look at what she's doing she directed people's attention she directed their wonder to the objects of her own attention although most of her verse was modeled on the classics these objects of her attention included the natural world and subjects of scientific concern she referred to Isaac Newton as creations boast and one of the favorites of the skies in a poem of blank verse directed to those privileged enough to study at Harvard Cambridge New England she exhorted them ye sons of science to unite the concerns of scientific inquiry with those of devotion she's exhorting the Harvard students as they were given opportunity to quote scan the heights above to traverse the ethereal space and mark the systems of revolving worlds she reminded them still more to contemplate Christ crucified for a fallen human race he deemed to die that they might rise again and share with him the sublime esc' eyes without self pity she marked well the injustice of her situation excluded from university education but then assumed the role of teacher to the students and offered a devotion that took astronomy scanned the heights up into a more transcendental vision sublime is disguise through devotion fascinating another poem treats nature very much in the spirit of other of Angelica 'ls it's called thoughts on the works of Providence and it's an ode not just to creation but to God is the creator and it stretches out over 131 lines of pentameter she begins seeking light divine mckenna muse to adequately praise the god of earth and sky as round it's center moves the rolling year she develops the imagery of rolling or revolving from the point of view of observational astronomy first the earth like a vast machine revolves around the Sun but then also from our point of view the celestial fear sphere of fixed stars rotates or whirls on the celestial access every 24 hours she says adored forever be the god unseen which round the Sun revolves this vast machine though to his eye it's mass a point appears adored the God that world surrounding spheres again that same idea like in James Harvey that is like a point in God's sight the word adored in this poem comes around again it revolves just as the earth comes round the Sun and the celestial sphere around the earth she recognizes like Harvey that all this immense machinery is a mere point in God's perspective and that the agency is divine as she continues she speaks of the Sun ruling the sky as in Psalm 19 but now she gives her first indication of incorporating new scientific detail as she describes the astronomical unit the Earth's Sun distance in terms quite accurate for her day saying of miles twice 40 millions is his height and yet his radiance dazzles more mortal sight what's the distance from the earth to the Sun anybody right so it depends on the aphelion or the perihelion and where you measure it from right in terms of modern measurements so in the 18th century the distance given by scientists is 86 million miles about 86 million 50 1398 miles and it's precisely the importance of this measurement that had scientists traveling the globe the first international scientific project sent scientists all over the world to measure the transit of Venus during Wheatley's lifetime she got the calculation right in terms of the science of her day incidentally the much more privileged much more educated Augustus toplady he's the author of the him Rock of Ages Calvinists in England got it wrong about the same time adding a zero to the millions decimal place more than once which meant he was only wait for it seven hundred and seventy four million miles off but she got it right but again her instinct is to respond with devotional sensibility to the scientific calculation of the Earth's Sun distance astonished that as such a distance still the Sun warms the earth and imparts a vigor to all the growth that the earth remains true to her course she exclaims Almighty in these wondrous works of thine what power what wisdom and what goodness shine and are thy wonders lured by men explored and yet creating glory an adored the real wonder for her is that these things might not lead scientists to astonished adoration of their creator it was finally not all about reason in the age of reason love answers to eternal love and invites wonder emphasized by the enjambment in the opening couplet of the final stanza with the word appears followed by the sage aura infinite love where ere we turn our eyes appears this every creatures wants supplies as for Julian of Norwich earlier it was love for Wheatley that was most fundamental to the universe all that remained she said was remand to shake off his ingratitude and swell the chorus of eternal praise Phyllis Wheatley demonstrates what a dynamic force evangelical devotion could be in relation to culture and science how readily this could be taken up and appropriated far far from the center of metropolitan power and privilege to sum up finally we have surveyed the response of several Evan Jellico's in the 18th century to the new science leading figures such as the Wesley brothers and Jonathan Edwards as well as these lesser known figures James Harvey and Steele and Phyllis Wheatley there are many others we could look at to be fun to look at such as the practicing scientist Isaac Milner at Cambridge who held the Lucasian chair Isaac Newton's old chair at Cambridge or the artist and amateur astronomer John Russell in London this is actually his pastel of the moon he used to study the moon with his Doland reflector telescope in his backyard in London six hours a night for two decades taking measurements with his micrometer of all the details of the moon and he produced this vast monumental pastel of the moon that hangs at the Museum of the history of science and Oxford there are many more figures we could look at but I focused tonight on these literary figures in the hope of capturing something of the religious sensibility that went along with the new science for evangelicals there was a sustained concern to perceive God's immediate presence in the natural world and respond with loving devotion along with the new physics had come a tectonic shift in metaphysics with enormous spiritual implications there was now literally nothing inside matter no apparent spirituality to the observable universe natural objects had been spiritually scoured this was Newton's v's inertia principle prior to the advent of materialist science the dominant outlook was instead to view nature as having an era star terms within itself a principle of motion there intelligible forms the shift away from this was spiritually troubling enough for Ralph Cudworth to write nearly a thousand pages from his college study in late 17th century Cambridge trying to expose the atheism implied by modern natural philosophy the word dead came up often in his description of the world described by the new science it was he said a world of dead and stupid matter a dead cadaverous thing cadaverous a great word a dead and wooden world and so on the of angelica's came on the scene after the initial ferment over all this had largely subsided after whole generation of religious thinkers had debated how best to respond to the Newtonian system if they paused to look around they might have observed it looks like we're all Newtonian now so how would the devotional instinct to look upon nothing as separate from God be maintained rat ciothes ination arguments from design were well and good but there had to be a more direct experience of God in nature an immediate consciousness of the presence of God this is what we have seen witnessed – in our case studies the devout response to Newtonian science was evident not only in the writings of John and Charles Wesley and John Jonathan Edwards but a chorus of other voices that take us to the end of the century when the natural sciences were becoming more defined so whether as poets painters medics chemists theologians philosophers have angélica holes in their various ways continued to offer up the praise of the mute creation scene now through a Newtonian lens or as Harvey said through an evangelical telescope thank you [Applause] well we now have the opportunity to interact with Bruce hindmarsh over his superb lecture on how evangelicals responded to the rise of science we'll begin with a short response before we open the floor to questions for 20 minutes so you can be formulating your questions now and again we plan to end at around nine ten let me turn now to introduce tonight's respondent Amanda Russell Jones is a sessional lecturer in history and interdisciplinary studies here at Regent Amanda holds a PhD in history from the University of Birmingham which is focused on the life and work of Josephine Butler here at Regents he teaches on several courses including women church and state as well as Technology wilderness and creation which he teaches from a sailboat around the Gulf Islands welcome Amanda evening I should perhaps Ayers respondent right at the beginning that I am the recipient of a dubious award for science at the age of fourteen I was given a school science prize for my performance in physics chemistry and biology in the end of your exams it was dubious not because I'd cheated but because the chemistry department made the mistake of setting exactly the same exam they'd set three years previously and my revision partner had a sister who has three years older and went on to medical school so we had the corrected answers in front of us we confess but the school decided to still give me the prize my father however who was a scientist said you're not a scientist you're a semi scientist he was always very encouraging about what I did but he was right and it was only years later that I reflected upon the fact that I came out of the bookstore with him having spent the prize holding a copy of the history of science in my hand I think he probably had more idea of where I was headed in terms of history and the queen of Sciences rather than anything to do with test tubes but thank you David and Bruce very much for all the work that's gone into putting this evening together in the very stimulating and fascinating lecture that we've just heard and I believe this is an important piece of work Karl Barth's once said that people should read with the Bible in one hand and the newspaper in the other but in more recent years scholars have realised that that was exactly what people were doing and have always done and are still doing today they were reading and applying the Bible in the light of what was going on in their contemporary culture they were bringing questions to the texts raised by the perception of how their culture was changing their beliefs challenging their beliefs their theology was a contextual theology and Bruce's work has given us windows into the 18th century Christian mind as men and women grappled with and reflected on the discoveries of Newtonian science and how it made them see the world in a new way indeed may them see God in a new light through your thorough knowledge of the period and your extensive research on this topic on both sides of the Atlantic you've made the case that the earlier angelical communities in the transatlantic awake Awakening were an important local context for negotiating the meaning and application of natural knowledge in the 18th century as you as you told us the new science literally landed on Jonathan Edwards desk in the library and demanded study and response but you've also shown us very clearly how the response of early evangelicals was one of doxology as they looked through the evangelical telescope and the evangelical microscope the work of God in creation and the work of God in Redemption as you say aroused the same religious emotions in the same response of doxology which the characters you have highlighted tonight expressed eloquently in prose poetry hymns and so on I was also pleased to find women in a lecture on the history of science until since it was not until 1945 that the Royal Academy of the sciences changed its rules so as a woman could become a fellow which was actually the same year that a woman could first study theology in Oxford I want to focus because I was invited to bring my own particular interests into this conversation on the interface between theology and science I want to focus on women the Bible slavery and women's biblical interpretation so there are two parts of your lecture that I want to consider which will become two questions that you may like to comment on before others bring their thoughts and questions so I want to comment on Phyllis Wheatley and and Wesley's tradesman as far as Wesley's tradesman can are concerned I question whether nine-tenths of people had the literacy or the money to appropriate physico theological doxology through Wesley's book but rather than the negatives could you say more about how these ideas were transmitted to the average weaver in the pew or to the man or woman who is working on the land I'm intrigued by the electricity machine and I was also intrigued by the children's books but turning to Phyllis Wheatley who's one of my heroes you draw attention to Wheatley's importance as you say her life stands uniquely at the crossroads of religion race gender and literature and you make clear that she appropriated 18th century literary culture creatively and intelligently in a way that very much reflected her own concerns I want to argue that the importance of Wheatley is expression of the Creator God and her response of wonder love and praise is even more significant than you claim and maybe the most profound example of Fisk o theological doxology we've come across tonight she holds the extraordinary place in American literary tradition of being the first published african-american and the first published african-american woman and she was and is an inspiration for those who follow not least because she had been interviewed by a panel of 18 men to verify that she wrote the poetry herself since is Bruce alluded there was a long-standing skepticism as to whether the African could ever create formal literature or master Arts and Sciences our focus is primarily transatlantic spiritual awakening and science but I want to turn up the volume on another element of this transatlantic world that evangelicalism was developing in ongoing transatlantic slavery so my comment is on transatlantic slavery and its relation to theology and science and ultimately to Wheatley's writing for part of the argument as to whether slavery was biblical hinged on whether the non-white person the African or the Savage was human or had a separate origin to adam polygenesis was more firmly on the agenda at the end of the 18th century and in the 19th than in the period you are considering nevertheless the ideas have been around since vaunts walter and you mentioned which field that you discussed with wickfield's and slavery in your book and I appreciate that time tonight did not permit as so much here but Edwards and Whitfield were slave owners who did not free their slaves upon their deaths indeed the scholar Eugene Genovese makes the point that Whitfield was a hero to the planters since he argued for the introduction of slave owning in the state of Georgia where it did not at that time exist and although he warned slave owners that they needed to treat their slaves better than they were doing he was nevertheless seduced we might say as were others by the notion of the godly slave-owner and the good that could result for the extension of God's kingdom through the labor and the profits of slavery and Bruce as you make clear in your book we comes and in your lecture Wheatley comes to the attention of the public through her elegy written on Whitfield's death and commentators have pointed out that she extolled Whitfield as a servant of God and evangelist and makes only a small reference to slavery so Wheatley says of Whitfield thou didst in strains of eloquence refined inflame the soul and captivate the mind and Wheatley quotes Whitfield are saying various things one of them being Whitfield saying of Christ take him ye Africans he longs for you impartial Savior is his title Jew if you will choose to walk in Grace's Road you shall be sons and kings and priests to God some have seen this as Wheatley exhorting those of African heritage to accept wickfield's Savior which I'm sure she was but I think there is more to it when Whitfield spoke of the impartial Savior presumably he was forensic Lee applying the work of Christ to the salvation of the slave as much as to the free person but as a slave owner he was not saying slave and free were equal in every other respect in God's eyes but I believe the impartiality of God meant more than that sweetly is she in common with other enslaved African Americans hearing the Word of God but hearing it in a different way drawing a distinction between the God of the Bible and the God of the white person the Church of God and the Church of America I would argue that her knowledge of the Bible and to a careful observational doxological study of God's second book the book of nature led her to the assurance that God was impartial . . the witness of the Spirit within confirmed her confidence that the God of all creation was her God she wrote that in bringing her from Africa God had brought her out of Egyptian darkness but that now she faced modern Egyptians in the promised land and I think she's conjuring a countercultural theology harvey's theological statements of faith is magnificent and weren't one longs to see it appreciated in contemporary church life including his sim is summary we should always view the visible system with an evangelical telescope and with an evangelical microscope so my question what then accounts for the 18th century evangelical blind spot on racial equality the long hours night after night systematically paying attention to the moon and the stars the rapt minut observations of drops of Jew and spider's webs and yet the complete inability to look at a black face and see it or rather him or her as a mirror of oneself and a co-equal image of God you say that these evangelicals quote nested the insights of materialist science within a larger vision of the universe in which God was emphatically present I believe that Wheatley did that and you say that their devotion did a kind of metaphysical work above its pay grade and I believe that Wheatley's devotion to God did indeed do a metaphysical work above its pay grade her devotion to the god of the Newtonian science that she so carefully referred to she too saw the human person renewed by God's Spirit could be conscious of God's presence everywhere in the material world but in her case everywhere including in the full humanity of black as well as white it this evening our primary concern has been Newtonian science and its influence on worldview including the theology of evangelicals in the transatlantic awakening but slavery and the fundamental question of the humanity of the African is part of that cultural context a second major debate at the interface of theology and science is at play at this time and in this transatlantic space maybe it is in Phyllis's January Thanksgiving to God for the life and work of Whitfield which led to her being the first published african-american that we see the mystery of the grace of God maybe just maybe it is Phyllis named after the ship that transported her into slavery who has the most significant physico theological doxology lost in wonder love and praise to the God of all creation in whose image as a female black enslaved poet she knew herself to be [Applause] [Applause] Thank You Amanda British tremendous just a couple comments in response I in terms of you know Wesley's sort of tradesman and so on I think there has been quite a lot of work done on literacy among the Methodists and it really is amazing so Vicki Vicki told her Burton's book on spiritual literacy and about reading among Methodists and so on there really is an entire educational program that is pushing out literacy and it is it's a it's it's a tremendous sight in terms of education literacy spiritual literacy and then people also then as Vicki tore Burton points out people then being able to speak for themselves not just to read for themselves so maybe maybe it wasn't 9/10 maybe Wesley exaggerated but maybe 7/10 people could read his work I don't know but I think but I you know he also said or somebody said that it was the most unique work of its kind the most compendium as kind I think it probably was true there was no other anthology like this and Wesley was brilliant at being able to speak as he said in plain words to plain people so it'd be interesting to see if we could find witnesses from ordinary people and that we're actually reading the book I'm not aware of aware of those and then in terms of Phyllis Wheatley you know I thank you for what you put how you've drawn all that out I think a couple a couple things I'm very happy to yield to her the place of most significant in terms of for the reasons you've said mostly what I wanted to do in a sense is just take her seriously and pay tribute to her ideas and just listen to her ideas on the specific point of Newtonian science but I think you bring up a really important point that in the age of reason there was a kind of inability to see what seemed to be unreasoned there was a this black spot on the age of reason that's that's race you know and I think even something about the Newtonian world in which truths and spaces are all interchangeable all the same and that things are everywhere the same and replaceable and an appreciation I think that that as you put it for things to be everywhere included it would include the full humanity at both black and white so Peter Choi many others have written about sort of Whitfield on slavery and about some of these issues there wasn't so much my focus tonight but I think you're absolutely right is in terms of the appropriation and the agency of Phyllis Wheatley like David George in among the Acadians and the black diaspora that were loyalists that were transported to Nova Scotia and then that were transported to Sierra Leone and as they marched ashore in Sierra Leone what did they sing the sang the song of Miriam the song of Exodus it was Exodus narratives that they were of liberation that they were appropriating and so I think it's really important to pay attention to the ways in which that kind of language is being appropriated so we have examples as we would say good and bad examples of sort of Angelica was responding to issues of race and issues of slavery and so on some that can see earlier and some later the egalitarian implications of the gospel and did I wrote a book about John Newton and from Moses life the author Amazing Grace he was blind to to these issues of race slavery chattel slavery and it's only late in his life when he came to see it that he did what he could that he joined in the efforts for abolition that he wrote against the slave trade and so on but the dawning of that consciousness for many came quite late it's a little bit like how people responded to the bloody code the Capitol legislations of the 18th century the initially there's that kind of spiritual language of egalitarianism and as spiritual solidarity with the prisoner that a sense that's the foundation later for John Howard and prison reform but I think often people can't initially see that and after summer comes much later so thank you of 15 minutes for further questions if you'd like to ask a question please proceed up to one of the microphones at the back row of seats and if you're with us on Facebook live via our live stream feel free to submit a question in the comment section there over to you sir Bruce wonderful lecture thank you so much in the year thirty seven fifty in a post-apocalyptic post nuclear holocaust world in the charred remains of the bottle in library a book of Phyllis weeklies writings was found in the year thirty seven fifty what did the text form and source critics make of the possibility the thought experiment 3750 you discover this like you'd never known it before I think I think there'd be a sense of wonder a sense of what would people say in the year thirty seven fifty I think I think surely there'd be a sense of delight that that somehow her her work found the light of day first looked by an african-american writer that finally such a thing appeared I don't know what to say beyond that but I like the thought experiment I used to do that my students imagined that a bomb went off and no book was left but this one what would you make of it so thank you very much Bruce for a wonderful lecture I keep thinking about how do we apply these principles to our our days now and sometimes I guess I'm old enough to feel a bit of humility around how we look back on people at and not that the things that they did like owning were good but when I look around me in the work that I do and you know we're friends and you know some of those things that that why is it that we there always seems to be in an age where we in whatever age or in that we don't see the in that glass the humanity of the other and in our age now I think the race issue and the foreigner that continues but also the very young people with disabilities people at the end of life are seen as lives that are not worth living and so how do we it strikes me that some of the ways that this was addressed in the age that you're talking about was through the arts and that was how they got to people so what lessons having really kind of stewed about this what lessons do you think there are for us today to take this out to a wider world thank you that's a great question it is haunting to think of what are those things in my culture that are written in letters too large to read what are those things where I am captive there's a kind of captivity to some of those ideas what can we do to be aware of that and how can we operate with a kind of humility what are we missing I think and you mentioned some examples of that one of the things that came to mind Margaret there was a paper I wrote a few years ago on spiritual formation interacting with social scientists and one of the things that I tried to draw upon I'm thinking of the spirit of wonder and how that can be operationalized in a sense both in terms of humility and a kind of chase nough sin what we know and I kind of epistemological reserve but also how that can lead us to treat other people so if if there's that sense that we'll the language that it can be used about when speaking about God we can talk about apophysis there's a point where we run out of words there's there's more to God than we can take in and so whether people call that negative theology or apophatic theology we reach a point naught of emptiness but we can't take in anymore and if creature is made in God God's image bear anything of the image of their creator I think the same is true of creation that wonder is being suspended and wonder in the face of all that is created and especially the analog yet personality ought us the analogy of persons that this is a person made in God's image means there's always more here than I can take in and I treat other people with a sense of wonder and a sense of awe and a sense of humility and I think so I think it has real practical implications wonder is not just being lost in a reverie somewhere but it actually means that we stand amazed in the presence of Jesus the Nazarene we stand amazed in the presence of all that God has made and we therefore treat it with a kind of love and reverence and attention love wants to explore every detail of the beloved love stands in awe of the beloved and all of these things I think invite a certain kind of response to people and to the world God's made for our next question let's go to the live stream and see if we have something there pastors artists reflecting on the discoveries of modern science for example relativity quantum physics what might hold them back what could open the door for more exemplars of modern Christian leaders and others reflecting on modern science I went to the lecture by Collins the geneticist here at UBC on on the human genome I think he was somebody who his response and him talking about his faith was it was deeply winsome giving it a Francis Collins giving it an account of the way in which his I think it was precisely and he was talking he used the language of finding it deeply satisfying the relationship of his faith to his his science that was one of the examples that comes to mind trying to think of the the cosmologists who wrote that little essay what do the heavens declare I think there are many examples I just can't think of them right now today as well as in the past but as a historian I would usually say not my period you know but not necessarily uniquely to them about all the darkness you know the cruelty from the industrial machine the cosmic machine people being drowned in storms and infants dying how did they relate all of that Providence and so on to this new order that they were observing like how did they incorporate it into their faith I suppose as well I I'm having difficulty formulating yeah what I'm trying to say yeah I'm trying to think actually in the in the advance of I mean industrialization and the developments in modernization in the 18th century there are people who were very much left behind there's the djinns crisis in the 1720s that led many people into addiction and they would sell their tools and and so on there was the enlargement of parishes outside of places like Bristol with early industrialization that leave left people without any kind of social systems and so on and one of the I think astonishing things is the way in which Wesley in particular his mission was to every one of those people it was to the outsider anybody who was outside anybody who felt too dirty to come to communion in the church you know they talked about seeing the gutters the tears flowing down the faces of the coal miners as they would hear some of the preaching and he genuinely Wesley was happy to meet at 5:00 in the morning with 10 people working class people in a parish and then move on and he set up the first micro-enterprise loans in England to help some of those people who were lost in addiction and suffering in London to regain their tools and to be able to go back to work and then paid back the micro-enterprise loan to give to somebody else he established the first Free Medical dispensaries in England he they gathered the Methodist class meetings began by gathering a penny a week from people to be able to give to some of those without the social supports and so on so I think I mean this is a really important movement that had this is not the great reversal wherever Angelica was just left behind any kind of social concern this was born in the midst of social concerns for the outsider Wesley is very last sermon that he gave when he spoke where his voice was cracking as a ninety-year-old manner somewhere around 90 years of age was about the wedding feast and it was about his appeal to the outsider and the last thing he said is still there is room and his voice broke still there is room so I think there is a very deep concern for those that have been left behind by whatever kind of advances we're going on in society I found your talk very very interesting and also it was like the response of the respondent I sorry I didn't get your name and it's very interesting exchange what I want to talk about is more how does the the Industrial Revolution the scientific revolution sorry get into the masses because we the most of well many of them are of course illiterate but and one way they do of course like in England I'm more my field is more Central Europe and but in England there's a fantastic plow it's not been invented the moldboard plow was developed in the 8th century already but it was an incredibly efficient plow that helped you cultivate lands that were otherwise very marshy and not wet and not dry enough to cultivate so so in that way even the illiterate masses would find out what science was accomplishing for them especially technology as a couple of instances fraud that I'd like to share from Central Europe where the masses are again being affected Joseph the second the Austrian Emperor called an enlightened monarch introduced what we would call today green burials he thought people were the poor people were wasting way too much on very expensive funerals and so and also worthy fertilizer so he had he simply passed a law he didn't have a parliament or a Supreme Court to bother him so he simply passed a law that from from now on all bodies would be wrapped in a burlap bag put under underground again six or eight feet whatever they are and then they throw and your sir if you could maybe summarize a little and pose a question just for the second time because I'm sure but thank you all right there in terms of how the Scientific Revolution reach the masses I think this is I'm it's a fascinating period in terms of the the culture the explosion of the periodical press so it was a magazines and newspapers and so on there's a lot that's being published there our popular science demonstrations especially with electricity all the tricks you can do with electricity for people but it's also a part of everyday life for people things like navigation this was the century of the hunt for the accurate way of calculating longitude at sea which had enormous implications in the age of the Merchant Marine for their ships crash or not you know whether they whether they found her or not and all the efforts this is Isaac Milner we didn't talk about he was a part of the longitude story in the 18th century of calculating and all the the new ways of calculating until a chronometer could be made that could be taken to sea of the accurate Greenwich Mean Time and so I want to keep track of all that and so in a maritime society they you know they are aware a lot of people are aware of different developments and sites but it's local it's contextual it's it's practical and it's it's sort of building up in a sense there wasn't sometimes it was actually the French philosophic aim up the idea that there was a Lockean Newtonian synthesis that just took over society just simply took over society and it was basically secularizing that's not what happened it wasn't that simple grain we have time for two final questions we'll start with ross harvey invoking the spirit as accounting for gravity i wondered if you could comment on what you know so one one tends to see and people such as you've been commenting on that the Christology there's lovely to see the Christology involved in creation do you is there a profound pneumatology around Providence for example in Edwards in Wesley I would direct people to a fine book by Ross Hastings on Jonathan Edwards which talks about his new mythology and in all seriousness I would and I think it's I think Edwards does that kind of work in many ways doesn't he in terms of Edwards is fascinating he develops some forget who it is we calls it the funky factor in Edwards but he develops a kind of philosophical idealism and by there's ways I think his whole program developed and he understands God's agency God's spiritual agency and the world actively every moment in the world and it actually begins I found this argument five times in his science notebooks by reflecting on what I think is Newton's query 32 in the optics on solidity and cohesion and Edwards reflects on what is it that make something solid mass II hard particle and makes it solid and indivisible and he he says it's not so much that it's a substance that's hard as it's a force that is resistance one part of space resists another part of space and then he inserts Ross knows all this this is occasional ISM that what happens when we consistently see something solid or we see something consistently happening like a law in nature is God is every moment doing it and that's his occasional ISM God is intervening at every moment and acting so consistently that we call it a law and in this kind of way and I don't know that directly sort of Trinitarian the spirit you would know better than me but he sees God's Spirit active in the world every moment I'm not sure I think most of the other figures I'm looking I think it is Christology it's more Colossians that they go to but and our final question here thank you for your lecture Bruce a little while ago is reading and came across a letter that John Wesley wrote very near his death I believe is within a year of his death encouraging William Wilberforce in his work of abolition and that was interesting to me listening to to Amanda's you know with with Whitfield and Wesley being exact you know well their lives overlapping constantly and their ministry is very similar but I'm curious if the difference between their positions on slavery has anything to do or if you know it has anything to do with Wesley's embrace of sort of an ancient if ik paradigm if you've come across any link there or or what you account for that difference between Wesley and and Whitfield on that issue it's a very good question I'm not sure I'm not sure in West which field would never be accused of overthinking a problem and and he he Whitfield basically didn't engage with science very much at all and a lot of it was practical exigencies in Georgia and so on and how he wanted to run the orphanage it was going into debt and so on and how he sort of backed into slavery in a sense even though as Amanda pointed out he was much loved in his rhetoric and so on appeal to many African Americans and slaves so I wonder if in part I mean the Quakers seem to be having more advanced views in terms of slavery if you want to call it that and and and but Wesley Wesley is one of those who did too exactly what his reasoning was how he got there it's an interesting case study to contrast Wesley in Whitfield it may be that one of the things and this alludes to Ross's comment about popular evangelism and episodes in the 20th century and so on it may be that this movement that arose that is has an educated leadership in the 18th century the relationship between if you like elite and popular educated and the kind of popular culture there's always danger in a popular movement that things can kind of go off into sheer populism and things can become unthoughtful or things can come reactionary and I think as a movement evangelism does require how is this for an advertisement a place like Regent College where people can be can be thoughtful about these things and I think Wesley was genuinely probably more thoughtful about these things

What is the Scientific Method? (1/2)

Views:4348|Rating:4.89|View Time:8:38Minutes|Likes:45|Dislikes:1
Video explaining the scientific method using examples from physics.

With this video I try to revert some common misconceptions about science, mainly by addressing what the words ‘theory’ and ‘evidence’ actually mean.
In the next episode I will follow up on Ockham’s razor and the issue of conspiracy theories in physics.

As promised, here are some links for further watching:

Also, I would like to thank Mike Gallis for allowing me to use his animation. Please check out his website for other cool physics animations:

Also, if you would like to do further reading on the actual original Cavendish experiment and what it was actually meant to find, please have a look at:

Thanks for watching!

Part 2 coming very soon

hello everyone and welcome to another episode South ask a physicist I'm finally back and ready to answer many more of you any interesting questions however before I get back to that there is something I would like to get out of the way a special episode if you will telling you about what physics and science in general really is about so I hope you enjoy this very special episode on physics and the scientific method well there have been many YouTube videos about this topic already and i welcome you to check them out I know it's him down my description however seeing that I come from a physics background I thought it would be good for me to give my two cents of the topic and you're there well in essence all scientific work including chemistry and biology is based on the scientific method while all sciences including chemistry and biology adhere to the scientific method it can be most clearly understood if you look at it from a physics perspective allow me to illustrate that in essence in science there are only four things we have observations theories predictions and evidence well so what are these was in essence assigned us for an observer process make an observation then he will come up with a theory to explain this process and to explain why it occurs then from that theory he will derive a prediction of other processes that should be happening or other effects relating to that process lastly you will test these predictions and if they do occur you will have evidence for that Siri or if they don't evidence against it and then all that science really is about coming up with theories to explain physical processes and then testing those theories to see as they help us to explain the world around us this method holds perfectly fine for all physical series so let me give you an example an example of an observation might be that when I let go of this pen it will drop to the ground yes always works now that we have motivation you need to come up with a theory to explain his hand now obviously nation times readers to assume that the earth was flat that every single is uniformity fall in one kingdom action this year we can easily this proven by the fact that two of us rounds and seems to in fact fall to all sorts of different directions depending where you are a much better theory must of course developed by Isaac Newton who gave us the theory of gravity and as I've mentioned before this theory states that all objects that have mass attract each other towards each other depending on the amount of the mass and the distance towards each other now this theory perfectly explains why the things we let go off drop to the ground I they are attracted to quartz the earth which is very massive so what predictions test this theory make well the series is that all objects at of mass attract each other so if I was taken two masses put them next to each other and given that forces which repels them are lower the car tation attraction between them they should slowly but surely move towards each other and well lastly testing this prediction it has been done and it's quite simple to do really the methods you take two heavy balls you suspended next to each other on two metal rods as shown in this graph and you attach a mirror on the pivoting point now you shine a laser pointer on the mirror making a projection on the wall now if the rod intestine bureau was to turn due to the gravitational connection of the heavy masses the laser projected by the mirror which move much more noticeably than number itself this even a small change in location of the metal balls will be indicated on the wall that you're projecting laser on and although the movement of the balls will be very subtle due to the laser you'll be able to see now this experiment has in fact been done and is being done many universities around the world for undergraduates to try out so and this is in fact evidence of illusions theory of gravity to run you some Rogan we ask the observation we have theory to explain it we have arrived the prediction from the cre and we have tested in thus providing evidence for Newton's theory a far more important protection perhaps is that was able to derive the motion of the planets around the Sun and the motion of the Moon run to earth from his theory and this predicted movement was in fact matched by the movement many astronomers at a time observed ie it matched Kepler's laws of planetary motion thus providing yet more evidence from in theory so by application of the scientific method we have already seen the dead to very credible pieces of evidence for a Newton theory one in fact they have been many more which would lead any sensible person to come to conclusion that this isn't that a valid theory explaining the makings of our universe however isolated from behind Stein it is not us was observed by the astrophysicist rning the gravitational force postulated by Newton does not only attract object at half-mast watch each other but showed that in light which was prudent to have no mass at all follows a curved trajectory when traveling near a heavy object like a star or planet this showed that Newton Syria was at least not entirely accurate even though it passed every test it was supposed to for that the correct theory that must develop out of which is of course general relativity has turned behind time but I don't want to dwell on that right now the important point GA said no matter how much evidence fight forgiven theory it only takes a single piece of card evidence to invalidate it now in fact while we think that general tivity is the answer now one day maybe we prove that wrong as well now seeing that any given theory could be disproven that simply you might wonder it's a is any certainty at all in the scientific results that we find and I suppose in essence they isn't but there is not the point while the theories for which we have evidence which we have not disproving yet may be wrong they are still on our best guess we have an understanding the universe you see the scientific method is not necessarily but give them a certainty what what the universe is like but it tells us our approach to finding out more about that me is the art of science

Feynman's Lectures on Physics – The Law of Gravitation

Views:140426|Rating:4.92|View Time:56:4Minutes|Likes:1700|Dislikes:29

“In this chapter we shall discuss one of the most far-reaching generalizations of the human mind. While we are admiring the human mind, we should take some time off to stand in awe of a nature that could follow with such completeness and generality such an elegantly simple principle as the law of gravitation. What is this law of gravitation? ”

“Richard Phillips Feynman (/ˈfaɪnmən/; May 11, 1918 – February 15, 1988) was an American theoretical physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics for which he proposed the parton model.”

ladies and gentlemen it's my privilege to introduce the messenger lecturer Oh Sir Richard P Fineman of the California Institute of Technology professor Fineman is a distinguished theoretical physicist and he's done much to bring order out of the confusion which is marked lots of the spectacular development in physics during the post-war period among his honors and awards I will mention only the Albert Einstein award in 1954 this is an award which has made every third year and which includes a gold medal and a substantial sum of money professor Fineman did his undergraduate work at MIT and his graduate work at Princeton he worked on the Manhattan Project at Princeton and later at Los Alamos he was appointed an assistant professor here at Cornell in 1944 although he did not assume residence until the end of the war I thought it might be interesting to see what was said about him when he was appointed at Cornell so I searched the minutes of our board of trustees and there's absolutely no record of his appointment there are however some 20 references to leaves of absence salary and promotions one reference interested me especially on July 31 1945 the chairman of the physics department wrote the Dean of the Arts College stating that dr. Fineman is an outstanding teacher and investigator the equal of whom develops infrequently the the Chairman suggested that an annual salary of $3,000 was a bit too low for a distinguished faculty member and recommended that Professor Fineman salary be increased $900 the Dean in an act of unusual generosity and with complete disregard for the solvency of the university crossed out the $900 and made it an even thousand you can see that we thought highly a professor Fineman even then Fineman took up residence here at the end of 1945 and spent five highly productive years on our faculty he left Cornell in 1950 and went to Caltech where he has been ever since before I let him talk I want to tell you just a little bit more about him three or four years ago he started teaching a beginning physics course that Caltech and the result has added a new dimension to his fame his lectures are now published in two volumes and they represent a refreshing approach to the subject in the purpose of the published lectures there's a picture of Fineman performing happily on the bongo drums my Caltech friends tell me that he sometimes drops in on the Los Angeles nightspots and takes over the work of the drummer but professor Fineman tells me that that's not so another of his specialties is safecracking one legend says that he once opened a locked safe in a secret establishment removed a secret document and left a note saying guess who I could tell you about the time that he learned Spanish before he went to give a series of lectures in Brazil but I won't this this gives me enough this gives you enough background I think so let me say that I'm delighted to welcome professor Fineman back to Cornell his general topic is the nature of physical law and his topic for tonight is the law of gravitation an example of physical law professor fine you saw it but in the infrequent occasions when I've been called upon in a formal place to play the bongo drums the introducer never seems to find it necessary to mention that I also do theoretical physics I believe that's probably that we respect the arts more than the sciences the artists of the Renaissance said that man's main concern should be for man and yet there are some other things of interest in the world even the artists appreciates sunsets and the ocean waves and the march of the Stars across the heavens and there is some reason then to talk of other things sometimes as we look into these things we get an aesthetic pleasure from them directly on observation but there's also a rhythm and a pattern between the phenomena of nature which isn't apparent to the eye but only to the eye of analysis and it's these rhythms and patterns which we call physical laws what I want to talk about in a series of lectures is the general characteristics of these physical laws that's even another level if you will of higher generality over the laws themselves and it's really I'm talking about is nature as seen as a result of detailed analysis but only the most overall general qualities of nature is what I mean we wish to speak about now such a topic as a tendency to become too philosophical because it becomes so general that person talks in such generalities that everybody can understand him and it's considered to be some deep philosophy if you however I would like to be very rather more special and I would like to be understood in an honest way rather than in a vague way to some extent and so if you don't mind I'm going to try to give instead of only the generalities in this first lecture an example of physical law so that you have at least one example of the things about which I am speaking generally in this way I can use this example again and again to give an instance to make a reality out of something which would otherwise be too abstract now I've chosen for my special example of physical law to tell you about the theory of gravitation the phenomena of gravity why I chose gravity I don't know I whatever I chose you would ask the same question actually it's a was one of the first great laws to be discovered and that has an interesting history you might say yes but then it's all had I would like to hear something about science more modern science more recent perhaps but not more modern modern science is exactly in the same tradition as the discoveries of the law of gravitation it is only more recent discoveries that we would be talking about and so I have no they do not feel at all bad about telling you of the law of gravitation because I am in describing its history in the methods the character of its discovery and its quality talking about modern science completely modern this law has been called the greatest generalization achieved by the human mind and you can get already from the by introduction I'm more interested not so much in the human mind as in the marvel of nature who can obey such an elegant and simple law as this law of gravitation so our main concentration will not be on how clever we are to have found it all out but on how clever she is to pay attention to it now what is this law of gravitation that they're going to talk about the law is that two bodies or bodies exert a force upon each other which is inversely as the square of the distance between them and varies directly as the product of their masses and the mathematician mathematically we can write that great law down and formula some kind of a constant I'm the product of the two masses divided by the square of the distance now if I add the remark that a body reacts to a force by accelerating well by changing its velocity every second to an extent inversely as its mass it reacts to changes velocity more if the mass is lower and so on inversely as the mass then I have said everything about the law of gravitation that needs to be said everything else is a consequence mathematical consequence of those two things that I said that's remarkable enough phenomenon itself that the next lecture will consider this in more detail now I know you're not all here I know some of you are but you're not all mathematicians and so you cannot all immediately see all of the consequences of these two remarks and so what I would like to do in this lecture is to briefly tell you the story of the discovery tell you what some of the consequences are what the effect of this discovery had on history of science what kinds of mysteries such a law entails something about the refinements made by Einstein and possibly the relation to other laws of physics the history of the thing you briefly is this that the ancients first observed the way the planets seemed to move about in the sky and concluded that they all went around we along with the earth went around the Sun this discovery was later made independently by Copernicus if they had forgotten that people have forgotten that had already been made now the next thing question that came up in the study was exactly how do they go around the Sun that is exactly what kind of motion do they go with the Sun at the center of a circle or do they go in some other kind of curve how fast they move and so on and this discovery took a longer to make the times after Copernicus four times in which there were great debates about whether the planets in fact went around the Sun along with the earth or whether the earth was at the center of the universe and so on and there were considerable arguments about this when a man named Tycho Brahe got an idea of an own way of answering the question he thought that it might perhaps be a good idea to look very very carefully and to record where the planets actually appear in the sky and then the alternative theories might be distinguished from one another this is the key of modern science and as the beginning of the true understanding of nature this idea that to look at the thing to record the mission to details and to hope that in the information thus obtained may lie a clue to one or another of a possible theoretical interpretation so Tycho was a rich man and owned I believe an island near Copenhagen outfitted his Island with great brass circles and special observing positions there's a situation chairs that you could look through little hold and record a night after night the position of the planets it's only through such hard work that we can find out anything when these all these data were collected they came into the hands of Kepler who then tried to analyze what kinds of motions the planets made around the Sun and he did this by a method of trial and error at one stage he thought he had it he assumed figured out that they went around the Sun in circles with the Sun off-center and noticed that one planet I think it was Mars but I don't know was eight minutes of Arc off and he decided this was too big for Tycho Brahe to have made an error and that this was not the right answer so because of the precision of experiments he was able to proceed and find that to go on to another trial and found in fact ultimately this three things first that the planets went in ellipses around the Sun with the Sun of the focus and ellipse is a curve you'd all artists know about because it's a foreshortened circle when children know about because somebody told them and if you take a string and tie it the to tacks and put a pencil in there it'll make an ellipse these two tacks are the foci and if the Sun is here the shape of the orbit of a planet around the Sun is one of these curves the next question is and going around the ellipse how does it go does it go faster when it's near the Sun slower when it's further from the Sun and so on we take away the other focus we have the Sun then and the planet going around and Kepler found the answer to this – he found this that if you put the position of the planet down in two at two times separated by some definite time let's say three weeks and then in another place in the orbit put the positions of the planets again separated by three weeks and draw lines from the Sun to the planet technically called radius radius vectors anyway lines from the Sun to the planet then the area that's enclosed in the orbit of the planet and the two lines that are separated by the finest position three weeks apart is the same no matter what part of the orbit the thing is on so that it has to go faster when it's closer in order to get the same area as it goes slower when it's further away and in this precise manner some several years later he found the third rule and that had not to do with the exactly of motion of a single planet around the Sun but related the various planets to each other and it said that the times that took the planet to go all the way around was related to the size of the orbit and that the times went as the square root of the cube of the size of the orbit and for the size of the orbit is the diameter all the way across the biggest distance on the ellipse so up he has these three laws which are summarized by saying it's an ellipse and that equal areas are swept in equal times and that the time to go around varies as a three half power of the size the square root of a to the side so as three laws Kepler which is a very complete description of the motion of the planets around the Sun the next question was what makes him go around how can we understand this in more detail or is there anything else to say in the meantime Galileo was investigating the laws of motion incidentally at the time of Kepler the problem of what drove the planets around the Sun was answered in suntanned served by some people by saying that there were angels behind you beating their wings and pushing the planet along around orbit as we'll see that that answer is not very far from the truth the only differences that the Angels sit in a different direction and the wings going but the point that the Angels sit the different direction is the one that I must now come to Galileo in studying the laws of motion and doing a number of experiments to see how balls roll down inclined planes and pendulous swung and so on discovered a idealization a great principle called a principle of inertia which is this that if a thing has nothing acting on it for object has nothing acting on and is going along at a certain velocity in a straight line it will go at the same velocity at exactly the same straight line forever unbelievable though that may sound to anybody who has tried to make a ball roll forever if the idealization did is correct and that that there were no influence is acting such as the friction on the floor and so on the thing would go at a uniform speed forever the next point was made by Newton who discussed the next question which is when it doesn't go in a straight line then what and the answered this way that a force is needed to change the velocity in any manner first for instance if you're pushing it in a direction that it moves it will speed up if you find that it changes direction then the force that must have been sideways and that the force can be measured by the product of two effects first how much does the velocity change in a small interval of time how fast is the velocity changing how much is it accelerating in this direction or how much is the velocity changing when it changes direction that's called the acceleration and when that's multiplied by a coefficient called the mass of an object or its inertia coefficient then that together the force one can measure the for instance if one has a stone on the end of a string and swings it in the circle overhead then one can measure every one finds one has to pull the reason is that the speed of this the velocity speed is not changing as it goes around the circle but it's changing its direction so there must be perpetually in in pulling force and this is proportional to the mass so that if we were to take two different objects first swing one and then swing another one at the same speed around the head and measured the force in the second one that second one the new force is bigger than the other force in the proportion that the masses are different this is a way of measuring the masses by how much how hard it is to change the speed now then Newton saw I firm this but for instance to take a simple example if a planet is going in a circle around the Sun no force is needed to make it go sideways tangentially if there were no force at all on it it would have just keep coasting this way but actually the planet doesn't keep coasting this way but finds itself later not out here where it would go if there were no force at all but further down toward this the Sun in other words its velocity its motion has been deflected toward the Sun so what the angels have to do is to beat their wings in toward the Sun all the time that the motion to keep it going in a straight line has no known reason the reason why things Coast forever has never been found out the law of inertia is no known origin so the angels don't exist but the continuation of the motion does but in order to obtain the falling operation we do need a force so it would became apparent that the origin of that the force was toward the Sun as a matter of fact Newton was able to demonstrate that the statement that equal areas are swept in equal times was a direct consequence of the simple idea that all of the changes in velocity are directed exactly to the Sun even in the elliptical case and maybe I'll have time next time to show you how that works in detail so from this law he would confirm the idea that the forces toward the Sun and for knowing how the periods of the different planets vary with the distance away from the Sun it's possible to determine how that force must weaken at different distances and he was able to determine that the force must vary inversely as the square of the distance now so far he hadn't said anything yes because he only said two things which Kepler said in a different language one is exactly equivalent to the statement that the forces toward the Sun and the other is exactly equivalent the statement that the law is inversely is the square of the distance but people had seen in telescopes the Jupiter's satellites going around Jupiter and it looked like a little solar system so the satellites were attracted to Jupiter and the moon is attracted to the earth and this goes around the earth is attracted the same way so it looks like everything's attracted everything else and so the next statement was to generalize this and to say that every object attracts every other object if so the earth must be pulling on the moon just as the Sun pulls on the planet but it's in known that the earth pulls on things because you're all sitting typing in your seats in spite of your desires to float out of the hall at this time the pole allowed for objects on the earth was well known in the phenomenon of gravitation and there was Newton's idea then that maybe the gravitation which held a moon in the orbit also applied was the same gravitation that pulled the objects toward the earth now it is easy to figure out how far the moon falls in one second because if it went in a straight line you know the size of the orbit it though it takes a month to go around and if you figure out how far it goes in one second you can figure out how far the circle of the moon's orbit has fallen below the straight line that it would have been in if it didn't go the way it does go and this distance is one twentieth of an inch now the moon is 60 times as far away from the Earth's center than we are with 4,000 miles away from the center in the moon is 240,000 miles away from the center so if the law of inverse-square is right an object that the Earth's surface should fall in one second by oh one twentieth of an inch times 3600 being a square of 60 because the force has been weakened by 60 times 64 the inverse square law in getting out there to the moon and if you multiply a 20th of an inch by 3600 you get about 16 feet and lo it is known already of Galileo's measurements that things fell in one second on the Earth's surface by 16 feet so this mean meant you see that he was on the right track there was no going back now because a new fact that was completely independent previously which is the period of the moon's orbit and his distance from the earth was connected to another fact which is how long it takes something to fall in one second so this was a dramatic test that everything's all right further he had a lot of other prediction he was able to calculate what the shape of the orbit should be if the law were the inverse square and found indeed that it was an ellipse so he got three for two as it were in addition a number of new phenomena had obvious explanations one was the tides the tides were due to the pull of the moon on the earth this had sometimes been thought of before with the difficulty that it was the pull of the moon on the earth the earth being here the water's being pulled up to the moon then the would only be one tide a day where that bump of water is under the moon but actually you know there are tides every 12 hours roughly and that's two tides a day but you must there was also another school of thought that had a different conclusion their theory was that it was the earth that was pulled by the moon away from the water so actually Newton was the first one to realize what actually was going on that the force of the moon on the earth and on the water is the same at the same distance and that the water here is closer to the moon and the water here is further from the moon than the earth then the rigid earth so that the water is pulled more toward the moon here and here is less toward the moon than the earth so there's a combination of those two pictures that makes a double pipe actually the earth does the same trick as the moon it goes around a circle it really I mean the force of the moon on the earth is balanced but by what by the fact that just like the moon goes in a circle the balance the Earth's force the right is also going in a circle actually the center of the circle is somewhere inside the earth it's also going in a circle to balance the moon so the two of them go around a common center here and if you wish this water is thrown off by centrifugal force more than the earth is and this water is attracted more than miss average of the earth at any rate the tides were then explained in it and the fact that they were to add a lot of other things became quite clear why the earth is round because everything gets pulled in and why it isn't round because it's spinning so that the outside gets thrown out a little bit not balancing and why the Sun and Moon around and so on now as the science developed and measurements were made ever more accurately the tests of Newton's law became much more stringent and the first careful tests involved the moons of Jupiter by careful observations of the way they went around over long period of time one could be very careful to check that everything was according to what Newton and turned out not to be the case the moons of Jupiter appeared to be first yet sometimes of eight minutes ahead of time and sometimes eight minutes behind ty schedule where schedule is the calculated values according to Newton's laws it was noticed that they were ahead of schedule when they were closed when Jupiter was close to the earth and behind schedule and was far away around a hard circumstance and mr. OMA having confidence in the law of gravitation came to an interesting conclusion that it takes light sometime to travel from the moon's to the earth and that what we're looking at when we see the moons and not how they are now but how they were the time ago that it took the light to get here now when Jupiter's near us it takes less time for the light to come and when Jupiter's further it takes a longer time so he had to correct observations for the differences in time and by the fact that they were this much too early or that much too late was able to determine the velocity of light this was the first demonstration that light was not an instantaneously propagating material I bring this particular matter to your attention because it illustrates something that when a law is right it can be used to find another one that by having confidence in this law if something is the matter it suggests perhaps some other phenomenon and with we had not known the law of gravitation we would have taken much longer to find the speed of light because we would not have known what to expect of Jupiter's satellites this process has developed into an avalanche of discoveries each new discovery permits the tools for much more discovery and this begin is the beginning of that avalanche which has gone on now for 400 years in a continuous process and we're still have a Lansing along at high speed at this time another problem came up the planet shouldn't really go in ellipses because according to Newton's laws they're not attracted only by the Sun but also they pull on each other a little bit only a little bit but a little bit is something and will alter the motion a little bit so Jupiter Saturn and Uranus were big planets that were known and the calculations were made as to how slightly different than the perfect ellipses of Kepler the planets ought to be going Jupiter Saturn and Uranus by the pull of one on each other and when they were finished the calculations I mean and the observations it was noticed that Jupiter and Saturn went according to the calculations but that Uranus was doing something funny another opportunity for Newton's laws to be found wanting but courage two men both who made these calculations atoms and laverre 'yeah independently and almost exactly the same time proposed that the motions of uranus were due to an unseen as yet new planet and they wrote letters to their respective observatories telling them to look turn your telescope and look there and you'll find a planet how absurd said one of the observatories that some guy sitting with pieces of paper and pencils can tell us where we look to find something new planet and the other Observatory was more or less well the administration was different and they found the Neptune more recently in the beginning of the 20th century it became apparent that the motion of the planet Mercury was not exactly right and this caused a lot of trouble and had no explanation until a modification of Newton's listed Altima T the Newton's laws were slightly off and that they had to be modified I will not discuss the modification in the tale it was made by Einstein now the question is how far does this law extend does it extend outside the solar system and so I show on the first slide evidence that the law of gravitation is on a wider scale than just the solar system here is a series of three pictures of a so-called double star as the third star fortunately in the picture so you can see that they're really turning around and that nobody just simply turned the frames of the pictures around which is easy to do on astronomical fiction but the stars are actually going around and by watching these things and plotting the orbit you see the orbit that they make on the next slide it's it's evident that they're attracting each other and that they're going around an ellipse according to the way expected these are succession of pictures going for all these different periods of time I think yes it goes around this way they didn't see it well when it was too close and here it is in nineteen five my slide is very old it's gone around maybe once more since and you'll be happy except when you notice if you would have noticed already that this Center is not and a focus of the ellipse puts quite a bit off so something the matter with the laudanol if God hasn't presented us with this orbit face on it's tilted at a funny angle and if you take an ellipse and Mark its focus and then hold the paper in an odd angle and look at it in projection which that the focus doesn't have to be at the focus of the projected image so it said because it's orbit is tilted in space that it looks that way that looks like it's not the right pattern but it's alright and you can figure everything out satisfactorily for that how about a different bigger distance there's forces between the stars does it go any further than these distances which are not more than two or three times the soul systems diameter here's something in the next slide that's a hundred thousand times as big as a solar system in diameter and this is a large number of stars tremendous numbers this white spot is not a solid white spot is just because of the failure of our instruments to resolve it but are very very tiny dots just like the other stars well separated from one another of not hitting each other each one falling through and back and forth through this great globular cluster it's one of the most beautiful things in the sky as good as sea waves and sunset and the distribution of this material it's perfectly clear that the thing that holds us together is the gravitational attraction of the stars for each other and the distribution of the material in the sense of how the stars peter out as you go out in distance permits one to find out roughly how what the law is a force between the stars and of course it comes out right that it is roughly the inverse square the accuracy of these calculations and measurements is not anywhere near as careful as in the solar system onward as gravity extends still further this is a little pinpoint inside of a big galaxy in the next slide shows a typical galaxy and it's clear that this thing again is held together somehow and the only candidate that's reasonable is gravitation but when we get to this can this size we have in any way any longer to check the inverse square law but there seems to be no doubt that these great agglomerations of stars and so these galaxies which are 50 to 100,000 light years across the solar system is welcome of the earth to the Sun is only eight light minutes this is 100,000 light years that the gravity is extending even over these distances and in the next slide as evidence extends even further here is what is called a cluster of galaxies there's a galaxy here and here and here galaxies here they're all in one lump of galaxies analogous to the cluster of stars but this time what's clustered are those big babies that I showed you in this previous slide now we this is as far as is about 1:10 to store well a hundreth maybe of the size of the universe and as far as we have any direct evidence that gravitational forces extend so the Earth's gravitation if we take the view has no edge as you may read in the newspapers when we planet get outside the field of the gravity it keeps on going ever weaker and weaker inversely as the square of the distance dividing by for each time you're twice as far away until it mingles with the strong fields and gets lost in the confusion of the strong fields of other stars but all together with the Stars in its neighborhood pulls the other stars to form the galaxies and all together they pull on other galaxies to make a pattern a cluster of galaxies so the Earth's gravitational field never ends but Peters out very slowly in a precise and careful law probably to the edges of the universe the law of gravitation is different than many of the others is very important in the economy or in the machinery of the universe there are many places where gravity has its practical applications as far as the universe is concerned but a typically among all the other laws of physics gravitation has relatively few practical applications I mean the new knowledge of the Lord has a lot of application keeps people in their seats on but it has few that knowledge of the law has few practical applications relatively speaking compared to the other laws this is one case in which I picked an atypical example it is impossible by the way by picking one example of anything to avoid picking one which is atypical in some sense that's the one though the world in it the only applications I could think of were first in some geophysical prospecting in predicting the tides nowadays more modern lee in working out the motions of the satellites and the Ender planet probes and so on that we send up and also modernly to calculate the predictions of the planets position which have great utility for astrologers to publish their predictions and horoscopes in the magazines that's the strange world we live in that all the advances in understanding are used only to continue the nonsense which is existed for 2,000 years now that that shows that gravitation extends to the great distances but Newton said that everything attracted everything else do I attract you excuse me I mean do I attract you I was going to say excuse me do I attract you physically I didn't what I mean is the rules it really true that two things attract each other can we make a direct test and not just wait for the planets and look at the planet to see if they attract each other and this experiment that the direct test was made by Cavendish on equipment which you see indicated on the next slide I got my slides right well I made a mistake I was talking about the dia car the importance of the gravitation I was overwhelmed with my clever remark about astrologers and forgot to mention the important places where gravitation does have some real effect in the behavior of the universe and one of the interesting ones is the formation of new stars in this picture which is a gaseous nebula inside our own galaxy and it's not a lot of stars but is gas there are places where the gas has been compressed or attracted to itself here it starts props by some kind of shockwaves to get collected but the reason remainder of the phenomenon is that gravitation pulls up the cloud of gas closer and closer together so big mobs of gas and dust collect and form balls which as they fall still further the heat generated by the falling lights them up and they become stars and we have in the next slide some evidence of the creation of new stars it is unfortunately harder to see than I thought it was when I looked at it before but is not exactly the same as this this bump here is further out than here and that this also has a new dot here there are I have found better examples but were unable to produce a slide there is one example of a star patch a light that grew in a place in 200 and 200 days so that when this there is in a making the same kind of condition of a gas cloud when the guest collects too much together by gravitation stars are born and this is the beginning of new stars so the stars belch out dirt and gases when they explode sometimes in the dirt and gases then collect back again and that new story sounds like perpetual motion I now turn to the subject I meant introduced which was the experiments on the small scale to see whether things really attract each other and I hope now that the next slide does indicate this is a second try ya Cavendish's experiment the idea was to hang by a very very fine quartz fiber a rod with two balls and then put two large LED balls in the positions indicated here next to it on the side then because of the attraction of the balls there would be a slight twist to the fiber it had to be done so delicately because the gravitational force between ordinary things is very very tiny indeed and there it was and it was possible then to measure the force between these two ball Cavendish called his experiment weighing the earth we're pedantic and careful today we wouldn't let our students say that we would have to say they measuring the mass of the earth but the reason he say that said that is the following by a direct experiment he was able to measure the force and the two masses and the distance and thus determine the gravitational constant you say yes but we have the same situation on the earth we know what the pull is and we know what the mass of the object pulled is and we know how far away we are but we don't know them either the mass of the Earth or the constant but only the combination so by measuring the constant and knowing the facts about the pull of the earth the mass of the earth could be determined so indirectly this experiment was the first to terminate of how heavy or how massive is the ball on which we stand it's a kind of amazing achievement to find that out and I think that's why Cavendish named his experiment that way instead of determining the constant in the gravitational equation weighing the earth he incidentally was weighing the Sun and everything else at the same time because the pull of the Sun is known in the same manner now one other test of the law of gravitation is very interesting and that is the question as to whether the the pull is exactly proportional to the mass if the pull is exactly proportional to the mass and the reaction to forces the motions induced by forces that changes in velocity are inversely proportional to the mass that means that two objects of different mass will change their velocity in the same manner in a gravitational field or two different things no matter what their mass in a vacuum will fall the same way toward the earth that's Galileo's old experiment from the Leaning Tower I took by sea on the son of two and a half – the Leaning Tower of Pisa and now every time a guest comes he says meaning Tala so anyhow it means for example that in a satellite I mean a man-made satellite an object inside will go around the earth in the same kind of an orbit as a satellite on the outside and thus float in the middle apparently so that this fact that the force is exactly proportional to the mass and that the reactions are inversely proportional of mass has this very interesting consequence the question is how accurate is it and it has been measured by an experiment by a man named vos in 1990 and more accurately by deke and it is known that one part in 10,000 million the mass is exactly proportional I mean the forces are perfectly proportional for the mass how it's possible to measure without accuracy I wish I had the time to explain but I'm afraid I I cannot it's a remarkably clever I'll give a hint Howard but give one hint that suppose that you wanted to measure whether it's true for the pull of the Sun you know the Sun is pulling us all it pulls the earth – but suppose you wanted to know whether that you a piece of lead here and the piece of copper here of polyethylene and lead it was first done with sandalwood now it's done with polyethylene whether the pull is exactly proportional to the to the inertia the earth is going around the Sun so these things are thrown out by inertia and they're thrown out to the extent that these two objects have inertia but they're attracted to the Sun to the extent that they have mass in the attraction law so if they're attracted to the Sun and the different proportion then they're thrown out by inertia one would be pulled toward the Sun and the other away and so hanging on another one of those Cavendish sports fibres the thing will twist for the Sun it doesn't twist that this accuracy so we know that the sun's attraction but these two objects is exactly proportional to the centrifugal effect which is inertia so the force of attraction on an object is exactly proportional to its coefficient of inertia in other words its mass I should say something about the relation of gravitation to other forces to other parts of nature other phenomena in nature and I'll have more to say of a general quality later but there is one thing that's particularly interesting that is that the inverse square law appears again it appears in the electrical laws for instance that electricity also exerts forces inversely as the square the distance this time between charges and one thinks perhaps inverse square the distance has some deep significance maybe gravity and electricity at different aspects that the same thing no one has ever succeeded in making gravity and electricity different aspects of the same thing today our theories of physics the laws of physics are a multitude of different parts and pieces that don't fit together very well we don't understand the one exactly in terms of the other we don't have one structure from which all is deduced we have several pieces that don't quite fit exactly yet and that's the reason why in these lectures instead of having the ability to tell you what the law of physics is I asked to talk about the things that are common to the various laws because we don't know we don't understand the connection between them but what's very strange is that there are certain things that are the same in both but now let's look again at the law of electricity the law goes inversely is the square of the distance but the thing that is remarkable is the tremendous difference in the strength of the electrical and gravitational laws people who want to make electricity and gravitation out of the same thing will find that electricity is so much more powerful in gravity that it's hard to believe they could both have the same origin now how can I say one thing is more powerful than another depends upon how much charge you have and how much mass you have I'm certainly well the you can't talk about how strong gravity is by saying I take a lump of such-and-such a size because you chose the size if we try to get something that nature produces her own pure number that has nothing to do with inches or years or anything to do with our own dimensions we can do it this way if we take the fundamental particles such as an electron any different ones will give different numbers but to get an idea the number take electrons two electrons a fundamental particle that's an object it's not something I can I don't have to tell you what units I measure in its to particle to the fundamental particles may repel each other inversing is a square that this is due to electricity and they attract each other inversely as the square that this is due to gravitation question what is the ratio of the gravitational force to the electrical force and that is illustrated on the next slide the ratio of the gravitational attraction to the electrical repulsion is given by a number with 42 digits and goes off here it's all this is written very carefully out so that's 42 digits now therein lies a very deep mystery where could such a tremendous number come from that means if you ever had a theory from which both of these things at the come how could they come in such disproportion from what equation has a solution which has for one two kinds of forces an attraction and a repulsion without fantastic ratio people have looked for such a large ratio in other places they looking for a large number they hope for example that there's another large number and if you want a large number why not take the diameter of the universe to the diameter of a proton amazingly enough it also is a number with 42 and so an interesting proposal is made that this ratio dependent is the same as a ratio of the size of the universe to the diameter of a proton but the universe is expanding with time and that would mean the gravitational constant is changing with time and although that's a possibility there's no evidence to indicate that it's in fact true and there are several difficulties were having partial indications that it doesn't that the gravitational constant has not changed in that way so this tremendous number remains a mystery I must say to finish about the theory of gravitation two more things one is that Einstein had to modify the laws of gravitation in accordance to his principle with his principles of relativity the first was one of the principles was that if X cannot occur instantaneously Y aligns the Newton's theory said that the force was instantaneous he has to modify Newton's laws they have very small effects these modifications one of them is all masses fall light has energy and energy is equivalent to Maher so light should fall and that should mean that light going near the Sun is deflected it is and also the force of gravitation is slightly modified in his theory so the laws slightly changed very very slightly and it is just the right amount to account for the slight discrepancy that was found in a movement of mercury finally with reconnection to the laws of physics on a small scale we have found that the behavior of not on a small scale obeys laws so different very different than things on a large kit and so the question is well does gravity how does gravity look on a small scale what is what is called a quantum theory of gravity there is no quantum theory of gravity today people have not succeeded completely in making a theory which is consistent with the uncertainty principles and the quantum mechanical principles I'll discuss these principles in another election now finally you will say to me yes you told us what happens but what is this gravity where does it come from and what is it do you mean to tell me that the planet looks at the Sun have sees how far it is takes the inverse of the square of the distance and then decides to move in accordance with that law and move in other words a law of state of the night mathematical law I'm giving you no clue as to the mechanism I will discuss the possibility of doing this in the next lecture which is the relation of the mathematics of physics but finally in this lecture I would like to discuss to remark just at the end here to emphasize some characteristics that the gravity has in common with the other laws that we have mentioned as we passed along the first is that it's mathematical in its expression the others are that way – we'll discuss that excellent second it's not exact Einstein how to modify it we know it isn't quite right yet because they have to put the quantum theory here that's the same with all other laws they're not exactly there's always an edge of mystery there's always a place that we have some fiddling around do yet that of course is not a property probably not a property it may or may not be a property of nature but it certainly is common with all the laws as we know them today it may be only a lack of knowledge but the most impressive fact is that gravity is simple it is simple to state the principle completely and have no left have not left any vagueness for anybody to change the ideas above it's simple and therefore it's beautiful it's simple in its pattern I don't mean it's simple in its action the motions of the various planets and the perturbations of one on another can be quite complicated to work out what to follow how all those stars in the globular cluster move is quite beyond our ability it's complicated in its actions but not in the basic pattern or the system underneath the whole thing is that's a simple thing that's common than all our laws they all turn out to be simple things or low complex in their actual actions finally comes the universality of the gravitational law the fact that it extends over such enormous distances that Newton in his mind worrying about the solar system was able to predict what would happen in an experiment of Cavendish where Cavendish's little model of the solar system the two balls attracting has to be expanded 10 million million times to become the solar system and then 10 million million times expanded once again and we find the galaxies attracting each other by exactly the same law nature uses only the longest threads the Weaver patterns so that each small piece of her pride of our fabric reveals the organization of the entire tapestry thank you changed

Gravitation part 2 Class 9th Science Chapter by CBSE Lectures

Views:2626|Rating:4.47|View Time:6:21Minutes|Likes:17|Dislikes:2
For DVD Purchase Call 7838942884 OR Click Here:
Buy Remaining Videos:

In our dairy life we have noticed things falling freely downwards towards earth when thrown upwards or dropped from some height.
Fact that all bodies irrespective of their masses are accelerated towards the earth with a constant acceleration was first recognized by Galileo (1564-1642).
The motion of celestial bodies such as moon, earth, planets etc. and attraction of moon towards earth and earth towards sun is an interesting subject of study since long time.
Now few question arises that are
(a) what is the force that produces such acceleration with which earth attract all bodies towards the center
(b) What is the law governing this force.
(c) Is this law is same for both earthly and celestial bodies.
Answer to this question was given by Newton as he declared that
Laws of nature are same for earthly and celestial Bodies.

The force between any object falling freely towards earth and that between earth and moon are governed by the same laws.

Kepler (1571-1631) Studied the planetary motion in detail and formulated his three laws of planetary motion, which were available Universal law of gravitation.
More About Gravitation

You must have observed that whenever you throw any object upwards it reaches a certain height and then falls downward towards the Earth. So these objects are acting under the gravitational pull of the Earth or gravitational forces which are forces of attraction.
Gravitational force or gravity of earth is responsible for pulling you and keeping you on earth.
Now each and every object in this universe that has mass exerts a gravitational force on every other mass and the size of that pull depends on how large or small are the masses of two objects under consideration.
So for smaller masses like two human beings the gravitational force of attraction is very small and is negligible because two peoples are not very massive
Now when you consider massive objects like planets, Sun, Earth, Moon or other celestial bodies, the gravitational pull becomes very strong.
So here you must note that gravitational force depends on how massive objects under consideration are.
Gravity is very important on earth . It is the gravitational pull of earth that keeps our planet orbiting round Sun.
The motion of moon is also affected by both Sun and Earth.
Why don’t Moon Fall down

You must wonder If gravitational force is a force of attraction then why does moon not fall into earth?
To understand this consider a person whirling a stone tied to a thread along a circular path as shown below in the figure.

If he releases the stone then it flies along the tangent , at that point on the circular path.
Before the release of thread it is centripetal force responsible for the motion of stone in the circular path where the stone moves with a certain speed and changes direction at every point.
During this motion the change in direction involves change in velocity which produces acceleration. This force which is called centripetal force , causes this acceleration and keeps the body moving along the circular path is acting towards the center.
Now when the thread is released the stone does not experience this force and flies off along a straight line that is tangent to the circular path.
The motion of the moon around the earth is due to the centripetal force. The centripetal force is provided by the force of attraction of the earth. If there were no such force, the moon would pursue a uniform straight line motion.
Universal Law of Gravitation

Universal Law of gravitation was formulated by Sit Issac Newton.
It states that
Every body in universe attracts every other body with a force which is directly proportional to the product of their masses and inversely proportional to the square of distance between them. The force acts along the line joining the two bodies

To understand that let us consider two objects of masses MM and mm that lie at a distance rr from each other as shown below in the figure

manish you come and sit on this chair have you noticed what had happened to the cushion where manish sat ma'am it was compressed downwards do you know what is this phenomena called no ma'am this is called pressure pressure is the force acting on a unit area pressure is equal to force divided by area in order to understand the concept of pressure I will show you an example two ladies with same build one is wearing pointed heel sandals and other is wearing flat sandals standing on the sand now the student you tell me whose feet would sink more ma'am the lady with pointed Hugh yes correct answer do you know why no ma'am lady with pointed he applies force on smaller area of sin whereas lady with flat sandals applies same force on a larger area of the sand the smaller the area on which course X greater the impact of force thus in considering the action of force on surface we must think of a force acting on a unit area rather than the total scores if two similar breaks are placed in two different ways the same force can produce different pressure depending on area over which it X force acts over large area produces less pressure force acts over small area produces more pressure therefore pressure is the force acting perpendicularly on a unit area of the object if force acting perpendicular is called the thirst therefore pressure is equal to first divided by area SI unit of pressure is Pascal one Pascal is equal to Newton divided by meter sphere that is PA is equal to and divided by m square one kilo Pascal is equal to one thousand Pascal's there are many applications to increase and decrease the pressure by decreasing and increasing the area smaller the area for example sharp Knight cuts easily pores of our hand holds over very small area producing more pressure number 2.8 nail sinks into the woods easily for large area tractor with number of wheels number three bridges with broad bases number four school bags with broad strips number of tiles of the heavy truck is more than coal which enable the tires to carry more weight and to prevent from bursting and sinking into the ground wooden sleepers kept below railway tracks spread the weight of passing train over large area which reduces pressure and prevent them from sinking into the ground pressure in fluids hello students yesterday we studied about the pressure exerted by solid objects today our topic is pressure exerted by Floyd's man what is food the substance which can flow easily are called fluids give me some examples of rates waterhole during milk yes but all liquids and gases are fluids fluids even exert upward pressure in all directions how do you feel when you swim we feel light yes it means you lose some weight and water pushes you up so whenever an object is immersed in water it appears to lose some weight and feels lighter this tendency of liquid to exert an upward force on an object placed in it is called buoyancy buoyant force the upward force acting on an object immersed in a liquid is called Varenne horse and it is also called up thirst causes off vine force let us take a block ABCD it is immersed in a liquid pressure exerted by water at the surface C D is equal to p1 it is in upward direction pressure exerted by the water at surface AV is equal to p2 it is in downward direction here p1 is greater than p2 and height H 1 is greater than height h2 course exerted on CD is equal to f1 which is is equal to p1 a force exerted on a B is equal to f2 is equal to p2 a here p1 is greater than p2 therefore F 1 is greater than F 2 the swine force acting on the block is equal to FB FB is equal to f1 minus f2 FB is equal to a within the whole bracket p1 minus see – does the upthrust on a body is due to the liquid arises on account of pressure difference between lower and upper surface of the body immersed in the liquid factors of acting up trusts number one volume the object immersed in the liquid number two density of the liquid

What is Calculus? (Mathematics)

Views:607536|Rating:4.87|View Time:9:14Minutes|Likes:11202|Dislikes:290
What is Calculus? In this video, we give you a quick overview of calculus and introduce the limit, derivative and integral.

We begin with the question “Who invented Calculus?” Next, we talk about the two main tools you’ll study: derivatives and integrals. To understand both of these you’ll first learn about limits. After you learn how to compute the derivative and integral for basic functions and apply them to real-world problems, you’ll move up to higher dimensions and study things like “partial derivatives” and “multiple integrals.”

What to watch next:
The Tangent Line & the Derivative

Product Rule for Derivatives

Quotient Rule

How to Study For a Test

How to Study Physics

How to Study Programming

Socratica Patreon

Socratica Paypal

We also accept Bitcoin! 🙂
Our address is: 1EttYyGwJmpy9bLY2UcmEqMJuBfaZ1HdG9

We recommend the following books:
Innumeracy: Mathematical Illiteracy and Its Consequences

The Art of Learning by Josh Waitzkin (Chess Prodigy)

Shop Amazon Used Textbooks – Save up to 90%

Written and Produced by Michael Harrison

Michael Harrison received his BS in math from Caltech, and his MS from the University of Washington where he studied algebraic number theory. After teaching math for a few years, Michael worked in finance both as a developer and a quantitative analyst (quant). He then worked at Google for over 5 years before leaving to found Socratica.

You can follow Michael on Twitter @mlh496

You can also follow Socratica on:
– Twitter: @socratica
– Instagram: @SocraticaStudios
– Facebook: @SocraticaStudios

what is calculus this isn't something we inherited from the ancient Greeks like geometry this subject was discovered much more recently in the late 1600s by Isaac Newton and Gottfried Leibniz they didn't work together they both discovered calculus on their own and as a result there was a huge argument over who should receive credit for its discovery but we're going to save that story for another video today let's talk about what they discovered in calculus you start with two big questions about functions first how steep is a function at a point second what is the area underneath the graph over some region the first question is answered using a tool called the derivative and to answer the second question we use integrals let's take a look at the derivative the tool that tells us how steep a function is at a point another way to think about the derivative is it measures the rate of change of a function at a point as an example let's use the function f of X equals x cubed minus x squared minus 4x plus 4 suppose we want to find the steepness of the graph at the point negative 1/6 how would we do it and what do we even mean by steepness in algebra you find the rate of change of a line by computing the slope the change in Y divided by the change in X but this is a curve not a line so we get a good look let's zoom in a bit here's the idea pick a second point nearby how about the point negative 0.8 6.0 4 8 next draw a line through these two points the slope of this line is a good approximation for the steepness of the curve at the point negative 1/6 if you compute the slope you get zero point two four this is a good approximation but we can do better what if we pick a different point that's even closer how about negative zero point nine six point zero six one if you compute the slope of the line through this point and negative 1/6 you get zero point six one if you keep picking closer and closer points and computing the slopes of the lines you'll get a sequence of slopes we start getting closer and closer to some number the lines are getting closer and closer to the tangent line and the slopes are approaching one so we say the slope of the curve at the point negative one six is one we call this number the derivative of f of X at the point where x equals negative one this is the slope of the tangent line through the point negative one six luckily you won't have to do this every time you want to measure the rate of change at a point in calculus you'll learn how to find a function which will give you the slope of any tangent line to the graph this function is also called the derivative next let's take a look at the integral this is the tool that lets you find areas under curves as an example let's look at the function G of x equals sine X what if we wanted to find the area under this curve between x equals 0 and x equals pi how would we do it we know how to find the area of simple shapes like rectangles and circles but this is much more curvy and complicated let's zoom in to get a closer look here is the idea slice the region into a bunch of very thin sections let's start with ten slices for each section find the area of the tallest rectangle you can fit inside there are ten thin rectangles the width of each rectangle is PI over ten and we can find the height using the function G of X next add up the areas of all ten rectangles we get a combined area of one point six six 936 this is a pretty good approximation to the area under the curve but we can do better what if we do this again but use twenty-five slices instead this time we get an approximate area of one point eight seven one nine five let's do this again and again using thinner and thinner slices fifty slices a hundred slices one thousand slices you get a sequence of areas that are getting closer and closer to some number it looks like the area is approaching two we call this area the integral of G of X from x equals zero to x equals pi so we have these two tools the derivative and the integral the derivative tells us about a function at a specific point while the integral combines the values of the function over a range of numbers but notice there is something similar to how we found the derivative and the integral in the case of the derivative we found two points that were close to each other then we let one point get closer and closer and closer to the point that we're interested in in the case of the integral we took the curve and we chopped it up into a bunch of rectangles to approximate the area under the curve then we took thinner and thinner rectangles to get better and better approximations in both cases we're using the same technique in the case of the derivative we're relating the points get closer to each other in the case of the integral we're letting the rectangles get thinner in both instances we're getting better and better approximations and we're looking at what number these approximations are approaching the number they're approaching is called the limit and because limits are key to computing both the derivative and the integral when you learn calculus you usually start by learning about limits a lot of your time in calculus will be spent computing derivatives and integrals you'll start with the essential functions polynomials trig functions sine cosine and tangent exponential functions and logarithmic functions these are the building blocks for most of the functions you'll work with next you'll make more complex functions by adding subtracting multiplying and dividing these functions together you'll even combine them using function composition in calculus there are a lot of rules to help you find derivatives and integrals of these more complex functions the derivative rules have names like the product rule quotient rule and chain rule the integral rules include a u-substitution integration by parts and partial fraction decomposition when you first start calculus your focus will be on basic functions functions with one input and one output but we don't live in a one-dimensional world our universe is much more complicated so once you've mastered calculus for basic functions you'll then move up to dimensions for example consider a function with two inputs and one output like f of X y equals e to the negative x squared plus y squared earlier we computed the derivative by computing slopes of tangent lines but in higher dimensions things are a bit more complex this is because on a surface instead of a tangent line you'll have a tangent plane to handle this you'll compute the derivative both in the X direction and in the Y direction we call these partial derivatives these two partial derivatives are what you need to describe the tangent plane we'll also need to generalize the integral the region below a surface is three-dimensional it has a volume not an area to compute the volume will approximate it using a bunch of skinny boxes to sum up all the volumes you'll need to use a double integral because the boxes are spread out in two dimensions but don't forget we live in three spatial dimensions so you'll also need to learn calculus for functions with three inputs XY and Z if a function has three inputs and one output we call it a scalar field an example would be a function returning the temperature at a given point in space and the outputs of functions don't have to be simply numbers that can also be vectors a function with three inputs and a vector output are called vector fields an example would be a function which gives the force vector due to gravity at every point in space to recap the two main tools you'll learn about in calculus are the derivative and the integral the derivative tells you about a function at a specific point namely tells you how quickly the function is changing at that point the integral combines the values of a function over a region you'll start your study of calculus by learning how to compute the derivatives and integrals of a wide variety of functions next you're going to take these tools and apply them to higher dimensions by using things called partial derivatives and multiple integrals and along the way you'll learn how to apply derivatives and integrals to solve real-world problems so now that you've seen the big picture it's time to start learning the details so let's get to work we'll be releasing many more calculus video soon the best way to find out when we release a new video is to text a friend each morning and asked if Socratic ax has published a new video and if you would like to help us grow in to release videos more quickly please consider supporting us on patreon

Lecture 1 | Modern Physics: Classical Mechanics (Stanford)

Views:857035|Rating:4.87|View Time:47:50Minutes|Likes:3347|Dislikes:90
Lecture 1 of Leonard Susskind’s Modern Physics course concentrating on Classical Mechanics. Recorded October 15, 2007 at Stanford University.

This Stanford Continuing Studies course is the first of a six-quarter sequence of classes exploring the essential theoretical foundations of modern physics. The topics covered in this course focus on classical mechanics. Leonard Susskind is the Felix Bloch Professor of Physics at Stanford University.

Complete playlist for the course:

Stanford Continuing Studies:

About Leonard Susskind:

Stanford University channel on YouTube:

this program is brought to you by Stanford University please visit us at stanford.edu classical mechanics is the basis for all of physics it's the basis of all of physics not only because it describes the motion of objects like particles and mechanical systems and so forth but because the basic framework the basic structure of all the physics is based on the principles of classical mechanics the conservation of energy the conservation of momentum the principles by which all systems evolve in nature is the same set of rules essentially exactly the same set of rules in a more abstract in the more general setting then the rules which govern how a simple particle moves for example under the influence of gravity but in order to understand that we have to understand the principles in a fairly general context let's begin with the very very simplest kinds of systems that we can think of systems that are so simple that in fact is simple than any real systems in nature laws of nature let's imagine laws of nature which are of the most primitive and simple kind acting on the most primitive and simple systems that we can imagine I want you to first of all suppose that time which evolves continuously under the my watch I see the second hand goes around and around and around that goes around continuously the time can evolve and be any real number t I want you to imagine it only occurs in beats a stroboscopic world in which you only look at the world at intervals of time which we could to be a thousandth of a second or it could be a millionth of a second let's just take it to be one second intervals and ask how in the stroboscopic world systems change with time let's also imagine a very very simple set of systems systems which are so simple that they only have a handful of configurations configurations mean everything we need know about the system to characterize it completely so the simplest system I can think of would be a system that has only two configurations heads or tails a coin lay according down on the table I don't have a coin so we'll take this coffee top here I could put it down heads or I could put it down tails at that if you can tell the difference from where you're sitting but I can tell the difference this is this is heads that's tails all right so we have a system then it's characterized by two states two states of being and we want to add to those two states a law a law of evolution in going from one instant of time to the next instant of time from one beat of the stroboscopic light to the next one what kind of laws are allowable what kind of laws the basic principles of physics allow and what kind of laws don't they allow that's going to be our first kind of question for tonight so our first concept is the space of states in this case just heads in tails it's just two points heads and tails so two points in an abstract space heads and tails it's called the phase space it's called the phase space of a system the space of possible states of a system and what do I mean by a state of a system I mean everything that you need to know in order to predict what happens next everything you need to know in order to be able to say with certainty what the next state of the system will be that's called the phase space of the system in this case just heads in tails what kind of laws can you imagine what kind of laws of nature can you imagine for this extremely simple world it is the simplest well I suppose you could imagine a slightly simpler world you can imagine a world with only one state heads not much can happen in that world there's only one law of physics heads goes to head since heads goes to heads goes to heads it's extremely boring nothing ever happens because there's only one state so how could anything happen but with two states has entails you can have a richer variety of possible laws of physics one law let's take the various varieties of laws of physics we can have one law might just say you stay the same if the state of the system is heads at one instant of time the next instant of time it will be heads if it starts tails it will stay tails that's a very boring law and let's graph that law by drawing an arrow if it starts heads then it stays heads let's just represent that by drawing an arrow from head to head and from tail to tail the meaning of this arrow you start at the tail end of the arrow and you follow it and you see that it comes around to the same point that stands for the rule that says that a heads stays a heads and in this case a tail stays a tail so a dull law of nature this stays that way for endless amounts of time and this stays that way for endless amounts of time another possible law of nature would be slightly more interesting situation if you look at it at one instant of time the Nexen since it's the opposite heads goes to tails because the heads goes to tails these are deterministic laws deterministic laws mean that if you know what is happening at one instant you know forever after you know everything about the system infinitely into the future completely deterministic classical mechanics has that nature to it that is completely deterministic so and in fact it is in a certain sense always reversible but we're coming for that all right so to draw the graph representing heads goes to tails and Tails goes to heads we draw an arrow from heads to tails and from tails – heads and we read that as saying that if you start at heads you go to tail if you start at pals you go to heads what is the evolution of a system under this law of nature if you start with heads it's heads tails heads tails heads tails heads tails forever after these are two consistent laws of physics in a world of only two states whether we can think of more laws yes we can but but for the moment these are two interesting ones now how can we generalize this we can generalize this first of all to systems with more States to States is not very many we could have a die die as in dice die has six states one two three four five and six and we could represent them as points six points now we have a large variety of different laws of physics that we could have for example we could simply have we could label these one two three four five and six which one is one and which one is six is not very important but the there are six of them we could have a law of physics which says 1 goes to 2 2 goes to 3 3 goes to 4 4 goes to 5 5 goes to 6 and 6 goes back to 1 this would be a complicated motion when thought of in terms of a dye you start with a 1 up and then it goes to the 2 up 3 up and so forth but it's a relatively simple picture when drawn in this manner here you can imagine more complicated laws of physics with six states well whether they're more complicated or not is in the eye of the beholder but there first of all we could have a similar law where instead of one going to two one could go to three three could go to to two could go to five one two three something like that that's really not very different than this each one goes to a neighboring one well not on neighboring one each one goes to a next one and they cycle around and one characteristic of such systems is they will just cycle around forever and ever and ever okay now you can have more complicated I don't know if they're more complicated different laws for example you could have a law which goes this way if you start at one you go to two if you go to two then you go to three and then you go back to one if you happen to start at three note this is three this is five right five six six six six one two three four five if you start at three you go to four if you start at four you go to five back to three in this case well that's somewhat similar to this you have two disconnected cycles to do got more complicated things here's another one in fact I think it would take a long time to draw all the possibilities one goes back to 1 2 goes to 3 3 goes to 4 all goes back to 2 five goes to six and six goes back to five right they're all acceptable laws of physics wherever you happen to be you know exactly where to go next so it's deterministic it's deterministic into the future meaning to say wherever you start you know where you will be arbitrarily into the future if you start here and you go a hundred thousand times you'll just wind up writing up some wherever wherever you start here you stay there so these are completely deterministic into the future but they're also completely deterministic into the past which means if you know where you are you know where you were before if you find yourself over here then you know in the previous cycle or the previous instant you were over here and so forth so you can trace your motion either into the future or into the past with complete confidence about where you'll be no matter how far you go that's the character in principle in principle if not in practice and practice things get jumbled up and you can't see them clearly enough and you and you miss detail but if you can see the infinitely small detail in a system and get every single bit of physics absolutely right for any classical system they are in this sense exactly deterministic both into the past and into the future now what kind of laws of physics do we not allow the kind of laws of physics that we don't allow I can best illustrate I think by drawing some some possibilities and explaining to you why they are not allowed here's a law of physics that is not allowed by the principles of classical physics three states three states is perfectly alright nothing wrong with that but let's draw some arrows – 3 2 goes to 3 and 3 goes back to 2 that would be perfectly alright by itself what about 1 well I could have one goes to one but I don't want one to go to one I want one to go to two that's completely deterministic into the future if I start at one I go to two I go to two I go to three I go three back two to two back to three I always know where to go in the next step I just follow the arrow wherever it happens to be but what does it fail it fails to be deterministic into the past supposing I know that I'm at – then where did I come from I could have come from one or I could have come from three so I cannot work my way backward with uniqueness I can work my way forward I cannot work my way backward that's a this is a law of physics which is irreversible it would not allow me to run the laws of physics backward it would lead to an ambiguity every time I were at to another law which is not allowed by the principles of classical mechanics or principles of classical physics would be basically the same thing but with the arrows turned in the opposite direction all arrows reversed here I have a problem not going into the past but I have a problem going into the future let's say yeah supposing now I find myself a – and I want to go into the future I don't know whether to go into the future by following this arrow to one or this arrow to three there are two arrows leading out of two over here one of them goes to three one of them goes to one nothing tells me which I wrote to follow so it's not deterministic into the future I might randomly decide to go from two to three or randomly decide to go from two to one these are rules each one of which is deterministic in one direction but not the other these are the sorts of things which are forbidden by the principles of classical mechanics why are they forbidden by the principal's yeah which one yeah that wouldn't come I wouldn't know where to go from three you get stuck at three yeah no good well you can't have a dangling one you gotta know where to go next now you could let's see we could try to put something like that in yeah okay so I would say we go from 1 to 2 from 2 to 3 2 3 2 3 2 3 now the problem with being going backward I think yeah yeah yeah right so one way or the other you get stuck over these rules all right how do we spot what's allowable and what's not allowable well it's very simple at every configuration 2 or 3 we should have one in and one out we should have a unique one in a unique arrow in and a unique arrow out one arrow in to tell us where we came from and one arrow out to tell us where we're going that's the character of classical physics uniqueness into the future uniqueness into the past when represented in terms of this very simple and analog world analog digital world of our finite number of states then the rules of physics as we know them would say every configuration has one in arrow in one out arrow now with this rule of course life is very boring because if you only have a finite number of states all it can happen to you is you cycle around endlessly among those number of States always in the same order you can have slightly more interesting situations there's no reason why the number of States has to be finite you could have a situation where there's an infinite number of possible state a state corresponding to every integer positive and negative a particle on a position on a line where the position could be any integer value and then a simple law of physics would be you go from one to the next wherever you are you go to the next one each point has one n in one out I can't draw them endlessly it'll take forever but each mark one in and one out this of course would also be a rather boring law of nature you just hop from one to the next to the next and next and next forever and ever and ever but at least you wouldn't be cycling around endlessly again we can have add some more states on to this we could add this on if we start on the line we simply move off and keep going forever and ever and ever but if we start over here we cycle around so we could have mixtures of both kinds of things some states we cycle around in other states we move off to infinity now notice that some of these laws of nature the phase space breaks up into different pieces which are connected among themselves but not connected to each other for example even with just two states we had two possibilities one like this and one like this in this case where it breaks up into more than one more than one piece we have something called a conservation law a conservation law is simply a memory of where we started a conservation law means that something is kept intact for all time some piece of knowledge is kept intact for all time and doesn't change in this case we could label this we could label a configuration over here with a plus one and a configuration over here with a minus one and then we could call we could invent a variable plus one over here and minus one over here it never changes if it's plus one that stays plus one if it's minus one it stays minus one that's a conservation law something which doesn't change with time on the other hand if we hop from plus 1 to minus 1 to plus 1 to minus 1 we don't have a conservation law conservation laws are always associated with these kind of closed families of different trajectories in the phase space which don't mix with each other which remember arm something about the system which might otherwise get mixed up if everything got all mixed together so there's all kinds of possibilities that are inherent in these deterministic laws but always the condition is one in line and one out line for every point that can also be called information conservation its information conservation in exactly the sense that that you never lose memory of where you started either into the past or into the future if you know where you are at any instant you know where you came from and you know where you'll be information about where you are is conserved never changes into the past in the future whereas if you have one of these bad laws laws which are forbidden by the rules of classical mechanics then you do lose information for example if you find yourself over here you don't know whether you came from here or from well no that's not quite right find yourself over here you don't know whether you came from here or over here so you lose information information conservation is perhaps the most fundamental law of basic classical physics that you don't lose information about now why that why is that so it's not written into the laws of physics why they are what they are maybe someday we'll understand Hyper laws physics or metal laws of physics or deeper laws of physics which will tell us why the laws of physics are what they are at the moment it's more or less an experimental fact that all the known laws of physics fit into this class of information conserving laws even those which are quantum mechanical but for our subject this quarter we're only interested in classical physics all right so that's the basic set up if we were interested only in this stroboscopic world of discrete time intervals the real world of course is more continuous than that supposing okay so let's make the law something like this if the last two states were heads heads then it stays heads if the last two states were heads tails then it goes to heads if the last two with heads tails heads then it goes to tails and if the last two with tails tails then it goes to tails I think that's a possible possible law then you would say I can't tell from the fact that it's heads where it goes next and indeed I can't but that would just be another way of saying that a specification of a heads by itself is not what you would call a state where you would call a state would be the specification of the previous last two entries because you need two entries to tell you what happens next now that raises the question that that's very very important in classical mechanics how much and what exactly do you need to know to say what happens next if the phase space is the space of things space of possibilities but always in such a way that they tell you exactly what happens next what is it that you do have to know next so that brings us to continuous physics let's take the motion of a particle let's take the motion of a particle is it enough to know where a particle is to say what happens next let's hypothesize that the generalization the continuous time is that we need to know the exact location of a particle along a line we've run out of ink I'm afraid not there is all right so let's imagine the motion of a particle along a line then you might think that the analog of a state is just a location of a particle where is it but is it enough to know where a particle is in order to say what happens next now what else do you need to know its velocity in order to know where a particle will be next you need to know not only where it is but how fast it's moving you need to know its velocity so that means the state in the same sense that I used it that which you need to know in order to know what happens next does not just consist of the location of a particle but you can say it two ways you need to know not just the location of the particle but you need to know also the previous location or better yet what causes for what is equivalent to knowing the previous location the velocity of velocity that tells you that the phase space the space of states the space of configurations is two-dimensional not one-dimensional it's not just a line it's a line that represents the position of the particle and a second line which represents its velocity either to the left or to the right positive velocity means moving to the right negative velocity means moving to the left supposing you're over here where do you move next you stay the same place why you're at the origin and you have no velocity what if you're over here so we could just we could draw a little loop here to say that you come back to the same place what if you're over here you stay the same place because you're moving with 0 velocity vertical axis is velocity so you come back to the same place what if you're over here where are you one second later let's chop time up into one-second intervals where are you next somewhere to the right huh what if you're over here now what if you're over here move twice as far or roughly twice as far to the right what is it down here you move to the left so we could fill up this space here with little tiny arrows to show where you move next but notice – no way you move next you have to know not only where you are in the sense of what X is but you also have to know the vertical component namely what the velocity is so that means the analog the analog of a point in the phase space is a point in the space not only of positions but also velocities yes why where that take place it means no it means exactly what the gentleman asked me before what if you had a law of nature which tells you in order to know where to move next you have to know your previous two entries all right now one way of saying it is all right in that case I need to know the previous two entries and it just doesn't fall into this class of things or you could say that the space of configurations doesn't just consist of a heads or a tails but it consists of a pair of entries a pair yeah that's right we meant we're not we model that that's right we need two axes that that's right we need an axis for the present configuration in the past one yeah that's exactly right well of course not because in classical physics the position of a particle and so forth is a real number a real number is something that you can never determine exactly ah and so there's always imprecision and that imprecision always represents a degree of uncertainty and where you will be next now it gets worse and worse in general it's likely to get worse and worse as you try to take larger and larger time intervals a given degree of imprecision in what you actually know can get magnified and get the magnified into worse and worse imprecision as time goes on so are in practice in practice classical systems don't really have the property that you can predict endlessly where they're going to be and exactly what they're going to do but you can always say given a time interval I want to be able to predict exactly for the next 30 seconds where every molecule in this room will be let's forget quantum mechanics now I want to be able to predict exactly where every mile kyoool there will be then there's a certain degree of precision in the present information at exactly one instant of time which will permit me to be able to extrapolate for 30 seconds okay it will not permit me to extrapolate for 40 seconds if I try to extrapolate for 40 seconds I will find the errors get magnified out of control if I want to predict correctly for 40 seconds I will have to do even better in my initial conditions and my knowledge of exactly what the state of the system is so in practice this idea of determinism is defective it's defective because in order to determine for a given length of time you'll have to have precision which is so good that it's way way beyond anything anybody can do but in principle given any length of time in classical physics there exist a degree of precision which allows you to extrapolate for that length of time does that answer the question yeah ah so some systems are very predictable some systems are less predictable and get out of control very quickly they're called chaotic systems but the principles are the same that it's just a degree of precision which you need to know in the beginning in order to extrapolate for a given length of time well you could decide I'm not sure what the difference between predictable and detail the way I use the terms I use them interchangeably so I can't say that okay okay yeah all right what's that yeah right the equations are deterministic if you know the initial conditions are deterministic are predictable infinitely predictable if you know the initial conditions with infinite precision you never do and therefore they're never completely predictable now the fact that you need to know both the position and the velocity in classical theory in the physics in order to predict what happens next reflects itself in the structure of the equations of mechanics specifically it tells you that the equations of motion Newton's equations in this case are what are called second-order equations instead of first-order equations let me illustrate it by starting with a first-order equation a first-order equation what a first-order equation means is that it only has quantities of first derivatives with respect to time in other words only contains velocities second-order means it contains not only first derivatives but second derivatives means it contains accelerations the equations of motion acceleration is the second derivative of the motion um we could write we know what the real equations of motion of Newton are F equals MA it contains acceleration let's write a phony equation let's write F equals mass times velocity force is F velocity is velocity and let's suppose that force just depends on where you are we have a particle that moves along a line it's subject to four which vary along the line force may be big here small there and so forth so the force depends on position and let's imagine this fake equation of motion that it's equal to mass times velocity and what is velocity velocity is the time derivative of the position the X by DT and I will use continuously throughout this course the notation that time derivative is just indicated by a dot that means time derivative all right what does this tell me what do I need to know in order to predict what happens next I say for this equation we only need to know the position of a particle if we know the position of a particle I can tell you what the velocity is just from the equation if I know that the position is a particular position then I know the force on it and from the equation that tells me what the velocity is does it tell me the acceleration how about the acceleration let's see if I can compute the acceleration to compute the acceleration from this equation we just differentiate it once more we write that the F by DT is equal to mass mass is just a constant times the second derivative of the position or just the acceleration so we have over here acceleration what about the F DT the time derivative of the force the force varies with time because the position varies with time so the F by DT is just a reflection of the fact that the position of the particle varies with time and we can write the F by DT using standard rules of calculus as just you did the change in F with respect to position times a change in position with respect to time in other words the velocity alright we've already figured out what the velocity is knowing the force knowing the position so we know the velocity and we can read off from this equation what the acceleration is we can figure out all of the derivatives of the motion if we know where the particle is by multiply differentiating this equation so what it tells us to make a short story out of it is it tells us that if we know where the position of the particle at any instant of time then we know where it's going to be in the next instant the next two instance the next three instants it completely predicts the motion but this is not the character of Newton's equations Newton's equations say F is equal to mass times acceleration mass times velocity so let's look at this equation F equals mass times acceleration can I predict from this what the velocity is no this is no equation for the velocity right if I know what the position is I know what the force is that tells me what the acceleration is but there's nothing in these equations which tell me what the velocity is that means I have to add in the velocity as a piece of information to begin with I have no choice I have to tell you in order to predict I have to tell you the position as well as the velocity then if I know the position and the velocity I can then predict the acceleration the next to a third derivative the fourth derivative and all of them so that tells me that in order to know where I am and where I'm going to be I have to know the position and the velocity the phase space is a two dimensional space so we see then that that classical mechanics does have this character of a configurate or a phase space of different configurations with a set of little arrows which tell you where to go next but the phase space itself has a position component to it and a velocity component to it we will go on and study the classical equations of motion and study them in a variety of different formulations but always the connecting link will always be conservation of information the idea that the laws of physics are completely deterministic and described by equations which tell you where you will be next that's the character of classical physics okay bifurcated it's one system but you need two pieces of informations that are warn you I'm not sure what you mean by bifurcated up here here you only need know now here you need one piece of information to say where you'll be next if you here you stay here if you're there you stay there yeah the stuff all there is is heads and tails that's all there is and if you know where you are you know where you'll be next no no we're in oh no no no no this one rule know this this is one rule heads goes two heads tails goes two tails that's a single rule and if you know that you're at heads then you know you'll be next at heads if you know you're at tails you know you but so you only need the piece of information heads or tails that does it that tells you everything now we I suggested a different does well yeah you don't need to know where you were before you only need to know where you are now to know where you'll be next so both of these laws require only knowledge of where you are at one instant to tell you where you'll be next you simply follow the if you start someplace you follow the arrow until you come back or you follow the arrow till you get to the next place and that tells you where you'll be next you don't need any more information than that the only question is what one of these points corresponds to does one of these cut points correspond to how much information does it correspond to a point in this space is it enough to know heads or tails to know what you'll do next or might you need to know the two previous things that's a different that's a different set up weighted stay with the heads and tails for a minute let's suppose that in order to say what happens next you need to know the previous two configurations let's do let's work out that example and let's write it in this form let's make up a law I think I had one down before heads heads goes – heads heads tails goes to watch tails tails heads goes to heads and tails tails goes to tails okay is there something wrong with that oh this only depends on Ed's coffee right you're right you're right right sorry sorry sorry sorry heads yeah we want this one to go to tails uh second one goes to heads good yeah what what what what say it again it's also true yeah yeah yeah yeah it's it also hard to work an example isn't it let's see hmm switch the second one this one make it tails okay I think that's perfectly all right you lose information why well okay here's a here's a set up here's a set up if you know the previous two then you know what happens next in fact let's let's now see if we can make a table out of this not a table but a set of points the set of points then there are four possibilities heads heads heads tails tails heads and tails tails okay let's see has heads heads heads tails tails heads and tails tails now supposing you go from heads heads to heads this is heads heads heads that means you go from heads heads to heads heads right you go from heads heads to heads heads now supposing you go from let's see what happens if you start with heads tails where do you go to so you go from heads head tails – tails tails okay okay what happens if you go from tails tails are we going to run to trouble aren't we tails tails goes to tails so tails tails goes to tails tails bad not allowable by the laws of physics and now tails heads goes where tails tails goes to tails tails great tails heads tails heads goes to tickles – heads tails so that comes here well it seems to me we go we should be able to make a consistent law whoa what uh this one this one Oh tails tails goes to tails tails goes to heads yeah okay so then tails tails goes to heads and that means that tails tails heads so I'm his tails tails goes to Ted – tails heads ah oh okay good good now we have a workable law now we have a workable law the only thing we had to do was to say that what we originally called a configuration namely a single heads or tails was not a complete specification information a complete specification of information involved two pieces of information and once we recognized that we were able to to to write this down as a law of physics which is deterministic and reversible okay so you don't know offhand you don't know to begin with what pieces of information you need to know in order to know what to do next but that's what a stead that's what the configuration space or that's what the phase space is it's the collection of all the things you need to know to know what happens next now of course you could go beyond this you could say I need to know the first three things in order to know what happens next you can do that you'll simply need more states to make a deterministic system what would it mean in classical mechanics to need more information than the positions and velocities suppose you need the positions velocities and accelerations that were in third order differential equations but what would it say about the phase space it would mean you would need positions velocities and accelerations to represent the phase space as it happens that is not the case for classical mechanics that's an experimental fact but it wouldn't stop us if we did if we did need the accelerations we would just write third order equations and we will make our phase space three-dimensional the preceding program is copyrighted by Stanford University please visit us at stanford.edu

Newton and the Scientific Method

Views:583|Rating:3.75|View Time:9:43Minutes|Likes:3|Dislikes:1
Isaac Newton, the Enlightenment and the development of the Scientific Method

what are the things we value most about our modern world freedom to enjoy ourselves to argue and discuss to express our views to find things out to read and write and the science and technologies that make these possible none of this happened by accident these freedoms were won through courage and with vision in an extraordinary period of human history it began in 17th century Europe it was called the Age of Enlightenment in the space of little more than a century religious faith yielded to reasoned argument and the power of aristocracy gave way to the power of knowledge but it was a hard-won battle against the fierce opposition of a powerful Church and ruthless monarchs it laid the foundations for the modern world our story of the Enlightenment begins in England in the late 17th century with Isaac Newton he was the founder of modern science which was to become the single greatest challenge to the power of the church at the beginning of this period most people's lives were based on obedience and belief they obeyed the church which told them to obey the king who it said had been chosen by God people believed what was in the Bible not what they found out for themselves and the Bible said that the earth and everything on it had been made by God only a few thousand years earlier and that the first humans had been thrown out of paradise for disobeying God's Word but Isaac Newton declared that knowledge was acquired not by blind faith but by observation and that it could be found not just in the Bible but in the world around us he laid down universal rules for what we now call science and one of those rules was that it was open to everybody there's one thing that everybody knows about Isaac Newton but was that he watched an Apple fall to the ground in a great flash of inspiration he suddenly conceived the theory of gravity and we can never know if that really happened or not but what we do know is that shortly before he died he told the same story to four different people and that suggests that he wanted people to believe that it was true and he sat in the orchard and he sat beneath this apple tree and he watched an apple fall off the tree to the ground something he'd seen hundreds of times before but this time was different he thought to himself why is it that the Apple always falls down why doesn't it go sideways or upwards mutants genius was to see things where most people would see nothing instead of empty space between the Apple and the ground Newton saw an active force pulling things towards the earth the force of gravity Newton said there's one single force the power of gravity that makes apples fall off the tree in wool salt garden next the mood and go around the earth and makes the earth go around the Sun Newton had discovered a vast and universal law was at this stage of his life in his 20s he didn't seem to care if anybody else knew about it he wrote up his discoveries in Latin in this book almost nobody could understand and he buried himself away at Cambridge University he seems as far as we can make out to have been a very withdrawn difficult prickly man he was an absolutely appalling lecturer some students saw him going by in the street and said there goes the man who lectures to the walls by which he meant that there was nobody in the room at all when Newton was lecturing but Newton's discovery that what happened in nature followed invisible rules and that those rules operated everywhere was too powerful to be hidden in a book Newton was asking questions about the world around him but he wasn't alone he was a founding member of the Royal Society the world's first scientific association dedicated to spreading the discoveries of the Enlightenment its members believed in the truth of the Bible but they no longer accepted that it was the only truth traditionally the way to find out about God was to read the Bible this was the book of God's holy word that he dictated it to Adam but Newton his contemporaries said there's a second book that we can learn from the book of nature and God has written book in a different alphabet and our job is to learn how to read that alphabet and it was this book of nature that Newton and his contemporaries started to explore using new more powerful microscopes and telescopes that they designed themselves these men were opening up an entirely new dimension to the world over the course of his life Newton seems to have changed his mind about the importance of publicizing his work his other great experiment is remembered not just for what he found out but for what he did with the discovery back at his boyhood home in 1666 he began to wonder where the colors in the spectrum come from every year a fair came to Starbridge in Cambridge in one year Isaac Newton went along and he bought a prison the prisons were being sold for children to use as toys people knew that when you throw light through a prism it makes a spectrum but Newton wanted to do something different he wanted to know if the colors were already in the light or were put there by the prism he may just ruin completely dark he just had a little teeny ray of sunlight coming through and he passed the ray of light through the prism and produced the rainbow on the opposite wall a spectrum he then isolated the red light from that spectrum and passed it through a second prism and the light coming out was still red he proved that the colors came not from the prism but the light itself but that wasn't what mattered most Newton had performed his experiment using a child's toy something anybody could buy and having done the experiment Newton made this sketch of it so this is Newton's original sketch then this diagram was published in English and his book the optics in the pas you had to be a very specialized scholar in order to understand what was in the Bible or what was in some esoteric Latin text but an experiment like this could be repeated anywhere in the world this was a new source of truth a new source of knowledge so ordinary people could use these scientific results to invent things to make machines to change the world to control the world so the people who understood and who were clever would take over from the people who'd been born into rich aristocratic wealthy families so what are they very very important things about the enlightenment as symbolized by experiments like this is it's not just about what knowledge was gained it's about who controls the knowledge Newton had established for the first time the scientific method how all science is carried out to this day that experiments must be published and must be able to be repeated anywhere in the world

Michio Kaku: The Universe in a Nutshell (Full Presentation)

Views:9418379|Rating:4.89|View Time:42:14Minutes|Likes:166915|Dislikes:3634
What if we could find one single equation that explains every force in the universe? Dr. Michio Kaku explores how physicists may shrink the science of the Big Bang into an equation as small as Einstein’s “e=mc^2.” Thanks to advances in string theory, physics may allow us to escape the heat death of the universe, explore the multiverse, and unlock the secrets of existence. While firing up our imaginations about the future, Kaku also presents a succinct history of physics and makes a compelling case for why physics is the key to pretty much everything.

Don’t miss new Big Think videos! Subscribe by clicking here:

Kaku’s latest book is The Future of the Mind: The Scientific Quest to Understand, Enhance, and Empower the Mind (

The Universe in a Nutshell: The Physics of Everything
Michio Kaku, Henry Semat Professor of Theoretical Physics at CUNY

The Floating University
Originally released September, 2011.

Directed / Produced by Jonathan Fowler, Kathleen Russell, and Elizabeth Rodd

BSc/MPhys Physics with Philosophy

Views:5447|Rating:5.00|View Time:2:28Minutes|Likes:11|Dislikes:0
Find out more about the BSc and MPhys Physics with Philosophy courses here at the University of Lincoln.

BSc Physics with Philosophy:

MPhys Physics with Philosophy:

the course of physics with philosophy at Lincoln is a full-fledged physics course with some elements of philosophy which permits the students to engage further with open questions in science and society physics is all that the scientific study of the material universe the world and the physical properties of the world but but it leaves some questions unanswered so it's a philosophy kind of complements physics in that sense say we want to probe more deeply into the nature of reality and ask what what reality actually is what do physical concepts actually mean so philosophy sheds some kind of a new light on the started physic curriculum and in particular it enables the trained physicists with this bit of philosophy to reflect on what physics is about and how does it work so we're looking for someone I think to study this joint course who is prepared to think independently and outside the box of normal scientific thought so it's a chance to explore maybe a more creative side in a more independent side then the straight kind of mathematical roots that the physics normally take they also opportunities where students are able during the summer to do proper research that they can showcase after that the coolest have small classes and a very small student or staff ratio which permits students to engage almost on a one-to-one basis with lecturing staff so that's definitely a plus here I'd encourage people to do this of course because it's interesting stimulating to study think a combination of a science subject with the humanities subject is an extremely unusual thing for people to graduate in and having those two perspectives and being able to integrate those two things makes you extremely employable in a whole range of future careers but you are given skills which a value onto job market but on top of that you've got an entirely new look on the side here as a whole and having that that scientific and that humanities perspective is something that offers you a unique selling point in the employment market

The Fourth Dimension [Metaphysical Science Lecture] (The Dimension Beyond Our Perception)

Views:6960|Rating:4.74|View Time:9:5:4Minutes|Likes:143|Dislikes:8
The Fourth Dimension [Metaphysical Science Lecture Audiobook] (by Charles Howard Hinton)

Einstein's General Theory of Relativity | Lecture 1

Views:2631093|Rating:4.85|View Time:1:38:28Minutes|Likes:17778|Dislikes:552
Lecture 1 of Leonard Susskind’s Modern Physics concentrating on General Relativity. Recorded September 22, 2008 at Stanford University.

This Stanford Continuing Studies course is the fourth of a six-quarter sequence of classes exploring the essential theoretical foundations of modern physics. The topics covered in this course focus on classical mechanics. Leonard Susskind is the Felix Bloch Professor of Physics at Stanford University.

Stanford Continuing Studies:

About Leonard Susskind:

Stanford University Channel on YouTube:

this program is brought to you by Stanford University please visit us at stanford.edu gravity gravity is a rather special force it's unusual it has different than electrical forces magnetic forces and it's connected in some way with geometric properties of space space and time but before and that connection is of course the general theory of relativity before we start tonight for the most part we will not be dealing with the general theory of relativity we will be dealing with gravity in its oldest and simplest mathematical form well perhaps not the oldest and simplest but Newtonian gravity and going a little beyond what Newton certainly nothing that Newton would not have recognized or couldn't have grasped Newton could grasp anything but some ways of thinking about it which will not be found in Newton's actual work but still lutonium gravity the Toney and gravity set up in a way that that is useful for going on to the general theory ok let's begin with Newton's equations the first equation of course is F equals MA force is equal to mass times acceleration let's assume that we have a frame of reference a frame of reference that it means a set of coordinates and as a collection of clocks and those frame and that frame of reference is what is called an inertial frame of reference an inertial frame of reference simply means 1 which if there are no objects around to exert forces on a particular let's call it a test object a test object is just some object a small particle or anything else that we use to test out the various fields force fields that might be acting on it the inertial frame is one which when there are no objects around to exert forces that object will move with you for motion with no acceleration that's the idea of an inertial frame of reference and so if you're an inertial frame of reference and you have a pen and you just let it go it stays there it doesn't move if you give it a push it will move off with uniform velocity that's the idea of an inertial frame of reference and in an inertial frame of reference the basic Newtonian equation number one I always forget which law is which there's Newton's first law second law and third law I never can remember which is which but they're all pretty much summarized by f equals mass times acceleration this is a vector equation I expect people know what a vector is a three vector equation will come later to four vectors where when space and time are united into space time but for the moment space is space and time is time and a vector means a thing which is like a pointer in a direction in space as a magnitude and that has components so component by component the X component of the force is equal to the mass of the object times the X component of acceleration Y component Z component and so forth in order to indicate that something is a vector equation I'll try to remember to put an arrow over vectors the mass is not a vector the mass is simply a number every particle has a mass every object has a mass and in Newtonian physics the mass is conserved that does not change now of course the mass of this cup of coffee here can change it's lighter now but it only changes because mass has been transported from one place to another so you can change the mass of an object by whacking off a piece of it and but if you don't change the number of particles change the number of molecules and so forth then the mass is a conserved unchanging quantity so that's first equation now let me write that in another form the other form we imagine we have a coordinate system an X a Y and a Z I don't have enough directions on the blackboard to draw Z I won't bother there's x y and z sometimes we just call them x1 x2 and x3 I guess I can draw it in x3 is over here someplace XY and Z and a particle has a position which means it has a set of three coordinates sometimes we will summarize the collection of the three coordinates x1 x2 and x3 incidentally x1 and x2 and x3 are components of a vector the components they are components of the position vector of the particle position vector of the particle I will often call either small R or large are depending on on the particular context R stands for radius but the radius simply means the distance between a point and the origin for example we're really talking now about a thing with three components XY and Z and it's the radial vector the radial vector this is the same thing as the components of the vector R alright the acceleration is a vector that's made up out of the time derivatives of XY and Z or X 1 X 2 and X 3 so for each component the compose for each component one two or three the acceleration which let me indicate well let's just call it a the acceleration is just equal the components of it are equal to the second derivatives of the coordinates with respect to time that's what acceleration is the first derivative of position is called velocity we can take this to be component by component x1 x2 and x3 the first derivatives velocity the second derivative is acceleration we can write this in vector notation I won't bother but we all know what we mean I hope we all know we mean buddies by acceleration and velocity and so Newton's equations are then summarized and summarized but rewritten as the force on an object whatever it is component by component is equal to the mass times the second derivative of the component of position so that's the summary of I think it's Newton's first and second law I can never remember which they are Newton's first law of course is simply the statement that if there are no forces then there's no acceleration that's Newton's first law equal and opposite right so this summarizes both the first and second law I never understood why there was a first and second law it seems to me there was just one F equals MA all right now let's begin even even previous to Newton with Galilean gravity gravity as Galileo understood it actually I'm not sure how much of this mathematics Galileo did or didn't understand he certainly knew what acceleration was he measured it I don't know that he had thee but he certainly didn't have calculus but he knew what acceleration was so what Galileo studied was the motion of objects in the gravitational field of the earth in the approximation that the earth is flat now Galileo knew the earth wasn't flat but he studied gravity in the approximation where you never moved very far from the surface of the earth and if you don't move very far from the surface of the earth you might as well take the surface of the earth to be flat and the significance of that is to twofold first of all the direction of gravitational forces is the same everywhere as this is not true of course if the earth is curved then gravity will point toward the center but in the flat space approximation gravity points down down everywhere is always in the same direction and second of all perhaps a little bit less obvious but nevertheless true then the approximation where the earth is Infinite and flat goes on and on forever infinite and flat the gravitational force doesn't depend on how high you are same gravitational force here as here the implication of that is that the acceleration of gravity since force apart from the mass of an object the acceleration on an object is independent the way you put it and so Galileo either did or didn't realize well he again I don't know exactly what Galileo did or didn't know but what he said was equivalent to saying that the force on an object in the flat space approximation is very simple its first of all has only one component pointing downward if we take the upward sense of things to be positive then we would say that the force is let's just say the component of the force in the X 2 direction the vertical Direction is equal to minus the minus simply means that the force is downward and it's proportional to the mass of the object times a constant called the gravitational acceleration now the fact that it's constant Everywhere's in other words mass times G doesn't vary from place to place that's this fact that gravity doesn't depend on where you are in the flat space approximation but the fact that the force is proportional to the mass of an object that is not obvious in fact for most forces it's not true for electric forces the force is proportional to the electric charge not to the mass and so gravitational forces are rather special the strength of the gravitational force on an object is proportional to its mass that characterizes gravity almost completely that's the special thing about gravity the force is proportional itself to the mass well if we combine F equals MA with the force law this is the law force then what we find is that mass times acceleration the second X now this is the vertical component by DT squared is equal to minus that's the minus M G period that's it now the interesting thing that happens in gravity is that the mass cancels out from both sides that is what's special about gravity the mass cancels out from both sides and the consequence of that is that the motion of an object its acceleration doesn't depend on the mass it doesn't depend on anything about the particle a particle object I'll use the word particle I don't necessarily mean the point the small particle or baseb as a particle an eraser is a particle a piece of chalk is a particle that the motion of the object doesn't depend on the mass of the object or anything else the result of that is that if you take two objects of quite different mass and you drop them they fall exactly the same way our Galileo did that experiment I don't know if they're whether he really threw something off the Leaning Tower of Pisa or not it's not important he yeah he did balls down an inclined plane I don't know whether he actually did or didn't I know the the the myth is that he didn't die I find it very difficult to believe that he didn't I've been in Pisa last week I was in Pisa and I took a look at the Leaning Tower of Pisa galileo was born and lived in Pisa he was interested in gravity how it would be possible that he wouldn't think of dropping something off the Leaning power tower is beyond my comprehension you look at that tower and you say that I was good for one thing dropping things off now I don't know maybe the Doge or whoever they call the guy at the time said no no Galileo you can't drop things from the tower you'll kill somebody so maybe he didn't but he must have surely thought of it all right so the result had he done it and had he not had to worry about such spurious effects as air resistance would be that a cannonball and a feather would fall in exactly the same way independent of the mass and the equation would just say the acceleration would first of all be downward that's the minus sign and equal to this constant G excuse me that mean yes now G is a number it's 10 meters per second per second at the surface of the earth at the surface of the Moon it's something smaller and the surface of Jupiter it's something larger so it does depend on the mass of the planet but the acceleration doesn't depend on the mass of the object you're dropping it depends on the mass of the object you're dropping it onto but not the mass of the object that's dropping that fact that gravitational motion is completely independent the mass is called or it's the simplest version of something that's called the equivalence principle why it's called the equivalence principle we'll come to later what's equivalent to what at this stage we could just say gravity is equivalent between all different objects independent of their mass but that is not exact were the equivalents an equivalence principle was about that has a consequence an interesting consequence supposing they take some object which is made up out of something which is very unwritten just a collection of point masses maybe maybe let's even say that not even they're not even exerting any forces on each other it's a cloud a varied a few diffuse cloud of particles and we watch it fall let's suppose we start each particle from rest not all at the same height and we let them all fall some particles are heavy some particles are light some of them may be big some of them may be small how does the whole thing fall the answer is all of the particles fall at exactly the same rate the consequence of it is that the shape of this object doesn't deform as it falls it stays absolutely unchanged the relationship between the neighboring parts are unchanged there are no stresses or strains which tend to deform the object so even if the object were held together by some sort of struts or whatever there would be no forces on those struts because everything falls together the consequence of that is the falling in a gravitational field is undetectable you can't tell that you're falling in a gravitational field by you when I say you can't tell certainly you can tell the difference between freefall and standing on the earth that's not the point the point is that you can't tell by looking at your neighbors or anything else that there's a force being exerted on you and that that force that's being exerted on you is pulling you down word you might as well for all practical purposes be infinitely far from the earth with no gravity at all and just sitting there because as far as you can tell there's no tendency for the gravitational field to deform this object or anything else you cannot tell the difference between being in free space infinitely far from anything with no forces and falling freely in a gravitational field that's another statement of the equivalence principle for example these particles could be equipped with lasers lasers and optical detectives of some sort what's that oh you could certainly tell if you was standing on the floor here you could tell that something was falling toward you but the question is from within this object by itself without looking at the floor without knowing the floor was it well you can't tell whether you're falling and it's yeah yeah if there was something that was calm that was not falling it would only be because there was some other force on it like a beam or a tower of some sort of holding it up why because this object if there are no other forces on and only the gravitational forces it will fall at the same rate as this all right so that's another expression of the equivalence principle that you cannot tell the difference between being in free space far from any gravitating object versus being in a gravitational field that we're going to modify this this is of course it's not quite true in a real gravitational field but in this flat space approximation where everything moves together you cannot tell that there's a gravitational field or at least you cannot tell the difference I will not without seeing the floor in any case the self-contained object here does not experience anything different than it would experience far from any gravitating hating object standing still or uniform in uniform motion no you're accelerating if you go up to the top of a high building and you close your eyes and you step off and go into freefall you will feel exactly the same you feel weird I mean that's not the way you usually feel because your stomach will come up and you know do some funny things you know you might you might lose it but uh but the point is you would feel exactly the same discomfort in outer space far from any gravitating object just standing still you feel exactly the same peculiar feelings one of those peculiar feelings due to they're not due to falling they do to not fall well they do to the fact that when you stand on the earth here there are forces on the bottoms of your feet which keep you from falling and if the earth suddenly disappeared from under my feet sure enough my feet would feel funny because they used to having those forces exerted on their bottoms you get it I hope so the fact that you feel funny in freefall is because you're not used to freefall and it doesn't matter whether you're infinitely far from any gravitating object standing still or freely falling in the presence of a gravitational field now as I said this will have to be modified in a little bit there are such things as tidal forces those tidal forces are due to the fact that the earth is curved and that the gravitational field is not the same in every same direction in every point and that it varies with height that's due to the finiteness of the earth but in the flat space surprise and the Flat Earth approximation where the earth is infinitely big pulling uniformly there is no other effective gravity that is any different than being in free space okay again that's known as the equivalence principle now let's go on beyond the flat space or the Flat Earth approximation and move on to Newton's theory of gravity Newton's theory of gravity says every object in the universe exerts a gravitational force on every other object in the universe let's start with just two of them equal and opposite attractive attractive means that the direction of the force on one object is toward the other one equal and opposite forces and the magnitude of the force the magnitude of the force of one object on another let's let's characterize them by a mass let's call this one little m think of it as a lighter mass and this one which we can imagine as a heavier object will call it begin all right Newton's law of force is that the force is proportional to the product of the masses making either mass heavier will increase the force or the product of the masses begin tons of little m inversely proportional to the square of the distance between them let's call that R squared let's call the distance between them are and there's a numerical constant this for this law by itself could not possibly be right it's not dimensionally consistent the if you work out the dimensions of force mass mass and R will not dimensionally consistent there has to be a numeric constant in there and that numerical constant is called capital G Newton's constant and it's very small it's a very small constant I'll write down what it is G it is equal to six or six point seven roughly times 10 to the minus 11th which is a small number so in the face of it it seems that gravity is a very weak force you might not think that gravity is such a weak force but to convince yourself it's a weak force there's a simple experiment that you can do week week by comparison with other forces I've done this for car classes and you can do it yourself just take an object hanging by a string and two experiments the first experiment take a little object here and electrically charge it electrically charge it by rubbing it on your sweater that doesn't put very much electric charge on it but it charges it up enough to feel some electrostatic force and then take another object of exactly the same kind rub it on your shirt and put it over here what happens they repel and the fact that they repel means that this string will shift and you'll see a shift take another example take your little ball there to be iron and put a magnet next to it again you'll see quite an easily detectable deflection of the of the string holding it next take a 10,000 pound weight and put it over here guess what happens undetectable you cannot see anything happen the gravitational force is much much weaker than most other kinds of forces and that's due to the or not due to but the not due to that the fact that it's so weak is encapsulated in this small number here another way to say it is if you take two masses each of one kilometer not one kilometer one kilogram kilogram is a good healthy mass right nice chunk of iron mm and you separate them by one meter then the force between them is just G and it's six point seven times ten to the minus eleven the you know the units being Newtons so it's very very weak force but weak as it is we feel that rather strenuously we feel it strongly because the earth is so darn heavy so the heaviness of the earth makes up for the smallness of G and so we wake up in the morning feeling like we don't want to get out of bed because gravity is holding us down Oh Oh the equal and opposite equal and opposite that's the that's the rule that's Newton's third law the forces are equal and opposite so the force on the large one due to the small one is the same as the force of the small one on the large one and but it is proportional to the product of the masses so the meaning of that is I'm not heavier than I like to be but but I'm not very heavy I'm certainly not heavy enough to deflect the hanging weight significantly but I do exert a force on the earth which is exactly equal and opposite to the force that they're very heavy earth exerts on me why does the earth excel if I dropped from a certain height I accelerate down the earth hardly accelerates at all even though the forces are equal why is it that the earth if the forces are equal my force on the earth and the Earth's force on me of equal why is it that the earth accelerates so little and I accelerate so much yeah because the acceleration involves two things it involves the force and the mass the bigger the mass the less the acceleration for a given force so the earth doesn't accelerate quickly I think it was largely a guess but there was certain was an educated guess and what was the key ah no no it was from Kepler's it was from Kepler's laws it was from Kepler's laws he worked out roughly speaking I don't know exactly what he did he was rather secretive and he didn't really tell people what he did but the piece of knowledge that he had was Kepler's laws of motion planetary motion and my guess is that he just wrote down a general force realized that he would get Kepler's laws of motion for the inverse-square law I don't believe he had any underlying theoretical reason to believe in the inverse-square law that's correct he asked a question for inverse square laws no no it wasn't the ellipse which was the the the orbits might have been circular it was the fact that the period varies is the three halves power of the radius all right the period of motion is circular motion has an acceleration toward the center any motion in the circle is accelerated to the center if you know the period in the radius then you know the acceleration toward the center okay or we could write let's let's do it anybody know what if I know the angular frequency the angular frequency of going around in an orbit that's called Omega you know a–they and it's basically just the inverse period okay Omega is roughly the inverse period number of cycles per second what's the what is the acceleration of a thing moving in a circular orbit anybody remember Omega squared R Omega squared R that's the acceleration now supposing he sets that equal to some unknown force law f of r and then divides by r then he finds Omega as a function of the radius of the orbit okay well let's do it for the real case for the real case inverse square law f of r is 1 over r squared so this would be 1 over r cubed and in that form it is Kepler's second law remember which one it is it's the law that says that the frequency or the period the square of the period is proportional to the cube of the radius that was the law of Kepler so from Kepler's laws he easily could have that that one law he could easily deduce that the force was proportional to 1 over R squared I think that's probably historically what what he did then on top of that he realized if you didn't have a perfectly circular orbit then the inverse square law was the unique law which would give which would give elliptical orbits so who's to say well then of course there are the forces on them for two objects are actually touching each other there are all sorts of forces between them that I'm not just gravitational electrostatic forces atomic forces nuclear forces so you'll have to my breaks down yeah then it breaks down when they get so close that other important forces come into play the other important forces for example are the forces that are holding this object and preventing it from falling these we usually call them contact forces but in fact what they really are is various kinds of electrostatic for electrostatic forces between the atoms and molecules in the table in the atoms and molecules in here so other kinds of forces all right incidentally let me just point out if we're talking about other kinds of force laws for example electrostatic force laws then the force we still have F equals MA but the force law the force law will not be that the force is somehow proportional to the mass times something else but it could be the electric charge if it's the electric charge then electrically uncharged objects will have no forces on them and they won't accelerate electrically charged objects will accelerate in an electric field so electrical forces don't have this Universal property that everything falls or everything moves in the same way uncharged particles move differently than charged particles with respect to electrostatic forces they move the same way with respect to gravitational forces and as a repulsion and attraction whereas gravitational forces are always attractive where where's my gravitational force I lost it yeah here is all right so that's that's Newtonian gravity between two objects for simplicity let's just put one of them the heavy one at the origin of coordinates and study the motion of the light one then Oh incidentally one usually puts let me let me refine this a little bit as I've written it here I haven't really expressed it as a vector equation this is the magnitude of the force between two objects thought of as a vector equation we have to provide a direction for the force vectors have directions what direction is the force on this particle well the answer is its along the radial direction itself so let's call the radial distance R or the radial vector R then the force on little m here is along the direction R but it's also opposite to the direction of R the radial vector relative to the origin over here points this way on the other hand the force points in the opposite direction if we want to make a real vector equation out of this we first of all have to put a minus sign that indicates that the force is opposite to the direction of the radial distance here but we have to also put something in which tells us what direction the force is in it's along the radial direction but wait a minute if I multiply it by r up here I had better divide it by another factor of R downstairs to keep the magnitude unchanged the magnitude of the force is 1 over R squared if I were to just randomly come and multiply it by r that would make the magnitude bigger by a factor of our so I have to divide it by the magnitude of our this is Newton's force law expressed in vector form now let's imagine that we have a whole assembly of particles a whole bunch of them they're all exerting forces on one another in pairs they exert exactly the force that Newton wrote down but what's the total force on a particle let's label these particles this is the first one the second one the third one the fourth one that I thought that thought this is the ithe one over here so I is running index which labels which particle we're talking about the force on the eigth article let's call F sub I and let's remember that it's a vector it's equal to the sum now this is not an obvious fact that when you have two objects exerting a force on the third that the force is necessarily equal to the sum of the two forces of the two are of the two objects you know what I mean but it is a fact anyway obvious on how obvious it is a fact that gravity does work that way at least in the Newtonian approximation with Einstein it breaks down a little bit but in Newtonian physics the force is the sum and so it's a sum over all the other particles let's write that J not equal to I that means it's a sum over all not equal to I so the force on the first particle doesn't come from the first particle it comes from the second particle third particle fourth particle and so forth each individual force involves M sub I the force of the ice particle times the four times the mass of the Jade particle product of the masses divided by the square of the distance between them let's call that R IJ squared the distance between the eigth article his I and J the distance between the earth particle and the J particle is RI J but then just as we did before we have to give it a direction but a minus sign here that indicates that it's attractive another R IJ upstairs but that's a vector R IJ and make this cube downstairs alright so that says that the force on the I've particle is the sum of all the forces due to all the other ones of the product of their masses inverse square in the denominator and the direction of each individual force on this particle is toward the other all right this is a vector sum yeah hmm the minus indicates that it's attractive excellent but you've got the vector going from like a J oh let's see that's a vector going from the J yes there is a question of the sine of this vector over here so yeah you know absolutely let's yeah I actually think it's yeah you're right you're absolutely right the way I've written that there should not be a minus sign here all right but if I put our ji there then there would be a minus sign right so you're right but in any case every one every one of the forces is attractive and what we have to do is to add them up we have to add them up as vectors and so there's some resulting vector some resultant vector which doesn't point toward any one of them in particular but points in some direction which is determined by the vector sum of all the others all right but the interesting fact is if we combine this this is the force on the earth particle if we combine it with Newton's equations let's combine it with Newton's equipped with Newton's F equals MA equations then this is F this on the ice particle this is equal to the mass of the I particle times the acceleration of the ice particle again vector equations now the sum here is over all the other particles we're focusing on number I I the mass of the ice particle will cancel out of this equation I don't want to throw it away but let's just circle it and now put it over on the side we notice that the acceleration of the ice particle does not depend on its mass again once again because the mass occurs in both sides of the equation it can be cancelled out and the motion of the ayth particle does not depend on the mass of the earth particle it depends on the masses of all the other ones all the other ones come in but the mass of the iPart achill cancels out of the equation so what that means is if we had a whole bunch of particles here and we added one more over here its motion would not depend on the mass of that particle it depends on the mass of all the other ones but it doesn't depend on the mass of the i particle here okay that's again the equivalence principle that the motion of a particle doesn't depend on its mass and again if we had a whole bunch of particles here if they were close enough together they were all moving the same way before before i discuss lumo mathematics let's just discuss tidal forces what tidal forces are once you set this whole thing into motion dynamic young we have all different masses and each part what's going to be affected by each one is every particle in there is going to experience a uniform acceleration oh no no no no no acceleration is not uniform the acceleration will get larger when it gets closer to one of the particles it won't be uniform anymore it won't be uniform now because the force is not independent of where you are now the force depends on where you are relative to the objects that are exerting the force it was only in the Flat Earth approximation where the force didn't depend on where you were okay now the force varies so it's larger when you're far away it's sorry it's smaller when you're far away it's larger when you're in close it changes in a vector form with each individual particles each one of them is changing position yeah and and so is the dynamics that every one of them is going towards the center of gravity of the fire not necessarily I mean they could be flying apart from each other but they will be accelerating toward each other okay if I throw this eraser into the air with greater than the escape velocity it's not going to turn around and fall back changing with what with respect to what time oh it changes with respect to time because the object moves moves further and further away it's not uniformly the radius is changing and it's yeah let's take the earth here's the earth and we drop a small mass from far away as that mass moves in its acceleration increases why does its acceleration increase the deceleration increases because the radial distance gets smaller so in that sense it's not the alright now once the gravitational force depends on distance then it's not really quite true that you don't feel anything in a gravitational field you feel something which is to some extent it different than you would feel in free space without any gravitational field the reason is more or less obvious here you are his is the earth now you're you or me or whoever it is happens to be extremely tall a couple of thousand miles tall well this person's feet are being pulled by the gravitational field more than his head or another way of saying the same thing is if let's imagine that the person is very loosely held together he's just more or less a gas of we are pretty loosely held together at least I am right all right the acceleration on the lower portions of his body are larger than the accelerations on the upper portions of his body so it's quite clear what happens to her he gets stretched he doesn't get a sense of falling as such he gets a sense of stretching being stretched feet being pulled away from his head at the same time let's uh let's all right so let's change the shape a little bit I just spend the week two weeks in Italy and my shape changes whenever I go to Italy and it tends to get more horizontal my head is here my feet are here and now I'm this way still loosely put together right now what well not only does the force depend on the distance but it also depends on the direction the force arm my left end over here is this way the force on my right end over here is this way the force on the top of my head is down but it's weaker than the force on my feet so there are two effects one effect is to stretch me vertically it's because my head is not being pulled as hard as my feet but the other effect is to be squished horizontally by the fact that the forces on the left end of me are pointing slightly to the right and the forces on the right end of me are pointing slightly to the left so a loosely knit person like this falling in freefall near a real planet or real gravitational object which has a real Newtonian gravitational field around it will experience a distortion will experience a degree of distortion and a degree of being stretched vertically being compressed horizontally but if the object is small enough or small enough mean let's suppose the object that's falling is small enough if it's small enough then the gradient of the gravitational field across the size of the object will be negligible and so all parts of it will experience the same gravitational acceleration all right so tidal for these are tidal forces these forces which tend to tear things apart vertically and squish them this way tidal forces tidal forces are forces which are real you feel them I mean certainly new the car the cause of the tides yeah I don't know to what extent he calculated what do you mean calculated the well I doubt that he was capable I'm not sure whether he estimated the height of the of the deformation of the oceans or not but I think you did understand this much about tides okay so that's the that's what's called tidal force and then under the tidal force has this effect of stretching and in particular if we take the earth just to tell you just to tell you why it's called tidal forces of course it's because it has to do with tides I'm sure you all know the story but if this is the moon down here then the moon exerting forces on the earth exerts tidal forces on the earth which means to some extent that tends to stretch it this way and squash it this way well the earth is pretty rigid so it doesn't it doesn't deform very much due to due to these two the moon but what's not rigid is the layer of water around it and so the layer of water tends to get stretched and squeezed and so it gets deformed into the a the form shell of water with a bump on this side and the bump on that side alright I'm not gonna go any more deeply into that that I'm sure you've all seen okay but let's define now what we mean by the gravitational field the gravitational field is abstracted from this formula we have a bunch of particles don't you have need some some coordinate geometry so that would you have the four kind of middle is being pulled by all the other guys on the side I'm not explaining it right it's always negative is that what you're saying doesn't know I'm saying so she's attractive all right so you have but what about the other guys that are pulling upon him a different direction here and we're talking about the force on this person over here obviously there's one force pressing this pushing this way and another force pushing that way okay no the cone no they're all opposite to the direction of the object which is pulling on that's what this – sorry instead well you kind of retracted the minus sign at the front and reverse the ji yeah so it's the trend we can get rid of a – like a RI j and our ji are opposite to each other one of them is the vector between I and J I and J and the other one is the vector from J to I so they're equal and opposite to each other the minus sign there look as far as the minus sign goes all it means is that every one of these particles is pulling on this particle toward it as opposed to pushing away from it it's just a convention which keeps track of attraction instead of repulsion yeah for the for the ice master that's my word you want to make sense but if you can look at it as a kind of an in Samba wasn't about a linear conic component to it because the ice guy affects the Jade guy and then put you compute the Jade guy when you take it yeah now what this what this formula is for is supposing you know the positions or all the others you know that then what is the force on the one additional one but you're perfectly right once you let the system evolve then each one will cause a change in motion and the other one and so it becomes a complicated as you say nonlinear mess but this formula is a formula for if you knew the position and location of every particle this would be the force something you need to solve some equations to know how the particles move but if you know where they are then this is the force on the particle alright let's come to the idea of the gravitational field the gravitational field is in some ways similar to the electric field of our of an electric charge it's the combined effect of all the masses Everywhere's and the way you define it is as follows you imagine an one more particle one more particle amount you can take it to be a very light particle so it doesn't influence the motion of the others and one more particle in your imagination you don't really have to add it in your imagination and ask what the force on it is the force is the sum of the forces due to all the others it is proportional each term is proportional to the mass of the sec strip article this extra particle which may be imaginary is called a test particle it's the thing that you're imagining testing out the gravitational field with you take a light little particle and you put it here and you see how it accelerates knowing how it accelerates tells you how much force is on it in fact it just tells you how it accelerates and you can go around and imagine putting it in different places and mapping out the force field that's on that particle or the acceleration field since we already know that the force is proportional to the mass then we can just concentrate on the acceleration the acceleration all particles will have the same acceleration independent of the mass so we don't even have to know what the mass of the particle is we put something over there a little bit of dust and we see how it accelerates acceleration is a vector and so we map out in space the acceleration of a particle at every point in space either imaginary or real particle and that gives us a vector field at every point in space every point in space there is a gravitational field of acceleration it can be thought of as the acceleration you don't have to think of it as force acceleration the acceleration of a point mass located at that position it's a vector it has a direction it has a magnitude and it's a function of position so we just give it a name the acceleration due to all the gravitating objects it's a vector and it depends on position here X means location it means all of the position components of position XY and Z and it depends on all the other masters in the problem that is what's called the gravitational field it's very similar to the electric field except the electric field and the electric field is force per unit charge it's the force on an object divided by the charge on the object the gravitational field is the force of their on the object divided by the mass on the object since the force is proportional to the mass the the acceleration field doesn't depend on which kind of particle we're talking about all right so that's the idea of a gravitational field it's a vector field and it varies from place to place and of course if the particles are moving it also varies in time if everything is in motion the gravitational field will also depend on time we can even work out what it is we know what the force on the earth particle is all right the force on a particle is the mass times the acceleration so if we want to find the acceleration let's take the ayth particle to be the test particle little eye represents the test particle over here let's erase the intermediate step over here and write that this is in AI times AI but let me call it now capital a the acceleration of a particle at position X is given by the right hand side and we can cross out BMI because it cancels from both sides so here's a formula for the gravitational field at an arbitrary point due to a whole bunch of massive objects a whole bunch of massive objects an arbitrary particle put over here will accelerate in some direction that's determined by all the others and that acceleration is the gravitation the definition is the definition of the gravitational field ok let's um let's take a little break we usually take a break in about this time and I recover my breath to go on we need a little bit of fancy mathematics we need a piece of mathematics called Gauss's theorem and Gauss's theorem involves integrals derivatives divergences and we need to spell those things out there a central part of the theory of gravity and much of these things we've done in the context of a lot of electrical forces in particular the concept of divergence divergence of a vector field so I'm not going to spend a lot of time on it if you need to fill in then I suggest you just find any little book on vector calculus and find out what a divergence and a gradient and a curl we don't do curl today what those concepts are and look up Gauss's theorem and they're not terribly hard but we're gonna go through them fairly quickly here since they we've done them several times in the past right imagine that we have a vector field let's call that vector field a it could be the field of acceleration and that's the way I'm gonna use it well for the moment it's just an arbitrary vector field a it depends on position when I say it's a field the implication is that it depends on position now I probably made it completely unreadable a of X varies from point to point and I want to define a concept called the divergence of the field now it's called the divergence because one has to do is the way the field is spreading out away from a point for example a characteristic situation where we would have a strong divergence for a field is if the field was spreading out from a point like that the field is diverging away from the point incidentally if the field is pointing inward then one might say the field has a convergence but we simply say it has a negative divergence all right so divergence can be positive or negative and there's a mathematical expression which represents the degree to which the field is spreading out like that it is called the divergence I'm going to write it down and it's a good thing to get familiar with certainly if you're going to follow this course it's a good thing to get familiar with but are they going to follow any kind of physics course past freshman physics the idea of divergence is very important all right supposing the field a has a set of components the one two and three component but we could call them the x y&z component now I'll use x y&z are X Y & Z which I previously called X 1 X 2 and X 3 it has components X a X a Y and a Z those are the three components of the field well the divergence has to do among other things with the way the field varies in space if the field is the same everywhere as in space what does that mean that would mean the field that has both not only the same magnitude but the same direction everywhere is in space then it just points in the same direction everywhere else with the same magnitude it certainly has no tendency to spread out when does a field have a tendency to spread out when the field varies for example it could be small over here growing bigger growing bigger growing bigger and we might even go in the opposite direction and discover that it's in the opposite direction and getting bigger in that direction then clearly there's a tendency for the field to spread out away from the center here the same thing could be true if it were varying in the vertical direction or who are varying in the other horizontal direction and so the divergence whatever it is has to do with derivatives of the components of the field I'll just tell you exactly what it is it is equal to the divergence of a field is written this way upside down triangle and the meaning of this symbol the meaning of an upside down triangle is always that it has to do with the derivatives the three derivatives derivative whether it's the three partial derivatives derivative with respect to XY and Z and this is by definition the derivative with respect to X of the X component of a plus the derivative with respect to Y of the Y component of a plus the derivative with respect to Z of the Z component of it that's definition what's not a definition is a theorem and it's called Gauss's theorem no that's a scalar quantity that's a scalar quantity yeah it's a scalar quantity so it's let me write it it's the derivative of a sub X with respect to X that's what this means plus the derivative of a sub Y with respect to Y plus the derivative of a sub Z with respect to Z yes so the arrows you were drawn over there those were just a on the other board you drew some arrows on the other board that are now hidden yeah those were just a and a has a divergence when it's spreading out away from a point but that there vergence is itself a scalar quantity oh let me try to give you some idea of what divergence means in a context where you can visualize it imagine that we have a flat lake alright just the water thin a a shallow lake and water is coming up from underneath it's being pumped in from somewheres underneath what happens that the water is being pumped in of course it tends to spread out let's assume that the height let's assume the depth can't change we put a lid over the whole thing so it can't change its depth we pump some water in from underneath and it spreads out okay we suck some water out from underneath and it spreads in it anti spreads it has so the spreading water has a divergence water coming in toward the towards the place where it's being sucked out it has a convergence or a negative divergence now we can be more precise about that we look down at the lake from above and we see all the water is moving of course it's moving if it's being pumped in the world it's moving and there is a velocity vector at every point there is a velocity vector so at every point in this lake there's a velocity vector vector and in particular if there's water being pumped in from the center here right underneath the bottom of the lake there's some water being pumped in the water will spread out away from that point okay and there'll be a divergence where the water is being pumped in okay if the water is being pumped out then exactly the opposite the the arrows point inward and there's a negative divergence the if there's no divergence then for example a simple situation with no divergence that doesn't mean the water is not moving but a simple example with no divergence is the waters all moving together you know the river is simultaneous the lake is all simultaneously moving in the same direction with the same velocity it can do that without any water being pumped in but if you found that the water was moving to the right on this side and the left on that side you'd be pretty sure that somebody is in between water had to be pumped in right if you found the water was spreading out away from a line this way here and this way here then you'd be pretty sure that some water was being pumped in from underneath along this line here well you would see it another way you would discover that the X component of the velocity has a derivative it's different over here than it is over here the X component of the velocity varies along the x direction so the fact that the X component of the velocity is varying along the direction there's an indication that there's some water being pumped in here likewise if you discovered that the water was flowing up over here and down over here you would expect that in here somewhere as some water was being pumped in so derivatives of the velocity are often an indication that the some water being pumped in from underneath that pumping in of the water is the divergence of the velocity vector now the the the the water of course is being pumped in from underneath so there's a direction of flow but it's coming from from underneath there's no sense of direction well okay that's that's what diverges just the diagrams you already have on the other board behind there you take say the rightmost arrow and you draw a circle between the head and tail in between then you can see the in and out the in arrow and the arrow of a circle right in between those two and let's say that's the bigger arrow is created by a steeper slope of the street it's just faster it's going fast it's going okay and because of that there's a divergence there that's basically it's sort of the difference between that's right that's right if we drew a circle around here or we would see that more since the water was moving faster over here than it is over here more water is flowing out over here then it's coming in over here where is it coming from it must be pumped in the fact that there's more water flowing out on one side then it's coming in from the other side must indicate that there's a net inflow from somewheres else and the somewheres else would be from the pump in water from underneath so that's that's the idea of oops could it also be because it's thinning out with that be a crazy example like the late guy young well okay I took all right so let's be very specific now I kept the lake having an absolutely uniform height and let's also suppose that the density of water water is an incompressible fluid it can't be squeezed it can't be stretched then the velocity vector would be the right thing to think about them yeah but you could have no you're right you could have a velocity vector having a divergence because the water is not because water is flowing in but because it's thinning out yeah that's that's also possible okay but let's keep it simple all right and you can have the idea of a divergence makes sense in three dimensions just as well as two dimensions you simply have to imagine that all of space is filled with water and there are some hidden pipes coming in depositing water in different places so that it's spreading out away from points in three-dimensional space in three-dimensional space this is the expression for the divergence if this were the velocity vector at every point you would calculate this quantity and that would tell you how much new water is coming in at each point of space so that's the divergence now there's a theorem which the hint of the theorem was just given by Michael there it's called Gauss's theorem and it says something intuitive very intuitively obvious you take a surface any surface take any surface or any curve in two dimensions and now suppose there's a vector field that the field points now think of it as the flow of water and now let's take the total amount of water that's flowing out of the surface obviously there's some water flowing out over here and of course we want to subtract the water that's flowing in let's calculate the total amount of water that's flowing out of the surface that's an integral over the surface why is it an integral because we have to add up the flows of water outward where the water is coming inward that's just negative negative flow negative outward flow we add up the total outward flow by breaking up the surface into little pieces and asking how much flow is coming out from each little piece yeah how much water is passing out through the surface if the water is incompressible incompressible means density is fixed and furthermore the depth of the water is being kept fixed there's only one way that water can come out of the surface and that's if it's being pumped in if there's a divergence the divergence could be over here could be over here could be over here could be over here in fact any ways where there's a divergence will cause an effect in which water will flow out of this region yeah so there's a connection there's a connection between what's going on on the boundary of this region how much water is flowing through the boundary on the one hand and what the divergence is in the interior the connection between the two and that connection is called Gauss's theorem what it says is that the integral of the divergence in the interior that's the total amount of flow coming in from outside from underneath the bottom of the lake the total integrated and now by integrated I mean in the sense of an integral the integrated amount of flow in that's the integral of the divergence the integral over the interior in the three-dimensional case it would be integral DX dy DZ over the interior of this region of the divergence of a if you like to think of a is the velocity field that's fine is equal to the total amount of flow that's going out through the boundary and how do we write that the total amount of flow that's flowing outward through the boundary we break up let's take the three-dimensional case we break up the boundary into little cells each little cell is a little area let's call each one of those little areas D Sigma these Sigma Sigma stands for surface area Sigma is the Greek letter Sigma it stands for surface area this three-dimensional integral over the interior here is equal to a two-dimensional integral the Sigma over the surface and it is just the component of a perpendicular to the surface let's call a perpendicular to the surface D Sigma a perpendicular to the surface is the amount of flow that's coming out of each one of these little boxes notice incidentally that if there's a flow along the surface it doesn't give rise to any fluid coming out it's only the flow perpendicular to the surface the component of the flow perpendicular to the surface which carries fluid from the inside to the outside so we integrate the perpendicular component of the flow over the surface that's through the Sigma here that gives us the total amount of fluid coming out per unit time for example and that has to be the amount of fluid that's being generated in the interior by the divergence this is Gauss's theorem the relationship between the integral of the divergence on the interior of some region and the integral over the boundary where where it's measuring the flux the amount of stuff that's coming out through the boundary fundamental theorem and let's let's see what it says now any questions about that Gauss's theorem here you'll see how it works I'll show you how it works yeah yeah you could have sure if you had a compressible fluid you could discover that all the fluid out boundary here is all moving inwards in every direction without any new fluid being formed in fact what's happening is just the fluid is getting squeezed but if the fluid can't squeeze if you cannot compress it then the only way that the fluid could be flowing in is if it's being removed somehow from the center if it's being removed by by invisible pipes that are carrying it all so that means the divergence in the case of water would be zero there was no water coming it would be if there was a source of the water divergence is the same as a source source of water is the source of new water coming in from elsewhere is right so in the example with the 2-dimensional lake the source is water flowing in from underneath the sink which is the negative of a source is the water flowing out and in the 2-dimensional example this wouldn't be a 2-dimensional surface integral it would be the integral in here equal to a one dimensional surface and to go coming out okay all right let me show you how you use this let me show you how you use this and what it has to do with what we set up till now about gravity I think hope a lifetime let's imagine that we have a source it could be water but let's take three dimensional case there's a divergence of a vector field let's say a there's a divergence of a vector field del dot a and it's concentrated in some region of space that's a little sphere in some region of space that has spherical symmetry in other words doesn't mean it doesn't mean that the that the divergence is uniform over here but it means that it has the symmetry of a sphere everything is symmetrical with respect to rotations let's suppose that there's a divergence of the fluid okay now let's take and it's restricted completely to be within here it does it could be strong near the center and weak near the outside or it could be weakened near the center and strong near the outside but a certain total amount of fluid or certain total divergence and integrated there vergence is occurring with nice Oracle shape okay let's see if we can use that to figure out what the field what the a field is there's a Dell dot a in here and now let's see can we figure out what the field is elsewhere outside of here so what we do is we draw a surface around there we draw a surface around there and now we're going to use Gauss's theorem first of all let's look at the left side the left side has the integral of the divergence of the vector field all right the vector field or the divergence is completely restricted to some finite sphere in here what is incidentally for the flow case for the fluid flow case what would be the integral of the divergence does anybody know if it really was a flue or a flow of a fluid it'll be the total amount of fluid that was flowing in per unit time it would be the flow per unit time that's coming through the system but whatever it is this integral doesn't depend on the radius of the sphere as long as the sphere this outer sphere here is bigger than this region why because the integral over that there vergence of a is entirely concentrated in this region here and there's zero divergence on the outside so first of all the left hand side is independent of the radius of this outer sphere as long as the radius of the outer sphere is bigger than this concentration of divergence iya so it's a number altogether it's a number let's call that number M I'm not Evan let's just Q Q that's the left hand side and it doesn't depend on the radius on the other hand what is the right hand side well there's a flow going out and if everything is nice and spherically symmetric then the flow is going to go radially outward it's going to be a pure radially outward directed flow if the flow is spherically symmetric radially outward direct directed flow means that the flow is perpendicular to the surface of the sphere so the perpendicular component of a is just a magnitude of AE that's it it's just a magnitude of AE and it's the same everywhere is on the sphere why is it the same because everything has spherical symmetry a spherical symmetry the a that appears here is constant over this whole sphere so this integral is nothing but the magnitude of a times the area of the total sphere if I take an integral over a surface a spherical surface like this of something which doesn't depend on where I am on the sphere then it's just I can take this on the outside the magnitude of the the magnitude of the field and the integral D Sigma is just the total surface area of the sphere what's the total surface area of the sphere just 4 PI 4 PI R squared oh yeah 4 PI R squared times the magnitude of the field is equal to Q so look what we have we have that the magnitude of the field is equal to the total integrated divergence divided by 4 pi the 4 pi is the number times R squared does that look familiar it's a vector field it's pointed radially outward well it's point the radially outward if the divergence is positive if the divergence is positive its pointed radially outward and it's magnitude is one over R squared it's exactly the gravitational field of a point particle at the center here that's why we have to put a direction in here you know this R hat this art will this R over R is it's a unit vector pointing in the radial direction it's a vector of unit length pointing in the radial direction right so it's quite clear from the picture that the a field is pointing radially outward that's what this says over here in any case the magnitude of the field that points radially outward it has magnitude Q and it falls off like 1 over R squared exactly like the Newtonian field of a point mass so a point mass can be thought of as a concentrated divergence of the gravitational field right at the center point mass the literal point mass can be thought of as a concentrated concentrated divergence of the gravitational field concentrated in some very very small little volume think of it if you like you can think of the gravitational field as the flow field or the velocity field of a fluid that's spreading out Oh incidentally of course I've got the sign wrong here the real gravitational acceleration points inward which is an indication that this divergence is negative the divergence is more like a convergence sucking fluid in so the Newtonian gravitational field is isomorphic is mathematically equivalent or mathematically similar to a flow field to a flow of water or whatever other fluid where it's all being sucked out from a single point and as you can see the velocity field itself or in this case the the field the gravitational field but the velocity field would go like one over R squared that's a useful analogy that is not to say that space is a flow of anything it's a mathematical analogy that's useful to understand the one over R squared force law that it is mathematically similar to a field of velocity flow from a flow that's being generated right at the center at a point okay that's that's a useful observation but notice something else supposing now instead of having the flow concentrated at the center here supposing the flow was concentrated over a sphere which was bigger but the same total amount of flow it would not change the answer as long as the total amount of flow is fixed the way that it flows out through here is also fixed this is Newton's theorem Newton's theorem in the gravitational context says that the gravitational field of an object outside the object is independent of whether the object is a point mass at the center or whether it's a spread out mass or there it's a spread out mass this big as long as you're outside the object and as long as the object is spherically symmetric in other words as long as the object is shaped like a sphere and you're outside of it on the outside of it outside of where the mass distribution is then the gravitational field of it doesn't depend on whether it's a point it's a spread out object whether it's denser at the center and less dense at the outside less dense in the inside more dense on the outside all it depends on is the total amount of mass the total amount of mass is like the total amount of flow through coming into the that theorem is very fundamental and important to thinking about gravity for example supposing we are interested in the motion of an object near the surface of the earth but not so near that we can make the flat space approximation let's say at a distance two or three or one and a half times the radius of the earth well that object is attracted by this point that's attracted by this point that's attracted by that point it's close to this point that's far from this point that sounds like a hellish problem to figure out what the gravitational effect on this point is but know this tells you the gravitational field is exactly the same as if the same total mass was concentrated right at the center okay that's Newton's theorem then it's marvelous theorem it's a great piece of luck for him because without it he couldn't have couldn't have solved his equations he knew he meant but it may have been essentially this argument I'm not sure exactly what argument he made but he knew that with the 1 over R squared force law and only the one over R squared force law wouldn't have been truth was one of our cubes 1 over R to the fourth 1 over R to the 7th with the 1 over R squared force law a spherical distribution of mass behaves exactly as if all the mass was concentrated right at the center as long as you're outside the mass so that's what made it possible for Newton to to easily solve his own equations that every object as long as it's spherical shape behaves as if it were appoint appointments so if you're down in a mine shaft that doesn't hold that's right but that doesn't mean you can't figure out what's going on you can't figure out what's going on I don't think we'll do it tonight it's a little too late but yes we can work out what would happen in the mine shaft but that's right it doesn't hold it a mine shaft for example supposing you dig a mine shaft right down through the center of the earth okay and now you get very close to the center of the earth how much force do you expect that we have pulling you toward the center not much certainly much less than if you were than if all the mass will concentrate a right at the center you got the it's not even obvious which way the force is but it is toward the center but it's very small you displace away from the centre of the earth a little bit there's a tiny tiny little force much much less than as if all the mass was squashed toward the centre so right you it doesn't work for that case another interesting case is supposing you have a shell of material to have a shell of material think about a shell of source fluid flowing in fluid is flowing in from the outside onto this blackboard and all the little pipes are arranged on a circle like this what does the fluid flow look like in different places well the answer is on the outside it looks exactly the same as if everything were concentrated on the point but what about in the interior what would you guess nothing nothing everything is just flowing out away from here and there's no flow in here at all how could there be which direction would it be in so there's no flow in here so the distance argument like if you're closer to the surface of the inner shell yeah wouldn't that be more force towards that no you see you use Gauss's theorem let's do count system Gauss's theorem says okay let's take a shell the field the integrated field coming out of that shell is equal to the integrated divergence in here but there is no divergence in here so the net integrated field coming out of zero no field on the interior of the shell field on the exterior of the show so the consequence is that if you made a spherical shell of material like that the interior would be absolutely identical to what it what it would be if there was no gravitating material there at all on the other hand on the outside you would have a field which would be absolutely identical to what happens at the center now there is an analogue of this in the general theory of relativity we'll get to it basically what it says is the field of anything as long as it's fairly symmetric on the outside looks identical to the field of a black hole I think we're finished for tonight go over divergence and all those Gauss's theorem Gauss's theorem is central there would be no gravity without Gauss's theorem the preceding program is copyrighted by Stanford University please visit us at stanford.edu