Lecture 1 | Modern Physics: Special Relativity (Stanford)

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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.

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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

NEET Physics – Motion in a Plane | IMPORTANT QUESTIONS | AIIMS | 2020 2021 Physics Video Lectures

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25 May, 19 – Motion in a Plane – Live Session and Revision – Abhishek Sir – 6:30 PM – NEET & AIIMS 2020 2021

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Powerful Shaolin Monk | Ultimate Proof Traditional Styles Work, Bruce Lee Philosophy

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Bruce Lee believed there was something to be learned from every Martial Art, finding what best suits who you are as an individual, and throwing away the rest… But amongst many martial artists, they believe traditional styles are not feasible… Why is this? Thank you for joining me as we take a closer glance.

Music: Nomadic Sunset
Music: [Nomadic Sunset] by Alexander Nakarada (www.serpentsoundstudios.com)
Licensed under Creative Commons BY Attribution 4.0 License

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Music used: Nomadic Sunset by Alexander Nakarada @ SerpentSound Studios

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Royal Institution Christmas Lectures 2008 – Lecture 1 – Breaking the Speed Limit

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did you know that a mobile phone has a hundred million switches on a chip the size of your fingernail we ever wondered why the laptops keep getting faster and cheaper whether you're a techno wizard or you don't know it's terrified through terror join me in the race to build the ultimate computer good evening and welcome to the Royal Institution all around the world scientists and engineers are locked in a fierce competition the outcome will decide the future of mobile phones laptops games consoles hundreds of other digital devices it's a race to build the ultimate computer and at its heart is the need for speed everyone wants to build faster computers now computers have not been around for very long but we can see just how much they've evolved already by looking at the world of computer games here's an example of a modern computer game and you can see the very realistic graphic something we've all become used to but if we go back just thirty years things looked very different this is a classic computer game from 1978 it's called space invaders and on the screen here you can see a little clip of space invaders and you see the graphics are very primitive at all two-dimensional very slow in fact them all in black and white and so the manufacturers of the console if you can just bring the camera and take a look at this they put strips of coloured plastic over the screen to try to make it look like it was a color display then in 1981 we had the BBC micro and here we can see a game running on the BBC micro quarterly that improved processing power of this computer allowed it to have a very primitive form of three-dimensional graphics this is called wireframe then in 1994 we had this the first playstation who can see it here running a game called Gran Turismo this is a racing name because of the increased processing power of this console we actually have solid three-dimensional objects and the movement the cars is based on physics however the own scape appears rather flat and unrealistic because the process is not fast enough to imitate real-world depth and light and finally here's a modern games console and here we see it running a game called fable – the graphics are now much more realistic fully three-dimensional we see reflections and shadows all rendered in real time at high resolution now the dramatic improvement in the graphics of the 30 year period since space invaders is a direct result of the huge growth in the processing speed of computers while space invaders could do 500,000 instructions per second a Playstation 3 or this Xbox 360 can do over 300 billion it's not just games consoles but I've been getting faster computers generally have been doubling in speed every two years and this has been happening for the last 50 years now in something doubles in a fixed time interval it leads to a very dramatic kind of growth we call it exponential growth the story for example that populations grow so here we see a population of bacteria every 20 minutes each bacterium splits in two so the total number of bacteria doubles every 20 minutes and we see it starts off rather slowly then it takes off in this very dramatic way now you can see for ourselves what exponential growth looks like by doing a little experiment if we can bring one of the handheld cameras in we can have a look at this so in this box I have 225 armed mouse traps and on each mousetrap we have a ping-pong ball now in a moment we're going to take one more ping pong ball and we're going to drop it in through this hole in the top and it's going to set off a chain and in this chain reaction the number of ping-pong balls flying through the air is going to grow exponentially okay so who would like to volunteer to come and set this off my goodness let's have you there yep yes you'd like to make your way along the enter row if you'd like to if you like to come and stand just there just just turn that way okay that's good and what's your name Mattie Matthew Matthew like to hold that no in a moment we're going to give you a three to one can ban okay when we get to go all I want you to do is to place the ball in through that little hole at the top you manage that yeah okay are we ready three two one go amazing excellent okay so so we have a high-speed camera that was looking at that sometimes we can just do a little action replay and see that in slow motion so there's the first ball setting off a couple more each of those is setting off several more and on that curve we can see the number of balls growing in this very dramatic way and what's really impressive about exponential growth is that the rate of growth is itself growing exponentially now the exponential growth of computer power is truly staggering it means for example that the computers that will be made in the next two years have as much processing power as all the computers ever made from the very first computer up to the present day if cars had improved at the same rate as personal computers then a typical family car today would travel 43,000 times faster than a Formula One racing car and it would go 200 times around the world on one liter of petrol now that amazing improvement in speed is made possible because of this this is a microprocessor and this is the microprocessor in its packaging what we see here are gold contacts which connect the microprocessor to the rest of the computer and on the other side is the processing chip itself now to me this is one of the most remarkable pieces of engineering ever created on this chip there are over 400 million components and tiny chips like this have changed the world we can see here a photograph of the surface of this chip this has been magnified a hundred times and it's showing just a fraction of the incredible complexity of the circuitry now as well as being very complex process is also very fast for instance they're very good at doing arithmetic so let's just see how good you are at doing arithmetic okay I'm going to bring up some arithmetic all questions and as soon as you know the answer I want you to shout it out as loudly as you can okay here comes the first one okay excellent next one excellent next one okay not quite so easy well a microprocessor like this can do a calculation like that and get the right answer in a nanosecond that's a billionth of a second now that sounds very fast but just how long is a nanosecond or to find out please welcome Hamish McLeod and it welcome to the Christmas lectures now you're a licensed firearms expert and you've bought a powerful looking gun with you can you tell us something about that yes yes this is a 0.22 air rifle it's probably one of most powerful air rifles available on the market today and certainly this gun would shoot a lead pellet just about as far as any air rifle could ok impressive what we're going to do that is we're going to time the pellet from this gun now to help us do that we've got some special apparatus over here in a moment we're going to mount the gun in this stand and we're going to fire it at this target now if we look here what we see is that the target consists of a circuit board on the circuit board we've edged a pattern a zigzag pattern of copper wire so in the pellet from the gun passes through this circuit board it will break an electrical circuit about 30 centimeters back we have a second target with the same pattern of copper wire and that's also connected to the same circuit so the circuit will tell us the time from the pellet passing through the first board to the pellet passing through the second board and the answer appear on this laptop so Hamish if you're all ready to load up certainly okay and once Hamish is all set we'll give it a 3 2 1 countdown okay are we ready everybody ok 3 2 1 go whoa okay well they look pretty much instantaneous to me but apparently it took just over two million nanoseconds that's two thousandths of a second for that pellet to travel 30 centimeters so in that time or microprocessor could have done just over 2 million complex arithmetic all calculations okay Hamish thank you very much so computers are extremely fast and they're getting faster every year and it's this amazing growth in power that's fueled the digital revolution it's transformed the nature of entertainment of communications of healthcare in fact almost every aspect of our lives has been touched or even revolutionized by the microprocessor and this is just the beginning if micro processes continue to improve in power their impact could be far greater still but a couple of years ago we had a big problem which threatens to stop further growth in the speed of micro processes now to understand what the problem is and what we're going to do about it please join me after the break [Applause] so let's look at why we might be reaching a speed limit for micro processors now to do this we need to understand something about how micro processors are made and how they work and for this I'm going to need a volunteer let's have you yes lights come on down if you'd like to stand just there what's your name yeah Joe okay so a microprocessor is a really big electrical circuit with hundreds of millions of tiny switches and to understand what they're all doing we've got a simple example so we're going to do here is going to make a milkshake and if we succeed this light will come on now to make a milkshake obviously we need milk so what I'd like you to do is to close this switch by moving that lever down okay now the light hasn't come on so obviously we don't have enough ingredients yet so let's suppose we also have some strawberries so you like to close that switch okay and the light comes on so if we have milk and strawberries we can make a strawberry milkshake okay let's suppose we don't have any strawberries can you open that switch for me let's say instead we have some bananas if you like to close that switch so again if we have milk and we have bananas we can make banana milkshake okay let's suppose we have strawberries and bananas so you want to close that switch again and the light stays on of course so what this says is that to make a milkshake we need milk and we need either strawberries or bananas or both now that kind of reasoning is called logic and computers are very good at it now the logic circuits in a microprocessor are vastly more complex than this but the principle is just the same they're based on switches which either on or off okay thank you very much now obviously mechanical switches such as those are much too slow we need a way to make fast electronic switches and the key to this is a remarkable substance called a semiconductor now semiconductor is something that's partway between a conductor like copper which allows electricity to flow very easily and an insulator such as plastic which doesn't allow electricity to flow at all and is that in-between property of semiconductors which allows them to be switched very quickly between being an insulator on the conductor and back again well in fact the very first semiconductor was discovered here in the Royal Institution by Michael Faraday back in 1833 he was experimenting with a material called silver sulfide this is actually quite a familiar material I have here a silver tankard as you can see it's all nice and shiny but if we leave this lying around for a few months it becomes covered in this black tarnish and that black tie is silver sulfide now Faraday discovered that silver sulfide is a semi conductor and in his notebook he described this discovery as very extraordinary but of course he had no idea that the huge practical impact this discovery would have over a hundred years later now in time other semiconductors were discovered such as germanium and silicon was a big breakthrough came in 1947 with invention of this this is called a transistor and we can think of this as a very fast electronic switch having no moving parts but as the transistor work well to find out let's see how we can make a switch using water so here I have a tank containing water and we're just going to build up a little bit of pressure in this tank – the water is now under pressure and it would like to flow up this tube along this tube at the top through this valve but this valve is closed at the moment and then down into this collection Tanner what I'm going to do is to allow some of the high-pressure water to flow into this cylinder and it will move the piston and then open the valve so let's see that happen so this is now flowing into the cylinder pushing on the piston opening the valve or we can see water flowing into the container and if I turn the tap back and release the pressure in the cylinder the valve closes again and the flow of water stops so we've used water pressure to control the flow of water now water pressure is a bit like voltage in an electrical circuit and the flow of water is a bit like electrical current so in a transistor we use a voltage to switch on and off an electrical current of course real transistors are much faster than this a real transistor can switch and the time it takes light to travel just a few millimeters now here we have a model a cross-section model of a transistor and this has been magnified ten million times so on the scale of this transistor the home processor would be the size of Greater London okay so how does this work well this is the silicon layer at the bottom here and on the top we have three copper electrodes this electrode is connected directly to the silicon and so electricity can flow in through this electrode through the silicon and then out through this electrode this is a layer of insulation on on top we have a third electrode now by applying a voltage to this middle electrode we can switch the silicon between being a conductor in which case electricity can flow from here across to here or being an insulator in which case no electricity flows insulation is important because it stops electrical current from flowing out of this middle electrode and in a modern microprocessor that insulation layer is just four atoms in thickness now first transistors were packaged individually I have here a circuit board on a computer was built in the early 1960s and you can see each of these silver cans is one separate transistor thank you now the next important development is called the integrated circuit and here we can see an example of an early integrated circuit in which four transistors have been manufactured on the same piece of silicon now in time people made integrated circuits with more and more transistors on that same piece of silicon first 10 then 20 and so on and that was done by making the transistors smaller and smaller now Gordon Moore who founded Intel notice that the number of transistors on a chip seemed to be doubling every two years and that's become known as Moore's law and that's continued to hold for the last 40 years the next generation of processor will have several billion transistors on each chip the micro processors are manufactured on the surface of a thin wafer of silicon I have one of them here I hope you can see this each of those little squares is a single microprocessor of the kind that we saw earlier now to me it seems incredible that something which is so tiny and so complex can be made at all it sounds almost impossible well to see how it's done we're going to have a go at doing something else which also sounds impossible and for this I'd like a volunteer we have someone from this side up at you would you like to come on down just tell me still sound here that's good what's your name Ronnie Riley okay what's your initials B what are your initials VG alright I'm going to give you a big marker pen there we go and I have here some rice just ordinary rice I'm gonna take a little grain of rice and I'm wondering do you think you could write your initials on the side of that grain of rice isn't that marker pen probably not now I think probably not either okay tell you what to make it a little bit easier and do something different just wait there a minute what I have here is just a sheet of plastic what I'd like you to do is to write your initials nice big writing nice big fat writing across the middle of that plastic for me okay excellent okay good so just go over that one more time make it really nice and big and fat that's it lots and lots of ink is really good for this excellent wonderful okay that's brilliant okay let's just pop the top back on there now what we're going to do with this if you'd like to come with me I'm over here like to stand just there we're going to take this and I'm going to pop it in this frame like this I'm going to switch on this light box so this is just a box with a bright light inside so lots of light is coming out it's passing through your initials and spreading out in all directions and over here we have a lens and the lens is collecting some of that light and focusing it down onto this grain of rice and also have a camera which is looking at that little grain of rice as well and if we can now take the feed if you can take the feed from that camera and bring it up on the screen if I keep out of the way of the light we should just there we can see there your initials written on a grain of rice okay okay I do one morphine hold out your hand like this and put it in front of your initials in front of the light box that's it and then hold it flat like that in front of the light box that's it and now just move it gently about we can see an image of your hand okay okay thank you very much indeed so to make a microprocessor we can lay out the design of the processor on a large scale and then use the projection technique just like this to shrink it down to a very small scale so we've seen how to project an image down to a small size but how can we use this to make a microprocessor well here I have a piece of a microprocessor that's part way through being manufactured in fact it's just the transistor that we saw earlier and we're going to see how to lay down how to create those three copper electrodes and to help us do this we have two workers from the microprocessor Factory okay and we'll find out in a minute why their dress did these strange costumes so how we going to make this microprocessor well at the bottom here we have the layer of silicon and we've already put down the insulation layer and the next stage is to lay down a complete layer of copper across the whole surface of the wafer and then on top of the copper we put down another layer of special material but sensitive to light the next thing we then do is to project an image of the copper wiring onto the top surface so if you can bring on the light please so this pattern of light is the pattern of copper that we want to create on the surface of the wafer now the light is causing a chemical change in the material in this top layer okay we can switch the light off now please and the next stage is to wash the wafer in a special chemical that dissolves away this green layer but it only dissolves the material that wasn't exposed to light so if we can just remove these two pieces now please excellent okay so the next stage is to wash the weight but wash the wafer in acid now acid dissolves copper but it only dissolves the copper where the copper is exposed the copper that's underneath these green regions is protected so let's add our acid and remove those two pieces then please good eye now the final stage is to use yet another chemical to remove all the remaining green layer so if you'd like to remove those two pieces and I'll give you a hand with this middle piece okay so now we've created our pattern of copper wiring if we just bring that light back on for a moment we can see that the pattern of copper corresponds exactly to the pattern of light and we can do this for the hundreds of millions of transistors on the microprocessor or at the same time now all of this has to be done under incredibly clean conditions in fact it's ten thousand times cleaner than an operating theater now to see why imagine that just one speck of dust got into the optical projection system we get an image of the speck of dust now on the scale of this transistor a speck of dust is a hundred meters across so if you get an image of a speck of dust instead of an image of the wiring we've ruined the circuit and that's why Alex here and Alain addressed in these rather strange-looking suits they're called bunny suits so hot in their boilers it's really hot so these suits are not there to protect the workers they're there to protect the microprocessor from a dust which they might bring with them into the factory okay we'll say thank you to our volunteers and join me again after the break [Applause] so we've seen micro processes are made of millions of switches called transistors and we've seen how to make a microprocessor by projecting light I now want to explain something very extraordinary at this way of making micro processes the cost of making a wafer such as this doesn't really depend very much on the exact details of what we make on the surface of it in particular if we make all the transistors half the size then we can fit four times as many transistors onto the wafer and so you can fit four times as many processes and that means each processor is now only a quarter of the cost now usually in life things which are cheaper are not as good as expensive things but in the case of micro processors something amazing happens when we make the transistors smaller they actually become faster if we think back to that water model earlier if we made it smaller the cylinder could fill up with water more quickly the same is true of transistors a smaller transistor can fill up with electrical charge more quickly now faster transistors means the computer is more powerful the rate at which transistors switch on and off is called the clock speed and if you've bought a computer recently it might have a clock speed of let's say three gigahertz that means the transistors are switching on and off three thousand million times a second so smaller transistors means they're faster and it means they're cheaper and also having lots of transistors to play with allows the designers of the microprocessor to use more sophisticated circuits and that also allows the processor to run faster now this extraordinary result explains why it is that computers have improved in such a dramatic way over the last 50 years but remember I said that a couple of years ago we hit the big problem which threatens to stop further growth in the speed of processes now to see what that problem is I'd like to try a little experiment and I'd like you all to join in please I'd like you to imagine that each of you is a transistor in a microprocessor now at the moment you're all switched off so you're all nice and relaxed and nice and cool okay what I'd like you to do it like you all stand up please okay now you're all transistors that are switched on but you're not using too much energy so you're still fairly cool okay I'd like you to sit down please and I'd like you to stand up again please okay sit down okay good or stand up please okay you can all sit down now thank you okay so by now you're probably feeling quite a bit hotter now imagine that we packed twice as many of you into the same space and asked you all to stand up and sit down again then you'll be generating twice as much heat and we'd need more ventilation to keep you cool now imagine we also asked you to stand up and sit down twice as fast then you generate even more heat and we need even more cooling well it's just like this for the transistors in a microprocessor when they're off they don't use much energy they don't produce much heat when they're on they also don't produce much heat it's only when they're switching between on and off that they use energy and produce heat now to see why it is that transistors only produce heat when they switch let's look again at our water model so this is just like the previous model that you saw the only difference is that we've added a second valve here now this valve is the opposite way round to the first one so in the first one is closed or off the second one is on or open and when this one is off the first one is on so let's see what happens when we allow high-pressure water into the cylinder so as the piston switches on this valve it allows water to flow through into the container but in a moment or two this second valve will close and so the flow of water stops now the flow of water remember is like the flow of electrical current so what the designers of micro processors do is they do a similar trick they use transistors in pairs so that when one of them is on there's another one which is off and that stops current flowing and the flow of current uses power and that generates heat so because the transistors only produce heat when they're actually switching it means that modern micro processors are very efficient in their use of energy so making transistor smaller makes them faster it makes them cheaper but it also means they produce more heat per unit area so all modern micro processors are very efficient as we approach a billion transistors on a chip they're going to be producing a lot of heat now to show you just how much heat we did the little experiment we took a standard personal computer which you see on the right here and then Ian and Nathan took it all apart and laid all the components out on a board and over here we see the end result here are the different parts of the computer laid out this is the main circuit board of the computer this is called the motherboard this houses many of the important components for example along here we have the memory chips and this is the power supply here this is the hard disk this stores all the data and the software and the processor itself is on the motherboard it's underneath here and usually on top of the processor we have a piece of metal like this which is called a heatsink and on top of the heatsink we have a little fan to keep it cool now we've removed the heatsink and we've replaced it with this copper dish this computer has been sitting here for a while doing a thousand million calculations a second and it's what I have here is an egg I'm just going to crack the egg and put it into that two copper dish and we'll come back to that in just a moment and see how that's getting on meanwhile this should have a look at a little photograph of a computer circuit board and this photograph has been taken with a special camera called a thermal imaging camera and so the different colors here represent different temperatures so the purples and the blues those are low temperatures and the yellows and reds are hotter temperatures the white is the hottest of all what we can see is that the hottest part of the computer is actually the microprocessor okay let's have a look and see how our eggs getting on and it's actually cooking very nicely in fact I think my stuff will be ready by the end of the lecture okay now really that's just a bit of fun but in fact heat has become one of the biggest problems that microprocessor designers have to face today in fact a microprocessor today is producing the same heat density as the hot plate on a cooker if we just continue to double the speed of processes every two years that in ten years time the heat density on a microprocessor will be the same as the heat density at the surface of the Sun now unless that heat can be taken away quickly the processor will overheat and fail so heat has become the main factor which limits the speed of modern micro processors so how are we going to solve this problem well instead of just switching the transistors on and off even faster manufacturers are trying a different approach a couple of years ago they started putting two computing engines called cos on the same chip we have here the photograph that we've seen earlier and you may have noticed that this part of the processor looks a lot like this part in fact this is just a copy of this part these are actually two separate processing cores on the same piece of silicon now with two cos running at the same speed that processor should be twice as fast we call this parallel processing there's a little snag however not all tasks can be shared for example it takes a woman nine months to make a baby but nine women can't make a baby in one month now fortunately many important computational problems can be shared efficiently across many cause so over the next few years we'll see processes with 4 cores 8 cores 16 cores and so on and this will allow the speed of processes to continue to double every two years for at least some time to come but is there a way to build vastly more powerful computers well here's one idea this is a model of what's called a carbon nanotube each of these is a carbon atom now if we can build a transistor out of carbon nanotubes it could fit could switch a thousand times faster than a transistor made of silicon now here's an even more exotic idea what about using DNA to store information and to process it a computer made of DNA could be 10 billion times more energy efficient than a computer made out of silicon and the DNA in one human body can store 10,000 times as much data as all the personal computers in the world put together now it's unlikely that general-purpose computers will ever be made out of DNA but one day we might be able to build intelligent drugs that use DNA computation to detect for example when a cell is about to become cancerous and to take the appropriate action now it might be many decades before ideas such as these become reality but in tomorrow's lecture we'll look at some of the amazing new technologies that are already changing the way in which humans and computers interact with each other join me then thank you you [Applause]

Why can't you go faster than light?

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One of the most counterintuitive facts of our universe is that you can’t go faster than the speed of light. From this single observation arise all of the mind-bending behaviors of special relativity. But why is this so? In this in-depth video, Fermilab’s Dr. Don Lincoln explains the real reason that you can’t go faster than the speed of light. It will blow your mind.

over the past hundred years or so scientists have pushed our understanding of the universe into some extreme conditions for example the world of the very small the realm of very high speeds and under the frigid conditions of near absolute zero while each of us have developed an intuition about how the world works it's very important to remember that this intuition only applies to a very limited set of conditions for instance there's absolutely no reason to expect that matter will act the same in the center of the Sun as it does here on earth on a bright and sunny day however that last statement is hard for some people to accept and judging by my email INBOX the extreme realm that causes people the most difficulty is what happens when things are going super fast in 1905 Albert Einstein published his theory of special relativity it predicts all sorts of mind-blowing things for instance distance shorten and clocks slowed down I made another video about how clocks act at high speed it turns out that all of those seemingly crazy implications originate from a single cause or maybe two if we take it slow so first let me tell you what this video isn't it doesn't tell you about the postulates that Einstein used to build this intuition and it certainly doesn't derive as equations instead this video tries to tell you the key insights that make it easier to develop a relativistic intuition I hope to teach you why it is impossible to go faster than the speed of light if you're not a physics groupie hearing that there's a maximum speed in the universe might surprise you but it's true and if you are a groupie you've probably heard that the reason that you can't go faster than light is due to the fact that mass increases when you speed up it turns out that the explanation of mass changing as you go faster is a wrong one I know that statement is going to confuse some people including those with fairly sophisticated understandings of relativity but it's true however that then leaves an open question just why is it that you can't go faster than speed of light it turns out to be due to a combination of a deep and fundamental property of the universe and fairly simple geometry so let me explain how that all works the first two the two crucial insights is that Einstein taught us the space and time were not separate entities but rather they are two components of a bigger idea called space-time I'll give you a helpful visual way to think about this in a moment but for right now just trust me on this then we need to combine that insight with the observation that everybody sees the speed of light to be the same no matter how fast they're moving with respect to one another let's start with an analogy and then come back to relativity to understand the analogy you need to imagine a car driving on a huge flat surface further you need to imagine that the car can only move at one speed say 60 miles per hour or so the comments don't fill up with a metric snobbery hate-mail 100 kilometers per hour now let's put a couple of arrows on the screen to point out north and east well we know the overall speed the car is going we don't know how much of it is in the east direction and how much of it is in the north direction so let's take a closer look at that the car can move entirely in the eastward direction which means that it has no motion in the northward direction or the car can move entirely northward and not at all eastward or we can live dangerously and move towards the Northeast in this case we see that the car is moving in both the east and north directions with neither direction getting all of the motion so that's the core analogy and hopefully it's very clear now let's bring in relativity and relativity we don't have the east and north directions instead we have space-time let's imagine that the horizontal direction of space and the vertical direction is time so suppose that there is a single and fixed speed that we can travel through space-time this happens to be true so it's not a ridiculous supposition we can therefore mix these ideas with our earlier analogy an object can move vertically in that case there moving through space and they're moving entirely through time that's probably what you're doing right now you're sitting and watching this video so your position in space isn't changing however you are experiencing time you aren't moving through space but you're moving through time on the other hand what happens as you start moving through space that's a fancy way to say that you've gained some velocity well we see here that what starts to happen is that as you begin to move through space you move less through time and eventually when you move only through space you don't move through time at all and this is basically what relativity says as you move faster and faster your clocks slowed down and as you get very close to the speed of light your clocks very nearly stopped we've scientifically proven that this is what happens and I direct you to my video on time dilation so you can see one way that we've tested that so this brings us to our fundamental realization of relativity the reason that we can't move through space faster than the speed of light is because we're constantly moving through space time at a single speed the speed of light if we aren't moving through space we experience time in the fastest way and if we start moving through space we experience time slower and slower finally since we're moving through space time at a single speed that means when we're only moving through space there's no more speed to gain we move through space at the speed of light and that's it this observation wasn't made by Einstein it was made by his mentor Hermann Minkowski Minkowski was one of Einstein's mentors and he was a better mathematician two years after Einsteins a seminal 1905 paper Minkowski appreciated the geometrical underpinnings of special relativity and had determined this deep and fundamental explanation why we can't travel faster than light through space there are two final important points first while Minkowski showed why Lightspeed is the maximum speed through space what he didn't explain was why we move only at one speed through time to this day nobody really knows it seems to be a fundamental property of space-time maybe it will take another person as smart as Einstein to figure out that particular conundrum the second point is more technical and I mention it only for the real physics nerds in my analogy I connected space and time as being similar to east and north and there's a lot of merit in that morphing from motion through time to motion through space was like turning a car from moving north to moving east however this analogy is also technically inaccurate from a mathematical point of view it uses the geometry of circles well the proper geometry is that of hyperbolas I only bring this up because I want you to know my analogy is imperfect and you shouldn't push it too far otherwise you might come to a numerically incorrect conclusion and think that you've made a new discovery if you want to dig into this more deeply be sure to use the full and proper Minkowski mathematics still even with the limitations I mentioned the core point is valid the reason that you can't move faster through space than the speed of light is because every object moves through space-time at one and only one speed the speed of light once you've embraced that central idea and the fact that space and time are just like two directions of space-time then all of those seemingly weird observations of relativity just click into place and special relativity makes total sense so I don't know about you but I think this insight about relativity is just about the coolest thing ever if you liked this video be sure to LIKE subscribe and share let's get those numbers up and let me know what you think in the comments I'll see you next time and keep on physics

Einstein's Theory Of Relativity Made Easy

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… 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.

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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

Experiment! How Does An Owl Fly So Silently? | Super Powered Owls | BBC

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Using sensitive sound equipment the team try to find out how an owl can fly so silently compared to other birds.

Taken from Super Powered Owls.

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How to Memorize Fast and Easily

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How to memorize fast and easily. Take this quick and easy challenge and discover the natural power of your memory.

Mind-blowing, right? Learn more at and discover how you can remember and recall this easily when you study – all the time.

It’s time to revolutionize the way you memorize 🙂

And if you’d like another cool memory challenge, check out the Ultra Challenge at

إذا كنت تريد الحفظ بمعدل 10 مرات أسرع ، فهذا الفيديو سيظهرلك بسهولة ويمكنك تطوير الاداء . تتذكر المعلومات بطريقتين رئيسيتين – كالكلمات ، باستخدام ذاكرتك اللفظية ، أو كالصور ، باستخدام الذاكرة البصرية الخاصة بك. إنها عمليات عقلية مختلفة وتحقق نتائج مختلفة بشكل كبير. لا يعتقد الناس أبدًا مدى الاختلاف المجنون تمامًا ، لذلك نحن نواجه تحديًا من أجلك وأنت تستطيع أن تثبت ذلك لنفسك. أولاً ، دعونا نختبر ذاكرتك اللفظية. سأقدم لك قائمة بعشرة كلمات وسنرى عدد الأشخاص الذين يمكنك تذكرهم. ها نحن ذا بيانو
فيل شاحنة
زجاجة كرة السله
كرسي أناناس
كلب لوحة
الترامبولين حسنًا ، أوقف الفيديو مؤقتًا واكتب جميع الكلمات التي يمكنك تذكرها. كيف ستذهب؟ إذا كنت مثل الشخص العادي ، فقد تمكنت من استدعاء حوالي من خمس إلى سبع كلمات ، ليس بالضرورة في الترتيب الصحيح. لذا كانت ذاكرتك اللفظية ، والآن دعنا نختبر ذاكرتك المرئية. سأقدم لك قائمة كلمات أخرى ، لكن هذه المرة ، سأقدم أيضًا قصة قصيرة ورسم صورة. لتنشيط الذاكرة البصرية ، فقط قم بإنشاء صورة ذهنية لكل شيء أصفه و ارسم يمكنك حتى إغلاق عينيك والاستماع فقط إلى صوتي. ها نحن ذا Ferrari – تخيل أنك تقود سيارة فيراري حمراء زاهية من الأعلى إلى الأسفل. الموسيقى تضخ فوق الهدير من المحرك ، ويحرك شعرك الريح دجاج – مع "الصراخ" بصوت عالٍ ، توجد دجاجات عملاقة في المقعد المجاور لك. إنه حجم شخص ضخم وأصفر. يجب أن يكون قد سقط من السماء. البطيخ – يفتح الدجاجة باب السيارة ويقفز على الطريق. كما أنها تقف هناك ، يتوغل البطيخ الأخضر الكبير فوقها ويستمر في الدوران على الطريق. باراك أوباما – أنت تشاهد لفة البطيخ على الطريق تمشي مباشرة إلى باراك اوباما ينقسم البطيخ إلى نصفين ويترك أوباما واقفًا هناك ، غارقا في عصير البطيخ عصير. Poodle – أوباما يلتقط كلبًا مهرًا ويستخدمه لمسح
عصير وجهه. كلب ذو لون أبيض ناصع ، لكن عندما يتغذى على عصير البطيخ ، يتحول ببطء إلى اللون الزاهي زهري. Flagpole – أوباما يرمي الكلب بعيدا ، ويطير في الهواء ويهبط على أعلى سارية العلم الطويلة. ثقل الكلب بالعصير يسبب انهيار سارية العلم ببطء. الكعك – مع "الصوت العالي" والفوضى تقع سارية العلم في الوسط الهائل في كعكة عيد الميلاد. الجليد والكريم والشموع يطيرون في كل مكان ، ويمطرون على الأشخاص المارة. Doll – دمية كبيرة من الكريمة تقع على رأس دمية باربي كبيرة الحجم. يخلق رد فعل كيميائي غريب والدمية يطلق النار في السماء مثل صاروخ فضائي ، شقراء الشعر زائدة وراءها. البيتزا – صاروخ الدمية يتصاعد ، وبمجرد أن يبدأ في الانخفاض ، تنفجر بيتزا كبيرة مفتوحة فوق رأسها مثل المظلة. يتم إرفاق البيتزا بالدميه عن طريق سلاسل طويلة من الجبن الذائب. الزرافة – تهبط البيتزا في النهاية على الأرض وتغطي الدمية والزرافة تمشي وتبدأ في أكل البيتزا ، وتثني العنق الطويل وتمتد لسانه ل لعق الجبن اللذيذ لوح التزلج – بعد تناول الكثير من البيتزا ، تقوم الزرافة بسحب لوح التزلج ، والقفز عليه ، ويبدأ في التزلق في الشارع ، وتتملص من العلامات الضوئيه وأضواء الشوارع أثناء دحرجته علي طول على طول. السيجارة – تبدأ لوح التزلج بالسعال وتتوقف وتستخدم إحدى عجلاتها لإشعال سيجارة. تصبح السيجارة غارقة في النيران و يرميها لوح التزلج بعيدا . تمثال الحرية – تطير السجائر المشتعلة في الهواء وتهبط على الشعلة ويصبح في ارتفاع عالي من تمثال الحرية. واشتعلت النيران في الشعلة أيضا. آيس كريم – تمثال الحرية أصبح على قيد الحياة ويدفع الشعلة إلى داخل دلو كبير من الآيس كريم. انها آيس كريم شوكولاتة الكرز التي تذوب وتبدأ في اظهار الفقاعة. الألعاب النارية – تنفجر الآيس كريم في الألعاب النارية ، وتضيء السماء فوق تمثال الحرية مع الألعاب النارية الملونة الزاهية التي تشكل عبارة "النهاية". حسنًا ، أوقف الفيديو مؤقتًا واكتب عدد الكلمات التي يمكنك التذكير باستخدام الذاكرة البصريه الحيلة هي إعادة إنشاء صورة في ذهنك لكل صورة في القصة. هل رأيت الفرق؟ وفعلت شيئًا مخادعًا ، أعطيتك خمسة عشر كلمة ، وليس عشرة ، لكن الشخص العادي سوف يفعل ذلك تمكنت من التذكير من عشرة إلى كل خمسة عشر كلمة ، ومعظمها بالترتيب الصحيح. اترك التعليق أدناه واسمحوا لي أن أعرف كيف سجلت ذاكرتك اللفظية ضد ذاكرتك البصريه ذاكرة. تقنيات الذاكرة المرئية كانت موجودة منذ آلاف السنين ، ولكن بالنسبة لبعضها غريب السبب ، معظم الناس لا يعرفون سوى تقنيات الذاكرة اللفظية. التقنيات اللفظية هي أشياء مثل الاختصارات وعلم الإيكولوجيات ، وجمعيات الكلمات والقوافي ، وحتى الأغاني ، وكلهم بحاجة إلى قطعة خطيرة من التكرار الممل. يمكن أن تكون رائعة لعدد قليل من الكلمات ، لكنها لا تصبح مذهلة بقوة الذاكرة البصرية الخاصة بك. إذا كنت ترغب في تنشيط القوة المذهلة للذاكرة البصرية لحفظ كاملها جدول دوري للعناصر ، وتوجه إلى MemorizePeriodicTable.com وتحقق من دورة فيديو الرسوم المتحركة خطوة بخطوة دورة. وإذا كنت ترغب في تعلم بعض تقنيات الذاكرة المرئية المدهشة ، فيمكنك التسجيل هناك لسلسلة تدريب الفيديو المجاني لدينا أيضا ، وتعلم كيفية تحويل نفسك إلى طالب رائع لا تنس أن تخبرني في التعليقات أدناه علاماتك للفعل والذاكرة البصرية ، وإذا كنت تعتقد أن هذا كان تحديا رائعا ، يرجى مشاركتها مع أصداقائك ونري هل ذاكرتهم رائعه مثل ذاكرتك .. شكرا للمشاهده مع السلامه ترجمة : محمود عبدالعزيز

Why Planes Don't Fly Faster

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1970s first class image courtesy British Airways and used under fair use guidelines
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“Prelude No. 7” and “Prelude No. 14” by Chris Zabriskie

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这是来自Wendover 出版商的视频,由Audible制作 最近我闲来无事,查看旧航班时刻表。 航空公司常常发行这些印有票价和飞行时间的时刻表小册子 当我看到美航1967年发行的时刻表时,
发现了一个很有意思的事 从时刻表上看,从纽约飞往洛杉矶的飞行时间是5小时43分钟 但可能并不是这样 美航3号航班,从中午12点从纽约肯尼迪机场起飞 并在下午2:43抵达洛杉矶。
其中存在3小时的时差 时至今日,
美航3号航班仍然在中午从纽约肯尼迪机场起飞 但不同的是,抵达洛杉矶的时间却是下午3:27 比1967年的时候还多了44分钟 其实,这是个全球性的普遍现象。
几乎所有的航班都比60年代时 的飞行时间要长一些。 通常来说,真正意义上的飞行时间,
也就是飞机真正在空中飞行的耗时是相同的。 但是现在,在机场和地面的拥挤和延误也已经算在飞行时间里面,才造成飞行时间的延长。 就算是这样,不管怎么说,
我们乘坐的飞机确实飞的更慢了。 在1967年,人类还没登月,
而且那时候的电脑看起来还像这样子占满整个房间。 但是那时我们无论飞往哪里都和现在一样快,
甚至比现在还要快。 在过去的50年为什么飞行速度都没提高,
反而变得更慢了呢? 先来看看三种常用的飞机发动机:
涡轮螺旋桨发动机(涡桨发动机),涡轮风扇发动机(涡扇发动机)和涡轮喷气发动机(涡喷发动机)。 每一种发动机都有一个最经济的飞行时速范围 你可以在大多数有螺旋桨的飞机上见到这种涡桨发动机 对于这种发动机来说,几乎所有的推力都来源于螺旋桨 用于驱动螺旋桨的涡轮虽然能够吸入并加速一部分空气的流动, 但是发动机排出的空气流速并不是很快,
所以发动机排气只能贡献不到10%的推力 就涡桨发动机而言,由于它的购买和运营成本很低,
所以很多短途支线通勤飞机 都使用涡桨发动机作为动力 当然,使用这种发动机的飞机都不能飞得很快。 涡桨发动机的最经济的飞行速度是
介于每小时325英里到每小时375英里之间。 如果需要飞得更快,那最好就用涡扇发动机 如今你到处都可以看到涡扇发动机 大多数商业飞机都是用涡扇发动机驱动的。 涡轮风扇,顾名思义,空气首先需要流过一个风扇 就像你在发动机前面看到的那样 然后,一部分空气会流进内部燃烧室 推动涡轮做功,用于驱动风扇。
还有一部分空气会从涡轮外围通过。 从涡轮外围通过的那一部分空气实际上是从外侧涵道通过的。
这一部分的空气也被加速了 而且涡扇发动机主要推力
就是从内涵道喷出的气流提供的 现代飞机飞行时速大多在每小时400-620英里之间,
这也是涡扇发动机最经济的时速 如果你想超音速飞行,也就是说飞行时速超过每小时767英里,那就需要涡轮喷气发动机(涡喷发动机)了。 涡喷发动机和涡扇发动机很像。不同点在于所有的空气都会流经涡轮。 没有外涵道 这样可以使飞机达到非常快的速度,
但代价是需要消耗大量的燃料 涡喷发动机在时速介于每小时1300-1400英里
才是经济时速 真正决定飞机发动机效率的,是一个数值,叫做“涵道比” 这是外涵道空气流量
和发动机核心机(内涵道)空气流量的比值 这是外涵道空气流量
和发动机核心机(内涵道)空气流量的比值 事情是这样的。实际上驱动大一点的风扇并不需要消耗太多的能源。 但如果需要更多的空气进入发动机内涵道
则需要增加非常多的燃油供应 也就是说,在消耗相同的能量下,如果能够让更多的空气流经外涵道,就可以获得更大的推力 所以,这像一条定律一样,即涵道比越大,发动机效率越高 看看这个通用电气GEnx发动机 这是一款用于波音787梦幻客机和波音747-8i洲际客机上的新型超高效率发动机。 你可以看到,风扇的尺寸比涡轮大得多 这是因为这款发动机拥有10:1的涵道比。
意味着流经涡轮周围(外涵道)的空气流量 是流进涡轮(内涵道)的10倍 相比较而言,CFM国际公司生产的,相对老旧的CFM56发动机的效率就低一些 你可以看到风扇的尺寸和涡轮比起来,差距没那么大了 因为这款发动机的涵道比只有5.9:1
但仍然属于大涵道比 如果和普惠公司的JT8D型发动机比起来
差距就会更明显 因为JT8D发动机的涵道比只有0.96:1
所以它的效率更加低下 但仍然比劳斯莱斯公司的奥林巴斯593发动机的效率
高得多 因为593发动机是涡喷发动机 我之前说过,就涡喷发动机而言,所有的空气必须经过涡轮 没有外涵道 所以,涡喷发动机的涵道比是0:1
也被称为“无外涵道发动机” 由于100%的空气会经过发动机内部并通过涡轮,这就导致 需要消耗更多燃料。燃料消耗比先前介绍的GEnx, CFM56, 甚至JT8D发动机都要高 协和式客机使用无涵道劳斯莱斯 奥林巴斯593发动机。
协和式飞机每飞行1英里(1.61公里) 就会烧掉21.25公斤的燃油。对比之下,
波音787梦幻客机使用涵道比为10:1的GEnx发动机 每飞行1英里(1.61公里)只烧掉8.5公斤的燃油。
但是,即使和波音787这样的中型飞机比起来,协和式飞机 也只能算是个小飞机 协和式飞机只能乘坐100人,
但787梦幻客机可以乘坐291人 这就可以算一下每个人消耗的燃油量
乘坐协和式飞机,每人每飞行100公里消耗17升的燃油 而乘坐波音787梦幻飞机,
每人每飞行100公里只消耗2.3升燃油 到头来,法航和英航
这仅有的两家运营协和式飞机的航空公司 也无力承担协和式飞机的飞行了 因为只有不到1/3的乘客
是真正买了机票搭乘协和式飞机的 剩下的乘客是使用累积旅程兑换
或者是从其他航班头等舱转过来的乘客 毕竟,搭乘协和式飞机至少花费相当于今天的7500美元才能买下一张从伦敦飞往纽约的单程票 飞行时间只有3小时。但是座位却是这样的 和当今飞机的经济舱的座位几乎没有区别 当协和式飞机刚问市的时候,
其他飞机的头等舱是这样的 虽然看起来很漂亮,
但是这只是加宽的经济舱座椅而已 在这样的座椅上睡觉确实比较困难 但是,当2003年协和式飞机退出运营时,
头等舱座椅却变成这样, 而且可以像一张床一样展开放平 所以很多人宁可选择在这样的座椅上度过7个小时,
而不是挤在协和式飞机那样狭小的座椅上 度过3个小时 在2000年,英航甚至在商务舱
也引入了这种能够完全平躺的座椅 但票价却比协和式飞机低得多。
在飞越大西洋上空时,旅客能够平躺着睡觉 所以乘坐协和式飞机已经不再是什么豪华享受了。 设计协和式飞机的初衷是为商务人士
提供一个高效的跨越大西洋的方式 但是,如果座椅能够完全平躺,那些飞往欧洲的乘客 就可以在晚上离开美国,在飞机上睡一觉,
醒来就到欧洲了 实际上并没有浪费时间 既不豪华又不高效,协和式飞机在2003年10月24日完成了最后一次商业飞行 并宣告了超音速商业飞行时代的结束 还有一点,航空公司真的不太在乎飞行速度 飞行速度只是吸引顾客的卖点而已 飞机本身的成本相对于总飞行成本来说
也只是一小部分。 所以航空公司并不会采用提高速度的办法
来增加飞机的使用次数 通常来说飞机的寿命和起降周期有关
就是说和飞机起飞和降落的次数有关 就波音787梦幻飞机可以承受44000次起降。
波音787的建议售价通常比实际销售价钱 高很多,每架飞机达到了2亿2460万美元 这就意味着每次起降就要花费超过5000美元。 再加上燃油费,从纽约到伦敦,燃油费也超过15000美元 因此,航空公司就让飞机尽可能在燃油最高效的速度飞行 这个速度几乎总是位于每小时500-550英里这一区间内 这样的后果就是飞行速度远低于音速–每小时767英里 为什么飞机不能贴近音速飞行呢? 这张图表显示飞机在不同速度下所受到的阻力 在0.8~1.2马赫之间属于“跨音速”范围
(1马赫相当于1倍音速) 在这个范围内,飞机周围的气流不完全是亚音速或超音速的。 本质上说,有些气流的流速已经超过音速,
但是也有一些气流仍然没达到音速 所以从0.8马赫的速度开始,部分气流已经超过音速,
这样会急剧增加阻力 并且降低飞机的操纵稳定性。
所以,以接近音速的速度飞行是非常危险的 你需要以远高出音速或远低于音速的速度飞行才安全 当飞机在跨音速飞行的时候, 你可以看到这样的现象 看这些条纹特别像照相机镜头上的划痕。
尽管这很难看到. 这些条纹实际上是微型的超音速激波 因为要破除气流相当大的阻力,所以在以0.8到1.2马赫的速度飞行时 实际上要比以1.2马赫以上的速度飞行消耗更多的燃油。这就是我们为什么有613.8英里/小时这个数字 这个数字几乎是现代亚音速喷气式飞机的速度极限 尽管搭乘超音速航班在3小时内横跨大西洋
是很炫酷很令人激动的事 但是就航空公司而言,真正想要的是快到像跳池塘那样
分分钟有100或200美金的利润进账 而且这在今天变得更加容易实现。 现今飞机的速度已经让我们在24小时之内飞去地球上的任何地方 这对大部分人来说已经足够快了 对大多数人来说,真正制约旅行的因素是成本,而不是速度。所以 飞机厂商和航空公司将继续致力于减少旅行成本,而不是减少旅行时间 最后再说一句。时间只是极少数特权人士的敌人。而成本才是大多数人的敌人 这个视频是由Audible做的。 Audible是领先的在线音频提供商,但我相信你之前已经听说过。 所以我只会告诉你我是怎么利用audible的 我喜欢阅读但是有很多时候我不能阅读纸质书籍 比如工作,做饭甚至编辑视频时。所以我下载了大量的音频书籍 通过Audible,我无论在干啥的时候都可以学习 如果你也像我这样喜欢在干活的时候听点什么,你就能把无聊的工作变得有趣一些 我最近开始听 Skyfaring,
这是一本由747飞行员Mark Vanhoenacker写的书 可以让你以飞行员的角度了解飞行,但是读起来更像读小说或回忆录 而不像读那些大量的专业书籍那样枯燥 最主要的是你现在可以免费听这本书。 Audible给了你30天免费试读的时间,可以登录Audible.com/Wendover这个链接领取 这是一本好书,而且Audible给了一个好看的链接,所以请去Audible.com/Wendover. 看看吧 除此之外,你也可以上atreon.com/wendoverproductions这个链接来支持 Wendover 出版商 在Twitter上关注我@WendoverPro,
我最新的视频是介绍美国总统如何出行的 也可以去下面这些栏目看看。
最重要的一点,请订阅该频道 这样就可以在未来新视频发布的时候第一时间收看 再次感谢您观看本视频,我们两周后见,

The Scientific Method, ACT Science Bootcamp #3

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— Learn more about the ACT Science Bootcamp here.

This is the third video of the ACT Science Bootcamp. Check out the other preview videos at the above link!

A deep understanding of the scientific method – how science WORKS – is key to understanding the ACT Science passages & answering the questions.

Check out more ACT Science videos here:

You can download five real practice ACTs here:

before we discuss the question types on experimental design an experimental reasoning we need to think a little sidebar and talk about the scientific method so really the science test is testing your understanding of how science works so how do scientists design their experiments how do they try to reach conclusions from the data what assumptions or considerations do they bring that's what it's all about so we're going to talk about some basic ideas about how science works how experiments are designed and then we're going to use these to answer questions eventually on experimental design on things like variables controls hypotheses all that good stuff so let's begin with a little thought experiment so imagine you want to study you want to study the effects of four factors on tree growth and you want to see how to each of these affect how well a tree grows so the amount of sunlight the tree receives per day the amount of water it receives per day average daily temperature how that affects tree growth and then the type of fertilizer used so what would you do well you probably set up an experiment and let's say you set this up so you set up four trees or force mini trees whatever and you gave them each different amounts of sunlight different amounts of water grew them at a different temperature a different fertilizer you gave to each one of them and then after a month or whatever you would measure their height and you would compare your results so you might say well look at this the one that grew the most is the one with the most Sun the most water and the highest temperature and with fertilizer D that's good and the one that grew the least was the one with the least amount of sunlight the least amount of water okay but then you begin to think a bit more what have I really learned for example if I wanted to know which fertilizer is best I actually really can't tell you can't say that it's d because maybe it was the eighth get 8 hours of sunlight that was responsible for this thing growing high maybe it was that it was at 40 instead of 10 that made this big difference we cannot extract or determine the individual effects of these individual factors on the height because we've mixed them up too much I mean it's not a surprise that the height was really slow for the tree that got barely any Sun barely any water and that was raised at a cool temperature but maybe a was really bad for the tree so that's why it didn't grow too high we just have no idea of telling what causes what what are the most important factors how do they impact the height what factors maybe don't even matter at all from this experiment as we've designed it it's a failure because we can't actually learn much of anything so we have to go back to the drawing board and after a little bit of thinking we come up with an experiment that's a bit more complicated but gets the job done what we actually have to do is four separate experiments and in each of the experiments we're going to hold the other variables constant and we'll call those the controls and then we're gonna vary one of the variables to see how the height changes so for example in experiment one the amount of water the temperature and the fertilizer is exactly the same these are being held constant and we actually call these controlled variables or just for short controls because they are held constant they are not changed on the other hand that the sunlight varies one tree gets no Sun all the way up to eight hours per day for these different trees and now when we go look at the heights suddenly we can learn what impact sunlight has on the height of the tree and we learn that sunlight is actually really important so important that if there's no sunlight we have absolutely no growth at all the tree doesn't grow and as we increase the amount of hours of sunlight the tree grows higher so from this experiment we see aa sunlight has this particular effect which we might be able to quantify an experiment – we hold sunlight temperature and fertilizer constant and now we vet vary the water so again if there's no water it doesn't grow at all interesting important point to know and as we get to one and two and three and four gallons it does grow better but notice once you get to about three gallons there's not much of a change so what we've kind of found at least partly is that three gallons is basically all you need any more water than this has no effect and there was no way for us if we look at in her experiment before there was no way for us to be able to tell from the three gallon the four gallon trials that this was an effect because everything else was getting in the way but now we can see ah okay once we get past three gallons it doesn't seem to make much of a difference in the third experiment we hold sunlight and water and fertilizer constant and we vary the temperature here is even more interesting because again if it's zero which is the freezing point for water that free dies it doesn't grow at all and ten degrees is a little bit too cold it stunts its growth but once we get to 20 temperature really has no effect there's no real change to the height of the tree so that's an important fact once we get to about 20 we don't have to raise the temperature any further that's good enough finally for sunlight for experiment four we keep sunlight water and temperature the same and what we vary is the fertilizer one trial gets none and then we give the other trials one of the four fertilizers and now we can see how well the fertilizers do compared to one another notice for example fertilizer see has no difference in height even though everything else is the same fertilizer seed basically had no effect even worse fertilizer D not only has no effect well no it has an effect it's a negative effect it actually hurt the tree the tree went from 12 inches when it had none to 8 when it had D and when you look at a and B clearly those make the tree grow higher than without so by doing these experiments by varying one variable and keeping the other three constant in these three experiments we can extract what actually has what actually matters and how it matters by how much so to get some definitions the control or the controlled variable are the factors that are held constant so for example in experiment 3 our controls were or controlled variables where sunlight water and the fertilizer another type of control these are this is kind of a difference between a controlled trial and a controlled variable but in any event the other type of control is when you have a variation of a particular factor that in one of those trials you don't expose the whatever you're looking at to any of the factor at all so let me just illustrate this more easily in the case of the fertilizer when we gave the fertilizer no we gave no fertilizer to one of the trees that tree was in itself a kind of control because if we didn't have this we wouldn't know what the baseline number is we wouldn't know that twelve is kind of a starting point and then the fertilizer has build it up from there so we do have to include and we do in a lot of these trials a none case so in the case of fertilizer the control would have been when we give them no fertilizer in the case of sunlight and water it was zero hours and zero gallons we need that to get a baseline to see number one what happens if we don't give them this water or this sunlight or this fertilizer what's kind of the baseline before we start giving to it and we also learn as we saw in the sunlight and water when you're give them nothing the tree doesn't grow at all whereas in the case of fertilizer you give them no fertilizer and the tree grows fine so you learn about which factors are essential in which aren't and what the net changed the net impact is of varying that factor the main control that you'll be dealing with is the one that's held constant but it is important to see that there is also a control in which you're not exposing a particular trial to and to the independent variable which by independent variable we mean the variable that's manipulated by the experimenter so in experiment one the independent variable was the sunlight that's the one we were changing into the independent variable is the water in three it's the temperature and in four it's the fertilizer and then finally the dependent variable is the result of the experiment it's basically what we're measuring what we're trying to learn as we do the experiment it's the factor that you might say depends on the changes in the independent variable so if we go to again our experiments the dependent variable in each case was the height that we were measuring that's what we were observing that's what we wanted to learn we want to learn how it compares to or is affected by the particular changes in the variables that we implemented with our independent variable so that's the basic idea behind experimental design you want to know for the purposes of your experimental design and reasoning questions you want to know what these three mean because very often they'll actually ask you direct questions about these you want to know that the control are the factors that are held constant or it's the trial that is not exposed to the particular independent variable you're working with not every experiment will have it a control group or control trial but some do most of the time though we're interested in the factors that are held constant the independent variable is the one that's controlled by the experimenter it's the one that's manipulated and the dependent variable is the one that is the result of the experiment so you definitely want to know those and you want to understand what's going on here at its core because this is experimental design this is experimental reasoning these are the kinds of things they're gonna ask you about in the questions we're gonna see in the next video

DK Yoo's Philosophy Explaining Bruce Lee's One Inch Punch

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Marconi Union – Weightless:


DK Yoo has studied under various disciplines and as result has developed a firm understanding of how to effectively deliver force. In this segment, I’ll demonstrate how DK and Bruce have a philosophy one in the same, and how after this video, you will too.

Music: [No Copyright Music] waterfalls of love – Muciojad
Saib. – Smooth

Script: “Bruce lee is descibing a relaxed and fluid Kinetic Chain, and the energy he describes is force. Force traveling through his body and into the point he wishes to strike with. That is basically the principle DK Yoo utilizes in all of his strikes. He emphasizes that hand speed should come from the ground up naturally, and when you have a fluid kinetic chain, you can definitely feel clean foot and hip movement effecting your overall handspeed. Here’s an experiment you can do to prove my point for yourself. Meditate for 5-10 minutes, just breathing in deep, exhaling slowly, relaxing while listening to marconi union – weightless, sceintifically proven as the most relaxing song in the world. I’ll leave a link down bellow… After you’re done, just loosely fail your arms with your hips as DK does. Then warm up by hopping forward a few times with your hips, just getting your brain muscle awarness warmed up… Now first loosely snap a punch with no body movement. Just let it flow out without any body. Now do a quick, clean, short hop forward, flow your hips into it, and feel that kinetic energy loosely flow from the ground up and into your shoulder. Hop forward, flow into hips and shoulder. Then as you feel that force in your hips and shoulder, snap out a punch. Instantly you should have felt a massive difference in speed, and some of you may have even injured your shoulder or elbow joint from elbow lockout. If thats the case, remember, it was for science. The entire motion should feel natural, fluid and relaxed. The faster you can master doing this, the more force you will deliver. That’s what DK Yoo is doing, and now understanding this philosophy, we can also see why he can push pedestrians over with his shoulder, and hips, overall using the fluid kinetic chain philosophy to transfer unexpected amounts of force from weird places. It’s interesting when you understand it, and even more so when you practice it yourself, but you know what’s really fascinating about all of this? What famous punch used this same philosophy? Bruce Lee’s one inch punch. ..”

po1 the more relaxed to the muscles are the more energy can flow through the body using muscular tensions to try to do the punch or attempting to use brute force to knock someone over will only work to opposite effect Bruce Lee Bruce Lee is describing a fluid kinetic chain and the energy he describes this force force traveling through his body and into the pointy which is to strike with that is the principle dku utilizes in all of the strikes he emphasizes that henskee should come from the ground up naturally and when you have a fluid kinetic chain you can definitely feel queen's foot and hip movement expecting your overall hand speed here's an experiment you can do to prove this point to yourself meditate for five to ten minutes just breathe in deep exhale slowly relaxing while listening to Marconi Union weightless scientifically proven as the most relaxing song in the world I'll leave a link down below after you're done just once we flare your arms their hips as DK does then warm up by hopping forward a few times with your hips just getting your brain muscle awareness warmed up now first loosely snap a punch with no body movement none just let it flow out with no body just get a feel for that now do it quick clean short hop forward floor your hips into it and feel that kinetic energy loosely flow from the ground up and into your shoulder forward hop flow into hips and shoulder then as you feel that force in your hips and shoulder snap out a punch instantly you should have felt a massive difference in speed and some of you may have even injured your shoulder or elbow joint due to elbow lockout if that's the case don't worry it was for science yes science the entire motion should feel natural fluid and relaxed the faster you can master doing this the more force you will deliver that's what dku is doing and now understanding this philosophy we can all also see why he can push pedestrians over with the shoulder and hips overall using the fluid kinetic changes will also be to transfer unexpected amounts of toys from weird places it's interesting when you understand it and even more so when you practice it yourself but you know what's really fascinating about all of this what famous punch used this same philosophy Saitama from one punch man no just kidding Bruce leaves one-inch punch people like the stairs all in the hand position and whatnot but look at his hips and look at it back lick then look at his Reed legs that's where the force is coming from he uses a fluid and relaxed kinetic chain to transfer force from the ground up from his feet to his legs into with hips letting that force travel all the way into is fist and that clean execution plus the overall quick snap the fast you are the more force you transfer is what created that powerful push the philosophy of fluidity is something both Bruce Lee and DK you both share and through it they both can generate unexpected amounts of force from weird places from weird angles try it out and experiment yourself let your friends in on it see how they stand with the philosophy and remember if you get injured thank you all for watching hope you enjoy we don't forget to Like and subscribe if you did your karma until next time

4 Things We Believed Before the Scientific Method | What the Stuff?!

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Before science brought us to this point, even some our most brilliant thinkers had some REALLY weird ideas…

10 Things We Thought Were True Before the Scientific Method:

What the Stuff?! episodes are available every Monday and Friday at Noon ET.

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Image Attribution:
Bloodletting tools

Music Attribution:
“Superconductivity” by Jahzzar

Science is having a great season. The scientific method has flown us to the moon, extended the average human lifespan and put the Internet in our pockets. But before we got here, even some brilliant thinkers had some really weird ideas.

Bodily humors! Feeling a little off? Hmmm… maybe you need to adjust your phlegm-to-black-bile ratio? Ancient physicians like Hippocrates thought that a person’s health and temperament was determined by four bodily fluids: blood, phlegm, choler (or yellow bile) and melancholy (or black bile). Amazingly, this idea still held influence in Western medicine through the Middle Ages and even after – you can thank this concept for enemas.

Spontaneous generation! This would be so creepy if it were true: Spontaneous generation was the widely-held belief that animals and other complex life forms would appear out of masses of lifeless matter in the right circumstances. For example, the 17th century Flemish physician Jan Baptista van Helmont made the keen observation that a sweaty shirt placed in a container of wheat would sprout mice within 21 days. How else could you possibly explain the presence of mice in a container of food left out for three weeks? He also claimed that basil pressed between two bricks would transform into a scorpion.

The Miasma Theory of Disease. When there were outbreaks of cholera in London in the mid-1800s, the English epidemiologist William Farr explained them with the conventional theory of the day: the miasma theory, which said that diseases were caused not by germs but by foul-smelling air and “night vapors.” Another English physician named John Snow was skeptical of this explanation, and the miasma theorists were all like ‘You know nothing John Snow,” until finally Snow conclusively traced the outbreaks to a public water pump that was drawing water chock full of cholera-infected raw sewage. Score one for germ theory.

Aristotelian Physics. Just one example: The ancient Greek philosopher Aristotle, one of the most revered intellectuals in history, wrote that a heavier object falls faster than a lighter object. Almost 2,000 years went by before Galileo Galilei smashed this hypothesis to pieces. When air resistance is put aside, all falling objects accelerate toward earth at the same constant rate. Legend says Galileo dropped cannonballs from the Leaning Tower of Pisa to test his idea, but nobody knows if this story is true. Either way, Aristotle fans ate crow.

What’s the weirdest obsolete science fact you’ve ever heard? Let us know in the comments and subscribe! And check out more crazy facts about the history of proto-science by reading 10 Things We Thought Were True Before the Scientific Method at HowStuffWorks!

look an alien watching cows that makes sense science is having a terrific season right now the scientific method has flown us to the moon extended the human lifespan and even put the Internet in our pockets but before we got to the point we're at right now a lot of really brilliant people have had some really bonehead ideas are you feeling a little under the weather maybe your phlegm the black bile ratio needs to be adjusted ancient physicians like Hippocrates used to believe that human temperament and health were the result of an interplay between the four bodily humors blood phlegm black bile also known as melancholy and yellow bile also known as Collard craziest part is this whole idea made it into the Middle Ages and beyond you can take the bodily humors idea for things like bloodletting and enemas this idea would be so creepy if it were true spontaneous generation was this widely held belief that higher complex organisms could arise spontaneously from lifeless and animate objects for example a 17th century Flemish physician named jean-baptiste upon Helmont had the idea that if you took a sweaty shirt and put it in a box of grain three weeks later you would sprout mice I mean how else are you going to explain the presence of mice in a box of grain left out for three weeks von Hellman also had the idea that if you took basil leaves and press them between two bricks you could make a scorpy miasma theory back in the mid 19th century London was plagued by cholera outbreaks and an epidemiologist of the day named William Farr chalked it up to miasma the idea that disease was spread through things like night vapors and foul-smelling gases but an English physician named John Snow questioned miasma everybody else is like you know nothing Jon Snow but Jon Snow won out he traced the cholera outbreaks back to a public water pump that was drawing water from cholera infected sewage mm-hmm score one for germ theory yes even Aristotle arguably one of the most brilliant people who has ever lived had a couple of missteps along the way take for example his idea that different objects will follow at different rates it makes sense in a way which is probably why it took two thousand years for Galileo Galilei to come along and smash the idea to bits we now know today that two objects regardless of their mass accounting for wind resistance will follow the earth at the same constant rate supposedly Galileo tested this by dropping cannonballs off the Leaning Tower of Pisa we don't know if that's true or not but it's a pretty great story what's the weirdest obsolete science fact you've ever heard let us know in the comment section below and don't forget to subscribe and check out more crazy facts about the history of proto science by reading 10 things we thought were true before the scientific method at howstuffworks.com

Imagining the Fourth Dimension

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imagining the fourth dimension here's where we start getting into some possible confusion because the same word can have many different meanings when people say that time is the fourth dimension what does that mean the fourth dimension adds a way for the third dimension to change this is obvious when we say the third dimension is space without time but the entropy-driven arrow of time that people associate with this concept is obviously not spatial because it behaves in ways that are different from the first three dimensions this is why some people prefer to say that the fourth dimension is a temporal dimension while the first three are spatial but the more we learn about space-time and general relativity the more we realize that time is not just an arrow the fourth dimension stretches it bends and quantum entanglement shows that it's possible for particles to make instantaneous connections within it even for there to be causality in times reverse direction and as mind-blowing as this may be to fathom the accepted definition for antimatter is that it's matter which is moving backwards in time this is why with this project I prefer to call the fourth dimension duration I ask people to accept that time is a direction not a dimension in the same way that up or forward are directions rather than dimensions two opposing directions can be used to describe a spatial dimension and time and anti-time are two words we can use to describe the fourth dimension but they're not the only words and this is important because all we're really trying to do here is come up with words that describe the dimension which is at right angles to the third dimension here's something important to remember none of these dimensions exist in isolation you can't make a 1d line without using points you can't make a 2d plane without lines you can't make a 3d space without planes and you can't have a 4d duration without multiple plank frames of space saying the fourth dimension is duration makes no more sense than saying the third dimension is depth if when we say those phrases we're thinking you can have duration without space or depth without length and width saying the fourth dimension is space-time then at least acknowledges that the fourth dimension encompasses the dimensions from which it is constructed and doesn't exist in isolation from the other dimensions let me say this again it doesn't matter what label you put on the fourth dimension or any additional dimension as long as you're thinking about how the new dimension is somehow at right angles to the ones before a rose by any other name still smells a sweet to paraphrase mr. Shakespeare so time is not really a dimension but no matter what to mention you're examining the direction of time is a word we can use for tracking change from state to state in art there really eleven dimensions I insist that it makes no sense to say that the first three dimensions are spatial and the fifth dimension and above our spatial or at very least space like but then to say that the fourth dimension isn't spatial if that were the case then the mental castle we're building here has a very rickety layer at the fourth dimension and the whole structure is prone to crashing down last entry we talked about how it's really impossible for us to see the third dimension because it takes a certain amount of time for the light from anything in the third dimension to breach our eye and that's just as true for our hand in front of our face as it is for a star ten light years away saying that a third dimensional object has length width and depth is a phrase we casually say but we have to keep in mind that discussing a third dimensional object like a cube is the same as discussing dragons or Flatlanders a 3d cube is an idea which we can freely discuss but without using the fourth dimension to view such an object it's only a concept likewise persons who talked about tesseracts as being four dimensional objects say that this is what the real fourth dimension is like but what we're really talking about with a tesseract or any other n-dimensional shape is the same as a cube it's an idea in order for a tesseract to really exist it has to have a duration within its dimension and when we watch an animation of a rotating tesseract we are visualizing how that structure could rotate and change from state to state over time when viewed from the third dimension likewise just as a cube represents a simple and idealized shape within the third dimension but there are the limitless range of other shapes that can exist within the third dimension the additional degree of freedom afforded by the fourth spatial dimension allows for an even larger number of other shapes which can exist within that dimension one word physicists used to describe the path an object takes within space-time is a world line another word for a fourth dimensional shape coined by author and futurist Bruce sterling is a spine with my imagining the tenth dimension project I asked people to visualize themselves in the fourth dimension as a long undulating snake which is a way to think about the data set that represents a person's length or duration within the fourth dimension from conception to death do you see how that snake is a spine depending upon your point of view though that snake could be much blurrier than what we show in the original animation every day our bodies are exchanging atoms with the outside world through the air we breathe the food we eat and the water we drink a constant cycle of repairs and replacement means the spine representing a person from conception to death is a much more wide-ranging and interconnected shape than what we might first imagined one of the 26 songs attached to this project called change and renewal is about this idea let's finish off by thinking about the point line plane postulate again which can be used to visualize any number of spatial dimensions the trick I've suggested you start with each time is to think of a point that encompasses the entire dimension then find a point that is outside of what that first point encompasses so a one-dimensional point in the largest version of its indeterminate state occupies the entire length of a line and some new point not found anywhere on that line allows us to visualize the second dimension a two-dimensional point in its largest version fills an entire plane and a point not within that plane gets us to the third dimension a third dimensional point at its largest version is like a single Planck unit sized slice of the entire universe and allows us to think about the possibility that Julian Barbour is pointed out that each of those 3d frames allows for the instantaneous quantum connections often deemed as supremely mysterious and unfathomable having said that though we still have to decode the mystery of how we can have a physical world made out of objects that are not infinitely large within the third dimension and this is why I say those quantum connections are at right angles to space-time so let's continue the point line plane postulates logic into the fourth dimension a 4d point at its largest version would encompass the universe not just in space but in space-time the point would reach from the beginning to the end of the universe in the same way that a photon traveling at the speed of light would perceive itself to be simultaneously emitted from a distance are and arriving at an observers retina this is an important concept we looked at in light has no speed it also ties nicely to something Einstein said a number of times there is a way of thinking about reality in which the separation between past present and future is only a stubbornly persistent illusion what's outside that largest possible 4d point we've just imagined well if you are a person who has been trained to believe that free will is also nothing more than a stubbornly persistent illusion you might well say that's as far as we need to go after all if the universe was set in motion at the Big Bang and anything we do is an inevitable outcome based on what has come before then the largest 4d point we can imagine accounts for all of that from the beginning to the end including the now that each of us is observing at this very instant but what if you believe in free will with this project that's where we start to think whoa the fifth dimension