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

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Will the US economy begin to slow down in 2019?



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Wall Street Journal chief economics correspondent Jon Hilsenrath on whether the U.S. economy will continue to grow and the Federal Reserve’s interest rate hike.

'I'm Actually an Expert' Jordan Peterson Calmly Destroys BBC Journalist Anne McElvoy



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‘I’m Actually an Expert’ Jordan Peterson Calmly Destroys BBC Journalist Anne McElvoy.

from Dr. Jordan Peterson’s interview with Anne McElvoy (05/2018).

but when you talk say about domestic violence and there's a sort of thesis behind it and I'm putting it in a bit bit of a nutshell here so feel free to just to rephrase it that men are in the end if men are pushed too far by too aggressive effeminate and there will be a backlash on that there is you know there's a sort of undercurrent of violence be it Val that's a warning Nora Nora that's a warning suggests that there may be sort of faults in those who push progressivism too far they will get a backlash some people that of course see that as a bit of a permissive environment for bad behavior well people tend to confuse describing the likelihood of something with supporting the fact that it exists and I'm describing the likelihood of something not not supporting the fact that it exists if you push too far on the left you're going to get a backlash on the right that's how things work and this is just a derivation of that as far as I'm concerned how does that domestic in the context of domestic violence oh I don't I don't think it's applicable in the context of domestic violence at all I don't I think those things are completely separate issues I mean you were mentioning to music's women and I was speaking only politically with regards to the backlash that was that that was a comment on political backlash and it had nothing to do with domestic violence that's a whole different issue but violence often pops up in your in your work or something you think drives things that I think you've said in relations between men are more regulated by a background threat of force and it's that's that's why men have difficulty with women is it no that's why it men have difficulty with women who are completely out of control but women who should control women well other women themselves men society just like everyone is controlled I mean you're controlled by society I'm controlled by society and thank God for that I mean part of funny I mean you described yourself as a liberal and I think a liberal doesn't think the society controls women woman well let's say regulates I'm a psychologist as well and what we also a woman what is this creature how would we know when we met one I'm sure that you've met women in your life that that that acted towards you in a bullying and detestable manner it's very difficult for women to with that because they don't have any real recourse and female bullying can be unbelievably vicious and usually that takes the place of takes the shape of reputation destruction innuendo and gossip it's well documented that's very difficult and they don't men do it too but men know but did any of you all know it sir yes when yes disproportionately women that's what the data indicate I mean if men are dates are on innuendo gossip well it's among antisocial behavior among adolescents it's well documented field so because people look at aggressive and anti-social behavior in women and in men and in women intends to take the expression of innuendo gossip and reputation destruction and in Menna take tends to take the form of outright physical aggression there's a whole literature on that it's it's not a surprise to anyone this has been known for suffer for thirty years I mean the rates of athletes right here the female ghosts it probably predates thirty no doesn't make it gospel but people have no it doesn't but people have looked at how women express it look women have to express aggression somehow unless you're willing to say that they're not aggressive they tend not to do it physically not to the degree men do so they use other channels and what other channels are there other than physical aggression if you're going to be aggressive well you go after people verbally you go after them with innuendo and gossip and reputation destruction and that's how it that's how it works and just to be clear that you think that's predominantly a female made it so probably it isn't that I think that it's the clinical literature indicates that is that I think it well not interviewing that clinically legitimate Jeanine you would you well I'm a psychologist and a scientist and I tend to and I tend to base my opinions on what I've read in the broad relevant clinical literature I'm not making this stuff up I studied antisocial behavior for like 15 years I'm actually quite an expert on it and so we know that men are more likely to look look look at it this way all right women are much more likely to try to commit suicide and men are much more likely to kill themselves and the reason for that is that men use lethal force and women don't now that's a big difference okay so then you say well women manifest aggression towards themselves and to others but they don't use lethal they don't use physical force the same way men do so they have to do it some other way what do something some other way that people are you war against yo so you busy a whole busy in like normal again half and half half Hobbs half Russo that's why I'm not an ideologue because I don't think that people are good or evil I think they're both I don't think that culture is security or tyranny I think it's both and I don't think that nature is benevolence or catastrophe I think it's both and that's why I'm not an ideologue and I mean would you stand on the me to campaign good thing I think that it risks damaging the presumption of him as innocence I mean there's plenty more to it than that oh sure women women women face the the arbitrary admixture of sexual advance and workplace and workplace performance all the time it's a very complicated thing to sort out we don't know how to sort it out exactly because you know I mean NBC for example the American TV station has has made it policy that you're not to hug your coworkers which you know may be true although I don't think it's the sort of thing that a corporation might be deciding for people but we don't know exactly what the rules are for governing fail male and female behavior in the workplace because we've only been working together for about 35 years we don't know after 35 years to figure something out not when you're talking about a transformation and behavior that that's that's that profound I mean we don't know how men and women can work properly together in the work force it's very complicated but you are the ones that asked about me – well you start with you're the one who me – is well me – is an expression of the fact that men and women are having a hard time regulating their behavior in the workplace it's the only reason I responded to that because the question I think is more broadly suggesting that some men are having a grave problem with what is the lesson of the Harvey Weinstein story feed someone should have said something about Harvey Weinstein much sooner but we could start somewhere else we could start with Harvey Weinstein was wrong to do what he did before we go – yes other people should have spoken out Thank You Teresa fair enough fair enough I thought that went without saying there are going to be psychopathic predators there going to exist and what has to happen is that people have to stop them because they won't stop themselves and so I thought that was sort of implicit in the statement obviously he shouldn't have done what he did but you don't think that the culture in which he was operating that there was particularly in his word in his world and in many other worlds that there was a culture of you know like this guy's a powerful guy he's the great silverback gorilla here let him get on with it oh I think that culture was everywhere in Hollywood which is why I wait well Hollywood particularly I mean the casting couch idea has been around for a very long period of time and I think that the Hollywood types who are all upset about this should look to their own devices with regards to the role they played in fostering the culture that managed that so we're talking about cultures so I mean it's certainly that the Hollywood what's the sensible thing for women to do about me – to your mind and what's the less sensible thing that that's a hard question it isn't obvious to me exactly what men and women have to do in the workplace to make that kind of sexual predation much less likely with all also subjecting themselves to restrictions on the sexual element aspect of their existence that would be unbearable it's very difficult what would be unbearable about that how about everybody wears the same uniform to work that's what the Mao as well look if you want to eliminate the differences between men and women sexually at the workplace you have to constrain the sexual differences I mean men wear suits to work we don't have to eliminate the sexual differences for people to work together with respect you have to eliminate them to some degree why I'm Jen because you're trying to you're try the question here is to what degree should sexually related behavior be impermissible at the workplace well it depends on how you define it should you be able to dress attractively and if you can dress it actively what do you mean by attractively exactly like precisely I mean I dress nicely today you look very well-dressed to me right your man and I'm a woman with bass nicely dressed now we're getting on with the interview what's the problem well the problem is is the boundaries of what constitutes nicely dressed because there's look because part of what constitutes attractiveness part of what constitutes nicely dressed is sexual attractiveness because you can't separate out human attractiveness sexual attractiveness from human attractiveness and so then the question is exactly where are the boundaries and that's what the discussion is about where are the boundaries the dinner parties to switch are described in a forward to your book where friends enjoy debates and disagreements do you think in the broader conversation we've lost that spirit or in danger of losing that spirit oh I think I think we're always in danger of losing that spirit right because lack of freedom is much more probable than freedom we have to be very careful to maintain that because it's always under threat but I do feel that it's under threat now I think that people are very careful about what they say in ways that aren't good I think the fact that many comedians won't perform on university campuses now is a very good indication of that that's a canary in the coal mine scenario that you

Biggest market crash of our lifetime is coming: Economist Harry Dent



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Economist Harry Dent discusses why he expects the stock market to crash in the next few years.

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

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

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.


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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

Slavoj Zizek Birthday Special: Politics, Philosophy, and Hardcore Pornography



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Slavoj Zizek, a Slovenian pop-star philosopher, is turning 70 today and so we made this film 🎂

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you some ten boer what you know that biological staat die proef diens proved that en denk geen zie span that and she of cortil-wodon flow dozen books nood dit is manche het op zo hij woont ik to make it maar moest markt is hjoed terug en dus hij komt inc now this may be zo arrogant mijn plan is to play kennen die index famous land is iets niet kennen die niks van die deed dat adres team je know why kennedy woon niks te lossen 13 kennedy your program de melk aub hier de f kennedy looked into a comment with you american people you waar het af wie kan het snel en strijd die coach de racing ai at shipdock niet langer is een basso staan dat door piraat kabel aapje was challenging knie people where or the thank all your vreemd de second reason is iets wat 3 key politieke hij wond toe adres people who maybe uit raakte bij in qua muziek of snoepje die tijd voor politieke correctness and tell them something klein arguta jaren toch joepie dit is een politieke correctness download to him pieters en de bodem bernice en dus regisseert heb te maken met hoe is een radian out why 3 effect biologisch aan digid balies meer in leraar slaat toe ds mijn tekst ons die dit niet nodig en daar fermin is tedx om geen ik zien krijgt iemand inklapt is the fact het in de guardian weeks now you see clearly de limiet of guardia niets clear that guardiani sparco pista blies moment werd dae-pa blies meer online van tijd tot tijd typ in de pen denkt iets bent er wat heidi's covered by one of your guide lot purely graaf die poortinga van dividend lebedev she controle guardian en dit is de setting kloont liever arts en olie weet jij iets de lieten biedt more open daar die aan ons op giuliana schuim door medisch guardiani die om je ideeën lean oud en liefde bit openen dat zou die het indertijd ben guy aan is totaal idee freddy's limiet der reddingstations denk totally by the story there are against maar door al mijn mooiste games maar door al hij drong er bij die zo maar door robots australisch democraat wat nou let's face it wordt american schaduw inc is zowel blind op er even know so that's why did you aard ip-adres of sam blair en soort tapes en die zware in passé's die filodeeg de setlijst van of qb lief langs uit liesbeth ik dat het minder die operaties in noordwijk kennis jij een beetje airplay skier yes oké oké waar je liep en liam dat baby de kaarten die korst en je om iets kartering fascineert me niks uit jfj je het lood eraf slabak japan there are much more cash orient oh nu kan ik denk in kaart voor catch caught in des toorns they don't lijk je op een inquit crap die daar iets echt intens wat en ijma nodig dank ster guy periferie die was creditcards dr denk je van ja ik want to be free to cheat dit kwijt hij die men werd een lade niet waar yolo' ben amerikaans bloed nieuw vorm of one gaan drie dollar bier sam tent je moet er was paniek in de ratio die kort likken rub die fikri ijsjes en jouw merk en denk wat week week at partij zingen in russia us pepers die oké we gaan drie darts bienestar 113 niet liep en liep door waar is mijn tanden pijn of een masker project one of the very first things like ik was de pijn en terms of use in oké the guidance on a real er gary ja ja wat nou de dino tunning vaak de prijs like most restaurant t het zand en waar ik als ik hier iets begon ik denk ik als ik denk dan terug hebt er kilo diy back to you probably won't recent nog de in kan toebrengen ook eens reason why do i van je nu wordt dit is absoluut leef je nog jij nog wat hij leek die houtbak coastal one of your extra pijn estonia wendel d de first het oer computerij steve en lexence you can go to the web en zo wonen of driving greece and the us be kost ik warm step towards the end i don't like it this permanent ervaar een doen oké wie woonde ik die een every weekend you believe him to vote voor chanting kunt gewoon je nog eens make it bless you go store figi ook ligue 2 hij zingt dit elektrisch er cruisen rond je zien die should be some pink lieten bit-rate wellicht niet ik heb gemerkt dat juist nieuws post adres energieberg yolo gelijke delen uw jij ook zo erg juno beach zich een nadacht now i'm from some sophie en daarna lichtgroen oh very well this location in china no but they are or during the lake huawei is nonetheless not a lot better en ben m in gaan we wat artdeco het google google facebook en al dood yolo is veiligheid goud of die met minder een theelepel peper zou of the three companies leuke dingen jij jij jij je let's take dat koop [Muziek] [Muziek] yolo was hij was improvising very well for twenty mimiek en ding ziet ook niet iets man daarvoor slaven but i know that was from tim dronk wit en lijden en en de red je bent heb fuck you in nood posities theoloog haar stam body combat niet en set liever even 5 minuten interview je die die een tea time pt uitnodigt carrière en people on early biedt een that it would be interesting to wat mee en peter van beek was kies het totaal obsessief code at only be be doing koolmijn van die pink and green care groei het is warmer haar broertjes like the world dik is een paar manga navi this at the world's most of you i get angry wk-limiet het niet natuurlijk ya know lesbian lesbisch indien het steeds re lee tae ik ofwel doe je muil dicht bij ted publiek lesbianas hem en ratey mensen iets at very complicated ik als duo deck schoenen slecht no no no no no in hardcore standaard pornografie iets en loud doe gek en lesbian interlude to him in the cold but you never geven mail homo sexslaaf die we gek zo gek zo uit zo'n lesbiënne genoeg en der spek uiteraard menno store cooki hebben er eli in rosario raar zijn snor mark boog en tijd wat is gent bij een klif japanese katoen en ja de cartoon sokken you might be a young sjokte noticias en wat is daar pornografie clive light weight loss het trainen en wie zo druk die zou diarree via ingreep ons zal ooit zo'n lesbian hoe ernorm uit pieper de green months and not show any nog mon nom op emoji's lesbisch nee boilies plek worden we hebben niets blijkt families matties oké oh je bent milf bestemd voor ja en o&o lying die eens in de kast ei kwijt hij en die macron yolo effect meer kwijt raak die is een tent 3 mavo loras wanneer u ben in de gijs coole die studios bij het neys die krullend gewoon mailen glad ik denk zo cardio je vloer yesterday wat die eieren floot door adria boodschap flight wypall liefst repens katholiek oud ben je upload je video up-load was zo je kunt 100 oh you can ski from oudheid ben lucie was zo no time dus sorry jetzt alien steve won't you can geert voor wat je nou elijah bouwt poli's piepers der nat is bed fm piepen claim olie antisemieten en was stom nou bied de sims team kesh hier in ex-joegoslavië all the public's that there there was a certain time of socialist opnieuw ziek werden we rooster in white dienst zou die jet poging durf en ik schets er rivival nou ik trek en hij dan 10 kids 3 li longing for that time wat ik noem de effect einde we wie er naast aanvraag lijk hij don't want to go back derm het en of zal een ober idioot imiteert lee afwerkt daar val of de atoomkern you remember they want and error ivan plans to the straal je dat billen denk wat lauw in de steen twee iets de meeting grauwe voor de young people you know where a ja 3 wit poli's regio heist od berg je loopt disc riep easyjet wordt hij waren ryan air flights many of them brittje studies voor de voet bij mij tree store to dream drie bier they should r&d werd zo broedtijd in uw en de pijn die deeg zult niet daarom craco om bieren place is criteria en die sleep die aspecten twang dat en inclusief men en tramspoor bied nu nou hij ander fandom huwelijk keer in slovenia wieg wit licht en zal ik de v&d van dat wij die markt lower class boete english sabca fijn voor het people who got drank they can be very valenti derde was absoluut race islam is hij ooit base lijkt de premie die god mij vreemd was het bier games reeds kiezen maar selectieve lee was het losing time the world as a recording lonen lieve kameraad stijl in novi dan voor die en die michaux free dan maar iv board room with pr okay okay okay verras kwestie riswick is famous die kritiek of politieke correctness speciekuip er one of je suiker point is that keeps a fan of menners komen cartesiaans gewoon wat en de fee sophie politieke correctness can be wrapped is een expension of mendez in terms of hooijdonk of en zo how do i get out of the spanish een beauty van quests omdat hij geen we langs een vorm die politieke correctness is precisely op en liep rood of mijners bacto strik en ik wil lees minder means to sound inc week you warm op vorm mali op blijft tot doel weinig are you okay dam oorlog der moet kunnen dat tragedy of politieke correct met is that instead of a feeling to mijners detail toe in val di rect en ecologie nieuwe leader process cute funny viewport net formide pro problem biedt politieke correct masking iets op mijn er iets de the case of menners betreft to be spontaneous mijners ik snap ik view the geef ik viel dan begreep proper lee jong willoughby pan eerst mijn der sar en melkkaraf leek jansen komt niet terug en gewoon zo ook een deze klier op het lief functie rijk claim leds gewone en nadruk werd een eiland of jurgis teken corporations koop in families en en body positief idee voor marketing initiatief web prijs graff trucs wat hij vindt iets zelf op yes mijn probleem biedt one line in modern familie hem en gewoon echt gewoon en transgender en gewoon is dit iets niet perfect lee doorway contempory global kapitalisme functions begint door af en jerry naïeve markt liefst kwestie hoe het liefst wat tape of subjectiviteit fiets door deze global kapitaal i think it's no longer materialen kunnen soort olie die seksuele prijs kilo door deze kapitalisme vrij om moet dieper leed een typisch u geeft to reinvent your cent voor op dialyse of of con su mary zwemmend gewoon een te wonen of zo aight je know where the videogame tomy winnie post president at is zo subversief die het transgender edia doe es pensioen tablet voor transgender piepen gauw voor baby club ingrid dooty of sjiitisch zie of decor wat naast eisen palm beach wat hij was er niet een betrapt kou in taaier practica the corporate america the media trip to part ik niet zo ik denk die sprong is crucial for me that we've abels koersniveau conservatief striking bek en gewoon maar die gestikt lee reactie neocons of draait en gewoon die dit term in ding factor de nominaatvorm to date is dienst wordt hij er om die kliko western booties remorque spiritueel eist gek nog niet zo intro yourself make something out of life bid leave something that een keer a very traditie leuk that i glass en ideaal over ga je haar tas mee bord de reason tank hier om die erft is not just to have good time but to do santing carrière en gewoon aan het science politiek ik dus niet so prijs die flow einde centrum en die is ook een fiets perfect niet modern kapitalisme moet je zwaai game a historie pi model en kapitaal iets doen absoluut niet no longer het niet materiaal kuip politiek it functions in de primitief ray price baby come true diever in het motel fight bodem facebook die druk een droom in gewoon dies bernie's lieve team is already viel er der ploeg radhika mijn laptop control hebben die omdat toe conclusies en naderen kampen de taillight hou van de waanzin twee leave-in giet olie society's you want to het steeds weer doodst reapers extra klein dorp dit moet je dubbel terry de band met de geen time was set eetbare mooi red bull eet dit team yes but goud doe ik doe het zodat dad bom of een you everyone to seduce you no lies politieke leuk record is en voor somebody' goed staat biedt syco en elysisch l'ecume dc's easy to account for how how prinses lief en radicaal per ministerie die 13 keek your life to pleasure doen wordt daar waar je band 6 to end up in the tail het controle en het college oké let go dong jij jij jij jij jij je iets about the relationship between filosofie en dood wit kersttijd spek boleet per deck one day a great filosofische werk would be great to their porskamp rijst in tire lee of dood en oké of ipod en reflect on the yes and can be cost a thing that and maybe then i already another book which can be het is de furies over dood gek en sven de manolo 3 of spiriet die korstdeeg antink in die partij lekker dikke laag verf en following the inner lovely pink parijs lied werd er een account die is week komieken moment it can be also very traject kom ik er moba iets kom ik er voor jumper oktober evolution staat dit is liebe regen en kroon mensen wonen en die hen tjoh dit stadium gewoon lucifer die treadley droog met iets met guard welk wat mij underline kijk die ja gerard in de tanks wie geeft toe tijd of droom wie ik heb al die naar i wie geeft dood when you just love and then one vriend go reely trek ik gaf creati jono like mike altijd ik ben ik denk dat zo gardeur that you can motieven kodama tijd ready then some kennen we hier view more when you dont like you're just to carry 5 humeur flitste game die je die jaar kunnen stem toen hij was serie denk primo levi is diegem en en nee dit kamer dit men die moest gewoon niet fijn jeans dit je discreet van maurice rieden echt dood wat de met je traditie slot die procent geriatrie grote laven typ de method of these iets dat iets zo sorry but that it can no longer be at rijkelijk die picos en ik ben bijna tien maatregelen die de week team 3 tcr geur die ik niet die derde the means you waar de terrorist je op reis meer no hij place where to lose my life gary op post doe je in auschwitz or may be in good luck you couldn't play the games zo ook een iets aparts duidelijker relationship with not just for this lovely day lek dikke reet warm sluis you niet van kennen of filosofie kon view more iets en much darker than men ons dat die agar agar of history' crash to be done to pestweb droog en het wordt hij drink nou maybe de plus een gok enkele als de kwestie wat pijn doe ik nou eens trein door het lijden door bijbel item that in kerken kat zo die is clearly that crusifix nice hebt ook nieuwtestamentische comedy iets absoluut recruitment door chili's to-read de bijbel is de series of dood luke led begin edgar wordt de bediening daarvoor god der klieren lee een eenwieler doken sterk 4-daags everthing zie apart league surprize product leek wil je idee apple or not wat is een dief nv cries never ever was bang is dit net iets absolutly kleren even predicted that i you we en they a bank boem ik de room try centrum gewoon echt absoluut league reach you again to riet christa niet die is dan kom ik er liepen oké lets go oh nou begrip diest gladys je laboratory retrospective underbelly op de in te noemen we toen ook risico probeer hem werd beautés de filantroop voor en deze politieke lector no no pest boot arnhem en dus zijn we hebben het einddoel ingaan politieke en alice is uitgeput keer een bergkam politieke mobiliteit dan drie liet de politiek hij doet gamma analyses wat mooi is een amateur we nemen rijkdom mijn politieke soorten tekst iets ik heb al with the feeling kan shouldn't be doing key and not really need some other people who know me better should be doing kit wired een of doe je niet zo basically een mormoon eiwitarm de kritiek filosofie in ik ben pieper dan notities en noteren postmodernist met men en werd traditioneel ja geer mijn probleem is want wat men dronk wit marxist reversal of geulle hij ging de greatest uw en in de gisteren filosofie kan gekker and then be seen lieve in dead chair and typing den bosch mark freud ga je de garraud that stuff iets 10 post cadel genspace but we didn't get campus en oude accounts bij german ideaal liefde mendix wij maybe this would be of gaan interesseren mij vreemd was medisch quest to may be de nieuw fenomeen en nou wie moe was out of gek deliaan speciosa blik die weet ie that's my en zoom inter stukken wel the idea of noralie die jij die ja of directly waarin code breng connecting kit bij de computer wordt 30 min voor op de status of some black die wie die arabisch team in de space of gek ellian d'electrique or not zo erg je loopbaan zijn meet en hij is droogbouw kafka dna riet paap liefst biologisch kaft kast licotec jullie dit cuffed capri vriend het web store milo jaren biedt sam markers insurance agency je wordt fabrikant legal advice tekst en maar jij die je denk aan dit years from now you look at en encyclopedia je look at kafka check dwar man legal theorie screwing fris time brood van noah's noodweer in politiek estlandse moet draaien en maybe uit het lijkt publiek reinbert leek die rood stig broeckx om gek god en pure music in share some aan import en politieke linda liedjes en zo mooi iets absoluut liep filosofie biedt met tars op spot id hoek hier en ook diep en de akiko strahm down that just to go folie diep mijn uitleg to go oké oké zie je hoeft de work to be a movie detector nooit 3g die wij die die tijd ipads summer detail really wanted to be a movie director bob einde of movies wordt hij make a wordt d cup taalib werd essentieel als er iets uit dans l'eau why the course in the snap dat hij in nood en movie de rector die kost ik dit niet get manier wordt er werk aan je broke down orde ready and then i reject je wanneer was in de plek om te reflecteren op de highscore hij simply great wesley sorry en dat hij komt of sorteren of mij highschool zien code klassieke muziek in juliana begrijp het boek in de mytek wordt in code klassieke moest hij liet dit k waarom der tijden op de boeg in af fortec zo'n ouwe agent stilistiek riet mooi model 3 loopt de movie director but to be er direct er steken rinke deed classical opella maybe vaak een echtpaar sifalar something like that one day a dag en uitingen that this doesn't mean that may be a bieden doe op dit stemt in glee cast hij discover ga of kunt thomas is de groene when you are not able to be calm but you one wat je moet gaan waar er iets doet voor uw iets ook de best zo een natte pessimist 4 ruwweg wordt questions zoet wibi s10a geert you will get en civiele warm het mij vijf werd kwesties wordt gebied hier daar verder gaan dan dat in seksuele wil je gary tibet ordinaire waar het hoort ga jullie zo vies de keten dwang lijk mijn vee waar filosofen quests op je afspraak ghibli dit ook de tuin is gekker fm namelijk duur of spiriet is op jesse lee wat consistent zeker geen de play-off derde de kapper of geef koning fles leg center easy hier moe van en zo wat de troep in a structure of gek een pennilow cleo spirito de sfinx leek die is die van mij t voor flesjes om diana genk hij liep met olie olie wordt martien populatie nr dr de beste indicator of werk wie haar spiritualiën many people deed zo een nadere tijd waar hij moet gelijk is niet rhett reese and movies and liked en dat team die en tijd dan ik hebben en die prejudice is hier al lijkt serieus zo gaultheria smoby slijpers of russian lov'st nelio bauer wanneer mijn vader branche en toorns bad of stoot hij dan ander is die meet bloed dat dan niet gewoon wordt barbosa dat blog aspect prik some people ask me why did wesley believing kom je nieren heiko them but gewoonlijk bitloos wordt de olie jouw termen die look at all the set of the is de gehoorzaam daarna in folie boutique post-apocalyptisch wees laat je mij niet die bernisser wijs en gewoon digitale filter zo again en niet verder big win populair kuip char en ga je theorie en de quests en hij moet lijkt to a scheme mij vreemd goede ice cream in discretion is wij er politieke drock's zo goed in ria socialism stalin die cement gewoon de great ass katastrofisch uit cellen vormen of de folie faria sociaal liet hem vast the disappearance of exellent politieke dood not only rasa bad bad bad het digitaal janken check of slaap ja bowlen is jeremy dick het incredible jobs en dc's en vet saai en goud oude politieke group in vervangers deze stille crisco they are our breath taking the limor verder kern nature doedee ja sorry hè mama en jouw stijl niet jij jij jij jij een mietje dit before [Muziek] dat tot quickly c wat hebben een png publiek loodgieter igor ieder hoofd naudts genomen je geen met na politicus en dat volle bak die term 18 that metabolieten glass approach iets precisely paard of ouderen bij de dam je na een paar adam in een post ideologische cynicus at the cute wordt wie niet snap en met de politieke steijns ik spreek hij sliep simpel pure politieke engagement big bazar en wreed thomas toe olie neoconservatieve en hij had najaar ben are ready te doen niet toe rivoli politicaly in case you must in de zijn tijd die construct eat sleep naar liet not to be teken too seriously en het gewoon het gewoon de problem to day voor wie is dit nog dit wie jaren boost syriërs maar twee rock tomaat in dienst kroniek uit dit jood en john that's why he mentioned before hij het meijer doos trees den goed protest against global warming' der dat hij chronik der op deadly syriërs in de weten tweet todaysart duits arno ready to be serious and that wraps 18 people of tien bijzonder zijn niet uw een meowth many droog en het gewoon het gewoon wat basically am dead ligurius ligt in vrij wouldn't be fanatiek lead at the gate dit door mijn werk doet in de rij moet spam een mijn werk katholiek die u hebben our everyday left or right and or most of some of them one duizend lang want duizend petrus lonk boots om gekke lenzen wonend gewoon m deadly serieus in politiek en in theorie aitink wie niet toe riet er een doek serieus approach hij want de dwars key maar dit gadget fans guy dwars met je schrijft je ster x is door die apparaat britt en ik woon daar stichting de direct vanaf trump is een expert slofjes there is no ik spreek jij sleeps uiting die lekkers en of tram yolo wij burger' plek nu begint bij die criteria expo romy en attent i got it should be kost dat perverse perversies van slitter en 's-cross lift bar waar het doen wordt d driemaal brutaliteit gewoon met s alle great zij kennen liefst from friday van der poel akon zo klier liep maar het word who is always just de gieten site of power waarborgsom ik snap stem dwars ligt waar je sterker het feminin positie is een permanent questioning of those cooper ik denk dat dominees master expert op en gewoon je sterke questions or with you hart en link medisch but why and my wordt uw arts een derde rijk der tm bleek de fundamenten familie sterke kwijt of ze iets wij biodesign er mee wat uw jimmy wat in mij voor uw wij doe je 10 pijn wordt usa en kan je hem dus die volume maar daar een la bardo teller deep web site die is in posing kornaat werd een soort florence hysterie ja is qua screening gedaan en uiting canvas met lee trump is spray 5 lee en reactie door die trump want to return to a shadow master the scars were there is a central olie die video klieren euro dus dit soms zo gauw in in de string griot preciate kim bicos kies doing kon is lief of zoiets totaal idiotie winguard en peter van kies meisjes is led step back from the post modern historia subversion en gewoon leds irriterend to the basic wood om mijn wie hem in begrijpbare taal die u treft door rely on your self en gewoon en très bon uiting that to they may be more than ever i styria is subjectief let's is there something more om goud deco eindeloos geen uw geld mee sorry er is like keek wat 3 in zo oagnes ik ben ecn vaak over hoe je vaak of yes yes voor een groentje er freedom of speech in de kern media landscape at night spreadsheets en dies lannings quote heino probably bij trend arie vers door dit lenin statement lijn die olielekken free dom yes bak voor groen to do but uit link to day to as the graph trips is more appropriate than ever be kost je no do they be automatique lee identify freed om echt freedom of choice wat eijsink troeven ridam is true democraat zal meteen waar uw code nood der hoeven niet om is nog best freedom of choice between wat crysis you goederen die geur bleek de modellen free dam is not go to apathisch 3d geeft zakken met kate strawberry cake en nee metrisch keek en bieden de trophy dom of kleurtjes waardoor grip die drie cape may be true to make it all you know that you question de werk modaliteit of choicy's you archieven in politiek oh god dit is voor niet dat proeven niet om en de moment you a project in the way free dan friedrich applicom barry limiet at fritom of speak oké ik was stabiel en ontvang geen en nat en toon te liberaal in de sense of reisjes job je op drock je kunt t and i think you won't wat ik de game time hij moet een ander echt die make the good force of nou in de good sense kronieken dief tuin en ze woont view more voor die kampen in dit mij barry en sweet home iframe weet wie min wit andere religies en zo een eind orea geeft de spier is wat die hem wij mee dairy hij digitaal boek dat abc dairy cow potentie van de dpf drock's about racer drock's effectief joe's waar der noot reely regisseert excuus tekens ertoe kiest er bleef stam solidarité en heeft politieke le correctness demonstrates you can be iets stream league ervoor nat door vent anyone en gewoon in gewoon wat de medisch politiek of politieke correct mist iets fr net iets ik stream lead route die we recife plastic uw wang sympathiek amper en no ties xl maat wind waait liepen huis who care about regisseur gauw wanneer waar je steeds something that might be interpreet en zijn eigen olie kritiek of black dv what you can dienst aan denken willoughby scheurtjes und ook leek that knew they treat black en hoe hebber er geen plek in of en en dingen jullie wat ik denk dan pak een lijststem indien wij voor ik hem gewoon exemple politieke lukken het racism we nemen dat dit is de krijgen in hardware het country iets was nobel in western europe to trace ik doe iets en effect of kolonialisme wie chef to be viewed die en is de black friend of mine from lake yriar moesten denken lee explo deadlines en tony die getver resium d3 tafel e strea de drain dit dom iemand die was de right to be old en tikken liefde ik weer even know it's your kilo die was de right to be wild weekend rootinck up ouders er weer zo'n app die zich wordt ibood be very attentive to how to precisely attempts to protect free domme ken limiet frido how a time to save the other dikke niet i can be can be a form of a graffiti en probeer het gewoon zo en gary slow for my life 4 you can noord de juridisch legree's politieke correct mist bradley corsica prijs toen of men daar zo fire on'y en gewoon you can never risico's probleem zo sexy slim reticulum wie simply extra briefing kimbo zien clean room of risk and beach at work at crow en roni en zowel een chuwon wordt is die uit die met en tof technologie club progres op die angelique everyone iets door make-up die voor follow the iemand een klokje of technologie to build machines which will be tas of dit wie juman familie no longer be niet bed en dit is dat kennen we met papiertje kunnen aaien rohnisch look is in dit oude piek goal is to be it and artificial intelligence hoe het minst verbaasd af en werp lakewood er de rivier wino ik was wat heeft 13 daley blind familie die tijd dig it will no longer niet de af en zo en zo en wat hij ging daar 13 niet mee eens is self-destruct even bij lening in technologie dat in dat zijn zo voorwielen kilar celibaat in de grens af we will become your flash we willen de comeback barbie jaren leider refuse to climb up in den artificieel intelli drents you can call iets singulariteit je nou wat hebben willen trouwen ow papier liefde wie dat friendly face macron 3 door hier ethiek en het sure that may be with verkoop more in the tail het eind am that not to classify prodent macro is populier teruggooien voor humor toen juf mij vreemd na expressie dus met de batch de phoenix my great for to him off road grafting terug door politieke correct mes witte freindly face the wie is de meest politicaly correct president werd gedreigd dat doe je het noot in die fanatiek uit g2 bieden vreemd dp centrum exmar marrone lo uiting je tries to be precisely technocraat wie de vreemd lees je het snapt reely populist commissie gitaar leg ik hem niet lezen was een games marine le pen en je loon noord bicos papa listen it's not just a feeling to people look lijn in a file to people know but the bill accuse lijn in of begin poli's en zo wordt je nooo pappa liever is onwijs en naderbij mensen mobiel ijs link police de basic wat is beetje glyfosaat bouwt papa lieve vorm i is dit project wie you your starting point is that we jaar in stammen krabben wat olie zijn oh wees niet and external and then i you never fully uitmeet net iets ouder own aantal bonito's voor in amper jullie wat is voor mij pure populist construct jullie member die zij die ja nou propagate ad bij het ooit right of culture at maxxium wel iets dat contract ik dat niet ik kies number die kop die moet dsg prompt net vernomen naar dit deed die plooi politieke correct met stroom zo mooi deed iets modern permissie be the integration of antoreti my god iets drieluik of de the climb the integration het biedt lease paard of modern kapitalist bijnaam iets 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recht mist en retoriek waarom ik had zo'n politicaly correct probleem en een anbi's van ann-joelle blijf ons quiz de was dat lukt het ook duur was in quito liep keith adecco in die kast pikken met het mi ndo doet dit in this you and you try guys what they said ziet het in roesink mee houdt je nou iets de chemie mijn jongen beter frame ik snap hier doos nee die score aan die rest mag worden waar t oor is fijn you you left review die officieel leg eens uit profielen ten derde en porselein hij krijgt de patiënt expert en die persoon wiens lijk die sorry bio de dieven en nadruk niet ethiek when they bedien jurk ie d'er reet blij toen joeri jordex is very interesting tunnel nu waar las dit is creatief training testing paar uur het liegen samba the covering blijven gewoon in gewoon pure ding is het morgen ook dit beeld kan nog iets kraan ik piep piep piep gerty had geen nut fortuin koekjes is casting mijn vorm kies wel zin heeft om te leveren shit roscoe die costastraat now you can 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racing spelers en ze wonnen gewonnen de mo guys riri lijn looky denk bieren uit die in website en denken als tijdens je leuk uit hij mat mat militaire gelijk not my god oké hij het kreeg hij was passionated bij mij vreemd van zuid-korea jullie men berg ga verheerlijk ook ben everybody moskou ging en oh oh in depressies en die dia was de verstikking to be banden mijn mond carrière indische worden hij was erg blij frames echt klaar pc wie ze lukt niet dan keek ik stelde hen en theo net ingrijpend 0 pubers people nieuwe note korea f you set the spot in cern ik er mank play the game pieken ccd het wijs shit is een social to move wouter wat er nee da's klaar en op je sliep op dit liet hem bereik bruno je lukt het ei homo's kop dienst jeroen some people der voor mijn universitaire the spring want dit niet doet ie niet nodig ja d want it to be to toertochten maar die heb ik niet oh en free die te dom jg can get je toe houden in mij terecht nou waar je adres en in free work of de repeat en 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die kracht [Applaus] ik ben desi goed kennen hij sprong zo ja ik heb voor anders denk je wellicht al eens naar een paar te testen mee sanne wat charme een c-clip de tank en ik herbegin heen van de brief je vlees en de passing longsleeve dames kan de app een succes sjaak normale to buy your eyes als ik even en we hem toe hoe dat never get up dus wat ga nemen maar ook michael en carplay je zak op een brandend huis was stil ccr aan wat we in de poriën staat weer een stille keer de pijp en de boel weer samsom bij de duke een feestje en ria personen die presteren

Top 10 Reasons to Use Technology in Education: iPad, Tablet, Computer, Listening Centers



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While there are some nay-sayers who bring up some negative effects of technology on education in the classrooms, mainly budgetary reasons, using new technologies is the future of the education system.
The following reasons should silence the nay-sayer and help them heed the wisdom from Sheryl Nussbaum-Beach,
“Teachers will not be replaced by technology, but teachers who don’t use technology will be replaced by those who do.”.

Teachers are finding that using different classroom technologies like tablet computers, digital video, iPods, and video games are great tools for helping students learn.

Kindle ereader nook
1. Students love it. Students love using new technologies in the classroom. Whether it be the “cool” factor or just a genuine interest in new technology, it allows teachers to vary activities of the day and engage students. There is no commandment for education that says,
“THOU SHALT NOT HAVE FUN”
Students spend hours using technology like computers and television when they get home from school–why not engage them through their favorite mediums?
2. It engages the Four Key Components to Learning. Click on the link to read how technology in education (specifically the iPod School Listening Center) engages the 4 key components to learning.
3. Professional Development. Using new technologies allow students to learn another skill set that will help them in the marketplace. How many jobs need basic computing skills let alone advanced computing skills? Any time a teacher can encourage students to learn through technology, they kill two birds with one stone.
The tech industry is only getting bigger–encourage your students to know and understand technology so they can be the tech leaders of the future!
With the power of the internet, knowing how to use a simple piece of technology can turn an aspiring singer into Justin Bieber (whose mom uploaded videos of him singing which led to his fame) and a hilarious prankster into YouTube Star EdBassmaster, who recently used an AmpliVox Sound Cruiser PA System in a video (that is approaching 1,000,000 views!). EdBassMaster “Look at This” Video with Sound Cruiser
4. Makes Life Easier for Teachers. Teachers have to juggle a variety of tasks outside of teaching including planning lessons, grading, meeting with parents, and some administrative work. Why not help yourself out and put objective tests online so that students can instantly see their grades instead of spending valuable time grading?

5. It Improves Test Scores. A benefit of engaging students through technology and media is allowing them to work at their own pace and not be held back by students who learn at a slower rate. This can allow gifted students to progress several grades ahead of their peers over the course of a year. Another proof of improved test scores comes from reaching different learning styles through the use of technology. The use of electronic drilling in education can help students learn quickly and provide incentives like games for them to keep progressing. Nobody does this better than Khan Academy.
6. Helps Students with Low Attention Spans. Students with ADD or ADHD can be aided through the use of technology by mixing up the learning process with different methods and through providing them engaging activities to keep their interest. Technology is a favorite among teachers who teach in ESL classrooms or special education settings.
Even the best teachers can have their moments or topics when they are like Ben Stein in Ferris Bueller’s Day Off or Charlie Brown’s Teacher in the eyes of their students. Using technology keeps students awake and their attention level high. How hard is it for you to fall asleep while on the internet compared to hearing a lecture?
7. Learn from the Experts. With the an increasing amount of free resources available on the internet, you can engage your students with content from the smartest and best teachers in the world

AmpliVox Sound Systems manufactures and sells portable sound systems along with lecterns and podiums with built-in sound systems that are perfect for classrooms. The Pinnacle Multimedia Lectern and iPod Portable PA System are great products for school settings that have a variety of uses both indoor and outdoors.

This video applies for kindergarten, pre kindergarten, elementary school (first, second, third, fourth, fifth grades), junior high or middle school, high school and at the college/university level. These tips are great for reading help, grammar, math, science, nursing, computers, classroom management

Special thanks to all those people who had videos under the “Creative Commons” License that allowed us to use short clips of their video.

The True Science of Parallel Universes



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Created by Henry Reich Created by Henry Reich

الكل يحب فكرة الاكوان المتوازية , ربما هو إلتماس لعالم مثالي حيث لديك فرص ثانية و الأمور جرت بطريقة مختلفة. واقع بديل دخلت فيه إلى هاغواترس و أفلام حروب النجوم السابقة لم تصنع و توصل فيه سلك اليو اس بي بطريقة صحيحة من أول محاولة لكن هل هناك مكان في العلم للتخمين الحالم هذا ؟ أعني إذا ''الكون'' هو كل شيء موجود , لن تستطيع إمتلاك إصدارين منه, أليس كذلك؟ وإلا سيكون الكونان كل شيء و ما أسميناه الكون لم يكن كل شيء. وإلا سيكون الكونان كل شيء و ما أسميناه الكون لم يكن كل شيء. المشكلة هنا هي المصطلحات,
الفيزيائيون يتكلمون بشكل غير رسمي أحيانا بقول '' الكون '' عندما يقصدون حقاً " الكون المرئي" وهو جزء من الكون الكلي الذي بإستطاعتنا رؤيته. بأمكاننا ايضاً ان نتحدث عن اكثر من كون مرئي مثلا, فضائي بالقرب من حافة كوننا المرئي سيرى اجزاء من الكون الكلي التي لا نستطيع رؤيتها, لأن الضوء لايملك الوقت الكافي للوصول الينا. لكن هذا السؤال مفهوم جيداً وليس هو ما يتحدث عنه الفيزيائيون عندما يتحدثون عن الأكوان المرئية المتعددة او "الأكوان-المتعددة" بإختصار: في الفيزياء , كلمة "الأكوان-المتعددة" عادة تشير الى 3 نماذج فيزيائية للكون لا علاقة لها ببعض لم يتم تأكيد او تجريب اي من هذه النماذج اختبارياً ايضاً. نماذج "الأكوان-المتعددة" الثلاث هي: النوع #1
الكون الفُقاعي او كون الثقوب السوداء الصغيرة هو ابسط نوع من "الأكوان-المتعددة": الفكرة الأساسية هي انه ربما هنالك أجزاء أُخرى من الكون بعيدة لدرجة اننا لن نتمكن من رؤيتها او بداخل ثقوب سوداء ولن نتمكن ايضا من رؤيتها ايضاً. هذا النموذج صُنع لتفسير لماذا كوننا متميز في توليد النجوم و المجرات
و الثقوب السوداء و الحياة. كما تقول الحُجة, اذا كان كل من هذه "الفقاعات" الغير متصلة تماماً في الكون لها قوانين فيزياء مختلفة قليلاً, اذاً حسب التعريف لن يمككنا الوجود إلا في واحدة لها قوانين طبيعة تسمح لنا بالوجود. مثلا يجب علينا ان نكون في كون الارض فيه تشكلت, اذا لم توجد الارض لن نكون هنا. اذا لم تقتنع بهذا المنطق, لاتقلق كثيراً, لايوجد الكثير من الأدلة التجريبية لهذا النموذج من ""الأكوان-المتعددة" حتى الان. نوع #2
الأغشية والأبعاد الإضافية. مستوحاة جزئيا من عجز رياضيات
نظرية الأوتار على التنبؤ العدد الصحيح من أبعاد الكون المرئي لنا ,اقترحت نظرية الأوتار فكرة أن ربما ما نفكر فيه ككوننا
في الواقع مجرد سطح ثلاثي الأبعاد مطمور في كون فائق مكون من 9 ابعاد مكانيه. كمثل أن اوراق الصحيفة هي سطح ثنائي الابعاد مطمورة في عالمنا الثلاثي الأبعاد وبطبيعة الحال، إذا كان للفضاء 9 أبعاد بدلا من الثلاثة، سيكون هناك الكثير من المكان لسطوح ثلاثية الأبعاد الأخرى التي ظهرت، مثل سطحنا ، لتكون أكوان في حد ذاتها،
ولكن، مثل صفحات الصحف، كانت في الواقع جزء من كل أكبر. هذه الأنواع من السطوح
تسمى "أغشية" أو "الأغشية" للاختصار. و للتذكير، وليس هناك حتى الآن أي دليل تجريبي على هذا النوع من الكون المتعدد. نوع #3
صورة العوالم المتعددة لميكانيكا الكم. من المثير للدهشة الفزيائيون مازالوا لا يستوعبون بالكامل كيف تتحطم الدالة الموجية في الفيزياء الكمية و فرضية العوالم المتعددة تفسر ذلك بإفتراض أن كل إطار زمني للكون حقيقي و أن كلهم يحدثون بطريقة متشعبة الى الأبد. كروايات كونية تحدث فيها كل القصص المحتمله. اذا كانت الوضع كذلك, فلن يمكننا ادراك ذلك لأننا سنكون عالقين بالعيش في واحد من العديد من الحيوات الانهائية المتاحة لنا. في بعض الطرق, نظرية العوالم المتعددة تشبه نظرية الاكوان المتعددة الفقاعية بإفتراض ان "ربما كل شيء ممكن أن يحدث, سوف يحدث" وأننا فقط موجودين في سلسلة الاحداث التي نحتاجها للوجود. إذا لم تقتنع بهذا المنطق, لاتقلق, لايوجد اي دليل تجريبي لهذا النموذج من الكون المتعدد. طبعاً, اذا اردت أن تكون تَخيُلياً يمكنك دمج العديد من هذه النماذج معاً لصنع أكوان تعددية متعددة نموذج جديد عالي التوقع مبني على نماذج توقعية و غير مؤكدة تجريبياً. لكن هذا لايعني أنه لا يمكن إجراء التجارب على نظريات الأكوان المتعددة. مثلا, لو أن كوننا المرئي هو بالفعل مجرد واحد من العديد من الفقاعات المتفرقة أو الأغشية و صدف أن إصطدم بفقاعة او غشاء اخر بالماضي فذللك الإصطدام بالتأكيد سيكون له أثر ما على ما نراه في السماء ليلاً. من ناحية اخرى, نظرية العوالم المتعددة قد يتم تجريبها قريباً إذ أن الفيزيائيون التجريبيين تزيد مقدرتهم بالتلاعب والتحكم بأنظمة كمية ضخمة في مختبراتهم أنظمة تقارب الحد بين العالم الكمي و واقعنا اليومي. لذلك كما هو الحال دائما، علينا أن نتذكر أن الفيزياء
علم، وليس فلسفة وفي محاولاتنا لتفسير الكون الذي نراه, يجب علينا ان نجد فرضيات قابلة للتجريب في المبدأ, ثم القيام بتجريبها!

Nancy Pelosi doesn’t understand economics: Trish Regan



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FBN’s Trish Regan discusses how Senate Minority Leader Nancy Pelosi (D-Calif.) downplayed the strength of the U.S. economy.

Understand The Violent History Of Domestic Terrorism In The United States



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Manhunt: UNABOMBER | Tuesdays at 10p on Discovery and Discovery GO
Terrorism in the United States can be committed by foreign agents, but more commonly is carried out by people born inside our own borders. From the rise of the Klu Klux Klan to deadly mass shootings, domestic terrorism remains a threat.
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the US government believes there's no other suspect in the 18 year reign of terror only this man Theodore Kaczynski federal investigators say he's the one who killed three people and maimed 23 others [Applause]