 In the previous video in this series, we saw the overwhelming experimental evidence that the speed of light is the same in all directions in all inertial reference frames, and that there is no evidence that light is the oscillation of some material. That terminology, inertial reference frame, keeps coming up, so just to review, that's a frame in which if I put something at rest, it stays at rest. If I push it, it moves with a uniform velocity, and that's true pretty much at the surface of the Earth in two directions corresponding to the ground, but not in the third direction corresponding to up and down. Of course, there is the force of gravity that makes that a non-inertial frame. Towards the end of making sense of this, a young Albert Einstein proposed a general principle of nature, the principle of relativity. The same laws of electrodynamics and optics will be valid for all frames of reference for which the equations and mechanics hold good. The goal of this video is to rigorously work through the implications of this, arriving at Einstein's so-called special theory of relativity, which he first presented in 1905 in a paper titled, at least the translation, is titled, On the Electrodynamics of Moving Bodies. But we're going to de-emphasize the math and try to work through things in a more graphical approach, and this is primarily due to a fellow named Minkowski. He presented these concepts in a 1908 presentation called Space and Time, and both of these papers are available in the excellent Dover Publications Book of Reprints of the original papers, at least the translations they were originally in German, available for about $10 from Dover Publications. The speed of light is central to relativity, and since it keeps coming up, it'll make our life a lot simpler if we take it to be one. In other words, if we choose units in which the speed of light is one. A simple way to do that is to keep the second as our unit of time, but instead of meters to use light seconds as the unit of distance. So a light second is approximately 300 million meters, which is almost the distance to the moon. If we do that, then the speed of light is one light second per second. Of course, this will make our day-to-day velocities very small. For example, a very large velocity to us is the speed of sound, roughly 330 meters per second. In this system of units, that becomes approximately one millionth of a light second per second. In other words, the speed of sound is about one millionth of the speed of light. Now back to the classical points of view that have us in a bind. We assume that we have two reference frames, one labeled with Latin letters, one with Greek letters. The Greek frame moves to the right relative to the Latin frame with a velocity v. We assume that everybody measures the same time, t or tau. Coordinates that are not in the direction of motion, that is, for example, y or z coming out of the board, would be the same, remain the same, but the coordinate in the direction of motion, x and psi, would drift. They drift because of the motion of the coordinate systems. So x would be equal to the psi coordinate of, say, the green dot plus the offset of the Greek coordinate system, v times time. Our dilemma is that experiment tells us that if we set off a pulse of light, both of these coordinate systems have to perceive themselves to be at the center of that circular wave. So for example, consider the blue point here, which would be perceived in the Greek frame as being on the center of that wave front. It would have traveled a certain distance in a certain amount of time, consistent with the speed of light being unity in the Greek frame. If our classical concepts are correct, in the Latin frame, it would have traveled a greater distance in the same amount of time. Hence it would be traveling at a greater velocity. The speed of light would be different in the two frames. This is readily apparent on a space-time diagram. If we imagine going along the horizontal axis four units of time and then going up four units of spatial displacement in the Latin system, that would be corresponding to the blue lines, we would go up to the blue point. And the slope in the blue coordinate system would be four units of space per four units of time, so a slope of one. That's the green line that goes through the blue point. In the Greek frame, though, we would go step off the same four units of time, but then we would have to go up four of the red coordinate lines. Those correspond to the Greek frame. And again, because the Greek frame is moving relative to the Latin frame, those lines are sloped. So we end up at the red point. That's psi is equal to four, at time t is equal to four. Now in the Greek frame, you would still have a speed of light, which would be four units of space in four units of time. But that same red point in the blue frame, in the Latin frame, of course, would be more than four units of space in four units of time. It would correspond to a larger velocity than unity. In other words, it has a larger slope. Looked at another way, we have two lines here, both of which are supposedly representing the same physical wave front. In the Latin frame, you have one unit of time, one unit of displacement, and that defines, from the blue triangle, one of those green lines. In the Greek frame, one unit of time, one unit of displacement defines the red triangle and the second green line. But the Michelson-Morley experiment tells us that these two lines have to coincide. Both of the frames of reference have to measure the same speed of light. Now the problem arises because of the slopes between the lines of constant x and constant psi. But this also suggests a very elegant, seemingly very simple solution, which nonetheless has profound implications. We simply need to introduce a corresponding drift or slope for the time coordinate. Now, one unit of spatial displacement over one unit of time gives us the same speed of light, the green line, whether we look at it in terms of the Greek frame or the Latin frame, in terms of the red or the blue coordinates. Now note that this is only true for the speed of light. If we look at other speeds, less than one, for example a speed of one half, say a displacement of two spatial units in four units of time, we'd get the two triangles shown here. And notice that in this case the two frames of reference would not agree that they were looking at the same velocity. The agreement comes only when the velocity is the speed of light, unity. For example in this case if we have four units of displacement in four units of time. To describe this mathematically we need to add the two terms shown underlined here in green. Before in the classical Galilean relativity we assumed everybody agreed upon time, but they disagreed upon spatial coordinates. Now if psi is equal to x minus vt, we have the symmetrical relationship that time tau is equal to t minus vx. Correspondingly if x is equal to psi plus v tau, then t is tau plus v psi. Those new terms create that slope between these lines of constant time. We have now two different time coordinates that are completely symmetric with respect to the spatial coordinates. Everything now is symmetric about the diagonal green line, which remember represents the speed of light. And we've gotten all this primarily by looking at pictures. Although we're not quite there yet, there's one little wrinkle we still have to fix. But before we move on, let's stop and see if this passes the giggle test. I mean could this possibly correspond to reality? Maybe mathematically this fixes our problem, but we've just said that different moving observers will have different time coordinates. Do we have any direct experience of that? Not at all. It seems completely absurd. But remember that this picture corresponds to those special units we chose to make the speed of light be one. We measure time in seconds, but spatial displacement in light seconds, about 300 million meters, almost the distance to the moon. That means our day-to-day life takes place in a little box that is extremely wide in the time coordinate, but very, very short in the spatial coordinate. We don't have daily experience with light second type displacements, but we certainly move through many hundreds of seconds even just watching this video. So our experience never takes us up towards the top parts of this plot where you would actually see the divergence between those T and tau axes. Now let's fix up the little detail I alluded to, and we do at this point have to do by necessity just a little bit of algebra, but it'll be worth it. Here are the transformations between space and time for the Latin and the Greek coordinates. And the terms with the red boxes around them are the new terms, the terms that deviate from the classical Galilean relativity that create that symmetry that causes everyone to see the same speed of light. Now if we start off with x and t values and transform those to Greek values, psi and tau, and then use the second set of equations to transform back to x and t, we should end up where we started from. We should have x equals x and t equals t. We substitute the first set of equations into the second. We end up with four terms, and x equals x and some other stuff, t equals t and some other stuff. Now the second and third terms that have the blue boxes around them cancel out, minus vt plus vt minus vx plus vx. Unfortunately the last terms, the fourth terms, do not cancel. And that leaves you with a v times a minus vx, and in the second equation a v times a minus vt. With the net result that your equations say x equals 1 minus v squared times x, which of course isn't equal to x, and likewise for t. Indeed, 1 minus v squared is something a little bit, at least a little bit, less than 1. So in transforming from one system to the other and back, we've had some shrinkage. Well the obvious way to counteract that would be with a little bit of expansion or dilation. So we now arrive at the final theory. We have a dilation factor, beta, is 1 over the square root of 1 minus v squared. And we simply multiply our transformations by that dilation or expansion factor. If we do that, then when we substitute the first set of equations into the second set, we end up with two factors of beta, or a beta squared, which would be 1 over 1 minus v squared. That will cancel out the 1 minus v squared that was causing us the problems. The shrinkage will be canceled out by the dilation. These then are the final equations for the transformation between space and time in special relativity. To be completely rigorous we should show all three spatial coordinates in time, but the other two spatial coordinates are kind of boring. The ones that don't correspond to the direction of motion don't have any unique transformation properties. They just stay the same. So now we have complete three dimensions of space and time for systems which are moving along their respective x or psi axes with velocity v. Just for giggles, let's go back and check out Einstein's original paper. There's his results. Our results? His results. Now, he didn't use units in which the velocity of light was 1. So c, the velocity of light, shows up in his equations. But notice if you make c equal to 1 in his equations, you get the equations that we just derived. Yeah, I know this Einstein kid. He passes. He's okay. No, we fact-checked him. Not a very elegant derivation he's got there, but I guess he couldn't handle pictures. We just had to, you know, keep an eye on the kid. Yeah, yeah, just all of today's work on YouTube. Yeah. Seriously though, we have in a few short minutes, we derived one of the great results in modern physics. And we have a rigorous basis now to push on in future videos to make predictions about how the world actually behaves and to compare those predictions to what's actually observed.