1 00:00:00,000 --> 00:00:05,454 [SOUND] So we're going to adopt Einstein's solution as verified by the 2 00:00:05,454 --> 00:00:10,270 Michelsonu-Morley experiment, and many others since, that claims the speed of 3 00:00:10,270 --> 00:00:13,210 light as measured by all observers as constant. 4 00:00:13,210 --> 00:00:17,277 Since we derived the velocity addition formula by simple logic, we're going to 5 00:00:17,277 --> 00:00:19,554 have to change simple logic and intuition. 6 00:00:19,554 --> 00:00:23,621 There are many approaches to doing this. I find that the most useful one is 7 00:00:23,621 --> 00:00:27,688 actually a little bit algebra heavy but on the other hand leads to clarity. 8 00:00:27,688 --> 00:00:31,810 We're going to derive the Lortentz transformations that are the replacement 9 00:00:31,810 --> 00:00:35,280 of Galilean transformations. And so the next clip is going to use a 10 00:00:35,280 --> 00:00:38,371 bit of algebra. If you're uncomfortable or if you feel 11 00:00:38,371 --> 00:00:42,304 that this drowns things out. You can, feel free to skip it, the next 12 00:00:42,304 --> 00:00:46,435 clip we'll present the result, and then, discuss it's consequences. 13 00:00:46,435 --> 00:00:50,756 But here, we're going to use only the assumption of the principle of 14 00:00:50,756 --> 00:00:54,886 relativity, as applied to electromagnetism therefore to the speed 15 00:00:54,886 --> 00:00:59,779 of light, to derive the corrections together on relativity, in the way that 16 00:00:59,779 --> 00:01:03,020 Einstein did. So what are the Lorentz transformations? 17 00:01:03,020 --> 00:01:07,932 They are derived by the following idea. We have the statement that the speed of 18 00:01:07,932 --> 00:01:12,005 light in constant as measured by observers no matter what their relative 19 00:01:12,005 --> 00:01:15,186 velocity to each other. So, we do the following think, thought 20 00:01:15,186 --> 00:01:18,598 experiment. We take our two observers, remember we 21 00:01:18,598 --> 00:01:22,868 had our two observers, the green axis and the blue axis. 22 00:01:22,868 --> 00:01:28,759 So, we have one observer x and t whose measurements will be represented in 23 00:01:28,759 --> 00:01:34,502 black, and we have another observer which was the blue observer who was moving 24 00:01:34,502 --> 00:01:37,816 relative to the first observer at some speed. 25 00:01:37,816 --> 00:01:41,940 So, this is the t prime axis. And, what we are going to do. 26 00:01:41,940 --> 00:01:48,280 Is that, at the same time as their, at the same instant as their meeting at t 27 00:01:48,280 --> 00:01:50,447 equals t prime equals x equals x prime equals zero. 28 00:01:50,447 --> 00:01:54,621 They both are at the same place, at the same time. 29 00:01:54,621 --> 00:02:01,202 At that same instant one of them, and it doesn't matter who, releases a pulse of 30 00:02:01,202 --> 00:02:07,864 light, moving say to the right at speed c and so, if you want you can also emit a 31 00:02:07,864 --> 00:02:12,520 pulse, to the left. the red line is therefore given by 32 00:02:12,520 --> 00:02:19,875 X, and as far as the, green observer measures it, the red line is described by 33 00:02:19,875 --> 00:02:27,417 the straight line X=CT because, the light is moving relative to the observer with 34 00:02:27,417 --> 00:02:31,701 speed C. But Einstein tells us it will also be 35 00:02:31,701 --> 00:02:37,939 described by X primeCT equals CT prime, and this is going to be inconsistent of 36 00:02:37,939 --> 00:02:42,874 course with Galilean relativity. What does it tell us? 37 00:02:42,874 --> 00:02:44,620 Well. Let's do some algebra. 38 00:02:44,620 --> 00:02:48,927 So we release this light pulse one goes to the left one goes to the right. 39 00:02:48,927 --> 00:02:53,466 x is plus or minus ct focusing on one of them will be quite enough we want to 40 00:02:53,466 --> 00:02:56,485 understand how. These two observers described this 41 00:02:56,485 --> 00:03:01,390 position and time of the same event, some birthday party that happened somewhere, 42 00:03:01,390 --> 00:03:06,112 and so I am going to claim that the position and the time as measured by the 43 00:03:06,112 --> 00:03:10,955 moving blue observer are going to be in some way dependent on the position and 44 00:03:10,955 --> 00:03:15,615 time as measured by the, stationary if you wish, the black observer, and these 45 00:03:15,615 --> 00:03:20,030 transformations have a few properties that I am going to insist upon. 46 00:03:20,030 --> 00:03:25,252 One is, of course that I want to insist that when x equals t equals zero. 47 00:03:25,252 --> 00:03:29,750 That was the instant that they met. At that time, at that event. 48 00:03:29,750 --> 00:03:32,930 Is also. The same. 49 00:03:32,930 --> 00:03:36,916 As x prime equals t prime equals zero. This is just a way of synchronizing their 50 00:03:36,916 --> 00:03:40,550 clocks and, indeed, these transformations have these properties. 51 00:03:40,550 --> 00:03:44,595 But this is not the most general functional dependence that has this 52 00:03:44,595 --> 00:03:47,350 property this is called a linear dependence. 53 00:03:47,350 --> 00:03:51,336 I am assuming that x prime is some number times x plus another number times t and, 54 00:03:51,336 --> 00:03:54,912 likewise for t prime. The reason I want to do this is because 55 00:03:54,912 --> 00:03:59,660 this has the property that straight lines in the xt plain will be transformed in 56 00:03:59,660 --> 00:04:02,473 the straight lines in the x prime t prime frame. 57 00:04:02,473 --> 00:04:06,460 We want this because we want physics to be preserved by these 58 00:04:06,460 --> 00:04:10,510 relativistic transformation. In other words, we want to know that an object 59 00:04:10,510 --> 00:04:14,833 that would be interpreted by the black observer as inertial, no forces act upon 60 00:04:14,833 --> 00:04:19,103 it, would also be interpreted by the blue observer as inertial the property of 61 00:04:19,103 --> 00:04:23,536 moving at a constant speed of having a straight world line needs to be preserved 62 00:04:23,536 --> 00:04:27,860 by our relativistic transformations so this is the most general transformation 63 00:04:27,860 --> 00:04:29,940 that has these two properties. And now, 64 00:04:29,940 --> 00:04:33,980 what are these numbers, A, B, C and D? Well, of course, in general, A, B, C and 65 00:04:33,980 --> 00:04:36,928 D will depend upon something. What will they depend on? 66 00:04:36,928 --> 00:04:40,969 Well, they relate the black observer's observations to the blue observer's 67 00:04:40,969 --> 00:04:43,863 observations, what relates these two observers is the 68 00:04:43,863 --> 00:04:48,121 relative velocity V, so A, B, C and D are arbitrary, initially, functions of V, and 69 00:04:48,121 --> 00:04:51,780 our job is to find these four functions of the relative velocity V. 70 00:04:51,780 --> 00:04:55,390 So let's do it. We expect that we think we know about the 71 00:04:55,390 --> 00:04:59,506 relation between the black guy's observations and the blue guy's 72 00:04:59,506 --> 00:05:02,673 observations. Well, the first is that if you said X 73 00:05:02,673 --> 00:05:06,283 prime equal to zero, that's the here of the blue observer. 74 00:05:06,283 --> 00:05:11,093 Well that here should be moving to the right, say, with respect with the black 75 00:05:11,093 --> 00:05:16,166 observer, x beat v, that's what defines the relative velocity v. And so setting x 76 00:05:16,166 --> 00:05:19,940 prime to zero means we should discover that x is equal to vt. 77 00:05:19,940 --> 00:05:26,523 Okay, put that into our equation, set x prime to zero, and what you see is that x 78 00:05:26,523 --> 00:05:31,273 prime equals zero means that ax plus bt is equal to zero. 79 00:05:31,273 --> 00:05:34,773 That's the equation x prime equal to zero. 80 00:05:34,773 --> 00:05:40,982 I can solve this and this says that x is equal to minus bt, divided by a. This 81 00:05:40,982 --> 00:05:46,298 indeed describes something that is moving at a constant velocity. 82 00:05:46,298 --> 00:05:49,459 That constant velocity is negative b over a. 83 00:05:49,459 --> 00:05:52,907 So, we discover that v must be negative b over a. 84 00:05:52,907 --> 00:05:58,797 The dependence of these coefficients on v is such that negative b over a is equal 85 00:05:58,797 --> 00:06:02,820 to v or more elegantly, b is equal to negative a times v. 86 00:06:02,820 --> 00:06:06,429 Okay, we learned something about these coefficients. 87 00:06:06,429 --> 00:06:11,838 If I want I can go back to my original equation and cross this guy out, and 88 00:06:11,838 --> 00:06:14,468 write minus A times V times T. Okay. 89 00:06:14,468 --> 00:06:18,676 What else do we know? Well, there's this relativity story. 90 00:06:18,676 --> 00:06:24,011 If the observer in black sees the observer in blue moving with speed V, 91 00:06:24,011 --> 00:06:29,572 then, of course, the observer in blue must see the observer in black moving 92 00:06:29,572 --> 00:06:33,780 with speed negative V. In other words, if you look at the 93 00:06:33,780 --> 00:06:39,717 position X equal to zero, then, that moves relative to the blue system with a 94 00:06:39,717 --> 00:06:44,577 velocity negative V. So lets plug that into our equations and 95 00:06:44,577 --> 00:06:48,098 see what it tells us. So, when I set x0, equal to zero, I can 96 00:06:48,098 --> 00:06:53,420 see from my equation, that when x0, equal to zero, I find that X prime is equal to 97 00:06:53,420 --> 00:06:57,268 B times T. When X is equal to zero, I find the T 98 00:06:57,268 --> 00:07:00,298 prime is equal to D times T. Remember, 99 00:07:00,298 --> 00:07:04,972 the idea is, any event, X equals zero at any time should be such 100 00:07:04,972 --> 00:07:11,046 that x prime is related to t prime. By this relation the world line of the 101 00:07:11,046 --> 00:07:15,560 black observer is of something moving with velocity minus v. 102 00:07:15,560 --> 00:07:22,044 So I set these two to satisfy that relation and I find that this means that 103 00:07:22,044 --> 00:07:27,694 x prime indeed is d over b imes t prime but that means that d over b should be 104 00:07:27,694 --> 00:07:31,700 negative v or, written more elegantly, b is negative d times v. 105 00:07:31,700 --> 00:07:37,137 And so that tells me, in fact, comparing these two, that d is the same as a. 106 00:07:37,137 --> 00:07:43,075 I can rewrite the equations implementing both of those results already and I see a 107 00:07:43,075 --> 00:07:47,010 simplified form. A appears twice three times in fact. 108 00:07:47,010 --> 00:07:51,946 we still haven't determined the coefficient c, but we haven't used 109 00:07:51,946 --> 00:07:55,952 relativity yet. We haven't used the idea that, that light 110 00:07:55,952 --> 00:08:00,240 pulse is moving with speed C. Notice so far, 111 00:08:00,240 --> 00:08:04,530 I haven't found any contradiction with Galilean relativity yet. 112 00:08:04,530 --> 00:08:07,100 But now I will, and 113 00:08:07,100 --> 00:08:12,900 what I'm going to find is the following story so I want to insist that the light 114 00:08:12,900 --> 00:08:18,907 pulse which is described remember by x is equal to ct that's that line moving world 115 00:08:18,907 --> 00:08:24,362 line of something moving with speed c is also described by x prime equal to ct 116 00:08:24,362 --> 00:08:31,201 prime with the same coefficient c. Hm, so now I need to plug all that into 117 00:08:31,201 --> 00:08:37,420 the current form of my equations. Let's try to do that. 118 00:08:37,420 --> 00:08:45,597 So if x is equal to ct, I find that x prime is equal to a times x of ct. 119 00:08:45,597 --> 00:08:52,056 Ct minus vt. And on the other hand, T prime is equal 120 00:08:52,056 --> 00:08:58,957 to c times x, which is ct, sorry for the double use of c at least it's clear in 121 00:08:58,957 --> 00:09:05,858 the type set version plus A times T. Now I'm not so interested in T prime, I 122 00:09:05,858 --> 00:09:13,290 want to study c times t prime to make the comparison, so let's put another c here. 123 00:09:13,290 --> 00:09:19,671 Multiplying this by C, I get a C here. And now I see that when I said this is 124 00:09:19,671 --> 00:09:26,556 equal, supposed to be equal to that the ACT terms agree so I can cross them out. 125 00:09:26,556 --> 00:09:32,434 The leftover pieces should agree with each other, so minus A times V times T 126 00:09:32,434 --> 00:09:37,640 should be C times the square of C minus the speed of light times T. 127 00:09:37,640 --> 00:09:43,770 Again I can cancel off the Ts and I find that C is equal to negative A times V 128 00:09:43,770 --> 00:09:45,385 over C squared. Aha! 129 00:09:45,385 --> 00:09:50,451 That nails everything in terms of this one coefficient A. 130 00:09:50,451 --> 00:09:55,529 So let's write all that out. And we can plug that back into these 131 00:09:55,529 --> 00:09:58,830 equations, and this is what we've discovered so far. 132 00:09:58,830 --> 00:10:04,138 And this is what we've discovered, I have put back the velocity dependence of a, 133 00:10:04,138 --> 00:10:09,316 now that it's not too cumbersome there's only one coefficient that contains all 134 00:10:09,316 --> 00:10:12,423 the information. Alright, how do I determine this, 135 00:10:12,423 --> 00:10:14,560 velocity dependence of A? Well, 136 00:10:14,560 --> 00:10:19,196 what I do here is I have here two equations determining x prime and t prime 137 00:10:19,196 --> 00:10:22,857 in terms of x and t. I can invert them just as I get for the 138 00:10:22,857 --> 00:10:27,494 Galilean transformation and solve for x and t as functions of x prime and t 139 00:10:27,494 --> 00:10:30,422 prime. Little bit of algebra, you're more than 140 00:10:30,422 --> 00:10:34,632 encouraged to do it yourself, and you find that these are the inverse 141 00:10:34,632 --> 00:10:38,659 transformations that follow. I mean, if x prime and t prime satisfy 142 00:10:38,659 --> 00:10:43,418 the, these two equations, you can solve for x and t and get these two equations. 143 00:10:43,418 --> 00:10:46,530 Now what do you expect? Remember x and t are the, 144 00:10:46,530 --> 00:10:52,480 position in time of a particular event, as measured by the black observer, for 145 00:10:52,480 --> 00:10:56,730 who the blue observer observes position x prime and 146 00:10:56,730 --> 00:11:00,100 time, T prime. And so, remember the black observer is 147 00:11:00,100 --> 00:11:05,387 moving relative to the blue observer with velocity minus V so we know, if Lorentz 148 00:11:05,387 --> 00:11:10,410 transformations, if our relativistic transformations are going to be correct. 149 00:11:10,410 --> 00:11:13,913 We know how to relate X and T to X prime and T prime. 150 00:11:13,913 --> 00:11:18,870 We basically repeat the calculation we get here but with relative velocity 151 00:11:18,870 --> 00:11:21,579 negative V. In other words, this should be the same 152 00:11:21,579 --> 00:11:26,338 as that with, the sign of V changed, so that's what we expect, that the 153 00:11:26,338 --> 00:11:31,655 transformation from X prime and T prime to X and T will be the same as this, with 154 00:11:31,655 --> 00:11:36,110 everything everywhere, V replaced by, relative velocity negative V. 155 00:11:36,110 --> 00:11:37,308 Okay. That's great, 156 00:11:37,308 --> 00:11:40,170 because now, lots of things look consistent. 157 00:11:40,170 --> 00:11:42,699 Look the, X prime plus vt prime cancels. 158 00:11:42,699 --> 00:11:46,626 That's, that's reasonable. The dependence here is reasonable. 159 00:11:46,626 --> 00:11:49,954 It's only a question of this, a of v and a of -v. 160 00:11:49,954 --> 00:11:52,617 What do we know about a of v and a of minus v? 161 00:11:52,617 --> 00:11:57,010 A of v is the coefficient corresponding to moving with velocity v, 162 00:11:57,010 --> 00:11:59,392 say to the right, or of speed v say to the right. 163 00:11:59,392 --> 00:12:03,046 If minus v is the coefficient corresponding to moving with that same 164 00:12:03,046 --> 00:12:06,245 speed to the left. The universe doesn't care if you're 165 00:12:06,245 --> 00:12:10,174 moving to the left and right. The universe as far as we know is 166 00:12:10,174 --> 00:12:12,919 isotropic, left and right are the same thing. 167 00:12:12,919 --> 00:12:16,598 So we in fact expect A of V to be the same as A of minus V. 168 00:12:16,598 --> 00:12:21,588 That's because which direction you're moving in should not change the nature of 169 00:12:21,588 --> 00:12:24,894 relativity. What that allows us is to put a plus sign 170 00:12:24,894 --> 00:12:28,698 here, because we don't care. Multiplying through by A of V, we find 171 00:12:28,698 --> 00:12:33,138 that A of V squared, therefore, is equal to 1 over 1 minus V squared over C 172 00:12:33,138 --> 00:12:38,274 squared, taking the square root we have our answer A of V is 1 over the square 173 00:12:38,274 --> 00:12:42,961 root of 1-V squared over C squared. Plugging that back in we have derived 174 00:12:42,961 --> 00:12:48,225 explicitly the Lorentz transformations, that form the core of the Theory of 175 00:12:48,225 --> 00:12:52,142 Special Relativity and we will, be studying them carefully. 176 00:12:52,142 --> 00:12:57,085 We will be using them everywhere, and note that they are not the equations of 177 00:12:57,085 --> 00:13:00,257 Galilean relativity. Of course they're not, they have the 178 00:13:00,257 --> 00:13:04,399 property that in when this change the speed of a pulse of light as measured by 179 00:13:04,399 --> 00:13:08,017 the two observers will be c and that violates the velocity addition. 180 00:13:08,017 --> 00:13:12,624 So, of course we made a change, but Newtonian and Galilean physics worked 181 00:13:12,624 --> 00:13:16,380 very well. what is, where is the limit in which we 182 00:13:16,380 --> 00:13:20,932 reproduce Newtonian physics? The limit is clearly that if V, the 183 00:13:20,932 --> 00:13:26,350 relative motion, is very small compared to C, then I can pretty much neglect 184 00:13:26,350 --> 00:13:30,324 these denominators. This is much smaller than one, These 185 00:13:30,324 --> 00:13:35,453 denominators are going to be approximately one and I find that X prime 186 00:13:35,453 --> 00:13:40,221 is indeed approximately X minus VT. And T prime would be approximately T 187 00:13:40,221 --> 00:13:43,870 because V over C is a small number. So, 188 00:13:43,870 --> 00:13:49,382 if I don't look too far away, and we'll talk about what that means in the next 189 00:13:49,382 --> 00:13:52,105 clip, then I am reproducing the galilean 190 00:13:52,105 --> 00:13:57,142 transformations that makes sense. the difference between relativity and 191 00:13:57,142 --> 00:14:01,328 galilean relativity shows up at. Velocities who which are not negligible 192 00:14:01,328 --> 00:14:05,144 to the speed of light the reason that common sense does not agree with what 193 00:14:05,144 --> 00:14:09,261 Einstein predicts is because common sense is the intuition that we have developed 194 00:14:09,261 --> 00:14:13,127 over the course of our life and over the course of our life we have not been 195 00:14:13,127 --> 00:14:17,094 walking around at speeds comparable to the speed of light we have no intuitive 196 00:14:17,094 --> 00:14:20,151 sense for what this means. So we'll have to use math, and Lorentz 197 00:14:20,151 --> 00:14:23,040 transformations are the math that will make everything clear.