WEBVTT 02:04.760 --> 02:07.130 Professor Ramamurti Shankar: So, 02:07.128 --> 02:09.148 let's begin now. First of all, 02:09.152 --> 02:12.622 I'm assuming all of you have some idea what special 02:12.622 --> 02:14.012 relativity means. 02:14.009 --> 02:16.099 There are two theories of relativity, one is the special 02:16.104 --> 02:17.594 theory and one is the general theory. 02:17.590 --> 02:21.310 The general theory is something that we won't do in any detail. 02:21.310 --> 02:23.840 Special theory is something we will do in reasonable detail. 02:23.840 --> 02:29.180 So, it's good to begin by asking some of you what is your 02:29.182 --> 02:34.622 present understanding of what the subject is all about. 02:34.620 --> 02:36.150 Yes, sir? The Yale cap, 02:36.147 --> 02:37.777 what do you think it's about? 02:37.780 --> 02:42.940 Student: It's about relative speed in two reference 02:42.944 --> 02:44.564 systems. Professor Ramamurti 02:44.557 --> 02:46.037 Shankar: Okay, it's about relative speed in 02:46.036 --> 02:46.836 two reference systems. 02:46.840 --> 02:48.990 I'll come to you; then I'll come to you. 02:48.990 --> 02:53.090 Student: It's based on the postulate that the laws of 02:53.094 --> 02:57.204 physics are the same in any two references moving in uniform 02:57.198 --> 03:01.018 motion relative to one another and the speed of light is 03:01.024 --> 03:03.324 constant in all references. 03:03.319 --> 03:04.619 Professor Ramamurti Shankar: Okay, 03:04.618 --> 03:05.688 I will take the last row there. 03:05.689 --> 03:09.229 Student: [inaudible] Professor Ramamurti 03:09.234 --> 03:12.564 Shankar: Okay, so what I've heard so far is 03:12.561 --> 03:17.191 that the laws of physics are the same for two people who are both 03:17.190 --> 03:21.530 in inertial frames of reference and the velocity of light's a 03:21.530 --> 03:25.180 constant. Right. 03:25.180 --> 03:27.590 That's certainly the way we understand the special 03:27.587 --> 03:28.567 relativity theory. 03:28.570 --> 03:30.140 But it's a very old one. 03:30.139 --> 03:33.149 It's been going on long before Einstein came. 03:33.150 --> 03:36.840 There was a theory of relativity at the time of Newton 03:36.841 --> 03:39.281 and that's where I want to begin. 03:39.280 --> 03:43.490 Relativity is not a new idea at all, it's an old one. 03:43.490 --> 03:48.420 And the old idea can be illustrated in this way and it 03:48.417 --> 03:51.947 will agree with your own experience. 03:51.949 --> 03:56.369 So, the standard technique for all of relativity is to get 03:56.372 --> 03:58.392 these high speed trains. 03:58.389 --> 04:01.019 I'm going to have our own high speed train; 04:01.020 --> 04:06.020 this is the top view of the train. 04:06.020 --> 04:09.010 And like in everything I do, we'll get away with the lowest 04:09.005 --> 04:11.995 number of dimensions, which happens to be this one 04:12.004 --> 04:15.054 spatial dimension and of course there is time. 04:15.050 --> 04:17.400 So, the train is moving along the x axis. 04:17.400 --> 04:20.980 You are in this train. 04:20.980 --> 04:27.060 You board the train and all the blinds are closed because you 04:27.060 --> 04:30.000 don't want to look outside. 04:30.000 --> 04:33.360 That's not because you're traveling through some parts of 04:33.360 --> 04:36.620 New Jersey; you don't want to look outside 04:36.617 --> 04:39.257 for this particular experiment. 04:39.259 --> 04:44.379 You get into the train, you settle down and you explore 04:44.379 --> 04:46.559 the world around you. 04:46.560 --> 04:50.050 You pour yourself a drink, you play pool, 04:50.046 --> 04:54.226 you juggle some ping pong balls, tennis balls, 04:54.230 --> 04:58.030 and you have a certain awareness of what's happening, 04:58.033 --> 05:01.913 namely, your understanding of the mechanical world, 05:01.910 --> 05:04.110 and then you go to sleep. 05:04.110 --> 05:10.210 When you are sleeping, some unseen hand gives to this 05:10.214 --> 05:15.384 train a large velocity, 200 miles an hour. 05:15.379 --> 05:18.619 The question is, "When you wake up, 05:18.623 --> 05:22.443 can you tell if you're moving or not?" 05:22.440 --> 05:24.800 That's the whole question. 05:24.800 --> 05:27.180 Will this speed, whatever I gave you, 05:27.175 --> 05:31.065 200 miles per hour--will it do anything to you in this train 05:31.069 --> 05:33.379 that will betray that velocity? 05:33.379 --> 05:37.839 So, when you wake up will you say I'm moving or not? 05:37.839 --> 05:40.939 Now, you might say, I'm not moving because I'm on 05:40.938 --> 05:44.358 Amtrak and I know this train is not going anywhere. 05:44.360 --> 05:47.160 That kind of sociological reason, by the way, 05:47.160 --> 05:49.960 there are many of them, you cannot invoke. 05:49.959 --> 05:51.839 You can only say, "I'm on this train. 05:51.840 --> 05:53.960 Is anything different?" 05:53.959 --> 05:57.159 And the claim is that nothing will be different. 05:57.160 --> 05:59.140 You just will not know you are moving. 05:59.139 --> 06:04.849 Now, if the train picks up speed, or slows down, 06:04.849 --> 06:08.129 you will know right away. 06:08.129 --> 06:10.519 If it picks up speed or accelerates, you find yourself 06:10.517 --> 06:13.217 pushed against the back of the seat or if the driver slams on 06:13.219 --> 06:15.869 the brake, you will slam into the front of 06:15.873 --> 06:17.573 the seat in front of you. 06:17.569 --> 06:20.809 No one is saying that when the motion is accelerated, 06:20.812 --> 06:22.062 you will not know. 06:22.060 --> 06:26.060 Accelerated motion can be detected in a closed train 06:26.060 --> 06:28.100 without looking outside. 06:28.100 --> 06:32.180 The question is uniform velocity, no matter how high, 06:32.184 --> 06:34.074 can that be perceived? 06:34.070 --> 06:35.350 Can that be detected? 06:35.350 --> 06:40.060 So, at the time of Galileo and Newton everybody agreed that you 06:40.057 --> 06:41.497 cannot detect it. 06:41.500 --> 06:46.450 Remember that if you started out and Newton's laws worked for 06:46.451 --> 06:50.001 you, you are called an inertial observer. 06:50.000 --> 06:52.880 One of the laws you want is, if you leave something, 06:52.875 --> 06:54.505 it should stay where it is. 06:54.509 --> 06:58.009 When the train is accelerating, that won't be true. 06:58.009 --> 07:00.689 You leave things on the floor when it's accelerating, 07:00.686 --> 07:02.226 things will slide backwards. 07:02.230 --> 07:04.110 So, with no apparent force acting on it, 07:04.110 --> 07:05.750 things will begin to accelerate; 07:05.750 --> 07:07.810 that's a non-inertial frame. 07:07.810 --> 07:09.250 We are not interested in that. 07:09.250 --> 07:12.400 You started out as an observer for whom the laws of Newton 07:12.401 --> 07:15.661 work, the laws of inertia work, F = ma, then you go to sleep 07:15.664 --> 07:18.514 and you wake up. So, when I said everything 07:18.514 --> 07:21.604 looks the same, I really meant that the laws of 07:21.602 --> 07:25.562 Newton continue to be the same because if the laws of Newton 07:25.561 --> 07:28.391 are the same, everything will look the same. 07:28.389 --> 07:31.089 That's what it means to say "everything looks the same." 07:31.089 --> 07:34.199 Our expectations of what happens when I throw it up or 07:34.201 --> 07:36.781 what happens when two billiard balls collide, 07:36.784 --> 07:39.724 everything is connected to the laws of Newton. 07:39.720 --> 07:42.960 So, the claim is, the laws of Newton will be 07:42.956 --> 07:46.716 unchanged when this velocity is added on to you. 07:46.720 --> 07:49.460 Now, we should be clear about one thing. 07:49.459 --> 07:53.469 If there is a train next to you in the beginning -- let me just 07:53.474 --> 07:56.844 put it on this side for convenience -- and you got in 07:56.840 --> 08:00.790 and you boarded this train but you looked at this train and it 08:00.790 --> 08:03.390 was not moving. If you lift the blind and look 08:03.391 --> 08:05.631 through, you'll see the other train and there's another 08:05.627 --> 08:08.067 passenger in the other train and you look at each other, 08:08.070 --> 08:10.120 you're not moving. 08:10.120 --> 08:14.530 When you wake up after this brief nap, you find when you 08:14.534 --> 08:19.514 look outside the other train is moving at 200 miles an hour. 08:19.509 --> 08:23.599 The question is, "Can you tell if it's you who's 08:23.603 --> 08:27.003 responsible for this relative motion, 08:27.000 --> 08:30.470 or maybe nothing happened to you and the other train is 08:30.466 --> 08:32.196 moving the opposite way?" 08:32.200 --> 08:37.420 And the claim of relativity is that you really cannot tell. 08:37.419 --> 08:40.709 You can tell there is motion between the two trains that 08:40.705 --> 08:42.015 wasn't there before. 08:42.019 --> 08:45.609 That's very clear if you look outside but there is no way to 08:45.612 --> 08:48.842 tell what actually happened when you were sleeping. 08:48.840 --> 08:51.530 Whether you were given the velocity of 200 to the right or 08:51.533 --> 08:54.233 the other train was given a velocity of 200 to the left or 08:54.226 --> 08:57.336 maybe a combination of the two, you just cannot tell. 08:57.340 --> 08:59.000 That's the word "relative." 08:59.000 --> 09:02.540 So far -- I didn't tell you -- if you have only one train, 09:02.543 --> 09:06.143 what I told you earlier, is that uniform velocity does 09:06.135 --> 09:09.605 not leave its imprint on anything you can measure. 09:09.610 --> 09:12.240 If you look outside, of course you can see the 09:12.235 --> 09:15.435 motion of the other train, but you still cannot tell who 09:15.443 --> 09:18.653 is moving. You cannot distinguish between 09:18.649 --> 09:20.639 different possibilities. 09:20.639 --> 09:24.409 So, you have every right to insist that you are not moving 09:24.409 --> 09:27.649 and the other train is moving the opposite way. 09:27.649 --> 09:32.109 Once again, you can make this argument only for uniform 09:32.105 --> 09:35.785 relative motion. If your train is accelerating, 09:35.789 --> 09:39.239 now I'm saying it as if it is an absolute thing, 09:39.236 --> 09:41.006 and it is. You cannot say, 09:41.007 --> 09:43.897 "I'm not accelerating, the other train is accelerating 09:43.897 --> 09:45.367 in the other direction." 09:45.370 --> 09:49.000 You cannot say that because you're the one who is barfing up 09:48.999 --> 09:52.259 and throwing up and slamming your head on the wall; 09:52.259 --> 09:53.889 nothing is happening to the other person. 09:53.889 --> 09:56.589 You cannot say "I'm still in the same frame, 09:56.587 --> 09:58.717 you are going the opposite way." 09:58.720 --> 10:00.720 If you are going the opposite way, why am I throwing up? 10:00.720 --> 10:05.130 Or if you are in a rocket and the rocket's taking off and the 10:05.130 --> 10:08.570 G forces are enormous, many times your weight, 10:08.571 --> 10:11.551 it is the astronauts who are going through the discomfort. 10:11.549 --> 10:14.769 At that time they cannot say we are at rest and everyone is 10:14.766 --> 10:17.816 going the opposite way because no one else is in danger, 10:17.817 --> 10:20.187 but they are. So, accelerated motion will 10:20.189 --> 10:22.909 produce effects. You cannot talk your way out of 10:22.908 --> 10:24.768 that. But uniform velocity will 10:24.774 --> 10:28.534 produce no effects on you and no effects on the other person. 10:28.529 --> 10:31.979 You can detect relative motion but you cannot in any sense 10:31.979 --> 10:35.849 maintain that you are moving and he's not or that he's moving and 10:35.851 --> 10:37.611 you are not. You can say, 10:37.608 --> 10:40.498 "I am at rest, things are the same as before, 10:40.497 --> 10:43.187 the train is moving the opposite way." 10:43.190 --> 10:47.650 Now, if you go in the Amtrak and you look outside and you 10:47.649 --> 10:50.909 don't see another train, but you see the landscape, 10:50.910 --> 10:53.480 you see trees and cows and everything, going at 200 miles 10:53.475 --> 10:55.165 an hour in the opposite direction, 10:55.169 --> 10:59.489 you have some reason to believe that probably the ground is not 10:59.489 --> 11:01.439 moving and you are moving. 11:01.440 --> 11:07.020 But that's just based on what I called earlier some sociological 11:07.015 --> 11:08.875 factors. In other words, 11:08.875 --> 11:12.515 it's completely possible to devise an experiment in which 11:12.519 --> 11:16.099 somebody puts the whole landscape on wheels and when you 11:16.097 --> 11:19.947 go to sleep the landscapes, cows and trees are made to move 11:19.945 --> 11:20.965 the opposite way. 11:20.970 --> 11:23.990 Not very likely, but that's because we know in 11:23.989 --> 11:27.809 practice no one is going to bother to do that just to fool 11:27.813 --> 11:29.523 you. But if that did happen, 11:29.517 --> 11:31.087 you won't know the difference. 11:31.090 --> 11:33.880 So, the reason we rule that out is we know some extraneous 11:33.884 --> 11:36.144 things not connected to the laws of physics. 11:36.139 --> 11:39.119 That's why we don't like to open the window and look at the 11:39.116 --> 11:41.166 landscape because then we have a bias. 11:41.169 --> 11:44.069 Open the window and look at another train and you just won't 11:44.066 --> 11:46.416 know. That is the principle of 11:46.416 --> 11:50.846 relativity, that uniform motion between two observers, 11:50.848 --> 11:54.358 both of whom are inertial, is relative. 11:54.360 --> 11:56.790 Each one can insist that he or she is not moving; 11:56.789 --> 11:59.299 the other person is the one who is moving. 11:59.299 --> 12:02.229 Of course, now, if the two, in reality, 12:02.233 --> 12:06.633 if the two trains were at rest--Let's imagine my train got 12:06.634 --> 12:09.544 accelerated. So, during the time it was 12:09.536 --> 12:12.996 accelerated, I would know, but if I was sleeping at the 12:12.997 --> 12:15.127 time, I don't know and when I wake up 12:15.132 --> 12:18.222 and the acceleration is gone and the velocity is constant, 12:18.220 --> 12:20.490 that's when I say, "I just cannot tell." 12:20.490 --> 12:25.690 All right. Now, let's show once and for 12:25.687 --> 12:31.127 all that the laws of Newton are not going to be modified. 12:31.129 --> 12:34.229 So, you find the laws of Newton before you go to sleep, 12:34.229 --> 12:36.069 you wake up, you find them again, 12:36.065 --> 12:38.825 you'll get the same laws; that's the claim. 12:38.830 --> 12:41.890 I hope you understand that all the mechanical things you see in 12:41.886 --> 12:44.446 the world around you come just from F = ma. 12:44.450 --> 12:48.000 We have seen projectiles and collision of billiard balls and 12:47.998 --> 12:50.358 rockets; they're all Newtonian mechanics. 12:50.360 --> 12:53.860 So, to say that things will look the same is to say the laws 12:53.856 --> 12:57.466 of Newton that you will deduce before and after waking up will 12:57.471 --> 13:00.341 be the same. So, let's show that. 13:00.340 --> 13:02.500 When you show that, you're really done with it once 13:02.504 --> 13:04.964 and for all. So, let's do the following. 13:04.960 --> 13:11.930 Here is the x axis and here is my frame of reference. 13:11.930 --> 13:12.950 This is my x axis. 13:12.950 --> 13:14.710 Let's call this the origin. 13:14.710 --> 13:18.150 The frame goes to negative and positive x values. 13:18.149 --> 13:22.789 Pick some object sitting at the point x. 13:22.790 --> 13:27.470 13:27.470 --> 13:31.150 Now, we are going to first define the notion of an event. 13:31.149 --> 13:35.749 An event is something that happens at a certain place at a 13:35.749 --> 13:40.879 certain time; that's called an event. 13:40.879 --> 13:43.239 For example, if there's a little firecracker 13:43.242 --> 13:46.372 going off somewhere at some time, the x is where it 13:46.373 --> 13:49.123 happened and the t is when it happened. 13:49.120 --> 13:52.750 So, this is space-time. 13:52.750 --> 13:56.030 Once again, space-time does not require Einstein coming in at 13:56.031 --> 13:57.631 all. We have known for thousands of 13:57.628 --> 14:00.078 years that if you want to set up a meeting with somebody, 14:00.080 --> 14:02.740 you've got to say where and you've got to say when and 14:02.742 --> 14:04.402 things do happen in space-time. 14:04.399 --> 14:06.479 The fact that you need x and t, 14:06.484 --> 14:08.904 or, if you're living in three spatial dimensions, 14:08.899 --> 14:11.119 the fact that you need x, y, 14:11.122 --> 14:13.022 z and t is not new. 14:13.019 --> 14:17.279 That is not the revolution Einstein created. 14:17.279 --> 14:19.999 The fact that you need four coordinates to label an event is 14:19.995 --> 14:22.955 nothing new. What he did that is new will be 14:22.957 --> 14:26.177 clear later. So, does everyone understand 14:26.181 --> 14:27.871 what an event means? 14:27.870 --> 14:30.160 Okay? An event is something that 14:30.157 --> 14:33.407 happens and to say exactly where and when it happened, 14:33.409 --> 14:36.489 in our world of one dimension, we give it an x and we 14:36.492 --> 14:37.592 give it a t. 14:37.590 --> 14:41.550 Now, that's me, and I'm going to give my frame 14:41.553 --> 14:44.463 of reference the name S. 14:44.460 --> 14:48.280 It turns out S is not just based on just my name. 14:48.279 --> 14:50.729 This is the canonical name for two observers, 14:50.725 --> 14:53.945 one is called S and one is called S prime. 14:53.950 --> 14:56.160 So, S prime, let's say, is you. 14:56.159 --> 15:01.549 So, your frame of reference is going to be taken to be sliding 15:01.550 --> 15:03.230 relative to mine. 15:03.230 --> 15:08.890 So, let's draw a y axis here. 15:08.889 --> 15:11.809 We don't really deal with the y coordinate but just to 15:11.805 --> 15:14.085 give you a feeling, this is my y axis; 15:14.090 --> 15:16.300 that's my x axis; this is my origin. 15:16.299 --> 15:18.679 y is not going to play a big role. 15:18.679 --> 15:23.209 Now, you are sliding this way, to the right, 15:23.211 --> 15:29.431 and your speed or velocity is always denoted by u. 15:29.430 --> 15:33.330 15:33.330 --> 15:36.130 Some number of meters per second, you are zooming to the 15:36.131 --> 15:37.711 right at some speed u. 15:37.710 --> 15:43.660 So, imagine now you are going past me. 15:43.659 --> 15:47.179 At some instant--I'm sitting here at the origin of 15:47.176 --> 15:51.186 coordinates, you cross me and then a little later you are 15:51.194 --> 15:54.154 somewhere there. So, that's your y frame 15:54.146 --> 15:56.136 axis; that's your origin. 15:56.139 --> 16:00.419 And the same event you say has a coordinate x prime. 16:00.420 --> 16:04.050 16:04.049 --> 16:07.589 We arrange it so that when you zoom past me, 16:07.594 --> 16:12.214 you set your clock to zero and I set my clock to zero. 16:12.210 --> 16:14.600 When you want to set the clock to zero is completely arbitrary. 16:14.600 --> 16:17.400 So, we will decide right when you pass me. 16:17.399 --> 16:19.379 I'm at the origin of my coordinates, you're at the 16:19.378 --> 16:20.548 origin of your coordinates. 16:20.549 --> 16:24.219 When you pass me, we'll click our stop watches 16:24.223 --> 16:27.083 and we will set the time to zero. 16:27.080 --> 16:28.920 So, here's an event. 16:28.920 --> 16:30.430 You and I crossed. 16:30.429 --> 16:32.589 What are the coordinates for that event? 16:32.590 --> 16:37.520 According to me, that event occurred at x 16:37.524 --> 16:41.834 = 0 and the time was chosen to be zero. 16:41.830 --> 16:45.280 According to you, your origin was also on top of 16:45.276 --> 16:49.746 my origin, so x prime was zero and the time is just the 16:49.750 --> 16:51.710 time. Everybody has a single time, 16:51.713 --> 16:53.113 and that time is called zero. 16:53.110 --> 16:55.080 That is one event. 16:55.080 --> 16:59.380 We made the coordinate of the event zero zero for both you and 16:59.377 --> 17:01.527 me. It is zero in space because my 17:01.529 --> 17:03.209 origin crossed your origin. 17:03.210 --> 17:05.780 That crossing took place at my origin; 17:05.779 --> 17:08.779 that's why my x is zero and took place at your origin, 17:08.783 --> 17:11.493 that's where your x prime is zero and the common 17:11.487 --> 17:13.637 time we chose to be zero by convention. 17:13.640 --> 17:17.650 Then, we want a second event. 17:17.650 --> 17:22.750 So, let's say the second event is some firecracker going off 17:22.754 --> 17:25.074 here. Here is something I should 17:25.068 --> 17:28.548 explain that I used to forget in the previous years. 17:28.549 --> 17:32.959 When I say I am moving, I imagine I am part of a huge 17:32.959 --> 17:36.689 team of people who are all moving with me. 17:36.690 --> 17:40.840 So, I've got agents all over the x axis who are my 17:40.835 --> 17:43.545 eyes and ears; they are looking out for me. 17:43.549 --> 17:46.309 So, even though I am here, if there's a firecracker 17:46.314 --> 17:48.474 exploding here, my guys will tell me. 17:48.470 --> 17:52.410 And you are carrying your own agents. 17:52.410 --> 17:55.550 Let's say at every point x you have a reporter, 17:55.547 --> 17:56.847 x = 1,2, 3,4; 17:56.849 --> 17:58.479 there are people sitting and watching. 17:58.480 --> 18:02.390 So, when I say I see something, I really mean me and my 18:02.390 --> 18:06.590 buddies, all traveling the same train at the same speed, 18:06.589 --> 18:09.659 all over space taking notes on what's happening. 18:09.660 --> 18:12.900 We'll pool our information later but we know this explosion 18:12.903 --> 18:16.363 took place here. I'll simplify it by saying I 18:16.363 --> 18:21.273 know an explosion took place here at location x at 18:21.270 --> 18:26.090 time t. So, this is our crossing. 18:26.090 --> 18:29.700 This event is when we crossed. 18:29.700 --> 18:32.410 Then, there is a firecracker. 18:32.410 --> 18:35.440 For the firecracker, I have to give some events 18:35.442 --> 18:37.752 [should have said "coordinates"]. 18:37.750 --> 18:40.700 I say it took place at location x at time t. 18:40.700 --> 18:44.420 18:44.420 --> 18:48.620 What do you say? You measure the distance from 18:48.618 --> 18:51.568 your origin, you call this some x prime, 18:51.567 --> 18:53.487 the time is still t. 18:53.490 --> 18:56.940 In Newtonian mechanics, the time is just the time. 18:56.940 --> 18:59.510 How many seconds have passed is the same for everybody. 18:59.509 --> 19:03.739 The question is, "What is the relation between 19:03.737 --> 19:06.647 x prime and x?" 19:06.650 --> 19:09.550 That's what we want to think about. 19:09.549 --> 19:12.569 So, you guys should think before I write down the answer. 19:12.569 --> 19:15.509 What's the relation of x prime to x? 19:15.509 --> 19:18.669 Well, this event took place at time t, 19:18.673 --> 19:22.563 so I know that your origin is to the right by an amount 19:22.556 --> 19:24.566 u times t. 19:24.570 --> 19:29.190 19:29.190 --> 19:32.480 So, the distance from your origin for this event, 19:32.484 --> 19:35.784 I maintain is x prime is x - ut. 19:35.780 --> 19:43.050 19:43.049 --> 19:45.109 Again, I want all of you to follow everything. 19:45.110 --> 19:47.020 These are all simple notions. 19:47.019 --> 19:52.009 Our origins coincided at zero time for the event that occurs 19:52.012 --> 19:53.622 at time t. 19:53.619 --> 19:56.199 Therefore, in a sense, you are rushing toward the 19:56.200 --> 19:57.740 event. You've gone a distance 19:57.736 --> 20:00.036 ut. Therefore, the distance from 20:00.044 --> 20:03.874 your origin to the event will be less than mine by this amount 20:03.865 --> 20:07.245 ut. This is the law of 20:07.250 --> 20:14.960 transformation of coordinates in Newtonian mechanics. 20:14.960 --> 20:19.110 If you have an event--if you want, formally, 20:19.108 --> 20:24.128 you can define it time, t prime for the primed 20:24.125 --> 20:26.565 observer. It goes without saying that 20:26.565 --> 20:28.405 t prime and t are the same. 20:28.410 --> 20:31.450 There's no notion of time for me and time for you. 20:31.450 --> 20:34.320 There's universal time in Newtonian mechanics. 20:34.320 --> 20:36.650 It just runs. We can call some time a zero. 20:36.650 --> 20:40.230 Once we have agreed, if you say you and I met at 20:40.234 --> 20:44.734 t = 0 and an explosion took place 5 seconds after our 20:44.734 --> 20:47.004 meeting, it's going to be 5 seconds 20:47.004 --> 20:50.134 after our meeting for me and it's going to be 5 seconds after 20:50.131 --> 20:51.331 the meeting for you. 20:51.329 --> 20:55.259 The time difference between two events is the same for all 20:55.259 --> 20:57.559 people. This is called the Galilean 20:57.562 --> 20:58.562 transformation. 20:58.560 --> 21:06.420 21:06.420 --> 21:09.750 What are the consequences of Galilean transformation? 21:09.750 --> 21:12.130 Well, let's look at the fact that x prime is x - 21:12.133 --> 21:12.513 ut. 21:12.510 --> 21:19.760 21:19.759 --> 21:21.789 Remember, everything is varying with time. 21:21.789 --> 21:24.879 So x prime is a function of time and x is a 21:24.879 --> 21:27.359 function of time, if you are watching a moving 21:27.362 --> 21:31.452 particle. Suppose this firecracker is not 21:31.451 --> 21:36.021 just one event, but it's a moving object. 21:36.019 --> 21:37.959 Let's give the object some speed; 21:37.960 --> 21:38.950 it's moving to the right. 21:38.950 --> 21:44.470 Then the velocity, according to me--I'm going to 21:44.466 --> 21:50.096 call v as dx/dt is the velocity. 21:50.099 --> 21:56.989 Let's just call it a bullet, according to S. 21:56.990 --> 22:02.720 Then w -- it's the standard name -- is the velocity 22:02.718 --> 22:07.138 of the bullet according to S prime. 22:07.140 --> 22:10.390 So, what I've done is, I first took one event and I 22:10.386 --> 22:14.216 gave it some coordinates and I told you how to transform the 22:14.218 --> 22:17.528 coordinate from one person to the other person. 22:17.529 --> 22:21.489 But now, take that point x not to be a fixed 22:21.492 --> 22:26.092 location but a moving object, so that as a function of time 22:26.088 --> 22:27.988 that body is moving. 22:27.990 --> 22:32.960 Then its velocity at any time is dx prime dt 22:32.962 --> 22:36.712 according to you; that is the dx/dt 22:36.708 --> 22:40.618 according to me, minus the derivative of this, 22:40.623 --> 22:42.453 which is u. 22:42.450 --> 22:46.230 Now, does that make sense? 22:46.230 --> 22:48.960 This should agree with common sense. 22:48.960 --> 22:52.290 For example, if that bullet is going at 600 22:52.292 --> 22:56.262 miles per hour to the right, that is 600 for me, 22:56.259 --> 23:00.509 and you are going to the right in your train at 200 miles per 23:00.512 --> 23:04.552 hour, you should measure the bullet speed to be reduced by 23:04.552 --> 23:06.752 200 and you should get 400. 23:06.750 --> 23:09.030 That's all it means. 23:09.029 --> 23:11.689 The two people will disagree on the velocity of the bullet 23:11.691 --> 23:13.981 because they are moving relative to each other. 23:13.980 --> 23:16.430 This is the way you will add velocities. 23:16.430 --> 23:20.770 But let's look at the acceleration. 23:20.769 --> 23:26.069 dw/dt is going to be dv/dt - 0 because 23:26.071 --> 23:28.671 u is a constant. 23:28.670 --> 23:33.240 23:33.240 --> 23:37.980 That means you and I agree on the acceleration of the body. 23:37.980 --> 23:40.370 We disagree on where it is. 23:40.369 --> 23:43.319 We disagree on how fast the bullet's moving. 23:43.319 --> 23:46.179 But we agree on the acceleration of the body because 23:46.177 --> 23:49.817 all I've done is add a constant velocity to everything you see. 23:49.819 --> 23:53.259 Therefore, if according to you the velocity of the body is not 23:53.259 --> 23:56.419 changing, according to me the velocity of the body is not 23:56.416 --> 23:59.056 changing, because the constant added will 23:59.062 --> 24:00.792 drop out of the difference. 24:00.789 --> 24:02.969 Or, if the body has an acceleration, 24:02.972 --> 24:06.342 we'll both get the same answer for the acceleration. 24:06.339 --> 24:08.009 So, that is the common acceleration a. 24:08.009 --> 24:10.729 So, if you like, a-prime is the same as 24:10.726 --> 24:11.386 a. 24:11.390 --> 24:15.000 24:15.000 --> 24:17.710 So, the acceleration of bodies doesn't change when you go from 24:17.713 --> 24:20.213 one frame of reference to another one going at a constant 24:20.205 --> 24:20.645 speed. 24:20.650 --> 24:23.700 24:23.700 --> 24:28.170 All right, so let's look at F = ma, 24:28.172 --> 24:34.612 which is md^(2)x/dt^(2) is equal to some force on the 24:34.608 --> 24:38.028 body. And you look at the body and 24:38.027 --> 24:43.337 you say d^(2)x prime over dt^(2) is the force on 24:43.338 --> 24:46.158 your body. First, I want to convince you 24:46.155 --> 24:49.055 we wanted to see that the left-hand sides are equal 24:49.062 --> 24:51.332 because the acceleration's the same. 24:51.329 --> 24:55.339 Then, I want to convince you that the right-hand sides are 24:55.340 --> 24:57.100 also going to be equal. 24:57.099 --> 25:01.389 I can take many examples but eventually you will get the 25:01.386 --> 25:04.396 point. Let us not consider one body, 25:04.402 --> 25:06.962 but let's consider two bodies. 25:06.960 --> 25:10.600 Two bodies are feeling a certain force due to, 25:10.602 --> 25:12.142 say, gravitation. 25:12.140 --> 25:15.190 And gravitation is, of course, a force in three 25:15.188 --> 25:19.228 dimensions but let's write the force in just one dimension. 25:19.230 --> 25:23.400 And let's say the force of gravity is equal to 1 over 25:23.395 --> 25:26.515 x_1 - x_2. 25:26.519 --> 25:31.169 Force on 1 due to 2 and the force on 2 due to 1 will be 25:31.166 --> 25:35.636 minus 1 over x_1 - x_2. 25:35.640 --> 25:39.760 The real forces are separation in three dimensions but this is 25:39.761 --> 25:41.181 a fictitious force. 25:41.180 --> 25:42.470 I want to call it gravity. 25:42.470 --> 25:45.810 It is any force that depends on the coordinates of the two 25:45.807 --> 25:47.727 particles. So, I will say 25:47.730 --> 25:51.480 m_1d^(2)x _1 over 25:51.479 --> 25:55.049 dt^(2) is 1 over x_1 - 25:55.054 --> 25:57.064 x_2. 25:57.059 --> 26:00.039 And m_2d^(2)x _2 over 26:00.041 --> 26:03.721 dt^(2) is minus 1 over x_1 minus--I 26:03.720 --> 26:06.320 have forgotten constants like g and 26:06.321 --> 26:09.621 m_1 and m_2. 26:09.619 --> 26:11.409 They don't matter for this purpose. 26:11.410 --> 26:13.420 So, here are two bodies. 26:13.420 --> 26:16.570 They feel a force for each other and I've discovered what 26:16.574 --> 26:18.604 the force is. It's 1 over x_1 - 26:18.601 --> 26:19.351 x_2. 26:19.349 --> 26:21.169 I don't care if it's 1 over x_1 - 26:21.170 --> 26:22.750 x_2 or (x_1 - 26:22.747 --> 26:25.227 x_2)^(2); that's not important. 26:25.230 --> 26:27.740 What's important is, it depends on x_1 26:27.736 --> 26:28.836 - x_2. 26:28.839 --> 26:32.669 You come along and you study the same two masses. 26:32.670 --> 26:34.540 What will you say is happening? 26:34.539 --> 26:37.279 You will say, m_1d^(2)x 26:37.275 --> 26:41.885 _1 prime over dt^(2) is equal to 1 over 26:41.887 --> 26:46.887 x_1 prime minus x_2 prime. 26:46.890 --> 26:48.540 Maybe I will--I'm sorry. 26:48.540 --> 26:50.070 Let me do it a little better. 26:50.069 --> 26:52.829 I can tell you what you will see. 26:52.829 --> 26:56.999 Given this is what I see, I can tell you what you will 26:56.998 --> 26:59.048 see. Let's do that in our head. 26:59.049 --> 27:02.659 We know that the acceleration is the same for any mass so I'm 27:02.663 --> 27:06.283 going to write this thing as m of dx over prime 27:06.276 --> 27:08.086 dt^(2). In other words, 27:08.094 --> 27:10.544 the acceleration according to me is the same as the 27:10.543 --> 27:12.163 acceleration according to you. 27:12.160 --> 27:14.970 Then, I'm also going to write the right-hand side as 27:14.967 --> 27:18.377 x_1 prime minus x_2 prime. 27:18.380 --> 27:20.230 Do you understand that? 27:20.230 --> 27:24.060 If there are two bodies feeling a force, if you see it from a 27:24.058 --> 27:26.678 moving train, the distance between the two 27:26.675 --> 27:29.095 bodies is the same for you and me, 27:29.099 --> 27:31.529 because x_1 prime is x_1 - 27:31.526 --> 27:34.076 ut and x_2 prime is x_2 - 27:34.084 --> 27:35.774 ut. Take the difference; 27:35.769 --> 27:38.339 the difference between the location of the particles is the 27:38.336 --> 27:39.306 same for you and me. 27:39.309 --> 27:42.599 Acceleration is the same, mass is postulated to be the 27:42.597 --> 27:46.317 same, so I know that you will get the same law that I get. 27:46.320 --> 27:50.590 You will get F = ma; your acceleration will be the 27:50.591 --> 27:52.621 same as mine; the force you attribute between 27:52.616 --> 27:54.166 the two bodies will also be the same. 27:54.170 --> 27:58.990 That is why I know that you will also deduce the same 27:58.991 --> 28:01.681 Newtonian laws that I will. 28:01.680 --> 28:04.740 You can also say it differently. 28:04.740 --> 28:09.020 If I woke up from my nap and I am now in a moving train and I 28:09.017 --> 28:13.147 examine the world around me, I'm going to get the same F 28:13.153 --> 28:16.203 = ma. Because as seen by a person on 28:16.196 --> 28:19.226 the ground, the masses obey F = ma. 28:19.230 --> 28:23.220 I am in this moving train now but I have the same acceleration 28:23.215 --> 28:25.955 for each mass and I have the same force. 28:25.960 --> 28:28.600 So, if you want, I'll complete the second 28:28.603 --> 28:30.703 equation. m_2d^(2) 28:30.703 --> 28:34.433 x_2 prime over dt^(2) will be minus 1 28:34.432 --> 28:38.162 over x_1 prime minus x_2 28:38.160 --> 28:41.120 prime. If this is a little difficult, 28:41.116 --> 28:43.126 we should talk about this. 28:43.130 --> 28:47.330 I'm telling you that if I deduce F = ma and the 28:47.327 --> 28:50.887 F depends on the separation between the 28:50.891 --> 28:53.911 particles, then I'm sure that you will 28:53.907 --> 28:58.077 find the same laws of motion because the acceleration is the 28:58.077 --> 29:02.457 same that I get because we have seen a-prime is the same 29:02.458 --> 29:05.228 as a. And the force will also be the 29:05.227 --> 29:08.097 same because the force depends on the separation between 29:08.100 --> 29:10.900 particles. And that doesn't depend on 29:10.899 --> 29:15.219 which train you're in or it's not affected by adding a 29:15.219 --> 29:19.049 constant velocity to the frame of reference. 29:19.049 --> 29:22.299 So, if you like, this is the way you prove in 29:22.303 --> 29:26.003 Newtonian mechanics the principle of relativity. 29:26.000 --> 29:29.590 So, not only is it something you observe by going on trains 29:29.587 --> 29:32.307 and what not, you can actually show that this 29:32.308 --> 29:35.028 is the reason everything looks the same. 29:35.029 --> 29:38.249 In other words, if the train was at rest on the 29:38.246 --> 29:42.366 platform and you and I were comparing notes and we both find 29:42.371 --> 29:45.081 F = ma, I go to sleep and I'm waking up 29:45.080 --> 29:47.140 and the train is going at a constant speed, 29:47.140 --> 29:49.910 if you can look through the window and look at the objects 29:49.911 --> 29:52.631 in my train, you will say they obey F = ma because 29:52.634 --> 29:54.194 nothing has happened to you. 29:54.190 --> 29:57.290 But you will predict I will also say F = ma because 29:57.291 --> 29:59.791 if you see an acceleration, I will see the same 29:59.794 --> 30:02.314 acceleration. If you see a distance between 30:02.313 --> 30:05.613 two masses to be one meter, I'll also think it's one meter. 30:05.609 --> 30:08.009 If the force is 1 over the square of the distance, 30:08.005 --> 30:10.105 we'll agree on the force, we'll agree on the 30:10.108 --> 30:12.258 acceleration, we'll agree on everything. 30:12.259 --> 30:15.029 And once you've proven F = ma is valid, 30:15.029 --> 30:17.919 it follows that every mechanical phenomenon will 30:17.922 --> 30:19.402 behave the same way. 30:19.400 --> 30:21.760 That's the reason things behave the same way. 30:21.760 --> 30:25.690 Yes? Student: If for some 30:25.693 --> 30:29.943 weird reason, suppose different frames of 30:29.936 --> 30:34.706 reference, the rule F = ma was to fail, 30:34.710 --> 30:37.150 what would happen? 30:37.150 --> 30:38.710 Professor Ramamurti Shankar: You mean if the 30:38.710 --> 30:39.740 rule failed in the other frame? 30:39.740 --> 30:40.740 Student: Hypothetically. 30:40.740 --> 30:41.280 Professor Ramamurti Shankar: Yes. 30:41.279 --> 30:42.899 Suppose, hypothetically, that happened. 30:42.900 --> 30:45.900 Then, it would mean that when you wake up in the train, 30:45.900 --> 30:48.680 you will look at the world around you, it will look 30:48.678 --> 30:50.788 different because F ≠ ma. 30:50.789 --> 30:52.879 You will conclude, when I went to sleep it was 30:52.877 --> 30:54.447 F = ma; when I got up, 30:54.449 --> 30:56.699 F ≠ ma, the train is moving. 30:56.700 --> 31:01.170 So, you will have to conclude that uniform velocity makes 31:01.172 --> 31:02.852 detectable changes. 31:02.849 --> 31:06.629 And if you look outside the train, to the other train, 31:06.625 --> 31:09.185 the other train's going backwards. 31:09.190 --> 31:12.000 You can now no longer say, "You're going the other way, 31:12.000 --> 31:14.340 I'm not moving", because the other person will 31:14.342 --> 31:16.582 say, "Hey, F = ma works for me." 31:16.580 --> 31:17.950 It doesn't work for you. 31:17.950 --> 31:19.500 So, you're the guy who's moving." 31:19.500 --> 31:23.080 So, you've lost the equal status with other inertial 31:23.076 --> 31:27.356 observers because those for whom F = ma worked will say 31:27.355 --> 31:30.595 they're not moving and for you, it doesn't work, 31:30.596 --> 31:32.756 so you will have to concede you are moving. 31:32.759 --> 31:35.439 So, uniform velocity, if it makes perceptible 31:35.441 --> 31:38.551 changes, can no longer be considered as relative. 31:38.549 --> 31:41.669 It's absolute and if you and I find each other moving, 31:41.673 --> 31:45.093 there may be a real sense in which I am at rest and you are 31:45.091 --> 31:47.121 moving, because for me F = ma 31:47.120 --> 31:48.600 works and for you it doesn't. 31:48.600 --> 31:50.800 Well, that's not what happens. 31:50.799 --> 31:52.989 In real life, you find it works for both of 31:52.985 --> 31:55.895 these and either of us can maintain we are not moving. 31:55.900 --> 32:01.850 So now, you've got to fast forward to about 300 years. 32:01.849 --> 32:05.509 This goes on, no problem with this principle 32:05.513 --> 32:08.583 of relativity and 300 years later, 32:08.579 --> 32:12.339 people have discovered electricity and magnetism and 32:12.341 --> 32:15.661 electromagnetism and electromagnetic waves, 32:15.660 --> 32:17.610 which they identify as light. 32:17.609 --> 32:22.489 And then, it was discovered that what you and I call light 32:22.487 --> 32:27.447 is just electric and magnetic fields traveling in space. 32:27.450 --> 32:29.680 You don't have to know what electro-magnetic fields are 32:29.683 --> 32:31.473 right now. They are some measurable 32:31.473 --> 32:33.683 phenomenon. They are like waves. 32:33.680 --> 32:37.700 And the waves have a certain velocity that Maxwell calculated 32:37.696 --> 32:41.846 and that velocity is this famous number 3 times 10 to the eight 32:41.847 --> 32:43.317 meters per second. 32:43.319 --> 32:46.939 And the question was, "For whom is this the 32:46.936 --> 32:48.846 velocity?" For example, 32:48.852 --> 32:52.162 you can do a calculation of waves on a string, 32:52.162 --> 32:55.622 something we'll be able to do in our course. 32:55.619 --> 32:58.819 Waves on a string will be some answer that depends on the 32:58.817 --> 33:02.187 tension on the string and the mass density of the string and 33:02.187 --> 33:05.837 that's the velocity as seen by a person for whom the string is at 33:05.841 --> 33:08.341 rest. But if you calculate the waves 33:08.341 --> 33:12.021 of sound in this room -- I talk to you, you hear me slightly 33:12.023 --> 33:15.773 later -- the time it takes to travel is the velocity of sound 33:15.766 --> 33:19.156 in this room. That is calculated with respect 33:19.164 --> 33:23.644 to the air in this room because the waves travel in the air. 33:23.640 --> 33:27.160 In fact, the fact that all of us are sitting on the planet, 33:27.156 --> 33:29.396 which itself is moving at whatever, 33:29.400 --> 33:31.600 1,100 miles per second, doesn't matter, 33:31.602 --> 33:34.622 because the air is being carried along so even if the 33:34.615 --> 33:37.745 Earth came to a sudden halt, as far as the velocity of sound 33:37.751 --> 33:39.571 in this room is concerned, it won't matter, 33:39.569 --> 33:41.559 because we are carrying the medium with us. 33:41.559 --> 33:46.889 So, people wanted to know what is the medium which carries the 33:46.886 --> 33:50.376 waves of light, electromagnetic waves? 33:50.380 --> 33:53.630 First of all, the medium is everywhere 33:53.625 --> 33:57.305 because--How do we know it's everywhere? 33:57.310 --> 33:58.920 Can anybody tell me? 33:58.920 --> 34:03.590 34:03.590 --> 34:05.240 Yes? Student: It travels 34:05.244 --> 34:06.224 through the vacuum of space. 34:06.220 --> 34:06.730 Professor Ramamurti Shankar: Right. 34:06.730 --> 34:07.940 It travels in the vacuum of space. 34:07.940 --> 34:09.830 We can see the Sun; we can see the stars, 34:09.831 --> 34:11.351 so we know the medium is everywhere. 34:11.349 --> 34:14.779 Then, you can sort of ask, "How dense is the medium?" 34:14.780 --> 34:18.100 It turns out that the denser the medium, the more rapidly 34:18.098 --> 34:20.708 signals travel, in most of the things that we 34:20.706 --> 34:22.116 know. For example, 34:22.122 --> 34:26.612 when we look at waves in sound and when we look at sound waves 34:26.610 --> 34:29.100 in a solid, or in iron, you find in a very 34:29.101 --> 34:31.441 dense material, that the velocity is very high. 34:31.440 --> 34:34.730 S,o this medium, which is called "ether," would 34:34.728 --> 34:38.018 have to be very, very dense to support waves of 34:38.016 --> 34:40.086 this incredible velocity. 34:40.090 --> 34:43.980 But then, planets have been moving through this medium for 34:43.979 --> 34:46.639 years and years and not slowing down. 34:46.640 --> 34:48.580 It's a very peculiar medium. 34:48.579 --> 34:52.189 But it has to be everywhere so we are all immersed in this 34:52.191 --> 34:55.991 medium because we are able to send light signals to different 34:55.992 --> 34:57.642 parts of the universe. 34:57.639 --> 35:02.019 And the question is, "How fast is the Earth moving 35:02.017 --> 35:04.427 relative to this medium?" 35:04.430 --> 35:09.280 You understand? This medium is all pervasive. 35:09.280 --> 35:12.230 We know that we can see the stars so it's going all the way 35:12.226 --> 35:13.696 up to the stars and beyond. 35:13.699 --> 35:17.239 And we are immersed in this and we are drifting around in space. 35:17.239 --> 35:19.969 What is our speed relative to the medium? 35:19.969 --> 35:21.639 That's the question that was asked. 35:21.639 --> 35:25.199 Well, to find the speed relative to the medium, 35:25.201 --> 35:30.081 you calculate the velocity in the medium by Maxwell's theory. 35:30.079 --> 35:33.939 So, here's the medium in which waves travel at a certain speed. 35:33.940 --> 35:37.410 This is planet Earth going around the Sun. 35:37.409 --> 35:40.169 At some instant, you will have a certain 35:40.172 --> 35:42.512 velocity with respect to ether. 35:42.510 --> 35:45.880 And therefore, the velocity of light as seen 35:45.884 --> 35:50.284 by you will be modified from c to c - V. 35:50.280 --> 35:54.500 In particular, suppose the waves are traveling 35:54.499 --> 35:56.749 to the right in ether. 35:56.750 --> 35:57.960 Let me draw it this way. 35:57.960 --> 36:00.470 The Earth is going at this instant at the speed V. 36:00.469 --> 36:04.439 We expect the speed to be c - V because part of the 36:04.441 --> 36:09.041 speed is neutralized because you are going along with the waves. 36:09.040 --> 36:10.580 You'll see a slower velocity. 36:10.580 --> 36:13.910 36:13.909 --> 36:16.979 So, Mr. Michelson and his assistant 36:16.980 --> 36:20.050 Morley--they did the experiment. 36:20.050 --> 36:25.720 And they got the answer equal to c. 36:25.720 --> 36:27.610 What does that mean? 36:27.610 --> 36:31.780 Student: The speed of light [inaudible] 36:31.782 --> 36:35.312 Professor Ramamurti Shankar: No, 36:35.305 --> 36:39.565 no, but you cannot jump to that right now. 36:39.570 --> 36:44.440 If you are following Newtonian physics, your expectation is, 36:44.439 --> 36:46.749 it should be c - V. 36:46.750 --> 36:50.570 Yes? Student: It means that 36:50.565 --> 36:52.615 there is no ether. 36:52.619 --> 36:53.799 Professor Ramamurti Shankar: Well, 36:53.804 --> 36:55.174 that's--not so fast, but it certainly means the 36:55.165 --> 36:56.845 following. Well, there's a simpler answer 36:56.853 --> 36:58.333 than that. Yes? 36:58.329 --> 37:00.819 Student: It means that the Earth is moving with respect 37:00.818 --> 37:01.958 to the ether. Professor Ramamurti 37:01.962 --> 37:02.642 Shankar: At what speed? 37:02.640 --> 37:03.860 Student: Zero. 37:03.860 --> 37:04.790 Professor Ramamurti Shankar: Zero. 37:04.789 --> 37:07.139 Because you don't have to--Look, you guys are ready to 37:07.139 --> 37:09.399 overthrow everything because you know the answer. 37:09.400 --> 37:14.420 But you've got to put yourself in the place of somebody in the 37:14.418 --> 37:16.918 early 1900s. There's no reason to overthrow 37:16.920 --> 37:18.400 anything. The answer is, 37:18.395 --> 37:20.735 you're going at the speed zero. 37:20.739 --> 37:24.719 Of course, you realize, that it is incredibly fortunate 37:24.720 --> 37:29.070 that on the one day Michelson wants to do the experiment, 37:29.070 --> 37:32.700 we happen to be at rest with respect to the ether. 37:32.700 --> 37:35.860 Fine. But we know that luck is not 37:35.860 --> 37:39.720 going to last forever because you are going around the Sun. 37:39.720 --> 37:42.860 On a particular day may be. 37:42.860 --> 37:46.160 But that velocity was such that on that day the Earth was at 37:46.156 --> 37:47.996 rest with respect to the ether. 37:48.000 --> 37:50.460 It's clear that six months later, when we are going the 37:50.460 --> 37:52.600 other direction, you cannot also be at rest with 37:52.601 --> 37:53.741 respect to the ether. 37:53.739 --> 37:57.729 But that's what you find when you do the experiment. 37:57.730 --> 38:01.360 You find every day you get the same answer and you jolly well 38:01.357 --> 38:02.987 know you are not at rest. 38:02.989 --> 38:04.609 You are moving around the Sun for sure. 38:04.610 --> 38:06.130 Yes? Student: Did they 38:06.130 --> 38:08.010 postulate a drag for the Earth turning [inaudible] 38:08.010 --> 38:09.660 Professor Ramamurti Shankar: Yes. 38:09.659 --> 38:11.619 So, people tried other solutions. 38:11.619 --> 38:14.949 But it is simply a fact that when you move one way or six 38:14.948 --> 38:18.278 months later in the opposite way, you get the same answer 38:18.277 --> 38:21.297 c. So, one possibility is, 38:21.302 --> 38:25.322 you don't want--Look, don't be ready to do 38:25.316 --> 38:28.446 revolutions, try to avoid it. 38:28.449 --> 38:31.499 So, one answer is, look at the speed of sound. 38:31.500 --> 38:35.380 You and I talked to each other then and we talked to each other 38:35.375 --> 38:38.235 six months from now; we get the same speed of sound. 38:38.239 --> 38:40.259 The speed of sound is published in textbooks, 38:40.262 --> 38:41.702 right? Seven hundred and something 38:41.703 --> 38:44.263 miles per hour. How come that doesn't change 38:44.262 --> 38:47.622 from day to day? Anybody here on this side can 38:47.624 --> 38:52.174 tell me why the speed of sound doesn't change from day to day 38:52.165 --> 38:54.355 even though we are moving? 38:54.360 --> 38:57.210 No one here can guess? 38:57.210 --> 39:00.170 Student: [inaudible] Professor Ramamurti 39:00.167 --> 39:02.157 Shankar: No, we are moving. 39:02.159 --> 39:04.739 Even six months from now, we get the same speed of sound 39:04.737 --> 39:06.627 in this room. When I talk to you, 39:06.631 --> 39:09.141 does it matter what time of year it is? 39:09.139 --> 39:12.749 Student: The medium is moving along with us? 39:12.750 --> 39:13.610 Professor Ramamurti Shankar: Yes, 39:13.613 --> 39:14.103 we are carrying air. 39:14.099 --> 39:16.709 As the Earth moves through space, it carries the air with 39:16.705 --> 39:19.585 you and the speed of the wave is with respect to the medium. 39:19.590 --> 39:22.130 If you can carry the medium with you, then it doesn't matter 39:22.125 --> 39:23.195 how fast you're moving. 39:23.200 --> 39:24.910 So, they tried that. 39:24.909 --> 39:28.989 They tried to argue that the Earth carries ether with it the 39:28.993 --> 39:31.003 way it carries air with it. 39:31.000 --> 39:34.820 Then, it's not an accident you are at rest with respect to the 39:34.819 --> 39:37.449 ether because you're taking it with you. 39:37.449 --> 39:40.389 But it's very easy to show by looking at distant stars that 39:40.390 --> 39:41.760 you cannot be doing that. 39:41.760 --> 39:44.190 I don't have time to tell you why that is true. 39:44.190 --> 39:51.090 So, you cannot take the ether with you and you cannot leave it 39:51.090 --> 39:56.520 behind, and that's the impasse people were in. 39:56.519 --> 40:00.989 So, it's as if there's a car that's going to the right at a 40:00.992 --> 40:02.922 certain speed c. 40:02.920 --> 40:06.530 You move to the right at some speed, maybe c/2. 40:06.530 --> 40:09.180 I expect you to get a speed c/2. 40:09.180 --> 40:11.520 But you keep getting c. 40:11.519 --> 40:14.349 You go three fourths of the velocity of light; 40:14.349 --> 40:15.949 you still get the velocity of light. 40:15.949 --> 40:20.119 That is very contrary to what we believe. 40:20.119 --> 40:23.509 In fact, that's in violent opposition to this law here. 40:23.510 --> 40:26.810 If this V were not a bullet but a light beam, 40:26.811 --> 40:30.311 suppose for me traveling at a speed c and you're 40:30.306 --> 40:33.216 traveling to the right at speed u, 40:33.220 --> 40:35.190 you should get c - u. 40:35.190 --> 40:37.970 That's the inevitable consequence of Newtonian 40:37.971 --> 40:40.381 physics. And you don't get that. 40:40.380 --> 40:43.220 And that was a big problem. 40:43.219 --> 40:47.619 So, people tried to fix it up by doing different models of 40:47.622 --> 40:49.942 ether, none of which worked. 40:49.940 --> 40:54.120 And nobody knew why light is behaving in this peculiar 40:54.119 --> 40:58.219 fashion, so that's when Einstein came in and said, 40:58.219 --> 41:00.939 "I know why light is behaving in this peculiar fashion. 41:00.940 --> 41:04.740 It is behaving in this way because if it didn't behave in 41:04.736 --> 41:09.206 this way, if the speed of light depended on how fast I'm moving, 41:09.210 --> 41:13.130 then when I wake up in this train, all I have to do is 41:13.133 --> 41:15.283 measure the speed of light. 41:15.280 --> 41:19.670 Originally, I got some number; now I get to get a different 41:19.666 --> 41:22.896 number and the difference will give me the speed of the train. 41:22.900 --> 41:26.210 So, it would have been possible to detect the velocity of the 41:26.209 --> 41:29.349 train without looking outside, just by doing an experiment 41:29.352 --> 41:32.532 with light. So, even though mechanical laws 41:32.526 --> 41:37.066 involving F = ma are the same, laws of electricity and 41:37.069 --> 41:41.459 magnetism would be such that somehow they would betray your 41:41.460 --> 41:44.330 velocity. And that would mean uniform 41:44.330 --> 41:48.420 velocity does make an observable change because it changes the 41:48.420 --> 41:51.370 velocity of light that you would measure. 41:51.369 --> 41:54.509 " But conversely, the fact that you keep getting 41:54.507 --> 41:58.307 the same answer means that electric and magnetic phenomena 41:58.312 --> 42:01.852 are part of the conspiracy to hide your velocity. 42:01.849 --> 42:05.249 Just like mechanical phenomena won't tell you how fast you're 42:05.251 --> 42:08.031 moving, neither will electromagnetic phenomena. 42:08.030 --> 42:11.100 Because to Einstein, it was very obvious that nature 42:11.099 --> 42:14.529 would not design a system in which mechanical laws are the 42:14.530 --> 42:17.360 same but laws of electricity are different. 42:17.360 --> 42:21.740 So, he postulated that all phenomena, whatever be their 42:21.740 --> 42:26.360 nature, will be unaffected by going to a frame at constant 42:26.364 --> 42:29.694 velocity relative to the initial one. 42:29.690 --> 42:33.200 That's a very brave postulate because it even applies to 42:33.196 --> 42:36.956 biological phenomena about which I'm sure Einstein knew very 42:36.958 --> 42:39.528 little. But he believed that natural 42:39.529 --> 42:42.799 phenomena will just follow either the principle of 42:42.800 --> 42:44.670 relativity or they won't. 42:44.670 --> 42:50.640 And that is something you should think about. 42:50.639 --> 42:52.489 Because that was the only reason he had. 42:52.489 --> 42:54.939 He just said, "I don't believe chapters 1 42:54.939 --> 42:58.189 through 10 in our book obey relativity and chapters 20 42:58.185 --> 43:00.875 through 30 where we do E&M doesn't." 43:00.880 --> 43:04.040 These are all natural phenomena that will obey the same 43:04.044 --> 43:07.034 principle, which says all observers that are uniform 43:07.033 --> 43:09.263 relative to motion are equivalent. 43:09.260 --> 43:12.360 Now, that's really based on a lot of faith and even though 43:12.355 --> 43:15.445 scientists generally are opposed to intelligent design, 43:15.449 --> 43:20.289 we all have some bias about the way natural laws were designed; 43:20.290 --> 43:21.560 there's no question about it. 43:21.559 --> 43:23.349 You can talk to any practicing physicist. 43:23.349 --> 43:26.629 We have a faith that underlying laws of nature will have a 43:26.625 --> 43:30.125 certain elegance and a certain beauty and a certain uniformity 43:30.131 --> 43:32.201 across all of natural phenomena. 43:32.200 --> 43:34.750 That is a faith that we have. 43:34.750 --> 43:37.790 It's not a religious issue; otherwise, I wouldn't bring it 43:37.789 --> 43:40.239 up in the classroom, but it is certainly the credo 43:40.239 --> 43:43.109 of all scientists, at least all physicists, 43:43.109 --> 43:47.739 that there is some elegance in the laws of nature and we put a 43:47.735 --> 43:51.335 lot of money on that faith, that the laws of nature will do 43:51.335 --> 43:52.485 this and will not do this. 43:52.490 --> 43:54.040 Who are we to say that? 43:54.039 --> 43:57.029 Who are we to say nature wouldn't have a system in which 43:57.032 --> 43:59.922 mechanics obeys the laws but electricity and magnetism 43:59.916 --> 44:02.316 doesn't? We haven't run into somebody 44:02.315 --> 44:04.695 called nature. We don't worship a certain 44:04.704 --> 44:07.894 deity called nature but we believe the laws of nature obey 44:07.892 --> 44:10.472 that. So, even though scientists or 44:10.470 --> 44:14.580 physicists in particular may not believe in design by any 44:14.584 --> 44:17.234 personal God, they do believe in this 44:17.232 --> 44:21.012 underlying, rational system that we are trying to uncover. 44:21.010 --> 44:24.000 You could be disproven, you could be wrong in making 44:23.997 --> 44:26.337 the assumption, but here it was right. 44:26.340 --> 44:30.350 It was really driven by this notion that all laws of physics 44:30.353 --> 44:33.553 should obey the same principle of relativity. 44:33.550 --> 44:37.790 So, Einstein's postulates are that light behaves in this way 44:37.790 --> 44:40.810 because if it didn't behave in this way, 44:40.809 --> 44:44.139 it would violate the principle of relativity, 44:44.141 --> 44:48.151 whereas we know mechanical phenomena do and electrical 44:48.153 --> 44:52.093 phenomena would not and that cannot be the case. 44:52.090 --> 44:53.140 You have a question? 44:53.139 --> 44:58.999 Student: Why would that not apply to the speed of sound? 44:59.000 --> 44:59.840 Professor Ramamurti Shankar: Yes. 44:59.840 --> 45:02.590 Because in the case of the speed of sound, 45:02.585 --> 45:04.925 you can take the medium with you; 45:04.929 --> 45:07.239 there is no such experiment you could perform. 45:07.239 --> 45:09.639 See, in the train, if you could carry the ether 45:09.642 --> 45:12.882 with you, there's no surprise you would get the same answer. 45:12.880 --> 45:15.120 But we know we cannot carry the medium with us, 45:15.116 --> 45:17.446 that comes from extra-terrestrial experiments. 45:17.449 --> 45:20.699 That's why the velocity of sound is not elevated to a 45:20.698 --> 45:24.008 fundamental velocity on which everybody will agree. 45:24.010 --> 45:26.550 So, the two great postulates--You've got to know 45:26.547 --> 45:28.327 where the postulates come from. 45:28.329 --> 45:33.959 Postulate Number 1 is simply a restatement of the relativity 45:33.955 --> 45:36.405 principle. I'll just say it in one 45:36.410 --> 45:39.420 sentence. Exact wording is not important. 45:39.420 --> 45:44.070 All inertial observers are equivalent. 45:44.070 --> 45:49.010 45:49.010 --> 45:53.380 "Equivalent" means each one of them is as privileged as any 45:53.380 --> 45:56.620 other one to discover the laws of nature. 45:56.619 --> 46:00.969 The laws of nature, we found, are not an accident 46:00.969 --> 46:03.959 related to our state of motion. 46:03.960 --> 46:06.580 If I find some laws, and you're moving relative to 46:06.579 --> 46:08.289 me, you'll find the same laws. 46:08.289 --> 46:10.699 And if you and I find each other in relative motion, 46:10.704 --> 46:13.644 you have as much right to claim you are at rest and I am moving 46:13.639 --> 46:16.389 and I have as much right to claim that I am at rest and you 46:16.385 --> 46:18.935 are moving. There is complete symmetry 46:18.941 --> 46:22.041 between observers in uniform relative motion. 46:22.039 --> 46:25.199 There is no symmetry between people in non-uniform [relative] 46:25.196 --> 46:27.146 motion. As I said, non-uniform motion 46:27.150 --> 46:30.110 creates effects which can destroy me and not destroy you. 46:30.110 --> 46:32.880 So, no one's trying to talk their way out of acceleration, 46:32.879 --> 46:35.599 whereas you can talk your way out of uniform velocity. 46:35.600 --> 46:37.170 That's the first principle. 46:37.170 --> 46:39.680 So this was there even from the time of Newton. 46:39.679 --> 46:43.759 What is true here is that all inertial observers are 46:43.756 --> 46:47.586 equivalent with respect to all natural phenomena, 46:47.592 --> 46:49.912 meaning all natural laws. 46:49.910 --> 46:53.130 46:53.130 --> 46:55.480 And that is a generalization, when we say "all" instead of 46:55.478 --> 46:56.218 just mechanical. 46:56.220 --> 47:00.360 47:00.360 --> 47:04.290 And the Second Postulate, you call it a postulate because 47:04.291 --> 47:06.961 there is just no way to deduce this, 47:06.960 --> 47:18.820 is that the velocity of light is independent of the state of 47:18.824 --> 47:24.604 motion of the source, of the observer, of everything. 47:24.600 --> 47:30.110 47:30.110 --> 47:32.700 If a light beam is emitted by a moving rocket, 47:32.699 --> 47:33.849 it doesn't matter. 47:33.849 --> 47:36.039 If a light beam is seen in a moving rocket, 47:36.044 --> 47:37.094 it doesn't matter. 47:37.090 --> 47:40.600 All people will get the same answer for the velocity of 47:40.599 --> 47:42.539 light. Student: Is there a 47:42.540 --> 47:44.890 reason why the speed of light is constant? 47:44.889 --> 47:45.439 Professor Ramamurti Shankar: No. 47:45.440 --> 47:47.060 That's why it is a postulate. 47:47.059 --> 47:49.179 You can show a few things later on. 47:49.179 --> 47:51.459 You can show that if there is any other speed, 47:51.463 --> 47:54.153 which is the same for everybody, that would have to be 47:54.152 --> 47:55.322 the speed of light. 47:55.320 --> 47:58.830 In the final theory of relativity, there are not two or 47:58.834 --> 48:02.484 three velocities that come out the same for everybody. 48:02.480 --> 48:05.840 There is only one velocity that can have the same answer for all 48:05.840 --> 48:08.200 people. That velocity is the velocity 48:08.201 --> 48:10.601 of light. By that I don't mean it has to 48:10.598 --> 48:12.158 be light itself. For example, 48:12.155 --> 48:14.505 gravitational waves travel at the speed of light. 48:14.510 --> 48:16.540 It's not just the light. 48:16.539 --> 48:19.929 It has to do with the velocity of light being a single number 48:19.925 --> 48:22.685 which has to have the same value for everybody. 48:22.690 --> 48:27.480 Okay, so it looks like he has solved a big problem because he 48:27.477 --> 48:30.507 has said why light behaves this way; 48:30.510 --> 48:33.860 light behaves this way because it is part of the big conspiracy 48:33.859 --> 48:35.209 to hide uniform motion. 48:35.210 --> 48:41.380 But you will see that you have made a terrible bargain because 48:41.384 --> 48:45.134 once you take these two postulates, 48:45.130 --> 48:48.210 you have restored the relativity principle to all 48:48.210 --> 48:50.050 phenomena. Okay. 48:50.050 --> 48:52.950 You've gone beyond mechanical phenomena to electro-magnetic 48:52.949 --> 48:55.649 phenomena. But you will find that you have 48:55.648 --> 48:59.308 to give up all the other cherished notions of Newtonian 48:59.305 --> 49:02.055 physics. Think about why. 49:02.059 --> 49:05.759 We are saying, here is a car going at 200 49:05.758 --> 49:08.808 miles an hour, according to me. 49:08.809 --> 49:12.519 You get into your own car and follow that car at 50 miles an 49:12.519 --> 49:15.689 hour. You should get 150 but you keep 49:15.693 --> 49:18.453 getting 200. This may not be true for cars 49:18.445 --> 49:21.925 going at the speeds I mentioned but when finally you are talking 49:21.934 --> 49:24.154 about a pulse of light, that is true. 49:24.150 --> 49:28.210 And you've got to agree that is really not compatible with our 49:28.205 --> 49:31.255 daily notions or with the formula I wrote down, 49:31.263 --> 49:32.663 w = v - u. 49:32.659 --> 49:36.589 When you put v = c, w has got to come out to 49:36.585 --> 49:39.045 be c. And that's not a property of 49:39.046 --> 49:40.646 the Newtonian transformation. 49:40.650 --> 49:44.530 So, what we are looking for is a new rule for connecting 49:44.528 --> 49:48.188 x and t and x prime and t 49:48.194 --> 49:50.924 prime, such that when the velocities 49:50.919 --> 49:54.469 are computed and applied to the velocity of light, 49:54.470 --> 49:55.840 you get the same answer. 49:55.840 --> 49:57.950 That's what we want to do now. 49:57.949 --> 50:01.009 So, here is how we are going to do this. 50:01.010 --> 50:03.360 Now, let's think about it. 50:03.360 --> 50:06.410 Let me send a pulse to the right at speed c. 50:06.409 --> 50:11.209 You are going to the right at three-fourths of c. 50:11.210 --> 50:14.740 My Newtonian expectation is, you should get the speed of the 50:14.737 --> 50:16.947 pulse to be one-fourth of c. 50:16.950 --> 50:19.020 But you insist it is c. 50:19.020 --> 50:20.760 So, what will I say to you? 50:20.760 --> 50:22.350 What will I accuse you of doing? 50:22.349 --> 50:26.389 I say you should get c/4 and you're getting c. 50:26.389 --> 50:29.869 And you're finding velocity by finding the distance it travels 50:29.874 --> 50:33.244 and dividing by the time so you are jacking up a number like 50:33.243 --> 50:35.703 one-fourth c to c itself. 50:35.700 --> 50:37.660 So, what could make you do that? 50:37.660 --> 50:41.440 Yep? Student: If I perceive 50:41.439 --> 50:44.959 that you're moving forward, or you're contracting your 50:44.964 --> 50:48.424 length, then you're going to measure 50:48.422 --> 50:54.402 velocity keeping in mind you should have had a greater length 50:54.404 --> 50:56.404 to begin with. Professor Ramamurti 50:56.402 --> 50:56.652 Shankar: Yes. 50:56.650 --> 50:58.930 So, one option--Let me repeat what he said. 50:58.929 --> 51:03.609 I will say your meter sticks are being somehow shorter. 51:03.610 --> 51:06.330 When you and I were buddies and were on the same train, 51:06.329 --> 51:07.889 we agreed on the meter stick. 51:07.889 --> 51:09.509 But now we have gone on the moving train. 51:09.510 --> 51:12.620 I will say there is something wrong with your meter stick. 51:12.620 --> 51:13.690 Not only something wrong. 51:13.690 --> 51:15.180 Specifically, I would say your meter sticks 51:15.182 --> 51:16.262 have shrunk. For example, 51:16.262 --> 51:18.402 if they have shrunk to one-fourth their size, 51:18.396 --> 51:21.256 it is very clear that you would get a velocity of four times 51:21.258 --> 51:25.678 what I expect. But there's another possibility. 51:25.679 --> 51:27.829 Student: Time may be running slow. 51:27.829 --> 51:29.169 Professor Ramamurti Shankar: Your clocks may be 51:29.170 --> 51:30.830 running slow. So, you let the light travel 51:30.828 --> 51:33.428 for four seconds and you thought it was only one second. 51:33.429 --> 51:35.319 That's why you got four times the answer; 51:35.320 --> 51:36.500 or it could be both. 51:36.500 --> 51:38.820 But something has to give. 51:38.820 --> 51:43.050 And that is why it is an amazing theory. 51:43.050 --> 51:46.840 That's why it is also amazing to me that somebody who was 26 51:46.839 --> 51:50.629 years old would simply follow the consequence of this theory 51:50.628 --> 51:54.098 and take it wherever it takes you but it is at the very 51:54.096 --> 51:57.946 foundations of space and time that you have to modify. 51:57.949 --> 52:00.449 So, even though you restored the relativity principle and 52:00.447 --> 52:03.577 brought it back to the front, the price you have to pay is to 52:03.582 --> 52:07.032 give up your notions that length is length and time is time. 52:07.030 --> 52:09.450 We used to think a meter stick is a meter stick and a clock is 52:09.452 --> 52:11.322 a clock. If I have a clock that ticks 52:11.317 --> 52:13.307 out one second, you take it on a train, 52:13.308 --> 52:16.348 I expect it to be ticking one second, but we're saying it's 52:16.348 --> 52:18.698 not. So, something has to give in 52:18.695 --> 52:21.905 length measurement or time measurement or both. 52:21.909 --> 52:24.449 And that's what we're going to find out. 52:24.449 --> 52:26.659 So, here's how you find that out. 52:26.659 --> 52:33.309 Let us say that--Maybe I'll do it all on one blackboard because 52:33.314 --> 52:37.934 this is the key to the whole calculation. 52:37.929 --> 52:42.379 You remember now, if there is an event here, 52:42.379 --> 52:47.759 you call it x prime and I call it x. 52:47.760 --> 52:51.540 52:51.539 --> 52:53.589 And according to me, you have traveled the distance 52:53.588 --> 52:54.078 ut. 52:54.080 --> 52:59.910 52:59.909 --> 53:08.809 So, x prime equals x - ut is what we used to say 53:08.812 --> 53:14.942 in the old days. And the converse of that is 53:14.940 --> 53:19.090 x = x prime + ut. 53:19.090 --> 53:21.580 But now, we'll admit the fact that maybe t and t 53:21.576 --> 53:22.836 prime are no longer the same. 53:22.840 --> 53:26.660 53:26.660 --> 53:29.190 But that's not all we will do. 53:29.190 --> 53:32.720 I will say, whenever you give me a length x prime, 53:32.715 --> 53:34.095 I just don't buy it. 53:34.099 --> 53:39.019 I take any length you give me and I jack it up by a factor of 53:39.020 --> 53:43.200 γ [correction: should have said 1/γ] 53:43.203 --> 53:46.323 to get the length according to me. 53:46.320 --> 53:50.190 And you will take any answer I give you and you will multiply 53:50.191 --> 53:51.741 it by the same factor. 53:51.739 --> 53:54.969 Don't worry about this yet!] In other words, 53:54.965 --> 53:57.585 we don't buy our units of length, 53:57.590 --> 54:01.360 so if you say it should be x prime + ut 54:01.357 --> 54:04.687 prime, that's your formula backwards for me. 54:04.690 --> 54:06.910 The coordinate of the event, according to you, 54:06.911 --> 54:09.381 in the old days, was x prime + ut. 54:09.380 --> 54:12.260 Now, we admit that t and t prime may not be the 54:12.263 --> 54:13.393 same. Then we say, 54:13.388 --> 54:17.068 but I will not take your lengths, I will multiply them by 54:17.066 --> 54:20.216 γ to get the lengths according to me. 54:20.219 --> 54:22.259 And you will not take my expectations, 54:22.259 --> 54:24.959 but you will multiply it by the same γ; 54:24.960 --> 54:28.460 that is the symmetry between the two observers. 54:28.460 --> 54:30.050 In other words, if I think your meter sticks 54:30.053 --> 54:31.983 are short and γ [correction: should have said 54:31.980 --> 54:33.500 1/γ] is a number less than 1, 54:33.500 --> 54:36.910 I'm allowing you to accuse me of having meter sticks which are 54:36.912 --> 54:38.532 short. It is very interesting. 54:38.530 --> 54:41.480 If I said your meter sticks are short and you say my meter 54:41.478 --> 54:44.788 sticks are longer than average, that's an absolute difference. 54:44.789 --> 54:48.249 But we both accuse each other of using shortened meter sticks, 54:48.248 --> 54:50.798 and so we use the same factor of γ. 54:50.800 --> 54:54.360 We are going to find this γ now. 54:54.360 --> 54:59.230 Student: How do you know that both the distance and time 54:59.232 --> 55:01.622 are different? Professor Ramamurti 55:01.616 --> 55:04.426 Shankar: We'll give it the possibility that they are 55:04.431 --> 55:07.401 different and then we will see that they are different. 55:07.400 --> 55:09.740 We know something's wrong with space and something's wrong with 55:09.736 --> 55:11.056 time. So, we'll not assume that 55:11.056 --> 55:12.616 t prime is equal to t. 55:12.619 --> 55:14.359 You have to open up the possibility. 55:14.360 --> 55:17.580 In the end, it may be that the nature will say t prime 55:17.579 --> 55:20.369 is t and something happens to length alone. 55:20.369 --> 55:22.959 But we'll find the answer is more symmetric. 55:22.960 --> 55:31.590 Student: Why do we say that the symmetry is the same? 55:31.590 --> 55:33.510 Professor Ramamurti Shankar: Because that is the 55:33.512 --> 55:34.832 symmetry between the two observers. 55:34.829 --> 55:39.059 If I want to say your meter sticks are short, 55:39.063 --> 55:41.953 why should you concede that? 55:41.949 --> 55:43.979 You should be able to accuse me of saying--The only difference 55:43.976 --> 55:45.836 between you and me is you are moving to the right and I'm 55:45.836 --> 55:46.596 moving to the left. 55:46.599 --> 55:49.849 Other than the sign of the velocity, each person says the 55:49.850 --> 55:53.330 other person is moving and so we will say that any length you 55:53.333 --> 55:56.123 give I'll discount by a factor of γ. 55:56.119 --> 55:57.739 By symmetry, anything I call a length, 55:57.737 --> 56:00.007 you will discount by the same factor of γ. 56:00.010 --> 56:02.640 Now, let's apply this. 56:02.639 --> 56:04.389 (x,t) was a certain event, 56:04.390 --> 56:06.480 right? Let's apply it to the following 56:06.479 --> 56:07.979 event. You have to follow this very 56:07.975 --> 56:10.005 carefully. When you and I crossed, 56:10.014 --> 56:13.224 remember that was at the origin of coordinates, 56:13.224 --> 56:16.994 x equals to t equals to zero and x 56:16.991 --> 56:20.551 prime equals to t prime equals to zero. 56:20.550 --> 56:23.230 At that instant when our origins touched, 56:23.231 --> 56:25.311 let us emit a flash of light. 56:25.309 --> 56:29.279 Okay, maybe when the origins touched, there's a spark, 56:29.281 --> 56:33.551 the light signal goes out and the light signal is detected 56:33.552 --> 56:35.802 here, by a light detector. 56:35.800 --> 56:41.770 That second event, detection of the pulse, 56:41.765 --> 56:49.035 it has a coordinates (x,t) for me; 56:49.039 --> 56:54.429 it has a coordinates (x prime, t prime) for you. 56:54.429 --> 56:58.049 The same event is given different coordinates. 56:58.050 --> 57:00.370 We've already used the fact that coordinates will be 57:00.365 --> 57:03.175 different but we are saying not only will x prime not be 57:03.181 --> 57:05.321 equal to x; t prime may not be equal 57:05.316 --> 57:06.046 to t either. 57:06.050 --> 57:10.650 57:10.650 --> 57:15.360 Okay, but now let's write down one important condition. 57:15.360 --> 57:17.540 What is the relation between x prime and t 57:17.542 --> 57:19.212 prime in this particular pair of events? 57:19.210 --> 57:24.430 57:24.430 --> 57:25.460 What is x prime? 57:25.460 --> 57:28.870 It is the location of the light pulse after certain time? 57:28.869 --> 57:32.899 So, what's the relation of the location of the detection of the 57:32.900 --> 57:35.630 light pulse and the time t prime? 57:35.630 --> 57:41.790 Yes? Student: [inaudible] 57:41.787 --> 57:45.437 Professor Ramamurti Shankar: He is saying 57:45.438 --> 57:49.088 x prime--Do you guys understand that? 57:49.090 --> 57:50.790 Do you agree with that statement? 57:50.790 --> 57:52.430 This is not a random event. 57:52.429 --> 57:55.659 The second event was the detection of a light pulse. 57:55.659 --> 57:59.269 Light pulse left the origin t prime seconds earlier 57:59.272 --> 58:02.342 and has come to this point, according to this guy, 58:02.337 --> 58:05.627 and the ratio of the distance to the time is the velocity of 58:05.628 --> 58:08.308 light. But it is also true that for 58:08.306 --> 58:12.966 me, the light went to distance x in a time t. 58:12.969 --> 58:17.379 So, that's the relation of x to t for me 58:17.377 --> 58:20.427 also. I'm going to use those two 58:20.427 --> 58:26.217 results and combine it with this to find this factor γ 58:26.217 --> 58:28.777 and we will do that now. 58:28.780 --> 58:32.760 I want you to multiply the left-hand side by the left-hand 58:32.756 --> 58:36.796 side and the right-hand side by the right-hand side of that 58:36.803 --> 58:39.773 equation. I hope you understand that in 58:39.770 --> 58:43.250 the Galilean days, in the old days--Let's see what 58:43.252 --> 58:45.712 you will say. I will say, x prime is 58:45.706 --> 58:48.016 x - ut because the origins have shifted by an 58:48.021 --> 58:48.931 amount ut. 58:48.929 --> 58:51.849 And you will say x is x prime plus ut 58:51.853 --> 58:52.933 with the same time. 58:52.929 --> 58:54.489 Now, I'm saying time is different. 58:54.489 --> 58:56.639 Not only that, I don't buy your length. 58:56.639 --> 58:59.909 If you expect me to have this length, I say "no." 58:59.910 --> 59:01.210 You exaggerated everything. 59:01.210 --> 59:03.970 I'll scale it down by γ and vice versa. 59:03.969 --> 59:07.119 So, if you multiply the left-hand side by the left-hand 59:07.115 --> 59:11.725 side, you get xx prime, the right-hand side you get 59:11.726 --> 59:17.416 γ^(2) times xx prime plus uxt prime 59:17.424 --> 59:23.634 minus ux prime t minus u^(2)tt prime. 59:23.630 --> 59:30.770 59:30.769 --> 59:35.629 Now, divide everything by xx prime. 59:35.630 --> 59:41.300 Then, you get 1 = γ^(2) times 1 + u. 59:41.300 --> 59:44.340 If you divide this by xx prime, you'll get t prime 59:44.338 --> 59:45.398 over x prime. 59:45.400 --> 59:51.010 If you divide this by xx prime, you'll get t/x. 59:51.010 --> 59:54.010 And here you get u^(2) times t/x, 59:54.012 --> 59:56.822 times t prime over x prime. 59:56.820 --> 1:00:07.320 1:00:07.320 --> 1:00:08.850 So, what does that mean? 1:00:08.849 --> 1:00:13.349 Well, t prime over x prime is 1/c and 1:00:13.350 --> 1:00:17.540 t/x is also 1/c because of what I 1:00:17.535 --> 1:00:19.305 wrote here. If you want, 1:00:19.309 --> 1:00:22.309 let me write this as t prime over x prime equals 1:00:22.305 --> 1:00:25.395 the c and t over x equals the c. 1:00:25.400 --> 1:00:26.970 So, they will cancel. 1:00:26.969 --> 1:00:31.569 And I'll get 1 = γ^(2) times 1 - u^(2) over 1:00:31.569 --> 1:00:36.009 c^(2) because this t/x is a 1/c and 1:00:36.007 --> 1:00:40.927 t prime over x prime is another 1/c. 1:00:40.930 --> 1:00:44.950 1:00:44.949 --> 1:00:50.779 So, that gives us the result that γ is 1 over square 1:00:50.776 --> 1:00:55.216 root of 1 - u^(2) over c^(2). 1:00:55.220 --> 1:01:00.890 1:01:00.889 --> 1:01:05.659 If you plug the γ back in, you will find x prime 1:01:05.660 --> 1:01:08.970 is x - ut divided by [square root of] 1:01:08.969 --> 1:01:11.969 1 - u^(2) over c^(2). 1:01:11.970 --> 1:01:22.320 1:01:22.320 --> 1:01:24.760 Now, once you do that, once you got the relation 1:01:24.758 --> 1:01:27.508 between x prime and x, you can go to the 1:01:27.509 --> 1:01:29.999 lower equation and solve for t prime. 1:01:30.000 --> 1:01:32.520 I don't feel like doing that. 1:01:32.519 --> 1:01:34.929 It's a simple algebraic equation once you've got 1:01:34.928 --> 1:01:37.438 x prime how to solve for t prime. 1:01:37.440 --> 1:01:40.130 Take the second of the two equations and solve for t 1:01:40.129 --> 1:01:42.829 prime. That detail I won't fill out, 1:01:42.832 --> 1:01:47.022 but you will get t prime is t - ux over 1:01:47.015 --> 1:01:49.585 c^(2) divided by this. 1:01:49.590 --> 1:01:52.890 So, I've not done every step but I've given you all the 1:01:52.889 --> 1:01:55.149 things you need to do the one step. 1:01:55.150 --> 1:01:57.650 There are equations up there that relate x and 1:01:57.650 --> 1:02:00.630 x prime to t and t prime so if you can get 1:02:00.631 --> 1:02:03.181 x prime in terms of x and t, 1:02:03.179 --> 1:02:05.409 the other equation that you solve will give you t 1:02:05.414 --> 1:02:07.124 prime in terms of x and t. 1:02:07.120 --> 1:02:08.540 And this is the result. 1:02:08.540 --> 1:02:12.420 1:02:12.420 --> 1:02:15.950 Okay. This guy deserves two boxes 1:02:15.952 --> 1:02:21.002 because it is the greatest result from relativity; 1:02:21.000 --> 1:02:23.440 it's called the Lorentz transformation. 1:02:23.440 --> 1:02:28.470 And we've been able to derive the Lorentz transformation with 1:02:28.474 --> 1:02:30.324 what little we know. 1:02:30.320 --> 1:02:32.180 And you can see, you can be a kid in high school 1:02:32.179 --> 1:02:33.049 and you can do this. 1:02:33.050 --> 1:02:35.950 There's no calculus or anything else involved, 1:02:35.951 --> 1:02:39.761 other than being open to the fact that the velocity of light 1:02:39.756 --> 1:02:41.816 behaves in this strange way. 1:02:41.820 --> 1:02:46.930 Yes? Student: How do you get 1:02:46.931 --> 1:02:48.921 t/x = c? Professor Ramamurti 1:02:48.917 --> 1:02:49.877 Shankar: Why is t/x = c? 1:02:49.880 --> 1:02:53.790 Oh, of course, you're quite right. 1:02:53.790 --> 1:02:56.720 So, you caught the mistake here. 1:02:56.719 --> 1:03:00.589 t/x is 1 over c and 1/c. 1:03:00.590 --> 1:03:02.960 I really meant to write x = ct. 1:03:02.960 --> 1:03:05.960 Yes? Student: [inaudible] 1:03:05.963 --> 1:03:09.453 Professor Ramamurti Shankar: No. 1:03:09.449 --> 1:03:13.809 If you define γ to be the absolute value by which you 1:03:13.810 --> 1:03:18.320 transfer lengths from you to me, then you can take the positive 1:03:18.315 --> 1:03:19.655 root. Student: [inaudible] 1:03:19.663 --> 1:03:20.693 Professor Ramamurti Shankar: Well, 1:03:20.690 --> 1:03:21.640 I can also tell you other reasons. 1:03:21.640 --> 1:03:25.060 Let's take this formula here. 1:03:25.059 --> 1:03:28.919 Let's take the case where the velocity is very small compared 1:03:28.922 --> 1:03:30.662 to the velocity of light. 1:03:30.659 --> 1:03:33.109 That u/c is a very small number. 1:03:33.110 --> 1:03:36.750 This number is almost 1 and I get x prime because x 1:03:36.751 --> 1:03:40.091 - ut, which I know to be the correct answer at lower 1:03:40.089 --> 1:03:42.749 velocities. If you pick the minus sign, 1:03:42.753 --> 1:03:46.133 I'll get minus of x - ut, that's not the right 1:03:46.131 --> 1:03:49.121 answer, not even close to the right answer. 1:03:49.119 --> 1:03:52.609 At low velocities, if you go to velocity u 1:03:52.608 --> 1:03:56.818 over c much less than 1, you have got to get back to 1:03:56.824 --> 1:03:58.864 Galilean transformation. 1:03:58.860 --> 1:04:01.860 You can see if u over c goes to zero. 1:04:01.860 --> 1:04:04.530 You can forget all about this factor here. 1:04:04.530 --> 1:04:07.380 You get x prime is x - ut and here, 1:04:07.376 --> 1:04:09.546 u over c can be neglected, 1:04:09.550 --> 1:04:12.600 forget all that, you get t prime equals 1:04:12.595 --> 1:04:14.875 t. So, this coordinate 1:04:14.880 --> 1:04:19.750 transformation would reduce to the Galilean transformation if 1:04:19.745 --> 1:04:24.205 the velocity between me and you is much smaller than the 1:04:24.205 --> 1:04:26.065 velocity of light. 1:04:26.070 --> 1:04:30.270 So, the formula really kicks in for velocities comparable to 1:04:30.267 --> 1:04:31.687 velocity of light. 1:04:31.690 --> 1:04:36.990 Yes? Student: [inaudible] 1:04:36.994 --> 1:04:38.434 What happens if u > c? 1:04:38.429 --> 1:04:39.669 Professor Ramamurti Shankar: Well, 1:04:39.674 --> 1:04:40.954 you start getting crazy answers, right? 1:04:40.949 --> 1:04:44.929 You can already see that the theory will not admit velocities 1:04:44.931 --> 1:04:47.321 bigger than the velocity of light. 1:04:47.320 --> 1:04:49.090 You can already see it in this formula. 1:04:49.090 --> 1:04:52.810 That tells you that the one single velocity that you wanted 1:04:52.805 --> 1:04:56.455 to be the same for everybody is also the greatest possible 1:04:56.457 --> 1:04:59.107 velocity; that no observer can move at 1:04:59.112 --> 1:05:02.762 this speed with respect to another observer that is equal 1:05:02.756 --> 1:05:05.746 to what is in excess of the speed of light. 1:05:05.750 --> 1:05:08.480 So, the speed of light, which came out to be a constant 1:05:08.481 --> 1:05:11.861 in the beginning of the theory, is also turning out to be the 1:05:11.858 --> 1:05:13.978 upper limit on possible velocities. 1:05:13.980 --> 1:05:17.460 That's the origin of the statement that no observer can 1:05:17.456 --> 1:05:21.376 travel at a speed bigger than light, but we'll discuss it more 1:05:21.383 --> 1:05:24.603 and more. But you have to understand what 1:05:24.596 --> 1:05:28.806 it is that is being derived, what is the meaning of this 1:05:28.813 --> 1:05:30.953 formula here. What is it telling you? 1:05:30.949 --> 1:05:34.699 If I say, this is called the Lorenz transformations, 1:05:34.696 --> 1:05:36.456 what do they tell you? 1:05:36.460 --> 1:05:40.820 What are these numbers and what's their significance? 1:05:40.820 --> 1:05:42.970 Would you like to try? 1:05:42.969 --> 1:05:45.819 Student: Well, one thing that they tell you is 1:05:45.824 --> 1:05:48.354 if u happens to be greater than c 1:05:48.350 --> 1:05:50.160 [inaudible] Professor Ramamurti 1:05:50.162 --> 1:05:52.872 Shankar: No, no, I don't mean what happens 1:05:52.871 --> 1:05:54.581 in the formula in special cases. 1:05:54.580 --> 1:05:55.820 What is it relating? 1:05:55.820 --> 1:05:59.440 What is xt and what is x prime t prime? 1:05:59.440 --> 1:06:04.000 Student: Okay, so x prime would 1:06:04.001 --> 1:06:10.221 be the distance that the person who's traveling at the higher 1:06:10.222 --> 1:06:12.092 speed experiences. 1:06:12.089 --> 1:06:16.029 And [inaudible] and so for that person, 1:06:16.029 --> 1:06:20.279 distance is going to seem shorter? 1:06:20.280 --> 1:06:22.340 Professor Ramamurti Shankar: No. 1:06:22.340 --> 1:06:25.120 See, I'm not even telling you to get consequences of the 1:06:25.123 --> 1:06:27.703 equation. What are the numbers x 1:06:27.700 --> 1:06:29.860 and t in this equation? 1:06:29.860 --> 1:06:31.920 And what are the numbers x prime and t 1:06:31.920 --> 1:06:32.930 prime in this equation? 1:06:32.929 --> 1:06:36.719 Student: x and t are distance and time 1:06:36.723 --> 1:06:40.193 for a person who is in an inertial frame of reference, 1:06:40.190 --> 1:06:41.630 who is not moving. 1:06:41.630 --> 1:06:42.410 Professor Ramamurti Shankar: Right. 1:06:42.409 --> 1:06:44.659 Student: And x prime and t prime are 1:06:44.660 --> 1:06:47.110 distance and time for the person who is moving at the speed of 1:06:47.112 --> 1:06:48.442 [inaudible] Professor Ramamurti 1:06:48.439 --> 1:06:50.609 Shankar: And when you say distance and time, 1:06:50.610 --> 1:06:52.740 what do you mean, distance and time? 1:06:52.739 --> 1:06:56.549 Student: The way that the length of the distance will 1:06:56.552 --> 1:06:58.622 seem to them that they travel. 1:06:58.619 --> 1:06:59.879 Professor Ramamurti Shankar: But what is 1:06:59.876 --> 1:07:03.156 happening at xt; it's the coordinates of what? 1:07:03.159 --> 1:07:04.919 Student: Of their location [inaudible] 1:07:04.920 --> 1:07:07.080 Professor Ramamurti Shankar: They are located at 1:07:07.079 --> 1:07:09.189 zero, zero. Right? 1:07:09.190 --> 1:07:10.500 What's happening at x and t? 1:07:10.500 --> 1:07:15.300 Yes? Student: It's observing 1:07:15.303 --> 1:07:17.093 the event. Professor Ramamurti 1:07:17.092 --> 1:07:17.532 Shankar: It's the event. 1:07:17.530 --> 1:07:19.710 The key I was looking for is an event. 1:07:19.710 --> 1:07:22.680 You've got to understand what the formula is connecting. 1:07:22.679 --> 1:07:26.969 Things are happening in space and in time, right? 1:07:26.970 --> 1:07:28.660 Something happens here. 1:07:28.659 --> 1:07:31.999 That something has a spatial coordinate and a time 1:07:31.997 --> 1:07:34.787 coordinate, according to two observers. 1:07:34.789 --> 1:07:38.259 The observers originally had their origins and their clocks 1:07:38.263 --> 1:07:41.513 coincide when they passed; that's how they're related. 1:07:41.510 --> 1:07:43.880 And one is moving to the right at speed u. 1:07:43.880 --> 1:07:47.850 Then, the claim is that if one event had a coordinate x 1:07:47.848 --> 1:07:51.238 and t for one person, for the other person moving to 1:07:51.235 --> 1:07:53.765 the right at speed u, the same event would have 1:07:53.772 --> 1:07:56.692 coordinates x prime and t prime and the relation 1:07:56.692 --> 1:07:59.562 between x and t and x prime and t 1:07:59.564 --> 1:08:03.534 prime is this. Yes? 1:08:03.530 --> 1:08:07.820 Student: The two observers [inaudible] 1:08:07.815 --> 1:08:12.485 don't they observe different laws and [inaudible] 1:08:12.491 --> 1:08:16.681 Professor Ramamurti Shankar: No. 1:08:16.680 --> 1:08:19.790 The fact that an event has different coordinates doesn't 1:08:19.792 --> 1:08:22.342 mean that you are observing different laws. 1:08:22.340 --> 1:08:24.930 For example, let's take that fire 1:08:24.927 --> 1:08:27.187 extinguisher. We look at it, 1:08:27.194 --> 1:08:29.834 it's obeying F = ma, right? 1:08:29.829 --> 1:08:32.589 The coordinates of the fire extinguisher with me as the 1:08:32.591 --> 1:08:35.201 origin is quite different from you as the origin. 1:08:35.199 --> 1:08:44.219 Student: [inaudible] Professor Ramamurti 1:08:44.224 --> 1:08:52.884 Shankar: You mean in these new equations? 1:08:52.880 --> 1:08:55.940 Yes, in these new equations, F = ma will not work; 1:08:55.940 --> 1:08:58.540 that's correct. Student: They are 1:08:58.540 --> 1:09:00.480 inertial references, even though they are moving? 1:09:00.479 --> 1:09:01.309 Professor Ramamurti Shankar: Ah. 1:09:01.310 --> 1:09:04.290 Yes. So, the point is the laws of 1:09:04.294 --> 1:09:07.354 Newton themselves have to be modified. 1:09:07.350 --> 1:09:11.700 F = ma will be modified in a certain way but the new 1:09:11.702 --> 1:09:16.432 modified laws will reduce to F = ma at low velocities, 1:09:16.430 --> 1:09:19.530 which is why in the old days it looked like F = ma. 1:09:19.529 --> 1:09:22.609 But there will be new laws, but they will also have the 1:09:22.607 --> 1:09:26.027 property that when I measure them I'll get the laws that will 1:09:26.026 --> 1:09:27.846 agree with what you measure. 1:09:27.850 --> 1:09:32.010 Yes? Student: Does individual 1:09:32.014 --> 1:09:36.404 values for time or distance still have to agree? 1:09:36.399 --> 1:09:38.379 Professor Ramamurti Shankar: The coordinates of 1:09:38.379 --> 1:09:40.059 an event will differ from person to person. 1:09:40.060 --> 1:09:43.880 That's not the same as saying the laws as deduced will be 1:09:43.881 --> 1:09:45.221 different. For example, 1:09:45.224 --> 1:09:47.634 there are two stars which are attracted to each other by 1:09:47.625 --> 1:09:49.895 gravitation and they are orbiting around their common 1:09:49.896 --> 1:09:51.356 center of mass. If I see them, 1:09:51.360 --> 1:09:53.600 I will find that they obey the law of gravitation with 1:09:53.596 --> 1:09:55.196 m_1m _2 over 1:09:55.199 --> 1:09:57.769 r^(2) where r is the distance between the points 1:09:57.772 --> 1:10:00.052 and the acceleration is whatever I think it is. 1:10:00.050 --> 1:10:03.070 You can go on a rocket and look at the same two stars. 1:10:03.069 --> 1:10:05.229 They will be in a more complicated motion, 1:10:05.227 --> 1:10:08.277 maybe the whole system will be drifting a little to you, 1:10:08.279 --> 1:10:11.449 but their acceleration will be the same as what I get and the 1:10:11.452 --> 1:10:14.682 force between them will also be the same as what I get and the 1:10:14.678 --> 1:10:17.738 laws that you would deduce by looking at that star would be 1:10:17.744 --> 1:10:19.864 the same laws that I would deduce. 1:10:19.859 --> 1:10:23.419 So, that's a difference between the laws being the same and the 1:10:23.423 --> 1:10:25.093 coordinates being the same. 1:10:25.090 --> 1:10:28.070 No one said x prime and x are the same in that 1:10:28.071 --> 1:10:29.911 equation there. They are different. 1:10:29.909 --> 1:10:31.599 We are looking at it from different vantage points. 1:10:31.600 --> 1:10:36.380 But the fact is that force is equal to mass times acceleration 1:10:36.376 --> 1:10:38.956 is the same for the two people. 1:10:38.960 --> 1:10:42.770 Okay. The laws will be the same but 1:10:42.767 --> 1:10:43.997 things won't look the same. 1:10:44.000 --> 1:10:45.860 For example, you can stand on your head, 1:10:45.861 --> 1:10:48.441 don't even have to go to another frame of reference; 1:10:48.439 --> 1:10:50.509 you can stand on your head, your z coordinate is the 1:10:50.514 --> 1:10:52.714 minus of my z coordinate; to every point I give a 1:10:52.713 --> 1:10:54.513 z, you will give a minus z. 1:10:54.510 --> 1:10:56.580 But the world, even though you are a little 1:10:56.580 --> 1:10:58.700 messed up and want to stand on your head, 1:10:58.699 --> 1:11:01.749 you have every right to do that and you will find that F = 1:11:01.752 --> 1:11:03.512 ma. So the point is, 1:11:03.513 --> 1:11:07.593 the way we see events may depend on their origin or 1:11:07.587 --> 1:11:10.447 coordinates, but the laws we deduce are to 1:11:10.452 --> 1:11:13.342 be distinguished from the perception that we have. 1:11:13.340 --> 1:11:15.470 Okay? For example, 1:11:15.472 --> 1:11:18.072 if I'm on the ground, I send a piece of chalk, 1:11:18.070 --> 1:11:19.860 it goes up and it goes down. 1:11:19.859 --> 1:11:22.319 If you see me from a moving train, you would think it went 1:11:22.319 --> 1:11:24.649 on a parabola. So, no one says the chalk will 1:11:24.654 --> 1:11:25.974 go up and down for you. 1:11:25.970 --> 1:11:30.480 For you, it will go like this but its motion will still obey 1:11:30.478 --> 1:11:33.228 F = ma, is what I'm saying. 1:11:33.229 --> 1:11:37.089 That's all you really mean by saying things look the same. 1:11:37.090 --> 1:11:41.320 So, what you have to understand is that Lorentz transformations 1:11:41.324 --> 1:11:45.084 are the way to relate a pair of events, given events. 1:11:45.080 --> 1:11:46.430 Here's a simple example. 1:11:46.430 --> 1:11:49.480 If you live in the xy plane, there's a point here. 1:11:49.479 --> 1:11:51.329 It's not an event, it's simply a point in the 1:11:51.325 --> 1:11:53.455 xy plane. You measure it this way and 1:11:53.457 --> 1:11:55.597 that way and you call it the coordinates. 1:11:55.600 --> 1:11:59.670 If somebody else picks a different coordinate system with 1:11:59.672 --> 1:12:02.852 an angle θ, that person will say that's 1:12:02.850 --> 1:12:06.090 x prime and that is y prime and x 1:12:06.089 --> 1:12:08.839 prime and y prime are not the same. 1:12:08.840 --> 1:12:11.270 I remind you, x prime is x cos> 1:12:11.272 --> 1:12:13.032 θ [delete "minus y"] 1:12:13.026 --> 1:12:15.736 plus y sin θ, etc., and y prime is 1:12:15.741 --> 1:12:18.641 something else. θ is the angle between 1:12:18.638 --> 1:12:19.718 the two observers. 1:12:19.720 --> 1:12:21.300 So, the point is the point. 1:12:21.300 --> 1:12:23.870 It certainly looks different to the two people, 1:12:23.869 --> 1:12:26.159 but the same point has two coordinates. 1:12:26.159 --> 1:12:29.549 Similarly, the same event, like the collision of two cars, 1:12:29.553 --> 1:12:32.533 will have different events for different people. 1:12:32.530 --> 1:12:34.120 That's not the new part. 1:12:34.119 --> 1:12:37.869 The new part is that the rules for connecting xt to 1:12:37.865 --> 1:12:41.475 x't', is quite different from the Galilean rules, 1:12:41.479 --> 1:12:43.679 new rules. It's what you guys have to 1:12:43.679 --> 1:12:45.059 understand. And finally, 1:12:45.060 --> 1:12:49.150 why did Einstein get the credit for turning the world into four 1:12:49.146 --> 1:12:51.186 dimensions instead of three? 1:12:51.189 --> 1:12:54.019 After all, x and t were present there, 1:12:54.017 --> 1:12:55.037 too. The point is, 1:12:55.038 --> 1:12:58.168 t prime is always equal to t no matter how you 1:12:58.166 --> 1:13:00.116 move, whereas in the Einstein theory, 1:13:00.124 --> 1:13:02.804 x and t get scrambled into x prime 1:13:02.797 --> 1:13:05.567 and t prime just the way x and y get 1:13:05.568 --> 1:13:08.388 scrambled into each other when you rotate your axis. 1:13:08.390 --> 1:13:12.070 So, to have time as another variable that doesn't transform 1:13:12.068 --> 1:13:15.618 at all is not the same as making it into a coordinate. 1:13:15.619 --> 1:13:18.419 The four-dimensional world of Einstein is four-dimensional 1:13:18.421 --> 1:13:21.471 because space and time mix with each other when you change your 1:13:21.469 --> 1:13:22.599 frame of reference. 1:13:22.600 --> 1:13:26.410 That's what makes t now a coordinate as previously it 1:13:26.414 --> 1:13:29.004 was something the same for all people.