1 00:00:01,320 --> 00:00:06,226 Before we go on to construct relativistic physics there's another important 2 00:00:06,226 --> 00:00:11,319 consequence of Lorentz invariance and other important way it teaches us to look 3 00:00:11,319 --> 00:00:14,114 at the universe that we need to come back to. 4 00:00:14,114 --> 00:00:18,275 And we need to understand this thing called the invariant interval. 5 00:00:18,275 --> 00:00:22,250 So these are our Lorentz transformations. When you study them, 6 00:00:22,250 --> 00:00:26,424 you notice that they have the following exciting property. 7 00:00:26,424 --> 00:00:29,879 That if you compute this combination of t and x, 8 00:00:29,879 --> 00:00:31,679 C pri-, c2 squared times t squared,2-x^2 minus X squared in one frame, 9 00:00:31,679 --> 00:00:35,061 or compute c2 squared times t2-x^2 squared in the other frame, you get the 10 00:00:35,061 --> 00:00:38,660 same answer. This is why it's called the invariant 11 00:00:38,660 --> 00:00:41,180 interval. All observers agree on it. 12 00:00:41,180 --> 00:00:47,095 Now this turns out that, to have great physical significance. 13 00:00:47,095 --> 00:00:51,999 In fact, you can derive much of relativity from just this property. 14 00:00:51,999 --> 00:00:58,390 Notice that there are, this breaks up the universe into, three pieces, when you 15 00:00:58,390 --> 00:01:03,146 have a particular event. So I'm going to pick a particular event 16 00:01:03,146 --> 00:01:03,740 to be xt=0. equals t equals zero. 17 00:01:03,740 --> 00:01:08,570 That's right here and now. And remember that we are talking about 18 00:01:08,570 --> 00:01:13,457 now over here and here, all up and down this axis, and what I 19 00:01:13,457 --> 00:01:17,270 notice is first of all, if I set s equal to zero, 20 00:01:17,270 --> 00:01:20,960 that means that C times T is either plus or minus X. 21 00:01:20,960 --> 00:01:26,525 Oh, X is either plus or minus C times T. That means these two trajectories of 22 00:01:26,525 --> 00:01:32,727 beams of light going either to the right or to the left emerging form the event 23 00:01:32,727 --> 00:01:37,457 that we call the x equals to zero or any particular event. 24 00:01:37,457 --> 00:01:43,426 Or, by the way, if you go to negativity, beams of light coming from the left or 25 00:01:43,426 --> 00:01:48,853 from the right, and impinging exactly at t equal zero at x equals zero. 26 00:01:48,853 --> 00:01:52,730 All of these are the regions in space at which 27 00:01:52,730 --> 00:01:58,280 S squared is zero, and of course, it's zero, and we said these light 28 00:01:58,280 --> 00:02:03,510 well, lines are preserved in both, in any, 29 00:02:03,510 --> 00:02:08,427 frame. So it's not surprising that S equals zero in one, frame means S equals 30 00:02:08,427 --> 00:02:12,805 zero in the other frame. these, these two lines are called the 31 00:02:12,805 --> 00:02:18,175 light cone, and they kind of look funny for a cone, but, you can imagine that 32 00:02:18,175 --> 00:02:22,908 they would be an actual cone if space were not one-dimensional but 33 00:02:22,908 --> 00:02:26,794 two-dimensional. Then, instead of minus x squared here, I 34 00:02:26,794 --> 00:02:32,516 would have to add minus y squared minus z squared, and here minus y prime squared 35 00:02:32,516 --> 00:02:37,744 minus z prime squared, and if I only had y and not z, then x squared plus y 36 00:02:37,744 --> 00:02:40,878 squared equals some constant times t2, squared is a cone, 37 00:02:40,878 --> 00:02:44,557 so that, the cone would be rotated this way in the, 38 00:02:44,557 --> 00:02:47,611 if you had a y axis coming out of the board. 39 00:02:47,611 --> 00:02:52,262 if you have z, then this circle should be replaced with a sphere. 40 00:02:52,262 --> 00:02:56,566 That's drawing in more dimensions than I even care to attempt. 41 00:02:56,566 --> 00:03:02,050 so this is what is called the light cone. And light cones are very important. 42 00:03:02,050 --> 00:03:07,432 they describe thrust as I said, the projecteries of beams of light that cross 43 00:03:07,432 --> 00:03:12,546 at some particular event, moving in all possible directions in one dimension 44 00:03:12,546 --> 00:03:17,929 neither move to the left or to the right. Here, s squared to zero along the light 45 00:03:17,929 --> 00:03:21,293 curve. the rest of the universe is divided into 46 00:03:21,293 --> 00:03:24,399 four regions. in this two dimensional case though. 47 00:03:24,399 --> 00:03:28,628 Remember that if the light cone, if we were in a higher dimension and we 48 00:03:28,628 --> 00:03:32,801 returned those circles, then there actually only three regions because you 49 00:03:32,801 --> 00:03:37,142 can move around over here to connect. The entire outside of the light cone is 50 00:03:37,142 --> 00:03:39,962 connected, but the inside of the light cone always 51 00:03:39,962 --> 00:03:43,457 breaks up into two pieces. The inside of the light cone is the 52 00:03:43,457 --> 00:03:47,686 region where C squared T squared is bigger than X squared than either T is 53 00:03:47,686 --> 00:03:52,310 positive or negative so there's a piece of the light cone here and a piece of the 54 00:03:52,310 --> 00:03:55,636 light cone there. Whereas the rest of the universe is where 55 00:03:55,636 --> 00:03:58,548 X squared is bigger than c squared t squared. 56 00:03:58,548 --> 00:04:02,780 So s squared is negative. This is negative, and s squared negative, 57 00:04:02,780 --> 00:04:05,800 in more than two dimensions is one big region. 58 00:04:05,800 --> 00:04:10,287 Okay, what do these mean? Well, let's think about it. 59 00:04:10,287 --> 00:04:15,148 So what it means is this. Let's look at a region where s2 squared 60 00:04:15,148 --> 00:04:19,936 is positive. remembering that s squared is c squared t 61 00:04:19,936 --> 00:04:26,763 squared minus x squared. this means that if I put some event over here at positive 62 00:04:26,763 --> 00:04:30,213 t. since ct is bigger than x, now I'm making 63 00:04:30,213 --> 00:04:35,731 everything positive so I can take the square roots, that means that x over t is 64 00:04:35,731 --> 00:04:40,481 less than the speed of light c. So I can construct an observer, I can 65 00:04:40,481 --> 00:04:45,790 thrown a rock or construct an observer that starts at this event and ends up 66 00:04:45,790 --> 00:04:48,197 here. This is an allowed motion for an 67 00:04:48,197 --> 00:04:52,936 observer, because all world lines of observers are, remember, constrained by v 68 00:04:52,936 --> 00:04:56,990 less than c to be within this region. Oh, so that means two things. 69 00:04:56,990 --> 00:05:01,854 One is, it means that if I want to say, cause something to happen here, I can do 70 00:05:01,854 --> 00:05:05,845 it by throwing a rock at it. I, if I throw the rock, okay, I may need 71 00:05:05,845 --> 00:05:10,100 to throw the rock very fast, but it doesn't need to go faster than light. 72 00:05:10,100 --> 00:05:14,532 The other thing is if instead of throwing a rock I throw a clock, then I have 73 00:05:14,532 --> 00:05:19,142 essentially established a T-prime axis, which goes through both this event, and 74 00:05:19,142 --> 00:05:22,569 that event there is some observer that is present at both. 75 00:05:22,569 --> 00:05:26,883 in that frame I can now draw the corresponding X-prime axis if I want. 76 00:05:26,883 --> 00:05:31,670 But the important thing is, that in that frame, the two events happened at exactly 77 00:05:31,670 --> 00:05:35,273 the same place. Oh, they happened at exactly the same 78 00:05:35,273 --> 00:05:40,125 place that means they both happened at along the t prime axis that's at x prime 79 00:05:40,125 --> 00:05:44,976 equals zero and so s squared is actually c squared time t prime squared where t 80 00:05:44,976 --> 00:05:49,040 prime is the length of time as measured by the clock I threw there. 81 00:05:49,040 --> 00:05:54,632 this is what we call proper time. It's the time between these two events as 82 00:05:54,632 --> 00:05:59,445 measured by a clock that is partis-, that is inertial, and is present at both 83 00:05:59,445 --> 00:06:02,347 events. And this is given in terms of the 84 00:06:02,347 --> 00:06:07,585 invariant interval, which must be positive for there to exist such a clock, 85 00:06:07,585 --> 00:06:13,248 as the root of S squared divided by C. We call that proper time and often denote 86 00:06:13,248 --> 00:06:20,023 it by the Greek letter tau. correspondingly, if I have an event, down 87 00:06:20,023 --> 00:06:24,890 here at negative T, with s squared positive, then that 88 00:06:24,890 --> 00:06:30,268 negative t is still bigger in magnitude than x squared, and that again, if you 89 00:06:30,268 --> 00:06:33,533 can just follow this down here. That means the event is sitting inside 90 00:06:33,533 --> 00:06:37,243 the light cone, and an observer from there, moving at less than the speed of 91 00:06:37,243 --> 00:06:40,063 light can make it to x equals to equal zero. 92 00:06:40,063 --> 00:06:45,640 And therefore you can throw a rock from here and hit somebody there and it again 93 00:06:45,640 --> 00:06:50,488 the time as measured by that rock. If it were a clock would be the copper 94 00:06:50,488 --> 00:06:53,179 time. So we have fancy names for this, these 95 00:06:53,179 --> 00:06:57,214 two parts of the light cone, and they indicate their significance. 96 00:06:57,214 --> 00:07:02,181 The upper part of the light cone with T bigger than zero and S squared positive, 97 00:07:02,181 --> 00:07:06,030 is called the future. It's called the future because these are 98 00:07:06,030 --> 00:07:10,500 all the places where I can throw something at it less than the speed of 99 00:07:10,500 --> 00:07:14,905 light, and if you include the boundary, which is all the plate, the events that 100 00:07:14,905 --> 00:07:18,243 can be reached at the speed of light from my current location. 101 00:07:18,243 --> 00:07:21,312 And if you remember that nothing moves faster than light, 102 00:07:21,312 --> 00:07:25,673 then all of the events here are all of the events on which a decision taken here 103 00:07:25,673 --> 00:07:29,765 could possibly have an in-, influence. I can decide whether or not to throw a 104 00:07:29,765 --> 00:07:33,318 rock that would hit here. I have no way to make a rock get to here. 105 00:07:33,318 --> 00:07:35,487 So, whether this is my future or my past 106 00:07:35,487 --> 00:07:38,391 does, doesn't matter. It's not my causal future, because no 107 00:07:38,391 --> 00:07:43,183 physical influence will be allowed. I cannot throw a rock faster than light, 108 00:07:43,183 --> 00:07:47,849 so I cannot make a rock that was here also be present at this position at that 109 00:07:47,849 --> 00:07:50,409 time. That requires moving faster than light. 110 00:07:50,409 --> 00:07:53,960 Similarly, this part over here is legitimately my causal past. 111 00:07:53,960 --> 00:07:58,674 These are all the events where someone taking the decision there could still get 112 00:07:58,674 --> 00:08:03,329 the information, or the causal effect, to meet at this position X equals T equals 113 00:08:03,329 --> 00:08:06,007 zero. So every event in space time, every point 114 00:08:06,007 --> 00:08:09,789 in this four dimensional thing has a light cone associated to it. 115 00:08:09,789 --> 00:08:12,699 which breaks the space time into three regions, 116 00:08:12,699 --> 00:08:14,593 the future, the past, and the rest. 117 00:08:14,593 --> 00:08:18,411 And remember these are actually connected, so they are really three 118 00:08:18,411 --> 00:08:20,747 regions rather than four. What's the rest? 119 00:08:20,747 --> 00:08:24,849 Well, let's see what the rest is. If s squared is negative, that means you 120 00:08:24,849 --> 00:08:29,351 are outside the light code well for one thing we said no observer can get from 121 00:08:29,351 --> 00:08:32,770 here to there so we're talking now about an event over here. 122 00:08:32,770 --> 00:08:38,387 I can't have an observer for which this is on the t axis, because that observer's 123 00:08:38,387 --> 00:08:42,900 t axis would be tilting more than 45 degrees from the vertical. 124 00:08:42,900 --> 00:08:49,437 But for that reason, of course I can find and observe, find an observer for which 125 00:08:49,437 --> 00:08:55,401 this event lies on his X prime axis. That means I can't make it fall at the 126 00:08:55,401 --> 00:09:01,443 same position as I did before for some observer, as x equals zero, but I can 127 00:09:01,443 --> 00:09:05,375 make it simultaneous. So I can always find some observer for 128 00:09:05,375 --> 00:09:08,213 any event, here in the outside of the light cone. 129 00:09:08,213 --> 00:09:12,588 There is some observer who thinks it's on the x prime axis, who think it's 130 00:09:12,588 --> 00:09:15,190 simultaneous with the event we started with. 131 00:09:15,190 --> 00:09:19,400 And that, for that observer, of course, the time of this event is t prime equals 132 00:09:19,400 --> 00:09:22,227 zero. Here's the speed with which that observer 133 00:09:22,227 --> 00:09:25,415 needs to move. And because T squared is bigger than X 134 00:09:25,415 --> 00:09:28,302 squared, that C squared, D squared is bigger than 135 00:09:28,302 --> 00:09:32,392 X squared, this is bigger. This is less than the speed of light, so 136 00:09:32,392 --> 00:09:35,941 that's legitimate. and, that means that, for that observer, 137 00:09:35,941 --> 00:09:39,370 since t prime is zero, S squared is just negative X prime 138 00:09:39,370 --> 00:09:42,016 squared. So, the root of negative S, S squared, 139 00:09:42,016 --> 00:09:46,347 when this makes sense, when S squared is negative, is the proper distance. 140 00:09:46,347 --> 00:09:49,650 It's the distance between these two events, 141 00:09:49,650 --> 00:09:55,972 as seen but measured by the observer for which they are simultaneous, of which 142 00:09:55,972 --> 00:10:00,733 that determines the speed wi- with which that observer moves. 143 00:10:00,733 --> 00:10:06,821 And now what happens if, instead of this green observer, I consider an observer 144 00:10:06,821 --> 00:10:10,880 moving even faster? So, his world line tilts from the 145 00:10:10,880 --> 00:10:16,700 vertical even more than that of the green observer, and his 146 00:10:16,700 --> 00:10:21,376 x double prime axis tilts from the horizontal even more. So for that 147 00:10:21,376 --> 00:10:26,809 observer notice, this point here, which had a positive value of T has a negative 148 00:10:26,809 --> 00:10:31,210 value of T double prime. the black observer thinks this event 149 00:10:31,210 --> 00:10:35,817 happened after, the exciting meeting of all the observers over here. 150 00:10:35,817 --> 00:10:41,113 The green observer thinks it happened at the same time, and the blue observer 151 00:10:41,113 --> 00:10:46,213 thinks it happened before they met. So it's a darned good thing that no 152 00:10:46,213 --> 00:10:50,928 decision taken here could impact something that happened there, 153 00:10:50,928 --> 00:10:55,792 because, if it could, then the blue observer effect preceded cause. 154 00:10:55,792 --> 00:11:02,452 You can only maintain, in the context of this crazy simul-, relative simultaneity 155 00:11:02,452 --> 00:11:03,650 story. The 156 00:11:03,650 --> 00:11:08,490 construct of a causal theory, if you allow that no material particle, we 157 00:11:08,490 --> 00:11:12,922 already talked about that. No observer, but also no information, no 158 00:11:12,922 --> 00:11:18,308 physical effect can move faster than light because moving faster than light is 159 00:11:18,308 --> 00:11:23,694 precisely tantamount to moving in an acausal way to effect proceeding cause. 160 00:11:23,694 --> 00:11:29,148 As long as you allow for its, it's effect preceding cause, as seen by someone 161 00:11:29,148 --> 00:11:32,830 moving sufficiently fast in the appropriate direction. 162 00:11:32,830 --> 00:11:37,856 The result is that we cannot allow in a relativistic theory any information to 163 00:11:37,856 --> 00:11:41,802 propagate faster than light. this is the origin of the idea of a 164 00:11:41,802 --> 00:11:46,765 cosmic speed limit, the speed of light, not only on material particles but also 165 00:11:46,765 --> 00:11:47,720 on information.