WEBVTT 00:00:00.035 --> 00:00:01.416 Special relativity begins 00:00:01.416 --> 00:00:06.363 with two statements that sound modest and turn out to reorganize space and time. 00:00:06.560 --> 00:00:11.262 We will keep one physical clock beside us while we follow their consequences. 00:00:12.059 --> 00:00:16.158 First, the laws of physics have the same form in every inertial frame. 00:00:16.262 --> 00:00:20.117 An inertial observer is simply one who is not accelerating. 00:00:21.169 --> 00:00:27.230 Second, every inertial observer measures the same speed c for light in a vacuum. 00:00:27.682 --> 00:00:30.281 The source may move, and the observer may move, 00:00:30.364 --> 00:00:32.849 but the measured light speed is still c. 00:00:33.646 --> 00:00:36.526 That second statement is the unfamiliar one. 00:00:36.722 --> 00:00:38.720 Ordinary speeds add. 00:00:38.917 --> 00:00:41.214 If you throw a ball forward from a train, 00:00:41.297 --> 00:00:45.280 someone beside the track combines the ball's speed with the train's speed. 00:00:45.732 --> 00:00:47.590 Light refuses that rule. 00:00:48.386 --> 00:00:52.079 We will not begin by guessing how clocks and rulers must change. 00:00:52.183 --> 00:00:56.618 We will ask what these two postulates force one very simple clock to do. 00:00:58.015 --> 00:00:59.432 Here is the clock. 00:00:59.745 --> 00:01:04.738 Two parallel mirrors face one another, separated by a fixed distance h. 00:01:04.935 --> 00:01:07.513 A pulse of light bounces between them. 00:01:08.124 --> 00:01:10.630 Let the pulse begin at the lower mirror. 00:01:10.713 --> 00:01:15.590 It travels upward at speed c, reaches the upper mirror, and returns. 00:01:15.786 --> 00:01:19.003 One complete up and down journey is one tick. 00:01:19.800 --> 00:01:24.430 The mirrors are separated by h, so the one-way journey has distance h. 00:01:24.513 --> 00:01:30.098 Speed is distance divided by time, which means the one-way time is h divided by c. 00:01:30.802 --> 00:01:33.798 A complete tick travels that distance twice. 00:01:33.994 --> 00:01:38.419 The clock therefore records delta tau equal to two h over c. 00:01:39.215 --> 00:01:41.582 I have called this interval delta tau 00:01:41.665 --> 00:01:46.066 because both endpoint events happen at the same place in the clock's own frame. 00:01:46.379 --> 00:01:48.467 The clock is there when the pulse leaves, 00:01:48.550 --> 00:01:51.453 and the same clock is there when the pulse returns. 00:01:52.250 --> 00:01:57.219 That is proper time: the time measured by a clock travelling with both events. 00:01:57.416 --> 00:01:59.797 Nothing relativistic has happened yet. 00:01:59.993 --> 00:02:02.768 We have only built a clock whose operation follows directly 00:02:02.768 --> 00:02:04.778 from the constant speed of light. 00:02:05.574 --> 00:02:07.987 Now let this entire clock move sideways. 00:02:08.070 --> 00:02:11.403 The passengers who carry it still see the same vertical pulse and the same 00:02:11.403 --> 00:02:16.477 tick delta tau. The interesting question is what someone beside the track must see. 00:02:17.635 --> 00:02:19.885 The passenger sees the clock at rest. 00:02:19.969 --> 00:02:24.765 The pulse goes straight up and straight down, and one tick lasts delta tau. 00:02:25.077 --> 00:02:28.851 The platform observer sees the whole apparatus move to the right. 00:02:29.648 --> 00:02:33.921 During the first half of the tick, the upper mirror moves sideways. 00:02:34.118 --> 00:02:36.414 The light still travels at speed c, 00:02:36.498 --> 00:02:39.598 so it follows a diagonal path to catch that mirror. 00:02:40.395 --> 00:02:41.588 During the second half, 00:02:41.672 --> 00:02:45.272 the clock keeps moving while the pulse returns to the lower mirror. 00:02:45.469 --> 00:02:48.612 The platform observer therefore sees two diagonal legs, 00:02:48.696 --> 00:02:50.926 not one vertical round trip. 00:02:51.722 --> 00:02:55.067 Both observers must use the same light speed c. 00:02:55.171 --> 00:02:57.850 The platform observer sees a longer light path, 00:02:57.934 --> 00:03:00.953 so that observer must assign more time to the tick. 00:03:01.266 --> 00:03:04.784 The geometry will tell us exactly how much more. 00:03:05.925 --> 00:03:08.291 Take only the upward half of the journey. 00:03:08.375 --> 00:03:11.020 The vertical separation of the mirrors is h. 00:03:11.104 --> 00:03:16.282 If the full platform time is delta t, this half lasts delta t over two. 00:03:17.195 --> 00:03:22.606 During that half tick, the clock moves horizontally by v times delta t over two. 00:03:22.919 --> 00:03:25.799 That is the bottom leg of a right triangle. 00:03:26.596 --> 00:03:28.710 Light covers the diagonal. 00:03:29.022 --> 00:03:33.981 Since its speed is c, the diagonal length is c times delta t over two. 00:03:35.195 --> 00:03:38.064 Now use only the Pythagorean theorem. 00:03:38.260 --> 00:03:41.766 The square of the diagonal equals the square of h plus the square 00:03:41.766 --> 00:03:43.485 of the horizontal leg. 00:03:44.282 --> 00:03:48.973 Multiply out the common factors and move the horizontal contribution to the left. 00:03:49.170 --> 00:03:53.942 Delta t squared over four multiplies c squared minus v squared. 00:03:54.739 --> 00:03:56.431 Solve for delta t. 00:03:56.515 --> 00:04:01.961 The answer is two h divided by the square root of c squared minus v squared. 00:04:03.358 --> 00:04:05.344 Carry that result forward. 00:04:05.540 --> 00:04:08.165 Factor c squared out of the square root. 00:04:08.478 --> 00:04:14.202 Two h over c is exactly the proper tick delta tau that the passenger measured. 00:04:14.998 --> 00:04:19.361 The remaining factor is one over the square root of one minus v 00:04:19.445 --> 00:04:21.280 squared over c squared. 00:04:21.895 --> 00:04:24.241 We call this factor gamma. 00:04:25.037 --> 00:04:28.416 So the platform time is gamma times the proper time. 00:04:28.613 --> 00:04:30.665 Since gamma is at least one, 00:04:30.749 --> 00:04:34.593 the moving clock accumulates less time between the same two meetings. 00:04:35.390 --> 00:04:40.499 Watch the triangle as the speed rises from forty percent to eighty percent of c. 00:04:40.811 --> 00:04:42.933 The mirror separation stays fixed, 00:04:43.017 --> 00:04:46.710 but the horizontal leg and the light path both grow. 00:04:47.762 --> 00:04:53.544 At low speed gamma is almost one, which is why ordinary experience hides relativity. 00:04:53.741 --> 00:04:57.457 At eighty percent of c, gamma is five thirds. 00:04:57.769 --> 00:05:02.228 Five platform seconds correspond to only three seconds on the moving clock. 00:05:03.025 --> 00:05:06.717 Time dilation is not a mechanical defect in the light clock. 00:05:06.914 --> 00:05:10.618 Every process carried with the moving observer follows the same spacetime 00:05:10.618 --> 00:05:15.495 geometry, including chemical reactions, heartbeats, and aging. 00:05:16.636 --> 00:05:20.039 Time dilation also forces a change in measured length. 00:05:20.235 --> 00:05:24.160 Turn a light clock so its pulse travels along the direction of motion. 00:05:24.957 --> 00:05:28.974 In the platform frame, call the moving mirror separation L. 00:05:29.171 --> 00:05:32.222 On the outward trip the front mirror runs away from the light, 00:05:32.306 --> 00:05:34.931 so the closing speed is c minus v. 00:05:35.727 --> 00:05:38.720 On the return trip the rear mirror moves toward the light. 00:05:38.804 --> 00:05:41.324 The closing speed is c plus v. 00:05:42.121 --> 00:05:44.118 Add the two travel times. 00:05:44.315 --> 00:05:47.984 The common denominator is c squared minus v squared. 00:05:48.781 --> 00:05:53.901 After simplifying, the platform round trip is two L gamma squared over c. 00:05:55.298 --> 00:05:57.714 Carry that platform result forward. 00:05:57.817 --> 00:06:02.621 In the clock's rest frame, the mirror separation is its proper length L zero, 00:06:02.705 --> 00:06:05.910 so the round trip is two L zero over c. 00:06:06.707 --> 00:06:12.234 Time dilation says the platform round trip equals gamma times the proper round trip. 00:06:12.337 --> 00:06:14.474 Equate those two expressions. 00:06:15.271 --> 00:06:17.373 Cancel the common factors. 00:06:17.570 --> 00:06:21.646 The moving length L equals the proper length divided by gamma. 00:06:21.842 --> 00:06:25.454 A moving object is shorter along its direction of motion. 00:06:26.250 --> 00:06:30.125 Length means the distance between two endpoints measured at the same time. 00:06:30.209 --> 00:06:32.509 That last phrase is essential. 00:06:32.706 --> 00:06:37.001 If the endpoints are moving, recording one endpoint now and the other endpoint 00:06:37.001 --> 00:06:39.567 later does not measure one length. 00:06:40.964 --> 00:06:45.748 To see why same time becomes delicate, put a lamp exactly at the centre of a train. 00:06:45.852 --> 00:06:49.672 The passengers know the rear and front walls are equally far from the lamp. 00:06:50.376 --> 00:06:51.979 The lamp flashes. 00:06:52.083 --> 00:06:56.388 Light travels left and right at the same speed c, through equal distances, 00:06:56.472 --> 00:07:00.199 so it strikes the rear and front simultaneously in the train frame. 00:07:00.996 --> 00:07:03.957 Now describe the same experiment from the platform. 00:07:04.270 --> 00:07:08.310 At emission, the lamp is between the moving rear and front walls. 00:07:08.414 --> 00:07:11.956 Both light pulses still travel at speed c in this frame. 00:07:12.753 --> 00:07:15.536 The rear wall moves toward the left-going pulse. 00:07:15.620 --> 00:07:18.686 The front wall moves away from the right-going pulse. 00:07:18.883 --> 00:07:22.378 Let the experiment run until the left pulse meets the rear. 00:07:23.175 --> 00:07:27.646 The rear has already been struck, but the right pulse still has farther to go. 00:07:27.842 --> 00:07:30.362 Remove the completed pulse and continue. 00:07:31.159 --> 00:07:34.198 Only later does the right pulse catch the front wall. 00:07:34.282 --> 00:07:38.509 The platform observer therefore says rear first, front second. 00:07:39.120 --> 00:07:43.254 This is not a disagreement about light taking time to reach someone's eyes. 00:07:43.450 --> 00:07:46.353 Each observer can correct for signal travel. They 00:07:46.353 --> 00:07:50.882 disagree about whether the two distant contact events share one time coordinate. 00:07:52.534 --> 00:07:54.621 The algebraic statement is this. 00:07:54.705 --> 00:07:55.796 The time separation 00:07:55.796 --> 00:08:02.182 in the moving frame equals gamma times delta t minus v delta x over c squared. 00:08:03.234 --> 00:08:08.065 If two events are simultaneous in one frame, delta t is zero. 00:08:08.378 --> 00:08:12.253 If they are separated in space, delta x is not zero, 00:08:12.337 --> 00:08:17.017 so another moving frame generally assigns a nonzero time separation. 00:08:18.069 --> 00:08:18.916 This is difficult 00:08:18.916 --> 00:08:23.271 because everyday objects move far too slowly for the disagreement to notice. 00:08:23.467 --> 00:08:24.443 It is also difficult 00:08:24.443 --> 00:08:28.495 because simultaneity feels like a fact about the events themselves. 00:08:28.808 --> 00:08:33.790 Relativity says it is a relation between the events and a chosen inertial frame. 00:08:34.493 --> 00:08:38.844 The train passengers and the platform observer agree on every event: 00:08:38.928 --> 00:08:42.342 the emission, the rear strike, and the front strike. 00:08:42.655 --> 00:08:46.379 What differs is the time coordinate assigned to separated events. 00:08:46.463 --> 00:08:49.831 A spacetime diagram makes that distinction visible. 00:08:50.915 --> 00:08:52.900 A spacetime diagram puts position 00:08:52.900 --> 00:08:57.707 on the horizontal axis and c times time on the vertical axis. 00:08:57.904 --> 00:09:01.736 Multiplying time by c gives both axes the same units. 00:09:02.533 --> 00:09:08.280 An object at rest stays at one x position, so its history is a vertical worldline. 00:09:08.477 --> 00:09:13.563 A moving object changes x as time passes, so its worldline tilts. 00:09:14.615 --> 00:09:16.125 Light is special. 00:09:16.438 --> 00:09:18.015 Since its speed is c, 00:09:18.098 --> 00:09:22.127 one unit of horizontal travel takes one unit of c times time. 00:09:22.742 --> 00:09:26.389 Light rays therefore follow these two forty-five-degree lines. 00:09:27.185 --> 00:09:31.006 Every inertial observer must draw light on those same lines. 00:09:31.458 --> 00:09:35.370 That shared light cone constrains how another observer's coordinate axes can sit 00:09:35.370 --> 00:09:36.753 in the diagram. 00:09:37.665 --> 00:09:42.366 The red c t prime axis is the moving observer's own worldline, 00:09:42.449 --> 00:09:44.528 x prime equal to zero. 00:09:44.632 --> 00:09:47.256 It tilts toward the right-going light ray. 00:09:48.053 --> 00:09:50.479 The red x prime axis is the set 00:09:50.479 --> 00:09:54.752 of events that the moving observer calls simultaneous with the origin. 00:09:55.205 --> 00:09:57.760 It tilts upward by the matching amount. 00:09:58.672 --> 00:10:02.864 So the moving observer's space and time axes tilt toward the light lines 00:10:02.864 --> 00:10:04.350 and toward each other. 00:10:04.802 --> 00:10:08.402 They are not a rotated pair of ordinary Euclidean axes. 00:10:08.599 --> 00:10:11.061 Their scaling is controlled by gamma. 00:10:11.765 --> 00:10:14.993 The coordinate equations are the Lorentz transformation. 00:10:15.190 --> 00:10:19.811 Position x prime equals gamma times x minus v t. 00:10:20.724 --> 00:10:23.941 The time equation mixes time with position. 00:10:24.137 --> 00:10:28.675 That mixed term, v x over c squared in ordinary time units, 00:10:28.758 --> 00:10:32.358 is the algebraic source of relative simultaneity. 00:10:33.155 --> 00:10:35.686 Set x prime equal to zero. 00:10:35.860 --> 00:10:40.388 The result is the tilted time axis, the moving observer's worldline. 00:10:41.162 --> 00:10:43.577 Set t prime equal to zero. 00:10:43.751 --> 00:10:46.222 The result is the tilted space axis, 00:10:46.305 --> 00:10:49.533 the moving observer's line of simultaneous events. 00:10:50.318 --> 00:10:55.648 Now choose two events E one and E two on one horizontal line. 00:10:55.845 --> 00:10:59.120 The stationary observer assigns them the same time. 00:10:59.916 --> 00:11:03.969 The moving observer's simultaneous line through E one is tilted. 00:11:04.073 --> 00:11:08.683 It reaches the same distant position at E three, not at E two. 00:11:09.480 --> 00:11:12.290 No event moves when we change coordinates. 00:11:12.603 --> 00:11:17.027 E one, E two, and E three stay where they are. 00:11:17.340 --> 00:11:22.228 What changes is which distant events one observer groups into a single present. 00:11:23.025 --> 00:11:28.166 The invariant quantity is not Euclidean distance on this page. Write it first. 00:11:28.249 --> 00:11:31.046 The time part is c delta t squared. 00:11:31.129 --> 00:11:33.577 The space part is delta x squared, 00:11:33.660 --> 00:11:37.642 and their difference has the same value for every inertial observer. 00:11:38.439 --> 00:11:44.384 For events on one clock's worldline, that invariant equals c delta tau squared. 00:11:44.581 --> 00:11:47.948 This is why different paths between the same meetings can contain different 00:11:47.948 --> 00:11:49.644 amounts of proper time. 00:11:50.764 --> 00:11:54.933 Two twins meet on Earth, synchronize clocks, and separate. 00:11:55.129 --> 00:11:56.802 One remains on Earth. 00:11:56.999 --> 00:12:00.875 The other travels outward at eighty percent of light speed, turns around, 00:12:00.958 --> 00:12:02.944 and returns at the same speed. 00:12:03.740 --> 00:12:07.468 The spacetime diagram makes the histories different immediately. 00:12:07.665 --> 00:12:11.111 The Earth twin follows one straight inertial worldline. 00:12:11.194 --> 00:12:15.688 The traveller follows one straight segment out and a different straight segment home. 00:12:16.485 --> 00:12:19.179 Let each leg last five Earth years. 00:12:19.376 --> 00:12:23.890 At speed zero point eight c, the turnaround is four light years away, 00:12:23.973 --> 00:12:26.877 and the reunion occurs after ten Earth years. 00:12:27.673 --> 00:12:31.296 For zero point eight c, gamma is five thirds. 00:12:32.093 --> 00:12:34.750 The Earth clock is present at departure and reunion, 00:12:34.833 --> 00:12:37.516 and it records ten years of proper time. 00:12:38.312 --> 00:12:43.129 On either travelling leg, five Earth-frame years correspond to five divided by gamma, 00:12:43.212 --> 00:12:45.210 which is three traveller years. 00:12:45.313 --> 00:12:48.077 Two legs therefore give six traveller years. 00:12:48.874 --> 00:12:51.824 At reunion the clocks can be compared at one place. 00:12:52.020 --> 00:12:56.061 The Earth twin has aged ten years and the travelling twin six. 00:12:56.258 --> 00:12:59.231 There is no dispute about that local comparison. 00:13:00.027 --> 00:13:01.815 The apparent paradox comes from saying 00:13:01.815 --> 00:13:05.694 that each inertial observer sees the other's clock run slowly. 00:13:05.890 --> 00:13:10.059 If the roles were perfectly symmetric, why would one twin be younger? 00:13:11.572 --> 00:13:15.253 The roles are symmetric only during any one inertial leg. 00:13:15.565 --> 00:13:19.163 Earth remains in one inertial frame throughout the experiment. 00:13:19.246 --> 00:13:22.393 The traveller changes inertial frames at the turnaround. 00:13:23.189 --> 00:13:27.730 Before turning, the traveller uses an outbound line of simultaneity. 00:13:27.926 --> 00:13:33.430 On these axes, one year of c times time has the same scale as one light year. 00:13:33.627 --> 00:13:36.182 Draw that line through the turnaround event. 00:13:36.978 --> 00:13:40.461 Where that line meets Earth's worldline, the outbound coordinates 00:13:40.461 --> 00:13:44.909 assign Earth a time of one point eight years at the distant turnaround moment. 00:13:45.821 --> 00:13:49.468 After turning, the traveller belongs to the inbound frame. 00:13:49.571 --> 00:13:52.545 Its line of simultaneity slopes the other way. 00:13:53.341 --> 00:13:57.127 That new line meets Earth's worldline at eight point two years. 00:13:57.231 --> 00:14:01.200 The inbound coordinates assign a much later Earth time to events simultaneous 00:14:01.283 --> 00:14:03.094 with the same turnaround. 00:14:03.891 --> 00:14:07.850 The difference between those coordinate assignments is six point four years. 00:14:08.047 --> 00:14:10.486 This is where the asymmetry lives: 00:14:10.590 --> 00:14:15.385 the traveller changes which distant Earth event belongs to the traveller's present. 00:14:16.182 --> 00:14:19.933 Earth does not physically age six point four years in an instant. 00:14:20.129 --> 00:14:22.962 Nothing discontinuous happens to Earth's clock. 00:14:23.159 --> 00:14:27.325 What changes abruptly, in an idealized instantaneous turnaround, 00:14:27.408 --> 00:14:31.008 is the traveller's coordinate rule for distant simultaneity. 00:14:31.805 --> 00:14:36.136 With a realistic gradual turn, the reassignment is gradual too. 00:14:36.240 --> 00:14:39.105 The final proper times remain essentially the same 00:14:39.188 --> 00:14:42.754 if the acceleration interval is short compared with the journey. 00:14:43.968 --> 00:14:47.171 Acceleration identifies which twin changes frames, 00:14:47.254 --> 00:14:51.399 but acceleration is not an extra aging penalty added by hand. 00:14:51.596 --> 00:14:55.451 The elapsed proper time comes from the complete spacetime path. 00:14:56.248 --> 00:15:00.346 Between the same departure and reunion events, the straight inertial worldline 00:15:00.346 --> 00:15:04.016 contains more proper time than the broken out-and-back path. 00:15:04.212 --> 00:15:06.140 Earth records ten years. 00:15:06.337 --> 00:15:08.288 The traveller records six. 00:15:09.085 --> 00:15:13.103 So the twin paradox is not a contradiction in time dilation. 00:15:13.207 --> 00:15:17.419 It is a warning that time dilation applies within one inertial comparison, 00:15:17.502 --> 00:15:21.775 while the traveller's full journey requires two different inertial frames. 00:15:22.901 --> 00:15:26.617 The last test comes from particles made high in the atmosphere. 00:15:26.814 --> 00:15:29.957 Muons are created roughly ten kilometres above the ground, 00:15:30.041 --> 00:15:34.152 but their proper mean lifetime is only two point two microseconds. 00:15:34.348 --> 00:15:37.135 Why do many reach detectors at sea level? 00:15:38.048 --> 00:15:41.358 Begin with the nonrelativistic expectation. 00:15:41.554 --> 00:15:44.492 Here is the ten-kilometre atmospheric depth. 00:15:44.689 --> 00:15:48.428 The muon's own lifetime is two point two microseconds. 00:15:49.225 --> 00:15:53.893 Even light travels only about zero point six six kilometres in that time. 00:15:54.089 --> 00:15:57.341 A slower particle should decay far above the ground. 00:15:58.044 --> 00:16:03.026 But atmospheric muons commonly travel near zero point nine nine eight c. 00:16:03.222 --> 00:16:06.694 At that speed gamma is about fifteen point eight. 00:16:07.607 --> 00:16:11.450 In Earth's frame, the moving muon's lifetime is dilated. 00:16:11.647 --> 00:16:13.737 Two point two microseconds multiplied 00:16:13.737 --> 00:16:17.639 by gamma becomes about thirty-four point eight microseconds. 00:16:18.435 --> 00:16:19.965 At almost light speed, 00:16:20.049 --> 00:16:24.276 that dilated lifetime corresponds to about ten point four kilometres. 00:16:24.472 --> 00:16:26.598 The muon can reach the ground. 00:16:27.510 --> 00:16:30.077 That is the Earth-frame explanation: 00:16:30.181 --> 00:16:34.256 the moving particle's internal clock accumulates less time while Earth clocks 00:16:34.256 --> 00:16:35.928 record the long flight. 00:16:37.743 --> 00:16:40.135 Now use the muon's own frame. 00:16:40.332 --> 00:16:41.757 The muon is at rest, 00:16:41.841 --> 00:16:46.475 so its lifetime is still the proper value, about two point two microseconds. 00:16:46.787 --> 00:16:49.238 The atmosphere and ground move toward it. 00:16:50.151 --> 00:16:54.726 The ten-kilometre atmospheric depth is moving, so it is length contracted. 00:16:54.922 --> 00:16:58.394 Divide ten kilometres by gamma fifteen point eight. 00:16:59.191 --> 00:17:03.788 The muon measures only about zero point six three kilometres between its creation 00:17:03.788 --> 00:17:05.809 event and the approaching ground. 00:17:06.606 --> 00:17:10.762 At zero point nine nine eight c, crossing that contracted distance 00:17:10.762 --> 00:17:14.862 takes about two point one microseconds of muon proper time. 00:17:16.076 --> 00:17:17.536 Let the ground approach. 00:17:17.620 --> 00:17:22.102 It covers the shortened distance before the muon's typical lifetime has elapsed. 00:17:22.899 --> 00:17:25.741 Earth says the muon's clock runs slowly. 00:17:25.825 --> 00:17:28.612 The muon says the atmosphere is short. 00:17:28.809 --> 00:17:31.174 These are not competing explanations. 00:17:31.258 --> 00:17:35.416 They are the time and space parts of one Lorentz transformation. 00:17:36.119 --> 00:17:39.057 Both descriptions predict the same local event: 00:17:39.161 --> 00:17:41.449 the muon and the detector meet. 00:17:41.553 --> 00:17:45.930 Experiments observe far more muons at the ground than a nonrelativistic 00:17:45.930 --> 00:17:48.136 lifetime calculation permits. 00:17:49.533 --> 00:17:51.426 Now collect the argument. 00:17:51.623 --> 00:17:55.757 First, every inertial observer measures the same light speed c. 00:17:56.460 --> 00:17:57.621 The moving light clock 00:17:57.621 --> 00:18:02.486 then forces moving clocks to accumulate less proper time between shared events. 00:18:03.283 --> 00:18:07.611 A longitudinal light clock and the same time-dilation law force moving lengths 00:18:07.695 --> 00:18:09.820 to contract along the motion. 00:18:10.617 --> 00:18:12.010 The train lamp shows the price 00:18:12.010 --> 00:18:15.192 of combining those results with constant light speed: 00:18:15.388 --> 00:18:18.768 distant simultaneity depends on the inertial frame. 00:18:19.472 --> 00:18:21.527 The spacetime diagram gathers all 00:18:21.527 --> 00:18:25.800 of this into tilted coordinate axes and invariant proper time. 00:18:25.996 --> 00:18:28.749 It also reveals the twin asymmetry: 00:18:28.945 --> 00:18:32.151 only the travelling twin changes inertial frames. 00:18:32.947 --> 00:18:37.731 Finally, atmospheric muons turn the geometry into an observation. 00:18:37.928 --> 00:18:40.610 Long lifetime in Earth's coordinates and short atmosphere 00:18:40.610 --> 00:18:44.616 in the muon's coordinates are two descriptions of the same arrival. 00:18:45.529 --> 00:18:49.640 Special relativity is not a story in which appearances deceive us. 00:18:49.743 --> 00:18:54.004 It is a precise rule for how different observers divide one spacetime into space 00:18:54.004 --> 00:18:59.218 and time, while agreeing on light, meetings, and every measurable event.