WEBVTT 00:00:00.035 --> 00:00:04.969 In nineteen oh five, Einstein published special relativity, and at its centre 00:00:04.969 --> 00:00:09.463 is a speed limit: no influence of any kind travels faster than light. 00:00:09.659 --> 00:00:12.807 Ten years later he published a theory of gravity. 00:00:13.003 --> 00:00:15.695 If you had to compress that theory into three words, 00:00:15.778 --> 00:00:18.356 they would be: not even gravity. 00:00:20.171 --> 00:00:22.053 This is where we are going. 00:00:22.249 --> 00:00:25.803 It is the equation Einstein arrived at in nineteen fifteen. 00:00:26.115 --> 00:00:31.769 The left hand side is a quantity that is zero when spacetime is flat and not zero 00:00:31.769 --> 00:00:33.674 when spacetime is curved. 00:00:33.777 --> 00:00:36.367 The right hand side is matter and energy. 00:00:37.164 --> 00:00:38.941 You do not have to read it yet. 00:00:39.138 --> 00:00:40.229 Everything between here 00:00:40.229 --> 00:00:44.848 and there is Newton, one coincidence that Newton himself noticed and shrugged at, 00:00:44.931 --> 00:00:48.008 and one change of mind about what a straight line is. 00:00:48.805 --> 00:00:49.906 Here is the Sun, 00:00:49.989 --> 00:00:54.088 here is the Earth, and this arrow is the pull the Sun exerts on it. 00:00:54.285 --> 00:00:57.395 Newton's law of gravity says that pull is his constant, 00:00:57.478 --> 00:01:02.726 times one mass, times the other, divided by the distance between them squared. 00:01:03.523 --> 00:01:05.172 Now jiggle the Sun. 00:01:05.624 --> 00:01:07.573 Read that formula literally: 00:01:07.656 --> 00:01:12.417 the instant the separation changes, the force out at the Earth changes with it. 00:01:12.729 --> 00:01:14.321 Not eight minutes later. 00:01:14.517 --> 00:01:15.482 Now. 00:01:16.394 --> 00:01:19.669 But light itself takes eight minutes to cross that gap. 00:01:20.121 --> 00:01:22.920 Watch a flash leave the Sun and reach us. 00:01:23.372 --> 00:01:25.427 Newton's force beat that flash across, 00:01:25.427 --> 00:01:28.179 which is exactly what the speed limit forbids. 00:01:29.579 --> 00:01:31.658 One of the two has to give. 00:01:31.762 --> 00:01:35.570 Einstein had spent years hunting down every way of sending a signal faster 00:01:35.570 --> 00:01:39.773 than light, and he was not about to allow a gravitational telephone. 00:01:40.086 --> 00:01:42.780 So the law that has to change is Newton's. 00:01:44.177 --> 00:01:47.788 There is a precedent for repairing an inverse square law. 00:01:47.985 --> 00:01:51.283 Electricity has one of exactly the same shape: 00:01:51.387 --> 00:01:56.856 a constant, times one charge, times the other, over the separation squared. 00:01:58.070 --> 00:02:00.950 Maxwell had already dressed that law up properly. 00:02:01.147 --> 00:02:02.585 Once the charges move, 00:02:02.668 --> 00:02:07.289 magnetic effects appear, and the whole package obeys the speed limit exactly. 00:02:07.602 --> 00:02:10.215 So copy the trick for gravity and be done. 00:02:11.127 --> 00:02:13.613 Two like charges push each other apart. 00:02:13.716 --> 00:02:15.935 Two masses pull each other together. 00:02:16.131 --> 00:02:18.800 Copy the electric calculation exactly 00:02:18.883 --> 00:02:23.121 and you get the electric answer, which is that the Sun would shove the Earth away. 00:02:23.918 --> 00:02:27.364 There is the difference, and it is one character wide. 00:02:27.447 --> 00:02:30.536 A plus on this side, a minus on that one. 00:02:30.849 --> 00:02:35.807 That single character is why gravity cannot simply be copied from electricity. 00:02:36.604 --> 00:02:38.938 The reason lies in the messenger. 00:02:39.135 --> 00:02:43.490 The electric force is carried by a particle of spin one, the photon. 00:02:43.802 --> 00:02:46.845 Gravity is carried by a particle of spin two. 00:02:47.158 --> 00:02:50.177 That difference is exactly the difference in sign. 00:02:50.973 --> 00:02:53.050 So Einstein could not copy. 00:02:53.133 --> 00:02:54.815 He had to find another route, 00:02:54.898 --> 00:02:59.613 and he had one clue: a coincidence sitting in the middle of Newton's own equations. 00:03:00.707 --> 00:03:04.643 Look again at the two laws, and at the letter m in each of them. 00:03:05.095 --> 00:03:06.429 In Newton's second law, 00:03:06.512 --> 00:03:11.273 the mass measures stubbornness: how hard the body fights being accelerated. 00:03:11.469 --> 00:03:13.745 That is the inertial mass. 00:03:14.658 --> 00:03:18.967 In the law of gravity, the mass measures something else entirely: 00:03:19.279 --> 00:03:22.055 how firmly gravity takes hold of the body. 00:03:22.251 --> 00:03:26.734 That is a completely different job, and it is called the gravitational mass. 00:03:27.530 --> 00:03:31.594 In electricity those two jobs are done by two unrelated numbers. 00:03:31.791 --> 00:03:34.811 A neutron is heavy and carries no charge at all. 00:03:35.007 --> 00:03:38.189 An electron is light and carries a full unit of it. 00:03:38.730 --> 00:03:40.518 So there is no relation whatsoever 00:03:40.518 --> 00:03:45.012 between how much a particle weighs and how strongly it feels an electric field. 00:03:45.627 --> 00:03:47.880 Now watch what gravity does. 00:03:48.421 --> 00:03:50.602 In gravity, the number that resists 00:03:50.685 --> 00:03:54.923 and the number that gets pulled are the same number. Not nearly the same. 00:03:55.120 --> 00:03:56.235 The same. 00:03:56.939 --> 00:04:00.864 That statement is the equivalence principle, and here is what it does. 00:04:01.061 --> 00:04:02.140 Drop a brick and a feather 00:04:02.140 --> 00:04:05.706 in a vacuum chamber, with no air to slow either one down. 00:04:06.502 --> 00:04:10.961 Gravity pulls the brick harder, because the brick has more gravitational mass. 00:04:11.065 --> 00:04:14.871 The brick also resists harder, because it has more inertial mass. 00:04:14.954 --> 00:04:19.146 The two effects cancel exactly, and the two bodies fall together. 00:04:20.360 --> 00:04:22.309 Newton noticed this himself. 00:04:22.392 --> 00:04:23.842 He ran experiments on it, 00:04:23.925 --> 00:04:27.664 and found the two masses agreeing to roughly one part in a thousand. 00:04:28.368 --> 00:04:31.942 By Einstein's day the agreement was one part in a billion. 00:04:32.025 --> 00:04:34.961 Today it is one part in ten to the fifteen, 00:04:35.044 --> 00:04:39.178 which makes it one of the most precisely tested statements in all of physics. 00:04:39.881 --> 00:04:42.912 In Newton's physics this is a pure coincidence. 00:04:43.016 --> 00:04:44.883 Two unrelated quantities, 00:04:44.966 --> 00:04:49.388 agreeing to fifteen decimal places, for no reason anybody could give. 00:04:49.471 --> 00:04:52.444 Einstein refused to accept that as an accident. 00:04:53.522 --> 00:04:55.961 Here is the other half of Einstein's clue. 00:04:56.064 --> 00:04:59.699 A bucket of water on a rope, swung in a vertical loop. 00:04:59.896 --> 00:05:02.532 Down at the bottom nothing is mysterious: 00:05:02.636 --> 00:05:05.237 gravity pulls the water into the base of it. 00:05:06.034 --> 00:05:10.713 Swing it fast enough and the water stays put at the top, upside down though it is. 00:05:11.765 --> 00:05:15.016 Ride along with the water and there is a second account of why. 00:05:15.213 --> 00:05:19.811 From inside, something presses the water outward, into the base of the bucket. 00:05:19.915 --> 00:05:22.168 We call that the centrifugal force. 00:05:22.547 --> 00:05:23.869 It is not a real force. 00:05:23.952 --> 00:05:25.181 Nothing is pushing. 00:05:25.264 --> 00:05:28.965 It appears only because the frame you chose to describe things in is spinning, 00:05:29.048 --> 00:05:32.764 and its size is your speed squared over the radius, times your mass. 00:05:33.561 --> 00:05:35.489 And look which mass that is. 00:05:35.686 --> 00:05:37.996 Not some new centrifugal charge. 00:05:38.100 --> 00:05:42.640 It is the inertial mass, the very same stubbornness from Newton's second law. 00:05:43.437 --> 00:05:45.760 Here, though, that is no coincidence. 00:05:45.863 --> 00:05:48.175 It could not have come out any other way. 00:05:48.371 --> 00:05:51.541 The reason you feel pressed outward is precisely that your body wants 00:05:51.541 --> 00:05:53.121 to keep going straight. 00:05:53.317 --> 00:05:57.591 Inertia is the whole cause, so inertia is the whole charge. 00:05:58.387 --> 00:06:00.954 So write two facts side by side. 00:06:01.150 --> 00:06:05.375 Any inertial force, by construction, couples to the inertial mass. 00:06:05.458 --> 00:06:09.218 And gravity, as we measured to fifteen decimal places, 00:06:09.301 --> 00:06:11.728 couples to the inertial mass as well. 00:06:12.339 --> 00:06:16.392 Which raises the question Einstein called the happiest thought of his life. 00:06:17.006 --> 00:06:19.596 What if gravity is an inertial force? 00:06:19.700 --> 00:06:24.148 Not a force at all, but the price of describing motion in the wrong frame. 00:06:24.851 --> 00:06:28.637 That guess is even permitted only because the two masses are equal. 00:06:28.741 --> 00:06:30.819 Try the same move on electricity and it 00:06:30.819 --> 00:06:36.253 collapses at once, because electric charge is nothing whatsoever like inertial mass. 00:06:37.049 --> 00:06:40.243 But it sounds mad, and here is exactly why. 00:06:40.555 --> 00:06:44.713 An inertial force is what you feel when you are not moving along a straight line. 00:06:44.909 --> 00:06:47.836 Move along a straight line and you feel nothing at all. 00:06:48.632 --> 00:06:52.369 So if gravity is one of those, then the astronaut floating in orbit, 00:06:52.452 --> 00:06:55.239 feeling nothing, is the one going straight. 00:06:55.691 --> 00:07:00.046 And you, held up by your chair, feeling that push in your back, are not. 00:07:00.842 --> 00:07:04.082 Which means we are about to be very wrong about straight lines. 00:07:05.150 --> 00:07:08.202 Here is a graph of height above the ground against time. 00:07:08.285 --> 00:07:09.874 This flat line is you, 00:07:09.957 --> 00:07:14.172 sitting in your chair, staying at the same height while time runs on. 00:07:14.713 --> 00:07:18.835 And this arc is a piece of chalk, thrown up and caught again. 00:07:19.032 --> 00:07:22.374 The chalk is in free fall the whole way, so it feels nothing, 00:07:22.457 --> 00:07:25.940 while you feel the chair pressing into your back the entire time. 00:07:26.644 --> 00:07:28.802 If gravity is an inertial force, 00:07:28.885 --> 00:07:32.914 then the chalk is the one travelling in a straight line, and you are not. 00:07:33.111 --> 00:07:35.909 Drawn like this, that is plainly absurd. 00:07:36.106 --> 00:07:40.298 One of them is straight, the other is bent, and I can see which. 00:07:41.211 --> 00:07:43.371 Unless the picture is lying to us. 00:07:43.567 --> 00:07:45.716 Which is a thing that pictures do. 00:07:46.741 --> 00:07:50.851 You have met this exact lie on an aeroplane. Here is the map on the seat 00:07:50.851 --> 00:07:55.020 in front of you, with San Francisco on the left and London on the right. 00:07:55.561 --> 00:07:59.231 The obvious shortest route is the dashed line straight across. 00:07:59.428 --> 00:08:02.804 But the aircraft flies the red one, up over Greenland, 00:08:02.887 --> 00:08:05.744 which on this map looks like a wasteful detour. 00:08:06.540 --> 00:08:10.163 Now put the same two cities on the actual Earth, which is a sphere. 00:08:10.960 --> 00:08:15.233 The shortest path between two points on a sphere is an arc of a great circle. 00:08:15.430 --> 00:08:16.531 Here is that arc, 00:08:16.614 --> 00:08:20.714 and it does run up over the north, exactly where the airline said it would. 00:08:21.255 --> 00:08:25.806 The detour was never a detour. The flat map bent it, because that map is trying 00:08:25.806 --> 00:08:30.706 to draw a curved Earth on a flat page, and something always has to give. 00:08:30.903 --> 00:08:34.062 What gives is your idea of which lines are straight. 00:08:35.276 --> 00:08:38.281 Now back to our graph, with the same accusation. 00:08:38.364 --> 00:08:41.268 That picture drew time and height on a flat grid. 00:08:41.580 --> 00:08:45.006 Spacetime is not flat: matter curves it. 00:08:45.319 --> 00:08:46.629 On the curved thing, 00:08:46.712 --> 00:08:50.904 the arc is the straight line, and the level line is the bent one. 00:08:51.956 --> 00:08:55.161 So the astronaut, feeling nothing, goes straight. 00:08:55.358 --> 00:08:57.028 You, held up by your chair, 00:08:57.111 --> 00:09:01.524 are being pushed off the straight line, and the weight you feel is the price. 00:09:02.138 --> 00:09:04.192 Gravity is not a force in the picture. 00:09:04.275 --> 00:09:06.249 It is a property of the picture. 00:09:06.558 --> 00:09:08.670 That leaves exactly one job: 00:09:08.753 --> 00:09:12.687 say precisely how much matter curves spacetime, and how much. 00:09:12.770 --> 00:09:17.113 It took Einstein eight further years, and this is the answer he arrived at. 00:09:17.596 --> 00:09:20.195 On the left, the curvature of spacetime. 00:09:20.278 --> 00:09:23.852 On the right, all the matter and all the energy there is. 00:09:23.935 --> 00:09:25.549 Read it as a slogan: 00:09:25.653 --> 00:09:31.145 matter tells spacetime how to curve, and curved spacetime tells matter how to move. 00:09:31.686 --> 00:09:34.030 Newton's law of gravity is gone. 00:09:34.113 --> 00:09:38.929 Newton's first law survives, and reads better than before: with no force on you, 00:09:39.012 --> 00:09:40.694 you travel in a straight line. 00:09:40.777 --> 00:09:42.856 We simply had the wrong lines. 00:09:43.949 --> 00:09:48.114 Curved spacetime has consequences, and the first is about clocks. 00:09:48.198 --> 00:09:53.482 Here is a mass, here is a clock sitting close to it, and here is one far away. 00:09:53.794 --> 00:09:55.455 Neither of them is moving. 00:09:56.252 --> 00:09:58.737 The one down low ticks slower. 00:09:58.841 --> 00:10:03.416 Not because of any motion, but because it sits deeper in the gravitational well. 00:10:03.868 --> 00:10:07.689 The whole effect is this square root, and it is less than one. 00:10:08.602 --> 00:10:10.956 This has been measured for seventy years. 00:10:11.040 --> 00:10:14.605 Two atomic clocks at two heights in one building disagree. 00:10:14.802 --> 00:10:17.098 Satellite navigation has to correct for it, 00:10:17.182 --> 00:10:20.422 or the position it reports would wander off within minutes. 00:10:21.636 --> 00:10:25.817 And notice that this is not the symmetric effect of special relativity. 00:10:25.920 --> 00:10:30.147 There, each of two moving observers sees the other one running slow. 00:10:30.344 --> 00:10:33.127 Here we both agree who is deeper in the well, 00:10:33.211 --> 00:10:36.057 so we both agree whose clock is the slow one. 00:10:36.853 --> 00:10:40.101 Now send a photon up from the low clock to the high one. 00:10:40.185 --> 00:10:41.625 Everything down there looks slow 00:10:41.625 --> 00:10:45.561 from up here, and that includes the oscillation of a light wave. 00:10:45.905 --> 00:10:49.621 So the light arrives with a lower frequency than it left with. 00:10:49.817 --> 00:10:53.205 Lower frequency means shifted toward the red end of the spectrum, 00:10:53.289 --> 00:10:55.635 and it means less energy carried. 00:10:55.831 --> 00:11:00.975 That is the gravitational redshift, and we see it in light climbing out of the Sun. 00:11:01.772 --> 00:11:02.768 Turn it around, 00:11:02.852 --> 00:11:08.019 and light falling down arrives blue, carrying more energy than it started with. 00:11:08.331 --> 00:11:11.673 Whatever the exchange rate is for time between two heights, 00:11:11.757 --> 00:11:14.451 it is also the exchange rate for energy. 00:11:15.155 --> 00:11:17.930 The third consequence is that light bends. 00:11:18.034 --> 00:11:23.468 Here is the Sun, and here is a star far off to the left, sending a ray past it. 00:11:23.572 --> 00:11:28.031 If gravity did nothing, that ray would run dead flat along the dashed line. 00:11:28.735 --> 00:11:29.939 It does not. 00:11:30.023 --> 00:11:34.749 The ray is pulled toward the Sun, and the observer over on the right, looking back 00:11:34.749 --> 00:11:39.765 along the direction it arrives from, sees the star shifted away from the Sun's edge. 00:11:40.562 --> 00:11:42.498 Treat light as a fast particle 00:11:42.582 --> 00:11:47.273 and Newton's gravity predicts a deflection of two G M over c squared b. 00:11:47.586 --> 00:11:49.789 Einstein's theory predicts four. 00:11:49.873 --> 00:11:51.952 Exactly twice as much. 00:11:52.563 --> 00:11:57.742 That factor of two was testable, if only you could see stars right beside the Sun. 00:11:57.938 --> 00:12:00.954 You cannot, except during a total eclipse, 00:12:01.038 --> 00:12:04.812 when the Moon covers the disc and the sky beside it goes dark. 00:12:05.609 --> 00:12:09.556 Eddington's expedition of nineteen nineteen measured it, came back 00:12:09.556 --> 00:12:14.131 with Einstein's number rather than Newton's, and made him famous overnight. 00:12:14.327 --> 00:12:18.960 A British expedition confirming a German theory, months after the war. 00:12:20.175 --> 00:12:23.125 Newton unified the falling apple and the planets. 00:12:23.321 --> 00:12:27.002 This unified the apple, the planets, and the light. 00:12:28.052 --> 00:12:30.444 Now take the whole argument to its limit. 00:12:30.896 --> 00:12:33.579 To leave a body you need escape velocity: 00:12:33.683 --> 00:12:36.852 enough kinetic energy to pay off the gravitational binding energy 00:12:36.852 --> 00:12:38.119 at its surface. 00:12:39.171 --> 00:12:43.433 Solve that for the speed and you get the square root of two G M over r. 00:12:43.629 --> 00:12:46.602 For the Earth it comes to eleven kilometres a second. 00:12:46.799 --> 00:12:49.865 Make the body more compact and the speed climbs. 00:12:50.777 --> 00:12:52.160 Keep squeezing. 00:12:52.356 --> 00:12:54.074 Here, for comparison, is the speed 00:12:54.074 --> 00:12:58.754 of light, and there is the moment the escape velocity matches it exactly. 00:12:59.551 --> 00:13:02.137 Set the two equal and solve for the radius. 00:13:02.221 --> 00:13:04.424 The mass of the escaping object cancels, 00:13:04.508 --> 00:13:09.397 and what is left is two G M over c squared, the Schwarzschild radius. 00:13:09.594 --> 00:13:13.750 People wrote that expression down in the seventeen hundreds and wondered. 00:13:13.947 --> 00:13:15.213 Now we know. 00:13:15.754 --> 00:13:20.480 Newton's law says nothing special happens at that radius. Einstein's does. 00:13:20.584 --> 00:13:24.148 For an observer trying to hover at rest, the pull is the inverse 00:13:24.148 --> 00:13:27.806 square law multiplied by one over that same square root. 00:13:28.603 --> 00:13:30.307 The grey curve is Newton. 00:13:30.391 --> 00:13:34.757 The red one is the truth, and it sits above the grey one everywhere. 00:13:34.953 --> 00:13:39.560 Gravity in general relativity is stronger at short range, not weaker, 00:13:39.644 --> 00:13:42.106 which is the opposite of what would rescue us. 00:13:43.019 --> 00:13:44.726 Now walk inward. 00:13:44.923 --> 00:13:50.461 Hover out here and you need a certain thrust. Closer in, more thrust. 00:13:50.658 --> 00:13:52.397 And at the Schwarzschild radius, 00:13:52.481 --> 00:13:56.789 the dashed line, the thrust you need runs away to infinity. 00:13:57.492 --> 00:14:01.592 So there is no hovering at that surface, or anywhere inside it. 00:14:01.904 --> 00:14:05.573 Fire your rocket as hard as you like and you still go in. 00:14:05.677 --> 00:14:08.186 That surface is the event horizon. 00:14:08.797 --> 00:14:12.896 It is not a wall, and nothing about crossing it feels violent. 00:14:13.092 --> 00:14:16.656 For a big enough hole the tidal stretching there is gentle and you sail 00:14:16.656 --> 00:14:18.642 through noticing nothing at all. 00:14:18.746 --> 00:14:22.415 What kills you is the singularity at the centre, much later. 00:14:23.212 --> 00:14:26.034 And from far away I never see you cross. 00:14:26.231 --> 00:14:31.293 Your clock, by that same square root, runs slower and slower as you approach. 00:14:31.490 --> 00:14:33.449 Your light stretches redder and redder. 00:14:33.533 --> 00:14:36.042 You fade out rather than arrive. 00:14:37.856 --> 00:14:40.214 Let us put the whole thing back together. 00:14:40.411 --> 00:14:43.113 Einstein's equation states it in one line, 00:14:43.197 --> 00:14:46.356 and the argument that got us there came in three steps. 00:14:47.268 --> 00:14:49.150 First, the speed limit. 00:14:49.347 --> 00:14:51.747 Nothing carries an influence faster than light, 00:14:51.831 --> 00:14:54.827 and Newtonian gravity plainly broke that rule. 00:14:55.624 --> 00:14:58.268 Second, the coincidence Newton shrugged at. 00:14:58.352 --> 00:15:01.104 The mass that resists being accelerated and the mass 00:15:01.104 --> 00:15:06.062 that gravity pulls on are one number, agreeing to fifteen decimal places. 00:15:06.859 --> 00:15:10.041 And third, the price of taking that seriously. 00:15:10.238 --> 00:15:12.046 Gravity is not a force at all. 00:15:12.130 --> 00:15:13.776 It is the shape of spacetime, 00:15:13.860 --> 00:15:17.681 and free fall is what a straight line looks like once you admit the shape. 00:15:18.477 --> 00:15:22.309 It took Einstein ten years, and it grew out of a coincidence that everyone else 00:15:22.309 --> 00:15:24.388 had already seen and let pass.