1 00:00:00,000 --> 00:00:03,901 So we have an understanding of the special theory of relativity, and we 2 00:00:03,901 --> 00:00:08,572 understand that the universe is Lorentz invariant, and we are now in the position 3 00:00:08,572 --> 00:00:11,375 that Albert Einstein finds himself in, in 1905, 1906. 4 00:00:11,375 --> 00:00:15,606 He has produced the special theory, and he now is trying to understand how to 5 00:00:15,606 --> 00:00:19,476 write relativistic physics. And he runs into the problem that gravity 6 00:00:19,476 --> 00:00:22,545 is not as we understand it, a good relativistic theory, 7 00:00:22,545 --> 00:00:25,046 and we can understand what is wrong with it. 8 00:00:25,046 --> 00:00:29,820 Look at what the fate familiar form of Newton's gravitational force law and 9 00:00:29,820 --> 00:00:33,798 everything about it is wrong. For example there is a denominator there, 10 00:00:33,798 --> 00:00:35,390 r squared. Who's measuring r? 11 00:00:35,390 --> 00:00:39,993 Remember if someone is moving, they would measure a different distance because of 12 00:00:39,993 --> 00:00:42,210 Lorentz contraction. Whose r do you use? 13 00:00:42,210 --> 00:00:46,637 Worse than that, what do you, when do you measure this distance? 14 00:00:46,637 --> 00:00:52,013 If I take the say the sum, at this point, and I somehow manage to move it, two 15 00:00:52,013 --> 00:00:56,036 astronomical units to the left. what happens to the force on Earth? 16 00:00:56,036 --> 00:01:00,059 Does Earth immediately respond? Because when do you measure this R? 17 00:01:00,059 --> 00:01:04,666 Earth is not allowed to immediately respond, because Earth is not allowed to 18 00:01:04,666 --> 00:01:09,739 know about the fact that the sun has moved until s- enough time, seven minutes 19 00:01:09,739 --> 00:01:14,112 or whatever it is, has passed for light to have arrived from the sun to the 20 00:01:14,112 --> 00:01:18,835 Earth, because no information is allowed to tr- to be transferred faster than the 21 00:01:18,835 --> 00:01:21,855 speed of light. Now, we know the resolution of this. 22 00:01:21,855 --> 00:01:26,147 Maxwell's equations, how do they do it? remember, Maxwell's equations, 23 00:01:26,147 --> 00:01:31,113 essentially, start with Coulomb's force law between charges, remarkably similar 24 00:01:31,113 --> 00:01:33,750 to Matt, Newton's force law between masses. 25 00:01:33,750 --> 00:01:36,376 And you know how this is resolved in Maxwell's theory. 26 00:01:36,376 --> 00:01:38,564 The way it's resolved in Maxwell's theory is, 27 00:01:38,564 --> 00:01:41,968 if you take a proton and an electron, and you move the proton suddenly, 28 00:01:41,968 --> 00:01:44,205 the electron does not respond instantaneously, 29 00:01:44,205 --> 00:01:47,366 because the electric field where the electron is has not changed. 30 00:01:47,366 --> 00:01:50,819 The motion of the proton generates an accelerating charge, generates an 31 00:01:50,819 --> 00:01:53,785 electromagnetic wave. That wave is a ripple in the field that 32 00:01:53,785 --> 00:01:57,287 changes it from the value associated to the first position to the value 33 00:01:57,287 --> 00:02:00,497 associated to the final position, and when this ripple gets to the 34 00:02:00,497 --> 00:02:04,192 electron, then the electron responds. And everything is nice and relativistic 35 00:02:04,192 --> 00:02:06,961 and Lowrance invariant. So perhaps we can take Maxwell's 36 00:02:06,961 --> 00:02:09,244 equations, apply them to gravity, and have a 37 00:02:09,244 --> 00:02:12,855 relativistic version of gravity. And this is, indeed, Einstein's first 38 00:02:12,855 --> 00:02:14,714 choice. The problem is a subtle one. 39 00:02:14,714 --> 00:02:18,696 what we need is a field theory. We need something that will carry at the 40 00:02:18,696 --> 00:02:21,830 speed of light the information from one place to the other. 41 00:02:21,830 --> 00:02:25,708 Well, Einstein's aware of this. he attempts to use Maxwell's equations, 42 00:02:25,708 --> 00:02:28,348 but he immediately knows that, that will not work. 43 00:02:28,348 --> 00:02:32,389 The difference between this equation and this equation is an important one. 44 00:02:32,389 --> 00:02:35,298 The quantity appearing here is, the electric charge. 45 00:02:35,298 --> 00:02:37,668 The electric charge is a conserved quantity, 46 00:02:37,668 --> 00:02:40,955 and Maxwell's equations imply in fact charge conservation, 47 00:02:40,955 --> 00:02:43,217 whereas the object appearing here is mass. 48 00:02:43,217 --> 00:02:45,750 And in Einstein's theory, mass is not conserved. 49 00:02:45,750 --> 00:02:47,048 Okay. We'll modify things. 50 00:02:47,048 --> 00:02:50,734 What is conserved, instead of mass. What is conserved as energy momentum? 51 00:02:50,734 --> 00:02:54,525 But energy momentum is a completely different beast than electric charge. 52 00:02:54,525 --> 00:02:57,121 Electric charge, remember, is Lorentz invariant. 53 00:02:57,121 --> 00:03:01,223 Every inertial observer measures the same amount of electric charge in a given 54 00:03:01,223 --> 00:03:03,871 region in space. On the other hand, mass is, or energy 55 00:03:03,871 --> 00:03:07,661 momentum is not Lorentz invariant. Different observers measure different 56 00:03:07,661 --> 00:03:11,867 energies and different momentum related, like T and X, by Lorentz transformations. 57 00:03:11,867 --> 00:03:14,048 This complicates the theory significantly. 58 00:03:14,048 --> 00:03:18,224 There's also philosophical question that a actually turns out to be a very 59 00:03:18,224 --> 00:03:21,548 important for the answer, which is, what is a gravitational force? 60 00:03:21,548 --> 00:03:23,541 So how do you measure a gravitational force? 61 00:03:23,541 --> 00:03:26,033 Well let's start. How do you measure the electric force? 62 00:03:26,033 --> 00:03:29,385 I want to measure the electric field at this point and space at this time. 63 00:03:29,385 --> 00:03:32,419 Well I bring along something charged and I measure the force on it. 64 00:03:32,419 --> 00:03:35,318 Divide the force by the charge. The answer is the electric field. 65 00:03:35,318 --> 00:03:38,942 You see none, perhaps you were mistaken. Perhaps what you were measuring was just 66 00:03:38,942 --> 00:03:42,339 the gravitational force on this object. I said, how, okay, I'll bring another 67 00:03:42,339 --> 00:03:45,420 object with the same mass as the first but with the opposite charge. 68 00:03:45,420 --> 00:03:48,640 Now, the gravitational force will be in the same direction, the electric force in 69 00:03:48,640 --> 00:03:51,177 the opposite direction. I'll look at the difference between the 70 00:03:51,177 --> 00:03:54,438 two, and the change is the elec, the one that changed between the positive and the 71 00:03:54,438 --> 00:03:56,210 negative particle. That's the electric force. 72 00:03:56,210 --> 00:03:58,392 Good. So now do the same for the elec, for the 73 00:03:58,392 --> 00:04:00,866 gravitational force. So you bring along some object. 74 00:04:00,866 --> 00:04:03,242 You measure the force that gravity applies to it. 75 00:04:03,242 --> 00:04:06,492 You divide that by its mass. You get the gravitational acceleration. 76 00:04:06,492 --> 00:04:08,820 Very good. Now bring another object upon oh wait, 77 00:04:08,820 --> 00:04:12,166 there is no object upon which gravity acts in the opposite direction. 78 00:04:12,166 --> 00:04:15,949 There's not even an object upon which gravity does not act that you can bring 79 00:04:15,949 --> 00:04:19,215 and replace it with. In fact all objects fall with the same 80 00:04:19,215 --> 00:04:23,981 acceleration there's this universality that suggests that thinking of gravity as 81 00:04:23,981 --> 00:04:28,776 a force is kind of difficult, because you can't even really operationally define 82 00:04:28,776 --> 00:04:32,212 how to measure it. Thinking along these lines, Einstein is 83 00:04:32,212 --> 00:04:36,913 working through it, and in 1907, he has what he considers and in writing, his 84 00:04:36,913 --> 00:04:40,349 happiest thought ever. And his happiest thought basically 85 00:04:40,349 --> 00:04:44,387 involved jumping off the roof. If you jump off the roof he realizes, 86 00:04:44,387 --> 00:04:47,100 there is no gravity until you hit the ground. 87 00:04:47,100 --> 00:04:52,245 This, the, the idea that Einstein is discovering Is and discovering the 88 00:04:52,245 --> 00:04:54,817 importance of is the principal of equivalence. 89 00:04:54,817 --> 00:04:58,932 We made a big deal, a few weeks ago, about the fact that when you are in free 90 00:04:58,932 --> 00:05:02,748 fall there is no gravity, and I emphasize that this is not some random idea. 91 00:05:02,748 --> 00:05:06,513 In space everything is in free fall. Stars, galaxies, planets, they're all in 92 00:05:06,513 --> 00:05:10,380 gravitational free fall, so they're not experiencing any gravitational force. 93 00:05:10,380 --> 00:05:14,012 The converse is, in fact, also true. I can play with the apparent 94 00:05:14,012 --> 00:05:17,240 gravitational force. By, accelerating, measuring, making 95 00:05:17,240 --> 00:05:20,873 measurements in an accelerating frame. Not in an inertial frame. 96 00:05:20,873 --> 00:05:24,851 Being an observer who is accelerating. For example, we've all had this 97 00:05:24,851 --> 00:05:27,849 experience. As the elevator starts going up, you feel 98 00:05:27,849 --> 00:05:31,654 as though your weight had increased. Why does your weight increase? 99 00:05:31,654 --> 00:05:36,208 Because now, the elevator floor has not only to hold you up against the pull of 100 00:05:36,208 --> 00:05:39,322 Earth's gravity, but also to supply the extra force to 101 00:05:39,322 --> 00:05:42,244 accelerate you up. And so the floor is pushing on you 102 00:05:42,244 --> 00:05:44,721 harder. That's the experience of weighing more, 103 00:05:44,721 --> 00:05:47,638 precisely. And so the gravitational acceleration at 104 00:05:47,638 --> 00:05:50,575 a given point, at any given time, depends on your 105 00:05:50,575 --> 00:05:53,950 acceleration, and by picking the suitable acceleration, 106 00:05:53,950 --> 00:05:56,763 ie letting go of everything and falling down, 107 00:05:56,763 --> 00:06:02,077 I can make the gravitational acceleration relative to me at any space or point in 108 00:06:02,077 --> 00:06:04,170 time be zero. How do I do that? 109 00:06:04,170 --> 00:06:07,051 Just let go, and when I free fall I don't experience gravity. 110 00:06:07,051 --> 00:06:10,893 The gravitational acceleration I then measure when I drop something like when I 111 00:06:10,893 --> 00:06:13,726 jumped off that chair. When I was jumping and I dropped the 112 00:06:13,726 --> 00:06:16,751 ball, I measured the gravitational acceleration of exactly zero. 113 00:06:16,751 --> 00:06:20,545 So Einstein realizes this, and then maybe the whole idea of gravity needs to be 114 00:06:20,545 --> 00:06:22,898 abolished. You don't need to think of gravity you 115 00:06:22,898 --> 00:06:26,212 just need to understand that we are working in an accelerating frame. 116 00:06:26,212 --> 00:06:30,006 So everywhere the surface of the earth should be thought of as accelerating up. 117 00:06:30,006 --> 00:06:33,223 I mean of course it's not getting anywhere, but accelerating up and 118 00:06:33,223 --> 00:06:36,579 relative to some rest, inertial frame, which is in free fall. 119 00:06:36,579 --> 00:06:40,756 And because the floor is accelerating up, that is why we experience weight. 120 00:06:40,756 --> 00:06:45,014 This would work well were it not for the tricky fact that Earth is a sphere. 121 00:06:45,014 --> 00:06:49,347 You can't accelerate the entire surface of Earth up, you can accelerate it out. 122 00:06:49,347 --> 00:06:53,846 But if it did accelerate out, then it would not only be pushing us up but also 123 00:06:53,846 --> 00:06:56,180 we'd be accelerating away from each other. 124 00:06:56,180 --> 00:06:59,414 We know how that happens. That has to do with tidal forces. 125 00:06:59,414 --> 00:07:03,206 What makes gravity, gravity, what distinguishes gravity from, just the 126 00:07:03,206 --> 00:07:07,389 result of measuring an accelerating frame, what, the actual gravitational 127 00:07:07,389 --> 00:07:11,739 force lies in, is in those tidal forces of which I made a big deal a couple of 128 00:07:11,739 --> 00:07:14,639 weeks ago. The fact that the acceleration of gravity 129 00:07:14,639 --> 00:07:17,148 is not the same everywhere in space and time, 130 00:07:17,148 --> 00:07:21,945 that is what the result of the fact that, that there is actually a gravitating body 131 00:07:21,945 --> 00:07:24,670 called Earth. If the gravitational acceleration were 132 00:07:24,670 --> 00:07:27,952 equal everywhere, then, you're actual in an accelerating elevator. 133 00:07:27,952 --> 00:07:31,799 Get out of the elevator, and you'll measure a gravitational acceleration of 134 00:07:31,799 --> 00:07:33,748 zero. So this idea proves to be key in 135 00:07:33,748 --> 00:07:37,800 Einstein's solution of the problem of finding a relativistic theory of gravity, 136 00:07:37,800 --> 00:07:42,006 but right at the top, we can use it, to good effect. So let's learn one important 137 00:07:42,006 --> 00:07:44,263 thing about gravity that will serve us well. 138 00:07:44,263 --> 00:07:47,333 So here's a aartoon version of the principle of equivalence. 139 00:07:47,333 --> 00:07:49,537 Here we have a space ship. It's out in space. 140 00:07:49,537 --> 00:07:52,441 There's no gravity. It's in the middle of nowhere in space. 141 00:07:52,441 --> 00:07:56,348 It's accelerating up at an acceleration that I choose to call G, and what that 142 00:07:56,348 --> 00:08:00,003 means is that if, if you sit here and throw a little rubber ball, well the 143 00:08:00,003 --> 00:08:03,859 rubber ball, of course, will, there's no, no gravity, there's nothing touching it. 144 00:08:03,859 --> 00:08:07,715 It will proceed to move in a straight line at a constant speed across the, 145 00:08:07,715 --> 00:08:10,069 across space. However, because the space ship is 146 00:08:10,069 --> 00:08:13,675 accelerating up, relative to the space ship, the ball is not accelerating. 147 00:08:13,675 --> 00:08:16,580 It's moving in a straight line. Relative to the space ship. 148 00:08:16,580 --> 00:08:20,807 The ball will appear to be accelerating down with acceleration g because the ball 149 00:08:20,807 --> 00:08:23,230 is not accelerating the spaceship is of course. 150 00:08:23,230 --> 00:08:26,595 Bring that same situation over to the right hand side, here, I have the same 151 00:08:26,595 --> 00:08:28,660 spaceship. It's sitting on the ground, it's not 152 00:08:28,660 --> 00:08:30,858 going anywhere but there's a gravitational force. 153 00:08:30,858 --> 00:08:33,327 A gravitational acceleration of g, as there is on Earth. 154 00:08:33,327 --> 00:08:36,603 And if I throw the rubber ball, well then it will fall on the ground with 155 00:08:36,603 --> 00:08:39,116 acceleration g. From the point of view of looking inside 156 00:08:39,116 --> 00:08:42,392 the spaceship, you cannot tell the difference between being on the ground 157 00:08:42,392 --> 00:08:45,533 and experiencing gravitational acceleration, or being in the middle of 158 00:08:45,533 --> 00:08:48,653 nowhere and accelerating up. Good, so that's the principle of 159 00:08:48,653 --> 00:08:50,458 equivalence. We know about that. 160 00:08:50,458 --> 00:08:52,671 How do we use, do something useful with it? 161 00:08:52,671 --> 00:08:57,038 Well, here's what Einstein says. Imagine putting a little blue light bulb 162 00:08:57,038 --> 00:09:01,813 at the tail end of our spaceship and let this little blue light bulb make a pulse 163 00:09:01,813 --> 00:09:05,015 of light. A little short wavelength pulse of blue 164 00:09:05,015 --> 00:09:07,344 light. Now when the pulse is emitted, the 165 00:09:07,344 --> 00:09:09,965 spaceship is moving maybe at some velocity v. 166 00:09:09,965 --> 00:09:14,082 What v is, is completely irrelevant. I can always pick an observer that is 167 00:09:14,082 --> 00:09:18,368 moving at a speed v and then for him the initial speed of this spaceship is zero. 168 00:09:18,368 --> 00:09:22,442 And were the spaceship moving at a constant speed, indeed then this pulse of 169 00:09:22,442 --> 00:09:26,727 light will travel at the exactly at the speed of light relative to the spaceship 170 00:09:26,727 --> 00:09:31,013 up to the top of the spaceship and if I put a detector at the top to absorb it I 171 00:09:31,013 --> 00:09:34,890 would observe that there was no Doppler shift, because detector and a, 172 00:09:34,890 --> 00:09:38,927 emitter are moving at zero relative velocity, like moves at the speed of 173 00:09:38,927 --> 00:09:41,458 light anyway. there would be zero redshift. 174 00:09:41,458 --> 00:09:45,977 However, in this case the spaceship is accelerating if the height of the 175 00:09:45,977 --> 00:09:51,096 spaceship is denoted by H, then it takes a time of the order of H divided by the 176 00:09:51,096 --> 00:09:56,065 speed of light, for this pulse of light to climb to the top of the space ship. 177 00:09:56,065 --> 00:10:01,356 If the space ship is accelerating with acceleration G, the, the change in its 178 00:10:01,356 --> 00:10:06,389 speed during the, this time, while the photon is moving up the space ship is of 179 00:10:06,389 --> 00:10:09,357 the order of its acceleration times that time. 180 00:10:09,357 --> 00:10:14,454 And so, when the photon is absorbed, the space ship is no longer moving at speed 181 00:10:14,454 --> 00:10:15,810 V. It's moving faster. 182 00:10:15,810 --> 00:10:21,058 The result will be of course, that, you can imagine V to be zero by picking an 183 00:10:21,058 --> 00:10:25,847 appropriate frame, you have something emitted, the observe-, the, reciever is 184 00:10:25,847 --> 00:10:28,734 not moving at the same speed as the emitter. 185 00:10:28,734 --> 00:10:33,130 The reciever is moving faster by this amount, and, we can use, are 186 00:10:33,130 --> 00:10:38,051 understanding of the Doppler formula to compute, the, degree by which, the, 187 00:10:38,051 --> 00:10:43,037 photon will reddened, because the receiver is, receiving from the observer, 188 00:10:43,037 --> 00:10:46,995 at this relative speed. And so dividing V by C, we find this is 189 00:10:46,995 --> 00:10:51,049 the non-relativistic expression for the Doppler shift were working with small 190 00:10:51,049 --> 00:10:55,154 age, and so in that approximation we can use the non-relativistic version we're 191 00:10:55,154 --> 00:10:57,856 not going to be doing anything superlativistic here. 192 00:10:57,856 --> 00:11:01,910 What's good about this is that now the principal of equivalence tells me that 193 00:11:01,910 --> 00:11:05,807 the exact same thing happen on the right if I put on the bottom floor of a 194 00:11:05,807 --> 00:11:10,068 building or the bottom of this space ship sitting on the ground, a blue flashlight 195 00:11:10,068 --> 00:11:13,862 and it makes a blue wave, that wave when absorbed at the top floor of the 196 00:11:13,862 --> 00:11:15,899 building. Will be red-shifted. 197 00:11:15,899 --> 00:11:20,893 And what this means is any light. Light is effected by gravity it has to be 198 00:11:20,893 --> 00:11:25,005 because of the principle of equivalence and, what we understand about special 199 00:11:25,005 --> 00:11:29,224 relativaty and, as you shine a light from the bottom floor to the top floor the 200 00:11:29,224 --> 00:11:33,176 light is reddened of course, the reddening is not significant, GH over C 201 00:11:33,176 --> 00:11:37,395 squared is not a large fraction, unless your building is exor, exorbitantly tall. 202 00:11:37,395 --> 00:11:41,507 But if you make a precision measurement you could do it and, this is not just 203 00:11:41,507 --> 00:11:45,459 about light remember, because right here is an atom, that is oscillating or 204 00:11:45,459 --> 00:11:49,792 whatever, a clock that is oscillating, with some Characteristic period as 205 00:11:49,792 --> 00:11:55,552 observed by someone at the top of the building clocks on the bottom floor runs 206 00:11:55,552 --> 00:12:00,438 slow the red shift remember is accompanied just as it was in the a 207 00:12:00,438 --> 00:12:04,740 relativistic red shift case it just reflects the change in. 208 00:12:04,740 --> 00:12:08,092 the way clocks work. So if you observe from the top floor a 209 00:12:08,092 --> 00:12:11,552 clock on the bottom floor, you will see that the clock runs slow. 210 00:12:11,552 --> 00:12:14,958 If you want to live a very long time, go live near the Dead Sea. 211 00:12:14,958 --> 00:12:17,662 You will age slower, albeit by a very small factor. 212 00:12:17,662 --> 00:12:18,797 Is this for real? Yes. 213 00:12:18,797 --> 00:12:23,056 This is definitely for real and in fact it was measured in a beautiful 214 00:12:23,056 --> 00:12:27,975 experimental tour de force by Pound Rebka in 1959 in this tower, over here on the 215 00:12:27,975 --> 00:12:32,348 left hand size of this building, the Jefferson Lab at Harvard University. 216 00:12:32,348 --> 00:12:36,843 The tower was designed initially to do some magnetism measurements in the 217 00:12:36,843 --> 00:12:40,183 nineteenth century. Pound Rebka re-purposed it. they put 218 00:12:40,183 --> 00:12:45,224 their emitter down on the bottom floor of this tower and the receiver at top floor 219 00:12:45,224 --> 00:12:49,325 of the tower and over this height. They found a very minuscule, but 220 00:12:49,325 --> 00:12:54,479 completely consistent with a Einstein's argument, redshift and then when they of 221 00:12:54,479 --> 00:12:59,759 course conversely went the admitted light at the top, it was glue shifted by the 222 00:12:59,759 --> 00:13:05,381 time it reached the bottom floor Everything is symmetric in that regard. 223 00:13:05,381 --> 00:13:08,206 But they needed, the, the, the, redshift was 224 00:13:08,206 --> 00:13:11,253 miniscule. They used a technique called Mossbauer 225 00:13:11,253 --> 00:13:15,530 spectroscopy to measure the wavelength of light with exquisite precision. 226 00:13:15,530 --> 00:13:20,745 These days, this gravitational red shift is all over the place the little GPS 227 00:13:20,745 --> 00:13:25,198 reciever in you cellphone knows about the gravitational red shift so it can 228 00:13:25,198 --> 00:13:29,534 translate the time of travel of the signals that it gets from various GPS 229 00:13:29,534 --> 00:13:34,456 satellites to their distance in a correct way taking into account that the receiver 230 00:13:34,456 --> 00:13:37,972 is here on the ground and the satellites are up in space.