1 00:00:01,100 --> 00:00:03,573 Okay. So we found some, at the, at the cost of 2 00:00:03,573 --> 00:00:06,666 a lot of time. We found some nice property of Newtonian 3 00:00:06,666 --> 00:00:09,139 mechanics. Well, invariant, under these 4 00:00:09,139 --> 00:00:13,469 transformations of Galilean relativity, as observed by one observer and as 5 00:00:13,469 --> 00:00:17,629 observed by another observer, moving relative to the first with a constant 6 00:00:17,629 --> 00:00:19,260 velocity. Physics is the same. 7 00:00:19,260 --> 00:00:23,273 What makes that a principle is what you do when it doesn't work. 8 00:00:23,273 --> 00:00:27,538 And when it doesn't work is when we introduce Maxwell's equations. 9 00:00:27,538 --> 00:00:31,614 Electrodynamics causes a problem. Remember, Maxwell wrote down his 10 00:00:31,614 --> 00:00:35,948 equations as a description of the interactions of electric currents and 11 00:00:35,948 --> 00:00:40,197 electric charges, and magnetic fields and so on. And he found when he wrote the 12 00:00:40,197 --> 00:00:44,446 right equations, that his equations had emitted wave solutions, and he computed 13 00:00:44,446 --> 00:00:48,749 from known measured quantities the speed of those waves, and discovered that it 14 00:00:48,749 --> 00:00:53,107 agreed with the measured speed of light, and therefore, had discovered that light 15 00:00:53,107 --> 00:00:59,050 was an electromagnetic phenomenon. But he measured the speed of light, 16 00:00:59,050 --> 00:01:01,960 and what that means is, now that you remember 17 00:01:01,960 --> 00:01:06,112 our velocity edition formula, was that the speed of light as measured by the 18 00:01:06,112 --> 00:01:10,210 black observer, or the speed of light as measured by the blue observer, they 19 00:01:10,210 --> 00:01:14,417 should get different results depending upon which way they're moving. If they 20 00:01:14,417 --> 00:01:18,679 both time the motion of a pulse of light, they should obtain different results. 21 00:01:18,679 --> 00:01:23,105 So Maxwell's equations can at most hold for one of those observers and the other 22 00:01:23,105 --> 00:01:27,531 observer must detect different electric and magnetic phenomena so that when they 23 00:01:27,531 --> 00:01:31,519 compute the speed of light they will get an answer consistent with their 24 00:01:31,519 --> 00:01:34,423 measurements. A technical way of saying this is that 25 00:01:34,423 --> 00:01:38,696 the electromagnetism, the equations, Maxwell's equations are not invariant, do 26 00:01:38,696 --> 00:01:43,195 not retain their form when you do the change of variables that we call Galilean 27 00:01:43,195 --> 00:01:46,850 relativity transformations, what we computed in the previous clip. 28 00:01:46,850 --> 00:01:49,325 So there are two possible solutions to this. 29 00:01:49,325 --> 00:01:53,142 I mean look, Newton, albeit with some error, computed the speed of sound 30 00:01:53,142 --> 00:01:57,532 already and sound moves with a speed that you can compute and it moves with that 31 00:01:57,532 --> 00:02:01,813 speed through air, and if you are moving through the air or if there is a brisk 32 00:02:01,813 --> 00:02:06,094 wind blowing, then of course sound will move at two different speeds, upwind and 33 00:02:06,094 --> 00:02:10,104 downwind, because it moves at a constant speed through the air and then in 34 00:02:10,104 --> 00:02:12,380 addition is carried on by the air's motion. 35 00:02:12,380 --> 00:02:15,556 So, the solution that Maxwell, in fact, adopt, suggested, 36 00:02:15,556 --> 00:02:19,527 is that there is something called, he, he called the luminiferous ether, 37 00:02:19,527 --> 00:02:21,966 some medium through which light propagates. 38 00:02:21,966 --> 00:02:26,221 And it's the properties of this, this ether that cause light to propagate at 39 00:02:26,221 --> 00:02:30,816 speed c, much as it is the properties of air that cause sound to propagate at the 40 00:02:30,816 --> 00:02:34,616 speed at which it propagates. And Maxwell's equations would then be 41 00:02:34,616 --> 00:02:37,510 valid for observers at rest relative to this ether. 42 00:02:37,510 --> 00:02:40,798 Just like the speed of sound would be measured correctly for an observer. 43 00:02:40,798 --> 00:02:44,042 If someone is, there is a brisk wind blowing, you would measure different 44 00:02:44,042 --> 00:02:47,500 properties of air, compared to what you would measure in a, 45 00:02:47,500 --> 00:02:51,819 steady, stationary air, and when you are moving relative to the ether, the 46 00:02:51,819 --> 00:02:56,253 velocity edition formula will tell you, you will measure a different speed of 47 00:02:56,253 --> 00:02:59,593 propagation of light. The other, much more radical solution, 48 00:02:59,593 --> 00:03:04,315 which will turn out to be experimentally, verifiably correct in fact, is that light 49 00:03:04,315 --> 00:03:07,080 propagates with a uniform speed C through space. 50 00:03:07,080 --> 00:03:12,533 there is no real ether, no material implementation of what the medium through 51 00:03:12,533 --> 00:03:15,902 which light propagates. The medium if you want is space. 52 00:03:15,902 --> 00:03:20,498 Maxwell's equations, in fact, hold for all inertial observers whatever their 53 00:03:20,498 --> 00:03:24,113 constant velocity. Remember an inertial observer is one for 54 00:03:24,113 --> 00:03:28,279 who Newton's first law holds. That means that all inertial observers 55 00:03:28,279 --> 00:03:31,772 relative to each other, are moving at constant velocities. 56 00:03:31,772 --> 00:03:36,130 We don't know who the rest observer is, but we know what the collection of all 57 00:03:36,130 --> 00:03:39,732 inertial observers are all those for whom Newton's First Law works. 58 00:03:39,732 --> 00:03:42,527 If Newton's First Law works in your, measurements. 59 00:03:42,527 --> 00:03:46,075 Then, when you measure the properties of electricity and magnetism, 60 00:03:46,075 --> 00:03:48,710 you will find that they obey Maxwell's equations, 61 00:03:48,710 --> 00:03:53,077 and therefore if you measured the motion of the path of light, you would find that 62 00:03:53,077 --> 00:03:57,444 it moves at the speed c, and the speed c will not depend on your velocity relative 63 00:03:57,444 --> 00:04:01,800 to another observer. This contradicts common sense, so let's 64 00:04:01,800 --> 00:04:07,824 see why it is that we believe it. Indeed the initial idea was that Maxel 65 00:04:07,824 --> 00:04:10,812 was right. There is an ether through which light 66 00:04:10,812 --> 00:04:15,853 propagates at speed C, and then your task is to measure your velocity relative to 67 00:04:15,853 --> 00:04:18,778 the ether. And you can do that by measuring the 68 00:04:18,778 --> 00:04:22,201 properties of light. And you do this with the following 69 00:04:22,201 --> 00:04:26,431 experimental apparatus. in Einsteinian terms it's called a light clock, in 70 00:04:26,431 --> 00:04:29,810 others an interferometer. What you're doing is you will set up two 71 00:04:29,810 --> 00:04:33,086 mirrors separated by distance L, as drawn over here on the right. 72 00:04:33,086 --> 00:04:36,107 And you will set up a beam of light to bounce between them. 73 00:04:36,107 --> 00:04:39,844 We know how to do this, this is technology that was well developed in the 74 00:04:39,844 --> 00:04:42,967 nineteenth century. And you can imagine that over here on the 75 00:04:42,967 --> 00:04:45,783 mirror on the right there is a counter that ticks over. 76 00:04:45,783 --> 00:04:49,828 We have a pulse of light bouncing back and forth, every time the pulse of light 77 00:04:49,828 --> 00:04:52,285 hits the mirror on the right, counter ticks over. 78 00:04:52,285 --> 00:04:56,279 You now have a clock and it's a good clock because the speed of light being a 79 00:04:56,279 --> 00:04:59,244 constant. The time that it takes light to bounce 80 00:04:59,244 --> 00:05:03,290 back and forth through this interferometer is twice the length 81 00:05:03,290 --> 00:05:06,951 divided by C because that is how far light has to travel. 82 00:05:06,951 --> 00:05:12,217 So this is true if light moves at a speed C, in other words, if you're at rest with 83 00:05:12,217 --> 00:05:16,262 respect to the ether. What happens if you take this whole clock 84 00:05:16,262 --> 00:05:21,272 and, assuming that the transparency is addressed with respect to the ether, you 85 00:05:21,272 --> 00:05:24,740 set this whole clock moving at a speed V to the right. 86 00:05:24,740 --> 00:05:29,666 Well, what that means is that when the light beam starts out here and is moving 87 00:05:29,666 --> 00:05:34,654 to the left, the mirror on the left is itself moving to the right, towards the 88 00:05:34,654 --> 00:05:38,021 light beam. The relative velocity between, the pulse 89 00:05:38,021 --> 00:05:42,449 with which the pulse of light is approaching the mirror is, of course, C 90 00:05:42,449 --> 00:05:47,375 plus V, when moving in this direction. Let me use red pen to indicate the motion 91 00:05:47,375 --> 00:05:50,796 of light relative to this moving, interferometer. 92 00:05:50,796 --> 00:05:54,390 The speed of light in this direction would be c plus v. 93 00:05:54,390 --> 00:05:58,068 This is exactly the addition, velocity addition formula we computed. 94 00:05:58,068 --> 00:06:02,296 On the other hand, when light is moving from the left mirror back towards the 95 00:06:02,296 --> 00:06:06,413 right mirror, then the right mirror is receding from it, and the light pulse 96 00:06:06,413 --> 00:06:09,213 will be closing on the mirror at a speed C minus V. 97 00:06:09,213 --> 00:06:13,496 And so, it'll take a shorter time for light to move from here to there than it 98 00:06:13,496 --> 00:06:17,120 does for light to move from here to there, but you can add the two. 99 00:06:17,120 --> 00:06:22,031 And you can obtain the, time that it will take light to move to the left and then 100 00:06:22,031 --> 00:06:25,949 back to the right by taking the time it takes to move to the right. 101 00:06:25,949 --> 00:06:28,581 plus the time it took to move to the left. 102 00:06:28,581 --> 00:06:31,446 And doing some algebra, you can write it this way. 103 00:06:31,446 --> 00:06:34,982 You see that the faster. The clock is moving at the end of the 104 00:06:34,982 --> 00:06:38,855 day, the longer it will take for a light pulse to move back and forth. 105 00:06:38,855 --> 00:06:43,402 if you think about it that's because it spends more time moving at the slower 106 00:06:43,402 --> 00:06:47,387 velocity because this is a larger number with a smaller denominator. 107 00:06:47,387 --> 00:06:50,980 Okay so moving the clock should slow it down so you can measure. 108 00:06:50,980 --> 00:06:55,633 yeah, but how do you know what the speed of the clock is when it's addressed, if 109 00:06:55,633 --> 00:06:59,833 you don't know what addressed is? Remember, it's hard to detect when you're 110 00:06:59,833 --> 00:07:02,614 addressed. well the trick is, that was what the 111 00:07:02,614 --> 00:07:05,508 clock will measure when it's moving along its axis. 112 00:07:05,508 --> 00:07:08,403 What happens if it moves perpendicular to its axis? 113 00:07:08,403 --> 00:07:13,000 So now we're taking the same light clock, except now the motion is with a speed v 114 00:07:13,000 --> 00:07:16,967 in the up direction. So we have those two mirrors and they're 115 00:07:16,967 --> 00:07:19,713 moving up. Of course, now if a light pulse bounces 116 00:07:19,713 --> 00:07:24,252 off the right mirror and moves directly to the left, it'll just go off into space 117 00:07:24,252 --> 00:07:28,343 because the left hand mirror will have moved up in the time that it took the 118 00:07:28,343 --> 00:07:32,771 light to transverse the, to traverse the clock. And so the light beam that we're 119 00:07:32,771 --> 00:07:37,086 interested in is a light beam that bounces back and forth between the moving 120 00:07:37,086 --> 00:07:41,457 mirrors. So it bounced off the red mirror here, hit the mirror on the left here, 121 00:07:41,457 --> 00:07:45,828 will bounce off and hit the right mirror back there, when the right mirror has 122 00:07:45,828 --> 00:07:47,621 moved up to this position. Right? 123 00:07:47,621 --> 00:07:50,098 So, and what we want to know is how long it 124 00:07:50,098 --> 00:07:54,600 takes light to transverse, traverse the clock, when the clock is moving in this 125 00:07:54,600 --> 00:07:57,024 direction. This is a geometric calculation. 126 00:07:57,024 --> 00:08:01,064 We're going to assume that despite the fact that there is a small motion 127 00:08:01,064 --> 00:08:06,027 relative to the, ether of the light beam, that this angle is small enough, or V is 128 00:08:06,027 --> 00:08:10,298 small compared to C, and to leading order, we're going to assume that light 129 00:08:10,298 --> 00:08:14,740 actually moves with a speed C. What that tells us is that if little T is 130 00:08:14,740 --> 00:08:19,838 the time it takes a light pulse to go from here to here, and by symmetry little 131 00:08:19,838 --> 00:08:24,807 T is also the time it takes it to get back so bit T's going to be just little T 132 00:08:24,807 --> 00:08:27,646 here is half the period of this light clock. 133 00:08:27,646 --> 00:08:30,760 Well we can do a calculation over the time little t. 134 00:08:30,760 --> 00:08:33,773 the light beam has traversed the distance ct. 135 00:08:33,773 --> 00:08:37,657 times t, in the same time the clock has moved up by a distance vt, times t. 136 00:08:37,657 --> 00:08:40,805 So that's this vertical blue segment over here. 137 00:08:40,805 --> 00:08:45,618 The distance between the mirrors is L. The, an application of the Pythagorean 138 00:08:45,618 --> 00:08:50,453 theorem shows us that l squared plus v squared t squared is the length square, 139 00:08:50,453 --> 00:08:54,692 of the length of a hypotenuse. We can solve this for t, and we find that 140 00:08:54,692 --> 00:08:58,273 t is given by this. Remember that the period of the clock is 141 00:08:58,273 --> 00:09:03,049 twice t so we write it down, and we see that moving gensferously also slows down 142 00:09:03,049 --> 00:09:06,870 the clock, but by a different factor. Notice the square root here. 143 00:09:06,870 --> 00:09:11,643 So the light clock slows down by one factor when its moving longitudinally by 144 00:09:11,643 --> 00:09:14,236 another factor when its moving transversely. 145 00:09:14,236 --> 00:09:18,950 If we spin the light clock around and assume that the Earth is not at rest with 146 00:09:18,950 --> 00:09:22,840 respect to the either we should find that we measure one 147 00:09:22,840 --> 00:09:28,171 That we measure a periodic change in the period of a light clock as its relative 148 00:09:28,171 --> 00:09:32,251 motion with respect to the either changes in its direction. 149 00:09:32,251 --> 00:09:37,517 And this measurement was essentially to a very high precision by Michaelson and 150 00:09:37,517 --> 00:09:41,125 Morely in 1887 and their result is that there is no change. 151 00:09:41,125 --> 00:09:45,438 There is no periodic change in the period of a light clock and so there are various 152 00:09:45,438 --> 00:09:49,803 theories advanced how to understand this perhaps just as if you measure the speed 153 00:09:49,803 --> 00:09:54,013 of sound on average north south and east west you get the same answer, despite the 154 00:09:54,013 --> 00:09:58,224 fact that the Earth is rotated to the east perhaps the either is being dragged 155 00:09:58,224 --> 00:10:00,862 along by the Earth. Lots of complicated ideas. 156 00:10:00,862 --> 00:10:06,170 the simplifying Gordian knot resolving solution of Einstein proposed in 1905 157 00:10:06,170 --> 00:10:09,304 because there is no such things as, as the ether. 158 00:10:09,304 --> 00:10:12,310 Maxwell's equations hold at any inertial frame. 159 00:10:12,310 --> 00:10:16,157 Consequences that you have to change your rules for velocity addition. 160 00:10:16,157 --> 00:10:20,499 Galilean relativity needs modification. Deep changes in what we think of as the 161 00:10:20,499 --> 00:10:24,566 obvious properties of space and time are necessary, but remember, Newtonian 162 00:10:24,566 --> 00:10:28,413 physics needs to still be valid, and hence, Galilean relativity, in some 163 00:10:28,413 --> 00:10:31,381 appropriate limit. So we're going to take Einstein's idea 164 00:10:31,381 --> 00:10:34,514 and see what it tells us, and then show why we believe it.