1 00:00:00,920 --> 00:00:04,781 So like it or not, intuitive or not, those are the rules of the games. 2 00:00:04,781 --> 00:00:09,325 The universe is invariant under Lorentz transformations and not under Galileo's 3 00:00:09,325 --> 00:00:12,392 transformations. If we're going to do physics, we're going 4 00:00:12,392 --> 00:00:16,879 to have to understand how to do physics in a way that is Lorentz invariant. 5 00:00:16,879 --> 00:00:21,308 So, one of the important things that we had discussed is our conservation laws. 6 00:00:21,308 --> 00:00:25,454 And, we run into trouble with our conservation laws, in the sense that the 7 00:00:25,454 --> 00:00:29,714 Newtonian conservation laws are not invariant under Lowen's transformation 8 00:00:29,714 --> 00:00:31,134 laws. What does that mean? 9 00:00:31,134 --> 00:00:33,690 That means that if you write momentum equal to mv, 10 00:00:33,690 --> 00:00:37,171 we can put little vectors there if you want to be fancy. 11 00:00:37,171 --> 00:00:41,523 And energy, kinetic energy for the moment, equal to half MV squared. 12 00:00:41,523 --> 00:00:45,813 those are not, not cons, not in Lorentz invariant quantities in the following 13 00:00:45,813 --> 00:00:48,672 sense. Of course momentum of the momentum of a 14 00:00:48,672 --> 00:00:53,397 given particle is different as observed by different observers because it's 15 00:00:53,397 --> 00:00:57,252 velocity is different. That was true in Galilean relativity as 16 00:00:57,252 --> 00:01:02,163 well, but in Galilean relativity you can show, very directly, that if momentum is 17 00:01:02,163 --> 00:01:06,181 conserved in some process in one frame, then momentum is conserved for any other 18 00:01:06,181 --> 00:01:09,533 observer moving at a constant velocity relative to the first observer who 19 00:01:09,533 --> 00:01:11,980 declared momentum to be conserved. And likewise energy. 20 00:01:11,980 --> 00:01:15,525 It's not trivial, but it's true. On the other hand we can use Roman 21 00:01:15,525 --> 00:01:17,866 transformations. You can find that one observer would 22 00:01:17,866 --> 00:01:20,560 claim that momentum is conserved and the other that it isn't. 23 00:01:20,560 --> 00:01:23,968 That's impossible. Momentum conservation thus cannot be a 24 00:01:23,968 --> 00:01:26,838 law of nature. Laws of nature have to be true as 25 00:01:26,838 --> 00:01:29,350 measured by an observer. On the other hand. 26 00:01:29,350 --> 00:01:33,653 The untonian conservation laws worked well for objects that were moving slowly 27 00:01:33,653 --> 00:01:37,411 relative to the speed of light. Then you can of course pick frames to 28 00:01:37,411 --> 00:01:41,779 observe them in, such that none of the objects move relative to you at close to 29 00:01:41,779 --> 00:01:44,561 the speed of light. Newtonian physics has to work. 30 00:01:44,561 --> 00:01:48,593 So we need and expect to find conservation laws that are extensions or 31 00:01:48,593 --> 00:01:52,908 modifications of the conservation of energy and momentum, but that are valid 32 00:01:52,908 --> 00:01:55,577 in a Lorentz invariant way, and in fact we do. 33 00:01:55,577 --> 00:01:59,949 The two quantities we find that extend it are a momentum and energy, or three 34 00:01:59,949 --> 00:02:04,264 components of momentum if you want. And the relativistic co-variant momentum 35 00:02:04,264 --> 00:02:08,749 is NV divided by the square root of one minus V squared over C squared, and the 36 00:02:08,749 --> 00:02:13,178 energy is MC squared divided by the square root of one minus V squared over C 37 00:02:13,178 --> 00:02:13,860 squared. Now, 38 00:02:13,860 --> 00:02:19,564 these are the objects, of course again it is not true that momentum is measured by 39 00:02:19,564 --> 00:02:23,587 each Lorentz, each, observer will be the same, but what is true is if these 40 00:02:23,587 --> 00:02:27,558 quantities are conserved, momentum and energy for one Lorentz observer, then 41 00:02:27,558 --> 00:02:29,910 they are conserved for any Lorentz observer. 42 00:02:29,910 --> 00:02:34,305 And therefore, the, the conservation of these is consistent with Lorentz 43 00:02:34,305 --> 00:02:37,275 invariance, and it is, in fact, the conservation law 44 00:02:37,275 --> 00:02:40,601 that obtains to nature. Now for the, we expect to recover 45 00:02:40,601 --> 00:02:44,640 Newtonian Conservation Laws when objects are not moving much faster. 46 00:02:44,640 --> 00:02:49,511 But at speeds comparable, to the speed of light, when they're moving much slower. 47 00:02:49,511 --> 00:02:54,322 And indeed, it's not too difficult to see that when v squared is much less than c 48 00:02:54,322 --> 00:02:58,540 squared in the case of momentum, we drop this factor and momentum at 49 00:02:58,540 --> 00:03:04,450 Velocities much smaller than the speed of light is mv and then we can use our 50 00:03:04,450 --> 00:03:09,567 Newton's approximation, to write remember the denominator. 51 00:03:09,567 --> 00:03:17,365 Here is one minus v squared over c squared to the power minus one half which 52 00:03:17,365 --> 00:03:22,895 is approximately one plus v squared times a half divided by2, c squared, the minus 53 00:03:22,895 --> 00:03:26,805 sign is cancelled. And so there is the first relativistic 54 00:03:26,805 --> 00:03:32,391 correction to, the Newtonian momentum conservation at velocities much smaller 55 00:03:32,391 --> 00:03:36,650 than the speed of light. Newtonian momentum will be conserved. 56 00:03:36,650 --> 00:03:40,630 At higher velocities, you have to consider this, factor, 57 00:03:40,630 --> 00:03:45,099 and remember, this approximation is valid when v2 squared over c2 squared is small. 58 00:03:45,099 --> 00:03:50,824 At velocities higher yet, you have to use this complete, relativistic expression. 59 00:03:50,824 --> 00:03:53,750 Very nice. So we have the, understanding of 60 00:03:53,750 --> 00:03:57,232 relativistic momentum. Notice that what this means is that as 61 00:03:57,232 --> 00:04:01,570 you accelerate a particle to speeds closer and closer to the speed of light, 62 00:04:01,570 --> 00:04:04,528 its momentum actually becomes closer and closer to infinity. 63 00:04:04,528 --> 00:04:08,504 It takes an infinite amount of momentum as you try to accelerate something to the 64 00:04:08,504 --> 00:04:10,153 speed of light. This is reasonable. 65 00:04:10,153 --> 00:04:13,790 It means if you go around giving something a kick and each kick you give 66 00:04:13,790 --> 00:04:17,428 it adds a certain quantity of momentum. You could ask well why don't I keep 67 00:04:17,428 --> 00:04:20,289 kicking it until its moving faster than the speed of light? 68 00:04:20,289 --> 00:04:24,404 The answer is that if each kick adds a certain amount of momentum that will only 69 00:04:24,404 --> 00:04:28,975 make its velocity asymptotically, as the number of kicks grows without bound 70 00:04:28,975 --> 00:04:33,728 become closer and closer to the speed of light, but never accelerated past the 71 00:04:33,728 --> 00:04:37,202 speed of light. This is sometimes stated as saying that 72 00:04:37,202 --> 00:04:41,408 you rewrite this as something times v, and interpret the something as some kind 73 00:04:41,408 --> 00:04:45,099 of relativistic mass. I don't like that point of view, and most 74 00:04:45,099 --> 00:04:48,450 physicists do not. We will not treat this as a change in the 75 00:04:48,450 --> 00:04:50,516 mass. We just treat it as the correct 76 00:04:50,516 --> 00:04:54,704 relativistic definition of momentum, so that the mass for us is going to be 77 00:04:54,704 --> 00:04:59,116 the object that's over there, and we'll see in a minute how we interpret it, and 78 00:04:59,116 --> 00:05:03,192 it's a property of the particle. So extracting Newtonian, the Newtonian 79 00:05:03,192 --> 00:05:07,716 limit of momentum conservation is a simple matter of setting v squared over c 80 00:05:07,716 --> 00:05:11,234 squared less than one. In the case of energy, this is a lot less 81 00:05:11,234 --> 00:05:12,686 obvious. What, what is this? 82 00:05:12,686 --> 00:05:16,260 when I drop the v squared over c squared I get nc squared. 83 00:05:16,260 --> 00:05:20,068 That doesn't look at all like the kinetic energy of anything. 84 00:05:20,068 --> 00:05:25,125 We were supposed to recover the kinetic energy conservation law, but if we follow 85 00:05:25,125 --> 00:05:30,057 through the Newton's approximation to the term, thing get a little better thus we 86 00:05:30,057 --> 00:05:32,680 have the same half V square over C square. 87 00:05:32,680 --> 00:05:37,060 And that ends up being, well expanding this the first term is mc squared. 88 00:05:37,060 --> 00:05:40,042 And the second term, holy moly, a half and v squared. 89 00:05:40,042 --> 00:05:44,119 And of course this is valid for v squared much less than c squared. 90 00:05:44,119 --> 00:05:47,161 For higher velocities you have to use all of this. 91 00:05:47,161 --> 00:05:51,360 But when you think about it this way, you see the kinetic energy here, 92 00:05:51,360 --> 00:05:56,633 and then this famous e equals mc sqared. The conservation law if mass is conserved 93 00:05:56,633 --> 00:06:01,392 and this is just a constant as long as the particle exists this is just a 94 00:06:01,392 --> 00:06:06,280 constant and remember shifting energy by a constant makes no difference but. 95 00:06:06,280 --> 00:06:11,842 The fact that in relativistic dynamics, it is passable 96 00:06:11,842 --> 00:06:15,680 because of the modified momentum conservation in fact. 97 00:06:15,680 --> 00:06:20,042 For mass to change the total mass of particles coming in a process need not be 98 00:06:20,042 --> 00:06:23,499 the sa, the total mass of the particles coming out of a collision. 99 00:06:23,499 --> 00:06:26,211 This in fact was not possible in Newtonian physics. 100 00:06:26,211 --> 00:06:30,935 you'll show it in the homework, but it is possible in relativistic physics, in a 101 00:06:30,935 --> 00:06:33,584 way that conserves both energy and momentum. 102 00:06:33,584 --> 00:06:38,459 If the mass changes, then we see here the rate of conversion of mass to energy, if 103 00:06:38,459 --> 00:06:43,154 the particle of mass end disappears, and it's mass becomes converted to energy. 104 00:06:43,154 --> 00:06:47,669 And this is the energy, and here is the kinetic energy contribution at slow 105 00:06:47,669 --> 00:06:51,040 velocities, and at higher velocities this is the more 106 00:06:51,040 --> 00:06:55,232 accurate description. So we have our relativistic conservation 107 00:06:55,232 --> 00:06:57,232 laws. few remarks about this. 108 00:06:57,232 --> 00:07:02,005 One is that if you compute just from the expressions [INAUDIBLE] wrote, 109 00:07:02,005 --> 00:07:06,520 E squared minus P squared C squared, you get M squared C to the fourth. 110 00:07:06,520 --> 00:07:10,455 It is this is the reason why. What is the mass of particle? 111 00:07:10,455 --> 00:07:15,486 No matter what momentum you give it the mass of a particle characterizes this 112 00:07:15,486 --> 00:07:18,646 property. To give it some momentum you'll have to 113 00:07:18,646 --> 00:07:22,569 give it some energy, and E squared can always be written as P 114 00:07:22,569 --> 00:07:25,707 squared C squared plus M squared C to the fourth. 115 00:07:25,707 --> 00:07:29,421 And this combination of ENP is a property of the particle. 116 00:07:29,421 --> 00:07:34,353 We call it the invariant mass, and that is what I will use to define the mass. 117 00:07:34,353 --> 00:07:38,260 this is a property of the particle and not of it's motion. 118 00:07:38,260 --> 00:07:41,154 Good. So a particle has a mass as long as it 119 00:07:41,154 --> 00:07:44,246 exists. The other thing to note, is I did, I will 120 00:07:44,246 --> 00:07:47,140 not prove it, but I said, of course, different 121 00:07:47,140 --> 00:07:52,008 observers will measure different momenta. Different observers will measure 122 00:07:52,008 --> 00:07:57,008 different, energies for the same system. But, in fact, these four quantities. 123 00:07:57,008 --> 00:08:02,008 E and then c2 squared times p, transform under Lorentz transmissions, in exactly 124 00:08:02,008 --> 00:08:05,100 the same way that t and x do. In other words, 125 00:08:05,100 --> 00:08:09,981 momentum along the direction of motion and energy get mixed in the same way 126 00:08:09,981 --> 00:08:14,369 precisely that t and x do. so you can compute east prime and p prime 127 00:08:14,369 --> 00:08:17,953 in terms of e and p, using the Lorentz transformations. 128 00:08:17,953 --> 00:08:22,649 And in particular, if the change in energy and momentum is zero in one frame, 129 00:08:22,649 --> 00:08:27,222 it's zero in any other frame. Because the Lorentz transformation takes 130 00:08:27,222 --> 00:08:28,891 xt=0 equals t equal zero, to x prime equals t prime equals zero. 131 00:08:28,891 --> 00:08:32,290 This is why energy momentum conservation is consistent. 132 00:08:32,290 --> 00:08:35,848 if you use this definition of energy. Okay. 133 00:08:35,848 --> 00:08:40,614 Now, an important canard, this E equals M C squared is sometimes 134 00:08:40,614 --> 00:08:45,911 invoked as the reason why nuclear, processes release so much energy. 135 00:08:45,911 --> 00:08:51,661 Oh, because they convert mass to energy, and C squared is such a large number. 136 00:08:51,661 --> 00:08:55,293 That's a canard. In fact, these conservation laws. 137 00:08:55,293 --> 00:08:59,530 Relativity was always true, even before we discovered it. 138 00:08:59,530 --> 00:09:05,888 if Lavoisier in 1777, when he made his ground breaking measurement, that said if 139 00:09:05,888 --> 00:09:11,676 you weigh the gas products of combustion, and the ashes, you recover the weight of 140 00:09:11,676 --> 00:09:16,955 the wood that you started burning. Had he measured very precisely, he would 141 00:09:16,955 --> 00:09:19,061 have discovered this is not true. Why? 142 00:09:19,061 --> 00:09:21,507 Because the wood, in burning, lost some heat. 143 00:09:21,507 --> 00:09:25,092 The heat radiated away. That means that the total energy of the 144 00:09:25,092 --> 00:09:28,676 system has been reduced. Converting by dividing by c squared, he 145 00:09:28,676 --> 00:09:31,408 would have found a very small reduction in mass. 146 00:09:31,408 --> 00:09:35,618 The point is that when wood burns, it doesn't produce that much energy, and 147 00:09:35,618 --> 00:09:39,885 therefore, it loses very little of its mass, because you are harnessing weak 148 00:09:39,885 --> 00:09:42,560 electric, magnetic forces at atomic radii. 149 00:09:42,560 --> 00:09:47,339 If you, on the other hand managed to harness the strong force, or forces that 150 00:09:47,339 --> 00:09:52,684 nuclear distances, the loss of the energy released is much larger. The change in 151 00:09:52,684 --> 00:09:58,029 the mass is actually measurable, and this is why the mass of four protons is lager 152 00:09:58,029 --> 00:10:01,677 than the mass of an alpha particle by a measurable amount. 153 00:10:01,677 --> 00:10:06,708 And, but, but this has, the statement that the sun is fuelled by the conversion 154 00:10:06,708 --> 00:10:12,036 of mass to energy is a canard, in the same way that the statement that my 155 00:10:12,036 --> 00:10:16,595 fireplace warms me up by converting mass to energy as a canard. 156 00:10:16,595 --> 00:10:22,383 The statement that is true is that the sun produces energy by harnessing a very 157 00:10:22,383 --> 00:10:26,741 strong nuclear force. the other thing is that the decay of a 158 00:10:26,741 --> 00:10:30,639 particle not conserving mass is consistent and it happens. 159 00:10:30,639 --> 00:10:33,735 We know of neutrons decaying to protons, electrons 160 00:10:33,735 --> 00:10:36,775 and neutrini. The sum of the masses of the constituents 161 00:10:36,775 --> 00:10:39,649 is not, is less than the sum, the mass of the neutron. 162 00:10:39,649 --> 00:10:42,523 Which is why they come off with some kinetic energy. 163 00:10:42,523 --> 00:10:45,397 Because of the conservation of, total energy mass. 164 00:10:45,397 --> 00:10:49,874 And, all our other conserved quantities. Electric charge, electron number and so 165 00:10:49,874 --> 00:10:53,301 on, are actually invariant. They are numbers, they are identical. 166 00:10:53,301 --> 00:10:57,723 The electric charge in a given region as measured by different observers is the 167 00:10:57,723 --> 00:10:59,933 same. So conservation of electric charge, 168 00:10:59,933 --> 00:11:02,255 electron number and so on, works very well. 169 00:11:02,255 --> 00:11:05,240 We don't have a problem. We have our conservation laws. 170 00:11:05,240 --> 00:11:09,660 Now let's write down the laws of physics. As I said electromagnetism is tooken care 171 00:11:09,660 --> 00:11:10,405 of, so to speak. 172 00:11:10,405 --> 00:11:13,138 It's taken care of. Electromagnetism, Maxwell's equations 173 00:11:13,138 --> 00:11:16,065 were Lorentz invariances. That's where we all started from. 174 00:11:16,065 --> 00:11:19,481 What about the nuclear interactions? Well, we know a nuclear, a, a way of 175 00:11:19,481 --> 00:11:23,287 writing the nuclear interactions, and though its in variant form, though that's 176 00:11:23,287 --> 00:11:25,630 not the way they were initially written. In fact. 177 00:11:25,630 --> 00:11:31,362 Are study of the nuclear interactions and of, electromagnetism, is often inherently 178 00:11:31,362 --> 00:11:36,690 in quantum, and by the 1940s, and re, for nuclear fie, interaction by the 1970s. 179 00:11:36,690 --> 00:11:41,681 A relativistic, version of, lumison variant version of, quantum mechanics 180 00:11:41,681 --> 00:11:46,537 called quantum field theory had been developed, and we have a quantum 181 00:11:46,537 --> 00:11:51,461 relavistic version, of electromagnetism, the strong nuclear force, the weak 182 00:11:51,461 --> 00:11:55,238 nuclear force. Physics, is very nicely consistent with, 183 00:11:55,238 --> 00:11:58,399 relativity, which leaves the question of gravity. 184 00:11:58,399 --> 00:12:03,041 Well turn, an, an, and this is the question with which Einstein was stuck. 185 00:12:03,041 --> 00:12:07,168 Newtonian gravity, our poster child for a theory of physics, the first one to 186 00:12:07,168 --> 00:12:10,355 really be understood the first fundamental force to be properly 187 00:12:10,355 --> 00:12:12,944 understood. the one that was our poster child for 188 00:12:12,944 --> 00:12:16,828 gallilaian invariance is not invariant as well will see and through lobeman's 189 00:12:16,828 --> 00:12:20,861 transformations it is not consistent with relatively Einstein understood this in 190 00:12:20,861 --> 00:12:23,750 1905 when he figured out the special theory of relativity. 191 00:12:23,750 --> 00:12:27,928 And it took a whole decade of Einstein to figure out the generalization. 192 00:12:27,928 --> 00:12:32,571 So we're going to spend some time trying to understand what's wrong, why gravity 193 00:12:32,571 --> 00:12:37,273 is not consistent with relativity, and how Einstein figured out how to resolve 194 00:12:37,273 --> 00:12:39,188 it. As for the quantum version of 195 00:12:39,188 --> 00:12:43,193 relativistically invariant gravity, that'll be left for your homework.