1 00:00:00,988 --> 00:00:01,976 [SQUEAKING] 2 00:00:01,976 --> 00:00:03,952 [RUSTLING] 3 00:00:03,952 --> 00:00:11,533 [CLICKING] 4 00:00:11,533 --> 00:00:12,950 SCOTT HUGHES: All right, so we are 5 00:00:12,950 --> 00:00:15,590 ready to move to the next lecture. 6 00:00:15,590 --> 00:00:17,010 I'm going to record today. 7 00:00:17,010 --> 00:00:19,100 So if we just quickly recap the highlights 8 00:00:19,100 --> 00:00:21,890 of what we did in our previous lecture, 9 00:00:21,890 --> 00:00:24,170 the goal of this lecture was to take all the tools 10 00:00:24,170 --> 00:00:27,867 that we have built and develop a theory of gravity. 11 00:00:27,867 --> 00:00:29,450 So the two ingredients to this-- first 12 00:00:29,450 --> 00:00:32,299 we're going to assert that we take a valid law of physics, 13 00:00:32,299 --> 00:00:35,290 something that is well-understood in a local 14 00:00:35,290 --> 00:00:39,010 Lorentz frames, which we take corresponding 15 00:00:39,010 --> 00:00:41,840 to a freely falling frame when gravity is invoked, 16 00:00:41,840 --> 00:00:43,880 and we will carry it over into a general form 17 00:00:43,880 --> 00:00:46,850 by rewriting that law in a tensorial form. 18 00:00:46,850 --> 00:00:48,590 The example I went through was that if I 19 00:00:48,590 --> 00:00:51,980 assert that an object is unaccelerated, then 20 00:00:51,980 --> 00:00:53,570 I can normally write that by saying 21 00:00:53,570 --> 00:00:57,800 that the proper derivative of the velocity is zero-- 22 00:00:57,800 --> 00:00:59,660 or if you like the second derivative, 23 00:00:59,660 --> 00:01:01,550 positional on the world line is zero-- 24 00:01:01,550 --> 00:01:08,093 that carries over to saying that the parallel transport of the 25 00:01:08,093 --> 00:01:10,010 of the four velocity, along the four velocity, 26 00:01:10,010 --> 00:01:11,180 is equal to zero. 27 00:01:11,180 --> 00:01:14,870 If I require conservation, local conservation of energy 28 00:01:14,870 --> 00:01:17,390 and momentum to be asserted in my local Lorentz 29 00:01:17,390 --> 00:01:19,490 frame, my freely falling frame in the form 30 00:01:19,490 --> 00:01:22,003 that it has zero divergence with a partial derivative, 31 00:01:22,003 --> 00:01:23,420 that [? carries ?] over in general 32 00:01:23,420 --> 00:01:29,420 to zero derivative with a covariant derivative. 33 00:01:29,420 --> 00:01:31,880 Our second ingredient is to try to understand 34 00:01:31,880 --> 00:01:36,090 how it is that I can build my theory of spacetime-- excuse 35 00:01:36,090 --> 00:01:36,590 me. 36 00:01:36,590 --> 00:01:41,670 How I can build my spacetime given a distribution of energy, 37 00:01:41,670 --> 00:01:44,120 matter, and momentum. 38 00:01:44,120 --> 00:01:46,940 We are guided by the Newtonian field equation, which tells me 39 00:01:46,940 --> 00:01:48,830 the Laplace operator on Newtonian 40 00:01:48,830 --> 00:01:52,130 gravitational potential is 4 pi G rho. 41 00:01:52,130 --> 00:01:56,510 By studying the equation of motion for Newtonian gravity 42 00:01:56,510 --> 00:02:00,800 and comparing it to the geodesic equation in the slow motion 43 00:02:00,800 --> 00:02:03,140 limit for a spacetime that deviates just 44 00:02:03,140 --> 00:02:05,960 slightly from the flat space on a special relativity, 45 00:02:05,960 --> 00:02:09,259 we noted that this phi is closely related to G00. 46 00:02:09,259 --> 00:02:12,560 It's precisely equal to up to a factor of 2. 47 00:02:12,560 --> 00:02:14,960 It's a bit of an offset from the negative 1 48 00:02:14,960 --> 00:02:18,950 that you get in the special relativity limit. 49 00:02:18,950 --> 00:02:21,380 We noted that this is not a healthy equation 50 00:02:21,380 --> 00:02:22,930 from the standpoint of being a ten-- 51 00:02:22,930 --> 00:02:26,150 of formulating things in terms of tensors, 52 00:02:26,150 --> 00:02:29,240 because rho is a particular component of the stress energy 53 00:02:29,240 --> 00:02:30,410 tensor. 54 00:02:30,410 --> 00:02:33,320 Doesn't make any sense to have a properly covariant eq-- 55 00:02:33,320 --> 00:02:35,040 if we want a properly covariant equation, 56 00:02:35,040 --> 00:02:38,390 it makes no sense to pick out one component of a tensor. 57 00:02:38,390 --> 00:02:41,750 So we upgraded rho to T mu nu. 58 00:02:41,750 --> 00:02:45,080 And we looked for, noting that two derivatives 59 00:02:45,080 --> 00:02:47,320 of the potential is something like two derivatives 60 00:02:47,320 --> 00:02:50,480 of the metric, which must be two derivatives of a curvature, 61 00:02:50,480 --> 00:02:52,850 we said, OK, it's got to be a curvature tensor 62 00:02:52,850 --> 00:02:56,150 on the left-hand side, and it's got to be a two index one, 63 00:02:56,150 --> 00:02:58,370 and it's got to be one that has no divergence 64 00:02:58,370 --> 00:03:02,780 so that this equation respects the local conservation 65 00:03:02,780 --> 00:03:04,760 of energy and momentum. 66 00:03:04,760 --> 00:03:08,710 And that led us by requiring that, 67 00:03:08,710 --> 00:03:10,970 in the appropriate limit of slow motion-- 68 00:03:10,970 --> 00:03:12,470 well, in this case, really it's just 69 00:03:12,470 --> 00:03:15,020 sort of a weak deviation from special relativity. 70 00:03:15,020 --> 00:03:18,380 By requiring this correspondence to hold, 71 00:03:18,380 --> 00:03:23,522 that force, the proportionality between my curvature tensor 72 00:03:23,522 --> 00:03:24,980 and my stress energy tensor will be 73 00:03:24,980 --> 00:03:28,250 8 pi G. If you want to work in units where the speed of light 74 00:03:28,250 --> 00:03:33,360 is not set equal to unity, it's 8 pi G over C to the 4th. 75 00:03:33,360 --> 00:03:34,860 So that's fine. 76 00:03:34,860 --> 00:03:37,230 This is, in fact, exactly how Einstein originally 77 00:03:37,230 --> 00:03:39,120 derived this. 78 00:03:39,120 --> 00:03:42,230 And it's a very physical, very-- 79 00:03:42,230 --> 00:03:45,270 it's a very physicist kind of an argument. 80 00:03:45,270 --> 00:03:47,880 There is a another route to the field equations, 81 00:03:47,880 --> 00:03:50,400 though, which I would like to talk about 82 00:03:50,400 --> 00:03:51,480 in this next lecture. 83 00:04:02,590 --> 00:04:05,680 And what's interesting is that, historically, this route 84 00:04:05,680 --> 00:04:08,800 was actually developed at almost the exact same time 85 00:04:08,800 --> 00:04:12,340 that Einstein developed this way of coming up with the field 86 00:04:12,340 --> 00:04:15,850 equations that are going to describe relativistic gravity. 87 00:04:15,850 --> 00:04:18,620 This particular route helps us to-- 88 00:04:18,620 --> 00:04:21,850 I'd say it clarifies some of the choices that 89 00:04:21,850 --> 00:04:24,490 are made that lead to this particular framework 90 00:04:24,490 --> 00:04:26,230 for the field equations. 91 00:04:26,230 --> 00:04:28,990 And as we'll see as we get to the end of this lecture, 92 00:04:28,990 --> 00:04:30,910 it helps to illustrate how it might 93 00:04:30,910 --> 00:04:35,470 be-- if general relativity is incomplete in some way, how 94 00:04:35,470 --> 00:04:37,888 we might systematically add a little bit 95 00:04:37,888 --> 00:04:39,430 of additional framework, a little bit 96 00:04:39,430 --> 00:04:41,440 of an additional structure. 97 00:04:41,440 --> 00:04:44,530 It gives us, what is the most natural way 98 00:04:44,530 --> 00:04:47,350 to introduce additional degrees of freedom, 99 00:04:47,350 --> 00:04:49,150 additional ways of modifying things? 100 00:04:49,150 --> 00:04:51,250 So the framework I'm going to talk about today 101 00:04:51,250 --> 00:04:55,270 is something that allows us to much more naturally move 102 00:04:55,270 --> 00:04:58,840 beyond what this might be, just in case-- 103 00:04:58,840 --> 00:05:01,120 Einstein, at some point, we're going 104 00:05:01,120 --> 00:05:03,490 to likely find that general relativity is not 105 00:05:03,490 --> 00:05:04,270 quite adequate. 106 00:05:04,270 --> 00:05:06,340 It doesn't describe everything to the degree 107 00:05:06,340 --> 00:05:08,800 that we are able to measure everything. 108 00:05:08,800 --> 00:05:13,500 This will give us the tools to go beyond that. 109 00:05:13,500 --> 00:05:16,120 So the particular route I'm going to [? derive ?] 110 00:05:16,120 --> 00:05:21,700 is found by varying a particular Lagrangian. 111 00:05:21,700 --> 00:05:23,080 We're going to essentially define 112 00:05:23,080 --> 00:05:27,086 an action principle for the gravitational interaction. 113 00:05:27,086 --> 00:05:31,450 So I will call this lecture the field equation 114 00:05:31,450 --> 00:05:40,136 via the Einstein-Hilbert action. 115 00:05:43,970 --> 00:05:46,850 That is the same Hilbert of Hilbert Spaces, David Hilbert, 116 00:05:46,850 --> 00:05:50,840 mathematician who was active at about the same time 117 00:05:50,840 --> 00:05:56,840 that Einstein was at his peak research productivity. 118 00:05:56,840 --> 00:05:59,600 As I said, this way of thinking about things 119 00:05:59,600 --> 00:06:03,500 and this formulation that allows us to re-derive this, 120 00:06:03,500 --> 00:06:05,660 it was developed almost in parallel 121 00:06:05,660 --> 00:06:09,260 to Einstein's formulation of general relativity, 122 00:06:09,260 --> 00:06:12,440 so much so that they practically overlapped. 123 00:06:12,440 --> 00:06:15,140 But it must be said that Hilbert gives priority 124 00:06:15,140 --> 00:06:16,130 to Einstein on this. 125 00:06:16,130 --> 00:06:17,760 Hilbert gave Einstein the priority 126 00:06:17,760 --> 00:06:21,350 because he felt that Einstein really clarified the physics. 127 00:06:21,350 --> 00:06:24,860 And it was his insights that led to this way of thinking 128 00:06:24,860 --> 00:06:27,177 about gravity, even though in some ways 129 00:06:27,177 --> 00:06:28,760 he came up with a way of understanding 130 00:06:28,760 --> 00:06:31,430 where the field equations come from that is somewhat more 131 00:06:31,430 --> 00:06:32,330 elegant. 132 00:06:32,330 --> 00:06:34,445 So let me first sketch schematically 133 00:06:34,445 --> 00:06:35,570 how we're going to do this. 134 00:06:41,810 --> 00:06:50,470 Schematically, what we do is we define an action 135 00:06:50,470 --> 00:07:03,645 as the integral of a Lagrange density over all of spacetime. 136 00:07:12,210 --> 00:07:15,560 So we're going to write our action as S, 137 00:07:15,560 --> 00:07:19,880 and we're going to write it as an integral over our spacetime 138 00:07:19,880 --> 00:07:26,560 coordinates of some Lagrange density. 139 00:07:26,560 --> 00:07:28,630 So this is not a lot in field theories. 140 00:07:28,630 --> 00:07:30,370 This Lagrange density is something 141 00:07:30,370 --> 00:07:35,290 that depends on the fields you are studying. 142 00:07:43,090 --> 00:07:45,820 The action must itself be a Lorentz scalar. 143 00:07:45,820 --> 00:07:47,530 If I'm working in a freely falling frame, 144 00:07:47,530 --> 00:07:51,280 it cannot be something that depends upon the particular 145 00:07:51,280 --> 00:07:52,990 frame that one is using. 146 00:07:52,990 --> 00:07:54,955 And so one often writes this. 147 00:07:58,610 --> 00:08:00,850 You explicitly call out the square root 148 00:08:00,850 --> 00:08:04,120 of minus G. That is necessary in order 149 00:08:04,120 --> 00:08:06,580 to make your little coordinate interval here 150 00:08:06,580 --> 00:08:09,950 be a proper volume integral. 151 00:08:09,950 --> 00:08:12,760 And just to make a little bit of a notational difference, 152 00:08:12,760 --> 00:08:16,930 let's call L hat the version of the Lagrange density 153 00:08:16,930 --> 00:08:19,210 that has that weighted in there, like so. 154 00:08:19,210 --> 00:08:21,810 So this will, in fact, be-- 155 00:08:21,810 --> 00:08:25,540 so this entire integral, both of these entire integrals, 156 00:08:25,540 --> 00:08:27,340 are Lorentz scalars. 157 00:08:27,340 --> 00:08:30,380 This is a proper element of volume, 158 00:08:30,380 --> 00:08:38,350 and so this is a proper Lorentz covariant well-defined Lagrange 159 00:08:38,350 --> 00:08:40,650 density. 160 00:08:40,650 --> 00:08:42,510 So whenever you have an action principle, 161 00:08:42,510 --> 00:08:50,180 you proceed via an extremization procedure. 162 00:08:50,180 --> 00:08:51,740 So suppose your Lagrangian-- 163 00:08:51,740 --> 00:08:53,532 I'm going to write this very schematically. 164 00:08:53,532 --> 00:08:57,130 Suppose it depends on some set of fields. 165 00:08:57,130 --> 00:09:14,310 When you extremize this, that amounts to our requirement 166 00:09:14,310 --> 00:09:18,990 that the action be stationary with respect 167 00:09:18,990 --> 00:09:20,450 to a variation of the fields. 168 00:09:39,210 --> 00:09:44,055 So if I vary my fields, I'm going to vary my action. 169 00:09:47,520 --> 00:09:50,010 And let's just schematically write this using 170 00:09:50,010 --> 00:09:52,650 the first form of things here. 171 00:09:52,650 --> 00:09:57,600 I will then say, however, my Lagrangian varies, 172 00:09:57,600 --> 00:10:07,970 as I vary whatever fields it depends on, this must be zero. 173 00:10:07,970 --> 00:10:10,340 That is my stationarity requirement. 174 00:10:10,340 --> 00:10:13,880 For this to occur, for this to be valid for arbitrary field 175 00:10:13,880 --> 00:10:24,648 variations, I will then deduce from this 176 00:10:24,648 --> 00:10:26,690 that the variation of the Lagrange density itself 177 00:10:26,690 --> 00:10:29,440 with those fields must be equal to zero. 178 00:10:29,440 --> 00:10:32,270 When I do this, this will then lead to Euler Lagrange 179 00:10:32,270 --> 00:10:33,710 equations for my fields. 180 00:10:49,240 --> 00:10:51,240 Now, many students in the class may have already 181 00:10:51,240 --> 00:10:53,110 seen things like this. 182 00:10:53,110 --> 00:10:56,610 But for those who have not, let me work through a fairly simple 183 00:10:56,610 --> 00:10:58,060 example. 184 00:10:58,060 --> 00:11:01,260 So let's imagine that I have a Lagrangian that just depends 185 00:11:01,260 --> 00:11:05,110 on a scalar field phi and derivatives of that scalar 186 00:11:05,110 --> 00:11:05,610 field. 187 00:11:12,400 --> 00:11:17,710 So L is going to depend on some phi 188 00:11:17,710 --> 00:11:21,460 and derivatives of that phi. 189 00:11:21,460 --> 00:11:24,130 In my notes actually, imagine there might be multiple scalar 190 00:11:24,130 --> 00:11:24,450 fields. 191 00:11:24,450 --> 00:11:25,742 Let's just keep it simple here. 192 00:11:25,742 --> 00:11:29,650 We'll just have one scalar field and its derivatives. 193 00:11:29,650 --> 00:11:31,270 When we do our variation, it's going 194 00:11:31,270 --> 00:11:34,090 to be important that we treat variations of the derivative 195 00:11:34,090 --> 00:11:37,630 differently from variations of the field. 196 00:11:37,630 --> 00:11:41,118 At the moment, I'm going to focus on flat spacetime. 197 00:11:41,118 --> 00:11:43,660 We'll see what happens when the spacetime's a little bit more 198 00:11:43,660 --> 00:11:45,077 interesting in just a few moments. 199 00:11:50,580 --> 00:11:54,740 And what I'm going to do is imagine 200 00:11:54,740 --> 00:12:13,237 that phi varies like so, and its derivative varies like so, OK? 201 00:12:13,237 --> 00:12:14,070 So let's apply this. 202 00:12:14,070 --> 00:12:16,590 I want to look at a variation of this thing with respect 203 00:12:16,590 --> 00:12:18,450 to my fields. 204 00:12:18,450 --> 00:12:25,990 So when I do my variation of the action S, 205 00:12:25,990 --> 00:12:34,150 I get one term that will arise in the variation in the field 206 00:12:34,150 --> 00:12:49,500 phi, and I will get another term that 207 00:12:49,500 --> 00:12:52,500 arises due to the variation in the derivative. 208 00:12:52,500 --> 00:12:56,340 That term is potentially a little bit problematic for us. 209 00:12:56,340 --> 00:12:58,260 I'm going to gloss over a couple of details. 210 00:12:58,260 --> 00:13:02,040 But essentially what you want to do is, at this point, 211 00:13:02,040 --> 00:13:04,470 apply integration by parts. 212 00:13:04,470 --> 00:13:06,030 When you apply integration by parts, 213 00:13:06,030 --> 00:13:09,350 you can move this derivative onto this term here. 214 00:13:23,020 --> 00:13:52,320 And in so doing, your delta S becomes something like this. 215 00:13:52,320 --> 00:13:54,360 Caution-- I'm being a little bit glib here. 216 00:13:54,360 --> 00:13:57,260 That is only true if-- 217 00:13:57,260 --> 00:14:01,020 when you do this properly, there is kind of a boundary term 218 00:14:01,020 --> 00:14:03,217 that I'm assuming can be discarded. 219 00:14:06,740 --> 00:14:08,740 I don't want to get lost in the weeds associated 220 00:14:08,740 --> 00:14:09,615 with that right here. 221 00:14:09,615 --> 00:14:12,168 Our goal is just to sort of sketch the main physical ideas. 222 00:14:12,168 --> 00:14:13,710 But I will say that this is something 223 00:14:13,710 --> 00:14:15,670 that should be treated with a little bit of care. 224 00:14:15,670 --> 00:14:17,170 And I highlight this because there's 225 00:14:17,170 --> 00:14:21,070 going to be a similar step that I'm going to sort of go over 226 00:14:21,070 --> 00:14:24,120 with a little less care than I would like as we move forward, 227 00:14:24,120 --> 00:14:26,970 and I'll comment on that at that time. 228 00:14:26,970 --> 00:14:29,280 Having done this and assuming that the discarding 229 00:14:29,280 --> 00:14:32,040 of the boundary term is safe for me here-- 230 00:14:34,872 --> 00:14:36,830 actually, let me just make one further comment. 231 00:14:36,830 --> 00:14:40,077 This is being done at over all of spacetime. 232 00:14:40,077 --> 00:14:41,660 And so when I do that, I'm essentially 233 00:14:41,660 --> 00:14:44,370 assuming that the boundary is the boundary of infinity, 234 00:14:44,370 --> 00:14:45,860 which doesn't really exist, OK? 235 00:14:45,860 --> 00:14:48,260 So that's one reason why you can generally get rid of it 236 00:14:48,260 --> 00:14:48,810 in this case. 237 00:14:48,810 --> 00:14:50,810 And it turns out to be fairly safe. 238 00:14:50,810 --> 00:14:52,663 I'm highlighting this point because it 239 00:14:52,663 --> 00:14:54,080 should be stated a little bit more 240 00:14:54,080 --> 00:14:56,990 carefully than I have it in my notes. 241 00:14:56,990 --> 00:14:59,293 And when we apply this to gravity, 242 00:14:59,293 --> 00:15:00,710 there's a little bit of a subtlety 243 00:15:00,710 --> 00:15:05,637 there that I will comment on at that time. 244 00:15:05,637 --> 00:15:07,220 For now, I'm just going to say, great. 245 00:15:07,220 --> 00:15:08,960 So that sort of works good. 246 00:15:08,960 --> 00:15:10,860 That works out very good, very well. 247 00:15:10,860 --> 00:15:16,170 And in order for this to be equal to zero, 248 00:15:16,170 --> 00:15:22,820 the term inside my square braces must vanish, 249 00:15:22,820 --> 00:15:38,410 and so I wind up with Euler Lagrange equations, which 250 00:15:38,410 --> 00:15:44,480 my field, my field phi, must satisfy given the Lagrangian L, 251 00:15:44,480 --> 00:15:45,700 OK? 252 00:15:45,700 --> 00:15:47,380 Now let's just take it one step further. 253 00:15:47,380 --> 00:15:50,890 So I haven't said yet where the Lagrangian comes from, OK? 254 00:15:50,890 --> 00:15:54,070 As I sort of indicated, when I sketch this over here, 255 00:15:54,070 --> 00:15:56,530 it's going to depend on the nature of the field you study. 256 00:15:56,530 --> 00:16:01,150 So it's very common when you're studying, say, a scalar field 257 00:16:01,150 --> 00:16:04,210 and flat spacetime that you imagine 258 00:16:04,210 --> 00:16:11,320 that your Lagrangian has what we call a kinetic term, which 259 00:16:11,320 --> 00:16:13,690 is proportional to something that's quadratic-- 260 00:16:17,626 --> 00:16:22,910 whoops-- in the derivatives of your field. 261 00:16:22,910 --> 00:16:27,710 If it is massive, you may have another term 262 00:16:27,710 --> 00:16:32,127 which sort of looks almost like a potential energy in which you 263 00:16:32,127 --> 00:16:34,460 get something that looks like the field squared coupling 264 00:16:34,460 --> 00:16:36,920 to a parameter with the dimensions of mass. 265 00:16:43,850 --> 00:17:12,230 Take these derivatives, so taking the derivative 266 00:17:12,230 --> 00:17:15,390 of this Lagrangian with respect to the field, 267 00:17:15,390 --> 00:17:16,260 that's quite simple. 268 00:17:21,440 --> 00:17:23,260 You just get m squared phi. 269 00:17:23,260 --> 00:17:31,830 Take the derivative with respect to the field's derivative, 270 00:17:31,830 --> 00:17:33,600 what pops out of this is-- 271 00:17:33,600 --> 00:17:34,578 pardon me. 272 00:17:40,940 --> 00:17:43,170 [INAUDIBLE] looks like this. 273 00:17:43,170 --> 00:17:57,580 Now, take one more derivative this guy, pardon me. 274 00:17:57,580 --> 00:17:58,360 Just one second. 275 00:17:58,360 --> 00:17:58,860 Yeah. 276 00:18:06,440 --> 00:18:07,285 My apologies. 277 00:18:07,285 --> 00:18:08,660 I was just looking over my notes. 278 00:18:17,050 --> 00:18:19,690 You just get the wave operator, the flat spacetime wave 279 00:18:19,690 --> 00:18:21,580 operator, acting on the field phi. 280 00:18:21,580 --> 00:18:36,130 And the Euler Lagrange equations give us 281 00:18:36,130 --> 00:18:38,290 the following wave equation governing 282 00:18:38,290 --> 00:18:40,300 this field in flat spacetime. 283 00:18:40,300 --> 00:18:43,000 This is a result known as the massive Klein Gordon equation. 284 00:19:01,400 --> 00:19:03,700 So that's great. 285 00:19:03,700 --> 00:19:06,610 How do we apply this to our theory of gravity? 286 00:19:14,190 --> 00:19:18,240 So there's two things that we need to decide. 287 00:19:22,020 --> 00:19:26,293 One is, what fields am I going to build my Lagrangian out of? 288 00:19:26,293 --> 00:19:28,710 And so I'm going to come back to this point, because there 289 00:19:28,710 --> 00:19:33,270 actually is a choice to be made here, which is-- 290 00:19:33,270 --> 00:19:34,590 it's a little non-obvious. 291 00:19:34,590 --> 00:19:37,060 And I'm going to have a brief aside 292 00:19:37,060 --> 00:19:42,210 where I discuss for a moment a slightly different way of doing 293 00:19:42,210 --> 00:19:43,992 it than what I'm going to cover here. 294 00:19:43,992 --> 00:19:45,450 But we're going to vary the metric. 295 00:19:45,450 --> 00:19:47,825 We're going to treat the metric as our fundamental field. 296 00:19:47,825 --> 00:19:50,347 And so when I write down the Lagrangian for gravity, 297 00:19:50,347 --> 00:19:52,680 I'm going to imagine that it's a function of the metric, 298 00:19:52,680 --> 00:19:54,347 and that would be the thing that I vary. 299 00:19:58,670 --> 00:20:03,690 The other issue is, what should I use for my Lagrangian? 300 00:20:19,880 --> 00:20:24,260 So the principle which I'm going to use to make the choice here 301 00:20:24,260 --> 00:20:25,710 applies to the gravity. 302 00:20:25,710 --> 00:20:27,300 So my question is-- pardon me. 303 00:20:27,300 --> 00:20:30,150 My writing and my brain are getting out of sync here-- 304 00:20:30,150 --> 00:20:32,490 how do we choose our Lagrangian? 305 00:20:32,490 --> 00:20:36,540 How do we choose L? 306 00:20:36,540 --> 00:20:42,270 So it must be a scalar, or a scalar density 307 00:20:42,270 --> 00:20:45,110 if I want to take in bearing in mind that factor of square root 308 00:20:45,110 --> 00:20:53,800 G, must yield a scalar action. 309 00:20:53,800 --> 00:20:55,800 And here's why I'm going to make a real decision 310 00:20:55,800 --> 00:20:57,300 with consequences. 311 00:20:57,300 --> 00:21:00,575 It must be built from curvature tensors. 312 00:21:12,650 --> 00:21:16,610 The reason I choose to make it something that is constructed 313 00:21:16,610 --> 00:21:21,170 from curvature tensors is that I want this Lagrange density 314 00:21:21,170 --> 00:21:25,040 to be something that I cannot get rid of by changing 315 00:21:25,040 --> 00:21:26,720 my reference frame. 316 00:21:26,720 --> 00:21:30,740 If it's the metric, I can make the metric flat at any point 317 00:21:30,740 --> 00:21:33,183 simply by going into a freely falling frame. 318 00:21:33,183 --> 00:21:35,350 It has something to do with the Christoffel symbols, 319 00:21:35,350 --> 00:21:36,830 or rather with my connection. 320 00:21:36,830 --> 00:21:40,370 I can make that vanish by going into a freely falling frame. 321 00:21:40,370 --> 00:21:42,810 I cannot get rid of the curvature. 322 00:21:42,810 --> 00:21:46,490 So this point two here, this second point, 323 00:21:46,490 --> 00:21:51,200 this follows because this cannot be something that is eliminated 324 00:21:51,200 --> 00:21:53,763 by changing my frame of reference. 325 00:22:11,530 --> 00:22:17,950 I claim, based on that, that the simplest action of all, 326 00:22:17,950 --> 00:22:20,770 the one that you should at least start with, 327 00:22:20,770 --> 00:22:30,100 the one that is least ornate and that encompasses these two 328 00:22:30,100 --> 00:22:41,150 principles, is I'm going to choose the L hat. 329 00:22:41,150 --> 00:22:43,290 in other words, the one I'm going to separate out, 330 00:22:43,290 --> 00:22:47,622 the factor of square root minus G. 331 00:22:47,622 --> 00:22:55,700 I want this to be the Ricci scalar. 332 00:22:55,700 --> 00:22:58,190 It's the simplest possible thing that I can make out 333 00:22:58,190 --> 00:23:01,790 of curvature tensors that is a curvature 334 00:23:01,790 --> 00:23:03,530 and is going to give me a scalar. 335 00:23:06,060 --> 00:23:08,510 One could make other choices, OK? 336 00:23:08,510 --> 00:23:12,060 And in essence, that is the reason I am doing this lecture, 337 00:23:12,060 --> 00:23:15,720 is I want to highlight the fact that this particularly 338 00:23:15,720 --> 00:23:18,900 simple choice, we're going to see it reproduces the Einstein 339 00:23:18,900 --> 00:23:19,650 field equation. 340 00:23:19,650 --> 00:23:23,160 And so, in that sense, general relativity, we're going to see, 341 00:23:23,160 --> 00:23:28,470 is really the simplest geometric theory of gravitation. 342 00:23:28,470 --> 00:23:31,170 One could make more complicated ones, OK? 343 00:23:31,170 --> 00:23:33,060 But this is the simplest one. 344 00:23:33,060 --> 00:23:35,293 And in the last couple minutes of this lecture, 345 00:23:35,293 --> 00:23:37,710 we'll sort of explore how we can make it a little bit more 346 00:23:37,710 --> 00:23:38,658 ornate. 347 00:23:38,658 --> 00:23:39,700 So let's begin with this. 348 00:23:39,700 --> 00:23:50,720 So I'm going to write S equals the integral of all 349 00:23:50,720 --> 00:23:51,980 these things. 350 00:23:51,980 --> 00:23:55,040 And let's be blunt. 351 00:23:55,040 --> 00:23:57,770 Because I'm allowed to cheat because I'm the professor, 352 00:23:57,770 --> 00:24:04,570 I'm going to throw in a proof factor of 1 over 16 pi G, OK? 353 00:24:04,570 --> 00:24:08,518 Now, I am going to regard my curvature tensor. 354 00:24:08,518 --> 00:24:10,810 Like I said, I'm going to regard this as something that 355 00:24:10,810 --> 00:24:13,840 is built from the metric. 356 00:24:21,420 --> 00:24:23,150 So this form in which I've written here, 357 00:24:23,150 --> 00:24:25,760 I'm just saying schematically that Ricci itself 358 00:24:25,760 --> 00:24:27,110 is built from the metric. 359 00:24:27,110 --> 00:24:33,580 The Ricci scalar is the trace of the Ricci curvature. 360 00:24:33,580 --> 00:24:42,710 When I vary this thing, what I would like 361 00:24:42,710 --> 00:24:44,813 is to see what happens-- 362 00:24:44,813 --> 00:24:46,745 whoops, forgot my integral. 363 00:24:56,890 --> 00:25:16,350 And I'm going to vary it in the following way, OK? 364 00:25:18,970 --> 00:25:21,250 let's just clean up my indices a little bit here. 365 00:25:25,435 --> 00:25:26,860 I like to keep things consistent. 366 00:25:26,860 --> 00:25:28,027 I'm going to confuse myself. 367 00:25:28,027 --> 00:25:29,200 Let's leave them like that. 368 00:25:29,200 --> 00:25:31,600 This is fine, OK? 369 00:25:31,600 --> 00:25:33,940 So here's what I want to evaluate. 370 00:25:33,940 --> 00:25:37,060 I'm going to do a variation of these things. 371 00:25:37,060 --> 00:25:39,300 I'm going to assert the variation in the-- 372 00:25:39,300 --> 00:25:40,550 well, I'm not going to assert. 373 00:25:40,550 --> 00:25:42,610 What I'm going to do is do this variation. 374 00:25:42,610 --> 00:25:45,250 That will give me the variation in the action. 375 00:25:45,250 --> 00:25:47,080 For this to be stationary, I'm going 376 00:25:47,080 --> 00:25:51,370 to require that this equals zero. 377 00:25:51,370 --> 00:25:57,010 So what this tells me is I need to work this variation of what 378 00:25:57,010 --> 00:25:58,822 is in square braces out. 379 00:25:58,822 --> 00:26:00,280 I need to work this out right here. 380 00:26:10,060 --> 00:26:15,300 All right, so this needs be done with some care, so let's do it. 381 00:26:40,730 --> 00:26:42,710 First, the thing which I'm going to be 382 00:26:42,710 --> 00:26:46,370 looking at variations of, basically, I'm 383 00:26:46,370 --> 00:26:49,640 looking at a variation of-- 384 00:26:49,640 --> 00:26:58,110 I want to know how this changes when I vary the metric. 385 00:26:58,110 --> 00:27:00,910 This is going to break into three pieces. 386 00:27:00,910 --> 00:27:06,390 I'm going to want the variation of the square root of minus 387 00:27:06,390 --> 00:27:08,850 determinant of the metric. 388 00:27:08,850 --> 00:27:14,190 And then this will come along for the ride. 389 00:27:14,190 --> 00:27:16,975 I'm going to want to look at-- 390 00:27:16,975 --> 00:27:18,600 well, I'm going to get one piece that's 391 00:27:18,600 --> 00:27:22,772 just the variation of the metric itself. 392 00:27:22,772 --> 00:27:24,480 Don't need to do too much more with that. 393 00:27:24,480 --> 00:27:26,190 That actually already has the factor 394 00:27:26,190 --> 00:27:30,850 that I want to factor out there. 395 00:27:30,850 --> 00:27:36,870 And then I'm going to have a term that 396 00:27:36,870 --> 00:27:43,680 looks like what I get when I vary my Ricci curvature, OK? 397 00:27:43,680 --> 00:27:46,440 So these are the three pieces that I need to worry about, 398 00:27:46,440 --> 00:27:47,280 really just two. 399 00:27:47,280 --> 00:27:49,320 This one's kind of trivial. 400 00:27:49,320 --> 00:27:52,410 This one's a little more complicated. 401 00:27:52,410 --> 00:28:01,200 All right, so when I vary my square root of minus G term, 402 00:28:01,200 --> 00:28:03,360 we discuss this a little bit in a previous lecture 403 00:28:03,360 --> 00:28:06,600 where I introduce some, what I call, party tricks involving 404 00:28:06,600 --> 00:28:08,430 the determinant of the metric. 405 00:28:08,430 --> 00:28:16,790 You go back and take a look at this, 406 00:28:16,790 --> 00:28:19,930 you'll see that this can be turned 407 00:28:19,930 --> 00:28:21,740 into the following calculation, OK? 408 00:28:24,480 --> 00:28:31,890 When you vary your Ricci tensor, by something of a miracle, 409 00:28:31,890 --> 00:28:36,200 you will discover that this can be written 410 00:28:36,200 --> 00:28:42,810 as a term that looks like the derivative, should 411 00:28:42,810 --> 00:28:47,660 be the difference rather, of two covariance derivatives 412 00:28:47,660 --> 00:28:54,705 of variations in your Christoffel symbols, OK? 413 00:28:54,705 --> 00:28:56,580 Now, I like to work out what those variations 414 00:28:56,580 --> 00:28:58,560 in the Christoffel symbols are. 415 00:28:58,560 --> 00:29:01,460 These are a little bit lengthy. 416 00:29:01,460 --> 00:29:03,210 Just bear with me while I write these out. 417 00:29:51,516 --> 00:29:55,280 OK, as promised, it's a little bit lengthy. 418 00:30:05,020 --> 00:30:09,040 Using those two terms, OK, using that term, 419 00:30:09,040 --> 00:30:10,510 setting the indices as appropriate 420 00:30:10,510 --> 00:30:14,380 and taking those covariant derivatives, what you will then 421 00:30:14,380 --> 00:30:20,890 find when you evaluate the important combination 422 00:30:20,890 --> 00:30:26,587 to give me that term that involves my Ricci variation 423 00:30:26,587 --> 00:30:27,295 looks as follows. 424 00:30:34,610 --> 00:30:37,120 So I'm going to take those things, 425 00:30:37,120 --> 00:30:41,110 put them together to make my delta Ricci tensor, 426 00:30:41,110 --> 00:30:42,340 and then trace over it. 427 00:30:45,630 --> 00:30:52,398 And you get a lot of cancellations when you do that, 428 00:30:52,398 --> 00:30:53,565 such that the final result-- 429 00:31:06,260 --> 00:31:08,390 Notice no free indices on this, OK? 430 00:31:08,390 --> 00:31:14,850 So I've got derivatives hooking onto the Gs here. 431 00:31:14,850 --> 00:31:17,190 The mu nu is hooking up to that mu nu there. 432 00:31:26,080 --> 00:31:29,090 And before I manipulate this further, 433 00:31:29,090 --> 00:31:32,850 I'm just going to note that this is in the form of-- 434 00:31:32,850 --> 00:31:35,830 you can write this as the divergence of some vector 435 00:31:35,830 --> 00:31:37,330 field. 436 00:31:37,330 --> 00:31:45,350 So I'm going to say that this is the divergence of VA, OK? 437 00:31:45,350 --> 00:31:45,850 Excuse me. 438 00:31:45,850 --> 00:31:46,600 V alpha. 439 00:31:46,600 --> 00:31:48,730 V alpha is implicitly defined. 440 00:31:48,730 --> 00:31:52,280 It's basically this beta covariant derivative 441 00:31:52,280 --> 00:31:56,870 of all this junk here, OK? 442 00:31:56,870 --> 00:31:58,550 You might think, well, wait a minute. 443 00:31:58,550 --> 00:32:00,350 I'm taking derivatives of the metric here. 444 00:32:00,350 --> 00:32:02,642 Why doesn't my covariant derivative metric just give me 445 00:32:02,642 --> 00:32:03,380 zero? 446 00:32:03,380 --> 00:32:05,172 We need to be little bit careful about that 447 00:32:05,172 --> 00:32:07,640 because we are varying the metric, OK? 448 00:32:07,640 --> 00:32:09,710 And when we're varying the metric, 449 00:32:09,710 --> 00:32:13,430 once we have finished our variational principle, 450 00:32:13,430 --> 00:32:15,780 we will settle down to a metric where that is the case. 451 00:32:15,780 --> 00:32:17,322 But in the middle of the calculation, 452 00:32:17,322 --> 00:32:20,390 you just happy a little bit more careful than that. 453 00:32:20,390 --> 00:32:25,150 All right, so now let's put all these ingredients together. 454 00:32:25,150 --> 00:32:37,680 And what you wind up with, combining all of these pieces-- 455 00:33:01,245 --> 00:33:02,220 Oops, pardon me. 456 00:33:02,220 --> 00:33:02,720 Pardon me. 457 00:33:02,720 --> 00:33:04,265 Pardon me. 458 00:33:04,265 --> 00:33:05,890 Actually, let me move to another board. 459 00:33:05,890 --> 00:33:06,988 I've run out of room. 460 00:33:13,593 --> 00:33:15,010 This is the length the expression. 461 00:33:15,010 --> 00:33:16,228 My apologies. 462 00:33:34,680 --> 00:33:38,622 OK, so this becomes-- 463 00:33:38,622 --> 00:33:40,080 make sure I have enough room to get 464 00:33:40,080 --> 00:33:41,247 the whole thing on the line. 465 00:34:05,520 --> 00:34:08,500 OK, so fair amount of analysis goes into-- you 466 00:34:08,500 --> 00:34:09,750 put all those pieces together. 467 00:34:09,750 --> 00:34:11,310 You combine them all. 468 00:34:11,310 --> 00:34:13,500 And our goal is to reassemble what 469 00:34:13,500 --> 00:34:17,980 I've got up there on the top of this middle board here. 470 00:34:17,980 --> 00:34:24,861 So what this tells me is varying the action-- 471 00:34:24,861 --> 00:34:50,290 whoops-- leads me to an equation of this form. 472 00:34:58,005 --> 00:34:59,880 OK, let's stop and look at this for a second. 473 00:34:59,880 --> 00:35:01,800 So variation of the action gives me 474 00:35:01,800 --> 00:35:06,280 something it looks like the Einstein tensor contracted 475 00:35:06,280 --> 00:35:09,520 on my variation of my metric, plus this divergence 476 00:35:09,520 --> 00:35:13,400 of this goofy vector field. 477 00:35:13,400 --> 00:35:18,300 If that divergence of the goofy vector field were not there, 478 00:35:18,300 --> 00:35:19,660 we would be home free. 479 00:35:19,660 --> 00:35:21,280 Imagine we could ignore it. 480 00:35:21,280 --> 00:35:24,340 I would want to set delta S equal to zero, 481 00:35:24,340 --> 00:35:26,797 and that would give me G alpha beta equals 0. 482 00:35:26,797 --> 00:35:28,630 Now, I haven't said anything about a source, 483 00:35:28,630 --> 00:35:30,880 so that, in fact, gives me exactly what 484 00:35:30,880 --> 00:35:35,050 I would need to develop my Einstein field equations, 485 00:35:35,050 --> 00:35:37,510 at least in the case where there's no stress energy 486 00:35:37,510 --> 00:35:38,260 tensor. 487 00:35:38,260 --> 00:35:40,360 In a few moments, we're going to put a stress energy tensor back 488 00:35:40,360 --> 00:35:40,900 into this. 489 00:35:40,900 --> 00:35:44,320 But that's sort of the first step, that-- that first line, 490 00:35:44,320 --> 00:35:46,390 if I did not have that divergence 491 00:35:46,390 --> 00:35:48,880 of V, that would give me exactly what I needed there. 492 00:35:51,800 --> 00:35:55,150 So how do I get rid of that irritating divergence 493 00:35:55,150 --> 00:35:55,900 of a vector field? 494 00:36:14,710 --> 00:36:19,580 Well, in Carroll, it's pointed out that, 495 00:36:19,580 --> 00:36:22,820 because this is a perfect divergence 496 00:36:22,820 --> 00:36:25,580 and it is a proper volume integral, 497 00:36:25,580 --> 00:36:28,700 one can invoke the divergence theorem. 498 00:36:47,920 --> 00:36:49,515 If you invoke the divergence theorem, 499 00:36:49,515 --> 00:36:50,890 and then imagine this integral is 500 00:36:50,890 --> 00:36:52,847 being done over all of spacetime, 501 00:36:52,847 --> 00:36:54,430 and you basically say, well, I imagine 502 00:36:54,430 --> 00:36:56,990 there's no boundary the same because this covers all of time 503 00:36:56,990 --> 00:37:03,903 and all of space, you would then presumably lose contributions 504 00:37:03,903 --> 00:37:04,570 at the boundary. 505 00:37:11,720 --> 00:37:17,660 And then only the G alpha beta equals zero term would remain. 506 00:37:25,060 --> 00:37:26,620 That's not a bad physical argument. 507 00:37:26,620 --> 00:37:29,507 It's somewhat glib, however, OK? 508 00:37:29,507 --> 00:37:31,090 Carroll himself actually noticed this. 509 00:37:31,090 --> 00:37:33,100 So in Carroll's textbook, he himself 510 00:37:33,100 --> 00:37:35,290 says that this is not the rigorous way 511 00:37:35,290 --> 00:37:36,340 you get rid of this, OK? 512 00:37:36,340 --> 00:37:40,950 So for those of you who are more mathematically purist, 513 00:37:40,950 --> 00:37:43,720 if you're feeling dissatisfied-- but it's good on you. 514 00:37:43,720 --> 00:37:46,020 That is the way you should feel about it. 515 00:37:46,020 --> 00:37:48,690 I as a physicist am reasonably comfortable with this, 516 00:37:48,690 --> 00:37:52,200 but I would like to do this a little bit more rigorously. 517 00:37:52,200 --> 00:37:57,690 There are, in fact, two rigorous approaches to fixing this up. 518 00:38:03,850 --> 00:38:08,010 One of them is, in fact, quite beautiful. 519 00:38:08,010 --> 00:38:14,070 And it was my goal to overhaul this lecture and present method 520 00:38:14,070 --> 00:38:17,160 one that I'm about to write down in this lecture 521 00:38:17,160 --> 00:38:19,400 that I'm doing right here, right now. 522 00:38:19,400 --> 00:38:25,980 And among the various tragedies associated with the COVID-19 523 00:38:25,980 --> 00:38:27,810 evacuation of our campus and everything 524 00:38:27,810 --> 00:38:31,928 moving into this online format, a very, very minor tragedy 525 00:38:31,928 --> 00:38:33,720 on the scale of everything that's going on, 526 00:38:33,720 --> 00:38:36,600 is that I just do not have the time 527 00:38:36,600 --> 00:38:38,970 to assemble the note to present what 528 00:38:38,970 --> 00:38:40,890 I was hoping to present to you. 529 00:38:40,890 --> 00:38:43,460 I hope by the end of the course to have enough bandwidth 530 00:38:43,460 --> 00:38:45,210 that I can maybe type some up and add them 531 00:38:45,210 --> 00:38:46,950 as supplemental notes to the course. 532 00:38:46,950 --> 00:38:49,043 But for now, they are not available. 533 00:38:52,670 --> 00:38:55,480 So let me just describe what I was going to present to you. 534 00:39:00,330 --> 00:39:06,487 Method one is you do what is called the Palatini variation. 535 00:39:10,870 --> 00:39:12,700 When you do a Palatini variation-- 536 00:39:17,510 --> 00:39:22,090 whoops-- you write the metric-- excuse me. 537 00:39:22,090 --> 00:39:38,010 You write the action like so, but you consider your Ricci 538 00:39:38,010 --> 00:39:42,510 curvature to be a function not of the metric 539 00:39:42,510 --> 00:39:45,734 but a function of the connection. 540 00:39:58,590 --> 00:40:01,740 Then when you do your variational principle, 541 00:40:01,740 --> 00:40:04,920 you don't just vary with respect to the metric. 542 00:40:04,920 --> 00:40:09,630 You vary it with both the metric and your connection 543 00:40:09,630 --> 00:40:10,805 coefficients. 544 00:40:23,430 --> 00:40:28,140 When you do this, if you do the calculation correctly, 545 00:40:28,140 --> 00:40:31,470 you, in fact, get both the Einstein equation, 546 00:40:31,470 --> 00:40:34,010 the Einstein-- so again, we've not introduced a source. 547 00:40:34,010 --> 00:40:37,550 I'm going to talk about source in just a moment. 548 00:40:37,550 --> 00:40:39,760 So the Einstein equation with no source 549 00:40:39,760 --> 00:40:53,410 emerges, and you find that your connection 550 00:40:53,410 --> 00:40:56,140 must be metric compatible. 551 00:41:04,120 --> 00:41:06,100 In other words, the connection is such 552 00:41:06,100 --> 00:41:10,960 that you are required to have covariant derivative 553 00:41:10,960 --> 00:41:13,027 of the metric equal to zero. 554 00:41:13,027 --> 00:41:14,860 It's actually a beautiful little calculation 555 00:41:14,860 --> 00:41:19,660 because we sort of argue that this was going to be the case-- 556 00:41:19,660 --> 00:41:21,580 when we were talking about how to define 557 00:41:21,580 --> 00:41:23,372 a covariant derivative, we noted that there 558 00:41:23,372 --> 00:41:25,930 are many ways to define a transport operation. 559 00:41:25,930 --> 00:41:27,790 This ended up being one that allowed us 560 00:41:27,790 --> 00:41:29,480 to define parallel transport. 561 00:41:29,480 --> 00:41:31,750 That became something that the physics drove us 562 00:41:31,750 --> 00:41:36,520 to selecting as a very natural choice for our connection. 563 00:41:36,520 --> 00:41:38,080 This variation doesn't necessarily 564 00:41:38,080 --> 00:41:39,850 know anything about that, but it shows 565 00:41:39,850 --> 00:41:42,190 that when you do your variational principle, treating 566 00:41:42,190 --> 00:41:45,430 metric and connection as separate degrees of freedom, 567 00:41:45,430 --> 00:41:49,840 that parallel transport-driven choice of the connection which 568 00:41:49,840 --> 00:41:54,350 gives us the Christoffel symbols emerges as the outcome of that. 569 00:41:54,350 --> 00:41:59,205 Now, this is not a terribly difficult calculation to do, 570 00:41:59,205 --> 00:42:01,330 but there's a lot of details you have to get right. 571 00:42:01,330 --> 00:42:05,740 And I've got about a fourth of it written up at this point 572 00:42:05,740 --> 00:42:08,590 and just haven't had a chance to get all the details completely 573 00:42:08,590 --> 00:42:09,940 straightened out. 574 00:42:09,940 --> 00:42:11,847 I really hope to have the bandwidth 575 00:42:11,847 --> 00:42:13,930 to finish it by the end of the term, in which case 576 00:42:13,930 --> 00:42:18,550 I will add it as supplemental notes to the course. 577 00:42:18,550 --> 00:42:21,910 The other thing you can do-- 578 00:42:21,910 --> 00:42:25,360 so there's absolutely nothing wrong with the approach 579 00:42:25,360 --> 00:42:28,480 that we started doing in which we vary the metric. 580 00:42:28,480 --> 00:42:30,760 And we just assume from the outset 581 00:42:30,760 --> 00:42:33,490 that my connection arises from the metric 582 00:42:33,490 --> 00:42:37,360 so that, the way we do it, the connection and the metric 583 00:42:37,360 --> 00:42:39,010 are not separate degrees of freedom, 584 00:42:39,010 --> 00:42:42,010 but the connection arises from the metric. 585 00:42:42,010 --> 00:42:43,430 And that is fine. 586 00:42:43,430 --> 00:42:46,173 But if we do that, then when you get down to this point, 587 00:42:46,173 --> 00:42:48,090 you just have to be a little bit more careful. 588 00:42:48,090 --> 00:42:51,340 You essentially take this argument that Carroll makes 589 00:42:51,340 --> 00:42:57,400 and you have to just do it with a lot more care. 590 00:42:57,400 --> 00:43:03,070 So what you need to do is define the boundary 591 00:43:03,070 --> 00:43:07,822 associated with the divergence integral very carefully-- 592 00:43:12,560 --> 00:43:13,060 whoops. 593 00:43:13,060 --> 00:43:14,560 Associated with divergence integral. 594 00:43:33,380 --> 00:43:48,200 In doing so, you have to very carefully treat 595 00:43:48,200 --> 00:43:50,045 the curvature on that boundary. 596 00:43:52,610 --> 00:43:55,337 So we are working in four dimensions. 597 00:43:58,470 --> 00:44:09,088 So it's a three dimensional boundary of our 4D spacetime. 598 00:44:13,148 --> 00:44:14,690 You can sort of think of the boundary 599 00:44:14,690 --> 00:44:16,400 as being a particular sort of slice 600 00:44:16,400 --> 00:44:18,563 through four dimensional space time. 601 00:44:18,563 --> 00:44:20,480 You pick a particular three dimensional slice, 602 00:44:20,480 --> 00:44:23,335 redefine that as the boundary region of this integral. 603 00:44:23,335 --> 00:44:24,710 And that requires us to introduce 604 00:44:24,710 --> 00:44:27,130 some additional concepts. 605 00:44:27,130 --> 00:44:30,680 In particular, once you pick out a particular slice 606 00:44:30,680 --> 00:44:34,250 of space time, some of the curvature associated 607 00:44:34,250 --> 00:44:36,920 with the metric on that slice isn't 608 00:44:36,920 --> 00:44:39,560 intrinsic to the spacetime itself, 609 00:44:39,560 --> 00:44:42,530 but it's imposed on the space time 610 00:44:42,530 --> 00:44:45,410 by the manner in which you have made that slice. 611 00:44:45,410 --> 00:44:48,320 Sort of in the same way that if I have a cylinder, 612 00:44:48,320 --> 00:44:50,510 the surface of a cylinder is intrinsically flat. 613 00:44:50,510 --> 00:44:53,030 If I put two geodesics on it and I parallel transport them 614 00:44:53,030 --> 00:44:56,870 around, they remain perfectly parallel forever and forever 615 00:44:56,870 --> 00:44:57,680 and forever. 616 00:44:57,680 --> 00:44:59,840 But it sure as hell looks curved. 617 00:44:59,840 --> 00:45:03,150 But that is because I have taken this flat surface 618 00:45:03,150 --> 00:45:04,670 and I have folded it up. 619 00:45:04,670 --> 00:45:07,460 I have embedded it in a higher dimensional space in such a way 620 00:45:07,460 --> 00:45:10,880 that there's extrinsic curvature that's been imposed on it. 621 00:45:10,880 --> 00:45:14,150 When you are carefully looking at the boundary of a four 622 00:45:14,150 --> 00:45:16,910 dimensional spacetime, you also need to very carefully treat 623 00:45:16,910 --> 00:45:19,220 the extrinsic curvature associated with how you put 624 00:45:19,220 --> 00:45:21,290 that boundary on top of things. 625 00:45:21,290 --> 00:45:26,388 When you do this correctly, you find 626 00:45:26,388 --> 00:45:28,430 you need to be a little bit more careful with how 627 00:45:28,430 --> 00:45:29,990 you define the Lagrangian. 628 00:45:47,662 --> 00:45:49,870 And when you, in fact, do that, it actually turns out 629 00:45:49,870 --> 00:45:53,800 you don't get that divergence term at all. 630 00:45:53,800 --> 00:46:03,470 You get sort of a counter term that cancels this guy away. 631 00:46:03,470 --> 00:46:05,950 So for students who are interested in seeing this 632 00:46:05,950 --> 00:46:08,350 explored and discussed in a lot more rigor 633 00:46:08,350 --> 00:46:11,530 than I am prepared to do, this is 634 00:46:11,530 --> 00:46:21,860 discussed in appendix E of Robert Wald's textbook, General 635 00:46:21,860 --> 00:46:23,342 Relativity. 636 00:46:29,670 --> 00:46:32,180 I will simply comment that Wald's textbook 637 00:46:32,180 --> 00:46:36,020 is one of the most mathematical textbooks on this topic. 638 00:46:36,020 --> 00:46:39,650 He does everything with great rigor and a lot of care. 639 00:46:39,650 --> 00:46:43,370 And the appendices is where he puts his particularly 640 00:46:43,370 --> 00:46:45,120 high-level, grotty details. 641 00:46:45,120 --> 00:46:47,150 So this really is something that is 642 00:46:47,150 --> 00:46:49,670 beyond the level of where we do a one semester 643 00:46:49,670 --> 00:46:52,370 course like 8.962. 644 00:46:52,370 --> 00:46:54,740 So it's unfortunate it sort of pops up here, 645 00:46:54,740 --> 00:46:56,240 which is why a textbook like Carroll 646 00:46:56,240 --> 00:46:58,790 does this in a somewhat glib way. 647 00:46:58,790 --> 00:47:02,318 I am personally a fan of doing this Palatini variation. 648 00:47:02,318 --> 00:47:03,860 And so, as I've emphasized, I'm going 649 00:47:03,860 --> 00:47:08,630 to try to clean that up and make that available. 650 00:47:08,630 --> 00:47:15,690 So what all of this ends up telling us-- 651 00:47:20,490 --> 00:47:22,240 so let's take the optimistic point of view 652 00:47:22,240 --> 00:47:28,420 here that we have either accepted the somewhat 653 00:47:28,420 --> 00:47:30,610 glib argument that we can discard that term, 654 00:47:30,610 --> 00:47:34,510 or we sat down and we slog through appendix E of Wald, 655 00:47:34,510 --> 00:47:37,840 or we wrote up the Palatini variation. 656 00:47:37,840 --> 00:47:40,540 Doing this variation in that case then leads us 657 00:47:40,540 --> 00:48:04,780 to G alpha beta equals zero. 658 00:48:04,780 --> 00:48:07,810 This is the Einstein equation when 659 00:48:07,810 --> 00:48:10,950 there is no source, when the stress energy tensor is zero. 660 00:48:10,950 --> 00:48:15,640 This is often called the vacuum Einstein equation. 661 00:48:25,740 --> 00:48:28,500 More generally, we should treat the action 662 00:48:28,500 --> 00:48:33,090 that we are varying as the Einstein-Hilbert action 663 00:48:33,090 --> 00:48:35,100 plus an action associated with matter. 664 00:48:35,100 --> 00:48:43,550 So a more general form would be to write 665 00:48:43,550 --> 00:48:55,510 S as the integral d4x square root minus G R over 16 pi 666 00:48:55,510 --> 00:48:58,870 G plus the Lagrangian for-- 667 00:48:58,870 --> 00:49:00,760 I'm just going to say m for matter. 668 00:49:00,760 --> 00:49:02,050 Could be matter and fields. 669 00:49:05,920 --> 00:49:18,680 If you do that, then when you look at your variation 670 00:49:18,680 --> 00:49:23,210 and enforce that the variation of the action equals zero, 671 00:49:23,210 --> 00:49:25,640 this leads you to an equation of the form-- 672 00:49:50,620 --> 00:49:53,150 that's given by this. 673 00:49:53,150 --> 00:49:59,380 And so up to a numerical factor, which I'll write out 674 00:49:59,380 --> 00:50:01,060 in just a second, you can basically 675 00:50:01,060 --> 00:50:04,210 see that this variation of my matter Lagrangian 676 00:50:04,210 --> 00:50:06,580 becomes the stress energy tensor. 677 00:50:16,345 --> 00:50:17,970 In particular, if you just look at this 678 00:50:17,970 --> 00:50:33,940 and say, oh, I am going to define T alpha beta as minus 2 679 00:50:33,940 --> 00:50:46,840 divided by square root of minus G, like this, 680 00:50:46,840 --> 00:50:48,670 then the Einstein-Hilbert variation 681 00:50:48,670 --> 00:50:54,640 gives me exactly the same Einstein equation 682 00:50:54,640 --> 00:50:57,690 as we had before. 683 00:50:57,690 --> 00:51:00,270 I have a set of exercises in the notes, which 684 00:51:00,270 --> 00:51:03,990 I'm not going to go through because they're 685 00:51:03,990 --> 00:51:05,347 slightly tedious. 686 00:51:05,347 --> 00:51:07,680 And I think I'm going to actually end a little bit early 687 00:51:07,680 --> 00:51:08,670 here. 688 00:51:08,670 --> 00:51:11,880 But I want to just describe one of the ways in which this 689 00:51:11,880 --> 00:51:13,200 can be used. 690 00:51:13,200 --> 00:51:28,060 So suppose I pick as my matter Lagrangian 691 00:51:28,060 --> 00:51:34,180 minus one fourth f mu nu, f mu nu, 692 00:51:34,180 --> 00:51:38,440 where f mu nu is the Faraday tensor that describes 693 00:51:38,440 --> 00:51:39,760 electric and magnetic fields. 694 00:51:59,530 --> 00:52:04,050 So what I want to do is show that, 695 00:52:04,050 --> 00:52:06,590 by applying this variational principle that I've 696 00:52:06,590 --> 00:52:08,730 defined over there to this-- 697 00:52:08,730 --> 00:52:11,460 but this gives me exactly the stress energy tensor 698 00:52:11,460 --> 00:52:15,120 that one expects for electromagnetic fields. 699 00:52:27,320 --> 00:52:33,250 OK, so defining my electromagnetic action, 700 00:52:33,250 --> 00:52:37,440 pull out the overall factor of minus one fourth, 701 00:52:37,440 --> 00:52:41,960 I integrate over my spacetime coordinates. 702 00:52:45,250 --> 00:52:49,270 And then notice the way this is set up. 703 00:52:49,270 --> 00:52:54,320 So I have f mu nu upstairs, f mu nu downstairs. 704 00:52:54,320 --> 00:53:04,210 Let's write my f mu nu upstairs as f alpha beta downstairs 705 00:53:04,210 --> 00:53:05,980 but with enough prefactors of the metric 706 00:53:05,980 --> 00:53:08,390 to raise those indices. 707 00:53:08,390 --> 00:53:10,100 So I do a variation on this. 708 00:53:16,660 --> 00:53:19,650 And the way I'm going to do it is I will vary the metric. 709 00:53:23,080 --> 00:53:40,140 When I do this, I will not vary the fields, OK? 710 00:53:40,140 --> 00:53:43,890 If one does variations of those things, you can do that, 711 00:53:43,890 --> 00:53:46,470 and you can use that to do things like define your field 712 00:53:46,470 --> 00:53:47,340 equations. 713 00:53:47,340 --> 00:53:49,800 But what I want to do is just see how, by varying 714 00:53:49,800 --> 00:53:52,830 how the metric couples do these things, 715 00:53:52,830 --> 00:53:55,250 the stress energy tensor emerges. 716 00:53:55,250 --> 00:53:58,640 So in my notes, I go through this calculation 717 00:53:58,640 --> 00:54:00,930 with some care. 718 00:54:00,930 --> 00:54:42,360 And what emerges at the end is something that looks like this. 719 00:54:46,330 --> 00:54:48,340 And in fact, it's not too hard to show. 720 00:54:48,340 --> 00:54:50,800 Dig into something, a textbook like Jackson, 721 00:54:50,800 --> 00:54:53,050 what is in square braces there is, in fact, 722 00:54:53,050 --> 00:54:56,920 the standard stress energy tensor associated 723 00:54:56,920 --> 00:54:59,030 with the electromagnetic field. 724 00:54:59,030 --> 00:55:01,780 So this technique, this definition I've put in here, 725 00:55:01,780 --> 00:55:07,130 it does work in all the cases that we know about. 726 00:55:07,130 --> 00:55:10,180 And so this variational principal 727 00:55:10,180 --> 00:55:11,973 root to the field equations-- 728 00:55:16,230 --> 00:55:21,180 so punchline is selecting L equals 729 00:55:21,180 --> 00:55:34,920 R yields the Einstein field equations 730 00:55:34,920 --> 00:55:36,450 upon variation of the metric. 731 00:55:44,100 --> 00:55:47,638 So I'll take a sip of water here. 732 00:55:47,638 --> 00:55:49,680 And then the question I want you to sort of think 733 00:55:49,680 --> 00:55:52,200 of at this point is, so what? 734 00:55:57,410 --> 00:56:02,300 I mean, it's nice that there is a second way to do this. 735 00:56:02,300 --> 00:56:05,382 But did we need it? 736 00:56:05,382 --> 00:56:06,840 Is it really important that we have 737 00:56:06,840 --> 00:56:09,390 the second way of doing this? 738 00:56:09,390 --> 00:56:11,870 Well, here I want to come back to a comment I 739 00:56:11,870 --> 00:56:17,150 made as I began this lecture, which was that our guiding 740 00:56:17,150 --> 00:56:23,360 principle for formulating this Lagrangian 741 00:56:23,360 --> 00:56:29,330 was that it has to be built out of curvature tensors, 742 00:56:29,330 --> 00:56:30,965 and it has to be a scalar. 743 00:56:35,730 --> 00:56:39,690 I would argue that the Ricci scalar 744 00:56:39,690 --> 00:56:43,130 is the simplest such construction that does that. 745 00:57:06,380 --> 00:57:10,880 But it is not the only scalar made from curvature tensors 746 00:57:10,880 --> 00:57:11,630 that will do this. 747 00:57:25,640 --> 00:57:30,790 Imagine you wanted to modify gravity, 748 00:57:30,790 --> 00:57:33,290 or you wanted to come up with a framework that 749 00:57:33,290 --> 00:57:35,660 was consistent with general relativity. 750 00:57:35,660 --> 00:57:40,340 And remember, L equals R gives me general relativity, 751 00:57:40,340 --> 00:57:43,720 gives me Einstein's field equations. 752 00:57:43,720 --> 00:57:47,210 It gives me Einstein's theory of gravity. 753 00:57:47,210 --> 00:57:51,110 Suppose I want to modify it in a way 754 00:57:51,110 --> 00:57:54,590 such that things look just like Einstein's gravity 755 00:57:54,590 --> 00:58:00,440 for most of the universe, but perhaps small differences 756 00:58:00,440 --> 00:58:04,070 kick in when the curvature is small in some sense that 757 00:58:04,070 --> 00:58:05,674 will need to be characterized. 758 00:58:42,270 --> 00:58:45,690 Well, one way you could imagine doing this 759 00:58:45,690 --> 00:58:48,410 is by adding a correction. 760 00:58:48,410 --> 00:59:04,700 Suppose we try L hat equals R minus alpha over R. That sort 761 00:59:04,700 --> 00:59:07,610 of suggests that this is going to lead 762 00:59:07,610 --> 00:59:21,360 to corrections to the framework of gravity 763 00:59:21,360 --> 00:59:30,910 when the curvature scale is less than or of order square root 764 00:59:30,910 --> 00:59:31,410 of alpha. 765 00:59:39,760 --> 00:59:43,610 It's actually a work of a couple pages of algebra. 766 00:59:43,610 --> 00:59:46,060 I can take this modified Lagrangian. 767 00:59:46,060 --> 00:59:49,240 I can go through exactly this variational principles 768 00:59:49,240 --> 00:59:50,470 that we spent today doing. 769 00:59:53,860 --> 01:00:01,390 And when I do so, I find a new field equation coupling 770 01:00:01,390 --> 01:00:05,425 spacetime to my sources. 771 01:00:13,120 --> 01:00:19,450 So vary the metric, enforce stationarity 772 01:00:19,450 --> 01:00:33,550 of the action, what you get is a field equation that 773 01:00:33,550 --> 01:00:34,300 looks like this. 774 01:01:17,300 --> 01:01:21,380 Notice, set that coupling alpha to zero 775 01:01:21,380 --> 01:01:27,410 and it says G alpha beta equals 8 pi newtons GT alpha beta. 776 01:01:27,410 --> 01:01:30,560 But introduce this little scale here, 777 01:01:30,560 --> 01:01:32,990 suddenly you've got all this other stuff emerging. 778 01:01:36,220 --> 01:01:38,640 This is an idea that was proposed and explored 779 01:01:38,640 --> 01:01:41,550 in a paper, if you want to read a bit more about this, 780 01:01:41,550 --> 01:01:46,230 by Sean Carroll et al, published in Physical Review 781 01:01:46,230 --> 01:01:49,770 D. It's now almost coming up on its second decade. 782 01:02:03,260 --> 01:02:08,360 So the reason why Sean and his collaborators developed this, 783 01:02:08,360 --> 01:02:10,670 I mentioned in my previous lecture 784 01:02:10,670 --> 01:02:15,110 that there's observations that have driven us to consider 785 01:02:15,110 --> 01:02:18,260 whether there is, in fact, a cosmological constant 786 01:02:18,260 --> 01:02:22,610 contributing to the large scale structure of our universe. 787 01:02:22,610 --> 01:02:26,300 And like good scientists, you're always 788 01:02:26,300 --> 01:02:29,060 looking for alternate hypotheses that 789 01:02:29,060 --> 01:02:32,060 could be tested or falsified. 790 01:02:32,060 --> 01:02:33,740 And so what Carroll et al pointed out 791 01:02:33,740 --> 01:02:41,840 was the universe's accelerated expansion, so the behavior that 792 01:02:41,840 --> 01:02:45,080 is driving us to consider the possibility the cosmological 793 01:02:45,080 --> 01:02:48,230 constant, it's coming on very, very large scales 794 01:02:48,230 --> 01:02:50,570 where the curvature is quite small. 795 01:02:50,570 --> 01:02:54,470 Laboratory and solar system and many other astrophysical tests 796 01:02:54,470 --> 01:02:56,330 of the foundations of general relativity 797 01:02:56,330 --> 01:02:57,800 aren't as strong there. 798 01:02:57,800 --> 01:03:00,740 So why not try something a little bit out of the box? 799 01:03:00,740 --> 01:03:04,250 And they said, let's think about a theory of gravity 800 01:03:04,250 --> 01:03:07,160 in which there are potentially important modifications when 801 01:03:07,160 --> 01:03:09,770 the curvature is small. 802 01:03:09,770 --> 01:03:10,940 This is what resulted. 803 01:03:10,940 --> 01:03:14,390 And they showed, actually, that by appropriately choosing 804 01:03:14,390 --> 01:03:16,690 the parameter alpha, actually, this 805 01:03:16,690 --> 01:03:18,440 is a point that's worth making. 806 01:03:18,440 --> 01:03:21,710 Notice this is introducing a scale, OK? 807 01:03:21,710 --> 01:03:24,410 The theory doesn't tell us what alpha is. 808 01:03:24,410 --> 01:03:27,380 From the standpoint of the analysis that motivated it, 809 01:03:27,380 --> 01:03:29,630 it was a phenomenological knob that they sort of 810 01:03:29,630 --> 01:03:32,510 tweaked to see whether they could explain observations. 811 01:03:32,510 --> 01:03:34,040 And indeed, they did find that they 812 01:03:34,040 --> 01:03:37,790 could get pretty good explanation of the accelerating 813 01:03:37,790 --> 01:03:40,670 expansion of the universe with this modified 814 01:03:40,670 --> 01:03:42,230 theory of gravity. 815 01:03:42,230 --> 01:03:46,070 But all good theories are testable. 816 01:03:46,070 --> 01:03:47,780 They're falsifiable. 817 01:03:47,780 --> 01:03:49,430 They give rise to hypotheses that you 818 01:03:49,430 --> 01:03:51,050 can compare against data. 819 01:03:51,050 --> 01:03:55,670 It turns out, if you tune alpha to make 820 01:03:55,670 --> 01:03:57,890 the expansion of the universe work, 821 01:03:57,890 --> 01:04:00,290 you get the orbits of the planets in our solar system 822 01:04:00,290 --> 01:04:01,670 wrong. 823 01:04:01,670 --> 01:04:04,400 So this ended up not being a viable theory, 824 01:04:04,400 --> 01:04:07,980 but it was a pretty cool thing to explore, OK? 825 01:04:07,980 --> 01:04:11,150 And I think it serves as a really wonderful example 826 01:04:11,150 --> 01:04:16,460 of how, in a theory of gravity, one formulates a hypothesis 827 01:04:16,460 --> 01:04:17,690 and then tests it. 828 01:04:17,690 --> 01:04:20,390 And then giving Sean and his colleagues 829 01:04:20,390 --> 01:04:22,243 all their due, cheerfully accepting 830 01:04:22,243 --> 01:04:24,410 the falsification when it turned out it didn't work. 831 01:04:26,832 --> 01:04:28,790 You know, it's a very, very nice piece of work. 832 01:04:28,790 --> 01:04:30,590 And I think it just is a really good idea 833 01:04:30,590 --> 01:04:34,160 of the power of this action-based method 834 01:04:34,160 --> 01:04:38,180 for considering modifications to your theory of gravity, OK? 835 01:04:38,180 --> 01:04:39,680 And it really highlights the fact 836 01:04:39,680 --> 01:04:42,830 that just choosing your Lagrange density 837 01:04:42,830 --> 01:04:46,340 as the Ricci scalar, that's the simplest possible thing it 838 01:04:46,340 --> 01:04:46,840 can make. 839 01:04:46,840 --> 01:04:48,980 And that is a sense in which general relativity 840 01:04:48,980 --> 01:04:52,820 is the simplest of all possible relativistic theories 841 01:04:52,820 --> 01:04:53,870 of gravity. 842 01:04:53,870 --> 01:04:55,658 Let me just give you another example of-- 843 01:04:55,658 --> 01:04:56,825 actually, two more examples. 844 01:05:00,648 --> 01:05:02,690 I'm not going to go through them in great detail, 845 01:05:02,690 --> 01:05:04,760 but just sketching the way people sometimes 846 01:05:04,760 --> 01:05:09,160 imagine modifying gravity. 847 01:05:09,160 --> 01:05:21,800 So suppose I choose as my gravitational action 848 01:05:21,800 --> 01:05:29,340 my normal Einstein-Hilbert term plus a term 849 01:05:29,340 --> 01:05:33,570 that goes as the Ricci curvature squared. 850 01:05:33,570 --> 01:05:37,950 This is something that you might expect to become important 851 01:05:37,950 --> 01:05:39,540 when the curvature is large. 852 01:06:06,528 --> 01:06:09,640 OK, so when R is greater than R of order, 853 01:06:09,640 --> 01:06:13,390 1 over beta, the second term, will become more important 854 01:06:13,390 --> 01:06:14,380 than the first term. 855 01:06:14,380 --> 01:06:16,780 Again, we've introduced a scale. 856 01:06:16,780 --> 01:06:20,400 There is some theoretical prejudice 857 01:06:20,400 --> 01:06:26,690 among the community that thinks about these kinds of models 858 01:06:26,690 --> 01:06:28,760 that perhaps this scale has something 859 01:06:28,760 --> 01:06:32,690 to do with where quantum gravity effects begin to kick in 860 01:06:32,690 --> 01:06:35,725 and classical general relativity must be modified. 861 01:06:35,725 --> 01:06:37,100 And so you can imagine, you might 862 01:06:37,100 --> 01:06:40,460 think that this beta is on the order of, 863 01:06:40,460 --> 01:06:44,261 when you work in units where G and C are equal to 1, 864 01:06:44,261 --> 01:06:45,900 H bar to the minus 1. 865 01:06:50,670 --> 01:06:54,140 Actions like this often emerge when you are studying-- 866 01:06:54,140 --> 01:06:56,580 they often show up when you are studying theories 867 01:06:56,580 --> 01:06:59,850 in which general relativity is itself an effective theory that 868 01:06:59,850 --> 01:07:04,392 comes about from averaging over small scale degrees of freedom. 869 01:07:04,392 --> 01:07:05,100 One more example. 870 01:07:19,310 --> 01:07:25,660 What if there were, in addition to the tensor, 871 01:07:25,660 --> 01:07:28,670 the metric tensor which describes 872 01:07:28,670 --> 01:07:30,980 the geometry of space time-- 873 01:07:30,980 --> 01:07:39,720 what if the manner in which the metric tensor coupled 874 01:07:39,720 --> 01:07:45,200 to its source varied depending on some additional field that 875 01:07:45,200 --> 01:07:46,808 filled all of spacetime? 876 01:08:21,170 --> 01:08:34,260 So you can imagine that your gravitational action took 877 01:08:34,260 --> 01:08:42,689 the usual form, but it was modified 878 01:08:42,689 --> 01:08:49,200 by some kind of a scalar field that might itself depend 879 01:08:49,200 --> 01:08:52,096 upon spacetime curvature, OK? 880 01:08:52,096 --> 01:08:56,640 And you can imagine then supplementing this 881 01:08:56,640 --> 01:09:00,000 with an action for the scalar field, 882 01:09:00,000 --> 01:09:02,399 which, up to an overall constant at least, I will write-- 883 01:09:19,189 --> 01:09:23,819 theories of this sort are called scalar tensor series. 884 01:09:23,819 --> 01:09:26,760 And what they essentially amount to is 885 01:09:26,760 --> 01:09:28,597 you imagine that there is some field 886 01:09:28,597 --> 01:09:30,180 in addition to the metric of spacetime 887 01:09:30,180 --> 01:09:31,310 which is affecting gravity. 888 01:09:31,310 --> 01:09:33,102 You can almost think of them as essentially 889 01:09:33,102 --> 01:09:38,850 saying that the effective gravitational constant varies 890 01:09:38,850 --> 01:09:41,729 depending upon the strength of the gravitational interaction. 891 01:09:41,729 --> 01:09:47,609 So perhaps down near a black hole 892 01:09:47,609 --> 01:09:52,590 the Newton constant is bigger or smaller or something than way 893 01:09:52,590 --> 01:09:53,700 out in empty space. 894 01:09:59,950 --> 01:10:03,310 Theories of this kind arise-- 895 01:10:03,310 --> 01:10:06,580 I should say terms of this kind and the action-- 896 01:10:06,580 --> 01:10:15,400 quite naturally in a lot of work in theoretical attempts 897 01:10:15,400 --> 01:10:18,525 to quantize gravity, things like that. 898 01:10:18,525 --> 01:10:20,650 And so people like to pay attention to these things 899 01:10:20,650 --> 01:10:23,770 and to see what kind of generic predictions they can make. 900 01:10:23,770 --> 01:10:27,340 These are known as scalar tensor theories. 901 01:10:32,290 --> 01:10:34,600 And if you're interested in seeing more about them, 902 01:10:34,600 --> 01:10:40,772 Sean Carroll's textbook has a full section describing them. 903 01:10:40,772 --> 01:10:41,980 A little bit out of date now. 904 01:10:48,070 --> 01:10:51,860 One thing that's kind of interesting as we sort of think 905 01:10:51,860 --> 01:10:53,860 about how the subject has evolved over the years 906 01:10:53,860 --> 01:10:55,960 that I've been teaching it is that when 907 01:10:55,960 --> 01:10:59,720 you have both a tensor component, in other words, 908 01:10:59,720 --> 01:11:02,260 a spacetime metric coupling to your stress energy 909 01:11:02,260 --> 01:11:05,290 tensor and you have a scalar field, 910 01:11:05,290 --> 01:11:07,540 you change the dynamics of the theory. 911 01:11:07,540 --> 01:11:10,870 And that ends up doing things like changing 912 01:11:10,870 --> 01:11:12,670 the character of gravitational radiation 913 01:11:12,670 --> 01:11:14,260 that is emitted from a system. 914 01:11:14,260 --> 01:11:17,860 And so, very recently, the most stringent limits 915 01:11:17,860 --> 01:11:21,430 on these possible modifications to general relativity 916 01:11:21,430 --> 01:11:24,520 have come about from things like LIGO observations 917 01:11:24,520 --> 01:11:27,520 which have looked at coalescences of black holes 918 01:11:27,520 --> 01:11:29,110 and neutron stars. 919 01:11:29,110 --> 01:11:32,080 And all of their measurements are 920 01:11:32,080 --> 01:11:34,840 consistent with there being no scalar 921 01:11:34,840 --> 01:11:36,268 modification to these things. 922 01:11:36,268 --> 01:11:37,810 And so it helps to set limits on many 923 01:11:37,810 --> 01:11:42,100 of these theoretical frameworks that introduce and look 924 01:11:42,100 --> 01:11:46,750 at modifications to GR which are motivated 925 01:11:46,750 --> 01:11:51,070 by certain considerations. 926 01:11:51,070 --> 01:11:54,910 This concludes the first half of 8.962. 927 01:11:54,910 --> 01:11:59,620 So if I'm just going to do a quick recap of where we are 928 01:11:59,620 --> 01:12:04,060 and where we are going, where I want you to sort of sit 929 01:12:04,060 --> 01:12:10,660 with things right now is that you have this arsenal which 930 01:12:10,660 --> 01:12:15,130 basically tells you that, given the metric of spacetime, 931 01:12:15,130 --> 01:12:16,840 you can compute the motion of bodies-- 932 01:12:21,680 --> 01:12:28,090 this is done using by geodesics-- 933 01:12:28,090 --> 01:12:33,250 you can characterize the curvature 934 01:12:33,250 --> 01:12:41,900 of that spacetime which tells you about tidal fields, 935 01:12:41,900 --> 01:12:46,640 and you now, as of these last two lectures, 936 01:12:46,640 --> 01:12:54,630 know how to get this by solving the Einstein field equation. 937 01:12:58,390 --> 01:13:00,640 What we will do in the remaining lectures 938 01:13:00,640 --> 01:13:02,290 that I'll be recording and distributing 939 01:13:02,290 --> 01:13:09,060 through this course's website is solving this equation 940 01:13:09,060 --> 01:13:11,550 and exploring the properties of these solutions. 941 01:13:11,550 --> 01:13:16,423 And so let me just sort of say this equation, 942 01:13:16,423 --> 01:13:17,840 if you go back to think about what 943 01:13:17,840 --> 01:13:20,270 goes into this-- at the beginning of my next lecture 944 01:13:20,270 --> 01:13:21,860 which I'll be recording tomorrow, 945 01:13:21,860 --> 01:13:23,570 we'll go through this in some care. 946 01:13:23,570 --> 01:13:27,230 But if you just think of this as a partial differential 947 01:13:27,230 --> 01:13:32,730 equation for the metric, it is a god-awful mess, OK? 948 01:13:32,730 --> 01:13:34,650 There's no nice way to put it. 949 01:13:34,650 --> 01:13:36,440 I mean, it's just a horrible mess 950 01:13:36,440 --> 01:13:40,840 of coupled, non-linear, partial differential equations. 951 01:13:40,840 --> 01:13:44,330 We are going to look at three ways of solving this. 952 01:13:55,050 --> 01:14:01,682 Method one, we will think about a kind of weak field expansion. 953 01:14:07,950 --> 01:14:12,330 I'm going to imagine that the spacetime corresponding 954 01:14:12,330 --> 01:14:21,670 to the sources we are studying looks like flat spacetime 955 01:14:21,670 --> 01:14:24,190 plus a small correction. 956 01:14:24,190 --> 01:14:27,520 That will allow me to take this horrible coupled, 957 01:14:27,520 --> 01:14:30,840 non-linear mess and linearize, OK? 958 01:14:30,840 --> 01:14:33,910 We will lay out the principles for doing that methodically 959 01:14:33,910 --> 01:14:37,400 in the beginning of the next lecture. 960 01:14:37,400 --> 01:14:47,710 Method two, we'll see what happens 961 01:14:47,710 --> 01:14:48,895 when we assert a symmetry. 962 01:14:52,730 --> 01:14:55,730 One of the reasons why those equations are such a mess 963 01:14:55,730 --> 01:14:57,770 is that, in general, there's just 964 01:14:57,770 --> 01:15:00,200 a tremendous number of terms that 965 01:15:00,200 --> 01:15:04,640 are coupling to one another, all sorts of things tying together, 966 01:15:04,640 --> 01:15:06,470 and it's just very, very complicated. 967 01:15:06,470 --> 01:15:09,200 But if we imagine that everything is, say, spherical 968 01:15:09,200 --> 01:15:11,600 symmetric, then that dramatically 969 01:15:11,600 --> 01:15:13,580 reduces the number of degrees of freedom 970 01:15:13,580 --> 01:15:17,750 that the spacetime could have even as a matter of principle. 971 01:15:17,750 --> 01:15:21,020 And so we will actually start by beginning 972 01:15:21,020 --> 01:15:24,680 to explore the most symmetric space times of all. 973 01:15:24,680 --> 01:15:27,200 And I will lay out exactly what that means. 974 01:15:27,200 --> 01:15:31,550 And that actually turns out, at least to within our ability 975 01:15:31,550 --> 01:15:33,580 to observationally probe this-- 976 01:15:33,580 --> 01:15:36,080 that appears to describe our universe on the largest scales. 977 01:15:36,080 --> 01:15:38,450 And so when we do the largest, the most symmetric things 978 01:15:38,450 --> 01:15:41,540 possible, that gives us spacetimes 979 01:15:41,540 --> 01:15:45,800 that worked very well for describing cosmology. 980 01:15:45,800 --> 01:15:48,560 Making things a little bit less symmetric but still highly, 981 01:15:48,560 --> 01:15:52,370 highly symmetric, that allows us to describe very strong field 982 01:15:52,370 --> 01:15:54,840 objects, things which I'm going to call-- 983 01:15:54,840 --> 01:15:56,750 I'm going to just call them stars. 984 01:15:56,750 --> 01:15:58,670 What I really mean by that are these 985 01:15:58,670 --> 01:16:01,550 are objects that are just self gravitating circle fluid 986 01:16:01,550 --> 01:16:03,650 balls, OK? 987 01:16:03,650 --> 01:16:06,290 And we will see how they can sort of hold themselves up 988 01:16:06,290 --> 01:16:08,878 against their own gravity, what their spacetime looks like. 989 01:16:08,878 --> 01:16:10,670 There's all sorts of interesting properties 990 01:16:10,670 --> 01:16:11,720 that emerge from them. 991 01:16:11,720 --> 01:16:14,090 And eventually we actually find that you 992 01:16:14,090 --> 01:16:18,290 reach a set of configurations where there is no-- 993 01:16:18,290 --> 01:16:22,550 no fluid can be allowed by the-- 994 01:16:22,550 --> 01:16:23,960 let me try that sentence again. 995 01:16:23,960 --> 01:16:27,890 You can find configurations such that no fluid allowable 996 01:16:27,890 --> 01:16:32,330 by the laws of physics can give you a stable object. 997 01:16:32,330 --> 01:16:35,060 They produce black holes. 998 01:16:35,060 --> 01:16:37,940 That will basically take us, essentially, 999 01:16:37,940 --> 01:16:39,400 to the end of the course, OK? 1000 01:16:39,400 --> 01:16:41,990 These two things have a tremendous amount 1001 01:16:41,990 --> 01:16:43,440 of content in them. 1002 01:16:43,440 --> 01:16:47,660 And this is where a huge amount of the literature 1003 01:16:47,660 --> 01:16:50,120 on general relativity has focused its attention 1004 01:16:50,120 --> 01:16:52,960 over the past several decades. 1005 01:16:52,960 --> 01:16:59,730 In my last lecture or so, I'm going to briefly describe 1006 01:16:59,730 --> 01:17:00,690 general solutions. 1007 01:17:06,240 --> 01:17:09,720 What do you do if there is no symmetry, 1008 01:17:09,720 --> 01:17:13,290 there is no weak expansion, no weak parameter 1009 01:17:13,290 --> 01:17:14,760 you can expand in? 1010 01:17:14,760 --> 01:17:19,320 You just have to deal with the non-linear, bloody mess 1011 01:17:19,320 --> 01:17:25,020 that these equations are and then try to solve it. 1012 01:17:25,020 --> 01:17:27,810 And what this will do is allow me to explore and tell you 1013 01:17:27,810 --> 01:17:32,700 a little bit about the field of numerical relativity, which 1014 01:17:32,700 --> 01:17:34,950 is what you do when you just want 1015 01:17:34,950 --> 01:17:38,580 to solve these things with no approximations, no assumptions 1016 01:17:38,580 --> 01:17:40,410 that simplify it whatsoever. 1017 01:17:40,410 --> 01:17:43,740 Just take Einstein's field equations, 1018 01:17:43,740 --> 01:17:46,380 put it into a computer, write a code, 1019 01:17:46,380 --> 01:17:50,590 hit Go, and see what happens. 1020 01:17:50,590 --> 01:17:52,305 So under the normal schedule of things, 1021 01:17:52,305 --> 01:17:53,680 where we would end today would've 1022 01:17:53,680 --> 01:17:55,920 been right before spring break. 1023 01:17:55,920 --> 01:17:59,100 And these lectures would start up right after spring break. 1024 01:17:59,100 --> 01:18:02,490 Under the system that we are operating now, 1025 01:18:02,490 --> 01:18:04,823 I do suggest you sort of save this for a little while. 1026 01:18:04,823 --> 01:18:06,740 Make sure that you understand what is going on 1027 01:18:06,740 --> 01:18:08,430 with these field equations. 1028 01:18:08,430 --> 01:18:12,600 And we will be ready to explore the solutions, 1029 01:18:12,600 --> 01:18:14,200 and not just compute these solutions 1030 01:18:14,200 --> 01:18:16,140 but also explore things like motion 1031 01:18:16,140 --> 01:18:18,210 in them and their properties. 1032 01:18:18,210 --> 01:18:24,190 And we'll begin that process immediately afterwards. 1033 01:18:24,190 --> 01:18:28,490 All right, and I will conclude this lecture here.