1 00:00:00,500 --> 00:00:02,410 [SQUEAKING] 2 00:00:02,410 --> 00:00:04,338 [RUSTLING] 3 00:00:04,338 --> 00:00:08,194 [CLICKING] 4 00:00:10,795 --> 00:00:12,420 SCOTT HUGHES: So we're switching gears. 5 00:00:12,420 --> 00:00:14,440 We're done with the large scale structure of the universe. 6 00:00:14,440 --> 00:00:16,079 Now we're done with cosmology. 7 00:00:16,079 --> 00:00:18,480 And I'm going to move into the topic that 8 00:00:18,480 --> 00:00:25,110 will dominate the last several lectures of 8.962. 9 00:00:25,110 --> 00:00:28,730 Let's look at how to construct the spacetime 10 00:00:28,730 --> 00:00:31,310 of a compact body. 11 00:00:31,310 --> 00:00:33,290 In particular, we're going to start by focusing 12 00:00:33,290 --> 00:00:36,560 on a compact spherical body. 13 00:00:44,310 --> 00:00:47,210 What I mean by compact-- so spherical, I hope, is clear-- 14 00:00:47,210 --> 00:00:51,030 but what I mean by compact is that the body occupies 15 00:00:51,030 --> 00:00:52,830 a finite spatial region. 16 00:01:06,030 --> 00:01:11,340 And it has a surface with an exterior that is vacuum. 17 00:01:15,890 --> 00:01:17,880 So what I'm going to do is design 18 00:01:17,880 --> 00:01:20,110 some kind of a spacetime. 19 00:01:20,110 --> 00:01:22,170 I'm imagining that there is a source that 20 00:01:22,170 --> 00:01:23,580 fills some compact region. 21 00:01:23,580 --> 00:01:26,430 I'm going to make it strictly symmetric. 22 00:01:26,430 --> 00:01:29,370 And the exterior of the source will be t mu nu equals 0. 23 00:01:34,330 --> 00:01:37,330 So if you think about this, this is telling you 24 00:01:37,330 --> 00:01:39,970 that if it's spherically symmetric, 25 00:01:39,970 --> 00:01:44,150 your metric can only depend on radius. 26 00:01:44,150 --> 00:01:46,840 And so the most general form, at least 27 00:01:46,840 --> 00:01:49,210 for a static spherically symmetric spacetime-- 28 00:01:49,210 --> 00:01:51,100 we'll talk a little bit about a few things 29 00:01:51,100 --> 00:01:54,190 related to non-static situations a little bit later. 30 00:01:57,280 --> 00:02:02,140 So the most general static form of a spherically symmetric 31 00:02:02,140 --> 00:02:12,410 compact body tells me that I can write my metric 32 00:02:12,410 --> 00:02:16,930 as some function of r, which for reasons of convenience 33 00:02:16,930 --> 00:02:23,220 I will write that is e to the 2 phi, t squared. 34 00:02:23,220 --> 00:02:28,780 Some other function of r, dr squared. 35 00:02:32,790 --> 00:02:36,330 And then some function of r times my angular sector. 36 00:02:41,758 --> 00:02:43,550 You might wonder why there's no cross-term. 37 00:02:43,550 --> 00:02:45,383 In particular, why can't I have a cross-term 38 00:02:45,383 --> 00:02:47,990 between my dt and my dr? 39 00:02:47,990 --> 00:02:50,325 I have a few lines on this in my notes. 40 00:02:50,325 --> 00:02:51,950 I won't go through them in detail here, 41 00:02:51,950 --> 00:02:53,300 but I'll just describe. 42 00:02:53,300 --> 00:02:55,070 Basically, what you find is that if there 43 00:02:55,070 --> 00:02:57,980 is a cross-term between, say, the t and the r pieces, 44 00:02:57,980 --> 00:03:01,100 you can get rid of it by a coordinate transformation. 45 00:03:01,100 --> 00:03:03,230 This way of doing things so that it 46 00:03:03,230 --> 00:03:08,530 is diagonal amounts to choosing a good time coordinate. 47 00:03:08,530 --> 00:03:10,730 We can simplify the angular sector 48 00:03:10,730 --> 00:03:13,419 by choosing a good radial coordinate. 49 00:03:24,400 --> 00:03:26,590 The word "good" is a little bit loaded here. 50 00:03:26,590 --> 00:03:27,490 I'm going to actually in a moment 51 00:03:27,490 --> 00:03:29,980 describe a slightly different choice of radial coordinate, 52 00:03:29,980 --> 00:03:32,350 which is also good, but for different purposes. 53 00:03:32,350 --> 00:03:35,500 The one I'm going to use is one where 54 00:03:35,500 --> 00:03:41,710 this function that gives length to the angular sector 55 00:03:41,710 --> 00:03:43,780 is just r squared. 56 00:03:43,780 --> 00:03:52,490 What this means is that the coordinate r, 57 00:03:52,490 --> 00:04:01,990 it labels nested spheres that have 58 00:04:01,990 --> 00:04:09,350 surface area 4 pi r squared. 59 00:04:09,350 --> 00:04:11,950 So if you imagine that in this spacetime 60 00:04:11,950 --> 00:04:16,899 you make a series of spheres, and each one of them 61 00:04:16,899 --> 00:04:22,120 has an area 4 pi r squared, that is the r that goes into this. 62 00:04:22,120 --> 00:04:25,090 Bear in mind-- [CLEARING THROAT] excuse me-- bear in mind, 63 00:04:25,090 --> 00:04:26,590 that does not mean-- 64 00:04:26,590 --> 00:04:32,050 let's say I've got one sphere of radius 4 pi r1 squared, 65 00:04:32,050 --> 00:04:37,390 and then outside of it a sphere of radius 4 pi r2 squared. 66 00:04:37,390 --> 00:04:40,180 That does not mean that the distance, the proper distance, 67 00:04:40,180 --> 00:04:42,790 between those two spheres is r2 minus r1. 68 00:04:56,830 --> 00:04:59,230 So this is unlikely to measure-- 69 00:04:59,230 --> 00:05:01,070 to label radial distance cleanly. 70 00:05:01,070 --> 00:05:02,530 In fact, it would only be the case 71 00:05:02,530 --> 00:05:05,600 that this cleanly labels radial distance if the function 72 00:05:05,600 --> 00:05:07,780 lambda were equal to 0. 73 00:05:07,780 --> 00:05:10,190 And as we'll see, that's generally not the case. 74 00:05:10,190 --> 00:05:13,453 So whenever we do things in a curved spacetime, 75 00:05:13,453 --> 00:05:15,620 things are always going to get a little bit messier. 76 00:05:15,620 --> 00:05:17,300 And what's nice about this is at least 77 00:05:17,300 --> 00:05:21,620 we do have a clean geometric meaning to the coordinate r. 78 00:05:21,620 --> 00:05:24,410 In fact, before I move on, this is 79 00:05:24,410 --> 00:05:33,680 called an areal radius, a radius simply related to surface area. 80 00:05:48,840 --> 00:05:52,320 So in the end, the spacetime that I 81 00:05:52,320 --> 00:05:57,540 will be working with in this lecture takes this form. 82 00:06:04,680 --> 00:06:07,590 And so this is a choice that is known 83 00:06:07,590 --> 00:06:10,016 as Schwarzschild coordinates. 84 00:06:23,140 --> 00:06:27,610 We will come back to the name Schwarzschild before too long. 85 00:06:27,610 --> 00:06:30,160 As I said, so this is a particularly good choice 86 00:06:30,160 --> 00:06:31,150 of radial coordinate. 87 00:06:31,150 --> 00:06:33,080 It'll be very convenient for what we do. 88 00:06:33,080 --> 00:06:36,250 It's not the only one. 89 00:06:36,250 --> 00:06:38,350 Other coordinate choices are possible. 90 00:06:38,350 --> 00:06:40,450 And another one that is quite useful for us-- 91 00:06:42,970 --> 00:06:46,670 we're not going to use it too much in this lecture-- 92 00:06:46,670 --> 00:06:49,723 is what are called isotropic coordinates. 93 00:06:54,460 --> 00:06:57,270 So this is a coordinate system in which 94 00:06:57,270 --> 00:07:17,770 you choose a radial coordinate which I will call r bar, 95 00:07:17,770 --> 00:07:19,900 such that your line element looks like this. 96 00:07:19,900 --> 00:07:21,790 What this does is this emphasizes 97 00:07:21,790 --> 00:07:25,600 the fact that we are working in a coordinate system-- 98 00:07:25,600 --> 00:07:29,800 or excuse me-- we're working in a spacetime in which all 99 00:07:29,800 --> 00:07:33,970 of your spatial slices are fundamentally isotropic. 100 00:07:33,970 --> 00:07:38,065 So get far enough away from this thing, and the three-- 101 00:07:38,065 --> 00:07:39,940 it's just emphasizing the fact that the three 102 00:07:39,940 --> 00:07:41,380 spatial directions are-- 103 00:07:41,380 --> 00:07:42,880 there is something special about it. 104 00:07:42,880 --> 00:07:46,660 There's presumably a body at r bar equals 0, in its vicinity, 105 00:07:46,660 --> 00:07:48,520 but other than that, there's nothing 106 00:07:48,520 --> 00:07:50,448 particularly special about-- 107 00:07:50,448 --> 00:07:52,240 if I go into a little freely falling frame, 108 00:07:52,240 --> 00:07:53,698 all three directions look the same. 109 00:07:53,698 --> 00:07:55,330 This helps emphasize this. 110 00:07:55,330 --> 00:07:58,990 We lose the areal interpretation of r here. 111 00:07:58,990 --> 00:08:07,500 I'll remind you that our weak field solution that we derived 112 00:08:07,500 --> 00:08:14,100 a few weeks ago, a few lectures ago rather, 113 00:08:14,100 --> 00:08:15,120 it looked like this. 114 00:08:28,700 --> 00:08:30,380 And voila, look at that. 115 00:08:30,380 --> 00:08:32,630 It's exactly the same form. 116 00:08:32,630 --> 00:08:37,100 So this is, in fact-- when we derive this, 117 00:08:37,100 --> 00:08:39,780 this is, in fact, actually in isotropic coordinates. 118 00:08:44,200 --> 00:08:46,280 What we found was that the mu that appears here 119 00:08:46,280 --> 00:08:53,810 is equal to minus phi with phi itself being very small. 120 00:08:53,810 --> 00:08:57,950 And the phi that came from this is 121 00:08:57,950 --> 00:09:01,370 Newton's gravitational-- the Newtonion gravitational 122 00:09:01,370 --> 00:09:02,232 potential. 123 00:09:06,020 --> 00:09:06,780 Let's switch back. 124 00:09:06,780 --> 00:09:10,040 We're going to use Schwarzschild coordinates. 125 00:09:10,040 --> 00:09:11,870 This is kind of an aside. 126 00:09:11,870 --> 00:09:14,480 We will use Schwarzschild shield coordinates 127 00:09:14,480 --> 00:09:16,007 for the bulk of our lecture. 128 00:09:16,007 --> 00:09:17,840 Actually, the last thing I'll say about this 129 00:09:17,840 --> 00:09:22,430 is that there is a problem. 130 00:09:22,430 --> 00:09:23,930 There's a problem on a problem set-- 131 00:09:23,930 --> 00:09:25,350 I might have made it optional. 132 00:09:25,350 --> 00:09:27,500 I don't quite recall-- 133 00:09:27,500 --> 00:09:30,770 in which for one of the spacetimes that you're 134 00:09:30,770 --> 00:09:35,390 going to be working with in about two more lectures, 135 00:09:35,390 --> 00:09:37,730 you convert between Schwarzschild coordinates 136 00:09:37,730 --> 00:09:39,050 and isotropic coordinates. 137 00:09:39,050 --> 00:09:41,640 I think it's an optional lec-- 138 00:09:41,640 --> 00:09:45,270 it's an optional problem at this point. 139 00:09:45,270 --> 00:09:47,420 But it's worth looking at. 140 00:09:47,420 --> 00:09:50,550 So let's build a bot. 141 00:09:50,550 --> 00:09:53,640 Let's build the spacetime of a compact spherical body. 142 00:10:01,710 --> 00:10:05,220 What we need are curvature tensors and matter terms. 143 00:10:13,800 --> 00:10:17,060 The curvature tensors, since I've given you-- 144 00:10:17,060 --> 00:10:17,560 whoops. 145 00:10:17,560 --> 00:10:18,280 Not there. 146 00:10:18,280 --> 00:10:18,400 No. 147 00:10:18,400 --> 00:10:19,000 The one at the top. 148 00:10:19,000 --> 00:10:19,660 I'm sorry. 149 00:10:19,660 --> 00:10:21,670 Since I've given you ds squared, you 150 00:10:21,670 --> 00:10:24,140 know all the metric elements. 151 00:10:24,140 --> 00:10:26,380 It's then just a matter of a little bit 152 00:10:26,380 --> 00:10:30,130 of having the algebraic stamina to run through and do 153 00:10:30,130 --> 00:10:31,420 the calculations. 154 00:10:31,420 --> 00:10:32,830 It's straightforward to construct 155 00:10:32,830 --> 00:10:36,400 all of the curvature tensor components. 156 00:10:36,400 --> 00:10:40,720 The GR tool Mathematica notebook that 157 00:10:40,720 --> 00:10:43,360 will soon be released on the 8.962 website, 158 00:10:43,360 --> 00:10:45,160 if it has not been released already, 159 00:10:45,160 --> 00:10:48,188 that will allow you to do this. 160 00:10:48,188 --> 00:10:50,230 I'm going to just sort of quote the results, what 161 00:10:50,230 --> 00:10:51,355 the results turn out to be. 162 00:10:51,355 --> 00:10:54,070 So for this spacetime, it's given in Carroll's textbook, 163 00:10:54,070 --> 00:10:57,280 a slightly different notation, but you can easily 164 00:10:57,280 --> 00:11:00,820 translate from the functions Carroll uses to the ones 165 00:11:00,820 --> 00:11:02,193 that I've written here. 166 00:11:02,193 --> 00:11:04,360 So we need to get those curvature tensor components. 167 00:11:04,360 --> 00:11:06,110 And we need matter. 168 00:11:06,110 --> 00:11:08,410 And so we will do what we generally 169 00:11:08,410 --> 00:11:10,090 always do in this class. 170 00:11:10,090 --> 00:11:15,160 We will treat our body as being made of a perfect fluid. 171 00:11:15,160 --> 00:11:22,300 So let's write out our curvature tensors that arise from this. 172 00:11:22,300 --> 00:11:24,970 I'm going to go straight to the Ricci tensor. 173 00:11:30,450 --> 00:11:33,270 If you want to work out all the complements to the Riemann 174 00:11:33,270 --> 00:11:35,580 tensor, knock yourself out. 175 00:11:35,580 --> 00:11:39,057 As I said, they are listed in Carroll's book. 176 00:11:39,057 --> 00:11:41,640 It's not terribly difficult to go and work all these guys out, 177 00:11:41,640 --> 00:11:43,790 though. 178 00:11:43,790 --> 00:11:45,560 Certainly not using Mathematica. 179 00:12:06,390 --> 00:12:11,820 So here's what you get for the rr complement of the Ricci. 180 00:12:11,820 --> 00:12:14,920 Notice we are seeing nonlinear terms in here. 181 00:12:14,920 --> 00:12:18,030 These are the kind of terms that in all of our calculations, 182 00:12:18,030 --> 00:12:21,780 when we were looking at bodies in a weak field spacetime, 183 00:12:21,780 --> 00:12:22,740 we set those to 0. 184 00:13:00,050 --> 00:13:01,460 So the form for the ttp. 185 00:13:01,460 --> 00:13:05,120 So notice it's quite similar to the form for rrp's. 186 00:13:05,120 --> 00:13:07,280 If you saw me hesitate while I was writing, 187 00:13:07,280 --> 00:13:09,470 I actually suddenly just got concerned I was writing 188 00:13:09,470 --> 00:13:12,230 the wrong line down, and was just verifying 189 00:13:12,230 --> 00:13:14,210 that I was writing the right thing. 190 00:13:14,210 --> 00:13:22,360 And two more non-trivial complements, or really 191 00:13:22,360 --> 00:13:24,660 one more non-trivial component, and then there's 192 00:13:24,660 --> 00:13:27,670 a fourth component that is simply related. 193 00:13:48,200 --> 00:13:51,590 So here is our Ricci tensor. 194 00:13:54,990 --> 00:13:57,820 We're going to solve it for a perfect fluid stress energy 195 00:13:57,820 --> 00:13:58,320 tensor. 196 00:13:58,320 --> 00:14:00,510 We will also solve it for-- 197 00:14:00,510 --> 00:14:02,940 sorry-- we will use this to make an Einstein tensor, 198 00:14:02,940 --> 00:14:05,280 equate it to a perfect fluid stress energy tensor, 199 00:14:05,280 --> 00:14:07,650 and use that to build the spacetime 200 00:14:07,650 --> 00:14:10,050 on the interior of this object. 201 00:14:10,050 --> 00:14:12,600 The exterior of the object is going to be treated as vacuum. 202 00:14:12,600 --> 00:14:14,502 That will have 0 stress energy tensor. 203 00:14:14,502 --> 00:14:15,960 And so it'll be a little bit easier 204 00:14:15,960 --> 00:14:18,030 when we construct the exterior solution just 205 00:14:18,030 --> 00:14:21,270 to set Ricci equal to 0 and solve it. 206 00:14:21,270 --> 00:14:23,940 Before I get into that, let's look at the source 207 00:14:23,940 --> 00:14:26,790 we're going to use to describe this compact spherical body. 208 00:14:29,660 --> 00:14:31,410 Like I said, we're going to treat this guy 209 00:14:31,410 --> 00:14:35,770 as a perfect fluid stress energy. 210 00:14:35,770 --> 00:14:38,073 So by now, you will have seen me written-- 211 00:14:38,073 --> 00:14:39,490 you'll have seen that I've written 212 00:14:39,490 --> 00:14:41,350 this form down multiple times. 213 00:14:56,480 --> 00:14:58,490 So these, the pressure and the density, 214 00:14:58,490 --> 00:15:00,500 are both functions only of radius, 215 00:15:00,500 --> 00:15:03,410 since this is spherically symmetric. 216 00:15:03,410 --> 00:15:09,780 And u denotes the four-velocity of fluid elements in this 217 00:15:09,780 --> 00:15:10,280 fluid-- 218 00:15:10,280 --> 00:15:12,362 in this body. 219 00:15:12,362 --> 00:15:14,195 I'm going to start calling this body a star. 220 00:15:16,612 --> 00:15:18,320 It doesn't necessarily have to be a star, 221 00:15:18,320 --> 00:15:20,240 but it's just a convenient shorthand. 222 00:15:20,240 --> 00:15:24,440 So I want this star's fluid to be static. 223 00:15:28,960 --> 00:15:35,260 So the four-velocity will have a timeline component, 224 00:15:35,260 --> 00:15:37,230 but there is no spatial motion. 225 00:15:37,230 --> 00:15:41,900 I want the star's fluid to be at spatial rest. 226 00:15:41,900 --> 00:15:47,160 I am going to require that this be properly normalized. 227 00:15:47,160 --> 00:15:49,380 u dot u equals minus 1. 228 00:15:49,380 --> 00:15:52,680 So if I do that in this spacetime, 229 00:15:52,680 --> 00:15:56,770 I'm led to the requirement that u upstairs 230 00:15:56,770 --> 00:16:00,150 t is e to the phi of r. 231 00:16:02,930 --> 00:16:03,590 Pardon me. 232 00:16:03,590 --> 00:16:05,720 e to the minus 5r. 233 00:16:05,720 --> 00:16:10,250 And u downstairs t is minus e to the 5r. 234 00:16:18,740 --> 00:16:20,800 Finally, we're going to assume-- 235 00:16:20,800 --> 00:16:24,450 so as I said, we're going to treat this as a compact body. 236 00:16:24,450 --> 00:16:27,830 So we're going to require that it have a surface. 237 00:16:27,830 --> 00:16:34,730 So we will assume rho of r equals 0, p of r 238 00:16:34,730 --> 00:16:40,520 equals 0 for r greater than or equal to r star. 239 00:16:40,520 --> 00:16:43,340 r star denotes the surface of this body. 240 00:16:51,300 --> 00:16:53,490 Let's attack. 241 00:16:53,490 --> 00:16:58,210 So it's a little bit easiest to do the exterior first. 242 00:17:05,500 --> 00:17:08,560 So we're going to consider the region outside of r star. 243 00:17:11,569 --> 00:17:13,819 t mu nu equals 0 there. 244 00:17:13,819 --> 00:17:23,170 And so my Einstein equations become Ricci equals 0. 245 00:17:23,170 --> 00:17:26,380 So I can just take every one of these things, 246 00:17:26,380 --> 00:17:31,510 set them equal to 0, and try to assemble solutions 247 00:17:31,510 --> 00:17:33,670 for phi and for lambda. 248 00:17:33,670 --> 00:17:36,640 Now a particularly convenient-- 249 00:17:36,640 --> 00:17:38,618 every one of these things has to equal 0 250 00:17:38,618 --> 00:17:40,660 so I can make various linear combinations of them 251 00:17:40,660 --> 00:17:42,160 to make things that are particularly 252 00:17:42,160 --> 00:17:43,560 convenient to work with. 253 00:17:43,560 --> 00:17:50,140 And if you sort of stare at the tt and the rr equations, 254 00:17:50,140 --> 00:17:55,600 one thing you see is that if you multiply the tt 255 00:17:55,600 --> 00:17:59,100 by this combination of metric functions, 256 00:17:59,100 --> 00:18:06,170 add the r r, what that does is it allows you to cancel out 257 00:18:06,170 --> 00:18:07,520 those nonlinear terms. 258 00:18:13,220 --> 00:18:15,210 And this combination tells you that-- 259 00:18:20,578 --> 00:18:23,370 whoops-- this combination tells you 260 00:18:23,370 --> 00:18:28,930 that it's an overall factor of 2/r, 261 00:18:28,930 --> 00:18:31,540 the radial derivative of phi equals the radial derivative 262 00:18:31,540 --> 00:18:33,430 of lambda. 263 00:18:33,430 --> 00:18:37,960 And so this tells me that at least in the exterior phi 264 00:18:37,960 --> 00:18:41,440 equals minus lambda, perhaps up to some constant 265 00:18:41,440 --> 00:18:44,350 of integration. 266 00:18:44,350 --> 00:18:51,300 If we go when we plug this into our metric, what that k does 267 00:18:51,300 --> 00:18:54,150 is it just gives me a rescaling, depending upon 268 00:18:54,150 --> 00:18:56,130 whether I want to attach it to the function phi 269 00:18:56,130 --> 00:18:59,400 or attach it to the function lambda. 270 00:18:59,400 --> 00:19:03,600 What we see is that K just essentially rescales. 271 00:19:03,600 --> 00:19:04,845 But let's do the following. 272 00:19:04,845 --> 00:19:06,720 Let's sort of say that I'm going to plug this 273 00:19:06,720 --> 00:19:10,470 in and say that I'm going to plug it in for phi. 274 00:19:10,470 --> 00:19:26,600 So if I plug this into the metric, 275 00:19:26,600 --> 00:19:31,310 I can capture the impact of the constant of integration k here 276 00:19:31,310 --> 00:19:33,564 by just rescaling my time coordinate. 277 00:19:36,255 --> 00:19:37,880 So we are free just to say, OK, there's 278 00:19:37,880 --> 00:19:38,922 no physics in this thing. 279 00:19:38,922 --> 00:19:40,220 So let's just set k equals 0. 280 00:19:46,860 --> 00:19:47,920 So that's already great. 281 00:19:47,920 --> 00:19:50,860 We have found that at least in the exterior 282 00:19:50,860 --> 00:19:53,020 phi is equal to minus lambda. 283 00:19:53,020 --> 00:19:56,170 We've reduced this metric from having two free functions 284 00:19:56,170 --> 00:19:58,570 to one. 285 00:19:58,570 --> 00:20:01,390 So we're feeling spunky now. 286 00:20:01,390 --> 00:20:05,500 Let's take a look at a couple more of these equations, 287 00:20:05,500 --> 00:20:07,000 and see what we can learn from them. 288 00:20:18,810 --> 00:20:24,120 So we still have some of these ugly things involving 289 00:20:24,120 --> 00:20:27,210 nonlinear stuff in the rr and the tt, 290 00:20:27,210 --> 00:20:32,180 but it's all linear in the theta theta equation. 291 00:20:32,180 --> 00:20:40,380 So let's look at r theta theta equals 0. 292 00:20:40,380 --> 00:20:45,890 So I have e to the minus 2 lambda r 293 00:20:45,890 --> 00:20:54,810 dr of lambda minus r dr of phi minus 1 plus 1. 294 00:20:54,810 --> 00:20:56,820 All this equals 0. 295 00:20:56,820 --> 00:20:59,250 So now you can either substitute for lambda or substitute 296 00:20:59,250 --> 00:21:00,750 for phi. 297 00:21:00,750 --> 00:21:03,720 I'm going to follow my notes here and substitute 298 00:21:03,720 --> 00:21:06,500 in lambda equals minus phi. 299 00:21:22,010 --> 00:21:24,290 Stare at this for a second, and you'll 300 00:21:24,290 --> 00:21:28,190 realize that you can rewrite this term on the left-hand side 301 00:21:28,190 --> 00:21:40,260 as dr of r e to the 2 phi equals 1. 302 00:21:40,260 --> 00:21:44,130 And this can be easily integrated up. 303 00:21:44,130 --> 00:21:50,100 And the solution is phi of r equals 1/2 304 00:21:50,100 --> 00:21:57,980 log 1 plus a/r, where a is some constant of integration. 305 00:22:06,197 --> 00:22:07,780 We're going to figure out a way to fix 306 00:22:07,780 --> 00:22:09,980 that constant of integration. 307 00:22:09,980 --> 00:22:12,220 And you can probably guess how I'm going to do that. 308 00:22:12,220 --> 00:22:14,290 But let's just hold that thought for now. 309 00:22:14,290 --> 00:22:17,230 So let's just take this solution and write down 310 00:22:17,230 --> 00:22:19,390 what our exterior spacetime looks like. 311 00:22:32,070 --> 00:22:34,100 So plugging this guy in, I get-- 312 00:22:48,281 --> 00:22:49,722 OK. 313 00:22:49,722 --> 00:22:52,537 So here's my line element. 314 00:22:52,537 --> 00:22:54,120 And we're going to do the usual thing. 315 00:22:54,120 --> 00:22:56,940 This is-- well, remember that this situation describes 316 00:22:56,940 --> 00:22:58,290 gravity. 317 00:22:58,290 --> 00:23:01,290 And gravity has a Newtonian limit, 318 00:23:01,290 --> 00:23:06,270 which these general relativistic solutions must capture. 319 00:23:06,270 --> 00:23:12,160 So let's make sure that this captures my weak field limit. 320 00:23:12,160 --> 00:23:19,180 So let's consider r gihugically greater than a. 321 00:23:19,180 --> 00:23:21,675 a must be some parameter with the dimensions of length. 322 00:23:24,725 --> 00:23:26,100 It might turn out to be negative. 323 00:23:26,100 --> 00:23:28,860 So let's take the absolute value of this thing. 324 00:23:28,860 --> 00:23:31,100 And let's consider a purely radial freefall. 325 00:23:41,850 --> 00:23:44,350 So I'm going to imagine-- 326 00:23:48,110 --> 00:23:50,070 when I work this guy out, I'm going 327 00:23:50,070 --> 00:23:52,730 to look at geodesics in this limit. 328 00:24:04,395 --> 00:24:06,950 I'm going to look at purely radial freefall. 329 00:24:06,950 --> 00:24:10,445 I'm going to consider the non-relativistic large r limit. 330 00:24:23,890 --> 00:24:30,410 So here I am imagining in my non-relativistic limit 331 00:24:30,410 --> 00:24:34,870 that dt d tau is much greater than the dr d tau. 332 00:24:34,870 --> 00:24:38,140 So when you work out all those components of the Ricci tensor, 333 00:24:38,140 --> 00:24:40,780 you will have computed all of the Christoffel symbols. 334 00:24:45,650 --> 00:24:47,770 And you go back to your handy dandy little table 335 00:24:47,770 --> 00:24:49,603 of these things that you will have computed. 336 00:24:56,830 --> 00:24:58,300 So here's what you get for that. 337 00:25:05,690 --> 00:25:08,830 Your phi is given by 1/2 log of 1 plus a/r. 338 00:25:20,130 --> 00:25:32,490 So-- taking the exponential and the proper derivatives, 339 00:25:32,490 --> 00:25:41,530 what this tells you is that this is negative 340 00:25:41,530 --> 00:25:49,258 a over 2r squared 1 plus a/r. 341 00:25:49,258 --> 00:25:51,550 And again, I'll emphasize, we are considering this sort 342 00:25:51,550 --> 00:25:54,640 of very large r limit. 343 00:25:54,640 --> 00:26:01,210 So this is approximately negative a over 2r squared. 344 00:26:01,210 --> 00:26:06,760 Dividing both sides by the factor of dt d tau squared, 345 00:26:06,760 --> 00:26:08,800 I finally get my equation of motion 346 00:26:08,800 --> 00:26:12,690 for non-relativistic weak field radial freefall 347 00:26:12,690 --> 00:26:19,800 to be that this thing's radial acceleration is a over 348 00:26:19,800 --> 00:26:21,268 2r squared. 349 00:26:21,268 --> 00:26:23,310 Now, we want this to capture the Newtonian limit. 350 00:26:31,530 --> 00:26:37,250 Newtonian freefall is given by dt dr 351 00:26:37,250 --> 00:26:40,640 equals the gravitational acceleration 352 00:26:40,640 --> 00:26:42,710 minus gm over r squared. 353 00:26:42,710 --> 00:26:44,510 They're exactly the same, provided 354 00:26:44,510 --> 00:26:47,975 I select a equals minus 2gm. 355 00:26:54,330 --> 00:26:58,510 So what emerges out of this analysis 356 00:26:58,510 --> 00:27:03,385 is this truly lovely solution. 357 00:27:18,370 --> 00:27:20,640 This is known as the Schwarzschild metric. 358 00:27:38,948 --> 00:27:40,740 This is a good point to tell a brief story. 359 00:27:43,350 --> 00:27:47,100 Carl Schwarzschild derived this solution 360 00:27:47,100 --> 00:27:51,750 shortly after the publication of Einstein's field equation. 361 00:27:51,750 --> 00:27:55,440 So Einstein's field equations appeared in late November-- 362 00:27:55,440 --> 00:27:57,510 excuse me-- late in the year 1915. 363 00:27:57,510 --> 00:28:00,000 I believe in November of 1915, but we 364 00:28:00,000 --> 00:28:01,870 should double-check that. 365 00:28:01,870 --> 00:28:06,120 Schwarzschild published this exact solution, 366 00:28:06,120 --> 00:28:10,500 a few months later, I believe in early 1916. 367 00:28:10,500 --> 00:28:12,510 Students of history will note that if you 368 00:28:12,510 --> 00:28:17,170 were in Germany in 1915 and 1916, 369 00:28:17,170 --> 00:28:20,440 you were likely to be involved in some rather 370 00:28:20,440 --> 00:28:24,670 all encompassing, non-academic activities at that time. 371 00:28:24,670 --> 00:28:27,520 And in fact, Carl Schwarzschild did this calculation 372 00:28:27,520 --> 00:28:29,260 while he was serving as an artillery 373 00:28:29,260 --> 00:28:32,200 officer in the German Army on the Western-- excuse me-- 374 00:28:32,200 --> 00:28:35,610 on the Eastern Front. 375 00:28:35,610 --> 00:28:40,535 He, in fact, was quite ill at the time he did this. 376 00:28:40,535 --> 00:28:41,910 I believe it was a lung infection 377 00:28:41,910 --> 00:28:45,390 that he had accumulated while serving in the trenches. 378 00:28:45,390 --> 00:28:49,630 And so while recuperating from this disease, 379 00:28:49,630 --> 00:28:52,470 he received Einstein's manuscripts 380 00:28:52,470 --> 00:28:55,230 because he was a professor in his civilian life, 381 00:28:55,230 --> 00:28:57,480 and just wanted to keep up with the literature. 382 00:28:57,480 --> 00:29:01,800 He was inspired by this relativistic theory of gravity, 383 00:29:01,800 --> 00:29:04,920 and decided that it would be worth a little bit of his time 384 00:29:04,920 --> 00:29:07,252 while he was resting in the hospital 385 00:29:07,252 --> 00:29:09,700 to see if he could do something with these exciting new 386 00:29:09,700 --> 00:29:15,220 equations that his colleague, Herr Doktor Professor Einstein, 387 00:29:15,220 --> 00:29:17,160 had worked out. 388 00:29:17,160 --> 00:29:20,290 And he came up with this, the first exact solution 389 00:29:20,290 --> 00:29:21,880 to the Einstein field equations. 390 00:29:21,880 --> 00:29:24,790 And indeed, a solution that continues 391 00:29:24,790 --> 00:29:30,700 to be of astrophysical and observational importance today. 392 00:29:30,700 --> 00:29:33,490 He succumbed to the disease that he was suffering 393 00:29:33,490 --> 00:29:37,960 and died several weeks after coming up with this solution 394 00:29:37,960 --> 00:29:38,710 and publishing it. 395 00:29:38,710 --> 00:29:41,590 This was a solution which stunned Einstein, who did not 396 00:29:41,590 --> 00:29:44,380 think that such a simple solution would ever be found, 397 00:29:44,380 --> 00:29:46,030 such a simple exact solution would ever 398 00:29:46,030 --> 00:29:48,400 be found to these horrendous equations 399 00:29:48,400 --> 00:29:50,740 that he had developed. 400 00:29:50,740 --> 00:29:55,450 I always find this story to be somewhat-- 401 00:29:55,450 --> 00:29:57,910 well, I'm not quite sure what to make of it. 402 00:29:57,910 --> 00:30:00,940 In these days of the coronavirus, 403 00:30:00,940 --> 00:30:03,478 where we are all isolating, we are-- 404 00:30:03,478 --> 00:30:05,020 many of us now are probably beginning 405 00:30:05,020 --> 00:30:07,810 to hear stories of people we know who may have the disease. 406 00:30:10,890 --> 00:30:12,870 Sometimes it's hard for me to crawl out 407 00:30:12,870 --> 00:30:14,700 of bed in the morning. 408 00:30:14,700 --> 00:30:15,990 Somehow I do so. 409 00:30:15,990 --> 00:30:18,150 And I sometimes think to myself, what 410 00:30:18,150 --> 00:30:20,190 would Karl Schwarzschild do? 411 00:30:20,190 --> 00:30:24,180 He didn't let the fact that he was dying of a lung infection 412 00:30:24,180 --> 00:30:25,927 from-- 413 00:30:25,927 --> 00:30:28,260 he did not let the fact he was dying of a lung infection 414 00:30:28,260 --> 00:30:30,870 prevent him from leaving an indelible stamp 415 00:30:30,870 --> 00:30:34,260 on the history of science. 416 00:30:34,260 --> 00:30:37,080 I do not aspire to such greatness. 417 00:30:37,080 --> 00:30:39,510 I do not think I am capable of such greatness. 418 00:30:39,510 --> 00:30:42,280 But it at least helps me to get out of bed in the morning. 419 00:30:46,380 --> 00:30:49,710 So like I said, this is a solution 420 00:30:49,710 --> 00:30:54,760 that is of observational importance today. 421 00:30:54,760 --> 00:31:00,840 And one of the reasons it has such significance 422 00:31:00,840 --> 00:31:04,150 is due to a result known as Birkhoff's theorem. 423 00:31:17,600 --> 00:31:30,420 Birkhoff's theorem teaches us that the exterior vacuum 424 00:31:30,420 --> 00:31:48,780 of any spherically symmetric body 425 00:31:48,780 --> 00:31:50,550 is described by the Schwarzschild metric. 426 00:32:06,670 --> 00:32:10,720 This is true even if the source is time varying. 427 00:32:10,720 --> 00:32:13,420 So in doing this derivation, I focused 428 00:32:13,420 --> 00:32:17,410 on a situation in which my spacetime was static. 429 00:32:17,410 --> 00:32:21,580 There is no time dependence whatsoever. 430 00:32:21,580 --> 00:32:25,030 It turns out that if it is not static, 431 00:32:25,030 --> 00:32:29,140 if it is a time-varying spacetime, as long as the time 432 00:32:29,140 --> 00:32:33,010 variations preserve spherical symmetry, 433 00:32:33,010 --> 00:32:35,818 the exterior is still described by Schwarzschild. 434 00:32:53,280 --> 00:32:54,890 It's even stronger than if-- 435 00:32:54,890 --> 00:32:57,620 only if. 436 00:32:57,620 --> 00:33:07,100 As long as the time variations preserve spherical symmetry. 437 00:33:07,100 --> 00:33:09,270 So for instance, it could be radial pulsations. 438 00:33:29,420 --> 00:33:34,100 If it is not a variation that preserves spherical symmetry, 439 00:33:34,100 --> 00:33:36,620 you're very likely to have a time-varying quadrupole moment, 440 00:33:36,620 --> 00:33:38,900 and then you produce gravitational waves. 441 00:33:38,900 --> 00:33:40,880 But if you have time variations that 442 00:33:40,880 --> 00:33:43,640 preserve spherical symmetry, spherical symmetry 443 00:33:43,640 --> 00:33:46,820 does not allow us to have a time-varying quadrupole moment. 444 00:33:46,820 --> 00:33:48,618 So no gravitational waves are produced. 445 00:33:48,618 --> 00:33:50,660 And what this tells you is that as long as you're 446 00:33:50,660 --> 00:33:52,452 in this vacuum region on the outside of it, 447 00:33:52,452 --> 00:33:56,210 your thing could be down there going [VOCALIZING],, 448 00:33:56,210 --> 00:33:58,220 as long as it does so in such a way that 449 00:33:58,220 --> 00:34:02,262 leaves it spherically symmetric, then that 450 00:34:02,262 --> 00:34:03,720 is the spacetime that describes it. 451 00:34:03,720 --> 00:34:07,740 You are completely ignorant of what the radial extent 452 00:34:07,740 --> 00:34:08,670 of the matter is. 453 00:34:08,670 --> 00:34:12,389 It only matters that you have a mass of m, 454 00:34:12,389 --> 00:34:15,719 and you are distance r away from its center, 455 00:34:15,719 --> 00:34:17,040 at least in these coordinates. 456 00:34:17,040 --> 00:34:19,679 In that sense, it's kind of like Birkhoff's theorem plays 457 00:34:19,679 --> 00:34:25,710 a role similar to Gauss's law in elementary electricity 458 00:34:25,710 --> 00:34:26,429 and magnetism. 459 00:34:26,429 --> 00:34:28,429 If you have a spherically symmetric distribution 460 00:34:28,429 --> 00:34:33,120 of charges, you are completely agnostic as to what that-- 461 00:34:33,120 --> 00:34:36,540 what the radial distribution of that spherical symmetry is. 462 00:34:36,540 --> 00:34:38,790 And in fact-- although, this isn't usually discussed-- 463 00:34:38,790 --> 00:34:43,139 if that were to be sort of a time variable kind of thing, 464 00:34:43,139 --> 00:34:45,260 as long as it was just sort of homologous 465 00:34:45,260 --> 00:34:47,730 in moving these guys in and out like that, 466 00:34:47,730 --> 00:34:52,050 you would still just get the E-field associated with a point 467 00:34:52,050 --> 00:34:53,980 charge at the origin. 468 00:34:53,980 --> 00:34:56,295 This spacetime is kind of the gravitational equivalent 469 00:34:56,295 --> 00:34:58,920 of a point charge at the origin, a point I'm going to come back 470 00:34:58,920 --> 00:35:00,820 to in a future lecture. 471 00:35:00,820 --> 00:35:03,780 Let me just give you a brief proof of Birkhoff's theorem. 472 00:35:03,780 --> 00:35:18,680 So imagine I want to consider a spacetime in which I 473 00:35:18,680 --> 00:35:22,280 allow my metric functions to vary with time. 474 00:35:36,530 --> 00:35:39,230 So go ahead and work all of these guys. 475 00:35:39,230 --> 00:35:40,710 Use that as your metric. 476 00:35:40,710 --> 00:35:44,600 Work out all of your curvature tensors. 477 00:35:44,600 --> 00:35:48,657 We are focusing on the exterior vacuum region of the star. 478 00:35:48,657 --> 00:35:50,990 So what we're going to do is set those curvature tensors 479 00:35:50,990 --> 00:35:52,760 equal to 0. 480 00:35:52,760 --> 00:35:54,740 What you find when you do this-- so 481 00:35:54,740 --> 00:35:57,380 when you work out your curvature tensors, in addition 482 00:35:57,380 --> 00:36:01,250 to those four that we had on the center boards earlier, 483 00:36:01,250 --> 00:36:04,370 the four components of Ricci, you 484 00:36:04,370 --> 00:36:06,200 find a new component enters. 485 00:36:16,750 --> 00:36:28,760 What you find is that there is a component r sub tr, 486 00:36:28,760 --> 00:36:30,680 which looks like-- oops, pardon me-- 487 00:36:30,680 --> 00:36:37,140 which looks like 2/r times the time derivative of your lambda. 488 00:36:37,140 --> 00:36:40,710 This you are going to set equal to 0 in the exterior. 489 00:36:40,710 --> 00:36:43,670 So this tells you that the time derivative of lambda is 0. 490 00:36:49,660 --> 00:36:54,010 This means that your assumed lambda of t and r 491 00:36:54,010 --> 00:36:55,210 is just a lambda of r. 492 00:36:55,210 --> 00:36:57,940 So there is no time dependence in that lambda. 493 00:36:57,940 --> 00:36:59,080 So you just killed that. 494 00:37:03,390 --> 00:37:06,930 You can then look at your other curvature components. 495 00:37:06,930 --> 00:37:09,690 And in particular, one of the things that you'll see 496 00:37:09,690 --> 00:37:11,640 is several of those curvature components. 497 00:37:11,640 --> 00:37:14,190 You'll see that they pick up a couple of additional terms 498 00:37:14,190 --> 00:37:17,910 involving d by dt as the various of the lambdas and the phis. 499 00:37:17,910 --> 00:37:20,082 You can now get rid of those. 500 00:37:20,082 --> 00:37:22,290 Where I think it's particularly clean to look at this 501 00:37:22,290 --> 00:37:25,740 is if you look at r theta theta. 502 00:37:29,400 --> 00:37:31,710 That turns out to be completely unchanged. 503 00:37:41,010 --> 00:37:44,860 So your analysis of the r theta theta term, 504 00:37:44,860 --> 00:37:46,545 it could proceed exactly as before. 505 00:37:46,545 --> 00:37:47,920 But the other thing you can do is 506 00:37:47,920 --> 00:37:49,180 you can say, hey, guess what. 507 00:37:49,180 --> 00:37:53,620 I'm now going to imagine that my phi is time dependent. 508 00:37:53,620 --> 00:37:55,480 My lambda is not time dependent. 509 00:37:55,480 --> 00:37:56,740 We just proved that. 510 00:37:56,740 --> 00:37:58,590 But my phi could be. 511 00:37:58,590 --> 00:38:02,470 Let's look at the time derivative of this Ricci term. 512 00:38:02,470 --> 00:38:05,170 So if our r theta theta is 0, the time derivative 513 00:38:05,170 --> 00:38:06,520 of r theta theta must be 0. 514 00:38:12,110 --> 00:38:14,020 So when I compute this, this leads 515 00:38:14,020 --> 00:38:19,140 to the requirement that the double derivative, 516 00:38:19,140 --> 00:38:21,352 one derivative in r, one derivative in phi-- 517 00:38:21,352 --> 00:38:23,310 excuse me-- one derivative in r, one derivative 518 00:38:23,310 --> 00:38:26,540 in t of the potential phi must be equal to 0. 519 00:38:37,470 --> 00:38:57,220 And so this in turn tells us for this condition to be true, 520 00:38:57,220 --> 00:38:58,350 it must be the case-- 521 00:39:04,270 --> 00:39:04,770 pardon me. 522 00:39:04,770 --> 00:39:05,460 Just one second. 523 00:39:05,460 --> 00:39:06,600 How does this follow? 524 00:39:06,600 --> 00:39:07,170 Oh, no. 525 00:39:07,170 --> 00:39:07,770 Yeah, yeah. 526 00:39:07,770 --> 00:39:09,720 So this tells me it must be the case 527 00:39:09,720 --> 00:39:20,290 that phi can be separated into a function of r 528 00:39:20,290 --> 00:39:22,690 in a function of t. 529 00:39:22,690 --> 00:39:26,080 If I take the t derivative and then the r derivative, I get 0. 530 00:39:26,080 --> 00:39:30,130 If I take the r derivative and then the t derivative, I get 0. 531 00:39:30,130 --> 00:39:37,480 So my line element, my most general line element 532 00:39:37,480 --> 00:39:54,650 now appears to be something that looks like this. 533 00:40:02,260 --> 00:40:04,780 But once I have this, I can actually just redefine 534 00:40:04,780 --> 00:40:05,800 the time coordinate. 535 00:40:05,800 --> 00:40:07,750 I mean, if I look at this thing I've got here, 536 00:40:07,750 --> 00:40:18,990 I can just absorb a function of t into my time coordinate, 537 00:40:18,990 --> 00:40:21,030 just absorb it in a time coordinate like so. 538 00:40:27,540 --> 00:40:31,600 And what happens is that I then recover the Schwarzschild form 539 00:40:31,600 --> 00:40:32,320 I originally had. 540 00:40:42,210 --> 00:40:43,440 So that's really nice. 541 00:40:43,440 --> 00:40:49,200 What this tells us is that any spherically symmetric vacuum 542 00:40:49,200 --> 00:40:54,120 spacetime is going to have a time-like Killing vector. 543 00:41:34,620 --> 00:41:36,840 And if it has a time-like Killing vector, 544 00:41:36,840 --> 00:41:42,060 that means that I can define a notion of conserved energy 545 00:41:42,060 --> 00:41:44,418 for objects that are moving in that spacetime. 546 00:42:06,100 --> 00:42:07,600 It's going to prove very useful when 547 00:42:07,600 --> 00:42:11,340 we start discussing orbits. 548 00:42:11,340 --> 00:42:15,630 So this punch line is this totally 549 00:42:15,630 --> 00:42:20,210 describes the exterior of my compact spherical body. 550 00:42:20,210 --> 00:42:21,210 What about the interior? 551 00:42:26,470 --> 00:42:30,620 What we're going to need to do is-- 552 00:42:30,620 --> 00:42:35,550 the interior has non-zero stress energy. 553 00:42:35,550 --> 00:42:38,810 So we're going to need to look at components of the Einstein 554 00:42:38,810 --> 00:42:47,000 equation and equate them to the source, my perfect fluid stress 555 00:42:47,000 --> 00:42:49,310 energy tensor, and see what develops. 556 00:42:57,280 --> 00:43:01,250 So what I'm going to do is write my Einstein equation 557 00:43:01,250 --> 00:43:05,540 in the form of an upstairs downstairs index. 558 00:43:05,540 --> 00:43:09,600 And I'm going to do it that way, because t upstairs mu 559 00:43:09,600 --> 00:43:14,540 downstairs nu, it is represented in the coordinate system I'm 560 00:43:14,540 --> 00:43:16,170 working in here. 561 00:43:16,170 --> 00:43:19,790 But assuming it's a diagonal of minus rho ppp. 562 00:43:25,800 --> 00:43:27,570 And I had all the Riemann-- 563 00:43:27,570 --> 00:43:29,820 excuse me-- all the Ricci components on the blackboard 564 00:43:29,820 --> 00:43:31,320 a few moments ago. 565 00:43:31,320 --> 00:43:36,930 From them, I can construct my Einstein tensor. 566 00:43:36,930 --> 00:43:41,040 So there's a couple components that turn out to be quite nice. 567 00:43:41,040 --> 00:43:45,630 One of them is the tt component. 568 00:43:45,630 --> 00:43:57,620 This turns out to be minus 1 over r squared d by dr r 1 569 00:43:57,620 --> 00:44:01,100 minus e to the minus 2 lambda. 570 00:44:04,030 --> 00:44:07,570 What's nice about this is notice it only depends on lambda. 571 00:44:10,100 --> 00:44:14,060 This is going to be equated to 8 pi g times 572 00:44:14,060 --> 00:44:17,810 t stress energy tensor tt, which is going to give me a minus rho 573 00:44:17,810 --> 00:44:18,990 there. 574 00:44:18,990 --> 00:44:22,397 So before I go and I do this, let me make a definition. 575 00:44:27,170 --> 00:44:34,000 I'm going to call e to the minus 2 lambda, 576 00:44:34,000 --> 00:44:41,600 I'm going to call that 1 minus 2gm of r over m. 577 00:44:41,600 --> 00:44:46,280 What I'm effectively doing is I am mapping my metric function 578 00:44:46,280 --> 00:44:50,120 lambda to a function m of r, which 579 00:44:50,120 --> 00:44:52,237 is a sort of mass function. 580 00:44:52,237 --> 00:44:54,320 Hopefully, you can kind of see that by doing this, 581 00:44:54,320 --> 00:44:59,130 remember, that this component of the metric, 582 00:44:59,130 --> 00:45:02,360 it's going to enter as e to the 2 lambda. 583 00:45:02,360 --> 00:45:04,640 And so when I match this, this is 584 00:45:04,640 --> 00:45:07,220 going to give me something that very nicely matches 585 00:45:07,220 --> 00:45:09,235 to my exterior spacetime. 586 00:45:23,810 --> 00:45:27,230 So plugging that in there, and valuing the derivatives, 587 00:45:27,230 --> 00:45:31,490 and equating this guy to the stress energy tensor 588 00:45:31,490 --> 00:45:37,840 is going to give me a very nice condition that the function 589 00:45:37,840 --> 00:45:40,630 m of r must obey. 590 00:46:12,560 --> 00:46:21,030 So plugging all this stuff in, I end up with minus 2g 591 00:46:21,030 --> 00:46:31,920 over r squared dm dr equals minus 8 pi g rho of r. 592 00:46:34,460 --> 00:46:36,290 Turning things around, I can turn this 593 00:46:36,290 --> 00:46:40,610 into an integral solution, which tells me that my mass function 594 00:46:40,610 --> 00:46:47,960 m of r is what I get when I integrate from 0 to r 4 pi 595 00:46:47,960 --> 00:46:49,310 rho of r prime. 596 00:46:55,470 --> 00:46:56,480 Two comments on this. 597 00:46:56,480 --> 00:46:59,063 You sort of look at this and go, ah, that makes perfect sense. 598 00:46:59,063 --> 00:47:01,230 This is what you would obviously get 599 00:47:01,230 --> 00:47:06,300 if you integrate up a density over a spherical volume. 600 00:47:06,300 --> 00:47:09,233 This is, hopefully, beautifully intuitive. 601 00:47:09,233 --> 00:47:10,650 And you kind of look at it and go, 602 00:47:10,650 --> 00:47:13,440 how could it be anything else? 603 00:47:13,440 --> 00:47:15,840 Well, it could be anything else. 604 00:47:15,840 --> 00:47:17,645 So two comments to make. 605 00:47:17,645 --> 00:47:19,020 First of all, I have set this up. 606 00:47:19,020 --> 00:47:21,219 I have imposed a boundary condition. 607 00:47:28,203 --> 00:47:29,870 I have imposed a boundary condition here 608 00:47:29,870 --> 00:47:33,915 that the mass enclosed at radius 0 is 0. 609 00:47:33,915 --> 00:47:35,540 You might look at that, and, well, duh. 610 00:47:35,540 --> 00:47:38,480 If I have a ball of radius 0, there can't be any mass in it. 611 00:47:38,480 --> 00:47:40,370 Of course, you're going to do that. 612 00:47:40,370 --> 00:47:42,860 Well, it's going to turn out that black holes are actually 613 00:47:42,860 --> 00:47:46,340 solutions that violate this boundary condition. 614 00:47:46,340 --> 00:47:47,600 Wah-wah. 615 00:47:47,600 --> 00:47:50,370 So there's that. 616 00:47:50,370 --> 00:47:55,080 The other thing to bear in mind is that 4 pi-- 617 00:47:55,080 --> 00:47:57,590 the reason why this probably makes intuitive sense 618 00:47:57,590 --> 00:48:15,270 is that in spherical symmetry in Euclidean space, 619 00:48:15,270 --> 00:48:22,500 4 pi r squared dr is the proper volume element for a Euclidean 620 00:48:22,500 --> 00:48:24,060 three-volume. 621 00:48:24,060 --> 00:48:27,040 We are not in Euclidean space. 622 00:48:27,040 --> 00:48:30,510 And so this is indeed the definition of m of r, 623 00:48:30,510 --> 00:48:34,950 but it is not what we would get if we did a proper volume 624 00:48:34,950 --> 00:48:40,210 integration of rho over the volume that 625 00:48:40,210 --> 00:48:41,260 goes out to radius r. 626 00:48:48,920 --> 00:49:02,370 If I were to define a quantity that is the proper volume 627 00:49:02,370 --> 00:49:13,410 integrated rho, let's call this m sub b. 628 00:49:13,410 --> 00:49:15,068 Let's make it lowercase. 629 00:49:21,542 --> 00:49:24,060 I'll define why I'm calling that b in just a moment. 630 00:49:27,690 --> 00:49:36,100 I could define this as what I get 631 00:49:36,100 --> 00:49:38,950 when I do a proper volume integration of this thing 632 00:49:38,950 --> 00:49:40,585 in my curved spacetime. 633 00:49:53,970 --> 00:49:56,310 You get an additional factor. 634 00:49:56,310 --> 00:50:00,113 And what you find is that this in general-- 635 00:50:00,113 --> 00:50:02,280 so remember, this is going to go into your spacetime 636 00:50:02,280 --> 00:50:03,000 as a function. 637 00:50:03,000 --> 00:50:04,375 It's going to look like something 638 00:50:04,375 --> 00:50:09,120 like 1 over that factor there. 639 00:50:09,120 --> 00:50:14,937 This in general is greater than m of r. 640 00:50:14,937 --> 00:50:17,270 So there is a homework exercise where you guys are going 641 00:50:17,270 --> 00:50:19,530 to explore this a little bit. 642 00:50:19,530 --> 00:50:29,240 m sub b is what I get when I count up, essentially 643 00:50:29,240 --> 00:50:33,920 including a factor of the rest mass of every little-- 644 00:50:33,920 --> 00:50:37,310 let's call it baryon that goes into this star. 645 00:50:37,310 --> 00:50:41,120 This counts up the total number of baryons 646 00:50:41,120 --> 00:50:43,030 present in my star in something that's 647 00:50:43,030 --> 00:50:46,070 called the baryonic mass. 648 00:50:46,070 --> 00:50:51,380 This m that appears here defines the amount of mass 649 00:50:51,380 --> 00:50:54,470 that generates gravity. 650 00:50:54,470 --> 00:50:57,260 They are not the same. 651 00:50:57,260 --> 00:51:03,250 The gravitational mass is generally somewhat less 652 00:51:03,250 --> 00:51:04,675 than the baryonic mass. 653 00:51:08,122 --> 00:51:10,080 You might sort of think, well, where did it go? 654 00:51:12,690 --> 00:51:16,650 The missing mass can be regarded as 655 00:51:16,650 --> 00:51:18,060 gravitational binding energy. 656 00:51:31,250 --> 00:51:33,260 It's really not that it's missing. 657 00:51:33,260 --> 00:51:35,600 It's just that gravitational energy 658 00:51:35,600 --> 00:51:37,080 holds this thing together. 659 00:51:37,080 --> 00:51:38,780 And whenever you bind something, you 660 00:51:38,780 --> 00:51:41,690 do that by putting it in a lower energy state. 661 00:51:41,690 --> 00:51:45,770 Gravity is essentially this sort of mb with this-- 662 00:51:45,770 --> 00:51:49,460 the way to think of my mb here is bear in mind 663 00:51:49,460 --> 00:51:51,360 that we're including factors-- 664 00:51:51,360 --> 00:51:53,810 let's just put factors of c squared back in here. 665 00:51:53,810 --> 00:51:56,840 This is adding up the rest energy 666 00:51:56,840 --> 00:51:59,920 of every particle in this star. 667 00:51:59,920 --> 00:52:01,730 But when I put them all together, 668 00:52:01,730 --> 00:52:03,400 gravity binds it together. 669 00:52:03,400 --> 00:52:05,150 And that actually takes away some of that. 670 00:52:05,150 --> 00:52:07,220 It puts it into a bound state. 671 00:52:07,220 --> 00:52:12,080 And so the difference between the baryonic mass, or energy, 672 00:52:12,080 --> 00:52:14,330 and the gravitational mass, and energy, 673 00:52:14,330 --> 00:52:17,030 tells you something about how strongly bound this object is. 674 00:52:21,410 --> 00:52:22,810 So that's one important equation. 675 00:52:29,692 --> 00:52:31,400 It looks like I might finish this lecture 676 00:52:31,400 --> 00:52:35,900 up a little bit on the early side, 677 00:52:35,900 --> 00:52:39,590 which is fine, because I could use a break. 678 00:52:39,590 --> 00:52:42,250 Another one that is important for us 679 00:52:42,250 --> 00:52:46,220 will be the rr component of the stress energy tensor. 680 00:52:46,220 --> 00:52:49,600 So this guy, when you work it out, 681 00:52:49,600 --> 00:52:54,780 it looks like e to the minus 2 lambda 2 over 682 00:52:54,780 --> 00:53:03,150 r d phi dr plus 1 over r squared. 683 00:53:06,290 --> 00:53:08,000 That's 1 over r squared. 684 00:53:08,000 --> 00:53:12,110 So let's insert our definition. 685 00:53:12,110 --> 00:53:15,050 This prefactor is minus-- excuse me-- 686 00:53:15,050 --> 00:53:26,150 1 minus 2gm of r over r 2 phi-- 687 00:53:26,150 --> 00:53:26,810 oh, shoot. 688 00:53:26,810 --> 00:53:28,320 Pardon me. 689 00:53:28,320 --> 00:53:29,880 Try that again-- 690 00:53:29,880 --> 00:53:40,400 2 over r d phi dr plus 1 over p square 1 over r squared. 691 00:53:40,400 --> 00:53:46,940 And we equate this to the appropriate component 692 00:53:46,940 --> 00:53:52,370 of the stress energy tensor, which is just 693 00:53:52,370 --> 00:53:54,660 going to be the pressure. 694 00:53:54,660 --> 00:53:56,900 This can be rearranged and give us 695 00:53:56,900 --> 00:54:00,862 an equation for a differential equation governing 696 00:54:00,862 --> 00:54:03,070 the metric potential, the trigonometric function phi. 697 00:54:28,620 --> 00:54:41,957 So what results of that exercise something 698 00:54:41,957 --> 00:54:54,532 that looks like this over this. 699 00:54:54,532 --> 00:54:56,990 So I would like to do something with this in just a moment. 700 00:54:56,990 --> 00:54:58,407 But it's worth commenting on this. 701 00:54:58,407 --> 00:55:03,140 So when one looks at the gravitational potential 702 00:55:03,140 --> 00:55:08,420 in a spherical fluid body in Newtonian gravity, 703 00:55:08,420 --> 00:55:13,070 you get gm of r over r squared. 704 00:55:13,070 --> 00:55:15,690 So there's two interesting things going on here. 705 00:55:15,690 --> 00:55:18,505 First, your m of r, which appears 706 00:55:18,505 --> 00:55:19,880 in the numerator of this thing is 707 00:55:19,880 --> 00:55:23,000 corrected by a term that involves the pressure. 708 00:55:23,000 --> 00:55:25,550 What you are seeing here is the fact 709 00:55:25,550 --> 00:55:29,130 that pressure in some sense reflects work that 710 00:55:29,130 --> 00:55:30,390 is being done in this body. 711 00:55:30,390 --> 00:55:31,338 It's being squeezed. 712 00:55:31,338 --> 00:55:32,880 Gravity is sort of squeezing it down, 713 00:55:32,880 --> 00:55:35,550 or your hands are pushing it down. 714 00:55:35,550 --> 00:55:37,560 When you give this body some pressure, 715 00:55:37,560 --> 00:55:39,420 you're doing work on it. 716 00:55:39,420 --> 00:55:43,830 Work adds to the energy budget of that body. 717 00:55:43,830 --> 00:55:45,990 Energy gravitates. 718 00:55:45,990 --> 00:55:49,080 In the denominator, you're getting this r minus 2-- 719 00:55:49,080 --> 00:55:50,590 instead of just having an r squared, 720 00:55:50,590 --> 00:55:54,600 you get a correction that involves this 2gm term in here. 721 00:55:54,600 --> 00:55:56,490 And this is just correcting for the fact 722 00:55:56,490 --> 00:56:00,930 that you are not working in a Euclidean geometry. 723 00:56:00,930 --> 00:56:04,150 So now I want to do a little bit more with this. 724 00:56:04,150 --> 00:56:09,430 And so let me borrow a result from an old homework exercise. 725 00:56:09,430 --> 00:56:11,570 I believe this was on P set 3. 726 00:56:11,570 --> 00:56:12,820 Although I don't quite recall. 727 00:56:15,610 --> 00:56:17,950 So on this thing you guys took a look 728 00:56:17,950 --> 00:56:25,330 at the equation of local stress energy conservation 729 00:56:25,330 --> 00:56:29,382 for a perfect fluid and hydrostatic equilibrium. 730 00:56:46,140 --> 00:56:54,120 And what you found was that this turns into a condition relating 731 00:56:54,120 --> 00:56:58,720 the pressure, density, and what you 732 00:56:58,720 --> 00:57:05,890 can kind of think of as the four-acceleration 733 00:57:05,890 --> 00:57:17,570 of the fluid elements. 734 00:57:17,570 --> 00:57:20,120 So we see sort of gradients of the pressure entering here, 735 00:57:20,120 --> 00:57:23,960 the density entering into here, and some things 736 00:57:23,960 --> 00:57:25,910 related to the behavior of the fluid elements. 737 00:57:28,770 --> 00:57:32,890 The spacetime that we are working with 738 00:57:32,890 --> 00:57:36,370 is one where we have constrained the behavior of that fluid. 739 00:57:52,150 --> 00:57:53,985 So we are going to require that our fluid-- 740 00:58:01,550 --> 00:58:02,550 let's write it this way. 741 00:58:12,500 --> 00:58:16,160 So our fluid only has a-- it's only the time-like component 742 00:58:16,160 --> 00:58:18,530 of its four-velocity is 0. 743 00:58:18,530 --> 00:58:22,100 And so when I shoot that all through the various things 744 00:58:22,100 --> 00:58:24,950 involved in there-- oh, and, of course, phi only depends on r. 745 00:58:30,120 --> 00:58:32,070 So what this equation then boils down 746 00:58:32,070 --> 00:58:42,020 to is rho plus p times this Christoffel symbol equals 747 00:58:42,020 --> 00:58:48,750 minus dp dr. When you expand this out, 748 00:58:48,750 --> 00:58:55,670 this turns into d phi dr. So what 749 00:58:55,670 --> 00:58:58,760 this tells me is that d phi dr is simply related 750 00:58:58,760 --> 00:59:00,290 to the pressure gradient. 751 00:59:12,680 --> 00:59:25,170 When you put all of this together, what you get 752 00:59:25,170 --> 00:59:29,490 is the following equation. 753 00:59:29,490 --> 00:59:31,950 This is sometimes called the equation 754 00:59:31,950 --> 00:59:36,390 of relativistic hydrostatic equilibrium. 755 00:59:36,390 --> 00:59:51,438 My pressure gradient must satisfy this equation. 756 00:59:51,438 --> 00:59:52,980 Let me write down two more, because I 757 00:59:52,980 --> 00:59:54,355 think I'm going to actually begin 758 00:59:54,355 --> 00:59:57,690 with this board in my next lecture. 759 00:59:57,690 --> 01:00:02,100 So I'm going to just repeat this equation governing phi. 760 01:00:14,520 --> 01:00:19,720 And m is defined as this integral. 761 01:00:32,030 --> 01:00:35,500 So here are three coupled differential equations, 762 01:00:35,500 --> 01:00:39,620 which I can now use to build a model 763 01:00:39,620 --> 01:00:41,810 for the interior of this body. 764 01:00:41,810 --> 01:00:43,170 We integrate this thing out. 765 01:00:43,170 --> 01:00:48,710 Basically, the way it works is you need to choose an initial-- 766 01:00:48,710 --> 01:00:50,090 some parameter that characterizes 767 01:00:50,090 --> 01:00:51,050 the initial conditions. 768 01:00:51,050 --> 01:00:54,710 You typically choose a density at the center. 769 01:00:54,710 --> 01:00:57,920 You choose rho at r equals 0. 770 01:00:57,920 --> 01:01:00,680 You need an equation of state that relates 771 01:01:00,680 --> 01:01:03,130 the pressure and the density. 772 01:01:03,130 --> 01:01:05,420 And then you just integrate these guys up. 773 01:01:05,420 --> 01:01:06,920 There's a matching that must be done 774 01:01:06,920 --> 01:01:09,712 at the surface of the star. 775 01:01:09,712 --> 01:01:11,170 Then once you've done that, that'll 776 01:01:11,170 --> 01:01:12,600 end up giving you sort of a notice 777 01:01:12,600 --> 01:01:15,150 that by integrating up this equation, 778 01:01:15,150 --> 01:01:18,000 you determined the function phi up 779 01:01:18,000 --> 01:01:20,400 to some constant of integration. 780 01:01:20,400 --> 01:01:22,440 When you match at the surface of the star, 781 01:01:22,440 --> 01:01:26,100 that lets you fix that constant of integration. 782 01:01:26,100 --> 01:01:26,640 Boom! 783 01:01:26,640 --> 01:01:28,770 You've made yourself a spacetime. 784 01:01:28,770 --> 01:01:39,420 This is known as the Tolman-Oppenheimer-Volkoff 785 01:01:39,420 --> 01:01:49,250 equation, universally abbreviated TOV. 786 01:01:49,250 --> 01:01:53,330 These are the way in which we make spherically 787 01:01:53,330 --> 01:01:57,802 symmetric fluid bodies in general relativity, 788 01:01:57,802 --> 01:01:59,510 spherically symmetric static fluid bodies 789 01:01:59,510 --> 01:02:02,120 in general relativity. 790 01:02:02,120 --> 01:02:03,620 In the next lecture-- so we're going 791 01:02:03,620 --> 01:02:06,080 to end this one a tiny bit early. 792 01:02:06,080 --> 01:02:09,990 This is actually a very natural place to stop. 793 01:02:09,990 --> 01:02:11,690 I am going to start with this. 794 01:02:11,690 --> 01:02:14,210 And what we're going to do is talk a little bit 795 01:02:14,210 --> 01:02:17,990 about how one solves these. 796 01:02:17,990 --> 01:02:21,278 And we're going to look at a couple-- 797 01:02:21,278 --> 01:02:23,570 we're going to look at one particular special solution, 798 01:02:23,570 --> 01:02:26,683 which is not very physical, but it's illustrative. 799 01:02:26,683 --> 01:02:28,100 And I'm going to talk a little bit 800 01:02:28,100 --> 01:02:33,110 about how to solve this for more realistic setups. 801 01:02:33,110 --> 01:02:37,100 That is in preparation for what is one of my favorite homework 802 01:02:37,100 --> 01:02:37,790 exercises. 803 01:02:37,790 --> 01:02:40,490 It's one where I give you an equation of state. 804 01:02:40,490 --> 01:02:42,920 And you guys just take these equations. 805 01:02:42,920 --> 01:02:45,590 And using a numerical integrator-- 806 01:02:45,590 --> 01:02:48,360 I hope everyone has access to something like Mathematica, 807 01:02:48,360 --> 01:02:50,810 or WolframAlpha, or something like this 808 01:02:50,810 --> 01:02:53,840 in your dispersed lives. 809 01:02:53,840 --> 01:02:56,330 All MIT students should be able to access the student 810 01:02:56,330 --> 01:02:57,575 license for Mathematica. 811 01:02:57,575 --> 01:02:59,450 So as long as you have the hardware for that, 812 01:02:59,450 --> 01:03:00,170 you should be OK. 813 01:03:02,790 --> 01:03:04,510 And this is-- actually, it's pretty cool. 814 01:03:04,510 --> 01:03:06,810 You can make relativistic stellar models. 815 01:03:06,810 --> 01:03:09,570 I've actually had in the past students 816 01:03:09,570 --> 01:03:12,220 use this as the basis for some research projects, 817 01:03:12,220 --> 01:03:15,180 because when you do things like this, 818 01:03:15,180 --> 01:03:19,990 this is how professionals make models of stars. 819 01:03:19,990 --> 01:03:21,640 So we will stop there. 820 01:03:21,640 --> 01:03:25,960 And those of you who are watching at home, 821 01:03:25,960 --> 01:03:28,680 you can go on to the next video, where we'll be doing this. 822 01:03:28,680 --> 01:03:31,500 As for me, I'm going to take a bit of a break, 823 01:03:31,500 --> 01:03:35,510 and start recording again in about a half an hour.