WEBVTT 00:00:00.956 --> 00:00:02.390 [SQUEAKING] 00:00:02.390 --> 00:00:03.824 [RUSTLING] 00:00:03.824 --> 00:00:07.170 [CLICKING] 00:00:09.707 --> 00:00:12.040 SCOTT HUGHES: So we'll pick up where we ended last time. 00:00:12.040 --> 00:00:16.720 We're looking at the spacetime of a compact spherical body, 00:00:16.720 --> 00:00:19.060 working in what we call Schwarzschild coordinates. 00:00:19.060 --> 00:00:22.810 We deduce that the line element describing this body 00:00:22.810 --> 00:00:26.100 is of the form ds squared equals negative e 00:00:26.100 --> 00:00:29.350 to the 2 phi dt squared plus tr squared divided 00:00:29.350 --> 00:00:34.780 by 1 minus 2g m of r over r plus r squared d 00:00:34.780 --> 00:00:38.800 omega, where d omega is the usual solid element, line 00:00:38.800 --> 00:00:40.270 element. 00:00:40.270 --> 00:00:41.050 And let's see. 00:00:45.970 --> 00:00:48.910 Yes, the body is described, interior, 00:00:48.910 --> 00:00:52.090 as a perfect fluid with particular density profile 00:00:52.090 --> 00:00:55.537 rho of r, a pressure profile p of r. 00:00:55.537 --> 00:00:56.620 That describes this thing. 00:00:56.620 --> 00:01:01.000 Everywhere inside some radius r star, which gives its surface. 00:01:03.660 --> 00:01:05.260 Sorry, I just distracted myself. 00:01:05.260 --> 00:01:09.335 I don't know why I've been calling this d omega. 00:01:09.335 --> 00:01:11.780 Should be d omega squared. 00:01:11.780 --> 00:01:12.280 Whatever. 00:01:16.440 --> 00:01:19.380 Fluff in that notation. 00:01:19.380 --> 00:01:23.040 OK, so in the exterior of this thing-- so for everywhere 00:01:23.040 --> 00:01:25.770 for r greater than our star, it is vacuum. 00:01:25.770 --> 00:01:26.700 There is no density. 00:01:26.700 --> 00:01:28.860 There is no pressure. 00:01:28.860 --> 00:01:31.440 Once we're outside this thing, the only mass you see 00:01:31.440 --> 00:01:32.730 is the mass of the star. 00:01:32.730 --> 00:01:35.890 So another way of saying this is that when-- you know what? 00:01:35.890 --> 00:01:38.500 I'll get to that in just a moment. 00:01:38.500 --> 00:01:42.870 And in the exterior, the e to the 2, 5 becomes 1 minus 2g, 00:01:42.870 --> 00:01:45.240 that same mass over r. 00:01:45.240 --> 00:01:51.250 Everywhere in the interior, the pressure, the function phi, 00:01:51.250 --> 00:01:53.660 and the mass r are governed by these equations. 00:01:53.660 --> 00:01:56.130 So the mass, as I described last time, 00:01:56.130 --> 00:01:58.913 it looks like a deceptively simple spherical integral. 00:01:58.913 --> 00:02:00.580 But just be aware that when you do this, 00:02:00.580 --> 00:02:02.523 you are not integrating over a proper volume. 00:02:02.523 --> 00:02:04.440 If you were to integrate over a proper volume, 00:02:04.440 --> 00:02:08.370 you would get a larger mass, and such a mass, in fact, 00:02:08.370 --> 00:02:10.139 does have meaning to it. 00:02:10.139 --> 00:02:14.025 And the difference between that mass and this mass 00:02:14.025 --> 00:02:15.900 tells you something about how gravitationally 00:02:15.900 --> 00:02:18.810 bound this object is. 00:02:18.810 --> 00:02:23.470 And these two equations actually have a Newtonian limit 00:02:23.470 --> 00:02:24.500 associated with them. 00:02:24.500 --> 00:02:27.280 So this is a relativistic version 00:02:27.280 --> 00:02:30.890 of the equation of hydrostatic equilibrium, 00:02:30.890 --> 00:02:34.776 and this has a simple Newtonian analog. 00:02:34.776 --> 00:02:36.450 Did they drop the minus sign? 00:02:36.450 --> 00:02:37.400 No. 00:02:37.400 --> 00:02:39.860 This has a simple Newtonian analog, 00:02:39.860 --> 00:02:41.810 describing the gravitational potential 00:02:41.810 --> 00:02:43.255 inside a fluid object. 00:02:43.255 --> 00:02:44.630 These whole things taken together 00:02:44.630 --> 00:02:47.390 are called the Tolman-Oppenheimer-Volkoff 00:02:47.390 --> 00:02:51.200 equations, or the TOV equations. 00:02:51.200 --> 00:03:01.100 So the comment I was making was that we require m of r star 00:03:01.100 --> 00:03:03.560 to equal m, the total mass of this object. 00:03:03.560 --> 00:03:05.330 So that's another condition on this. 00:03:05.330 --> 00:03:08.155 Once you integrate up, then you switch over 00:03:08.155 --> 00:03:09.530 to the mass, the total mass, that 00:03:09.530 --> 00:03:13.080 is used in the exterior of the star. 00:03:13.080 --> 00:03:20.490 So if you wish to solve these things, 00:03:20.490 --> 00:03:23.370 what you basically do is just choose a central density. 00:03:27.930 --> 00:03:40.940 You have to choose an equation of state, 00:03:40.940 --> 00:03:44.690 which allows you to relate the pressure to the density. 00:03:44.690 --> 00:03:48.830 We'll talk about that a little bit more later in this class. 00:03:48.830 --> 00:03:52.610 And then you just start integrating. 00:03:52.610 --> 00:04:01.510 So you basically then integrate until you find 00:04:01.510 --> 00:04:04.540 that the pressure equals 0. 00:04:04.540 --> 00:04:09.070 The radius at which this occurs defines the star surface. 00:04:25.320 --> 00:04:27.810 As you integrate along, this allows you to build up. 00:04:27.810 --> 00:04:30.990 You then build the mass profile of the star, the pressure 00:04:30.990 --> 00:04:33.090 profile of the star. 00:04:33.090 --> 00:04:37.830 You are building up how the phi changes as you integrate along 00:04:37.830 --> 00:04:39.180 the star. 00:04:39.180 --> 00:04:40.980 So note, if you integrate this up, 00:04:40.980 --> 00:04:44.320 this is only defined up to a constant of integration. 00:04:44.320 --> 00:04:45.750 And so what you then need to do is 00:04:45.750 --> 00:04:48.997 once you have reach the surface of the star, you know m of r. 00:04:48.997 --> 00:04:50.580 And so what you're going to need to do 00:04:50.580 --> 00:04:54.030 is adjust the phi that you found in order 00:04:54.030 --> 00:04:58.440 to match with the exterior solution 1 minus 2gm over r. 00:04:58.440 --> 00:05:00.750 This is an exercise you will do on an upcoming homework 00:05:00.750 --> 00:05:01.250 assignment. 00:05:01.250 --> 00:05:03.780 This is one of my favorite assignments in the class. 00:05:03.780 --> 00:05:06.600 It's a really good chance to actually see 00:05:06.600 --> 00:05:10.540 the way we solve these equations. 00:05:10.540 --> 00:05:13.170 And this kind of an exercise, it's 00:05:13.170 --> 00:05:15.900 done all the time as part of modern research, 00:05:15.900 --> 00:05:18.300 and you will do this for a particularly simple kind 00:05:18.300 --> 00:05:21.170 of equation of state. 00:05:21.170 --> 00:05:24.250 So I want to look at what some examples of objects like this 00:05:24.250 --> 00:05:27.010 actually look like. 00:05:27.010 --> 00:05:31.230 So what we're going to do is consider 00:05:31.230 --> 00:05:34.620 an unrealistic but instructive idealized limit. 00:05:41.650 --> 00:05:49.750 So imagine a star that has rho equal constant. 00:05:53.270 --> 00:05:55.880 That is something where no matter how hard you squeeze it, 00:05:55.880 --> 00:05:58.280 you cannot change its density. 00:05:58.280 --> 00:06:00.050 The only way that that can happen 00:06:00.050 --> 00:06:02.360 is if you have an object that is infinitely stiff. 00:06:05.490 --> 00:06:13.200 This corresponds to a speed of sound, which is defined 00:06:13.200 --> 00:06:20.130 as dpd rho that is infinite. 00:06:20.130 --> 00:06:21.510 Now, of course, no speed of sound 00:06:21.510 --> 00:06:24.160 can actually exceed the speed of light. 00:06:24.160 --> 00:06:28.000 So this is a somewhat pathological object. 00:06:28.000 --> 00:06:31.350 Nonetheless, it's useful for us for the simple reason 00:06:31.350 --> 00:06:33.960 that the mass function that emerges for this 00:06:33.960 --> 00:06:34.710 is quite trivial. 00:06:49.910 --> 00:06:54.150 Rho is constant, so m of r just goes as the volume inside 00:06:54.150 --> 00:06:55.710 radius r times rho. 00:07:02.560 --> 00:07:10.620 The star then has a total mass of 4/3 pi r, 00:07:10.620 --> 00:07:13.370 to be pi rho r star cubed. 00:07:16.680 --> 00:07:19.290 So we know how its density behaves, it's a constant, 00:07:19.290 --> 00:07:21.720 we know how the mass function behaves. 00:07:21.720 --> 00:07:25.205 The challenge is solving for the pressure profile. 00:07:30.710 --> 00:07:36.560 So the pressure is governed by taking this differential 00:07:36.560 --> 00:07:41.800 equation and basically plug in that mass, 00:07:41.800 --> 00:07:45.130 plug in this mass function, and see what you get. 00:07:45.130 --> 00:08:17.220 So if I go and I throw this guy in here, 00:08:17.220 --> 00:08:18.990 this is what you end up getting. 00:08:18.990 --> 00:08:26.816 So bear in mind as you go through this-- 00:08:26.816 --> 00:08:28.150 yeah, I see what I did here. 00:08:28.150 --> 00:08:33.650 So what I do as I just pulled out my factor of 4/3, 00:08:33.650 --> 00:08:38.043 4/3 pi rho r cubed-- excuse me, my 4/3 pi r cubed. 00:08:38.043 --> 00:08:39.460 I factor that out, and then cancel 00:08:39.460 --> 00:08:43.070 it an overall factor of r squared in the denominator. 00:08:43.070 --> 00:08:45.980 So this is simply what I have over here 00:08:45.980 --> 00:08:48.230 with this mass function defined. 00:08:48.230 --> 00:08:50.450 We want to solve this to find p. 00:08:53.560 --> 00:08:55.420 This is one of those rare moments 00:08:55.420 --> 00:09:01.990 where nature and analysis conspire and a miracle occurs. 00:09:08.480 --> 00:09:14.240 OK, it's a somewhat messy looking kind of solution, 00:09:14.240 --> 00:09:45.760 but nonetheless, it turns out that the pressure profile 00:09:45.760 --> 00:09:48.130 is determined by-- you can manipulate this equation. 00:09:48.130 --> 00:09:49.480 You can integrate it up. 00:09:49.480 --> 00:09:52.660 And this ends up describing what the pressure 00:09:52.660 --> 00:09:59.950 profile looks like, p sub c is the pressure at r equals 0. 00:10:04.570 --> 00:10:06.790 We define the surface of the star 00:10:06.790 --> 00:10:11.050 as being the location at which the pressure goes to 0. 00:10:11.050 --> 00:10:17.845 So by using that, so by exploiting that condition, 00:10:17.845 --> 00:10:20.650 we can use this equation to make a mapping 00:10:20.650 --> 00:10:24.700 between the radius of the star and the central pressure. 00:10:24.700 --> 00:10:26.680 So it's a spherically symmetric thing. 00:10:26.680 --> 00:10:29.680 Once we have chosen what this density is, 00:10:29.680 --> 00:10:32.720 then there is essentially a one parameter family of solutions. 00:10:32.720 --> 00:10:37.180 We-- if we choose a central pressure, that determines 00:10:37.180 --> 00:10:38.500 what the radius would be. 00:10:38.500 --> 00:10:40.737 Conversely, if we want to have a particular radius, 00:10:40.737 --> 00:10:42.820 that determines what the central pressure must be. 00:10:48.473 --> 00:10:49.890 Let's just do a little of analysis 00:10:49.890 --> 00:10:51.240 that follows from this. 00:10:51.240 --> 00:10:57.480 So p equals 0 at r equals r star. 00:10:57.480 --> 00:11:00.870 That defines the surface. 00:11:00.870 --> 00:11:04.290 So putting all that together, you 00:11:04.290 --> 00:11:06.210 can manipulate this equation to find 00:11:06.210 --> 00:11:19.690 that the radius of the star is determined 00:11:19.690 --> 00:11:26.290 from the central pressure, like so. 00:11:26.290 --> 00:11:27.760 If you prefer, you can write this 00:11:27.760 --> 00:11:30.490 as an equation for the central pressure in terms 00:11:30.490 --> 00:11:31.630 of the radius of the star. 00:12:08.870 --> 00:12:11.510 I've written this in terms of m tot 00:12:11.510 --> 00:12:13.430 but m tot is, of course, simply related 00:12:13.430 --> 00:12:16.650 to this radius-- excuse me, to this density 00:12:16.650 --> 00:12:18.545 and the star's radius cubed. 00:12:24.895 --> 00:12:26.270 So what's the importance of this? 00:12:26.270 --> 00:12:32.510 This is, as I've emphasized, a fairly idealized problem. 00:12:32.510 --> 00:12:34.280 Nature will never give us an object 00:12:34.280 --> 00:12:36.110 that has constant density. 00:12:36.110 --> 00:12:38.810 Any object, if you give it a little bit of a squeeze, 00:12:38.810 --> 00:12:40.850 the density will change. 00:12:40.850 --> 00:12:43.340 Causality requires that the speed of sound 00:12:43.340 --> 00:12:46.470 be less than the speed of light. 00:12:46.470 --> 00:12:51.420 So clearly, this is a somewhat fictional limit. 00:12:51.420 --> 00:12:53.870 But we can learn something very interesting about this. 00:12:53.870 --> 00:12:59.660 Notice that this formula for the central pressure, 00:12:59.660 --> 00:13:04.160 it diverges for a particular compactness of the star. 00:13:10.310 --> 00:13:11.973 So p usb c goes to infinity. 00:13:11.973 --> 00:13:13.640 Perhaps a little bit more easily to see, 00:13:13.640 --> 00:13:31.840 the denominator goes to 0, for a certain compactness, m over r. 00:13:31.840 --> 00:13:35.860 So let's look at the value at which the denominator goes 00:13:35.860 --> 00:13:36.360 to 0. 00:13:48.050 --> 00:13:49.420 So let's see. 00:13:49.420 --> 00:13:50.700 Move my 1 to the other side. 00:13:50.700 --> 00:13:51.640 Divide by 3. 00:13:51.640 --> 00:13:52.230 Square it. 00:14:02.960 --> 00:14:03.720 Rearrange terms. 00:14:17.480 --> 00:14:24.960 So if the ratio of this star's total mass to its radius 00:14:24.960 --> 00:14:33.130 is such that gm over r star exceeds 4/9 00:14:33.130 --> 00:14:36.165 than the pressure diverges. 00:14:40.510 --> 00:14:51.640 What this tells me is that I cannot construct a physically 00:14:51.640 --> 00:14:57.490 allowable static object, even using this stuff as we can 00:14:57.490 --> 00:15:02.050 imagine fluid, a fluid that has an infinite sound speed. 00:15:02.050 --> 00:15:05.560 I cannot make a star more compact than that. 00:15:09.010 --> 00:15:13.090 Making it that compact requires infinite pressure at the core. 00:15:15.720 --> 00:15:19.080 This implies that stars have a maximum compactness. 00:15:39.560 --> 00:15:59.510 We cannot have physically realizable pressure profiles 00:15:59.510 --> 00:16:00.410 if-- 00:16:00.410 --> 00:16:01.900 and let's turn that around-- 00:16:01.900 --> 00:16:09.290 if the ratio of the radius to g times 00:16:09.290 --> 00:16:14.840 the total mass is smaller than 9/4. 00:16:14.840 --> 00:16:18.740 Now, this holds for the stiffest possible fluid that we can even 00:16:18.740 --> 00:16:24.030 imagine, one which the laws of physics actually do not permit. 00:16:24.030 --> 00:16:26.390 And so one infers from this, actually, 00:16:26.390 --> 00:16:29.600 that this bound basically tells me 00:16:29.600 --> 00:16:34.760 that given any physical fluid, any physically realizable star 00:16:34.760 --> 00:16:37.820 that I can construct, I must have 00:16:37.820 --> 00:16:39.590 a maximum allowed compactness. 00:16:42.420 --> 00:16:46.230 Putting this a little bit more precisely 00:16:46.230 --> 00:16:51.740 brings us to a result that is known as Buchdahl's theorem. 00:17:10.236 --> 00:17:11.819 Buchdahl's theorem tells me that there 00:17:11.819 --> 00:17:33.600 is no stable spherical fluid configuration 00:17:33.600 --> 00:17:41.390 in which the configuration's radius is 00:17:41.390 --> 00:17:45.260 smaller than 9/4 of gm total. 00:17:48.220 --> 00:17:51.250 For those of you who like to put factors of c in there, 00:17:51.250 --> 00:17:54.400 divide by c squared, and this tells you something 00:17:54.400 --> 00:17:57.520 that you can convert to SI units, which 00:17:57.520 --> 00:18:00.490 tells you how small you are allowed to make an object. 00:18:04.390 --> 00:18:05.620 So this is clear. 00:18:05.620 --> 00:18:09.460 It emerges, and very nicely, in this idealized but unphysical 00:18:09.460 --> 00:18:10.660 fluid limit. 00:18:10.660 --> 00:18:13.510 But it can be proven more generally. 00:18:13.510 --> 00:18:16.700 You just have to make a few assumptions about the way 00:18:16.700 --> 00:18:20.980 that the pressure profile is not singular in any place. 00:18:24.840 --> 00:18:31.230 Proof of this can be found in the beautiful, old textbook 00:18:31.230 --> 00:18:39.700 by Weinberg, section 11.6. 00:18:39.700 --> 00:18:42.340 Well, when you hear something like 00:18:42.340 --> 00:18:44.000 that, you gotta think to yourself, 00:18:44.000 --> 00:18:48.970 well, suppose I made a star with some kind of a fluid 00:18:48.970 --> 00:18:55.150 and I gave it a radius of 10/4 gm total over c squared. 00:18:55.150 --> 00:18:57.280 And I just came along and squeezed it. 00:18:57.280 --> 00:18:59.730 What would happen? 00:18:59.730 --> 00:19:04.750 Well, notice the word "stable" in this definition. 00:19:04.750 --> 00:19:07.210 You're free to do that, and should you 00:19:07.210 --> 00:19:11.110 do so, you would simply no longer have a stable object. 00:19:11.110 --> 00:19:14.170 So remember, part of what went into this analysis 00:19:14.170 --> 00:19:18.580 is we were assuming the spacetime and the fluid that 00:19:18.580 --> 00:19:20.020 is the source of the spacetime, we 00:19:20.020 --> 00:19:21.540 were assuming everything is static. 00:19:21.540 --> 00:19:24.460 OK, we're making sure everything just sits still. 00:19:24.460 --> 00:19:26.920 This is telling us you can't do that if you 00:19:26.920 --> 00:19:28.840 want to have a star as compact as that. 00:19:28.840 --> 00:19:31.930 So if you were to do this, it would collapse. 00:19:31.930 --> 00:19:34.630 It would become dynamical and the spacetime 00:19:34.630 --> 00:19:38.820 would transition into something else. 00:19:38.820 --> 00:19:40.425 What that something else might be 00:19:40.425 --> 00:19:44.668 will be a topic that we get into a little bit more 00:19:44.668 --> 00:19:46.710 after we have developed some additional material. 00:19:52.270 --> 00:19:54.010 So let's talk a little bit. 00:19:54.010 --> 00:19:57.270 This will help you with the homework assignment, where 00:19:57.270 --> 00:20:01.860 you guys are going to construct relativistic stellar models. 00:20:06.570 --> 00:20:12.600 Let's talk a little bit about how we describe real objects. 00:20:12.600 --> 00:20:20.600 They are not of constant density, 00:20:20.600 --> 00:20:26.910 and they instead have some p that 00:20:26.910 --> 00:20:30.600 is a function of the density. 00:20:30.600 --> 00:20:34.890 It's worth noting that in an even more general case-- 00:20:34.890 --> 00:20:36.570 this is actually worth a brief aside-- 00:20:42.480 --> 00:20:46.950 the equation of state relates the pressure to local density 00:20:46.950 --> 00:20:49.140 and s, where s is the entropy. 00:20:55.210 --> 00:20:58.210 For the kind of applications where general relativity tends 00:20:58.210 --> 00:21:01.660 to be important, for instance, when we're studying 00:21:01.660 --> 00:21:05.860 the stellar structure of a neutron star, 00:21:05.860 --> 00:21:09.445 the fluid ends up being so cold that you don't need 00:21:09.445 --> 00:21:10.570 to worry about the entropy. 00:21:10.570 --> 00:21:17.680 And where that comes from is that when 00:21:17.680 --> 00:21:21.470 I revisit my first law of thermodynamics 00:21:21.470 --> 00:21:26.620 and I include temperature and entropy effects, so the term 00:21:26.620 --> 00:21:30.040 that I left out in my earlier accounting, TDS. 00:21:30.040 --> 00:21:33.100 If t is small, I can ignore that term, 00:21:33.100 --> 00:21:34.600 and it ends up being something where 00:21:34.600 --> 00:21:38.920 my local energy, or energy density, 00:21:38.920 --> 00:21:40.960 ends up only depending on the pressure. 00:21:40.960 --> 00:21:43.180 But "cold" is a wiggle word. 00:21:43.180 --> 00:21:45.880 I have to define a scale to say whether an object is 00:21:45.880 --> 00:21:48.970 cold or hot. 00:21:48.970 --> 00:21:52.360 There's a few notes laying this out right in my notes. 00:21:52.360 --> 00:21:53.695 Let me just sketch the key idea. 00:21:57.170 --> 00:22:03.100 Cold depends on the fluids, and it's worth 00:22:03.100 --> 00:22:05.350 noting that the kind of fluid you're playing with here 00:22:05.350 --> 00:22:07.070 tend to be made out of fermions. 00:22:07.070 --> 00:22:10.260 And so it depends on the fluids' Fermi temperature. 00:22:16.840 --> 00:22:20.620 So the Fermi temperature is defined 00:22:20.620 --> 00:22:27.700 as the Fermi energy normalized to a Boltzmann factor. 00:22:27.700 --> 00:22:29.770 You get the Fermi energy by looking 00:22:29.770 --> 00:22:35.500 at how the energy levels are filled in your fermion fluid 00:22:35.500 --> 00:22:36.976 here. 00:22:36.976 --> 00:22:39.310 I have a few additional notes to lay out 00:22:39.310 --> 00:22:43.540 the way in which you can relate this to the local density, 00:22:43.540 --> 00:22:47.790 the mass of each particle that goes into this Fermi fluid. 00:22:47.790 --> 00:22:55.080 The punchline is that for neutron stars, one 00:22:55.080 --> 00:22:57.190 of the cases where we do, in fact, 00:22:57.190 --> 00:23:01.800 make dense general relativistic fluid stars, 00:23:01.800 --> 00:23:05.430 the Fermi temperature tends to be on the order of 10 00:23:05.430 --> 00:23:08.970 to the 13 Kelvin, so about 10 trillion Kelvin. 00:23:08.970 --> 00:23:17.940 When we actually observe these objects, 00:23:17.940 --> 00:23:23.320 they are on the order of 10 to the 6 to 10 to the 9 Kelvin. 00:23:23.320 --> 00:23:27.630 So they're a factor of about 10 of the 7, 10 to the 4 to 10 00:23:27.630 --> 00:23:30.810 to the 7 times colder than the Fermi temperature. 00:23:30.810 --> 00:23:34.170 Even though they may be a billion Kelvin, they are cold. 00:23:34.170 --> 00:23:37.380 So we're going to use what are called cold equations of state 00:23:37.380 --> 00:23:38.340 to describe these guys. 00:23:42.910 --> 00:23:48.560 So with that out of the way, let's talk a little bit 00:23:48.560 --> 00:23:51.620 about the kinds of equations of state that we will tend to use. 00:23:58.160 --> 00:24:01.700 So people who study the physics of dense matter, 00:24:01.700 --> 00:24:06.050 a lot of their lives is really down to understanding 00:24:06.050 --> 00:24:08.998 what the equation of state of that cold matter looks like. 00:24:08.998 --> 00:24:10.790 Some of them are concerned about hot matter 00:24:10.790 --> 00:24:13.040 as well, in which case they might be actually worrying 00:24:13.040 --> 00:24:15.433 about things at tens of trillions of Kelvin. 00:24:15.433 --> 00:24:17.600 But if you're looking at astrophysical applications, 00:24:17.600 --> 00:24:20.970 you're generally interested in the cold matter. 00:24:20.970 --> 00:24:24.950 And so they end up putting the other very complicated models 00:24:24.950 --> 00:24:31.100 using QCD and effective field theories to try 00:24:31.100 --> 00:24:36.350 to understand how it is that a particular fluid 00:24:36.350 --> 00:24:38.690 of dense matter, how its pressure and its density 00:24:38.690 --> 00:24:39.590 are related. 00:24:39.590 --> 00:24:41.420 And what I'm sort of wheeling around here 00:24:41.420 --> 00:24:44.600 is that you generally do not have a simple analytic form. 00:24:44.600 --> 00:24:47.480 You wind up with some kind of a fairly complicated function 00:24:47.480 --> 00:24:51.230 that emerges from a numerical calculation. 00:24:51.230 --> 00:24:52.610 It often ends up being-- 00:24:52.610 --> 00:24:55.143 if you are a user of this equation of state-- 00:24:55.143 --> 00:24:56.810 it ends up being in the form of a table. 00:24:56.810 --> 00:24:58.100 So they might actually just give you 00:24:58.100 --> 00:24:59.642 a file that's got a bunch of numbers, 00:24:59.642 --> 00:25:02.510 which says if the density is this, 00:25:02.510 --> 00:25:03.710 then the pressure is this. 00:25:03.710 --> 00:25:06.770 And you can fit little functions to that 00:25:06.770 --> 00:25:10.500 that allow you to look things up and do your calculations. 00:25:10.500 --> 00:25:12.260 But it's not in the form of a clean thing 00:25:12.260 --> 00:25:15.528 that you can write down on the blackboard. 00:25:15.528 --> 00:25:17.820 I want something clean I can write down the blackboard. 00:25:17.820 --> 00:25:25.090 So I'm going to introduce an approximation, which 00:25:25.090 --> 00:25:33.190 is useful for testing things out, test cases, 00:25:33.190 --> 00:25:34.920 and for pedagogy. 00:25:34.920 --> 00:25:41.560 What we do is we take the pressure 00:25:41.560 --> 00:25:53.200 to be a power law of the density. 00:25:53.200 --> 00:26:00.410 So what we do is we write p equals k rho 00:26:00.410 --> 00:26:04.930 0 to the gamma, where k and gamma are constants. 00:26:11.760 --> 00:26:14.730 A form that looks like this, this is called a polytrope. 00:26:21.720 --> 00:26:24.690 Now, the thing which I particularly want to highlight, 00:26:24.690 --> 00:26:27.990 and for those of you who are going to do this highly 00:26:27.990 --> 00:26:30.360 recommended homework exercise, please 00:26:30.360 --> 00:26:33.750 pay attention at this point. 00:26:33.750 --> 00:26:39.420 This rho 0 is not-- oh, I erased it. 00:26:39.420 --> 00:26:42.140 This rho 0 is not the rho-- 00:26:42.140 --> 00:26:43.050 oh, there it is-- 00:26:43.050 --> 00:26:46.770 it's not the rho that appears in the equation of state. 00:26:46.770 --> 00:26:48.690 It's slightly different. 00:26:48.690 --> 00:26:52.200 Rho 0 is not the rho that appears, for instance, 00:26:52.200 --> 00:26:54.398 in the t of e equations. 00:26:58.890 --> 00:27:05.130 Rho 0 is what is called the rest mass density. 00:27:10.550 --> 00:27:14.750 It does not take into account the fact that if I take a big-- 00:27:14.750 --> 00:27:17.540 let's say I've got a big bucket of nuclear fluid. 00:27:17.540 --> 00:27:21.620 So I take my bucket here and I squeeze down on it. 00:27:24.140 --> 00:27:26.360 When I squeeze down, its density is 00:27:26.360 --> 00:27:28.310 going to increase, first of all, because I 00:27:28.310 --> 00:27:30.560 have decreased the volume. 00:27:30.560 --> 00:27:33.200 So the number of particles remains fixed, 00:27:33.200 --> 00:27:35.730 but I decrease the space in there. 00:27:35.730 --> 00:27:38.930 But I have also done work on it, because this thing exerts 00:27:38.930 --> 00:27:41.540 a pressure that opposes my squeezing. 00:27:41.540 --> 00:27:43.940 And I need to take into account the fact 00:27:43.940 --> 00:27:48.050 that the work I do in squeezing this fluid 00:27:48.050 --> 00:27:50.030 increases the density rho. 00:28:26.740 --> 00:28:29.320 So when you write out your t of e equations, 00:28:29.320 --> 00:28:31.460 rho is energy density. 00:28:31.460 --> 00:28:35.812 All forms of energy gravitate. 00:28:35.812 --> 00:28:37.520 This is just the way people traditionally 00:28:37.520 --> 00:28:38.690 write the equation of state. 00:28:38.690 --> 00:28:42.260 This is when one is doing nuclear physics. 00:28:42.260 --> 00:28:43.760 There's good reasons for doing this, 00:28:43.760 --> 00:28:45.980 but it's not the most convenient form 00:28:45.980 --> 00:28:48.410 for the kind of calculations that we want to do, 00:28:48.410 --> 00:28:54.280 and that you are going to want to do, in the problem set. 00:29:00.470 --> 00:29:02.892 Fortunately, it's not too difficult to convert, so let 00:29:02.892 --> 00:29:04.350 me describe to you how you do that. 00:29:19.150 --> 00:29:23.760 So we are going to use the first law of thermodynamics 00:29:23.760 --> 00:29:25.932 in a form in which I've written now a couple times. 00:29:25.932 --> 00:29:28.140 Interestingly, it showed up in our cosmology lecture. 00:29:30.960 --> 00:29:39.210 So my first law tells me du equals minus pressure dv. 00:29:39.210 --> 00:29:42.080 So this is my total energy in a fluid element 00:29:42.080 --> 00:29:45.202 and this is the work done on a fluid element. 00:29:45.202 --> 00:30:01.154 So rho is equal to the amount of energy in a fiducial volume. 00:30:04.060 --> 00:30:07.525 My rest energy-- excuse me, my rest density-- 00:30:11.230 --> 00:30:15.210 is the rest energy of every little body that 00:30:15.210 --> 00:30:20.485 goes into this per unit volume. 00:30:25.530 --> 00:30:41.940 This means that I can write du as d rho over rho 0, 00:30:41.940 --> 00:30:48.133 provided I throw in an extra factor of m rest 00:30:48.133 --> 00:30:49.300 to get the dimensions right. 00:30:51.900 --> 00:31:01.920 And I can write d volume as d1 over rho 0, 00:31:01.920 --> 00:31:09.190 provided I throw in that factor of m rest 00:31:09.190 --> 00:31:11.400 to get the dimensions right. 00:31:11.400 --> 00:31:15.900 I know this looks weird but, it's perfectly valid. 00:31:15.900 --> 00:31:21.930 So I'm going to rewrite my first law of thermodynamics 00:31:21.930 --> 00:31:33.550 as d rho over rho 0 equals minus pd 1 over r 0. 00:31:47.750 --> 00:31:49.510 Let's manipulate that right-hand side. 00:31:52.660 --> 00:31:55.370 So I'm going to assume this polytropic form. 00:31:55.370 --> 00:31:59.400 I'm going to use p equals k rho 0 to the gamma. 00:32:03.870 --> 00:32:05.700 But I'm going to switch that around. 00:32:05.700 --> 00:32:13.220 I'm going to write this as rho 0 equals p over k 00:32:13.220 --> 00:32:14.580 to the power of 1 over gamma. 00:32:17.110 --> 00:32:25.320 So when I do that, I get d rho over rho 0 00:32:25.320 --> 00:32:39.780 equals kappa 1 over gamma over kappa pdp over p to the 1 00:32:39.780 --> 00:32:42.330 plus 1 over gamma. 00:32:56.887 --> 00:32:57.970 Pardon me just one moment. 00:32:57.970 --> 00:32:59.710 I did something clever in my notes here 00:32:59.710 --> 00:33:02.620 and I'm just trying to make sure I understand 00:33:02.620 --> 00:33:04.470 what the hell I actually did. 00:33:09.670 --> 00:33:11.768 So I'm going to level with you. 00:33:11.768 --> 00:33:13.310 I've gone through this several times. 00:33:13.310 --> 00:33:16.070 There's a step in the calculation that at this point, 00:33:16.070 --> 00:33:18.475 I, for some stupid reason, didn't write down. 00:33:18.475 --> 00:33:20.600 I'm going to trust I knew what I was doing, though, 00:33:20.600 --> 00:33:22.730 because I know the final result was right. 00:33:22.730 --> 00:33:26.790 You can integrate up both sides here. 00:33:26.790 --> 00:33:28.910 Oh, I think I see what I did. 00:33:28.910 --> 00:33:29.410 OK. 00:33:29.410 --> 00:33:33.440 So you integrate up both sides here. 00:33:33.440 --> 00:33:35.440 And what you find is this becomes rho 00:33:35.440 --> 00:33:42.480 equals p over gamma minus 1 plus a constant. 00:33:46.740 --> 00:33:48.510 Yeah, not 100% sure how I actually did 00:33:48.510 --> 00:33:50.010 that, so my apologies on that. 00:33:52.957 --> 00:33:54.790 I'm going to assume I knew what I was doing. 00:33:54.790 --> 00:33:58.078 I will try to fix this and I may post an addendum here. 00:33:58.078 --> 00:34:00.120 The next step, actually, is you want to determine 00:34:00.120 --> 00:34:01.120 what that constant is. 00:34:01.120 --> 00:34:03.090 So the way you determine the constant 00:34:03.090 --> 00:34:08.380 is you take advantage of the fact that rho goes to rho 0. 00:34:08.380 --> 00:34:11.320 The energy density becomes the rest energy density 00:34:11.320 --> 00:34:12.719 if there is no pressure exerted. 00:34:16.429 --> 00:34:19.070 And so this gives us our final relationship here, 00:34:19.070 --> 00:34:26.210 which is that rho equals rho 0 plus p over gamma minus 1. 00:34:36.800 --> 00:34:41.239 OK, I will double check how I went from line 2 line 3 there, 00:34:41.239 --> 00:34:46.170 but the final thing that I have boxed online for 00:34:46.170 --> 00:34:49.560 is, indeed, exactly what you need to do in order 00:34:49.560 --> 00:34:52.147 to build a stellar model. 00:34:52.147 --> 00:34:54.480 I guess I've been emphasizing this is something you will 00:34:54.480 --> 00:34:57.480 do on an upcoming problem set. 00:34:57.480 --> 00:34:58.980 Let me just sketch the recipe. 00:34:58.980 --> 00:35:01.025 I've said this verbally, but let me just write it out explicitly 00:35:01.025 --> 00:35:01.525 here. 00:35:04.920 --> 00:35:09.990 So I will give you an equation of state. 00:35:09.990 --> 00:35:18.740 You then need to pick rho 0 at r equals 0. 00:35:21.650 --> 00:35:24.500 Using your equation of state and using that relationship 00:35:24.500 --> 00:35:28.370 between rho and rho 0, this will give you 00:35:28.370 --> 00:35:30.650 rho at the center, pressure at the center. 00:35:33.550 --> 00:35:44.760 Set m of r at the center to 0. 00:35:44.760 --> 00:35:47.470 I emphasize, again, that this may seem obvious, 00:35:47.470 --> 00:35:50.418 but it is somewhat important that you get it right. 00:35:50.418 --> 00:35:51.960 When you do this homework assignment, 00:35:51.960 --> 00:35:53.502 I'll give you a little hint as to how 00:35:53.502 --> 00:35:55.130 to build that in smoothly. 00:35:55.130 --> 00:35:56.130 It can be a little bit-- 00:35:56.130 --> 00:35:59.370 I don't want to say tricky, but it's worth 00:35:59.370 --> 00:36:00.900 thinking about a little bit. 00:36:00.900 --> 00:36:10.880 Then what you do is integrate your equations for the pressure 00:36:10.880 --> 00:36:15.930 in the mass from r equals 0. 00:36:15.930 --> 00:36:18.660 And this cannot be done analytically. 00:36:18.660 --> 00:36:20.650 You have to use a numerical integrator. 00:36:34.380 --> 00:36:36.660 If you have never used one of these before, 00:36:36.660 --> 00:36:39.090 I will give you a Mathematica notebook that 00:36:39.090 --> 00:36:40.740 demonstrates how to use it. 00:36:40.740 --> 00:36:42.690 This is a skill that is worth knowing. 00:36:42.690 --> 00:36:46.800 The plain truth of the matter is that the class of problems 00:36:46.800 --> 00:36:51.473 that are amenable to purely analytic solutions, those 00:36:51.473 --> 00:36:52.140 are interesting. 00:36:52.140 --> 00:36:53.015 They're illustrative. 00:36:53.015 --> 00:36:54.720 They're good to work with. 00:36:54.720 --> 00:36:57.780 But they tend to be unphysical and they're just 00:36:57.780 --> 00:37:00.660 not the ones that are of interest for many things 00:37:00.660 --> 00:37:03.915 that we study in science. 00:37:03.915 --> 00:37:07.770 So show while you're doing this-- 00:37:07.770 --> 00:37:10.170 this is not necessary to make your model, 00:37:10.170 --> 00:37:11.630 but it's very useful to do this. 00:37:16.210 --> 00:37:25.960 You can also integrate, whoops, d5 er from the center. 00:37:31.010 --> 00:37:47.690 So a caution is that you do not know phi at r equals 0. 00:37:47.690 --> 00:37:50.480 So what you should do is just temporarily set it equal to 0. 00:37:50.480 --> 00:37:52.855 And what you're going to be doing then when you integrate 00:37:52.855 --> 00:37:55.640 this up is you will calculate the delta phi that 00:37:55.640 --> 00:38:01.680 describes your model from the center to the surface, which 00:38:01.680 --> 00:38:02.850 brings me to step 4. 00:38:18.860 --> 00:38:27.240 When you find p equals 0, you've hit the surface. 00:38:38.190 --> 00:38:43.340 So what we do is we use the fact that p of r 00:38:43.340 --> 00:38:51.950 equals 0 defines the star's radius r star. 00:38:51.950 --> 00:39:03.080 Once you've done that, you now know the total mass 00:39:03.080 --> 00:39:05.190 and the radius. 00:39:05.190 --> 00:39:07.400 So you will find, when you're doing this, 00:39:07.400 --> 00:39:11.750 that your numerical integrator is not super well-behaved 00:39:11.750 --> 00:39:13.310 as you approach the surface. 00:39:13.310 --> 00:39:16.610 This is a feature, not a bug. 00:39:16.610 --> 00:39:20.420 What's going on is that as you begin to approach the surface, 00:39:20.420 --> 00:39:23.270 the gradient and the pressure gets quite steep. 00:39:23.270 --> 00:39:25.370 And so the way one numerically integrates 00:39:25.370 --> 00:39:29.441 a set of couple equations like this is by, essentially, 00:39:29.441 --> 00:39:34.490 if you take advantage of the fact that an integral, 00:39:34.490 --> 00:39:36.560 it's what you get by sort of dividing things up 00:39:36.560 --> 00:39:41.052 into tiny little pieces and add up like little rectangles. 00:39:41.052 --> 00:39:43.010 And when you're solving a differential equation 00:39:43.010 --> 00:39:45.470 like this, you're essentially taking the continuum solution-- 00:39:45.470 --> 00:39:47.637 that you guys have learned how to do in many cases-- 00:39:47.637 --> 00:39:49.940 and you're approximating it by a series of smaller 00:39:49.940 --> 00:39:53.300 and smaller finite steps. 00:39:53.300 --> 00:39:55.550 Because the gradient in the pressure 00:39:55.550 --> 00:39:58.640 gets large as you approach the surface, 00:39:58.640 --> 00:40:00.410 numerical integrators typically try 00:40:00.410 --> 00:40:04.340 taking an infinite number of infinitesimal steps, which 00:40:04.340 --> 00:40:07.550 makes the CPU sad, and so it's likely to exit with an error 00:40:07.550 --> 00:40:08.780 condition. 00:40:08.780 --> 00:40:11.570 Generally, when that has happened, 00:40:11.570 --> 00:40:13.520 you've gotten an answer that's probably 00:40:13.520 --> 00:40:15.800 good to within a part in a million, 00:40:15.800 --> 00:40:17.240 or something like that. 00:40:17.240 --> 00:40:19.400 Fine for our purposes. 00:40:19.400 --> 00:40:21.650 If you need to do something a little bit more careful, 00:40:21.650 --> 00:40:25.760 that's a subject for a numerical analysis class. 00:40:25.760 --> 00:40:27.710 For us, I will give you some hints 00:40:27.710 --> 00:40:32.100 on this when you begin exploring these solutions. 00:40:32.100 --> 00:40:38.890 So you have an additional boundary condition. 00:40:38.890 --> 00:40:48.100 You know by Birkhoff's theorem that the Schwarzschild metric 00:40:48.100 --> 00:40:49.200 describes the exterior. 00:40:55.620 --> 00:41:07.500 That means gtt is minus 1 minus, given by this for everywhere 00:41:07.500 --> 00:41:09.600 greater than r star. 00:41:09.600 --> 00:41:18.860 This gives us a boundary condition that phi of r star 00:41:18.860 --> 00:41:30.300 must be 1/2 log 1 minus 2m total over r star. 00:41:30.300 --> 00:41:32.290 By enforcing this boundary condition, 00:41:32.290 --> 00:41:34.620 you can go back to your solution for phi 00:41:34.620 --> 00:41:36.660 and you can figure out what the value at r 00:41:36.660 --> 00:41:41.040 equals 0 should have been to give you a continuous function 00:41:41.040 --> 00:41:42.975 that matches at the surface. 00:41:46.000 --> 00:41:51.960 So that's it for spherical stars. 00:41:51.960 --> 00:41:55.488 I look forward to you doing these exercises. 00:41:55.488 --> 00:41:57.030 My own biases are perhaps coming out, 00:41:57.030 --> 00:41:58.155 but these are a lot of fun. 00:42:00.600 --> 00:42:03.970 The one thing which I will do for you, 00:42:03.970 --> 00:42:07.020 and I regret that my notes didn't really have this, 00:42:07.020 --> 00:42:10.855 is I will try to figure out how on earth I went from line 2 00:42:10.855 --> 00:42:15.720 to line 3 in this calculation over here, 00:42:15.720 --> 00:42:18.900 going from the rest mass density, the rest energy 00:42:18.900 --> 00:42:21.330 density to the energy density. 00:42:21.330 --> 00:42:23.460 My apologies that that's not there. 00:42:23.460 --> 00:42:26.370 All I can say is that there are many distractions these days 00:42:26.370 --> 00:42:28.512 and I overlooked that when I was reviewing my notes 00:42:28.512 --> 00:42:29.970 and preparing for today's lectures. 00:42:43.440 --> 00:42:48.780 I'd like to take this moment to take a little bit of a detour. 00:42:48.780 --> 00:42:53.250 Let's imagine that we have a spacetime that 00:42:53.250 --> 00:42:56.140 is Schwarzschild everywhere. 00:43:07.770 --> 00:43:25.780 In other words, it has this form for all r, not simply 00:43:25.780 --> 00:43:27.580 the exterior of some object. 00:43:30.370 --> 00:43:34.415 We already know that this spacetime is a vacuum solution. 00:43:43.590 --> 00:43:49.440 I know that t mu nu equals 0. 00:43:49.440 --> 00:43:50.410 Back up for a second. 00:43:50.410 --> 00:43:54.040 If I generate the Einstein tensor for this, 00:43:54.040 --> 00:43:58.240 I will get identically 0, which implies 00:43:58.240 --> 00:44:01.120 that this corresponds to a solution, which has t 00:44:01.120 --> 00:44:01.750 mu equals 0. 00:44:04.480 --> 00:44:07.690 I also know, though, that if I examine 00:44:07.690 --> 00:44:11.020 the behavior of radial geodesics in the weak field 00:44:11.020 --> 00:44:27.050 of this spacetime, I find that they 00:44:27.050 --> 00:44:29.000 fall towards this like an object that 00:44:29.000 --> 00:44:30.950 is falling towards a mass m. 00:44:46.090 --> 00:44:48.430 So this spacetime appears to be something 00:44:48.430 --> 00:44:51.160 that is everywhere vacuum. 00:44:51.160 --> 00:44:53.890 There is nothing in this spacetime, 00:44:53.890 --> 00:44:56.050 and that nothing has a mass of m. 00:45:19.570 --> 00:45:21.490 I hope that bothers you. 00:45:21.490 --> 00:45:24.670 That is among the sillier things that has been 00:45:24.670 --> 00:45:26.320 said in the name of physics. 00:45:26.320 --> 00:45:29.020 That sure sounds silly. 00:45:29.020 --> 00:45:32.230 But let me remind you that we, in fact, have seen something 00:45:32.230 --> 00:45:37.180 very similar in a much less complicated theory of physics. 00:45:50.590 --> 00:45:57.290 So if I look at the electric field of a point charge 00:45:57.290 --> 00:45:58.070 at the origin-- 00:46:03.902 --> 00:46:08.210 so that's the three vector e is just q displacement 00:46:08.210 --> 00:46:09.560 factor over r cubed. 00:46:25.390 --> 00:46:28.240 If I compute the divergence of this electric field-- 00:46:33.470 --> 00:46:36.430 the divergence, of course, tells me about the charge density-- 00:46:36.430 --> 00:46:37.250 and I get 0. 00:46:41.060 --> 00:46:45.110 So this is an electric field that has no charge density 00:46:45.110 --> 00:46:47.750 anywhere, but that lack of charge density 00:46:47.750 --> 00:46:48.890 has a total charge of q. 00:47:02.840 --> 00:47:05.750 This was something that we easily learn how to resolve. 00:47:05.750 --> 00:47:08.360 Usually at the MIT curriculum, this often 00:47:08.360 --> 00:47:11.540 shows up when you take a course like 8.07. 00:47:11.540 --> 00:47:14.150 What we do is we say, oh, all that's going on here 00:47:14.150 --> 00:47:25.080 is that I have a singular point charge at r equals 0. 00:47:25.080 --> 00:47:27.460 So yeah, I've got no charge density, 00:47:27.460 --> 00:47:29.060 but I have a total charge. 00:47:29.060 --> 00:47:29.560 Fine. 00:47:29.560 --> 00:47:31.450 We were happy with that. 00:47:31.450 --> 00:47:33.760 I want you to think of the Schwarzschild metric 00:47:33.760 --> 00:47:37.900 as doing something similar for gravity. 00:47:37.900 --> 00:47:42.010 There is no source anywhere, but there is mass. 00:47:42.010 --> 00:47:44.680 Maybe there's just something singular and a little 00:47:44.680 --> 00:47:46.360 funny going on at r equals 0. 00:48:03.928 --> 00:48:05.470 You might be concerned about what's 00:48:05.470 --> 00:48:06.800 happening there at r equal 0. 00:48:09.817 --> 00:48:11.275 When I say it plays a similar role, 00:48:11.275 --> 00:48:14.260 it plays a similar role to the pull on point charge. 00:48:28.478 --> 00:48:30.520 So there'll be nothing there, but perhaps there's 00:48:30.520 --> 00:48:32.710 something funny going on at r equals 0. 00:48:32.710 --> 00:48:34.348 And by the way, the field equations 00:48:34.348 --> 00:48:36.640 that govern gravity, my relativistic theory of gravity, 00:48:36.640 --> 00:48:37.520 they're non-linear. 00:48:37.520 --> 00:48:40.100 So when I say there's something funny going on at r equals 0, 00:48:40.100 --> 00:48:42.550 it could be really funny. 00:48:42.550 --> 00:48:46.050 So we're not going to get too worked up about that, 00:48:46.050 --> 00:48:49.210 but we're just going to bear in mind this is odd. t mu equals 0 00:48:49.210 --> 00:48:51.350 but it has mass. 00:48:51.350 --> 00:48:55.460 So let's look at the spacetime itself. 00:48:55.460 --> 00:49:02.480 Just staring at this, we can see two radii where it appears 00:49:02.480 --> 00:49:04.690 something odd is going on. 00:49:18.380 --> 00:49:21.430 So you can see right away, lots of stuff kind of blows up 00:49:21.430 --> 00:49:26.200 and behaves badly at r equals 0. 00:49:26.200 --> 00:49:30.400 And you can also see that your gtt and your grr, 00:49:30.400 --> 00:49:37.270 they are behaving in a way that is potentially problematic when 00:49:37.270 --> 00:49:39.397 the radius is 2gm. 00:49:39.397 --> 00:49:41.230 So you look at that and think, yeah, there's 00:49:41.230 --> 00:49:43.280 two radii there that look sick. 00:49:43.280 --> 00:49:46.710 I am worried about this spacetime. 00:49:46.710 --> 00:49:50.130 Well, we should be cautious. 00:49:50.130 --> 00:49:53.010 One of the parables that we learned about when we studied 00:49:53.010 --> 00:49:57.210 linearized gravity is that we can sometimes 00:49:57.210 --> 00:50:01.565 put ourselves into a coordinate system that confuses us. 00:50:01.565 --> 00:50:02.940 When we study linearized gravity, 00:50:02.940 --> 00:50:05.250 we found a solution that looked everywhere. 00:50:05.250 --> 00:50:07.860 It looked like the entire spacetime metric was radiated. 00:50:07.860 --> 00:50:10.140 And it turned out only two of those 10 components 00:50:10.140 --> 00:50:10.975 were radiative. 00:50:10.975 --> 00:50:12.600 That turned out to be something that we 00:50:12.600 --> 00:50:16.103 were able to cure by introducing a gauge transformation. 00:50:16.103 --> 00:50:17.770 Doing that here's a little bit trickier, 00:50:17.770 --> 00:50:20.490 but we're going to need to think about, 00:50:20.490 --> 00:50:23.220 how can I more clearly call out the physical content 00:50:23.220 --> 00:50:25.060 of this spacetime? 00:50:25.060 --> 00:50:29.670 So one of the lessons that I hope 00:50:29.670 --> 00:50:31.740 has been imparted in this class so far 00:50:31.740 --> 00:50:33.900 is that if you really want understand 00:50:33.900 --> 00:50:37.320 the nature of gravity, you want to go from the metric 00:50:37.320 --> 00:50:38.650 to the curvature. 00:50:38.650 --> 00:50:45.900 So what I'm going to do is assemble an invariant scalar 00:50:45.900 --> 00:50:47.262 from my curvature. 00:50:59.570 --> 00:51:01.290 And I'm going to use the Riemann tensor 00:51:01.290 --> 00:51:03.750 because I know Ricci vanishes in this spacetime, 00:51:03.750 --> 00:51:07.800 so that wouldn't give me anything interesting. 00:51:07.800 --> 00:51:10.320 So what I'm going to do is assemble an object. 00:51:10.320 --> 00:51:16.640 I'm going to call it capital I. And that's just Riemann 00:51:16.640 --> 00:51:19.700 contracted into Riemann. 00:51:19.700 --> 00:51:20.860 This actually has a name. 00:51:20.860 --> 00:51:24.200 It is known as the Kretschmann scalar. 00:51:24.200 --> 00:51:25.580 And you can go in. 00:51:25.580 --> 00:51:27.590 You can work out all these components. 00:51:27.590 --> 00:51:32.690 The gr tool that is posted to the 8.962 website 00:51:32.690 --> 00:51:35.930 is something you can explore with us. 00:51:35.930 --> 00:51:38.600 And this is just a number. 00:51:38.600 --> 00:51:47.510 Turns out to be 48 g squared m squared over r to the sixth. 00:51:47.510 --> 00:51:48.590 What does this guy mean? 00:51:48.590 --> 00:51:55.570 Well, in an invariant way, it's kind of Riemann squared. 00:51:55.570 --> 00:51:58.430 Riemann tells me to go back and think about things 00:51:58.430 --> 00:51:59.670 like geodesic deviation. 00:51:59.670 --> 00:52:01.640 It tells me about the strength of tides. 00:52:01.640 --> 00:52:12.080 So roughly speaking, square root I is an invariant way 00:52:12.080 --> 00:52:14.105 of characterizing tidal forces. 00:52:29.380 --> 00:52:35.020 So if you're sitting around in the Schwarzschild spacetime 00:52:35.020 --> 00:52:38.020 and you want to give yourself an estimate of what 00:52:38.020 --> 00:52:42.040 kind of tidal forces are likely to act on you, 00:52:42.040 --> 00:52:45.520 compute the Kretschmann scalar, take its square root, 00:52:45.520 --> 00:52:48.370 and that'll give you an idea of how strong they typically 00:52:48.370 --> 00:52:48.870 tend to be. 00:52:53.860 --> 00:52:58.680 So notice, when we look at this, this tells us r equals 2gm. 00:52:58.680 --> 00:53:02.308 If you plug r equals 2gm in there, 00:53:02.308 --> 00:53:03.350 nothing special about it. 00:53:14.390 --> 00:53:15.990 It's a radius just like any other. 00:53:15.990 --> 00:53:24.280 As you go from 2.001gm gm to 1.99999gm, 00:53:24.280 --> 00:53:25.420 it increases a little bit. 00:53:25.420 --> 00:53:27.210 Of course, it's got the 1 over r to the sixth behavior, 00:53:27.210 --> 00:53:29.190 but it's not like there's a sudden transition, 00:53:29.190 --> 00:53:34.070 or anything particularly special happens right at that radius. 00:53:34.070 --> 00:53:40.010 But it is hella singular at r equals 0. 00:53:47.070 --> 00:53:51.270 So sure enough, r equals 0 is a place where 00:53:51.270 --> 00:53:53.700 tidal forces blow up. 00:54:01.380 --> 00:54:02.340 OK, fine. 00:54:02.340 --> 00:54:04.590 We're going to need to do a little bit more work then, 00:54:04.590 --> 00:54:07.560 because I still want understand, yeah, OK, r equals 2gm. 00:54:07.560 --> 00:54:10.710 There's no diverging tidal forces there, 00:54:10.710 --> 00:54:13.230 but that metric still looks wacky at that point. 00:54:13.230 --> 00:54:14.970 So what is going on there? 00:54:19.167 --> 00:54:21.250 So let's think about the geometry of the spacetime 00:54:21.250 --> 00:54:23.190 in the vicinity of 2gm. 00:54:30.920 --> 00:54:33.080 So let's imagine. 00:54:33.080 --> 00:54:36.020 Let's do the following exercise. 00:54:36.020 --> 00:54:51.150 Suppose I draw a circle at some radius r that's 00:54:51.150 --> 00:54:56.250 in the theta equals pi over 2 plane. 00:54:56.250 --> 00:55:00.220 So I'm just sweeping around in phi. 00:55:00.220 --> 00:55:01.320 I'm making this like so. 00:55:05.140 --> 00:55:11.220 So here is my r cosine phi axis. 00:55:11.220 --> 00:55:15.310 Here is my r sine phi axis. 00:55:15.310 --> 00:55:20.830 Here is my circle of radius r. 00:55:20.830 --> 00:55:24.490 And let's ask, what is the surface 00:55:24.490 --> 00:55:29.090 area that this guy sweeps out as an advance forward in time? 00:55:29.090 --> 00:55:32.970 So as this thing goes forward in time, 00:55:32.970 --> 00:55:39.730 it sort of sweeps out a cylinder in a spacetime diagram. 00:55:39.730 --> 00:55:45.070 Let's compute the proper area associated with this cylinder 00:55:45.070 --> 00:55:47.800 that this circle is sweeping out as it moves forward in time. 00:55:51.180 --> 00:55:54.900 So the surface area of my tube, I 00:55:54.900 --> 00:55:59.580 integrate from some start time to some end time. 00:56:05.810 --> 00:56:09.800 I'm going to integrate around in phi, 00:56:09.800 --> 00:56:12.020 and then the proper area element that I 00:56:12.020 --> 00:56:18.258 need to do this is going to be gtt g5 phi, the 1/2. 00:56:18.258 --> 00:56:20.800 There's actually a minus sign in there to get the sign right. 00:56:20.800 --> 00:56:21.980 Let's write it like this. 00:56:25.190 --> 00:56:28.540 To remind you how I do this, think of this area element 00:56:28.540 --> 00:56:29.680 as a 2 volume. 00:56:29.680 --> 00:56:32.080 Go back to some of our earlier discussion of defining 00:56:32.080 --> 00:56:37.180 integrals in spacetime, and this is the proper area associated 00:56:37.180 --> 00:56:39.880 with a figure that has some extent in time and extent 00:56:39.880 --> 00:56:40.420 in angle. 00:56:50.207 --> 00:56:51.290 So let's compute that guy. 00:57:08.650 --> 00:57:11.450 So I take my Schwarzschild metric. 00:57:11.450 --> 00:57:17.040 g5 phi in the theta equals pi over 2 plane is just r. 00:57:17.040 --> 00:57:25.110 gtt is the square root of 1 minus 2gm over r. 00:57:25.110 --> 00:57:27.840 So the area of my tube is going to be 00:57:27.840 --> 00:57:37.840 r, integrate from my start time to my end time, dt. 00:57:54.080 --> 00:57:54.920 So this is easy. 00:57:54.920 --> 00:58:05.790 So I get a 2 pi 2 pi r square root 1 minus 2g m over r, 00:58:05.790 --> 00:58:08.660 and let's just say my interval is delta t. 00:58:12.550 --> 00:58:15.550 Notice what happens as I take the radius 00:58:15.550 --> 00:58:17.590 of this thing down to 2gm. 00:58:22.140 --> 00:58:25.176 This goes to 0. 00:58:25.176 --> 00:58:28.230 This r goes to 2gm. 00:58:28.230 --> 00:58:32.910 If I go inside 2gm, I don't even want to compute that. 00:58:32.910 --> 00:58:36.950 Something has gone awry. 00:58:36.950 --> 00:58:40.590 But look, I can draw this thing just fine. 00:58:40.590 --> 00:58:42.450 Clearly, there's a surface there. 00:58:42.450 --> 00:58:44.447 It's got to have an area associated with it. 00:58:44.447 --> 00:58:46.530 Why are you telling me that the area of this thing 00:58:46.530 --> 00:58:49.860 is 0 in that limit and is a nonsense integral if I 00:58:49.860 --> 00:58:50.940 go inside this thing? 00:58:53.664 --> 00:58:57.460 Well, what's happening is we have uncovered 00:58:57.460 --> 00:59:00.070 a coordinate singularity. 00:59:00.070 --> 00:59:12.870 The time coordinate is badly behaved as we-- 00:59:12.870 --> 00:59:22.215 not well-- as we approach this radius, r equals 2gm. 00:59:28.950 --> 00:59:34.180 Let me give you an analogy that describes, essentially-- 00:59:34.180 --> 00:59:41.180 it's something that is very similar to that tube, 00:59:41.180 --> 00:59:43.220 that world tube that I just drew. 00:59:43.220 --> 00:59:45.785 But let me do it in a, perhaps, more familiar context. 00:59:48.370 --> 00:59:54.430 Suppose I want to draw a sphere, and all 00:59:54.430 --> 00:59:59.620 that I know about a sphere is that it has got two 00:59:59.620 --> 01:00:04.830 coordinates to cover it, an angle phi and an angle theta. 01:00:04.830 --> 01:00:14.420 And so I could say, OK, here is my sphere. 01:00:20.280 --> 01:00:23.550 Here is theta equals 0, phi equals 0. 01:00:23.550 --> 01:00:25.170 Here is phi equals pi over 2. 01:00:25.170 --> 01:00:27.180 Here's phi equals pi. 01:00:27.180 --> 01:00:29.025 Pi equals 3 pi over 2. 01:00:29.025 --> 01:00:32.214 Phi equals 2 pi. 01:00:32.214 --> 01:00:33.990 Theta equals pi over 2. 01:00:33.990 --> 01:00:34.980 Theta equals pi. 01:00:38.130 --> 01:00:38.880 There's my sphere. 01:00:42.020 --> 01:00:47.650 So this chart that I've just drawn here, it's true. 01:00:47.650 --> 01:00:53.090 This does represent the coordinate system, 01:00:53.090 --> 01:01:01.695 but it's a horrible rendering of a sphere's geometry. 01:01:06.980 --> 01:01:09.440 What I didn't realize when I wrote this down here is 01:01:09.440 --> 01:01:16.070 that, in fact, at theta equals 0 and theta equals pi, 01:01:16.070 --> 01:01:19.850 every phi value should be collapsed to a single point. 01:01:36.770 --> 01:01:40.450 This is reflecting the fact that if you look at a globe, 01:01:40.450 --> 01:01:45.160 all lines of longitude cross the north pole and the south pole. 01:01:45.160 --> 01:01:49.780 Every value of the azimuthal angle 01:01:49.780 --> 01:01:51.370 on the surface of the earth, they 01:01:51.370 --> 01:01:53.980 become singular at the north pole and the south pole. 01:01:53.980 --> 01:01:57.400 This drawing, well, it's like one of those, 01:01:57.400 --> 01:01:59.650 I forget the names of them, but the various renderings 01:01:59.650 --> 01:02:01.240 of a map that try to take the earth 01:02:01.240 --> 01:02:03.460 and write it on a flat space, and you wind up 01:02:03.460 --> 01:02:06.970 with Greenland being three times the size of Africa, 01:02:06.970 --> 01:02:08.090 or something like that. 01:02:08.090 --> 01:02:12.225 And it's because there should be zero area at the top here. 01:02:12.225 --> 01:02:13.600 As you approach the top the area, 01:02:13.600 --> 01:02:15.058 it should be getting much stronger. 01:02:15.058 --> 01:02:17.800 And when you do representation of your map like this, 01:02:17.800 --> 01:02:19.950 you're spreading everything way, way out. 01:02:22.530 --> 01:02:23.700 What is going on? 01:02:23.700 --> 01:02:27.050 And why this idea of drawing this world tube that 01:02:27.050 --> 01:02:29.670 is swept out by my circle of radius 01:02:29.670 --> 01:02:32.250 r as it advances forward in time? 01:02:36.580 --> 01:02:40.690 That drawing does not account for the fact 01:02:40.690 --> 01:02:46.660 that the Schwarzschild time coordinate is singular 01:02:46.660 --> 01:02:50.700 as you approach r equals 2gm. 01:02:50.700 --> 01:03:04.880 It's going to turn out all times t map to a single sphere, 01:03:04.880 --> 01:03:07.530 and r equals 2gm. 01:03:10.890 --> 01:03:14.580 To get some insight into what's going on here, 01:03:14.580 --> 01:03:16.194 let's do a little thought experiment. 01:03:27.810 --> 01:03:30.390 What I'm going to do is imagine I'm at rest 01:03:30.390 --> 01:03:32.610 in the Schwarzschild spacetime. 01:03:32.610 --> 01:03:35.310 So let's say that I'm at some finite radius r. 01:03:35.310 --> 01:03:36.570 I am not in a weak field. 01:03:36.570 --> 01:03:39.870 OK, maybe I am at something like r equals 01:03:39.870 --> 01:03:42.760 4gm, or something like that. 01:03:42.760 --> 01:03:48.765 And what I'm going to do is drop a little rock, drop a particle. 01:03:58.820 --> 01:04:03.810 So I'm going to drop a particle from r equals r0. 01:04:03.810 --> 01:04:06.560 I'm going to integrate the geodesic equation, 01:04:06.560 --> 01:04:09.500 and I'm going to parameterize what its radial motion looks 01:04:09.500 --> 01:04:10.820 like as a function of "time." 01:04:35.890 --> 01:04:38.578 I put "time" in quotes here because you 01:04:38.578 --> 01:04:40.120 should be saying at this point, well, 01:04:40.120 --> 01:04:42.910 you just told me that time is doing something kind of funny 01:04:42.910 --> 01:04:43.410 here. 01:04:43.410 --> 01:04:45.420 What do you mean by that? 01:04:45.420 --> 01:04:47.880 I'm actually going to do this for two different notions 01:04:47.880 --> 01:04:48.380 of time. 01:04:51.810 --> 01:05:00.330 I'm going to do this for the coordinate time t, 01:05:00.330 --> 01:05:06.510 and I'm also going to do this for proper time, 01:05:06.510 --> 01:05:14.980 tau, as measured along that world, that infall. 01:05:21.240 --> 01:05:22.990 So I'm not going to go through the details 01:05:22.990 --> 01:05:23.830 of this calculation. 01:05:23.830 --> 01:05:27.100 It's a straightforward, moderately tedious exercise. 01:05:30.308 --> 01:05:32.350 I would just quote to you what the result ends up 01:05:32.350 --> 01:05:33.106 looking like. 01:05:52.470 --> 01:05:54.950 So let's first write down what the solution looks like, 01:05:54.950 --> 01:05:58.530 parameterized by the proper time. 01:05:58.530 --> 01:06:03.593 So this is most easily written as tau, proper time, and 2gm. 01:06:03.593 --> 01:06:05.010 Essentially, I'm just going to use 01:06:05.010 --> 01:06:06.900 it to set a system of units. 01:06:06.900 --> 01:06:10.710 I write this thing as a function of r. 01:06:10.710 --> 01:06:12.150 My solution turns out to be-- 01:06:27.734 --> 01:06:29.940 it looks like this. 01:06:29.940 --> 01:06:33.230 So if I were to make a plot of what this thing's motion looks 01:06:33.230 --> 01:06:46.020 like as a function of time, so here's our 0. 01:06:46.020 --> 01:06:50.030 Here is r of tau. 01:06:52.930 --> 01:06:56.520 And let's just put in, for fun, let's say this is 2gm. 01:07:02.090 --> 01:07:04.550 Zoom, fallen. 01:07:04.550 --> 01:07:10.630 You reach r equals 0 in finite proper time. 01:07:10.630 --> 01:07:14.350 The parable of the Kretschmann scalar is that as you do so, 01:07:14.350 --> 01:07:17.470 the tidal forces acting on you are diverging. 01:07:17.470 --> 01:07:21.340 So if you have any last wishes, send them out 01:07:21.340 --> 01:07:23.290 because you're not going to have a lot of time 01:07:23.290 --> 01:07:26.340 to tell people about them. 01:07:30.710 --> 01:07:34.480 Let's now write it as a function of coordinate time t. 01:07:38.620 --> 01:07:42.457 This ends up being-- 01:07:42.457 --> 01:07:45.040 bear with me while I write this out, this is slightly lengthy. 01:08:23.399 --> 01:08:25.210 OK, so what I mean on this last line 01:08:25.210 --> 01:08:27.069 is if you want to get the complete solution, 01:08:27.069 --> 01:08:28.736 just write both of these functions down. 01:08:28.736 --> 01:08:33.279 Again, subtract them off and place the r's with r0. 01:08:33.279 --> 01:08:41.870 When you look at this, here's what you see. 01:08:45.870 --> 01:08:51.412 The motion expressed in times of the coordinate time t 01:08:51.412 --> 01:08:55.609 asymptotically approaches the radius 2gm, 01:08:55.609 --> 01:09:00.069 but it never quite reaches it. 01:09:00.069 --> 01:09:03.700 As t goes to infinity, it eventually reaches-- 01:09:03.700 --> 01:09:06.359 so r, you can see it appearing in the behavior 01:09:06.359 --> 01:09:07.979 of this natural log. 01:09:07.979 --> 01:09:11.258 r gets to 2gm as t goes to infinity. 01:09:15.569 --> 01:09:22.399 So as measured by clocks on the infalling body, 01:09:22.399 --> 01:09:24.920 it rapidly reaches r equals 0. 01:09:24.920 --> 01:09:26.630 According to this coordinate time, 01:09:26.630 --> 01:09:28.550 it never even crosses r equals 2gm. 01:09:31.180 --> 01:09:34.540 What the hell is going on with that? 01:09:34.540 --> 01:09:44.649 Well, to give a little bit of insight into this, 01:09:44.649 --> 01:09:47.450 it's useful to stop for a second and ask ourselves, 01:09:47.450 --> 01:09:52.370 what is that coordinate time t actually measuring? 01:09:52.370 --> 01:09:58.375 So let me write down the Schwarzschild metric 01:09:58.375 --> 01:09:59.500 and let's think about this. 01:10:20.480 --> 01:10:24.990 So kind of hard to see what t means in this, 01:10:24.990 --> 01:10:26.480 but let's consider a limit. 01:10:26.480 --> 01:10:33.260 Suppose I consider observers who are very far away. 01:10:33.260 --> 01:10:37.220 If I look at people who are at r, much, much larger than 2gm. 01:10:44.110 --> 01:10:55.810 For such observers, spacetime looks like this, 01:10:55.810 --> 01:11:00.850 and this is nothing more than flat spacetime in spherical 01:11:00.850 --> 01:11:02.980 coordinates. 01:11:02.980 --> 01:11:05.740 This is what we call an asymptotically flat spacetime. 01:11:05.740 --> 01:11:09.040 As you get sufficiently far away from the source, 01:11:09.040 --> 01:11:11.770 it looks just like flat spacetime. 01:11:11.770 --> 01:11:14.153 essentially, special relativity rules apply. 01:11:14.153 --> 01:11:15.820 And that gives us some insight into what 01:11:15.820 --> 01:11:18.370 this coordinate t means. 01:11:18.370 --> 01:11:21.220 The t that we are using in the Schwarzschild coordinate 01:11:21.220 --> 01:11:32.200 system, this is time as measured by distant observers. 01:11:38.760 --> 01:11:42.750 Tau is time, as measured by this infalling observer. 01:11:42.750 --> 01:11:46.200 So what we are seeing here is the infalling observer 01:11:46.200 --> 01:11:52.170 crosses 2gm, reaches r equals 0, and has a very short life. 01:11:52.170 --> 01:11:55.900 But those who are using clocks, adapted to things very, 01:11:55.900 --> 01:11:59.850 very far away, never even see it cross 2gm. 01:11:59.850 --> 01:12:01.890 Why is that? 01:12:01.890 --> 01:12:04.710 Well, we will pick this up in the next lecture, 01:12:04.710 --> 01:12:07.620 but let me remind you that when we initially 01:12:07.620 --> 01:12:11.550 began working on this subject, one of the very first lectures, 01:12:11.550 --> 01:12:14.280 we talked about something called the Einstein synchronization 01:12:14.280 --> 01:12:17.400 procedure, where what we did was we imagined spacetime 01:12:17.400 --> 01:12:20.310 was filled with a conceptual lattice of measuring 01:12:20.310 --> 01:12:21.960 rods and clocks. 01:12:21.960 --> 01:12:25.020 And we synchronized all of those clocks 01:12:25.020 --> 01:12:31.530 by requiring that the time delay between different clocks 01:12:31.530 --> 01:12:33.390 is synchronized according to the time it 01:12:33.390 --> 01:12:37.090 takes for light to travel from one to the other. 01:12:37.090 --> 01:12:40.110 This is telling us we are actually working-- 01:12:40.110 --> 01:12:41.700 when we use Schwarzschild time, we 01:12:41.700 --> 01:12:45.150 are working in a system that reflects an underlying 01:12:45.150 --> 01:12:47.340 inheritance from special relativity. 01:12:47.340 --> 01:12:50.880 These are clocks that have been synchronized by the Einstein 01:12:50.880 --> 01:12:53.290 synchronization procedure. 01:12:53.290 --> 01:12:55.680 And so the pathological behavior that we 01:12:55.680 --> 01:13:00.360 see here, it must ultimately owe to the behavior of these clocks 01:13:00.360 --> 01:13:02.490 that we use to define our coordinate system, 01:13:02.490 --> 01:13:04.560 and the behavior of those clocks is linked 01:13:04.560 --> 01:13:07.410 to the behavior of light. 01:13:07.410 --> 01:13:11.370 So in order to get insight as to what is going on with this, why 01:13:11.370 --> 01:13:14.430 is it that if I use a clock adapted to the infalling body, 01:13:14.430 --> 01:13:17.370 I see painful death, but if I use a clock adopted 01:13:17.370 --> 01:13:19.410 to someone very far away, I don't even 01:13:19.410 --> 01:13:23.195 see it approach that dangerous r equals to a radius. 01:13:23.195 --> 01:13:24.570 In order to resolve that mystery, 01:13:24.570 --> 01:13:29.100 I'm going to need to examine what the motion of light looks 01:13:29.100 --> 01:13:31.170 like in this spacetime. 01:13:31.170 --> 01:13:34.340 We'll pick that up in the next lecture.