WEBVTT 00:00:00.090 --> 00:00:01.670 The following content is provided 00:00:01.670 --> 00:00:03.820 under a Creative Commons license. 00:00:03.820 --> 00:00:06.550 Your support will help MIT OpenCourseWare continue 00:00:06.550 --> 00:00:10.160 to offer high quality educational resources for free. 00:00:10.160 --> 00:00:12.700 To make a donation or to view additional materials 00:00:12.700 --> 00:00:16.620 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.620 --> 00:00:17.327 at ocw.mit.edu. 00:00:21.820 --> 00:00:24.690 PROFESSOR: OK, starting again. 00:00:24.690 --> 00:00:26.650 I want to begin with a quick review 00:00:26.650 --> 00:00:28.210 of what we said last time. 00:00:28.210 --> 00:00:29.740 This time will be quicker than usual 00:00:29.740 --> 00:00:31.940 because we're not really continuing from there, 00:00:31.940 --> 00:00:34.300 we'll be starting a completely new topic today. 00:00:34.300 --> 00:00:36.320 But I thought I'd remind you, nonetheless, 00:00:36.320 --> 00:00:38.870 that we had a last lecture-- the universe was not 00:00:38.870 --> 00:00:41.300 created between then and now. 00:00:41.300 --> 00:00:43.640 And what we talked about last time 00:00:43.640 --> 00:00:47.850 was the spacetime geodesic equation, 00:00:47.850 --> 00:00:49.170 which is written this way. 00:00:49.170 --> 00:00:52.090 It's a completely identical to the equation 00:00:52.090 --> 00:00:56.470 we derived for geodesics in pure spatial situations. 00:00:56.470 --> 00:00:59.490 The only difference really, is a difference in notation. 00:00:59.490 --> 00:01:01.390 Instead of using i and j the tradition 00:01:01.390 --> 00:01:05.890 is to use mu, nu, et cetera for the spacetime indices which 00:01:05.890 --> 00:01:11.780 are sum from 0 to 3, where x super 0 is identical to t. 00:01:11.780 --> 00:01:14.970 And in addition, the parameter which is sometimes called s 00:01:14.970 --> 00:01:17.840 when we're talking about space, is called tau 00:01:17.840 --> 00:01:19.760 when we're talking about time because what 00:01:19.760 --> 00:01:22.370 the parameter refers to is the proper time 00:01:22.370 --> 00:01:26.000 along the trajectory of the object whose geodesic we're 00:01:26.000 --> 00:01:28.810 calculating. 00:01:28.810 --> 00:01:31.560 We then introduced the Schwarzschild metric, 00:01:31.560 --> 00:01:35.350 which we did not derive, but we claimed 00:01:35.350 --> 00:01:38.190 it describes the metric for any spherically symmetric mass 00:01:38.190 --> 00:01:40.490 distribution as long as you're talking 00:01:40.490 --> 00:01:44.910 about the region outside where the masses are located. 00:01:44.910 --> 00:01:47.290 And here M is the total mass of the object, which 00:01:47.290 --> 00:01:49.840 is the only thing the metric really depends on, 00:01:49.840 --> 00:01:54.520 G is Newton's constant, and C is of course the speed of light. 00:01:54.520 --> 00:01:56.350 The metric has an interesting feature 00:01:56.350 --> 00:01:58.770 that the coefficients of both the dt 00:01:58.770 --> 00:02:03.750 squared term and to dr squared term becomes singular either 0 00:02:03.750 --> 00:02:06.770 or infinity, depending on which one you're looking at, 00:02:06.770 --> 00:02:09.620 a particular value of the radius called the Schwarzschild 00:02:09.620 --> 00:02:14.380 radius, given by this formula 2 GM divided by c squared. 00:02:14.380 --> 00:02:17.160 The bigger the mass the bigger the Schwarzschild radius, 00:02:17.160 --> 00:02:19.380 they're proportional to each other. 00:02:19.380 --> 00:02:21.900 The metric is singular at those points, 00:02:21.900 --> 00:02:24.630 but I mentioned but did not prove, 00:02:24.630 --> 00:02:27.250 that that particular singularity is 00:02:27.250 --> 00:02:30.900 in fact what is referred to as a coordinate singularity. 00:02:30.900 --> 00:02:33.530 It's a singularity that's there only because of the way 00:02:33.530 --> 00:02:35.495 the coordinates were chosen. 00:02:35.495 --> 00:02:37.870 So there are other ways of choosing the coordinates where 00:02:37.870 --> 00:02:40.650 that singularity disappears. 00:02:40.650 --> 00:02:43.209 There's also singularity at r equals zero, 00:02:43.209 --> 00:02:44.750 and that singularity is real, there's 00:02:44.750 --> 00:02:46.860 no way to remove that singularity 00:02:46.860 --> 00:02:48.185 by a change of coordinates. 00:02:51.690 --> 00:02:56.470 However, although r sub s is not a true singularity, 00:02:56.470 --> 00:02:58.470 it is a horizon. 00:02:58.470 --> 00:03:04.320 And by that we mean that if any particle or even a light beam 00:03:04.320 --> 00:03:07.720 gets inside the Schwarzschild radius, it can never get out. 00:03:07.720 --> 00:03:13.225 There's no geodesic which will take it out of the horizon. 00:03:13.225 --> 00:03:14.950 And it's not even a matter of geodesics, 00:03:14.950 --> 00:03:17.660 there are no time-like paths even 00:03:17.660 --> 00:03:20.330 if you have a rocket which would then not follow geodesic. 00:03:20.330 --> 00:03:23.950 There's no way to get out from inside a black hole. 00:03:23.950 --> 00:03:28.140 We didn't show that, but that fact is claimed. 00:03:28.140 --> 00:03:32.660 Then we calculated the geodesic for our radially falling 00:03:32.660 --> 00:03:33.300 object. 00:03:33.300 --> 00:03:35.260 We solve the problem of an object that 00:03:35.260 --> 00:03:38.360 is released from rest at some initial value r sub 0, 00:03:38.360 --> 00:03:41.590 and just let fall straight down towards the center 00:03:41.590 --> 00:03:42.580 of the sphere. 00:03:42.580 --> 00:03:47.660 And the the equation describing the geodesic 00:03:47.660 --> 00:03:51.020 is just a special case of the general equation 00:03:51.020 --> 00:03:52.680 that we had a few slides ago. 00:03:52.680 --> 00:03:54.990 And we need only look at the radial component 00:03:54.990 --> 00:03:57.630 if we want to track how the radius changes with time. 00:03:57.630 --> 00:04:01.270 So there was a free index mu in the generic form 00:04:01.270 --> 00:04:05.700 of the equation, we're setting mu equal to the r variable. 00:04:05.700 --> 00:04:08.420 And then the equation reduces to this form 00:04:08.420 --> 00:04:12.450 and we know what these G sub t t's and G sub r r's are, 00:04:12.450 --> 00:04:15.060 they come from the equation for the Schwarzschild metric 00:04:15.060 --> 00:04:16.910 on the previous slide. 00:04:16.910 --> 00:04:18.750 And that equation can be manipulated 00:04:18.750 --> 00:04:21.125 and eventually it simplifies to something extraordinarily 00:04:21.125 --> 00:04:22.050 simple. 00:04:22.050 --> 00:04:25.530 It's just the statement that d squared r d tau squared is 00:04:25.530 --> 00:04:29.920 equal to minus GM over r squared, which looks exactly 00:04:29.920 --> 00:04:32.380 like the Newtonian equation for something falling 00:04:32.380 --> 00:04:35.107 in a spherically symmetric gravitational field. 00:04:35.107 --> 00:04:36.690 But it's not really the same equation, 00:04:36.690 --> 00:04:39.356 it just looks like it's the same equation, because the variables 00:04:39.356 --> 00:04:41.180 both a different meanings. r and tau 00:04:41.180 --> 00:04:43.840 both have different meanings from the r and t that 00:04:43.840 --> 00:04:46.950 would have appeared in the Newtonian calculation. 00:04:46.950 --> 00:04:49.200 The r variable that appears here is not really 00:04:49.200 --> 00:04:50.707 the distance from the origin. 00:04:50.707 --> 00:04:52.790 If you wanted to know the distance from the origin 00:04:52.790 --> 00:04:55.160 you'd have to integrate the metric singular even-- 00:04:55.160 --> 00:04:57.600 if not even a well defined distance to the origin 00:04:57.600 --> 00:05:00.510 because the origin singular. 00:05:00.510 --> 00:05:04.000 And the tau that appears here is a time variable, 00:05:04.000 --> 00:05:07.064 but it's not the time that would be read on any fixed clock, 00:05:07.064 --> 00:05:08.480 rather it's the time that would be 00:05:08.480 --> 00:05:11.070 read on the wristwatch of the person falling 00:05:11.070 --> 00:05:14.180 into the spherically symmetric object, which we might consider 00:05:14.180 --> 00:05:15.000 to be a black hole. 00:05:18.010 --> 00:05:22.250 And we were able to solve this equation 00:05:22.250 --> 00:05:24.660 by using essentially conservation of energy 00:05:24.660 --> 00:05:27.261 techniques, or at least what would be called conservation 00:05:27.261 --> 00:05:29.260 of energy if we were doing the Newtonian version 00:05:29.260 --> 00:05:31.420 of the problem, which is the same equation even 00:05:31.420 --> 00:05:34.740 though the variables have a different interpretation. 00:05:34.740 --> 00:05:37.240 So we were able to calculate not r as a function of tau, 00:05:37.240 --> 00:05:40.570 but at least tau a function of r. 00:05:40.570 --> 00:05:43.770 And we got that equation, which is a little complicated, 00:05:43.770 --> 00:05:45.320 but the interesting thing about it 00:05:45.320 --> 00:05:49.960 is that it gives finite answers for every value of r going 00:05:49.960 --> 00:05:53.020 all the way down to r equals zero. 00:05:53.020 --> 00:05:55.810 So it means that in a finite amount of time, 00:05:55.810 --> 00:05:59.770 as seen by the person falling into the black hole, 00:05:59.770 --> 00:06:02.740 the person would reach r equals zero, at which point 00:06:02.740 --> 00:06:04.710 he would disappear into the singularity. 00:06:04.710 --> 00:06:06.780 He'd actually be ripped apart as he approached 00:06:06.780 --> 00:06:08.950 the singularity because of tidal forces 00:06:08.950 --> 00:06:12.480 which pull more strongly on the front part of him 00:06:12.480 --> 00:06:16.360 than on the back part of him, stretching the object out 00:06:16.360 --> 00:06:19.440 in the radial direction. 00:06:19.440 --> 00:06:23.830 However, curiously, if one calculates 00:06:23.830 --> 00:06:25.820 what this trajectory looks like as a function 00:06:25.820 --> 00:06:29.300 of the coordinate time, t, we did actually 00:06:29.300 --> 00:06:31.240 do that calculation but we looked 00:06:31.240 --> 00:06:35.230 at how it would behave in the limit 00:06:35.230 --> 00:06:37.720 as you approached-- as the particle approached 00:06:37.720 --> 00:06:39.010 the Schwarzschild horizon. 00:06:39.010 --> 00:06:41.850 And we discovered it would take an infinite amount of time, 00:06:41.850 --> 00:06:44.980 as seen from the outside, for the in falling object 00:06:44.980 --> 00:06:47.700 to reach the horizon, let go through the horizon 00:06:47.700 --> 00:06:49.500 and get to r equals zero. 00:06:49.500 --> 00:06:52.280 So from the outside, it looks like the object never 00:06:52.280 --> 00:06:54.210 actually falls into the black hole, 00:06:54.210 --> 00:06:57.220 but just gets closer and closer and closer as t approaches 00:06:57.220 --> 00:06:58.240 infinity. 00:06:58.240 --> 00:07:01.830 So it's an example of a very highly distorted spacetime, 00:07:01.830 --> 00:07:04.400 where you can see very different pictures depending 00:07:04.400 --> 00:07:08.250 on which observer you're trying to describe 00:07:08.250 --> 00:07:09.350 the observations of. 00:07:12.130 --> 00:07:15.220 And I think that's it. 00:07:15.220 --> 00:07:17.350 Any questions about any of that? 00:07:17.350 --> 00:07:20.510 I guess I'll put that back up. 00:07:23.680 --> 00:07:24.400 OK. 00:07:24.400 --> 00:07:27.030 On your homework you'll be applying this geodesic equation 00:07:27.030 --> 00:07:32.560 to a model universe, to Robertson-Walker universe, 00:07:32.560 --> 00:07:35.645 and this will serve only as an example for those calculations. 00:07:35.645 --> 00:07:38.270 I guess there's also a homework problem about the Schwarzschild 00:07:38.270 --> 00:07:40.460 metric, that orbits in the Schwarzschild metric 00:07:40.460 --> 00:07:42.389 that you'll be working at. 00:07:42.389 --> 00:07:43.930 It's all in principal straightforward 00:07:43.930 --> 00:07:46.950 if you just look at equations and follow 00:07:46.950 --> 00:07:49.030 what the equations tell you, thinking carefully 00:07:49.030 --> 00:07:52.570 about what the variables mean. 00:07:52.570 --> 00:07:53.560 OK. 00:07:53.560 --> 00:07:58.240 In that case, let's get started on today's work, 00:07:58.240 --> 00:07:59.944 which is a change of gear. 00:07:59.944 --> 00:08:01.860 We're now going to be talking about black body 00:08:01.860 --> 00:08:05.060 radiation and its effect on the universe. 00:08:05.060 --> 00:08:07.890 I should say that my original plan was 00:08:07.890 --> 00:08:09.990 to get into this set of lectures notes-- 00:08:09.990 --> 00:08:12.710 lecture notes six, which have not been handed out yet-- 00:08:12.710 --> 00:08:15.222 to get into those last time and to finish them today. 00:08:15.222 --> 00:08:16.930 I don't think that's going to be possible 00:08:16.930 --> 00:08:18.638 because I didn't get into them last time, 00:08:18.638 --> 00:08:21.720 and I don't think I'll be able to finish them today. 00:08:21.720 --> 00:08:24.710 But I would like today to be sort of the closing for what's 00:08:24.710 --> 00:08:27.010 needed for the problem set due Monday and for the quiz 00:08:27.010 --> 00:08:28.370 next week. 00:08:28.370 --> 00:08:30.710 So, shortly after today's lecture 00:08:30.710 --> 00:08:32.230 I will send you an email telling you 00:08:32.230 --> 00:08:34.630 where the cutoff is as far as the reading 00:08:34.630 --> 00:08:35.740 and the lecture notes. 00:08:35.740 --> 00:08:40.059 And I also hope to post the lecture notes by tomorrow. 00:08:40.059 --> 00:08:42.919 And I also will be posting a set of review problems 00:08:42.919 --> 00:08:44.770 as we had for quiz one. 00:08:44.770 --> 00:08:47.180 And I hope to get that done by tomorrow. 00:08:47.180 --> 00:08:50.130 You may have noticed that not all of my hopes are filled, 00:08:50.130 --> 00:08:54.310 but I do my best, and I'll try. 00:08:54.310 --> 00:08:56.270 OK are there any logistic questions or anything 00:08:56.270 --> 00:08:57.760 before we go on? 00:08:57.760 --> 00:08:59.680 Yes. 00:08:59.680 --> 00:09:01.520 AUDIENCE: Can you post the solutions 00:09:01.520 --> 00:09:04.901 to the previous problem set, the one 00:09:04.901 --> 00:09:07.310 that we turned in on Friday? 00:09:07.310 --> 00:09:11.310 PROFESSOR: Oh, um, yes I can, I should. 00:09:11.310 --> 00:09:14.530 OK I will. 00:09:14.530 --> 00:09:17.010 OK, that's a third item I should try to get done today. 00:09:17.010 --> 00:09:19.020 Thanks for reminding me. 00:09:19.020 --> 00:09:21.849 And the solutions to the problem set 00:09:21.849 --> 00:09:23.390 that you'll be handing in Monday will 00:09:23.390 --> 00:09:25.264 be posted very shortly after you hand them in 00:09:25.264 --> 00:09:27.080 so that people can start talking about them 00:09:27.080 --> 00:09:29.070 and prepare for the quiz. 00:09:29.070 --> 00:09:29.570 Yes. 00:09:29.570 --> 00:09:33.285 AUDIENCE: Do we have any day at which the videos might be up? 00:09:33.285 --> 00:09:35.660 PROFESSOR: Ah, I've request about that and all I was told 00:09:35.660 --> 00:09:38.250 was that they're doing their best. 00:09:38.250 --> 00:09:43.510 So, I did look into it, but I don't know the answer. 00:09:48.370 --> 00:09:51.867 I hope that you'll have all the videos available to study 00:09:51.867 --> 00:09:54.200 for the quiz, but I don't know if that's going to happen 00:09:54.200 --> 00:09:54.699 or not. 00:09:59.060 --> 00:10:00.920 OK. 00:10:00.920 --> 00:10:26.287 in that case, the new topic is Blackbody Radiation 00:10:26.287 --> 00:10:27.870 and the Early History of the Universe. 00:10:39.290 --> 00:10:41.750 So far we've dealt with a universe which 00:10:41.750 --> 00:10:44.330 contains only non-relativistic matter 00:10:44.330 --> 00:10:46.460 and that, as we said from the beginning, 00:10:46.460 --> 00:10:50.160 describes our universe for the bulk of its history. 00:10:50.160 --> 00:10:51.890 But in the early period, the universe 00:10:51.890 --> 00:10:53.580 was in fact dominated by radiation, 00:10:53.580 --> 00:10:55.390 as we will now be calculating. 00:10:55.390 --> 00:10:57.550 And in the more recent period, the universe 00:10:57.550 --> 00:10:59.129 is dominated by dark energy, which 00:10:59.129 --> 00:11:00.920 we'll be talking about immediately after we 00:11:00.920 --> 00:11:04.090 finish talking about radiation. 00:11:04.090 --> 00:11:08.070 So the important point here is that even though we don't think 00:11:08.070 --> 00:11:11.220 of light as having mass, light certainly 00:11:11.220 --> 00:11:13.640 has energy and relativistically we 00:11:13.640 --> 00:11:16.200 know that energy and mass are equivalent. 00:11:16.200 --> 00:11:20.470 The key equation that actually dominates today's lecture 00:11:20.470 --> 00:11:24.610 is perhaps the most famous equation physics, 00:11:24.610 --> 00:11:30.260 E equals MC squared, energy and mass are equivalent. 00:11:30.260 --> 00:11:32.689 And the numbers-- I'll just give you 00:11:32.689 --> 00:11:33.980 some numbers for this equation. 00:11:33.980 --> 00:11:35.930 You've probably already aware that the numbers 00:11:35.930 --> 00:11:38.560 are kind of out of sight. 00:11:38.560 --> 00:11:45.190 One kilogram, a point to that equation, 00:11:45.190 --> 00:11:50.390 is equivalent to-- I don't have any figures to give you-- 00:11:50.390 --> 00:11:56.860 8.9876, in case you really want to know it accurately, times 10 00:11:56.860 --> 00:12:00.850 to the 16th, most important to know the exponent there, 00:12:00.850 --> 00:12:01.350 joules. 00:12:04.010 --> 00:12:07.284 And it's also perhaps interesting to translate this 00:12:07.284 --> 00:12:08.700 into the kind of energy units that 00:12:08.700 --> 00:12:12.100 are use when we talk about power consumption 00:12:12.100 --> 00:12:14.350 in practical situations. 00:12:14.350 --> 00:12:33.240 It corresponds to 2.497 times 10 to the 10th kilowatt hours, 00:12:33.240 --> 00:12:35.720 which is a lot of kilowatt hours. 00:12:35.720 --> 00:12:38.390 An interesting comparison, is the total power consumption 00:12:38.390 --> 00:12:41.040 of the world, which turns out to be 00:12:41.040 --> 00:12:44.320 comparable to a kilograms worth of things. 00:12:44.320 --> 00:12:47.410 I looked things up in the Wikipedia, 00:12:47.410 --> 00:12:49.460 and it told me that in 2008-- which 00:12:49.460 --> 00:12:51.300 is the most recent year it had numbers for, 00:12:51.300 --> 00:12:54.940 which is a little surprising, that's so far in the past-- 00:12:54.940 --> 00:12:57.300 the total world power consumption 00:12:57.300 --> 00:13:14.040 for the year equaled about, I'm rounding off here, 00:13:14.040 --> 00:13:17.020 about 150 petawatt hours. 00:13:24.760 --> 00:13:26.820 Now, I always have to look up peta when I see it, 00:13:26.820 --> 00:13:28.653 I haven't quite learned what peta means yet. 00:13:28.653 --> 00:13:36.250 But they translated this, this means 150 trillion kilowatt 00:13:36.250 --> 00:13:37.430 hours. 00:13:37.430 --> 00:13:45.930 So 150 times 10 to the 12th kilowatt hours. 00:13:45.930 --> 00:13:48.370 And if we divide this by the number of hours in the year-- 00:13:48.370 --> 00:13:50.475 to ask how much is per hour, which 00:13:50.475 --> 00:13:52.850 is I think a natural thing to think about if you're using 00:13:52.850 --> 00:13:57.610 kilowatt hours to measure the power, especially the energy-- 00:13:57.610 --> 00:14:05.290 the power that goes with this is about 17 times 10 00:14:05.290 --> 00:14:08.900 to the ninth kilowatts. 00:14:08.900 --> 00:14:18.250 And therefore 17 times 10 to the 9th kilowatt hours per hour. 00:14:18.250 --> 00:14:20.080 And if you compare these two numbers, 00:14:20.080 --> 00:14:27.910 it means that if you could convert one kilogram per hour 00:14:27.910 --> 00:14:31.110 into pure energy, that would be about 00:14:31.110 --> 00:14:51.680 equal to 1.5 times the world's power usage in 2008. 00:14:51.680 --> 00:14:55.370 So if you could convert matter completely to energy, 00:14:55.370 --> 00:14:57.780 as they do on Star Trek, it would 00:14:57.780 --> 00:14:59.590 mean that you could fill up your tank 00:14:59.590 --> 00:15:03.290 of your typical American car and power 00:15:03.290 --> 00:15:04.515 the world for about two days. 00:15:07.476 --> 00:15:08.850 But of course we can't undo this, 00:15:08.850 --> 00:15:12.670 that's the important fact concerning power. 00:15:12.670 --> 00:15:15.620 Only a small fraction of the mass 00:15:15.620 --> 00:15:18.010 of the uranium in a nuclear reactor 00:15:18.010 --> 00:15:22.050 is actually converted as power, a fraction of a percent. 00:15:22.050 --> 00:15:23.730 So you don't get nearly as much power 00:15:23.730 --> 00:15:25.340 as this calculation would indicate, 00:15:25.340 --> 00:15:26.790 but in principle this much energy 00:15:26.790 --> 00:15:31.620 is contained in the matter that we have around us all the time. 00:15:49.730 --> 00:15:50.230 OK. 00:15:50.230 --> 00:15:51.896 I want to introduce a few formulas which 00:15:51.896 --> 00:15:53.660 we'll be using sooner or later concerning 00:15:53.660 --> 00:15:57.190 the relativistic treatment of momentum and energy, 00:15:57.190 --> 00:15:58.940 which is what we're getting into here. 00:15:58.940 --> 00:16:03.070 And we're not trying to derive relativity in this course, 00:16:03.070 --> 00:16:07.110 so I'm just trying to quote the results that we'll be needing. 00:16:07.110 --> 00:16:16.910 So, it is useful to introduce an energy momentum four-vector, 00:16:16.910 --> 00:16:22.680 which has a 0-th component and an i-th component, 00:16:22.680 --> 00:16:27.280 where i refers to the spatial indices 1, 2, and 3. 00:16:27.280 --> 00:16:32.130 And sometimes I might write this as p zero p 00:16:32.130 --> 00:16:35.050 with a vector sign over it, where the vector sign indicates 00:16:35.050 --> 00:16:36.750 the three components 1, 2, 3. 00:16:39.610 --> 00:16:43.340 The momentum here is the momentum. 00:16:43.340 --> 00:16:45.415 It differs in its relationship to velocity 00:16:45.415 --> 00:16:47.040 from what we have from Newton, and I'll 00:16:47.040 --> 00:16:48.880 write that in a minute, but this momentum 00:16:48.880 --> 00:16:52.330 is the conserved physical momentum of an object. 00:16:52.330 --> 00:16:55.080 And p0 is also conserved. 00:16:55.080 --> 00:17:05.140 p0 is just an abbreviation for the energy divided by C. 00:17:05.140 --> 00:17:07.630 And this quantity forms a four dimensional 00:17:07.630 --> 00:17:09.530 vector in special relativity. 00:17:09.530 --> 00:17:11.589 And when we say that it's a four-vector, 00:17:11.589 --> 00:17:13.609 we're actually making a definite statement 00:17:13.609 --> 00:17:17.519 about how it transforms from one Lorentz frame to another. 00:17:17.519 --> 00:17:20.190 a four-vector is something which transforms 00:17:20.190 --> 00:17:24.890 in exactly the same way as x super mu, four spatial and time 00:17:24.890 --> 00:17:25.975 coordinates transform. 00:17:29.140 --> 00:17:31.410 And in particular, we learned that there 00:17:31.410 --> 00:17:35.040 was an invariant associated with spacetime transformations this 00:17:35.040 --> 00:17:38.530 s squared, which was x squared plus y squared plus z squared 00:17:38.530 --> 00:17:41.150 minus C squared t squared. 00:17:41.150 --> 00:17:45.170 And the same thing will happen here. 00:17:45.170 --> 00:17:56.270 p squared, which means the Lorentz invariant 00:17:56.270 --> 00:18:08.000 square of the four-vector is the Lorentz [INAUDIBLE] again here, 00:18:08.000 --> 00:18:12.880 and it's again equal to the sum of the squares of the 3 00:18:12.880 --> 00:18:19.450 spatial components minus the square of the time component. 00:18:24.790 --> 00:18:34.280 And that can be written out as the square of the momentum-- 00:18:34.280 --> 00:18:38.000 spatial momentum-- minus E squared C squared. 00:18:42.460 --> 00:18:44.925 And the claim is that this is also Lorentz invariant. 00:19:01.990 --> 00:19:04.670 And we could figure out what Lorentz invariant quantity 00:19:04.670 --> 00:19:08.511 it's equal to by having knowledge 00:19:08.511 --> 00:19:10.010 that this is the same in all frames. 00:19:10.010 --> 00:19:12.550 We can evaluate it in the simplest frame. 00:19:12.550 --> 00:19:15.165 And the simplest frame would be the rest frame of the object. 00:19:15.165 --> 00:19:17.290 In the rest frame of the object the p would be zero 00:19:17.290 --> 00:19:20.670 and this would then just be minus E squared over C squared. 00:19:20.670 --> 00:19:25.800 E squared would be M squared C to the fourth, 00:19:25.800 --> 00:19:36.810 so that implies that this is equal to minus M0 C squared 00:19:36.810 --> 00:19:43.960 squared where M0 is often called the rest mass. 00:19:49.670 --> 00:19:53.030 And when I say it's often called the the rest mass, what I mean 00:19:53.030 --> 00:19:54.950 is that nobody ever mistakes the word "rest 00:19:54.950 --> 00:19:57.840 mass" to mean anything else, if anybody says "rest mass," 00:19:57.840 --> 00:20:00.150 and he knows what he's talking about, he means this. 00:20:00.150 --> 00:20:02.230 But this is often sometimes just called the mass 00:20:02.230 --> 00:20:04.440 because sometimes people only talk about masses 00:20:04.440 --> 00:20:06.714 as being rest masses. 00:20:06.714 --> 00:20:08.130 But I'll try to call this the rest 00:20:08.130 --> 00:20:10.780 mass because I will use the word mass in other ways. 00:20:19.490 --> 00:20:23.130 We could also relate this momentum to the velocity, 00:20:23.130 --> 00:20:27.220 and in doing that we will again encounter this factor of gamma 00:20:27.220 --> 00:20:29.900 that we found kinematically earlier. 00:20:29.900 --> 00:20:31.083 AUDIENCE: Yes, question. 00:20:31.083 --> 00:20:34.190 Did you forget to divide by C squared there? 00:20:34.190 --> 00:20:42.420 PROFESSOR: Um, yes. 00:20:42.420 --> 00:20:45.330 There's too many C's here. 00:20:45.330 --> 00:20:46.220 Absolutely. 00:20:46.220 --> 00:20:48.380 It should have units of momentum. 00:20:48.380 --> 00:20:52.030 So it should have units of mass times velocity squared. 00:20:52.030 --> 00:20:52.920 Thank you. 00:20:52.920 --> 00:20:53.420 Thank you. 00:20:59.030 --> 00:21:04.769 So, the quantity gamma, is the same quantity 00:21:04.769 --> 00:21:06.560 we encounter at the beginning of the course 00:21:06.560 --> 00:21:08.650 when we talked about time dilation and Lorentz 00:21:08.650 --> 00:21:12.070 contraction, it just depends on the velocity and approaches 00:21:12.070 --> 00:21:14.830 infinity as the velocity approaches the speed of light. 00:21:14.830 --> 00:21:19.360 And the physical momentum of a particle, relativistically 00:21:19.360 --> 00:21:23.750 is equal to gamma times M sub 0 times V. 00:21:23.750 --> 00:21:27.421 Where V is the ordinary velocity-- this 00:21:27.421 --> 00:21:29.170 is special relativity, we're not concerned 00:21:29.170 --> 00:21:32.800 with coordinate velocity versus physical velocity yet-- 00:21:32.800 --> 00:21:34.920 and gamma is that factor. 00:21:34.920 --> 00:21:39.040 So the momentum is larger than what you would get a la Newton. 00:21:39.040 --> 00:21:43.260 The energy can also be written down. 00:21:43.260 --> 00:21:48.480 And this formula is one expression 00:21:48.480 --> 00:21:51.285 we can use to find the energy in terms of the momentum. 00:21:59.380 --> 00:22:04.570 The energy in terms of the momentum 00:22:04.570 --> 00:22:15.420 is M0 C squared squared plus p squared C squared. 00:22:15.420 --> 00:22:18.830 And it also can be written in terms of the velocity 00:22:18.830 --> 00:22:22.610 as just gamma times M0 C squared where the velocity appears 00:22:22.610 --> 00:22:23.220 in the gamma. 00:22:39.390 --> 00:22:44.430 And a special case of this is when the particle is at rest. 00:22:44.430 --> 00:22:49.660 Might as well write this, the energy of a particle at rest, 00:22:49.660 --> 00:22:54.180 which we might call E0, is just M0 C squared, which gets us 00:22:54.180 --> 00:22:58.350 back to where we started with E equals MC squared. 00:23:01.624 --> 00:23:03.040 Now, I might just say a quick word 00:23:03.040 --> 00:23:05.946 about where these formulas come from, what idea underlies them. 00:23:05.946 --> 00:23:07.320 I'm not going to make any attempt 00:23:07.320 --> 00:23:10.080 to derive them because we just don't have time. 00:23:10.080 --> 00:23:13.510 It'd be easy to drive them, but we want to do other things. 00:23:13.510 --> 00:23:15.550 But logically, where they come from 00:23:15.550 --> 00:23:20.650 is simply the observation, by Einstein originally, 00:23:20.650 --> 00:23:24.290 that if one has the Lorentz transformations relating 00:23:24.290 --> 00:23:29.260 what one inertial observer sees to another inertial observer, 00:23:29.260 --> 00:23:32.425 if one used those transformations but used 00:23:32.425 --> 00:23:35.820 the Newtonian definitions of energy and momentum, 00:23:35.820 --> 00:23:38.430 then you would find immediately that if energy and momentum 00:23:38.430 --> 00:23:41.770 were conserved in one frame you then know how to calculate what 00:23:41.770 --> 00:23:44.210 happens in other frames by using the transformations, 00:23:44.210 --> 00:23:47.220 you'd find it would not be conserved other frames. 00:23:47.220 --> 00:23:49.710 So if the conservation of energy and momentum 00:23:49.710 --> 00:23:53.150 are to be a universal principle of physics, which Einstein 00:23:53.150 --> 00:23:57.080 wanted to maintain, it would be necessary to redefine 00:23:57.080 --> 00:23:58.630 energy and momentum. 00:23:58.630 --> 00:24:01.220 Now they're defined in ways so that they approach 00:24:01.220 --> 00:24:04.110 the Newtonian values for small velocities, 00:24:04.110 --> 00:24:07.030 but for velocities of the order of the speed of light 00:24:07.030 --> 00:24:08.000 they're different. 00:24:08.000 --> 00:24:10.734 And they have the property-- not completely obvious from what 00:24:10.734 --> 00:24:13.150 we wrote, well it is actually completely obvious from what 00:24:13.150 --> 00:24:14.780 we wrote-- they have the property 00:24:14.780 --> 00:24:17.770 that if it's conserved in on frame, 00:24:17.770 --> 00:24:19.420 it's conserved in all frames. 00:24:19.420 --> 00:24:21.291 And what makes it obvious-- maybe 00:24:21.291 --> 00:24:23.290 the connections between these different formulas 00:24:23.290 --> 00:24:25.400 are not obvious-- but I did tell you 00:24:25.400 --> 00:24:29.900 that p super mu transforms as a four-vector, 00:24:29.900 --> 00:24:33.100 meaning it transforms the same way as x super mu transforms, 00:24:33.100 --> 00:24:35.030 which are linear transformations. 00:24:35.030 --> 00:24:38.517 And that's enough to guarantee that if p mu is conserved 00:24:38.517 --> 00:24:40.600 in one frame, it has to be conserved in all frames 00:24:40.600 --> 00:24:43.210 because delta p mu, the change in p mu, 00:24:43.210 --> 00:24:45.815 would also be a four-vector, and if a four-vector vanishes 00:24:45.815 --> 00:24:47.705 in one frame, it vanishes in all frames. 00:24:56.940 --> 00:24:57.440 OK. 00:24:57.440 --> 00:24:59.773 Well I wanted to give you sort of a quick example of how 00:24:59.773 --> 00:25:00.925 this works in practice. 00:25:13.170 --> 00:25:15.775 So I to just talk about the energetics of a hydrogen atom. 00:25:25.770 --> 00:25:30.820 A hydrogen atom consists of a proton, 00:25:30.820 --> 00:25:38.582 with a mass that we'll call M sub p, and an electron, 00:25:38.582 --> 00:25:40.040 with the mass that we'll call M sub 00:25:40.040 --> 00:25:46.680 E. And, as, if you imagine starting with the electron 00:25:46.680 --> 00:25:50.890 and proton arbitrarily far apart and bring them together, 00:25:50.890 --> 00:25:53.492 what you can discover experimentally-- of if you know 00:25:53.492 --> 00:25:54.950 quantum mechanics you can calculate 00:25:54.950 --> 00:25:58.960 theoretically-- energy is released, 00:25:58.960 --> 00:26:01.160 because you're releasing potential energy as you 00:26:01.160 --> 00:26:03.430 bring the electron into the atom. 00:26:03.430 --> 00:26:08.490 And the amount of energy released 00:26:08.490 --> 00:26:20.070 is 13.6 electron volts. 00:26:20.070 --> 00:26:22.220 And the important E equals MC squared 00:26:22.220 --> 00:26:24.010 implication which I want to point out here 00:26:24.010 --> 00:26:26.544 is that loss of energy. 00:26:26.544 --> 00:26:28.460 This energy would be extracted from the system 00:26:28.460 --> 00:26:30.192 as you made the hydrogen atom. 00:26:30.192 --> 00:26:32.400 The fact that you've extracted energy from the system 00:26:32.400 --> 00:26:34.530 means that now the system should have less energy 00:26:34.530 --> 00:26:36.314 than it had to start with. 00:26:36.314 --> 00:26:38.230 Initially it had the rest energy of the proton 00:26:38.230 --> 00:26:42.760 and the rest of electronic, M sub p C squared and M sub E C 00:26:42.760 --> 00:26:43.620 squared. 00:26:43.620 --> 00:26:45.530 Now it has less energy by delta E, 00:26:45.530 --> 00:26:48.250 and that means it also has to have less mass. 00:26:48.250 --> 00:26:50.120 So the mass of a hydrogen atom is not 00:26:50.120 --> 00:26:52.190 the sum of the mass of the proton and electron, 00:26:52.190 --> 00:26:54.300 as it would be in Newtonian mechanics, 00:26:54.300 --> 00:26:56.940 but is less by an amount proportional 00:26:56.940 --> 00:27:01.717 to this energy given off, delta E, 13.6 electron volts. 00:27:01.717 --> 00:27:03.300 And just putting in the C squares in I 00:27:03.300 --> 00:27:07.120 hope the right places, the mass of a hydrogen atom 00:27:07.120 --> 00:27:09.780 will be equal to the mass of a proton 00:27:09.780 --> 00:27:12.040 plus the mass of an electron, but then 00:27:12.040 --> 00:27:18.800 minus the binding energy expressed in mass units-- delta 00:27:18.800 --> 00:27:19.760 E over C squared. 00:27:36.420 --> 00:27:36.920 OK. 00:27:36.920 --> 00:27:38.900 So I guess, just probably one more topic 00:27:38.900 --> 00:27:41.940 I want to talk about in terms of just basic special relativity, 00:27:41.940 --> 00:27:45.840 and this actually get's into general relativity. 00:27:45.840 --> 00:27:48.190 I wanted to find the relativistic mass 00:27:48.190 --> 00:27:53.570 of any system just being its energy divided by C squared. 00:27:58.970 --> 00:28:01.310 And this means that the relativistic mass of a particle 00:28:01.310 --> 00:28:03.960 increases with its velocity. 00:28:03.960 --> 00:28:06.060 The energy of a single moving particle 00:28:06.060 --> 00:28:08.280 is gamma times M0 C squared. 00:28:08.280 --> 00:28:09.940 That would say the by this definition 00:28:09.940 --> 00:28:15.390 the relativistic mass of that particle is gamma times M0. 00:28:15.390 --> 00:28:18.390 And I might mention that this concept of relativistic mass 00:28:18.390 --> 00:28:22.410 is disparaged in many books on special relativity. 00:28:22.410 --> 00:28:24.650 It's certainly a concept that you can do without, 00:28:24.650 --> 00:28:27.840 so people who emotionally are bothered by it 00:28:27.840 --> 00:28:29.720 can get along without it because it is just 00:28:29.720 --> 00:28:31.350 the energy divided by C squared. 00:28:31.350 --> 00:28:34.750 And in fact a lot of work in special relativity 00:28:34.750 --> 00:28:37.150 is done in units where C is equal to 1 00:28:37.150 --> 00:28:39.060 and then it is just the energy. 00:28:39.060 --> 00:28:41.000 Especially if you use C equals 1 you 00:28:41.000 --> 00:28:44.245 could dispense with this concept of relativistic mass. 00:28:44.245 --> 00:28:45.870 We're not going to be using C equals 1, 00:28:45.870 --> 00:28:48.940 so the phrase relativistic mass will 00:28:48.940 --> 00:28:50.650 allow us to abbreviate E divided by E 00:28:50.650 --> 00:28:53.460 squared in a convenient way. 00:28:53.460 --> 00:28:55.650 But the important thing is not the definitions, 00:28:55.650 --> 00:28:57.233 the important thing is what properties 00:28:57.233 --> 00:28:59.400 does this relativistic mass have, 00:28:59.400 --> 00:29:01.710 whether or not one chooses to call it relativistic mass 00:29:01.710 --> 00:29:03.920 or E divided by C squared. 00:29:03.920 --> 00:29:07.030 And it as an important property concerning 00:29:07.030 --> 00:29:10.980 the gravitational field created by matter. 00:29:10.980 --> 00:29:15.420 Now the gravitational field of a single moving object 00:29:15.420 --> 00:29:17.130 is complicated. 00:29:17.130 --> 00:29:19.880 If we were talking about, say, a moving star 00:29:19.880 --> 00:29:22.440 that was moving at a velocity large enough 00:29:22.440 --> 00:29:26.947 so we care about relativity, the way we calculate that actually 00:29:26.947 --> 00:29:28.947 as we start with the Schwarzschild metric, which 00:29:28.947 --> 00:29:32.190 will describe the metric of the star outside of the matter 00:29:32.190 --> 00:29:33.310 if it were stationary. 00:29:33.310 --> 00:29:35.101 And then you can just make a transformation 00:29:35.101 --> 00:29:35.960 to a moving frame. 00:29:35.960 --> 00:29:37.501 You're allowed to use any coordinates 00:29:37.501 --> 00:29:39.730 you want in general relativity, so transforming 00:29:39.730 --> 00:29:43.680 to coordinates that describe the moving frame is no problem. 00:29:43.680 --> 00:29:47.480 But it distorts the field in a complicated way, nonetheless, 00:29:47.480 --> 00:29:49.330 a way that you can deal with. 00:29:51.940 --> 00:29:53.690 And what you find, of course, is that what 00:29:53.690 --> 00:29:56.590 you get would be asymmetric once you transform to the moving 00:29:56.590 --> 00:30:00.520 frame, it would show the signs of the velocity 00:30:00.520 --> 00:30:03.500 that you used to transform from the original spherically 00:30:03.500 --> 00:30:06.597 symmetric Schwarzschild metric to the new frame. 00:30:06.597 --> 00:30:09.180 So the bottom line is that the gravitational field of a moving 00:30:09.180 --> 00:30:13.330 object is not isotropic, it's more complicated than that, 00:30:13.330 --> 00:30:16.310 just as the electric field would be. 00:30:16.310 --> 00:30:19.860 But if we have a gas of particles, 00:30:19.860 --> 00:30:24.160 which is pretty much what we have in the early universe. 00:30:24.160 --> 00:30:35.290 If we have a gas of particles in a box moving every which way, 00:30:35.290 --> 00:30:38.575 then if we thought of this box as being an object 00:30:38.575 --> 00:30:40.700 that we're only going to look at from the outside-- 00:30:40.700 --> 00:30:43.880 a black box in the classic use of the phrase 00:30:43.880 --> 00:30:51.320 black box-- the mass of the black box 00:30:51.320 --> 00:30:56.352 really would just be the sum of the relativistic masses 00:30:56.352 --> 00:30:57.060 of the particles. 00:31:00.250 --> 00:31:05.300 And the isotropy of the metric that any one particle would 00:31:05.300 --> 00:31:08.670 generate would be canceled by averaging or summing 00:31:08.670 --> 00:31:11.230 over all the particles going every which direction, because 00:31:11.230 --> 00:31:17.162 on average the velocity of particles inside the box is 0. 00:31:17.162 --> 00:31:19.620 So this relativistic mass, when you're talking about a gas, 00:31:19.620 --> 00:31:22.780 really is the mass per particle. 00:31:22.780 --> 00:31:24.970 And if you divide that by the volume 00:31:24.970 --> 00:31:26.820 really you do get the mass density, 00:31:26.820 --> 00:31:28.720 which is a relevant mass density in terms 00:31:28.720 --> 00:31:30.960 of talking about how this matter would 00:31:30.960 --> 00:31:33.300 generate gravitational fields. 00:31:33.300 --> 00:31:33.800 Yes. 00:31:33.800 --> 00:31:35.299 AUDIENCE: So why do some people have 00:31:35.299 --> 00:31:37.447 emotional problems with it? 00:31:37.447 --> 00:31:40.030 PROFESSOR: I think some people have emotional problems with it 00:31:40.030 --> 00:31:42.860 because when one thinks about pedagogy in a course, 00:31:42.860 --> 00:31:45.900 for example, one worries about people confusing it 00:31:45.900 --> 00:31:49.067 with the rest mass. 00:31:49.067 --> 00:31:50.150 I think that's the reason. 00:31:54.390 --> 00:31:56.950 And I guess there are other possible sources of confusion, 00:31:56.950 --> 00:31:59.020 so your question is a good one. 00:31:59.020 --> 00:32:02.340 Another source of confusion is that this mass does not 00:32:02.340 --> 00:32:04.679 fit into an F equals ma equation. 00:32:04.679 --> 00:32:06.220 So in calling this the mass you might 00:32:06.220 --> 00:32:09.920 suggest to students who aren't paying attention to every word 00:32:09.920 --> 00:32:12.880 that you say, you might go ahead and put F equals 00:32:12.880 --> 00:32:14.440 ma for this mass, that does not work. 00:32:18.420 --> 00:32:20.600 So it has some of the properties of a mass, 00:32:20.600 --> 00:32:22.369 but not-- by no means all of them. 00:32:27.719 --> 00:32:29.260 But in particular for us is important 00:32:29.260 --> 00:32:31.634 because we're going to be interested in the gravitational 00:32:31.634 --> 00:32:33.710 field of a gas and then it really 00:32:33.710 --> 00:32:35.730 is the mass that determines that. 00:32:40.640 --> 00:32:42.950 In the more formal language in general relativity, 00:32:42.950 --> 00:32:46.182 it's the mass-- it's the energy density, that 00:32:46.182 --> 00:32:48.640 appears in the equations that produce gravitational fields, 00:32:48.640 --> 00:32:50.395 and then this really is the energy density 00:32:50.395 --> 00:32:54.230 except for a factor of C squared. 00:32:54.230 --> 00:32:55.020 OK, any questions? 00:32:55.020 --> 00:32:57.680 Because now I'm going to leave this formalism 00:32:57.680 --> 00:33:02.921 and get into what role this radiation could 00:33:02.921 --> 00:33:04.045 play in the early universe. 00:33:32.800 --> 00:33:37.106 So now I'd like to talk about radiation in particular, 00:33:37.106 --> 00:33:39.450 and for now I mean electromagnetic radiation 00:33:39.450 --> 00:33:42.530 just ordinary photons. 00:33:42.530 --> 00:33:46.330 And we're not accustomed to thinking of light 00:33:46.330 --> 00:33:47.980 as having mass, but we know light 00:33:47.980 --> 00:33:51.740 has energy and energy is related to mass by a factor of C 00:33:51.740 --> 00:33:52.560 squared. 00:33:52.560 --> 00:33:57.000 So we can write down the formula that 00:33:57.000 --> 00:34:01.330 says that rho is equal to u divided by C squared. 00:34:01.330 --> 00:34:02.940 Where u is the energy density, which 00:34:02.940 --> 00:34:04.525 we know electromagnetic fields have. 00:34:10.699 --> 00:34:13.670 And rho will be the mass density of radiation. 00:34:52.710 --> 00:34:57.810 Now, photons have zero rest mass, 00:34:57.810 --> 00:35:01.630 so if we apply for example this formula for a photon 00:35:01.630 --> 00:35:04.469 we would set M sub 0 equal to 0. 00:35:04.469 --> 00:35:06.510 And we said that photons have zero rest mass what 00:35:06.510 --> 00:35:09.930 we mean is that there's no lower limit 00:35:09.930 --> 00:35:11.961 to the energy a photon can have. 00:35:11.961 --> 00:35:13.460 In general, the rest mass determines 00:35:13.460 --> 00:35:15.640 the lowest possible energy a particle could have, 00:35:15.640 --> 00:35:17.020 which is when it's at rest. 00:35:17.020 --> 00:35:19.360 Photon can never be rest and there's 00:35:19.360 --> 00:35:21.990 no lower limit to what its energy can be. 00:35:27.350 --> 00:35:36.390 So for photons, M0 is equal to 0 and that 00:35:36.390 --> 00:35:41.860 implies that the energy of the photon 00:35:41.860 --> 00:35:44.790 is just C times the magnitude of its momentum. 00:35:51.372 --> 00:35:53.955 And this formula one can derive just for electromagnetic waves 00:35:53.955 --> 00:35:56.130 is purely classical EM-- you don't 00:35:56.130 --> 00:35:58.057 have to be talking about photons, 00:35:58.057 --> 00:36:00.390 but since it's true for a classical electromagnetic wave 00:36:00.390 --> 00:36:02.890 it had better also be true for photons because we think 00:36:02.890 --> 00:36:05.210 of this classical electromagnetic waves 00:36:05.210 --> 00:36:09.340 as really being made out of photons. 00:36:09.340 --> 00:36:11.520 So the energy that exists in the universe 00:36:11.520 --> 00:36:13.970 in the form of electromagnetic radiation 00:36:13.970 --> 00:36:18.230 will have an energy density, which we know how to calculate. 00:36:18.230 --> 00:36:21.540 And we know to calculate the momentum of any given photon. 00:36:21.540 --> 00:36:24.260 Now that will average to zero, but nonetheless 00:36:24.260 --> 00:36:28.010 if we imagine talking about a box of photons, which 00:36:28.010 --> 00:36:29.745 are bouncing off the walls, the fact 00:36:29.745 --> 00:36:31.120 that each photon carries momentum 00:36:31.120 --> 00:36:34.030 means that there will be a pressure on the walls. 00:36:34.030 --> 00:36:36.450 And we will be interested in that pressure, 00:36:36.450 --> 00:36:40.040 we'll be calculating a formula for it in a minute. 00:36:40.040 --> 00:36:42.910 So, what we now want to talk about 00:36:42.910 --> 00:36:48.555 is what happens when we put a photon gas into the universe 00:36:48.555 --> 00:36:49.930 and allow the universe to expand. 00:37:24.326 --> 00:37:26.200 What happens to the radiation energy density, 00:37:26.200 --> 00:37:29.960 or equivalently mass density, as the universe expands? 00:37:35.860 --> 00:37:38.260 This turns out to be a very easy question 00:37:38.260 --> 00:37:40.770 to answer if we think of the energy 00:37:40.770 --> 00:37:43.960 as being made out of photons. 00:37:43.960 --> 00:37:47.690 We would get an equivalent identical answer 00:37:47.690 --> 00:37:52.057 if we use classical Maxwell's equations 00:37:52.057 --> 00:37:53.640 and talked about how energy density is 00:37:53.640 --> 00:37:55.954 a [INAUDIBLE] of electromagnetic fields behaved. 00:37:55.954 --> 00:37:57.870 It would be more work actually do it that way, 00:37:57.870 --> 00:38:00.140 but we would get the same answer. 00:38:00.140 --> 00:38:02.210 In terms of photos, we could simply 00:38:02.210 --> 00:38:09.010 notice that the number density of photons-- photons 00:38:09.010 --> 00:38:11.690 are not going to disappear as the universe expands, 00:38:11.690 --> 00:38:13.350 the number density will just keep 00:38:13.350 --> 00:38:16.160 the same number of photons, but as the universe expands 00:38:16.160 --> 00:38:19.172 those photons will occupy a larger volume. 00:38:19.172 --> 00:38:20.630 So it's exactly the same as what we 00:38:20.630 --> 00:38:23.380 said about non-relativistic matter, 00:38:23.380 --> 00:38:26.840 the number density of photons will fall like 1 00:38:26.840 --> 00:38:31.380 over the cube of the scale factor, which just 00:38:31.380 --> 00:38:33.035 says that photons are conserved. 00:38:41.200 --> 00:38:46.520 And the volume of any region grows like a cubed. 00:38:46.520 --> 00:38:49.477 I should also mention that I'm using gamma here. 00:38:49.477 --> 00:38:51.060 Gamma of course also sometimes meant 1 00:38:51.060 --> 00:38:53.435 over the square-root of 1 minus V squared over C squared, 00:38:53.435 --> 00:38:55.570 but besides that use, gamma is also 00:38:55.570 --> 00:38:57.340 just a label that means photons. 00:39:00.920 --> 00:39:02.820 It comes from the idea of gamma rays, 00:39:02.820 --> 00:39:05.180 but it's actually used in this context 00:39:05.180 --> 00:39:08.092 for any kind of a photon no matter 00:39:08.092 --> 00:39:09.550 what its frequency is, whether it's 00:39:09.550 --> 00:39:13.010 a gamma ray or an x-ray or visible light, or infrared. 00:39:13.010 --> 00:39:13.510 Yes. 00:39:13.510 --> 00:39:15.930 AUDIENCE: Is this assumed for time average? 00:39:15.930 --> 00:39:18.350 Because photons can be absorbed, right? 00:39:18.350 --> 00:39:20.286 I mean, at least like in a small [? slice ?] 00:39:20.286 --> 00:39:23.680 can't there be a decided non-relativistic matter? 00:39:23.680 --> 00:39:25.090 PROFESSOR: Is this true on the--? 00:39:25.090 --> 00:39:28.650 Well, the validity of this formula-- you're 00:39:28.650 --> 00:39:32.130 right this formula's not exact, photons can be absorbed. 00:39:32.130 --> 00:39:34.940 But in terms of what happens as the universe evolves, 00:39:34.940 --> 00:39:37.450 that's a very, very, very minor process, 00:39:37.450 --> 00:39:39.700 especially when we're talking about the early universe 00:39:39.700 --> 00:39:43.180 when there isn't really anything around to absorb them. 00:39:43.180 --> 00:39:46.055 So, especially for the early universe and even pretty well 00:39:46.055 --> 00:39:48.180 today, if we're talking about the cosmic background 00:39:48.180 --> 00:39:51.380 radiation, which is the bulk of the photons, 00:39:51.380 --> 00:39:55.390 this formula's a very good approximation. 00:39:55.390 --> 00:39:55.890 Yes. 00:39:55.890 --> 00:39:59.320 AUDIENCE: Even though the photons in the early universe 00:39:59.320 --> 00:40:01.280 created a lot of massive particles, 00:40:01.280 --> 00:40:05.710 [INAUDIBLE] didn't that affect the expansion? 00:40:05.710 --> 00:40:09.200 PROFESSOR: Are you're asking, will the photons 00:40:09.200 --> 00:40:11.540 be important if there's a lot of mass of particles? 00:40:11.540 --> 00:40:12.210 Is that your question? 00:40:12.210 --> 00:40:13.835 AUDIENCE: Would the photons, won't they 00:40:13.835 --> 00:40:17.060 decay into matter antimatter? 00:40:17.060 --> 00:40:20.040 PROFESSOR: Will the photon decay into matter antimatter? 00:40:20.040 --> 00:40:22.860 No, not really. 00:40:22.860 --> 00:40:26.670 It is, in principle, possible for two photons 00:40:26.670 --> 00:40:30.750 to collide and produce an electron positron pair. 00:40:30.750 --> 00:40:34.700 It's actually a rather small cross section for that. 00:40:34.700 --> 00:40:38.510 And all of these processes in the early universe 00:40:38.510 --> 00:40:40.400 will rapidly reach an equilibrium, 00:40:40.400 --> 00:40:42.880 which we'll be talking about more a little later, where 00:40:42.880 --> 00:40:45.640 there'll be just as many photons converting into E plus E 00:40:45.640 --> 00:40:48.700 minus pairs as there will be E plus E minus pairs colliding 00:40:48.700 --> 00:40:51.220 and making photons. 00:40:51.220 --> 00:40:55.360 So the early universe is assumed to reach equilibrium 00:40:55.360 --> 00:40:58.817 very quickly, and all the description we'll be giving 00:40:58.817 --> 00:41:00.650 will be a description of the universe that's 00:41:00.650 --> 00:41:03.070 in thermal equilibrium with these processes will 00:41:03.070 --> 00:41:05.110 tend to iron out. 00:41:05.110 --> 00:41:06.610 We will learn that they don't always 00:41:06.610 --> 00:41:08.693 cancel each other because the universe is cooling, 00:41:08.693 --> 00:41:11.935 and that means it can't be exactly in thermal equilibrium. 00:41:11.935 --> 00:41:14.310 And there are some cases where the effect of that cooling 00:41:14.310 --> 00:41:16.700 is significant and we'll be talking about those. 00:41:16.700 --> 00:41:18.750 But for the most part, if the cooling is slow, 00:41:18.750 --> 00:41:22.769 which it is for the most part compared to other processes, 00:41:22.769 --> 00:41:24.435 everything stays in thermal equilibrium. 00:41:28.990 --> 00:41:29.490 OK. 00:41:29.490 --> 00:41:30.190 Those are good questions. 00:41:30.190 --> 00:41:31.820 We've gotten a little ahead of what I wanted to talk about. 00:41:31.820 --> 00:41:33.882 So for now I'm just imagining a free photon gas, 00:41:33.882 --> 00:41:35.340 which is an excellent approximation 00:41:35.340 --> 00:41:36.964 for the early universe. 00:41:36.964 --> 00:41:38.630 And those photons just continue to exist 00:41:38.630 --> 00:41:41.300 as the universe expands, so their number density 00:41:41.300 --> 00:41:44.401 falls off as 1 over a cubed. 00:41:44.401 --> 00:41:45.900 But there's another affect that goes 00:41:45.900 --> 00:41:47.880 on which is that the photons are redshifting. 00:41:47.880 --> 00:41:50.050 And we already know about that, but now we're 00:41:50.050 --> 00:41:53.720 going to take it into account in terms of the energy balances. 00:41:53.720 --> 00:42:00.710 So we know that the frequency of a photon at some time t2 00:42:00.710 --> 00:42:05.410 divide by its frequency at some time t1, 00:42:05.410 --> 00:42:16.530 and here nu equals frequency, is just 00:42:16.530 --> 00:42:19.820 diluted by the expansion of the universe. 00:42:19.820 --> 00:42:22.040 This ratio is 1 over 1 plus z, where 00:42:22.040 --> 00:42:27.050 z is the redshift between these two times. 00:42:27.050 --> 00:42:30.670 But written out in more detail which 00:42:30.670 --> 00:42:32.880 is a formula we'll actually be using, 00:42:32.880 --> 00:42:36.240 is just a of t1 divided by a of t2. 00:42:39.660 --> 00:42:42.290 When the scale factor doubles, all the frequency is half. 00:42:45.180 --> 00:42:47.200 And that means that all the photons are lowering 00:42:47.200 --> 00:42:52.026 in frequency, and we also know that photons are quantized. 00:43:03.160 --> 00:43:05.850 The energy of a photon can't be any old thing, 00:43:05.850 --> 00:43:12.620 but in fact, the energy of a photon is equal to h times nu. 00:43:35.550 --> 00:43:37.705 Where little h is what's called Planck's constant. 00:43:46.770 --> 00:43:51.420 And numerically there are various units 00:43:51.420 --> 00:43:56.950 you could express it in, but it's 4.136 times 00:43:56.950 --> 00:44:01.875 10 to the minus 15th electron volt seconds. 00:44:05.360 --> 00:44:07.620 So if you measure a frequency an inverse seconds, 00:44:07.620 --> 00:44:09.875 you get an energy in electron volts from that formula. 00:44:17.040 --> 00:44:19.475 The important thing for now though is 00:44:19.475 --> 00:44:24.440 that this says that the energy of each photon-- being 00:44:24.440 --> 00:44:26.690 proportional to its frequency, and the frequency being 00:44:26.690 --> 00:44:29.760 proportional to 1 over the scale factor-- 00:44:29.760 --> 00:44:34.950 the energy of each photons is proportional to 1 over a of t. 00:44:39.820 --> 00:44:41.620 And then the total energy density 00:44:41.620 --> 00:44:44.610 of photons in photon on gas which 00:44:44.610 --> 00:44:48.360 I'll call u sub gamma, the energy density of the gas, 00:44:48.360 --> 00:44:51.940 can be thought of as the number density of photons 00:44:51.940 --> 00:44:55.440 times the energy of each photon. 00:44:55.440 --> 00:44:57.150 And the number density is falling off 00:44:57.150 --> 00:45:01.180 like 1 over a cubed, the energy is falling off like 1 over a. 00:45:01.180 --> 00:45:04.170 And therefore, this is proportional to 1 00:45:04.170 --> 00:45:05.500 over a to the fourth. 00:45:12.780 --> 00:45:16.020 So as the universe expands, the density 00:45:16.020 --> 00:45:19.230 of non-relativistic matter falls off like 1 over a cubed-- 00:45:19.230 --> 00:45:22.060 as we've been talking about some time now-- 00:45:22.060 --> 00:45:24.550 but the energy of radiation falls off faster, 00:45:24.550 --> 00:45:26.110 like 1 over a to the fourth because 00:45:26.110 --> 00:45:27.568 of the red shifting of the photons. 00:45:55.820 --> 00:46:00.170 OK, now once we know this, we can ask ourselves 00:46:00.170 --> 00:46:03.920 what happens if we look at our universe going backwards, 00:46:03.920 --> 00:46:07.440 knowing where we are now where we come from? 00:46:07.440 --> 00:46:10.340 And if the energy density of photons 00:46:10.340 --> 00:46:15.630 is falling off faster than the energy density of matter, 00:46:15.630 --> 00:46:17.750 it would mean that the ratio is getting smaller 00:46:17.750 --> 00:46:19.722 as we go forward in time. 00:46:19.722 --> 00:46:21.430 But that of course implies that the ratio 00:46:21.430 --> 00:46:24.397 gets larger as we go backwards in time. 00:46:24.397 --> 00:46:26.230 So as we go backwards in time, the radiation 00:46:26.230 --> 00:46:28.610 becomes more and more important, and there actually 00:46:28.610 --> 00:46:32.590 is going to be a time when the radiation will equal the matter 00:46:32.590 --> 00:46:36.390 and at earlier times the radiation will dominate. 00:46:36.390 --> 00:46:41.300 Today I'll just give you a number for now, 00:46:41.300 --> 00:46:42.990 we'll learn later how to calculate it, 00:46:42.990 --> 00:46:46.440 but for today the total radiation energy 00:46:46.440 --> 00:46:54.100 density in the universe is equal to 7.01 times 10 00:46:54.100 --> 00:46:57.885 to the minus 14 joules per meter cubed. 00:47:07.162 --> 00:47:09.245 And this actually includes two kinds of radiation, 00:47:09.245 --> 00:47:14.980 it includes photons and also neutrinos, 00:47:14.980 --> 00:47:17.220 which at least in the early universe 00:47:17.220 --> 00:47:20.057 behaved just like radiation. 00:47:20.057 --> 00:47:22.640 And we'll be talking more about neutrinos later so don't worry 00:47:22.640 --> 00:47:25.820 if you don't have any idea what a neutrino is. 00:47:29.240 --> 00:47:32.090 But for now it's just another contribution to the radiation, 00:47:32.090 --> 00:47:34.494 and we can measure basically this is all 00:47:34.494 --> 00:47:35.910 based on measuring the temperature 00:47:35.910 --> 00:47:38.007 of the cosmic microwave background radiation-- 00:47:38.007 --> 00:47:40.090 and we'll learn later how to make that conversion. 00:47:40.090 --> 00:47:41.140 But once you measure the temperature 00:47:41.140 --> 00:47:42.530 of the cosmic background radiation 00:47:42.530 --> 00:47:45.010 and have a theory about how many neutrinos there should be, 00:47:45.010 --> 00:47:46.718 that's actually all theoretical and we'll 00:47:46.718 --> 00:47:48.320 talk about that later as well. 00:47:48.320 --> 00:47:49.730 One can determine what the energy 00:47:49.730 --> 00:47:54.310 density of that radiation is. 00:47:54.310 --> 00:48:02.270 And it corresponds to a mass density just dividing by C 00:48:02.270 --> 00:48:13.030 squared of 7.80 times 10 to the minus 31 kilograms per meter 00:48:13.030 --> 00:48:14.234 cubed. 00:48:14.234 --> 00:48:15.650 And when I think of mass densities 00:48:15.650 --> 00:48:19.020 I always like to think of in centimeters per-- excuse me, 00:48:19.020 --> 00:48:21.684 grams per centimeter cubed because I'm 00:48:21.684 --> 00:48:24.100 used to the density of water being one gram per centimeter 00:48:24.100 --> 00:48:26.420 cubed and I like to be able to make that comparison. 00:48:26.420 --> 00:48:31.220 So just making that conversion, usually I use SI units, 00:48:31.220 --> 00:48:35.090 but some things just seem to make more sense in other units. 00:48:35.090 --> 00:48:41.026 So it's 10 to the minus 34 grams per centimeter cubed, 00:48:41.026 --> 00:48:44.270 so 10 to the minus 34, or maybe 10 to the minus 33, 00:48:44.270 --> 00:48:47.260 times the density of water. 00:48:47.260 --> 00:48:50.130 And this is incredibly low even compared 00:48:50.130 --> 00:48:53.260 to the critical density of our universe, 00:48:53.260 --> 00:48:54.710 and the actual density we know is 00:48:54.710 --> 00:48:57.020 very near this critical density. 00:48:57.020 --> 00:48:59.800 Let me remind you that the critical density 00:48:59.800 --> 00:49:03.180 we derived a formula for, and when we put numbers 00:49:03.180 --> 00:49:05.050 into that formula we found that was 00:49:05.050 --> 00:49:10.310 equal to 1.88 times little h0 squared, 00:49:10.310 --> 00:49:13.670 which is Hubble's units-- Hubble's constant in units 00:49:13.670 --> 00:49:17.820 of 100 kilometers per second per megaparsec-- 00:49:17.820 --> 00:49:21.830 I'll write that in a second-- times 10 00:49:21.830 --> 00:49:25.605 to the minus 29 grams per centimeter cubed. 00:49:45.610 --> 00:49:48.770 So just writing down the equation for little h sub 0 00:49:48.770 --> 00:49:54.390 is where capital H sub 0, Hubble's expansion rate, 00:49:54.390 --> 00:50:02.320 is equal to 100 times little h sub 0 00:50:02.320 --> 00:50:07.150 kilometers per second per megaparsec. 00:50:15.410 --> 00:50:17.298 Yep. 00:50:17.298 --> 00:50:20.868 AUDIENCE: What's the, I guess, what's 00:50:20.868 --> 00:50:23.010 the motivation of normalizing the Hubble 00:50:23.010 --> 00:50:24.930 constant in this way? 00:50:24.930 --> 00:50:26.900 PROFESSOR: Well, I think the real motivation is 00:50:26.900 --> 00:50:30.780 that the astronomers like these peculiar units of kilometers 00:50:30.780 --> 00:50:34.230 per second per megaparsec, but if your favorite unit is called 00:50:34.230 --> 00:50:36.740 a kilometer per second per megaparsec 00:50:36.740 --> 00:50:39.670 you don't want to have to say those units very often. 00:50:39.670 --> 00:50:42.600 So H0 is dimensionless, so it's a dimensionless way 00:50:42.600 --> 00:50:44.160 of talking about the Hubble constant. 00:50:49.006 --> 00:50:50.630 But that's the only importance, there's 00:50:50.630 --> 00:50:52.930 no real-- no deep significance to it. 00:50:52.930 --> 00:50:56.750 But it's a standard notation, so it's worth knowing. 00:50:56.750 --> 00:51:02.820 And then finally, we can write down 00:51:02.820 --> 00:51:10.320 how much the radiation contributes to omega-- omega 00:51:10.320 --> 00:51:14.510 sub r, r is going to indicate radiation. 00:51:14.510 --> 00:51:17.164 The notation, by the way, will be-- 00:51:17.164 --> 00:51:19.390 and I realized I've already violated that notation. 00:51:19.390 --> 00:51:22.551 This really should have been r. 00:51:22.551 --> 00:51:26.410 Well, I'm not going to change it, it'll get too messy. 00:51:26.410 --> 00:51:29.080 But I'm going to start using the notation where gamma indicates 00:51:29.080 --> 00:51:32.199 photons and little r indicates radiation. 00:51:32.199 --> 00:51:34.740 And the difference is that there are other kinds of radiation 00:51:34.740 --> 00:51:37.130 besides photons, in particular we've already 00:51:37.130 --> 00:51:41.020 added in neutrinos in part of what we're calling radiation. 00:51:41.020 --> 00:51:44.190 So omega sub r, which now includes photons and neutrinos, 00:51:44.190 --> 00:51:47.560 is just defined to be the mass density radiation divided 00:51:47.560 --> 00:51:49.650 by the critical density. 00:51:49.650 --> 00:51:52.830 And that turns out, when you combine these numbers, 00:51:52.830 --> 00:52:00.035 to be 4.15 times 10 to the minus 5 little h0 to the minus 2. 00:52:03.870 --> 00:52:13.230 And then for h0 equals 0.67, which is the Planck satellite 00:52:13.230 --> 00:52:17.460 value for little h0, we finally get 00:52:17.460 --> 00:52:28.270 omega sub r is equal to 9.2 times 10 to the minus 5. 00:52:28.270 --> 00:52:30.500 So roughly 10 to the minus 4. 00:52:30.500 --> 00:52:33.110 The fraction of the mass density, or energy density 00:52:33.110 --> 00:52:39.020 today is about 10 to the minus 4 fraction and radiation. 00:52:39.020 --> 00:52:41.620 And actually as I write this, this actually 00:52:41.620 --> 00:52:46.000 calls to mind another reason for defining little h sub 0, which 00:52:46.000 --> 00:52:48.920 is that if you write formulas in terms of little h sub 0, 00:52:48.920 --> 00:52:51.640 they remain valid between one year and next year. 00:52:51.640 --> 00:52:55.100 The observational value of the Hubble parameter 00:52:55.100 --> 00:52:58.510 is still floating around and differs, 00:52:58.510 --> 00:53:00.800 for example, every time I teach this course. 00:53:00.800 --> 00:53:02.640 So the formulas in terms of little h sub 0 00:53:02.640 --> 00:53:06.380 stay, and then you plug in the current value of little h sub 0 00:53:06.380 --> 00:53:12.250 to get the best value that one can currently write down. 00:53:12.250 --> 00:53:16.135 Now we know the Hubble constant to within a few percent, which 00:53:16.135 --> 00:53:18.960 is much better than it used to be, but it's still floating. 00:53:18.960 --> 00:53:22.340 The Planck value was somewhat lower than the previous except 00:53:22.340 --> 00:53:28.520 a value, which was about 0.70. 00:53:28.520 --> 00:53:29.430 OK. 00:53:29.430 --> 00:53:34.400 So we know enough information to extrapolate backwards 00:53:34.400 --> 00:53:37.120 and calculate when this radiation would've 00:53:37.120 --> 00:53:39.960 equaled the energy density of matter. 00:53:55.667 --> 00:53:57.250 Because we know how the ratio changes. 00:53:57.250 --> 00:53:59.740 It changes by a factor of a, the scale factor, 00:53:59.740 --> 00:54:02.130 because the energy density of non-relativistic matter 00:54:02.130 --> 00:54:03.892 is falling off like 1 over a cubed. 00:54:03.892 --> 00:54:06.100 The energy density of radiation is flowing off like 1 00:54:06.100 --> 00:54:07.240 over a to the fourth. 00:54:07.240 --> 00:54:09.340 So the ratio between them just changes 00:54:09.340 --> 00:54:15.260 by a factor of a, decreasing as a gets larger. 00:54:15.260 --> 00:54:23.390 So we can write RHO radiation of t divided by RHO matter of t. 00:54:26.740 --> 00:54:28.750 The m there means non-relativistic matter. 00:54:41.740 --> 00:54:46.640 This is just equal to the current value. 00:54:46.640 --> 00:54:52.290 And the current value is gotten by taking that number 00:54:52.290 --> 00:55:02.320 and, where I forgot to write down is that omega matter today 00:55:02.320 --> 00:55:09.780 is about 0.30. 00:55:09.780 --> 00:55:16.960 So the ratio of the two, omega radiation over omega matter, 00:55:16.960 --> 00:55:24.000 which is the same as RHO radiation over RHO matter, 00:55:24.000 --> 00:55:35.380 is about 3.1 times 10 to the minus 4. 00:55:38.700 --> 00:55:42.160 And that number is about to appear in this equation. 00:55:42.160 --> 00:55:44.280 If we want the ratio as a function of time, 00:55:44.280 --> 00:55:51.500 we start with this value today, 3.1 times 10 to the minus 4, 00:55:51.500 --> 00:55:56.050 and then we can just multiply that by the scale factor 00:55:56.050 --> 00:55:59.710 today divided by the scale factor at the time 00:55:59.710 --> 00:56:02.410 that we want to know it, because we know it falls off as 1 00:56:02.410 --> 00:56:03.420 over the scale factor. 00:56:08.080 --> 00:56:10.090 And by putting an a of t0 here, this just 00:56:10.090 --> 00:56:11.840 guarantees that if we let t equal t0, 00:56:11.840 --> 00:56:14.630 we get this number, which is the right ratio for today. 00:56:21.880 --> 00:56:22.380 OK. 00:56:22.380 --> 00:56:25.250 Now it's just a matter of arithmetic. 00:56:25.250 --> 00:56:37.090 We also know that for a matter-dominated universe. 00:56:37.090 --> 00:56:42.910 And for now, we're going to estimate when the radiation 00:56:42.910 --> 00:56:44.881 energy will equal the matter density. 00:56:44.881 --> 00:56:46.130 This will only be an estimate. 00:56:46.130 --> 00:56:47.713 We're going to estimate it by assuming 00:56:47.713 --> 00:56:50.230 that we can approximate the universe as matter dominated 00:56:50.230 --> 00:56:55.500 between now, all the way back until that time. 00:56:55.500 --> 00:56:58.110 There are two errors in that calculation. 00:56:58.110 --> 00:57:01.600 We're not taking into account here the era of acceleration 00:57:01.600 --> 00:57:03.600 where dark energy is playing a significant role. 00:57:03.600 --> 00:57:05.110 We'll learn later how to do that. 00:57:05.110 --> 00:57:08.110 And also, as we approach this time when they're 00:57:08.110 --> 00:57:10.960 making equal contributions, we will run into a regime 00:57:10.960 --> 00:57:13.050 where the contribution of the radiation 00:57:13.050 --> 00:57:14.930 itself will be relevant. 00:57:14.930 --> 00:57:17.310 So this is only an estimate. 00:57:17.310 --> 00:57:20.257 But what we're going to do is we're 00:57:20.257 --> 00:57:21.840 going to assume that we can treat this 00:57:21.840 --> 00:57:25.530 as a matter-dominated universe with a of t proportional 00:57:25.530 --> 00:57:28.524 to the 2/3. 00:57:28.524 --> 00:57:30.440 And then we can plug numbers into this formula 00:57:30.440 --> 00:57:32.960 and ask, when was the ratio 1? 00:57:32.960 --> 00:57:34.720 And when the ratio was 1, we call 00:57:34.720 --> 00:57:39.100 that the time of equality, using the subscript EQ for equality 00:57:39.100 --> 00:57:43.110 to indicate anything having to do with that crossing point. 00:57:43.110 --> 00:57:49.590 And what we find is that the z of equality, 00:57:49.590 --> 00:57:57.085 and z is just the ratio of the a's, is-- according 00:57:57.085 --> 00:58:01.120 to this calculation, it would be 3.1 times 10 00:58:01.120 --> 00:58:05.230 to the minus 4 minus 1. 00:58:05.230 --> 00:58:07.604 I fibbed when I said that z is the ratio of the a's. 00:58:07.604 --> 00:58:08.770 It's offset by a little bit. 00:58:08.770 --> 00:58:11.470 It's 1 plus z that's the ratio of the a's. 00:58:11.470 --> 00:58:14.440 And that's why there's a minus 1 there. 00:58:14.440 --> 00:58:19.900 And numerically, this is about equal to 32,000. 00:58:19.900 --> 00:58:22.670 So if we look back in the history of the universe, 00:58:22.670 --> 00:58:26.870 we can define looking back in terms of the redshift. 00:58:26.870 --> 00:58:31.650 If you look back to a redshift of 3,200, we get to the time 00:58:31.650 --> 00:58:35.210 when matter and radiation had the same energy density. 00:58:38.140 --> 00:58:41.916 And we can know what time that is if we assume t 00:58:41.916 --> 00:58:44.966 to the 2/3, which again is only a crude approximation. 00:58:44.966 --> 00:58:47.340 We don't necessarily expect to get the right answer here. 00:58:47.340 --> 00:58:49.423 But we expect to get the right order of magnitude. 00:58:57.410 --> 00:59:02.500 So t-equality, according to this situation, 00:59:02.500 --> 00:59:21.100 would be about 75,000 years after the Big Bang-- just 00:59:21.100 --> 00:59:23.150 converting the scale factor that we just 00:59:23.150 --> 00:59:26.823 calculated to a time using that formula, treating this as t 00:59:26.823 --> 00:59:29.000 to the 2/3. 00:59:29.000 --> 00:59:32.080 So this says that about 75,000 years after the Big 00:59:32.080 --> 00:59:35.341 Bang, the energy densities of matter and radiation 00:59:35.341 --> 00:59:35.840 were equal. 00:59:35.840 --> 00:59:37.256 In the earlier times, the universe 00:59:37.256 --> 00:59:38.420 was radiation dominated. 00:59:38.420 --> 00:59:41.521 The radiation exceeded the matter in its energy density. 00:59:41.521 --> 00:59:42.020 Yes? 00:59:42.020 --> 00:59:45.020 AUDIENCE: It seems like zEQ is relatively about 31 00:59:45.020 --> 00:59:45.520 [INAUDIBLE]. 00:59:52.404 --> 00:59:54.320 PROFESSOR: You might be right if we would just 00:59:54.320 --> 00:59:55.278 look to these formulas. 00:59:55.278 --> 00:59:57.460 When I calculated this at home, I 00:59:57.460 --> 00:59:59.170 kept more decimal places all the way 00:59:59.170 --> 01:00:03.120 and rounded off each answer to one significant figure. 01:00:03.120 --> 01:00:05.750 And that's not the same as taking the answer to one 01:00:05.750 --> 01:00:07.380 significant figure and calculating 01:00:07.380 --> 01:00:09.320 and then rounding off to ones in every figure. 01:00:09.320 --> 01:00:13.610 So I think there's always an ambiguity of, roughly speaking, 01:00:13.610 --> 01:00:15.794 1 in the last decimal place whenever 01:00:15.794 --> 01:00:16.960 you're rounding numbers off. 01:00:16.960 --> 01:00:18.932 AUDIENCE: So we just plug that [INAUDIBLE] 01:00:23.370 --> 01:00:25.572 PROFESSOR: 3,220-- starting with this 01:00:25.572 --> 01:00:27.780 or starting with a more accurate number than the 3.1. 01:00:27.780 --> 01:00:28.780 AUDIENCE: Just 1/3.1. 01:00:28.780 --> 01:00:29.660 PROFESSOR: 1/3.1. 01:00:29.660 --> 01:00:30.160 OK. 01:00:30.160 --> 01:00:31.370 So, OK? 01:00:31.370 --> 01:00:34.256 AUDIENCE: Sorry. 01:00:34.256 --> 01:00:37.238 AUDIENCE: Wait no, [INAUDIBLE] like we have 1 01:00:37.238 --> 01:00:38.585 over 3 times 10 negative 4. 01:00:38.585 --> 01:00:40.040 That's 3 times 10-- 01:00:40.040 --> 01:00:41.156 PROFESSOR: It's 3.1. 01:00:41.156 --> 01:00:42.560 AUDIENCE: You have 1 over 3.1. 01:00:42.560 --> 01:00:43.476 PROFESSOR: 1 over 3.1. 01:00:43.476 --> 01:00:44.747 So you have to divide. 01:00:44.747 --> 01:00:46.118 AUDIENCE:I do that all the time. 01:00:46.118 --> 01:00:48.414 AUDIENCE: [INAUDIBLE] 01:00:48.414 --> 01:00:50.080 PROFESSOR: So apparently it's even right 01:00:50.080 --> 01:00:52.100 if you just calculate with that. 01:00:52.100 --> 01:00:54.300 But there is actually some ambiguity. 01:00:54.300 --> 01:00:57.630 The numbers I'm giving you probably 01:00:57.630 --> 01:01:00.620 have some uncertainty of 1 in the last digit, 01:01:00.620 --> 01:01:03.880 depending on how you calculate. 01:01:03.880 --> 01:01:05.610 But the number they give you, I think 01:01:05.610 --> 01:01:08.600 they're the ones that you get if you start 01:01:08.600 --> 01:01:13.524 using this and the 0.67 and from then on do everything 01:01:13.524 --> 01:01:14.940 to large numbers in decimal places 01:01:14.940 --> 01:01:16.280 and round off at each stage. 01:01:16.280 --> 01:01:18.639 You'll get the numbers I've given you. 01:01:18.639 --> 01:01:20.430 In any case, all these are really, at best, 01:01:20.430 --> 01:01:21.450 order of magnitude estimates. 01:01:21.450 --> 01:01:24.110 So worrying about whether or not the last figure is accurate 01:01:24.110 --> 01:01:26.570 is not a big deal. 01:01:26.570 --> 01:01:27.737 Yes? 01:01:27.737 --> 01:01:29.195 AUDIENCE: I don't really understand 01:01:29.195 --> 01:01:32.093 how you got t [INAUDIBLE] without telling was a of t dot 01:01:32.093 --> 01:01:33.059 is [INAUDIBLE]. 01:01:35.626 --> 01:01:36.500 PROFESSOR: I'm sorry. 01:01:36.500 --> 01:01:38.166 There is actually a piece of information 01:01:38.166 --> 01:01:39.520 I used I forgot to write here. 01:01:39.520 --> 01:01:41.140 You're absolutely right. 01:01:41.140 --> 01:01:46.654 I used t0 is equal to 13.8 times 10 to the nine years. 01:01:46.654 --> 01:01:48.320 And then everything can be related to t0 01:01:48.320 --> 01:01:51.214 if you know how things are proportional to t. 01:01:51.214 --> 01:01:52.130 You're absolute right. 01:01:52.130 --> 01:01:53.910 I did not give you all the information 01:01:53.910 --> 01:01:55.201 necessary for that calculation. 01:01:57.970 --> 01:01:58.940 Now I think I have. 01:01:58.940 --> 01:02:01.000 I haven't done the arithmetic for you. 01:02:01.000 --> 01:02:04.361 But otherwise it's all there. 01:02:04.361 --> 01:02:04.860 OK. 01:02:04.860 --> 01:02:10.960 Now I might mention that in Ryden's book, 01:02:10.960 --> 01:02:14.610 she does the calculation taking into account everything-- 01:02:14.610 --> 01:02:17.660 matter, radiation, cosmological constants. 01:02:17.660 --> 01:02:28.100 And her number for t-equality is 47,000 years, 01:02:28.100 --> 01:02:30.980 which verifies that we have the right order of magnitude. 01:02:30.980 --> 01:02:32.480 And actually, the biggest difference 01:02:32.480 --> 01:02:34.840 between her number and my number is not 01:02:34.840 --> 01:02:37.090 that she's taken into account these more sophisticated 01:02:37.090 --> 01:02:39.330 things, but rather that she used a different value 01:02:39.330 --> 01:02:41.604 for the Hubble expansion rate than I'm using. 01:02:41.604 --> 01:02:44.020 She's using a value that was current at the time she wrote 01:02:44.020 --> 01:02:47.350 her book, which was like '72, I think. 01:02:47.350 --> 01:02:50.339 h0 equals 0.72 instead of [INAUDIBLE] 0.67. 01:02:50.339 --> 01:02:52.380 And that does make a significant difference here. 01:02:55.020 --> 01:02:57.700 But either of these numbers are, I think, probably 01:02:57.700 --> 01:03:00.420 within the range of uncertainty of when it really happened. 01:03:00.420 --> 01:03:05.450 But it's on that scale, on the scale of 50,000 years, 100,000 01:03:05.450 --> 01:03:08.980 years, something of that order. 01:03:08.980 --> 01:03:11.260 So there was a significantly long period 01:03:11.260 --> 01:03:13.260 compared to human lifetime when the universe was 01:03:13.260 --> 01:03:14.630 radiation dominated. 01:03:14.630 --> 01:03:17.340 But it's a very small fraction of the overall history 01:03:17.340 --> 01:03:18.970 of the universe, but nonetheless does 01:03:18.970 --> 01:03:23.340 have important features that happened during that time 01:03:23.340 --> 01:03:23.970 period. 01:03:23.970 --> 01:03:25.860 Now, if we want to understand those features, 01:03:25.860 --> 01:03:29.290 we have to understand how a radiation-dominated universe 01:03:29.290 --> 01:03:31.830 evolves, which is what we're going to get to next. 01:03:48.641 --> 01:03:49.140 OK. 01:03:49.140 --> 01:03:50.724 The next little chapter than I'm going 01:03:50.724 --> 01:03:56.600 to be talking about-- the dynamics 01:03:56.600 --> 01:03:58.295 of a radiation-dominated universe. 01:04:09.895 --> 01:04:11.520 This is a chapter that you more or less 01:04:11.520 --> 01:04:13.487 get to work out the equations for yourself 01:04:13.487 --> 01:04:14.820 on one of the homework problems. 01:04:14.820 --> 01:04:16.780 That's part of this week's set. 01:04:16.780 --> 01:04:21.460 So I will try here to outline the logic. 01:04:21.460 --> 01:04:24.550 But because all the calculations are in the homework, 01:04:24.550 --> 01:04:28.300 I will basically skip the calculations themselves, 01:04:28.300 --> 01:04:30.340 and let you do them for yourselves 01:04:30.340 --> 01:04:32.460 as part of the homework. 01:04:32.460 --> 01:04:41.670 But where we start is we have written down Friedman equations 01:04:41.670 --> 01:04:43.730 for the matter-dominated case. 01:04:43.730 --> 01:04:46.095 And I'll start by reminding us what those were. 01:05:09.690 --> 01:05:12.870 And then in addition to these two equations, which 01:05:12.870 --> 01:05:16.120 describe our expanding universe, which we've derived sometime 01:05:16.120 --> 01:05:19.540 ago for-- and to remind us here, this 01:05:19.540 --> 01:05:23.035 is for a matter-dominated universe. 01:05:27.100 --> 01:05:30.070 And matter-dominated means non-relativistic matter 01:05:30.070 --> 01:05:31.400 dominated. 01:05:31.400 --> 01:05:34.230 And going along with these equations, 01:05:34.230 --> 01:05:38.970 we also know that RHO of t for non-relativistic matter 01:05:38.970 --> 01:05:42.340 falls off like one over a cubed of t. 01:05:47.190 --> 01:05:50.740 This can be converted into a differential equation for RHO. 01:05:50.740 --> 01:05:53.840 That is, we can calculate Rho dot from this equation. 01:05:53.840 --> 01:05:57.270 And the way to see that is probably most easily 01:05:57.270 --> 01:06:06.410 to start on a new blackboard and write that equation not as 01:06:06.410 --> 01:06:08.620 a proportionality, since it's hard to differentiate 01:06:08.620 --> 01:06:10.575 a proportionality, but we can write it 01:06:10.575 --> 01:06:13.430 in an arbitrary constant of proportionality. 01:06:13.430 --> 01:06:15.490 And then it becomes an equality. 01:06:15.490 --> 01:06:18.080 So I'm going to write the equation as RHO of t 01:06:18.080 --> 01:06:23.680 is equal to some constant, b, divided by a cubed of t. 01:06:23.680 --> 01:06:25.890 And this we know how to differentiate. 01:06:25.890 --> 01:06:29.850 We can write Rho dot is equal to minus b over a 01:06:29.850 --> 01:06:33.050 to the fourth of t times a dot. 01:06:36.810 --> 01:06:41.060 And that is equal to minus 3. 01:06:41.060 --> 01:06:43.360 I'm sorry--there's a 3 here. 01:06:43.360 --> 01:06:48.130 Minus 3 times a dot over a times the original RHO. 01:06:52.130 --> 01:06:53.860 So we can forget the intermediate steps, 01:06:53.860 --> 01:06:56.860 and we just arrived at the equation 01:06:56.860 --> 01:07:01.756 that RHO dot is equal to minus 3 a dot over a times RHO. 01:07:01.756 --> 01:07:03.380 And we can think of that as going along 01:07:03.380 --> 01:07:05.630 with equations one and two. 01:07:05.630 --> 01:07:07.850 Maybe I'll even give that a number. 01:07:07.850 --> 01:07:10.770 Equation three will be RHO dot is 01:07:10.770 --> 01:07:15.285 equal to minus 3 a dot over a times RHO. 01:07:18.850 --> 01:07:21.970 Now for radiation, there will be a 4 here. 01:07:21.970 --> 01:07:24.540 The 4 will arrive the same way as the 3 arrives there. 01:07:24.540 --> 01:07:28.740 It's just the power that appeared in the factor of a. 01:07:28.740 --> 01:07:30.540 So for radiation, this last formula we know 01:07:30.540 --> 01:07:33.480 is going to be modified, which is the key point. 01:07:33.480 --> 01:07:35.920 Note that these three formulas are not 01:07:35.920 --> 01:07:38.750 independent of each other. 01:07:38.750 --> 01:07:42.050 If we know, for example, equation one, which 01:07:42.050 --> 01:07:44.006 is an equation for a dot, we could 01:07:44.006 --> 01:07:45.630 differentiate that with respect to time 01:07:45.630 --> 01:07:47.980 and get an equation for a double dot. 01:07:47.980 --> 01:07:51.120 When we do that, everything has to be differentiated. 01:07:51.120 --> 01:07:53.390 So it involves differentiating a, 01:07:53.390 --> 01:07:57.170 but that just expresses things in terms of derivatives of a. 01:07:57.170 --> 01:07:59.980 But the new quantity that gets introduced is RHO. 01:07:59.980 --> 01:08:02.570 If we wanted to differentiate this equation with respect 01:08:02.570 --> 01:08:06.300 to time, we have to know what RHO dot is. 01:08:06.300 --> 01:08:07.130 But we do. 01:08:07.130 --> 01:08:09.850 That's what equation three tells us. 01:08:09.850 --> 01:08:13.620 So we can differentiate equation one, use equation three, 01:08:13.620 --> 01:08:15.661 and we can derive an equation for a double dot. 01:08:15.661 --> 01:08:17.244 And if these equations are consistent, 01:08:17.244 --> 01:08:18.500 it'd better be equation two. 01:08:18.500 --> 01:08:19.279 And it will be. 01:08:19.279 --> 01:08:21.500 You can check it. 01:08:21.500 --> 01:08:23.790 And actually, I think any two of these equations 01:08:23.790 --> 01:08:25.760 can be used to derive the third. 01:08:25.760 --> 01:08:27.414 Those equations just are-- really 01:08:27.414 --> 01:08:28.830 a set of two independent equations 01:08:28.830 --> 01:08:30.640 and one dependent equation. 01:08:30.640 --> 01:08:35.210 You can shuffle it any way you want. 01:08:35.210 --> 01:08:39.340 But, now what we want to do is to consider 01:08:39.340 --> 01:08:40.779 a different kind of matter. 01:08:40.779 --> 01:08:42.380 Instead of non-relativistic matter, 01:08:42.380 --> 01:08:45.569 we're considering photon matter. 01:08:45.569 --> 01:08:47.210 And in particular, we know that it's 01:08:47.210 --> 01:08:48.615 going to change equation three. 01:08:56.939 --> 01:09:01.220 So for radiation, 3 gets modified 01:09:01.220 --> 01:09:10.620 into 3 prime, which is the equation that 01:09:10.620 --> 01:09:16.890 says that RHO dot is equal to minus 4 a dot over a times RHO. 01:09:36.630 --> 01:09:41.010 So how are we going to fix these equations? 01:09:41.010 --> 01:09:43.080 Now they're inconsistent. 01:09:43.080 --> 01:09:46.521 If we change three and don't change either one or two, 01:09:46.521 --> 01:09:48.520 we know that we're inconsistent, because any two 01:09:48.520 --> 01:09:51.649 of those equations can be used to derive the third. 01:09:51.649 --> 01:09:53.210 So we're in trouble. 01:09:53.210 --> 01:09:55.230 Either equations one or two will also 01:09:55.230 --> 01:09:58.260 have to be modified if we're going to modify equation three. 01:10:15.660 --> 01:10:16.160 OK. 01:10:16.160 --> 01:10:20.040 Before we go on, I'd like to say a little more about why 01:10:20.040 --> 01:10:23.032 this equation is different from that equation. 01:10:23.032 --> 01:10:24.490 One might think that it should just 01:10:24.490 --> 01:10:26.412 be governed by the conservation of energy. 01:10:26.412 --> 01:10:28.120 After all, we just write down an equation 01:10:28.120 --> 01:10:32.540 for RHO dot-- how energy density changes with time. 01:10:32.540 --> 01:10:35.030 Shouldn't conservation of energy determine that? 01:10:35.030 --> 01:10:36.850 It does. 01:10:36.850 --> 01:10:39.660 But there is an extra element to conservation of energy 01:10:39.660 --> 01:10:42.950 that we need to take into account, 01:10:42.950 --> 01:10:45.540 and that is the pressure of the gas affects 01:10:45.540 --> 01:10:48.510 what happens to its energy as it expands. 01:10:48.510 --> 01:10:50.470 So before we get back to the early universe, 01:10:50.470 --> 01:10:52.680 I just want to consider a gas in a piston chamber. 01:10:56.180 --> 01:11:01.990 And I'm going to let the piston have an area a, and inside 01:11:01.990 --> 01:11:05.750 we're going to have a volume v. Just to define our notation. 01:11:09.260 --> 01:11:11.340 If we have a gas inside a piston chamber 01:11:11.340 --> 01:11:15.390 and let the piston chamber enlarge by pulling out 01:11:15.390 --> 01:11:21.330 on the piston, the gas has a pressure, in general, 01:11:21.330 --> 01:11:24.570 and that pressure will exert a force on the piston. 01:11:24.570 --> 01:11:27.910 So if I allow the piston to move to the right, 01:11:27.910 --> 01:11:30.150 that gas will be exerting a force 01:11:30.150 --> 01:11:32.680 on the piston in the direction that it's moving. 01:11:32.680 --> 01:11:35.920 And that means the gas will be doing work on the piston. 01:11:35.920 --> 01:11:39.080 So, by our ordinary notions of Newtonian conservation 01:11:39.080 --> 01:11:43.310 of energy, we would know that the gas would lose energy, 01:11:43.310 --> 01:11:47.490 and we can even calculate how much energy it loses. 01:11:47.490 --> 01:11:51.870 And the formula is easy enough to get in a Newtonian context. 01:11:51.870 --> 01:11:55.280 It's just du is equal to minus the pressure 01:11:55.280 --> 01:11:58.970 of the gas times the change in volume. 01:11:58.970 --> 01:12:00.550 A famous formula. 01:12:00.550 --> 01:12:02.550 And this just comes about by saying 01:12:02.550 --> 01:12:06.040 that the work that's done is the force times the distance. 01:12:06.040 --> 01:12:08.850 The force is the pressure times the area. 01:12:08.850 --> 01:12:12.710 And the volume is the area times the distance. 01:12:12.710 --> 01:12:14.460 And putting those things together, 01:12:14.460 --> 01:12:16.204 you get this formula immediately. 01:12:16.204 --> 01:12:18.120 Now this formula is actually much more general 01:12:18.120 --> 01:12:21.310 than the quasi derivation that I just showed. 01:12:21.310 --> 01:12:23.750 It works no matter what the shape of the gas is. 01:12:23.750 --> 01:12:25.740 If you put a gas in any kind of a container 01:12:25.740 --> 01:12:31.560 and let that container enlarge, even in an irregular way, 01:12:31.560 --> 01:12:33.770 the work that the gas will do will always 01:12:33.770 --> 01:12:35.620 be equal to minus the pressure of the gas 01:12:35.620 --> 01:12:39.144 times the change in the volume. 01:12:39.144 --> 01:12:40.810 We can apply this to the early universe. 01:12:40.810 --> 01:12:43.780 It actually works. 01:12:43.780 --> 01:12:45.455 The difference between our two cases 01:12:45.455 --> 01:12:48.696 is that our non-relativistic matter has no pressure at all. 01:12:48.696 --> 01:12:50.820 We're just talking about particles sitting at rest. 01:12:50.820 --> 01:12:52.420 They're not bouncing off of any walls. 01:12:52.420 --> 01:12:54.249 They're not creating any pressure, 01:12:54.249 --> 01:12:56.290 while the photons are moving around all the time. 01:12:56.290 --> 01:12:57.820 And if you imagine a box of them, 01:12:57.820 --> 01:12:59.820 they'd be hitting against the walls of that box, 01:12:59.820 --> 01:13:00.680 exerting a pressure. 01:13:05.990 --> 01:13:08.170 And we're now in a position to relate the pressure 01:13:08.170 --> 01:13:11.460 to the difference between the 3's and the 4's that 01:13:11.460 --> 01:13:15.955 appear in those two equations for RHO dot in the two cases. 01:13:22.000 --> 01:13:27.190 To apply this naive idea to a piece of the universe, 01:13:27.190 --> 01:13:36.800 we can imagine choosing-- we're going to choose some fixed 01:13:36.800 --> 01:13:42.370 volume in our co-moving coordinate system. 01:13:53.842 --> 01:13:55.800 So our box, the volume that we're talking about 01:13:55.800 --> 01:13:57.633 will actually be expanding with the universe 01:13:57.633 --> 01:13:59.900 but be fixed in co-moving coordinates. 01:13:59.900 --> 01:14:08.410 And the physical volume therefore of our box 01:14:08.410 --> 01:14:13.590 will be a cubed of t times the coordinate volume 01:14:13.590 --> 01:14:18.560 of the box-- the volume and not just cubed. 01:14:18.560 --> 01:14:20.610 And this volume will be independent of time. 01:14:20.610 --> 01:14:21.911 There's time dependents there. 01:14:21.911 --> 01:14:23.160 There's time dependents there. 01:14:23.160 --> 01:14:25.225 The physical volume of our box will be enlarging. 01:14:28.870 --> 01:14:33.540 The total energy in our box, the total gas energy, 01:14:33.540 --> 01:14:35.840 which I'll call capital U, will just 01:14:35.840 --> 01:14:40.180 be the physical volume times the energy density. 01:14:40.180 --> 01:14:42.530 The energy density is energy per physical volume. 01:14:46.380 --> 01:14:50.660 And we can now apply this formula 01:14:50.660 --> 01:14:55.000 using this U and this v. 01:14:55.000 --> 01:14:57.076 And here again is one of these cases where 01:14:57.076 --> 01:14:58.950 I'm going to be skipping steps because you're 01:14:58.950 --> 01:15:01.160 going to be doing it in detail on the homework. 01:15:01.160 --> 01:15:03.140 But by putting these equations together, 01:15:03.140 --> 01:15:08.680 what you'll find is that d dt of a cubed times RHO times c 01:15:08.680 --> 01:15:14.080 squared-- this is just d dt of a cubed times the energy 01:15:14.080 --> 01:15:18.170 density-- basically, the left hand side 01:15:18.170 --> 01:15:25.890 of that equation divided by dt and divided by v coordinates-- 01:15:25.890 --> 01:15:32.881 is equal to minus p times d dt of a cubed. 01:15:32.881 --> 01:15:35.130 And this is just the PDV term from the right hand side 01:15:35.130 --> 01:15:38.117 of that equation rewritten in terms of the variables. 01:15:38.117 --> 01:15:39.700 And you'll be doing this for homework. 01:15:39.700 --> 01:15:42.609 I'm just getting straight the factors 01:15:42.609 --> 01:15:43.900 to make sure I have them right. 01:15:58.021 --> 01:15:58.520 OK. 01:15:58.520 --> 01:16:00.311 Reshuffling that equation-- and again, this 01:16:00.311 --> 01:16:02.140 is a homework problem-- you can turn that 01:16:02.140 --> 01:16:04.590 into an equation for RHO dot. 01:16:04.590 --> 01:16:09.120 And what you'll get is minus 3 a dot 01:16:09.120 --> 01:16:15.420 over a times RHO plus p over c squared. 01:16:24.690 --> 01:16:27.110 And now we can see how our two cases emerge. 01:16:27.110 --> 01:16:29.134 If the pressure is 0, we get minus 3 a dot 01:16:29.134 --> 01:16:30.550 over 8 times RHO, which is what we 01:16:30.550 --> 01:16:32.700 had for non-relativistic matter. 01:16:32.700 --> 01:16:34.660 And the photon gas is going to have a pressure, 01:16:34.660 --> 01:16:36.285 and we could read off from this formula 01:16:36.285 --> 01:16:39.370 to know what the pressure has to be to turn the 3 into a 4. 01:16:39.370 --> 01:16:44.290 The pressure has to be a factor of a third. 01:16:44.290 --> 01:16:46.810 So you determine for this that the pressure for light 01:16:46.810 --> 01:16:53.050 is 1/3 of the energy density, or 1/3 times RHO c squared. 01:16:53.050 --> 01:16:55.869 And that's what you need to turn the 3 into a 4. 01:16:55.869 --> 01:16:57.910 So we now have indirectly calculated the pressure 01:16:57.910 --> 01:17:00.076 of light, and this agrees with any other calculation 01:17:00.076 --> 01:17:03.090 for the pressure of light that you might do. 01:17:03.090 --> 01:17:05.670 It's by no means the only way to calculate it. 01:17:05.670 --> 01:17:09.060 And now finally, we're in a position-- 01:17:09.060 --> 01:17:11.200 and we'll just do this quickly to decide 01:17:11.200 --> 01:17:13.920 how to modify these equations. 01:17:13.920 --> 01:17:16.690 Now, we're not in a position to determine that rigorously. 01:17:16.690 --> 01:17:20.410 It can be determined rigorously by doing general relativity, 01:17:20.410 --> 01:17:23.800 which we're not doing at that level. 01:17:23.800 --> 01:17:26.690 But we can still motivate the answer. 01:17:26.690 --> 01:17:28.360 One of these two equations is going 01:17:28.360 --> 01:17:30.850 to have to be changed to accommodate 01:17:30.850 --> 01:17:34.470 a more general expression for RHO dot. 01:17:34.470 --> 01:17:36.740 The top equation we know is really 01:17:36.740 --> 01:17:39.030 an equation for conservation of energy. 01:17:39.030 --> 01:17:41.930 That's how we got it in the Newtonian case 01:17:41.930 --> 01:17:46.110 where little k ended up being partial to the energy 01:17:46.110 --> 01:17:47.640 in the Newtonian case. 01:17:47.640 --> 01:17:49.140 But this is basically a conservation 01:17:49.140 --> 01:17:50.440 of energy equation. 01:17:50.440 --> 01:17:52.120 And that's what you expect, just given 01:17:52.120 --> 01:17:54.290 your general notion of mechanics as well. 01:17:54.290 --> 01:17:57.560 If you have a second order equation, 01:17:57.560 --> 01:17:59.930 a second order differential is with respect to time. 01:17:59.930 --> 01:18:01.110 That's the force equation. 01:18:01.110 --> 01:18:02.860 And if you have a first order differential 01:18:02.860 --> 01:18:06.010 equation with respect to time, 1/2 mv squared plus v of r 01:18:06.010 --> 01:18:07.070 equals constant. 01:18:07.070 --> 01:18:09.300 That's energy conservation. 01:18:09.300 --> 01:18:10.250 Same thing here. 01:18:10.250 --> 01:18:13.910 And we know that energy cannot suddenly change. 01:18:13.910 --> 01:18:18.780 If we imagine-- I guess the first experiment I want to do 01:18:18.780 --> 01:18:20.730 is imagining somehow there's an explosion 01:18:20.730 --> 01:18:22.920 throughout all of space. 01:18:22.920 --> 01:18:27.670 I imagine putting pieces of TNT throughout space 01:18:27.670 --> 01:18:30.560 and arranging for, at the same cosmic time, for all of them 01:18:30.560 --> 01:18:31.200 to be ignited. 01:18:31.200 --> 01:18:34.014 And that would suddenly change the pressure of the universe, 01:18:34.014 --> 01:18:35.805 but it would not change the energy density. 01:18:35.805 --> 01:18:38.662 The energy density would be conserved. 01:18:38.662 --> 01:18:40.620 So the bottom line is that pressures can change 01:18:40.620 --> 01:18:43.950 discontinuously, but energy densities cannot. 01:18:43.950 --> 01:18:48.640 And since this equation is the conservation of energy 01:18:48.640 --> 01:18:51.660 equation, we'd expected that nothing can change suddenly 01:18:51.660 --> 01:18:54.050 here, that the pressure term cannot contribute here, 01:18:54.050 --> 01:18:56.574 because if it did, the pressure term would changed suddenly. 01:18:56.574 --> 01:18:58.740 Nothing else in this equation would change suddenly. 01:18:58.740 --> 01:19:01.350 There would be no way the equation could be satisfied. 01:19:01.350 --> 01:19:05.130 But if we added a pressure term to the second equation, 01:19:05.130 --> 01:19:08.900 that would allow the pressure to change discontinuously 01:19:08.900 --> 01:19:10.135 as the TNT went off. 01:19:10.135 --> 01:19:12.260 And that would change a double dot discontinuously. 01:19:12.260 --> 01:19:14.530 And there's nothing wrong with a double dot changing 01:19:14.530 --> 01:19:15.350 discontinuously. 01:19:15.350 --> 01:19:17.780 If you suddenly the apply a new force to a particle, 01:19:17.780 --> 01:19:21.350 you suddenly change its second derivative of its motion. 01:19:21.350 --> 01:19:23.500 You suddenly change its acceleration. 01:19:23.500 --> 01:19:24.510 So that's OK. 01:19:24.510 --> 01:19:26.690 So any pressure to this term make sense. 01:19:26.690 --> 01:19:29.490 Adding pressure to this equation does not make sense. 01:19:29.490 --> 01:19:31.370 And then we can just ask, what do you 01:19:31.370 --> 01:19:33.910 have to do to this equation if we're going to add a pressure 01:19:33.910 --> 01:19:36.280 term to make all three equations now 01:19:36.280 --> 01:19:40.220 consistent with the new equation for RHO dot? 01:19:40.220 --> 01:19:42.970 It's your homework problem to answer that question, 01:19:42.970 --> 01:19:44.949 but the homework tells you the answer, 01:19:44.949 --> 01:19:46.990 and I'll write the answer on the board right now, 01:19:46.990 --> 01:19:50.830 and then we'll consider today's lecture over. 01:19:50.830 --> 01:19:57.840 The bottom line is that equation number one 01:19:57.840 --> 01:20:02.750 has to be modified into one prime, which 01:20:02.750 --> 01:20:05.930 says that equation number two has to be modified. 01:20:05.930 --> 01:20:06.960 What am I talking about? 01:20:13.570 --> 01:20:17.890 And the new equation is a double dot 01:20:17.890 --> 01:20:23.600 is equal to minus 4 pi over 3 G times RHO 01:20:23.600 --> 01:20:28.880 plus 3p over c squared times a. 01:20:34.140 --> 01:20:37.070 And now we have a consistent set of Friedman equations, 01:20:37.070 --> 01:20:38.570 and these are the Friedman equations 01:20:38.570 --> 01:20:40.790 that we would have gotten if we had done everything 01:20:40.790 --> 01:20:43.330 using general relativity from the beginning. 01:20:43.330 --> 01:20:44.870 And we'll stop there. 01:20:44.870 --> 01:20:49.380 And we will meet again next Tuesday. 01:20:49.380 --> 01:20:51.170 And I'll send you an email about-- there 01:20:51.170 --> 01:20:54.575 will be at least one homework problem on the problems set 01:20:54.575 --> 01:20:58.446 that will have to be held over to the following problems set. 01:20:58.446 --> 01:20:59.820 I'll send you an email about that 01:20:59.820 --> 01:21:01.960 and post it on the website.