Ph 236a: General Relativity 31 October 2007WEEK 5: RELATIVISTIC STARS — STATIC AND SLOWLY ROTATING;LOCAL INERTIAL FRAMES AND PROPER REFERENCE FRAMES;SYMMETRIES AND CONSERVATION LAWSRecommended Reading:1. Relativistic Stars:a. MTW Chapter 23 plus sections 24.1 and 24 .2, and Exercise 26.1.b. Blandford and Thorne, sections 25.1 and 25.3 of version 0625.1.K.pdf of chapter25, available at http://www.pma.caltech.edu/Courses/ph136/yr2006/ .[For a solution to this exercise, see Sections III and IV of the paper “SlowlyRotating Relativi stic Stars. I. Equations of Structure” by James B. Hartle, TheAstrophysical Journa, 150, 1005–1029 (1967). I do not know of any textbooktreatment of this, though such a treatment should exist.]2. Local Inertial Frames and Proper Reference Framesa. MTW Section 13.6b. Section 23.5 of chapter 23 (Version 0623.1.K.pdf ) of Blandford and Thorne. Thisis a more elementary treatment, and from a more familiar, less sophisticatedviewpoint.3. Symmetri es and Conservation Lawsa. MTW Secs. 25. 2, 25.3, 25.4, and Box 19.1.[Note: Box 19.1 g ives a more general discussion of the conservation law for thesource’s a ngular momentum than I developed in my Wednesday lecture. But thebasic idea i s the same as in my Wednesday lecture. Box 19.1 omits the key ideathat the conservatio n laws for the source’s ma ss M and angular momentum~S (orJ in my lectures) owe their existence to the symmetries of spacetime at radialinfinity.]Possible Supplementary Reading:4. Relativistic Starsa. Schutz (A First Course in General Relativity): Chapter 10. This covers the samematerial as MTW Chapter 23.c. Wald (General Relativity): Sections 6.1 and 6 .2. This covers some of that samematerial, but in a quicker and more mathematicall y sophisticated way.b. Carroll Spacetime and Geometry, pp. 230–236.c. Hartle Gravity: Chap. 24. This discusses the construction of relativistic stellarmodels.d.. For a detailed discussion of neutron stars: Chapter 9 of Stuart L. Shapiro andSaul A. Teukolsky, Black Holes, White Dwarfs, and Neutron Stars (Wiley, NewYork, 1 983).4. Local Inertial Frames and Proper Reference Frames:a. Regretably, most textbooks omit any discussion of lo cal Lorentz frames or properreference frames along the world line of some observer. The closest they come isto construct local inertial frames in the vicinity of a single event in spacetime;1these are often called Riemann normal coordinates. See, e. g., Carroll, pp. 111–113and Wald p. 42.b. Hartle’s undergraduate textbook is an exception. It does discuss l ocal Lorentzframes all along the world line of a freely falling observer, and calls them “freelyfalling frames”. Another, technical t erm is “Fermi Normal Coordinates.” SeeHartle Sec. 8.4.5. Symmetri es and Conservation Laws:a. Hartle, Sec. 8.2. This is an elementary discussion.b. Wald, Sec. C.3 (pp. 44 1-444). This is a sophisticated discussion. It includes abrief introduction t o Lie derivatives.c. Carroll, Sec. 3.8 on Killing vectors and asso ciated conservation laws; AppendixB on Lie derivatives.Note Concerning Computer-Aided Tensor Analysis: Some of the exercises involvecalculations of curvature tensors that would be long and tedious if carried out by hand. Isuggest you use tensor-manipulation computer software. Sources for such software are givenon our course home page, http://www. pma.caltech.edu/∼ph236/yr2007/] . I especiallyrecommend the Mathematica programs associat ed wit h H artle’s textb ook; see the link onour home page). They are easy to understand, and with them as templates, you can writeother tensor manipulation software problem set in which you are asked to do computer-aided t ensor calculations. There are links to other relevant software packages on theGRTensor web site: http://grtensor.phy.queensu.ca/************************************************************ProblemsNote: The point values of each problem is shown after its title. As usual, the maxi-mum number of points that will be given for this set is 50.Problems Related to My Material on Relativistic Stars1. Flat Friedman Universe. [15 points]This exercise is designed to give you practice at building a model for a physical systemin the same way as I did in my lecture on Monday. I dealt with a static, sphericalstar. You will deal with a simple model for our universe and its evolution.Idealize our universe as being spatially homogeneous and Euclidean. In other words,through every event there passes a 3-dimensional spacelike hyp ersurface that has flat,Euclidean geometry. C onsider any specific such surface; call it S. Consider the familyof observers on S, who see S as their 3-spaces of simultaneity. Call them homogeneousobservers. When these homogeneous observers make measurements of their surround-ings, they all see ident ically the same things—the same density, the same pressure,the same temperature, the same spacetime curvature, etc.2a. Use these physical features of the universe to construct coordinates {t, x, y, z} i nwhich t he spacetime metric takes the flat Friedman formds2= −dt2+ a(t)2(dx2+ dy2+ dz2) . (2)Explain in detail, step-by-step, how the coordinat es are constructed, and showthat your construction leads to a metric of the form (2).b. What are the basis vectors (orthonormal tetrad) ~eˆαof each homogeneous o b-server’s proper reference frame? Express them in terms o f the coordinate basisvectors.c. This universe is filled with perfect fluid, that is at rest with respect to the homo-geneous observers. What are the components of the fluid’s stress-energy tensorin these observer’s proper reference frames?d. These observers a pply t heir laws of local energy conservation and local momentumconservation to this fluid. What do t hose laws say about the evolution of thefluid’s total density of mass-energy ρ and pressure P , in terms of the evolution ofa(t).e. Set Hˆαˆβ= Gˆαˆβ− 8πTˆαˆβ, where G is the Einstein t ensor and T the stress-energytensor. Energy-momentum conservation and the contracted Bia nchi identitiesguarantee that~∇ · H = 0 before the Einstein equations have been imposed. Showthat this, plus the homogeneity i mply that if we impose just one of the 10 Einsteinequations Hˆ0ˆ0= 0 , all t he other Ei nstein equations are guaranteed to be satisfied.f. Compute Gˆ0ˆ0(using computer softwa re — e.g., if using Hartle’s softwa re, bycomputing the coordinate components G00= 0 and then transforming to theorhonormal basis.g. Show that the