Ph 236a: General Relativity 26 October 2005WEEK 4: SPACETIME CURVATURE, LAWS OF PHYSICSIN CURVED SPACETIME, AND EINSTEIN FIELD EQUATIONPoint Values for Problems Have Been Changed as of 29 October; no other changesRecommended Reading:1. MTW:a. Secs. 8.6 and 8.7b. Secs. 1 6.2 and 16.3c. Secs. 17.1 — 17.4 and 17.62. Blandford and Thorne [at http://www.pma.caltech.edu/Courses/ph136/yr200 6/ ]:Secs. 24.2–24.8 and 24.9.1 (of chapter 24).Possible Supplementary Reading:3. MTW Chapters 11 and 13 (but not Sec. 13.6 which we will study later), on GeodesicDeviation and Spacetime Curvature. This is an in-depth treatment of these topics,initially (Chap. 11) without a metric and then (Chap. 13) with a metric.4. MTW: The remainder of Chapters 16 and 17 (portions not contained in items 1.b and1.c).5. Hartle (Gravity), Chaps. 6, 7, 21 and 22. Chapters 6 and 7 are a leisurely, elementaryintroduction; 21 and 22 add the mathematical detail.5. Schutz (A First Course in General Relativity): Chapters 6, 7 and 8. This is anelemetary-to-intermediate-level treatment.6. Carro ll (Spacetime and Geometry), Sections 3.6–3.11 and Chapter 4. This is anintermediate-level treatment, some of it from a different point of view than my own,and with some emphasis on the connections to field theory and to some contemporarytopics.7. Wald (Genera l Relativity): Chaps. 3 and 4, but not Sec. 4.3b. This is a concise,mathematically sophisticated treatment.8. MTW Chapter 7. This describes what happens when one tries to construct a relativis-tic theory of gravity as a linear field theory in flat spacetime, and why such theoriesdon’t work. For a sophisticated follow up on this chapter’s tensor theory of gravity i nflat spacetime, see Section 5 of Box 17.2 — which describes, in brief, how the attemptto make the tensor theory self consistent leads to general relativity.9. Richard P. Feynman, Fernando B. Morinigo , and William G. Wagner, Feynman Lec-tures on Gravitation (Addison Wesley, 1995). Pages x–xvi of the forward t o this book(by Preskill and Thorne) outline Feynman’s approach to the derivation of the Ei nsteinfield equation, an approach closely related to that sketched in item 4 above. Chapters1–6 of this book present Feynman’s derivation.1Problems. All problems are worth 15 poi nts unless ot herwise indicated. The maximumnumber of points that will be given to anyone on this set is 50.1. Formula for Components of the Riemann tensor in an Arbitrary Basisa. Blandford and Thorne, Exercise 24.8b. If you prefer a more sophisticated route to this result: Read MTW Exercise 11.3;then do MTW Exercise 11.4.2. “Flat” Friedman Universe: Local Lorentz Frame and CurvatureThe spacetime metric for a flat Friedman Universe isds2= −dt2+ a2(t)(dx2+ dy2+ dz2) = −dt2+ a2(t)δjkdxjdxk. (2)a. Verify that a particle at rest in the (t, x, y, z) coordinate system [i.e., a particlewith world line (x, y, z) = constant] moves along a geodesic of spacetime, andthus is freely falling.b. Consider a freely falling particle that is at rest at the spatial ori gin, ( x, y, z) =(0, 0, 0). In the vi cinity of this particle’s world line introduce new coordinatesT = t +12˙aaa2(x2+ y2+ z2) , X = ax, Y = ay, Z = a z , (3)where the dot means d /dt. Show that these new (T, X, Y, Z) coordinates are alocal Lorentz frame of the freely falling particle — i.e., show that in this coordinatesystem, up to corrections that are second order in spatial distance from the spatialorigin, gµν= ηµν.c. The freely falling observer at (x, y, z) = (0, 0, 0) measures the motion of anearby freely fall ing particle whose world line in the original coordinate systemis (x, y, z) = (δx, δy, δz). The observer studies the particle in the observer’s lo-cal Lorentz coordinate system {xˆα} ≡ (T, X, Y, Z). By comparing the particle’smotion with the equation of geodesic deviation in t he local Lorentz frame, showthat the tide-producing local Lorentz components of the Riemann tensor areRˆjˆ0ˆkˆ0= −¨aaδjk. (4)d. There are also nonzero purely spatial compo nents o f Riemann, Rˆjˆkˆl ˆm, whichcannot be computed in the above manner. Use symmetry arguments to showthat they must have the formRˆjˆkˆl ˆm= R(t)(δˆjˆlδˆk ˆm− δˆj ˆmδˆkˆl) , (5)where R(t) is some function of t. The remainder of this exercise is directed towardverifying Eq. (4) and computing t he R(t) of Eq. (5).d. Compute the connection coefficients in the (t, x, y, z) coordinate basis, either byhand or using tensor-manipulation software.†† For details on where to find the relevant computer softwa re, see our course home page,http://www.pma.caltech.edu/∼ph236/yr2007/] . I especially recommend the Mathemat-ica programs associated w ith Ha rtle’s textbook (see the link on our home page). They areeasy to understand, and with t hem as templates, you can write other tensor manipulationsoftware.2e. Compute the components of the Riemann tensor in the (t, x, y, z) coordinate basis,either by hand or using tensor-manipulation software.f. Transform the components of Riemann to the local Lorentz frame of the freelyfalling observer. Thereby verify that the only nozero components are those ofEqs. (4) and (5), and deduce the value of R(t).3. Geometrized Units. [5 Points]a. Compute the values of t he following quantities in geometrized units (units wi thG = c = 1); express your answers in centimeters or some power thereof: Planck’sconstant, ¯h; the charge of t he electron, e; the fine structure constant, e2/(¯hc);your mass; the mass of the sun; the mass of the Earth.b. Use dimensional considerations to restore the factors of G and c in all the num-bered equations in this problem set (below). You might want to wait and do t hisafter you have studied all of the problems that contain numbered equations.4. Precession of the Equinoxes.As an exploration of curvature coupling effects in general relativi ty, do MTW Exercise16.4. [Note: intrinsic angular momentum is discussed, in any Lorentz frame (globalor local), in Box 5.6.]5. Quantum-Gravity-Induced Curvature Coupling in Maxwell’s Equations[Problem due to Walter Goldberger with modificati ons by Kip] [NOTE: In my lectureon Wednesday I forgot about the possibility of the scalar curvature R generating cur-vature coupling, so my argument that the Maxwell equations cannot have curvaturecoupling was wrong. Here is an important counter example. - Kip]In Box 16.1 of MTW it is argued (via the equivalence principle) that, because