236a: Ph General Relativity Yanbei Chen
31 October 2008 Caltech
WEEK 5: EINSTEIN EQUATION AND PHYSICS IN CURVED SPACETIME Recommended Reading. 1. MTW: Chapter 11.6 (Riemann normal coordinates), Chapter 13.6 (proper reference frame), 2. MTW: Chapter 16 (Equivalence Principle), Chapter 17 (Einstein Equation), Chapter 23 (Spherical Stars) 3. Carroll, Spacetime and Geometry, Sec. 5.2. This outlines the construction of the most general form of metric for spherically symmetric spacetimes. [Here you will encounter the concept of a Killing vector field, which we will cover next week. These few pages do not have much technical calculations involving Killing vectors.] Alternatively, Box 23.3 of MTW provides a less detailed explanation. 2. Blandford and Thorne, Secs. 25.125.3 Possible Supplementary Reading Covering Similar Material: 3. Schutz (A First Course in General Relativity): Sec. 8.1-8.2, Chapter 10. 5. Carroll (Spacetime and Geometry): Sec. 4.14.7, 5.15.2.
Problems The maximum number of points that will be given for this problem set is 50. The point value of each problem is stated at its beginning. Maximize your learning to grunge ratio by working only those problems that are useful to you. Skip problems that are too easy or obvious. Although you might be able to find homework solutions in previous years, you are not supposed to consult with them before you hand in your homework! On the other hand, you may encounter solutions of some of these problems if you read certain textbooks. In that case, you should not simply copy the solutions from those books. If you understand that derivation right away, then there is really no point doing the problem anymore. But if you feel you could learn something by writing down that solution on your own, then you can still hand in your own writing of that solution. (This situation is similar to having asked the TA, or other experts, for help on your homework.) In any case, these exercises are designed to help you understand the materials better. Please do whatever that helps you learn.
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There are a large number of problems this week, because lectures this week involved several different topics. Please only choose the most interesting problems for you. Problems on curvature tensors 1. [5 Points] Weyl tensor, MTW Exercise 13.13. 2. [5 Points] Riemann normal coordinates, MTW Exercise 11.9. 3. [5 Points] Curvature on the surface of a sphere, BT Exercise 24.9 4. [5 Points] Geodesic deviation on the surface of a sphere, BT Exercise 24.10 5. [10 Points] Constant negative curvature space in 2 dimensions. Redo the above two problems, switching to negative constant curvature. (Change "sin" into "sinh") Problems on proper reference frames 6. [10 Points] Inertial and Coriolis forces, MTW Exercise 13.14. 7. [10 Points] Read Exercise 23.9 of BT, understand the concept of expansion, shear and rotation of a congruence of time-like geodesics, and derive the Raychauduri Equation: d 1 = - 2 - + - R u u d 3 Hint: first write d/d = u P u;
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then (i) use the fact that P ; = 0, (ii) commute covariant differentiations on u to get a Riemann term, and (iii) note that u; u = -u; u ; Problems on the formulation of the theory of gravity 8. [10 Points] Newtonian Limit of General Relativity. [Note: This problem helps you get insight into the relationship between general relativity and Newton's description of gravity.] a. MTW Exercise 17.6 b. MTW Exercise 16.1. Note: In the spacetime metric (16.2a) of this exercise we see two places that the Newtonian potential enters: the time-time part of the metric, g00 = -1-2 (the "correspondence relation" dealt with in Exercise 17.6), and the space-space part of the metric, gjk = (1 + 2)jk . Show that only the time-time (correspondence-relation) term influences the motion of the fluid. Show this by prepending some coefficient to the space-space part of the metric, i.e. by writing the metric as ds2 = -(1 + 2)dt2 + (1 - 2)(dx2 + dy 2 + dz 2 ) , (4)
and by then demonstrating that does not enter into the fluid's equation of motion. As we shall see later, the term in the space-space part of the metric is actually a relativistic "post-Newtonian" correction to the Newtonian theory of gravity. We will discuss post-Newtonian corrections more systematically later in this course.
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9. [10 Points] Stress Tensor for Newtonian Gravitational Field. [Note: This problem is also relatively easy; it is in part an exercise in index manipulation, and in part a vehicle for understanding Newtonian gravity more deeply.] Define a stress tensor for the Newtonian gravitational field as follows: TN jk = 1 4 1 ,j ,k - gjk ,l ,l 2 . (5)
This stress tensor lives in the ordinary, flat, Euclidean of space Newtonian physics. a. Explain why, if expression (5) is truly a stress tensor for the Newtonian gravitational field, the Newtonian equation of motion for a perfect fluid [the second of Eqs. (16.3a) of MTW] should be reexpressible as dvj = -(TN jk + pgjk ),k dt (6)
in a Cartesian coordinate system. Verify that this equation is, indeed, equivalent to (16.3a) of MTW. b. Explain why you would expect the relativistic law of momentum conservation to reduce to the following in the Newtonian limit: (vj ),t + (TN jk + pgjk + vj vk ),k = 0 . (7)
Explain the physical meaning of each of the terms in this equation. Verify that this equation is, in fact, just a rewritten form of Eq. (6). c. For further insight into a full stress-energy tensor for gravity in the Newtonian limit, see Box 12.23 of Chapter 12 (version 0612.2.K.pdf) of Blandford and Thorne. 10. [10 Points] Nordstrm's Theory of Gravity [Note: It is often said that Einstein's o theory can be obtained uniquely through general covariance, i.e., the postulate that physical laws must be written in frame-independent ways. However, it is possible to insert specific geometric structures into a gravitational theory, but write them in a frame-independent way. Sec. 17.6 of MTW discusses the difference between frameindependent formulation and the non-existence of a priori structures.] Read Sec. 17.6 of MTW, and do Exercise 17.8 Problems on curvature coupling 11. [10 Points] Precession of the Equinoxes. As an exploration of curvature coupling effects in general relativity, do MTW Exercise 16.4. [Note: intrinsic angular momentum is discussed, in any Lorentz frame (global or local), in Box 5.6.] 12. [10 Points] Quantum-Gravity-Induced Curvature Coupling in Maxwell's Equations [Problem due to Walter Goldberger with modifications by Kip Thorne] 3
In Box 16.1 of MTW it is argued (via the equivalence principle) that, because the electromagnetic field tensor is physically measureable and the Maxwell equations expressed in terms of it involve only first derivates, not second, there is unlikely to be any curvature coupling in those Maxwell equations; and, in particular, the Maxwell equations should read in curved spacetime, as in flat. (Here the notation is F F ; .) a. Show that conservation of charge follows from the first of these equations; that is, show that J = 0. [(1 + R)F ] = 4J , F = 4J , F + F + F = 0 (W G.1)
b. As will be explained below, quantum gravity might induce a curvature coupling of the following form: F + F + F = 0 ; (W G.2)
here is some constant and R = R is the scalar curvature of spacetime. Like Eq. (WG.1), these reduce to the familiar Maxwell equations in flat spacetime (since R = 0 there). The fact that the second of these equations is unmodified means that we can still write F in terms of a vector potential, F = A - A , and the theory is still gauge invariant. Show that this version of the Maxwell equations, like the more conventional version, implies charge conservation. [Hint: first show that antisymmetry of F implies [(1 + R)F ] = (1 + R) F , then use that antisymmetry to show that this expression entails commutation of covariant derivatives, and show that the curvature terms produced by that commutation give a vanishing result.] c. Quantum gravity is known to introduce curvature couplings into the laws of h physics, with coupling constants that involve the Planck length, lP . Reexpress this Planck length in cgs units by restoring the appropriate factors of G and c, and evaluate it numerically (you should get an extremely small number). By dimensional considerations, estimate the numerical value of the coupling constant which quantum gravity might induce. d. Perform a 3+1 split of these modified Maxwell equations in the local Lorentz frame of some observer--e.g., an experimenter in the space shuttle; i.e., rewrite them in terms of the electric and magnetic fields E and B, and the charge and current densities and j that the observer measures. Discuss the physical manifestations of the curvature coupling that these 3+1 equations predict; pay attention to the fact that R vanishes in vacuum, and that at the surface of some solid body R will have a delta-function behavior. You might want to consider, for example, Gauss's law which usually expresses the total charge inside a body as a surface integral of the electric field sticking out of the body. Estimate the dimensionless magnitude of these physical effects. [Answer: They are so tiny that there is no hope at all to measure them with forseeable technology.] Problems on the interior of stars 11. [10 Points] Derive components of the Einstein tensor within a static spherical star. Exercise 23.4 of MTW. 4