Physics 325: General Relativity Spring 2012 Final Review Problem Set Date: Friday 4 May 2012 Instructions: This is the third of three review problem sets in Physics 325. It will count for twice as much as a regular problem set. In addition to the formulae provided, you may consult the textbooks and your notes from lecture. However, in contrast to the regular weekly problem sets, you may not discuss this review problem set with anyone else. Note that you have a choice of problems in Part I, and a choice of subproblems in Part II. Show all of your work and indicate clearly which of the problems in Part I and subproblems in Part II you would like considered. This review problem set is due by the end of the day on Friday 4 May 2012.
Physics 325, Spring 2012: Final Review Problem Set p. 2 Useful formulae Metric: Proper time:: for a timelike worldline. ds 2 = g αβ (x dx α dx β. dτ 2 = ds 2, 4-velocity: u α = dxα dτ u α = dxα dλ where λ is an affine parameter. Dot product: (for a massive particle, (for light, a b = g αβ a α b β. Dot product of the four velocity with itself: { 1 (for a massive particle, u u = 0 (for light. Schwarzschild metric: ( ds 2 = 1 2M r ( dt 2 + 1 2M r 1dr 2 + r 2 (dθ 2 + sin 2 θdφ 2. Schwarzschild metric in Kruskal-Szekeres coordinates: ds 2 = 32M 3 e r/2m( dv 2 + du 2 + r 2 (dθ 2 + sin 2 θdφ 2. r Relation between Schwarzschild and Kruskal-Szekeres coordinates: { ( r ( t 2M 1 e r/2m = U 2 V 2 V/U r > 2M,, tanh = 4M U/V r < 2M. Schwarzschild metric in Eddington-Finkelstein coordinates: ( ds 2 = 1 2M dv 2 + 2dvdr + r 2 (dθ 2 + sin 2 θdφ 2. r
Physics 325, Spring 2012: Final Review Problem Set p. 3 Minkowski metric of flat 4D spacetime: ds 2 = dt 2 + dx 2 + dy 2 + dz 2. Minkowski metric of flat 2D spacetime: ds 2 = dt 2 + dx 2. 2D Rindler metric: ds 2 = X 2 dt 2 + dx 2. Christoffel symbols: g αδ Γ δ βγ = 1 2 ( gαβ x γ + g αγ x β g βγ x α. Covariant derivatives of vectors: β V α = V α x + β Γα βγv γ (in the direction of the coordinate x β, ( V t V α = t β α x + β Γα βγv γ (in the direction of the vector t. Four-acceleration: a α = u u α. Killing vectors and conserved quantities: The metric ds 2 = g αβ (xdx α dx β has a Killing vector η if the transformation x α x α + ɛη α is a symmetry of the metric for constant ɛ. In this case, the quantity is conserved along geodesics. u α = dx α /ds for spacelike geodesics. K = η u Here u is the 4-velocity for timelike or null geodesics, and Identities and an integral involving hyperbolic functions: cosh 2 ϑ sinh 2 ϑ = 1, 1 tanh 2 ϑ = sech 2 ϑ, du a 2 u = 1 u 2 a tanh 1 a
Physics 325, Spring 2012: Final Review Problem Set p. 4 Functional derivatives and Euler-Lagrange equations: The functional F [x α (σ] = σb σ A f(x α, ẋ α, σ dσ is extremized when its functional derivatives with respect to the functions x α (σ all vanish, 0 = δf δx α (σ = f x d ( f, α dσ ẋ α where ẋ α = dx α /dσ. These equations are known as the Euler-Lagrange equations. Geodesic equation: The geodesic equation is d 2 x α dτ 2 + dx β dx γ Γα βγ dτ dτ = 0, where the Γ α βγ are the Christoffel symbols. Surface gravity: The surface gravity κ at a horizon is defined by the equation ξ ξ α horizon = κξ α horizon, where ξ is Killing vector that (i is timelike in a region of space bounded by the horizon and (ii becomes null at the horizon.
Physics 325, Spring 2012: Final Review Problem Set p. 5 Part I. Short problems on Kruskal coordinates. 1 choose two of Problems (ii, (iii, and (iv. Please answer Problem (i, then i. Kruskal diagram (5 pts. Draw a Kruskal diagram (i.e., the U, V plane. Indicate the horizons, lines of constant t, curves of constant r and the singularities. ii. Timelike worldlines in Kruskal coordinates (5 pts. Please evaluate u u for the Schwarzschild metric in Kruskal coordinates. Express your answer in terms of dv/dτ, du/dτ, dθ/dτ and dφ/dτ. What is the normalization of u u for a timelike worldline? Use this fact to show that ( 2 dv > dτ ( 2 du, (1 dτ and hence that dv/du > 1 for a timelike particle worldline even if it is moving nonradially. iii. The other side of the Kruskal extension (5 pts. Suppose that the black hole in the center of our galaxy were really described by the maximal Kruskal extension instead of having been produced by collapsing stars. Using a Kruskal diagram, explain why it would not be possible to traverse from one asymptotic region of the Kruskal extension to the other. Is it possible for an observer, initially starting outside of the horizon, to eventually see light from stars on the other side of the extension? iv. Two observers in Kruskal coordinates (5 pts. Two observers in two rockets are hovering above a Schwarzschild black hole of mass M. They hover at a fixed radius R such that ( 1/2 R 2M 1 e R/4M = 1 2, (2 and fixed angular position. The first observer leaves this position at t = 0 and travels into the black hole on a straight line in a Kruskal diagram until destroyed in the singularity at the point (U, V = (0, 1. The other observer continues to hover at R. (a On a Kruskal diagram, sketch the worldlines of the two observers. At which spacetime point (U, V does the first observer leave the radius R? (b Does the observer who falls into the black hole follow a timelike worldline? 1 Three of these problems were taken from Hartle: Problem (ii is Hartle 12.21, (iii is Hartle 12.25 and (iv is an abridged version of Hartle 12.22.
Physics 325, Spring 2012: Final Review Problem Set p. 6 Part II. Problems 1 and 2 each have seven subproblems, labeled (a through (g. For each problem, please do five of the seven subproblems. If you like, you may do additional subproblems for up to 1 point extra credit each. 1. Acceleration in Rindler spacetime (15 pts. In this problem, unprimed indices denote Minkowski coordinates and primed indices denote Rindler coordinates. (a Consider the transformation from 2D Minkowski coordinates x α = (t, x to 2D Rindler coordinates x α = (T, X, t = X sinh T, x = X cosh T. (3 Starting from the 2D Minkowski metric (given in the formula reference, please use Eqs. (3 to derive the 2D Rindler metric, ds 2 = X 2 dt 2 + dx 2. (4 (b Consider a particle moving at fixed Rindler coordinate X = 1/g. What is the infinitesimal proper time dτ in terms of dt? Compute the Minkowski components of the 4-velocity u α = dx α /dτ by differentiating Eqs. (3 with respect to T and then multiplying by dt/dτ. Repeat this procedure to compute the acceleration a α = du α /dτ. Note that this expression for the acceleration is valid only in Minkowski space! (Otherwise a α = u u α. Finally, verify the result a a = g 2, (5 for this worldline. (c Using the explicit formula for the Christoffel symbols in terms of derivatives of the metric, derive the following Christoffel symbols for the Rindler metric (4: Γ X T T = X, Γ T T X = Γ T XT = 1/X. (6 Explain why these are the only nonvanishing Christoffel symbols. (d Consider again a particle moving at fixed Rindler coordinate X = 1/g. Which component of u α = (u T, u X is zero? Using the metric (4 and normalization condition on u for a massive particle, determine the remaining component of u α. Then using the expression for acceleration in general coordinates (given in the formula reference, show that the Rindler
Physics 325, Spring 2012: Final Review Problem Set p. 7 components of the acceleration vector are a α Eq. (5 again holds. = (a T, a X = (0, 1/X = (0, g. Confirm that (e Show that for timelike worldlines dt/dx > 1/X. What happens to the lightcones as X 0? (f Identify a Killing vector of the metric (4. Is this vector timelike, null or spacelike, and why? You should find that it is timelike away from X = 0, but null at X = 0. A horizon can be defined as a surface in spacetime at which a timelike Killing vector become null, so this indicates that X = 0 is a horizon. Can you think of a Killing vector with this property for the Schwarzschild black hole? (g Using the identity cosh 2 T sinh 2 T = 1, solve Eqs. (3 for X 2 in terms of Minkowski coordinates t, x. What is the equation for the horizon X = 0 in terms of Minkowski coordinates? From the Minkowski spacetime point of view, is there a reason why massive particles can only travel in one direction through this horizon? 2. Surrounded by a horizon: de Sitter spacetime (20 pts. 2 We have already seen two useful forms of the metric of de Sitter spacetime (the uniformly accelerating universe. In this problem we will explore a third, which was not always recognized as the same space. 3 In static coordinates, the de Sitter metric is where R is a constant. ds 2 = (1 r2 dt 2 + (1 r2 1 dr 2 + r 2 (dθ 2 + sin 2 θdφ 2, (7 R 2 R 2 (a By analogy to the Eddington-Finkelstein coordinates of Schwarzschild spacetime, let us set t = u + F (r, where the derivative F (r is determined by the condition that g rr = 0 in the resulting metric. Please solve for F (r and show that the resulting metric is ds 2 = (1 r2 du 2 2dudr + r 2 (dθ 2 + sin 2 θdφ 2. (8 R 2 If you encounter a ± sign, choose the value that gives F (r > 0 for r < R. Then, integrate F (r to determine F (r. (A useful integral can be found among the reference formulae for hyperbolic functions. (b Using the expression (8 determine the following possible slopes of radial light rays at 2 This problem was inspired by Hartle 15.10. 3 Historically it was known as the Einstein static universe.
Physics 325, Spring 2012: Final Review Problem Set p. 8 each point in the u, r plane: u = const, or dr du = 1 ( r 2 2 R 1. (9 2 Show that r = R is a horizon of the opposite sort than the ones we are used to: light rays starting at r > R cannot enter the r < R region. The interpretation of this is that due to the accelerating expansion of the universe, objects at r > R are swept away faster than light can return. (c Define t = u + r and determine d t/dr for each of the radial light rays of Eq. (9. Use these slopes to sketch the light rays on an Eddington-Finkelstein ( t, r diagram. Be sure to label the horizon. (d,e (counts as two parts Restricting to the equatorial plane θ = π/2, use the metric (8 to write an integral expression for the proper time along a timelike worldline u = u(σ, r = r(σ, φ = φ(σ, (10 connecting two points A and B in de Sitter space. Here σ is an arbitrary parameter along the worldline. Express your answer as a functional τ AB [u(σ, r(σ, φ(σ] = σb σ A f( u, r, ṙ, φ dσ, (11 where a dot denotes d/dσ. As indicated, you should find that f = f(u, r, ṙ, φ, with no explicit dependence on u, φ. Then, derive the Euler-Lagrange equations, ( d ( 1 r2 du dτ R 2 dτ + dr = 0, dτ d ( du + r ( du 2 ( dφ 2 + r = 0, dτ dτ R 2 dτ dτ d ( r 2 dφ = 0. dτ dτ Finally, identify two Killing vectors and interpret the first and the third equations as the corresponding conservation laws. (f Carry out the derivatives in Eqs. (12. The second equation is the d 2 u/dτ 2 equation. It can be used to eliminate d 2 u/dτ 2 from the first equation to give the d 2 r/dτ 2 equation. The (12
Physics 325, Spring 2012: Final Review Problem Set p. 9 third equation is the d 2 φ/dτ 2 equation. Show that these equations reduce to d 2 u dτ + r ( du 2 ( dφ 2 + r = 0, 2 R 2 dτ dτ d 2 r ( dτ 1 r2 r ( du 2 2r du dr ( ( 2 R 2 R 2 dτ R 2 dτ dτ 1 r2 dφ 2 r = 0, R 2 dτ d 2 φ dτ + 2 dr dφ 2 r dτ dτ = 0, and then, read off the nonvanishing Christoffel symbols from these equations. (13 (g Using the Γ α uu Christoffel symbols and the Killing vector Γ u uu = r R 2, Γr uu = r R 2 ( 1 r2 R 2, (14 ξ = u, ξα = (1, 0, 0, 0, (15 compute the surface gravity κ, defined by the equation evaluated at the horizon r = R. ξ ξ α horizon = κξ α horizon, (16