Final exam Due: 1:00pm, Tuesday April 19. Hand in your completed exam at my office, RPHYS rm. 309. Instructor: I. Morrison TAs: G. Franzmann Y. Gobeil Instructions This is the final exam for PHYS 514: General Relativity (Winter 2016). The exam is worth 20% of your total grade in the course. You have 24 hours to complete this exam. You may consult any standard references on GR that you like, including course materials, textbooks, and reliable internet sources. You may use Mathematica and similar software. You may NOT consult research articles (i.e., anything published in a journal or available on a pre-print archive) or special-topics textbooks. You may NOT consult any person about this exam (regardless of whether they are in the class), nor should you troll internet chat rooms and comment threads. Given these restrictions, you will not find the solutions to these problems online, and I do not recommend you spend your limited time scouring the internet for clues. You have the skills and experience you need to complete these problems. Submit all your work in paper. Please place your solutions in numerical order. You do not need to show evidence of computing curvature quantities, as these you can do with the Mathematica routine provided on the course website. This exam has 6 problems. For full credit you must submit solutions for 3 problems of your choosing. Each problem is worth 10 points for a total of 30 points. You may submit one additional solution for extra credit determined as follows: the 3 solutions with the highest score will be graded out of 10, and the extra solution will be graded out of 5. Note that with this scheme 3 complete solutions are better than 4 partial solutions. I am available to provide clarifying comments about the exam questions. The best way to reach me during the exam is by submitting a post to the discussion FINAL! on the course s mycourses platform. I will moderate questions and provide answers. Alternatively, you may contact me at: office: RPHYS rm. 309 email: imorrison@physics.mcgill.ca mobile: +1 (514) 442-9523 Please remember to HAVE FUN and enjoy this exam. Final exam Page 1 of 9
1. A sluggish star Consider an isolated, slowly rotating neutron star. The resulting spacetime is axi-symmetric and stationary. In a convenient set of coordinates (t, r, θ, φ) the perfect fluid composing the star has 4-velocity u µ = u t (1, 0, 0, Ω) µ, (1) where Ω = u φ /u t is the angular velocity of the fluid. Assume that Ω 1 (with geometrized units G = c = 1) and consider the spacetime to order O(Ω 1 ). a) Argue that the line element describing the spacetime of the star may be written ( ds 2 = e 2Φ(r) dt 2 + 1 2m(r) ) 1 dr 2 2ΩH(r, θ)dtdφ + r ( 2 dθ 2 + sin 2 θdφ 2) + O(Ω 2 ), r (2) where Φ(r), m(r), and H(r, θ) are O(Ω 0 ). b) Now assume that Derive an equation of motion for ω(r) which is of the form H(r, θ) = ω(r)r 2 sin 2 θ. (3) ω (r) + A(r)ω (r) + B(r)ω(r) + C(r) = 0, (4) where A(r), B(r), and C(r) are functions of ρ(r), p(r), m(r), and Φ(r) but not their derivatives. c) In order to fully determine ω(r) one must supplement (4) with 2 boundary conditions. Outside the star, the spacetime is an axi-symmetric, stationary solution to vacuum Einstein s equations. Under reasonable conditions, it follows that this portion of the spacetime agrees with the Kerr solution. Matching (2) to the Kerr metric in Boyer-Lindquist coordinates (t, r, θ, φ) (Carroll (6.70)) so that the metric is continuous provides a boundary condition for ω(r). What is a reasonable boundary condition for ω (r)? The moment of inertia I is a constant which relates the fluid angular velocity Ω to the total angular momentum of the spacetime J via J = IΩ. The moment of inertia contains information about the stellar equation of state, and as a result, independent measurements of I and Ω could provide new information about the equation of state of dense matter. d) Compute I as a function of ω(r), ρ(r), p(r), m(r), and Φ(r). You may leave your expression in terms of an integral over r. Hint: In the Newtonian limit the moment of inertia is given by I Newtonian = 8π 3 R where r = R is the radius of the surface of the star. 0 dr r 4 ρ(r), (5) Final exam Page 2 of 9
2. Resistance is futile The Borg is an alien civilization whose spacecraft are cube-shaped. A Borg ship of total mass M and sides of length L rotates about a principal axis with angular frequency ω (see Fig. 1). a) Compute the resulting gravitational radiation observed at a distance R L far from the ship. I.e., give the linear metric perturbation h µν (x) far from the ship. Make sure to state your choice of gauge. b) What is the power emitted by the Borg vessel due to gravitational radiation? Figure 1: A Borg vessel rotating about a principal axis. Final exam Page 3 of 9
3. The whole universe Consider the (2 + 1)-dimensional line element ds 2 = dt 2 + cos 2 (t/κ) ( dx 2 + κ 2 cosh 2 (x/κ) dφ 2), t ( κ π 2, π ) x, R, φ φ + 2π. (6) 2 κ Here κ is a constant of dimensions length. This line element describes a cosmology with equal-time surfaces which have topology R S 1. But not all is as it seems. a) Show that the spacetime described by (6) is not geodesically complete. In particular, consider a null geodesic which is outgoing, i.e., which satisfies dx dt > 0 for x > 0, or dx dt < 0 for x < 0. (7) Show that such a geodesic, if originating at some finite t, x, reaches the boundary of the chart (6) in finite affine parameter. Hint: You only need one example geodesic to show a spacetime is not geodesically complete. It may be simplest to consider a geodesic originating at the origin t = x = 0. b) Extend this spacetime by finding a coordinate chart which covers the region described by (6), as well as a larger region. Show that the outgoing null geodesic you considered in (a) requires infinite affine parameter to reach the boundary of your extended spacetime. (Do not worry about ingoing geodesics.) c) Draw a Penrose (a.k.a. Penrose-Carter, a.k.a. conformal) diagram of the extended spacetime. Hint: This spacetime is not asymptotically flat, and the Penrose diagram of the extended spacetime is not a diamond. d) What is this spacetime, really? Final exam Page 4 of 9
4. Who needs Killing vectors? In this problem you will show that on any asymptotically flat spacetime there exist conserved charges analogous to energy, momentum, and rapidity in Minkowski space no Killing vectors required! Consider a (3 + 1)-dimensional spacetime whose metric may be written g µν = η µν + h µν, (8) where η µν is the usual Minkowski metric and h µν is O(r 1 ) at asymptotically large radius r but need not be small at finite radius. In general, such metrics will satisfy Einstein s equation with some non-vanishing matter source Tµν matter, and will not admit any Killing vectors. Consider expanding the Einstein tensor in a power series in h: G µν [g] = G µν (n) [h], G (n) µν [h] = O(h n ). (9) n=1 Then Einstein s equation may be written G (1) µν [h] = 8πTµν total = 8π ( ) Tµν matter + Tµν grav, (10) where the graviton stress tensor is defined to be Tµν grav := 1 G (n) µν [h]. (11) 8π Recall that, in Cartesian coordinates such that η µν = diag( 1, 1, 1, 1) µν, the linearized Einstein tensor may be written G (1) µν [h] = 1 ( ) λ λ h µν λ µ h νλ λ ν h µλ + η µν α β h αβ, (12) 2 n=2 where h µν = h µν 1 2 η µνh. No gauge choice has been imposed. a) Show that, when Einstein s equation holds, the total stress tensor T total µν transforms like a rank 2 tensor under Lorentz transformations. b) Show that, when Einstein s equation holds, D µ T total µν = 0, where D µ denotes the covariant derivative compatible with the Minkowski metric. It follows from these results that we may regard (10) as describing a rank 2 tensor matter field h µν on a fixed Minkowski space. We may construct charges in the usual way we do on Minkowski space: namely, for any Minkowski Killing vector field ξ µ the charge Q[ξ] := d 3 x g Σ n µ ξ ν Tµν total (13) Σ is independent of the Minkowski Cauchy surface Σ on which it is evaluated. Here g Σ is the determinant of the induced metric on Σ and n µ is the future-pointing normal vector to Σ. Final exam Page 5 of 9
c) Show that, for any Minkowski KVF, the integrand in (13) is a total derivative, and thus one may use Stokes Theorem to recast the integral as an integral over Σ at any fixed time. You do not need to simplify your expression for Q[ξ] as much as possible, but it should be clear that your final expressions are linear in h µν. Hint: For energy you should find E := Q[ 0 ] = 1 16π where r k is the outward-pointing normal vector to Σ. Σ d 2 x g Σ r k ( j h j k kh j j), (14) Hint: If considering a general Minkowski KVF is difficult, you may instead consider translations, rotations, and boosts separately. Final exam Page 6 of 9
5. An uncanny resemblance Consider a (4 + 1)-dimensional spacetime with coordinates (x µ, θ), where the index µ = 0, 1, 2, 3, and the line element is ds 2 = g µν (x)dx µ dx ν + (dθ + A µ (x)dx µ ) 2, θ θ + 2πl. (15) Here g µν (x) and A µ (x) are functions of x µ but not θ. a) Show that under 4-dimensional coordinate transformations x µ y µ (x) (16) the metric components g µν (x) and A µ (x) transform as 4-dimensional tensors of rank 2 and 1 respectively. b) Show that under coordinate transformations θ θ + f(x), (17) the components g µν (x) are unchanged and A µ (x) transforms as A µ (x) A µ (x) µ f(x). (18) c) Consider a massive, freely-falling particle traveling in the 5-dimensional spacetime. Note that θ θ + constant is an isometry. Find a constant of motion L associated with this isometry. d) Show that the particle s equation of motion may be written ẍ µ + Γ µ αβẋα ẋ β = q m F µ νẋ ν. (19) Here dots denote derivatives with respect to an affine parameter. In this expression all 4-dimensional indices are raised/lowered with g µν (x) and its 4-dimensional inverse, Γ µ αβ is the Christoffel symbol associated with g µν (x), F µν (x) = µ A ν (x) ν A µ (x), and q/m is a constant related to L. e) In general, the Einstein-Hilbert action for 5-dimensional gravity is S 5 = 1 d 4 xdθ g 5 (x, θ) R 5 (x, θ), (20) 16πG 5 where G 5 is the 5-dimensional gravitational constant, g 5 (x, θ) is the determinant of the 5-dimensional metric, and R 5 (x, θ) is the Ricci scalar constructed from the 5-dimensional metric. In light of (19), it should not surprise you to learn that, upon restricting attention to metrics of the form (15), this action may be written S 5 = 1 d 4 x g(x) R(x) κ d 4 x g(x) F µν F µν (x), (21) 16πG 4 2 where G 4 is the effective 4-dimensional gravitational constant, R(x) is the Ricci scalar constructed from g µν (x), and κ is a constant. Take (21) as given and determine G 4 and κ in terms of G 5 and l. Final exam Page 7 of 9
6. Roll with it Consider a k = 0 FRW cosmology whose matter content is a scalar field with action S matter = d 4 x ( g(x) 1 ) 2 µ φ µ φ(x) V (φ). (22) The scalar field potential V (φ) is a functional of φ but not the metric. Due to homogeneity and isotropy, scalar field configurations which produce FRW cosmologies have no dependence on FRW spatial coordinates, so φ = φ(τ). a) Record the equation of motion for φ in FRW coordinates. (You do not need to derive this from first principles.) Note that this equation of motion is similar to that of a classical particle in a 1-dimensional potential. In this analogy φ 2 is the usual kinetic term, V (φ) is the potential, and H φ is a friction term with a time-dependent coefficient of friction. The scalar field will want to minimize its potential energy, and thus will seek out configurations which minimize V (φ). We often say that φ is like a ball rolling down a hill. b) Show that the stress tensor takes the form of a perfect fluid and determine the effective energy density ρ and pressure p as functions of φ and V (φ). Depending on the form of the potential V (φ), a scalar field can take on several different effective equations of state p = wρ. Consider the potential shown in Fig. 2. For each regime below, determine the effective equation of state parameter w and compute φ(τ) and a(τ). Figure 2: An example potential. c) Potential-dominated regime: Suppose φ is initially has a value of (i) with φ 0. In this regime the potential dominates over kinetic terms, i.e., 1 2 φ 2 V (φ). (23) Final exam Page 8 of 9
d) Kinetic-dominated regime: Eventually φ will reach value (ii). Due to the steep gradient of V (φ), here the kinetic term dominates over the potential: 1 2 φ 2 V (φ). (24) e) Oscillatory regime: When φ reaches a local minimum in V (φ) it will oscillate about the minimum, losing energy due to Hubble friction. Near (iii) the potential may be Taylor expanded, V (φ) m2 2 φ2, m > 0. (25) Consider the ansatz 3 3 H φ = H sin θ, φ = cos θ, (26) 4π 4π m where θ is an unknown function of τ. Note that this ansatz satisfies the Friedmann equation. Derive two 1st order ODEs for H and θ. Using these, argue that at sufficiently late times (mτ 1) θ mτ. (27) Obtain the equation of state, φ(τ), and a(τ) at lowest order in 1/(mτ). f) Final state: What is the final state of this universe? What would be the final state if we instead considered the potential V (φ) V (φ) + C for some constant C? Final exam Page 9 of 9