Write your CANDIDATE NUMBER clearly on each of the THREE answer books provided. Hand in THREE answer books even if they have not all been used.

Transcription

1 UNIVERSITY OF LONDON BSc/MSci EXAMINATION May 2007 for Internal Students of Imperial College of Science, Technology and Medicine This paper is also taken for the relevant Examination for the Associateship GENERAL RELATIVITY For Third - and Fourth- Year Physics Students Monday 21st May 2007: to Answer THREE questions. All questions carry equal marks. Marks shown on this paper are indicative of those the Examiners anticipate assigning. General Instructions Write your CANDIDATE NUMBER clearly on each of the THREE answer books provided. If an electronic calculator is used, write its serial number in the box at the top right hand corner of the front cover of each answer book. USE ONE ANSWER BOOK FOR EACH QUESTION. Enter the number of each question attempted in the horizontal box on the front cover of its corresponding answer book. Hand in THREE answer books even if they have not all been used. You are reminded that the Examiners attach great importance to legibility, accuracy and clarity of expression. University of London / 2 / 60 Turn over for questions

2 The Minkowski metric is η μν = diag(1, 1, 1, 1). As usual, Latin indices i, j, k run over 1, 2, 3 and Greek indices α,β... run over 0, 1, 2, 3. The line element is defined by ds 2 = g μν dx μ dx ν. The Christoffel symbol is given by Ɣ α βγ = 1 ( ) 2 gασ g σβ,γ + g σγ,β g βγ,σ, where commas denote ordinary differentiation, i.e. g σγ,β = β g σγ, etc. The Riemann tensor is R β αμν = μ Ɣ β να νɣ β μα + Ɣσ να Ɣβ μσ Ɣσ μα Ɣβ νσ. The Ricci tensor is R αν = R μ αμν

3 1. The Newtonian equations governing the motion of a test particle in a gravitational field, and relating that gravitational field to the distribution of matter are, respectively, and where ρ M is the mass density. d 2 x + ϕ = 0 (1.1) dt2 2 ϕ = 4πGρ M, (1.2) (i) Why does Eq. (1.1) not depend on the mass of the particle? Without going into any experimental detail explain the principle behind the very accurate experiments verifying that this is indeed the case. In general relativity the equation of a geodesic for a material particle is d 2 x μ dτ 2 + dx α dx β Ɣμ αβ dτ dτ = 0. State clearly the conditions which apply in the weak-field, non-relativistic, static limit, when it is appropriate to write g μν = η μν + h μν. Show that in this limit Ɣ i 00 = 1 2 ih 00, so that the spatial part of the geodesic equation reduces to d 2 x i dt c2 i h 00 = 0. (iv) (v) By comparison with Eq.(1.1) show that h 00 = 2ϕ/c 2 in this limit. The equation relating the metric to the energy-momentum tensor is R μν = 8πG c 4 (T μν 1 2 g μνt σ σ ), [1 mark] where T μν = (ρ M + p/c 2 )U μ U ν (p/c 2 )g μν, and U μ is the fluid four-velocity. Show that in the same limit R 00 = λ Ɣ λ 00 and the R 00 equation correctly reproduces Eq. (1.2). [7 marks] Please turn over

4 2. (i) Show that the flat space-time Minkowski metric ds 2 = c 2 dt 2 dx 2 dy 2 dz 2 (2.1) takes the form ds 2 = c 2 dt 2 dρ 2 ρ 2 dφ 2 dz 2 (2.2) in cylindrical polar coordinates in which ρ 2 = x 2 + y 2 and φ is the azimuthal angle about the z axis. [3 marks] Now consider the metric ) ds 2 = (1 2 c 2 (x2 + y 2 ) c 2 dt (ydx xdy)dt dx 2 dy 2 dz 2. (2.3) Show that, in cylindrical polars, it takes the form ds 2 = ) (1 2 c 2 ρ2 c 2 dt 2 2 ρ 2 dφdt dρ 2 ρ 2 dφ 2 dz 2. [3 marks] By making the substitution φ = t show that the metric (2.3) describes Minkowski space-time as seen in a rotating frame. (iv) Derive the geodesics in x, y and z for the metric (2.3). [6 marks] (v) Show that, in the non-relativistic limit, they reduce to the usual equations of Newtonian mechanics for a free particle in a rotating frame exhibiting centrifugal and Coriolis forces

5 3. (i) The covariant derivative D μ V ν = μ V ν + Ɣ ν μα V α is constructed so as to transform as a tensor. What is the geometrical significance of the second term? Write down the covariant derivative of the second rank tensor T λ ν. [1 mark] One way of defining the Riemann tensor is by the non-commutativity of the covariant derivative when acting on a vector: [D μ,d ν ]V λ = R λ αμν V α. Using the expressions of (i) and show that R λ αμν = μɣ λ να νɣ λ μα + Ɣσ να Ɣλ μσ Ɣσ μα Ɣλ νσ. (iv) The tensor R βαμν is defined as R βαμν = g βλ R λ αμν. What are the properties of g μν and the Ɣ λ μα in a free-fall (or local inertial) frame? Show that, in such a frame ) F R βαμν = ηβλ ( μ Ɣ λ να νɣ λ μα F = 1 ) (g βν,αμ g αν,βμ g βμ,αν + g αμ,βν. 2 (v) Hence show that R βαμν is (a) antisymmetric in α β, (b) symmetric in (βα) (μν), (c) cyclic in (αμν), i.e. R βαμν + R βμνα + R βναμ F = 0. [6 marks] (vi) Explain why these relations are true in a general frame Please turn over

6 4. The Schwarzschild metric governing the motion of particles around a black hole is (i) ( ds 2 = c 2 1 a ) dt 2 dr2 r 1 a/r r2 (dθ 2 + sin 2 θdϕ 2 ). Label particle geodesics by the particle proper-time τ, for which ( ) ds 2 L = = c 2. dτ In the equatorial plane θ = π/2 show that the geodesic equations for t and φ take the form ( 1 a ) ṫ r = γ r 2 φ = J, where γ and J are constants of the motion and derivatives are with respect to τ. Suppose that a = 0 (no black hole). Show that, in Cartesian co-ordinates m 2 L = m 2 c 2 = E 2 /c 2 p 2, where E = mc 2 γ and p = mγ v, v = dx/dt. E is the energy of the particle. [3 marks] When a = 0, define E by ( E mc 2 = γ = 1 a ) ṫ. r By rearranging the terms in L, show that where ( E mc 2 V(r, J ) = ) 2 = ṙ2 + V(r, J ), c2 ( 1 a )(1 + J 2 ) r c 2 r 2. (iv) (v) Sketch V(r, J ) for fixed J and a. Show that, for fixed J, V(r, J ) has a minimum at r = r + = J 2 [ ac a2 c 2 ] J 2. As J varies, V(r, J ) is minimised when J 2 = 3a 2 c 2. Show that, for this value, the minimum value of E is E/mc 2 = 8/9. How do you understand this, and what implications does it have for the observation of black holes?

7 5. This question is concerned with linearised gravity, in particular with gravitational waves in empty space-time. The metric is decomposed as g μν = η μν + h μν, where η μν is the Minkowski metric, and h μν the small fluctuations around it. (i) Using the definitions given at the beginning of this exam paper show, to lowest order in h μν and their derivatives, that (h α β,γ + hα γ,β h βγ, α ) Ɣ α βγ = 1 2 and R βδ = 1 2 [ ] h α δ,βα h δβ, α α +h αβ, α δ hα α,βδ. Consider plane wave solutions of the form h μν (x) = Re [ α μν e ik.x], where the components x μ are (ct, x) and the components k μ are (ω/c, k). Show that h μν,ρσ = k ρ k σ h μν, and similarly for other derivatives. Use these results to show that the Ricci tensor R βδ [ k 2 h βδ k β δ k δ β ]. where α = h γ α k γ 1 2 hσ σ k α. Consider the solutions to the Einstein equations in empty space, R βδ = 0. If k 2 = 0, but k μ = 0, for all μ, show that R βδ = 0 implies α = 0. [Hint: use the fact that, k 2 = 0 ω = 0.] (iv) We satisfy this by taking h β α k β = 1 2 hσ σ k α = 0 for each α. In addition, we impose the conditions h 0i = 0, i = 1, 2, 3. By considering a wave in the 3 direction, for example, show that the h μν describe waves travelling at the speed of light, with only two of the amplitudes a μν independent. [7 marks] Please turn over

8 6. This question is concerned with Fermat s principle of least time. Consider a medium with a position-dependent refractive index n(x). That is, the speed of light at x is c/n(x). Fermat s principle states that light moves from a point A to a point B in the medium so as to minimise the time of transit. (i) Show that this is the geodesic in space (not spacetime) from A to B that follows from the metric ds 2 = n(x)ds 2, where ds 2 is the flat-space metric ds 2 = dx 2 + dy 2 + dz 2. [3 marks] Consider a medium with spherical symmetry and metric (in spherical polar coordinates) ds 2 = n(r) [ dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ) ] whose refractive index n(r) only depends on radial distance r, such that n(r) 1 as r. You are asked to calculate the deviation (from straight lines) of light rays that pass near the origin. Without loss of generality we can take rays parallel to the z axis (i.e. fixed φ). (a) Show that, for such rays, n(r)r 2 θ = J, where J is a constant. (b) Thereby, on eliminating θ from the Lagrangian condition L = 1 (where all derivatives are with respect to s), show that 1 n(r) =ṙ2 + J 2 n 2 (r)r 2. (c) Using the relation dr/dθ =ṙ/ θ, show that, if u = 1/r, then ( ) du 2 + u 2 = n(1/u) dθ J 2. (d) If n(r) = 1 + (b/r 2 ) show that this equation reduces to ( ) du 2 = u2 0 u2 dθ (1 + bu 2 0 ), where r 0 = 1/u 0 is the distance of closest approach. (e) Define = θ/ 1 + bu 2 0. By recognising the known result that, for constant n(r) = 1, θ changes by π, show that changes by π, and thus that the deflection of light is δθ = π( 1 + bu 2 0 1). [3 marks] End