Mathematical Tripos Part III Exam: General Relativity and Gauge-Invariant Perturbations, Exams of Mathematics

Download Mathematical Tripos Part III Exam: General Relativity and Gauge-Invariant Perturbations and more Exams Mathematics in PDF only on Docsity! MATHEMATICAL TRIPOS Part III Thursday 31 May 2001 1.30 to 4.30 PAPER 68 GENERAL RELATIVITY Attempt any THREE questions. The questions are of equal weight. Candidates may make free use of the information given on the accompanying sheet. Information The signature is ( + − − − ), and the curvature tensor conventions are defined by Rikmn = Γikm,n − Γikn,m − ΓipmΓpkn + ΓipnΓpkm . You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator. 2 1 Define the concept of a linear connection ∇ on a manifold M. If M possesses a metric tensor g whose components with respect to a local coordinate system are gab, show that there exists a unique symmetric Levi-Civita or metric connection such that ∇g = 0, and determine its components Γabc. Next suppose that gab are the components of another symmetric tensor g on M, and let Γ abc be the components of that symmetric connection for which ∇ g = 0. Show that Sabc = Γ abc − Γabc are the components of a tensor S. Finally suppose that the connections Γ and Γ have the same geodesics. Show that there exists a covector Vc (to be determined) such that Sabc = 2δa(bVc). 2 Write an essay on gauge-invariant, linearized, vacuum perturbations of flat Minkowski spacetimes. [You may use any information from the lecture handout included with this examination paper.] 3 Using standard notation the action S for the Brans-Dicke theory of gravity is given by 16πGS = ∫ √ −g [ RΦ + ωΦ,aΦ,a Φ + 16πGLmatter ] dΩ, where Φ is a scalar field, ω is a coupling constant and Lmatter is the Lagrangian of the matter content. Derive the field equation ΦGab + (gacgbd − gabgcd)Φ;cd + ωΦ−1(gacgbd − 12g abgcd)Φ,cΦ,d = −8πGT abmatter, where Gab is the Einstein tensor, and show that the field equation for Φ can be written in the form Φ = 8πG 3 + 2ω gcd T cd matter. [You may use freely the following results for variations: (a) δgcd = −gacgbdδgab, (b) δ √ −g = 12 √ −ggabδgab, (c) δRbc = ( δΓaba ) ;c − ( δΓabc ) ;a. ] Paper 68