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Information about the mathematical tripos part iii exam held on may 31, 2001, focusing on general relativity and gauge-invariant perturbations. Three questions, one of which requires writing an essay on gauge-invariant, linearized vacuum perturbations of flat minkowski spacetimes. The document also provides the action for the brans-dicke theory of gravity and asks to derive the field equation for the scalar field. Additionally, there are questions on the definition of a linear connection and its relationship to the metric tensor, as well as a comparison of toy models for de sitter and anti-de sitter spacetimes.
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Thursday 31 May 2001 1.30 to 4.
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.
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
16 πGS =
−g
ωΦ,aΦ,a Φ
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^ − 12 gabgcd)Φ,cΦ,d = − 8 πGT (^) matterab ,
where Gab^ is the Einstein tensor, and show that the field equation for Φ can be written in the form
Φ =
8 πG 3 + 2ω
gcd T (^) mattercd.
[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
