Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 37B (General Relativity)

Table of contents
- a 37B
  - Solution a
- b 37B
  - Solution b

a

Solution

The Levi-Civita connection is

Γ^{α}_{β γ} = \frac{1}{2} g^{α δ} (g_{δ β, γ} + g_{δ γ, β} - g_{β γ, δ}) .

For

g_{α β} = η_{α β} + h_{α β}

, retain only first-order terms in

h

. Substitution into the curvature formula and contraction gives the linearized Ricci tensor and scalar

2 R_{α β} = h_{α}^{γ}_{, γ β} + h_{β}^{γ}_{, γ α} - h_{γ}^{γ}_{, α β} - h_{α β, γ ρ} η^{γ ρ} .

Hence

A = 1, B = 1, C = - 1, D = - 1.

b

Solution

For the given metric, direct substitution of the stated connection coefficients into the curvature formula leaves

R_{uu} = - \frac{1}{2} (H_{, xx} + H_{, yy}) .

The inverse metric has

g^{uu} = 0

, and every other Ricci component vanishes, so

R = g^{α β} R_{α β} = 0.

Thus the full, nonlinear vacuum Einstein equation is precisely the transverse Laplace equation from a plane-fronted gravitational wave:

H_{, xx} + H_{, yy} = 0.

The dependence on

u

is unrestricted.

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