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Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 310 / 2 / b / Solution Created 2026-10-03 Updated 2026-10-06

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The same invariant interval can be written as

g_{μν} d x^{μ} d x^{ν} = g_{α β} d x^{α} d x^{β}

. Applying the chain rule to the differentials proves the metric transformation law

g_{μν} (x) = \frac{\partial x ^{α}}{\partial x ^{μ}} \frac{\partial x ^{β}}{\partial x ^{ν}} g_{α β} (x) .

(1)

For first-order scalar cosmological perturbations and

ξ^{μ} = (T, \partial^{i} L)

, compare perturbations at the same background coordinate label. Expanding the Jacobians and the shifted background gives the linear metric gauge-transformation law

δ g_{μν} = δ g_{μν} - L_{ξ} \overset{g}{ˉ}_{μν} .

(2)

For the background

\overset{g}{ˉ}_{00} = - a^{2}

\overset{g}{ˉ}_{ij} = a^{2} δ_{ij}

, the Lie derivative components are

(L_{ξ} \overset{g}{ˉ})_{00} = - 2 a^{2} (T^{'} + H T), (L_{ξ} \overset{g}{ˉ})_{0 i} = a^{2} \partial_{i} (L^{'} - T),

(3)

(L_{ξ} \overset{g}{ˉ})_{ij} = 2 a^{2} H T δ_{ij} + 2 a^{2} \partial_{i} \partial_{j} L, H = \frac{a ^{'}}{a} .

(4)

Here primes denote conformal time and

H

is the conformal Hubble parameter. These expressions establish all four alternatives, including any chosen pair.

The

00

component is

δ g_{00} = - 2 a^{2} A

, immediately giving the lapse function perturbation transformation. The

0 i

component is

a^{2} \partial_{i} B

, giving the shift vector scalar-potential transformation. The trace of the spatial perturbation is

6 a^{2} C

, so tracing the spatial Lie derivative gives the transformation of

C

. Its trace-free scalar part is

2 a^{2} (\partial_{i} \partial_{j} - δ_{ij} \nabla^{2} /3) E

, giving the transformation of

E

. Thus

A B C E = A - T^{'} - H T, = B + T - L^{'}, = C - H T - \frac{1}{3} \nabla^{2} L, = E - L .

(5)

As usual, identifying scalar potentials from their derivatives uses the standard boundary conditions, or nonzero Fourier modes, to remove homogeneous ambiguities.

At first order, tensor cosmological perturbations are spatial transverse-traceless tensors. The coordinate-generated spatial perturbation consists of a trace term and symmetrized derivatives of the displacement. In Fourier space, the latter terms carry a factor

k_{i}

k_{j}

. The transverse-traceless projector removes these longitudinal terms and the trace, including the derivative of a transverse vector displacement. Therefore the tensor perturbation is gauge invariant at linear order around the homogeneous background. This is a first-order statement, not a claim of automatic invariance at arbitrary perturbative order.

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