Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 357 / 1 / a

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- Solution a

Solution

The inverse metric condition is

(η^{α ρ} + k^{α ρ}) (η_{ρβ} + h_{ρβ}) = δ^{α}_{β} .

Keeping only terms linear in the metric perturbation gives

k^{α}_{β} + h^{α}_{β} = 0.

Thus the linearized inverse metric is

k^{α β} = - h^{α β}, g^{α β} = η^{α β} - h^{α β} + O (ϵ^{2}),

where indices on

h_{α β}

are raised with the Minkowski metric.

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