Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 311 / 4 / b / i

Table of contents
- Solution i

Solution

For this metric,

- g = a^{2}

, while

g^{ηη} = - a^{- 2}

,

g^{zz} = a^{- 2}

and

g^{ij} = δ^{ij}

. The massless Klein-Gordon equation is consequently

□ ϕ = \frac{1}{a ^{2}} (- \partial_{η}^{2} + \partial_{z}^{2}) ϕ + δ^{ij} \partial_{i} \partial_{j} ϕ = 0.

Insert the spatial Fourier transform given in the question. Each wavenumber mode then satisfies

Φ_{k_{z} k}^{''} + ω_{k_{z} k}^{2} (η) Φ_{k_{z} k} = 0, ω_{k_{z} k}^{2} (η) = k_{z}^{2} + a (η)^{2} ∣ k ∣^{2} .

Thus every field mode is a simple harmonic oscillator with a time-dependent angular frequency.

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