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

For this metric, $\sqrt{-g}=a^2$, while $g^{\eta \eta }=-a^{-2}$, $g^{zz}=a^{-2}$ and $g^{ij}={\delta }^{ij}$. The massless Klein-Gordon equation is consequentlyInsert the spatial Fourier transform given in the question. Each wavenumber mode then satisfiesThus every field mode is a simple harmonic oscillator with a time-dependent angular frequency.