 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2004/iii/paper-59/2/b/solution

 Top articles  Latest articles New article in topic

Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 59 / 2 / b / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-14

The massless scalar stress tensor in mixed-index form is

T^{μ}_{ν} = g^{μα} \partial_{α} ϕ \partial_{ν} ϕ - \frac{1}{2} δ^{μ}_{ν} g^{α β} \partial_{α} ϕ \partial_{β} ϕ .

In conformal Newtonian gauge,

g^{00} = - a^{- 2} (1 - 2Φ)

and

g^{ij} = a^{- 2} (1 + 2Ψ) δ^{ij}

to first order. Spatial gradients of the background vanish, so products of two spatial perturbation gradients are second order. Consequently

g^{α β} \partial_{α} ϕ \partial_{β} ϕ = - a^{- 2} (ϕ_{0}^{'2} + 2 ϕ_{0}^{'} δ ϕ^{'} - 2Φ ϕ_{0}^{'2}) .

Substitution yields

a^{2} T^{0}_{0} = - \frac{1}{2} ϕ_{0}^{'2} - ϕ_{0}^{'} δ ϕ^{'} + Φ ϕ_{0}^{'2}, a^{2} T^{0}_{i} = - ϕ_{0}^{'} \partial_{i} δ ϕ .

The potential

Ψ

multiplies only second-order gradient products in these components. The same reasoning makes the off-diagonal spatial stress vanish at first order, consistent with

Φ = Ψ

.

 Read the full article

 New to topics? Read the docs here!