Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 310 / 3 / c

Table of contents
- Solution c

Solution

For a spatial index

j

, the background Christoffel symbols give

\overset{ˉ}{Γ}^{μ}_{μ 0} = 3 H, - \overset{ˉ}{Γ}^{i}_{0 j} δ T^{0}_{i} = - Hδ T^{0}_{j}, - \overset{ˉ}{Γ}^{0}_{ij} δ T^{i}_{0} = - a^{2} Hδ T^{j}_{0} .

Together with the

3 Hδ T^{0}_{j}

trace term, the first two contributions combine to

2 Hδ T^{0}_{j}

. Varying the Levi-Civita connection and contracting it with

\overset{ˉ}{T}^{μ}_{ν} = diag (- \overset{ρ}{ˉ}, \overset{ˉ}{P}, \overset{ˉ}{P}, \overset{ˉ}{P})

gives

δ Γ^{μ}_{μ j} = \frac{1}{2} \partial_{j} (h^{μ}_{μ}),

and the surviving combination reduces to

- (\overset{ρ}{ˉ} + \overset{ˉ}{P}) (\frac{1}{2} \partial_{j} h_{00} - H h_{j 0}) .

Consequently the spatial momentum equation is

\partial_{0} δ T^{0}_{j} + \partial_{i} δ T^{i}_{j} + 2 Hδ T^{0}_{j} - a^{2} Hδ T^{j}_{0} - (\overset{ρ}{ˉ} + \overset{ˉ}{P}) (\frac{1}{2} \partial_{j} h_{00} - H h_{j 0}) = 0 .

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