Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 357 / 2 / e

Solution

Since

α n = \partial_{t} - β

and

j_{μ} n^{μ} = 0

, the scaling rule for a covector Lie derivative has no extra term:

α L_{n} j_{μ} = L_{\partial_{t} - β} j_{μ} = \partial_{t} j_{μ} - L_{β} j_{μ} .

The acceleration is the spatial lapse gradient,

a_{μ} = D_{μ} lo g α = \frac{D _{μ} α}{α} .

Multiplying the momentum equation by

α

therefore gives

\partial_{t} j_{μ} = L_{β} j_{μ} - α D_{ν} S^{ν}_{μ} - S^{ν}_{μ} D_{ν} α + α K j_{μ} - ρ D_{μ} α .

For spatial components, the shift term can be written

L_{β} j_{i} = β^{k} D_{k} j_{i} + j_{k} D_{i} β^{k},

so explicitly

\partial_{t} j_{i} = β^{k} D_{k} j_{i} + j_{k} D_{i} β^{k} - α D_{k} S^{k}_{i} - S^{k}_{i} D_{k} α + α K j_{i} - ρ D_{i} α .

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