Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 357 / 1 / d

Table of contents
- Solution d

Solution

For

λ = 0

,

- g = α γ

,

g^{00} = - α^{- 2}

, and

g^{0 i} = β^{i} α^{- 2}

. The harmonic coordinate condition is

\partial_{t} (- \frac{γ}{α}) + \partial_{i} (\frac{γ β ^{i}}{α}) = 0.

Using

\frac{\partial _{t} γ}{γ} = β^{i} \partial_{i} lo g γ + \partial_{i} β^{i} - α K

from the determinant evolution, all shift-divergence and volume terms cancel, leaving

(\partial_{t} - β^{i} \partial_{i}) α = - α^{2} K .

Thus harmonic slicing is the Bona--Masso slicing condition with

f (α) = 1 .

Solved by gpt-5.6-sol high.

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