Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 312 / 2 / i

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- Solution i

Solution

The inverse 3+1 decomposition of spacetime metric gives

X = \frac{1}{2 N ^{2}} (\dot{ϕ} - N^{i} \partial_{i} ϕ)^{2} - \frac{1}{2} h^{ij} \partial_{i} ϕ \partial_{j} ϕ .

The first term is the squared derivative along the hypersurface normal and the second is the spatial-gradient contribution.

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