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

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- Solution ii

Solution

At fixed

N

and

h_{ij}

, varying the shift gives

δ K_{ij} = - \frac{1}{2 N} (^{(3)} \nabla_{i} δ N_{j} +^{(3)} \nabla_{j} δ N_{i})

and

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

Integration by parts in the gravitational term turns the variation of

K_{ij} K^{ij} - K^{2}

into the spatial divergence of

K_{i}^{j} - δ_{i}^{j} K

. Since the shift has no time derivative, its Euler-Lagrange equation is the ADM momentum constraint for a P(X, phi) scalar field

^{(3)} \nabla_{j} (K_{i}^{j} - δ_{i}^{j} K) + \frac{P _{, X}}{M _{Pl}^{2} N} \partial_{i} ϕ (N^{j} \partial_{j} ϕ - \dot{ϕ}) = 0 .

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