 Momentum equation in a 3+1 decomposition

ID: momentum-equation-in-a-3-plus-1-decomposition

With

K_{μν} = - ⊥^{ρ}_{μ} \nabla_{ρ} n_{ν}

and

a_{μ} = n^{ρ} \nabla_{ρ} n_{μ}

, spatial projection of

\nabla_{μ} T^{μν} = 0

gives

L_{n} j_{α} = - D_{μ} S^{μ}_{α} - S^{μ}_{α} a_{μ} + K j_{α} - ρ a_{α} .

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