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3+1 decomposition of spacetime

3+1 decomposition of the stress-energy tensor

With $K_{\mu\nu}=-\perp^\rho{}_\mu\nabla_\rho n_\nu$ and $a_\mu=n^\rho\nabla_\rho n_\mu$, spatial projection of $\nabla_\mu T^{\mu\nu}=0$ gives
$$
\mathcal L_nj_\alpha=-D_\mu S^\mu{}_\alpha-S^\mu{}_\alpha a_\mu+Kj_\alpha-\rho a_\alpha.
$$


Momentum equation in a 3+1 decomposition

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

Momentum equation in a 3+1 decomposition

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