= Komar angular momentum with inner boundaries
{c}
With zero <cosmological constant> and the component <Hodge star operator> convention $(\star j)_{abc}=\epsilon_{abcd}j^d$, the <Einstein field equations> and <Killing equation> give $d\star dm=-16\pi\star(T_{ab}m^b-\tfrac12Tm_a)$. If $m$ is tangent to a spacelike <hypersurface>, the trace term pulls back to zero. For $\partial\Sigma=S_\infty-\bigcup_iS_i$, <Stokes theorem> yields
$$
J_\infty-\sum_iJ_i=-\int_\Sigma\star(T_{ab}m^b\,dx^a).
$$
The inner <Komar charges> cannot be dropped for a vacuum <Kerr black hole>, whose <angular momentum> is nonzero although its <stress-energy tensor> vanishes. A matter-only volume formula requires a regular filling with no inner boundaries.
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