= Komar charge
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= Komar integral
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{synonym}
A <Komar charge> is an oriented surface integral constructed from a <Killing vector field> and the <metric tensor>. Let $k$ be the metric-dual <differential one-form> of a stationary <Killing vector field>. In four dimensions, with <geometrized units> $G=c=1$ and the standard future/outward orientation, its mass normalization is
$$
M_K=-\frac1{8\pi}\int_S\star dk,
$$
where $\star$ is the <Hodge star operator> and $S$ is a closed spacelike two-surface. The normalization of the <Killing vector field> fixes the <time> scale: for <Schwarzschild spacetime>, unit time translation at infinity gives the black-hole mass parameter.
The <Killing equation> relates the divergence of $dk$ to a contraction of the <Ricci tensor>. Consequently, homologous enclosing surfaces give the same mass in a vacuum region with zero <cosmological constant>. Matter or a nonzero cosmological constant introduces a volume contribution; an arbitrary finite-radius surface integral need not equal the asymptotic mass. Axial symmetry instead gives <Komar angular momentum> with a different normalization.
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