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Komar angular momentum is the Komar charge associated with an axial Killing vector field in four-dimensional general relativity. In geometrized units its normalization is $(16\pi {)}^{-1}{\int }_S\star dm$, where $m$ is the metric-dual differential one-form. With zero cosmological constant, vacuum Einstein field equations make $d\star dm=0$, so homologous surrounding surfaces in the vacuum region give the same charge.

With zero cosmological constant and the component Hodge star operator convention $(\star j{)}_{abc}={ϵ}_{abcd}{j}^d$, the Einstein field equations and Killing equation give $d\star dm=-16\pi \star (T_{ab}{m}^b-\frac{1}{2}T{m}_a)$. If $m$ is tangent to a spacelike hypersurface, the trace term pulls back to zero. For $\partial \Sigma =S_{\infty }-{\bigcup }_i{S}_i$, Stokes theorem yieldsThe 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.

Komar mass is the Komar charge associated with a stationary timelike Killing vector, using the normalization appropriate to gravitational mass.

Use geometrized units $G=c=1$. For a stationary asymptotically flat spacetime, normalize the timelike Killing vector field $\xi$ to have squared norm $-1$ at infinity. The Komar mass isHere the oriented binormal is chosen so that an ordinary positive-mass static solution gives a positive integral. Equivalently, on a static slice with lapse $N=\sqrt{-{\xi }^2}$ and outward spatial unit normal $s^i$,This second expression fixes the orientation unambiguously and is convenient for the Majumdar–Papapetrou solution.

The spatial metric is ${\gamma }_{ij}=H^2{\delta }_{ij}$ and $N=H^{-1}$. On a large coordinate sphere, $s^i=H^{-1}{\hat{r}}^i$ and $dA=H^2{r}^{2}d\Omega$. ThusThe original PDF has the Euclidean norm in each source denominator, which is missing in the TeX aid. With that norm, the harmonic function has asymptoticsThereforeIf $M_i$ are retained as length parameters while $G$ is restored, the physical mass is ${\sum }_i{M}_i/G$. This is the charge at infinity. A Komar integral is not generally surface-independent here, since Einstein-Maxwell theory has the electromagnetic stress-energy tensor between the surfaces; in particular it should not be confused with a sum of vacuum horizon Komar charges.

For a Killing vector field $\xi$, set $K^{\mu \nu }={\nabla }^{\mu }{\xi }^{\nu }=-K^{\nu \mu }$. Choose surface orientation conventions so that the Komar charge, up to overall normalization, isThis is the Komar current from trace-reversed stress-energy construction. The equality is the spacetime Stokes theorem. With $[{\nabla }_{\mu },{\nabla }_{\nu }]V^{\rho }=R^{\rho }{}_{\sigma \mu \nu }{V}^{\sigma }$, the Killing equation and ${\nabla }_{\nu }{\xi }^{\nu }=0$ give $J_{\xi }^{\mu }=R^{\mu }{}_{\nu }{\xi }^{\nu }$. In four dimensions, at zero cosmological constant, the Einstein field equations give the currentReversing orientation changes the common overall sign, not conservation. The contracted Bianchi identity givesAn isometry preserves scalar curvature, and the second term contracts symmetric and antisymmetric tensors. Equivalently stress-energy conservation, the Killing equation, and ${\mathcal{L}}_{\xi }T=0$ cancel the matter expression; its trace derivative needs this last observation. Hence $\boxed{{\nabla }_{\mu }{J}_{\xi }^{\mu }=0}$. Nonzero cosmological constant adds $\Lambda {\xi }^{\mu }$ to the geometric current, also divergence-free.