Komar charge

A Komar integral constructs a conserved gravitational charge from the exterior derivative of the one-form dual to a Killing vector. In four dimensions, a stationary mass is conventionally normalized as

A Killing vector field gives an antisymmetric tensor ${\nabla }^{\mu }{\xi }^{\nu }$ whose divergence is this current, up to overall surface-orientation convention. The four-dimensional Einstein field equations express it as $8\pi G(T^{\mu }{}_{\nu }-\frac{1}{2}T{\delta }^{\mu }{}_{\nu }){\xi }^{\nu }$ at zero cosmological constant. The contracted Bianchi identity and Killing invariance of scalar curvature prove its divergence is zero. Stress conservation alone does not cancel the trace derivative without Killing invariance of the trace.

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.

For an axial Killing vector $\xi$ and zero cosmological constant, set $F^{ab}={\nabla }^b{\xi }^a$. The Killing equation and Ricci identity give ${\nabla }_b{F}^{ab}=-R^a{}_b{\xi }^b$. In a vacuum region this vanishes, so Stokes theorem makes the Komar angular momentum equal on homologous enclosing surfaces. The region may contain gravitational waves with nonzero Weyl curvature: Ricci flatness, rather than complete flatness, is what removes the flux. This proves absence of axial angular-momentum transport by exactly axisymmetric gravitational radiation, without assuming that a radiating spacetime is stationary.

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.

For future static-slice normal $u$ and outward radial normal $n$, take $d{\Sigma }_{ab}=(n_a{u}_b-n_b{u}_a)dA$, with both antisymmetric components summed. For the Schwarzschild-Tangherlini metric the flux is $-{\Omega }_{D-2}{r}^{D-2}{f}^{\prime }=-(D-3){\Omega }_{D-2}\mu$. The displayed dimension-dependent coefficient then reproduces the ADM mass. Reversing orientation reverses the flux; including a factor of one-half in the binormal convention also changes its numerical value. A four-dimensional normalization cannot be carried unchanged into other dimensions.

For $d{s}^2=-f(r)d{t}^2+d{r}^2/f(r)+R(r{)}^{2}d{\Omega }^2$ and $K={\partial }_t$ normalized at infinity, the Komar mass on a round sphere is $R^2{f}^{\prime }/(2G)$. The future unit normal to a static slice is $f^{-1/2}K$, and the outward unit radial normal is $f^{1/2}{\partial }_r$; their contraction with ${\nabla }_a{K}_b$ gives $f^{\prime }/2$, whose integral over the sphere yields the formula. The asymptotic mass is its limit at infinity. In the presence of matter, the finite-radius integral need not be constant.

For a spacetime with a non-flat reference background, replacing $k$ by $k-\bar{k}$ removes the background contribution and can make the Komar surface integral finite.