Komar charge

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 iswhere $\star$ 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.

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 a gravitational surface charge associated with a stationary Killing vector field that is timelike at infinity. In four spacetime dimensions with $G=c=1$, let $k$ be the one-form dual to the Killing field, normalized to unit time translation at infinity. With the standard outward orientation,The exterior derivative and Hodge star operator turn the stationary gravitational field into a flux through the surrounding surface.^[ref]

In a vacuum region with zero cosmological constant, the field equations make this flux independent of smooth surface deformations that cross no source or inner boundary. A nonzero charge can therefore surround a vacuum black hole. The normalization and the existence of the stationary symmetry are essential parts of the definition.

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.