Komar mass ($M_K$)

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.