 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.