Static spherical Komar mass in reciprocal radial gauge ($M_K(r)=R(r{)}^2{f}^{\prime }(r)/(2G)$)

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.