Komar mass

Komar mass is the Komar charge associated with a stationary timelike Killing vector, using the normalization appropriate to gravitational mass.

Table of contents
- Komar mass normalization in higher dimensions Komar mass
- Static spherical Komar mass in reciprocal radial gauge Komar mass

Komar mass normalization in higher dimensions ( $M = - \frac{D - 2}{16 π G _{D} ( D - 3 )} \int \nabla^{a} k^{b} d Σ_{ab}$ )

 0  0

For future static-slice normal

u

and outward radial normal

n

, take

d Σ_{ab} = (n_{a} u_{b} - n_{b} u_{a}) d A

, with both antisymmetric components summed. For the Schwarzschild-Tangherlini metric the flux is

- Ω_{D - 2} r^{D - 2} f^{'} = - (D - 3) Ω_{D - 2} μ

. 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.

Static spherical Komar mass in reciprocal radial gauge ( $M_{K} (r) = R (r)^{2} f^{'} (r) / (2 G)$ )

 0  0

For

d s^{2} = - f (r) d t^{2} + d r^{2} / f (r) + R (r)^{2} d Ω^{2}

and

K = \partial_{t}

normalized at infinity, the Komar mass on a round sphere is

R^{2} f^{'} / (2 G)

. 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^{'} /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.