The Komar mass of a system is one of several formal concepts of mass that are used in general relativity. The Komar mass can be defined in any stationary spacetime, which is a spacetime in which all the metric components can be written so that they are independent of time. Alternatively, a stationary spacetime can be defined as a spacetime which possesses a timelike Killing vector field. It is named after Arthur Komar who developed the concept in 1962. The following discussion is an expanded and simplified version of the motivational treatment in Wald (1984, p. 288).
Motivation Consider the Schwarzschild metric. Using the Schwarzschild basis, a frame field for the Schwarzschild metric, one can find that the radial acceleration required to hold a test mass stationary at a Schwarzschild coordinate of r is:
a r ^ = m r 2 1 − 2 m r c 2 {\displaystyle a^{\hat {r}}={\frac {m}{r^{2}{\sqrt {1-{\frac {2m}{rc^{2}}}}}}}}
Because the metric is static, there is a well-defined meaning to "holding a particle stationary". Interpreting this acceleration as being due to a "gravitational force", we can then compute the integral of normal acceleration multiplied by area to get a "Gauss law" integral of:
4 π m 1 − 2 m r c 2 {\displaystyle {\frac {4\pi m}{\sqrt {1-{\frac {2m}{rc^{2}}}}}}}
While this approaches a constant as r approaches infinity, it is not a constant independent of r. We are therefore motivated to introduce a correction factor to make the above integral independent of the radius r of the enclosing shell. For the Schwarzschild metric, this correction factor is just g t t {\displaystyle {\sqrt {g_{tt}}}} , the "red-shift" or "time dilation" factor at distance r. One may also view this factor as "correcting" the local force to the "force at infinity", the force that an observer at infinity would need to apply through a string to hold the particle stationary. To proceed further, we will write down a line element for a static metric.
d s 2 = g t t d t 2 + q u a d r a t i c f o r m ( d x , d y , d z ) {\displaystyle ds^{2}=g_{tt}\,dt^{2}+\mathrm {quadratic\ form} (dx,\,dy,\,dz)}
where g t t {\displaystyle g_{tt}} and the quadratic form are functions only of the spatial coordinates x, y, z and are not functions of time. In spite of our choices of variable names, it should not be assumed that our coordinate system is Cartesian. The fact that none of the metric coefficients are functions of time makes the metric stationary: the additional fact that there are no "cross terms" involving both time and space components (such as d x d t {\displaystyle dxdt} ) make it static. Because of the simplifying assumption that some of the metric coefficients are zero, some of our results in this motivational treatment will not be as general as they could be. In flat space-time, the proper acceleration required to hold station is d u / d τ {\displaystyle du/d\tau } , where u is the 4-velocity of our hovering particle and τ {\displaystyle \tau } is the proper time. In curved space-time, we must take the covariant derivative. Thus we compute the acceleration vector as:
a b = ∇ u u b = u c ∇ c u b {\displaystyle a^{b}=\nabla _{u}u^{b}=u^{c}\nabla _{c}u^{b}}
a b = u c ∇ c u b {\displaystyle a_{b}=u^{c}\nabla _{c}u_{b}}
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