The Komar mass (named after Arthur Komar) 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.
The following discussion is an expanded and simplified version of the motivational treatment in (Wald, 1984, pg 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:
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:
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 , 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. (Wald, 1984).
To proceed further, we will write down a line element for a static metric.
where gtt 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 dx dt) 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 , where u is the 4-velocity of our hovering particle and tau is the proper time. In curved space-time, we must take the covariant derivative. Thus we compute the acceleration vector as:
where ub is a unit time-like vector such that ub ub = -1.
The component of the acceleration vector normal to the surface is
where Nb is a unit vector normal to the surface.
In a Schwarzschild coordinate system, for example, we find that
as expected - we have simply re-derived the previous results presented in a frame-field in a coordinate basis.
We define
so that in our Schwarzschild example:
We can, if we desire, derive the accelerations ab and the adjusted "acceleration at infinity" a infb from a scalar potential Z, though there is not necessarily any particular advantage in doing so. (Wald 1984, pg 158, problem 4)
We will demonstrate that integrating the normal component of the "acceleration at infinity" a inf over a bounding surface will give us a quantity that does not depend on the shape of the enclosing sphere, so that we can calculate the mass enclosed by a sphere by the integral
To make this demonstration, we need to express this surface integral as a volume integral. In flat space-time, we would use Stokes theorem and integrate over the volume. In curved space-time, this approach needs to be modified slightly.
Using the formulas for electromagnetism in curved space-time as a guide, we write instead.
where F plays a role similar to the "Faraday tensor", in that We can then find the value of "gravitational charge", i.e. mass, by evaluating and integrating it over the volume of our sphere.
An alternate approach would be to use differential forms, but the approach above is computationally more convenient as well as not requiring the reader to understand differential forms.
A lengthy, but straightforward (with computer algebra) calculation from our assumed line element shows us that