The following discussion is an expanded and simplified version of the motivational treatment in (Wald, 1984, pg 288).
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.
, 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.
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
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
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.
(Wald 1984, pg 158, problem 4) We will demonstrate that integrating the normal component of the "acceleration at infinity"
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.
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
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 Thus we can write In any vacuum region of space-time, all components of the Ricci tensor must be zero.
This demonstrates that enclosing any amount of vacuum will not change our volume integral.
It also means that our volume integral will be constant for any enclosing surface, as long as we enclose all of the gravitating mass inside our surface.
By using Einstein's Field Equations letting u=v and summing, we can show that
This allows us to rewrite our mass formula as a volume integral of the stress–energy tensor.
where To make the formula for Komar mass work for a general stationary metric, regardless of the choice of coordinates, it must be modified slightly.
We will present the applicable result from (Wald, 1984 eq 11.2.10) without a formal proof.
While it is not necessary to choose coordinates for a stationary space-time such that the metric coefficients are independent of time, it is often convenient.
When we chose such coordinates, the time-like Killing vector for our system
Evaluating the "red-shift" factor K based on our knowledge of the components of
If we chose our spatial coordinates so that we have a locally Minkowskian metric
we know that With these coordinate choices, we can write our Komar integral as While we can't choose a coordinate system to make a curved space-time globally Minkowskian, the above formula provides some insight into the meaning of the Komar mass formula.
Essentially, both energy and pressure contribute to the Komar mass.
We also wish to give the general result for expressing the Komar mass as a surface integral.