Komar Mass - Komar Mass As Volume Integral - General Stationary Metric

General Stationary Metric

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.

where V is the volume being integrated over

T_ab is the Stress-energy tensor

ua is a unit time-like vector such that ua u_a = -1

is a Killing vector, which expresses the time-translation symmetry of any stationary metric. The Killing vector is normalized so that it has a unit length at infinity, i.e. so that at infinity.

Note that replaces in our motivational result.

If none of the metric coefficients are functions of time,

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 becomes a scalar multiple of a unit coordinate-time vector, i.e. . When this is the case, we can rewrite our formula as

Because is by definition a unit vector, K is just the length of, i.e. K = .

Evaluating the "red-shift" factor K based on our knowledge of the components of, we can see that K = .

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. Furthermore, the contribution of local energy and mass to the system mass is multiplied by the local "red shift" factor