Komar Mass - Motivation

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 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 make 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 du/d tau, 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 u_b = -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 a_b and the adjusted "acceleration at infinity" ainf_b 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" ainf 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 lengthly, 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. Because Stokes theorem guarantees that our surface integral is equal to the above volume integral, our surface integral will also be independent of the enclosing surface as long as the surface encloses all of the gravitating mass.

By using Einstein's Field Equations

letting u=v and summing, we can show that R = -8 π T.

This allows us to rewrite our mass formula as a volume integral of the stress-energy tensor.

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