Komar mass in stationary spacetimes

A non-technical definition of a stationary spacetime is a spacetime where none of the metric coefficients ${\displaystyle g_{\mu \nu }\,}$ are functions of time. The Schwarzschild metric of a black hole and the Kerr metric of a rotating black hole are common examples of stationary spacetimes.

By definition, a stationary spacetime exhibits time translation symmetry. This is technically called a time-like Killing vector. Because the system has a time translation symmetry, Noether's theorem guarantees that it has a conserved energy. Because a stationary system also has a well defined rest frame in which its momentum can be considered to be zero, defining the energy of the system also defines its mass. In general relativity, this mass is called the Komar mass of the system. Komar mass can only be defined for stationary systems.

Komar mass can also be defined by a flux integral. This is similar to the way that Gauss's law defines the charge enclosed by a surface as the normal electric force multiplied by the area. The flux integral used to define Komar mass is slightly different from that used to define the electric field, however – the normal force is not the actual force, but the "force at infinity". See the main article for more detail.

Of the two definitions, the description of Komar mass in terms of a time translation symmetry provides the deepest insight.

ADM and Bondi masses in asymptotically flat space-times

If a system containing gravitational sources is surrounded by an infinite vacuum region, the geometry of the space-time will tend to approach the flat Minkowski geometry of special relativity at infinity. Such space-times are known as "asymptotically flat" space-times.

For systems in which space-time is asymptotically flat, the ADM and Bondi energy, momentum, and mass can be defined. In terms of Noether's theorem, the ADM energy, momentum, and mass are defined by the asymptotic symmetries at spatial infinity, and the Bondi energy, momentum, and mass are defined by the asymptotic symmetries at null infinity. Note that mass is computed as the length of the energy–momentum four-vector, which can be thought of as the energy and momentum of the system "at infinity".

The ADM energy is defined through the following flux integral at infinity.^[1] If a spacetime is asymptotically flat this means that near "infinity" the metric tends to that of flat space. The asymptotic deviations of the metric away from flat space can be parametrized by

${\displaystyle g_{\mu \nu }=\eta _{\mu \nu }+h_{\mu \nu }}$

where ${\displaystyle \eta _{\mu \nu }}$ is the flat space metric. The ADM energy is then given by an integral over a surface, ${\displaystyle S}$ at infinity

${\displaystyle P^{0}={1 \over 16\pi G}\int \left(\partial ^{k}h_{jk}-\partial ^{j}h_{kk}\right)d^{2}S_{j},}$

where ${\displaystyle S_{j}}$ is the outward-pointing normal to ${\displaystyle S}$. The Einstein summation convention is assumed for repeated indices but the sum over k and j only runs over the spatial directions. The use of ordinary derivatives instead of covariant derivatives in the formula above is justified because of the assumption that the asymptotic geometry is flat.

Some intuition for the formula above can be obtained as follows. Imagine that that we take the surface, S, to be a spherical surface so that the normal points radially outwards. At large distances from the source of the energy, r, the tensor ${\displaystyle h_{ij}}$ is expected to fall off as ${\displaystyle r^{-1}}$ and the derivative with respect to r converts this into ${\displaystyle r^{-2}}$ The area of the sphere at large radius also grows precisely as ${\displaystyle r^{2}}$ and therefore one obtains a finite value for the energy.

It is also possible to obtain expressions for the momentum in asymptotically flat spacetime. To obtain such an expression one defines

${\displaystyle H^{\mu \alpha \nu \beta }=-{\bar {h}}^{\mu \nu }\eta ^{\alpha \beta }-\eta ^{\mu \nu }{\bar {h}}^{\alpha \beta }+{\bar {h}}^{\alpha \nu }\eta ^{\mu \beta }+{\bar {h}}^{\mu \beta }\eta ^{\alpha \nu }}$

where

${\displaystyle {\bar {h}}_{\mu \nu }=h_{\mu \nu }-{1 \over 2}\eta _{\mu \nu }h_{\alpha }^{\alpha }}$

Then the momentum is obtained by a flux integral in the asymptotically flat region

${\displaystyle P^{\mu }={1 \over 16\pi G}\int \partial _{\alpha }H^{\mu \alpha 0j}d^{2}S_{j}}$

Note that the expression for ${\displaystyle P^{0}}$ obtained from the formula above coincides with the expression for the ADM energy given above as can easily be checked using the explicit expression for H.

The Newtonian limit for nearly flat space-times

In the Newtonian limit, for quasi-static systems in nearly flat space-times, one can approximate the total energy of the system by adding together the non-gravitational components of the energy of the system and then subtracting the Newtonian gravitational binding energy.

Translating the above statement into the language of general relativity, we say that a system in nearly flat space-time has a total non-gravitational energy E and momentum P given by:

${\displaystyle E=\int _{v}T_{00}dV\qquad P^{i}=\int _{V}T_{0i}dV}$

When the components of the momentum vector of the system are zero, i.e. Pⁱ = 0, the approximate mass of the system is just (E+E_binding)/c², E_binding being a negative number representing the Newtonian gravitational self-binding energy.

Hence when one assumes that the system is quasi-static, one assumes that there is no significant energy present in the form of "gravitational waves". When one assumes that the system is in "nearly-flat" space-time, one assumes that the metric coefficients are essentially Minkowskian within acceptable experimental error.

The formulas for the total energy and momentum can be seen to arise naturally in this limit as follows.^[1] In the linearized limit, the equations of general relativity can be written in the form

${\displaystyle \partial _{\alpha }\partial _{\beta }H^{\mu \alpha \nu \beta }=16\pi GT^{\mu \nu }}$

In this limit, the total energy-momentum of the system is simply given by integrating the stress-tensor on a spacelike slice.

${\displaystyle P^{\mu }=\int T^{\mu 0}d^{3}x}$

But using the equations of motion, one can also write this as

${\displaystyle P^{\mu }={1 \over 16\pi G}\int \partial _{\alpha }\partial _{\beta }H^{\mu \alpha 0\beta }d^{3}x={1 \over 16\pi G}\int \partial _{\alpha }\partial _{j}H^{\mu \alpha 0j}d^{3}x}$

where the sum over j runs only over the spatial directions and the second equality uses the fact that ${\displaystyle H^{\mu \alpha \nu \beta }}$ is anti-symmetric in ${\displaystyle \nu }$ and ${\displaystyle \beta }$. Finally, one uses the Gauss law to convert the integral of a divergence over the spatial slice into an integral over a Gaussian sphere

${\displaystyle {1 \over 16\pi G}\int \partial _{\alpha }\partial _{j}H^{\mu \alpha 0j}d^{3}x={1 \over 16\pi G}\int \partial _{\alpha }H^{\mu \alpha 0j}d^{2}S_{j}}$

which coincides precisely with the formula for the total momentum given above.