In GR, the effective total mass of a static system can be obtained by integrating not only the energy density terms but also the diagonal terms of the stress-energy tensor, which correspond to pressure in each direction, giving the "Komar mass". This means effectively that the total mass is given by the volume integral of the energy density plus three times the pressure (at least in the case where the pressure is the same in each direction). The energy density terms as affected by the gravitational potential within the system add up to the mass of the original material minus twice the Newtonian potential energy, and the pressure term then adds back in the Newtonian potential energy, so the overall result is consistent with Newtonian theory.(adsbygoogle = window.adsbygoogle || []).push({});

I've been wondering for a while how this works in Newtonian terms, and how to extend it to cover non-static cases, but it turns out to be much simpler (and unfortunately less interesting) than I had hoped.

The potential energy relative to infinity of a small component of mass m due to any other mass M within the system is given by -GmM/r, where r is the distance between the masses.

I've only recently noticed that this value can also be considered as the product of the gravitational force -GmM/r^{2}and the distance r to the other mass. This means we can express the potential energy of the system as the sum (or integral) of the force times the distance for each pair of masses making up the system, except that we also need to divide by two in order to avoid counting each pair both ways.

If the system is static, then the forces are in equilibrium, so to evaluate the potential energy between any two parts of the system we can simply consider the component of the force in the plane perpendicular to the line joining those parts. That force component is equal to the partial pressure in the plane due to those parts multiplied by the area of that plane. If we integrate that component of the pressure in the direction of the distance vector, giving the volume integral of the perpendicular pressure component, this is equivalent to multiplying the force by the distance between the parts in question. As we are only considering the force once for each pair of parts, no factor of 1/2 is needed. The sum of these volume integrals of the pressure components in each direction, as used in the Komar mass calculation, equals the sums of the forces between the parts times their distances, giving the potential energy, as described above.

If however the system is not static, the original description in terms of the sum of the forces times the distances still holds anyway, giving the potential energy, which still needs to be added back to the energy density to compensate for the double loss of potential energy due to time dilation.

In the non-static case, it is clear that the calculation merely gives the original potential energy and does not suggest any location for it. This therefore also suggests that even though the Komar mass expression involves integrating the pressure, this does not mean that the pressure acts as a local gravitational source in this case. For example, consider two gravitating objects held apart by a light rigid rod, which is then gently pushed aside, abruptly reducing the pressure in the rod to zero. The total effective gravitational source strength should surely be unchanged by this, at least initially, but the moment the rod is no longer under pressure it no longer contributes to the stress-energy tensor.

This suggests that both the Komar mass calculation from the pressure components and the alternative force times distance calculation describe an effective adjustment to the gravitational source strength to account for gravitational energy without assigning it a location or a physical mechanism, even though the Komar mass expression appears at first glance to treat the pressure as a local source term.

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# Newtonian generalization of Komar mass

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