I've been wondering about the effects of the gravitational interaction of a body on itself according to the laws of general relativity. The general relativity courses that I have taken in the past haven't touched on this issue, and I wonder if anyone on this forum would be able to help. In classical electrodynamics, an electrically charged object interacts with itself so that its effective mass is greater than its bare mass. For point particles, this leads to problems as the contribution to the effective mass due to electromagnetic interactions is infinite. Are there similar issues with general relativity? Does gravitational self-interaction lead to a change in the effective mass of an object? If there is such an effect, what sign is it, and is this "gravitational mass" always finite? By a naive comparison with electrodynamics I'd expect the gravitational field to reduce the mass, because the different parts of the body attract each other rather than repel.
You best bet at answering this would be to assume the the problem is "non-relatavistic" conditions i.e. weak field approximation and slow motion (valid is most cases unless you happen to be speeding in the vicinity of a black hole, also called the newtonian limit), so that you may invoke the corresponcende principle, in which case you recover the Poisson equation for the gravitonic potential: [tex] \nabla^2\Phi=4\pi G\rho [/tex] Thus, as electrodymical systems and gravitonic systems are governed by the exact same equation, it will necessarily yield the same solutions and same conclusions. So in the newtonian limit, the answer would be yes. Wouldn't dare take a guess at the true relativistic limit though.
The problem with that limit is that gravitational and electromagnetic forces between bodies are instantaneous and there is no adjustment to the bare mass in either case. I think maybe using linearized gravity (but not the low speed limit) with the relevant gauge choice would be the way to go though.
Using the weak field approximation given here, it is possible to derive the following for the "energy density" of a grativational field. [tex] \frac{-1}{8\pi G}(B^2+E^2) [/tex] Dividing this by [tex]c^2[/tex] would give the contribution to the effective mass, so as I was thinking, the effective mass should be reduced by the presence of the gravitational field. This is only the weak field approximation though. I don't know whether the effective mass of a black hole is finite.
The issues of mass in GR can get rather subtle. There are actually several, slightly different concepts of mass in GR, all of which are closely related but different in detail from each other and from the concept of mass in SR. This has cause some authors to write that there is "no good defintion of mass", but in my opinion, the problem is basically that there are too many defintions of mass, where one ideally would only want one. Commonly used defintions of mass are Komar mass, ADM mass, and Bondi mass. ADM mass is one of the more powerful concepts, but Komar mass is probably the easiest to use. GR can be interpreted in terms of an action principle using Hilbert's action, so Noethers theorem applies. In fact, Noether's theorem was specifically developed to solve some of the issues that arise with mass in GR. By Noether's theorem, any system with a time translation symmetry will have a conserved energy. This energy can be divided by c^2 and interpreted as a mass. This particular form of mass is the Komar mass. Your intuition is generally correct in the Newtonian limit (gravitational self-interaction does lower the mass of a system). To go beyond the Newtonian limit with any degree of rigor, you need to reformulate the whole concept of mass in a non-newtonian manner. You might want to look at: http://en.wikipedia.org/w/index.php?title=Mass_in_general_relativity&oldid=177903144 and http://en.wikipedia.org/w/index.php?title=Komar_mass&oldid=170982833 Wald, "Genreal Relativity", has a good discussion of Komar mass. http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html has a good non-technical overview. Aside from Noether's theorem, there are other ways of approaching the issue of mass - ADM mass was originally derived from a Hamiltonian formulation, and other common approaches use the idea of psuedotensors. Noether's theorem is IMO still one of the best ways to deal with the fundamental issues (as compared to the other approaches). YMMV.
Thanks Pervect, that's very interesting. I'm familiar with Noether's theorem, but had no idea that it was developed especially to deal with issues in GR. I'll have a read through your links. Btw, if the weak field approximation for the energy density in my previous post is valid, I just worked out that a spherical body of mass M and radius R should have a gravitational correction to its mass of [itex]-3GM^2/ 5Rc^2[/itex] which for the Earth is [itex]2.5\times 10^{15}\textrm{kg}[/itex], reducing its mass by about 40 billionths of a percent.
If you're already familiar with Noether's theorem, you might find this link helpful. (This link is present inside the wikipedia article, but buried a bit deeply where it might take you a while to stumble across it). It discusses some of the history of Noether, Hilbert, and the concept of energy in GR.
That's a "coulomb-like" approximation where the original masses are assumed to be unaffected by interactions and the field energy is negative. A somewhat more relativistic but still semi-Newtonian approximation says that the effective mass of an object in energy units is given by its proper mass multiplied by the time dilation factor of the metric at its location, typically 1-Gm/rc^{2} for the usual central case, which is effectively a multiplicative form of the potential. That gives a result which was somewhat surprising to me but I was subsequently told is a quite conventional approach in quantum approaches to gravitational field theory, which is to use a model that assumes the total energy of a system is made up of the integral of the proper mass-energy density multiplied by the multiplicative potential at that location (which is decreased by DOUBLE the potential energy relative to the original rest mass, because of the effect of masses on themselves and one another) plus a positive effective field energy with the same magnitude as above (1/2 of field^2) which is equal to the potential energy (plus any kinetic energy). This means that in a static configuration (such as a single central mass), the total energy is equal to the original energy of the component masses minus the potential energy, as in Newtonian theory, giving the same result as the Komar mass, but in a different way.