jtbell said:
Assuming we're in a domain where GR effects are negligible, is it more complex than simply including interaction energies, or energy and momentum carried by e-m fields and the like, in E_{total} and {\vec p}_{total}?
I can tell you for sure that in nearly Minkowskian space-time (i.e. a flat metric) with orthogonal cartesian coordinates (x,y,z,t) one can find the mass of an isolated system from the stress-energy tensor in the center of momentum frame by taking the intergal
<br />
\int_V \left(T^{00}+T^{11}+T^{22}+T^{33}\right) \,dx dy dz<br />
While I could be mistaken on the following point, I do not believe that this expression is equivalent to the intergal of the energy density of particles and fields, which I would expect to be simply \int T^{00} dV, without the pressure terms T^{11},T^{22},T^{33}.
In many cases, the difference does not matter. For instance, if you have a *small* spherical pressure vessel containing pressurized gas, (small meaning that the metric coefficients are still close to unity everywhere including the center of the pressure vessel) there is no net contribution to the mass of the system from the pressure terms, for the positive pressure in the interior of the sphere is exactly balanced out by the negative pressure (tension) in the exterior of the sphere. To give one a general idea of it's applicability, this approximation is pretty good even for a sphere of gas as large as our sun (if one considers errors measured in units of parts per million as being "pretty good").
One thing that makes it difficult to construct an example is that classically, systems of positve and negative charges do not posses tensile strength, making it hard to construct a purely electromagnetic example.
Unfortunately, I do not see any simple way to motivate the inclusion of the pressure terms in the formula for mass, though it is clear from various sources (such as Wald) or Carlip's paper on kinetic energy in GR
http://arxiv.org/abs/gr-qc/9909014
that these pressure terms are needed. I have a suspicion that Einstien's full field equations are needed to motivate these terms (that the equivalence principle alone is not sufficient). Someday I hope to understand at a more fundamental level why these terms are needed - at the moment, I simply have to point to the sources that say they are needed.
Meanwhile, though the presence of the pressure terms may be somewhat puzzling, I also have to point out that the 4-vector approach still does work to find the invariant mass of extended systems via the formula E^2-p^2 (c == 1) when one has necessary pre-conditions (asymptotically flat and/or static space-times) needed to define energy and momentum.
So is the energy of a extended system the sum of the energies of its parts? Probably not, definitely not if it's a large system (i.e. one where the metric coefficients become non-Minkowskian). Is the invariant mass of an extended system still given by E^2 - p^2 when c=1? Yes.
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