So, a given closed system will always posses the same amount of mass and the same amount of energy, in the ratio given by e=mc^2. A hot dumbbell has more mass than an otherwise identical cold one, a spring that is compressed has more mass than one that is relaxed, etc.
Let’s say I have a system composed solely of two planets, orbiting each other at some velocity. I add energy to the system by moving the planets farther apart while keeping their velocity the same, thus adding solely gravitational potential energy. According to e=mc^2, the mass of the system has now increased. How does this increased mass manifest itself?
The short answer is that it does make a difference. If you had a moon orbiting this double planet (far enough away that we could treat the two planets as some kind of composite object), then the moon would “see” a different mass in the second case than in the first case. I’m pretty sure that the “effective mass” would be larger in the second case, but don’t hold me to that.
The long answer is that it gets really complicated. One of the bêtes noires of General Relativity is that it’s fiendishly difficult to assign a “local” energy to a gravitational field — it’s not really possible to say something like “at this point there is 1 kg/m[sup]3[/sup] of hydrogen, the equivalent of 0.02 kg/m[sup]3[/sup] of electromagnetic field energy, and the equivalent of 0.0001 kg/m[sup]3[/sup] of gravitational energy.” The first two of those are well-defined notions, but the third one turns out not to be. It is possible, however, to define a notion of the total energy (rather than a local energy density) of a system, and that’s what I was referring to in my first paragraph.
The formula for a mass parameter in a stationary asympotically flat spacetime as an inetrgal taken over any topological two-sphere (S) in the vacuum surrounding the mass is given by Wald as:
M = -1/8π ∫[sub]S[/sub] ϵ[sub]abcd[/sub]∇[sup]c[/sup]ξ[sup]d[/sup]
So where mass can sensibly be defined, it isn’t a simple case of E= mc[sup]2[/sup]
Well, sure, but hidden in those timelike KVFs is information about what the metric is doing, and the metric responds to the central mass. I would expect that in the near-Newtonian regime you could expand this out in the form of (total mass) + (corrections), where corrections would include things like contributions from in the energy density due to the motion, gravitational binding energy, and so forth.
In the near-Newtonian (i.e. linearized gravity) you essentially ignore the non-linear terms such as those due to ‘GPE’.
It would make a difference, but you could only really quantify that difference when the spacetime is stationery and asympotically flat (using the formula above).
Or consider the nonlinear terms only to first order, or nth order for some other n, depending on just how close to Newtonian you are, and just how precise an answer you want.
In other words, you’re defining “a decent distance away from the Newtonian limit” to mean “close enough that all of the nonlinear terms are negligible”. Which is fine, but there’s still always a regime where you do use those higher-order terms.
I think once you can’t accurately approximate gravity as a field over a flat background (a la the weak field approximation) your a decent distance away from the Newtonian limit.
How accurate is “accurate”, though? The Newtonian limit is a good approximation for the gravitational field of the Sun, but Mercury’s orbit deviates by small but noticeable amounts from its orbit due to these corrections. This is the whole point of “Post-Newtonian” corrections, which is a pretty well-developed notion in GR (see, for example, Part 3 of this paper.)
So to get back to the topic, the formula mentioned above can be found at the bottom of the Wikipedia article on Komar Mass. Basically, it’s complicated and you need a lot of knowledge about the details of general relativity to make out the details.
Yep, I quoted the exact same formula earlier. It comes from the asympotic behaviour of the gravitational field, if you like it’s ameasure of how much attraction an observer who is very far away from the source(s) feels.