“Proper mass” and “gravitational binding energy” in general relativity
There are some issues with proper mass. I think the best way to remind yourself of this is to say to yourself proper mass isn't proper at all.
I'm reading Robert Wald's "General Relativity" and after the discussion of the Schwarzschild Solution it goes on to talk about interior, static, spherically symmetric solutions. Wald says that "M" in the Schwarzschild Solution comes from the Newtonian analogue, but it's misleading in that it doesn't represent the "proper mass" associated with integrating using the volume element associated with the metric.
IMHO he could have found a way to explain things more clearly. Imagine you drop a 511keV electron into a black hole. Now, you know that gravity is not a force in the Newtonian sense. And you know about conservation of energy. So, after the electron has fallen in, you should know that the black hole mass increase is 511keV/c². It doesn't increase by the 511keV rest mass of the electron plus the mass-equivalence of the electron's considerable kinetic energy. Because at all points during its fall, these two things added up to 511keV/c². When you catch a falling body and dissipate the kinetic energy as radiation, you're left with a mass deficit. The invariant mass has reduced. A radiating body loses mass. The same happens in the hydrogen atom, which weighs less than an unbound proton plus an unbound electron. This difference is the binding energy, and it is negative. Because the mass-energy is reduced. Because gravity converts internal kinetic E=mc² mass-energy into macroscopic external kinetic energy. Provided this didn't get dissipated in our scenario, the black hole mass increase is the full 511keV/c².
He then says that the difference between the "M" from the Schwarzschild metric and the "proper mass" can be interpreted as Gravitational Binding Energy. How accurate of an interpretation is this?
I think it's accurate. In our scenario the black hole mass M has increased by the full 511keV/c², but on the way in the electron suffered a mass deficit. Its proper mass reduced. And the mass-deficit link I gave you was in the Wikipedia Binding energy article
And also, if we consider the vacuum metric of a spherically symmetric collapsing star (which by Birkhoff's theorem should be the Schwarzschild Solution), would "M" in the Schwarzschild metric increase as the star becomes more dense since "gravitational binding energy" should increase.
No. The M represents the amount of energy present. Conservation of energy tells you it stays the same. Note that there is no actual negative energy present. In the typical bound system, there's just less positive matter-energy present because gravity converted some of it into kinetic energy which gets radiated away. Black holes are atypical because they don't radiate.
or is "M" an invariant constant for the system?
Yes, see above.
Also, would the "proper mass" remain invariant (as intuitively it should)?
Remember that mass deficit. Invariant mass varies. And proper mass isn't proper at all.