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I am reading through Wald chapter 6 section 2 on the interior Schwarzschild solution. In it he states that matching the interior solution to the exterior (Schwarzschild) solution gives a Schwarzschild mass of $$M=4\pi\int_0^R \rho(r)r^2 d$$ This would presumably be the same mass ##M## that appears in the Schwarzschild metric. However, Wald then notes that the proper mass is actually $$M_P=4\pi\int_0^R \rho(r)r^2\left[1-\frac{2m(r)}{r}\right]^{-1/2} dr$$ Where ##m(r)=4\pi\int_0^r \rho(r')r'^2 dr'##.

Wald then goes to say that ##M_P\gt M## and that the difference ##M_P-M## can be interpreted as a "gravitational binding energy".

This interpretation seems screwy to me since I would have expected the "gravitational binding energy" to source gravitation as well, and I would have expected the Schwarzschild mass to account for this. Indeed, one usually talks about the contribution of "potential energy" to the "rest mass" of an object. In Astrophysical contexts, we talk about how, for example, a supernova releases 99% of the gravitational binding energy of the core into the shock wave and that this energy accounts for ~10% of the rest mass of the core. In light of this, why does the "Schwarzschild mass" which dictates gravitation not include the "gravitational binding energy" of the mass?

I would have expected the situation to be exactly the opposite. That the "proper mass" would only account for the "mass due to the integrated local mass density", while the Schwarzschild mass would include the gravitational binding energy.

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# Proper mass vs Schwarzschild Mass

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