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For a thin spherical shell of mass $M$ collapsing to a Schwarzschild black hole, the Birkhoff's theorem ensures that outside is Schwarzschild metric. Then the ADM mass of this spacetime is just constant $M$ during the whole collapse. However, physically one expects the infalling mass shell have decreasing gravitational potential energy and hence the total energy (mass of the shell+gravitational potential energy) of spacetime should decrease. How do we resolve the tension? Does it implies as the collapse goes, the shell is getting heavier and heavier?

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Ignoring the last stage of a black hole forming, basically in all forms of gravitational collapse, forming stars from a thin cloud of hydrogen etc., you have a conversion of potential energy into velocity. This kinetic energy is radiated away as heat loss as the collapse continues.

So you start off with a cooler less dense object, and after a small "compression" you have smaller somewhat hotter object which have the same gravitational impact on the outside world. As the object cools and radiates away heat it will lose mass, the outside world will perceive it as having less mass.

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  • $\begingroup$ I see you point from the physical perspective. However, I wonder mathematically how should one model this transfer of energy. If exact spherical symmetry is assumed. One will not see radiation for example and the ADM mass will be unchanging. $\endgroup$ – Shadumu May 28 '19 at 14:47

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