Bruce J. answered • 08/30/19

Caltech/Johns Hopkins Grad with 15+ years of tutoring experience

Great question. First off, I'd like to say that a particle's energy in special relativity is only equal to mc^{2} in its *rest frame*. More generally it is equal to mc^{2}/(√(1-(v/c)^{2})), where v is the velocity of the particle relative to the observer.

Two things come up for me when I ponder this question. The first is that the formula E = mc^{2} is a result of special relativity, which assumes a flat spacetime structure. If we can say one thing about black holes, it is that they are *not* associated with flat spacetimes, at least locally. Therefore we have no reason to assume that the relation E = mc^{2} will hold inside a black hole. The second is that we haven't discussed what type of energy we're talking about. Are we talking about mass-energy, as in special relativity? Or are we talking about radiation-energy, such as that associated with black bodies and black holes (i.e. Hawking radiation)? In the event that we're talking about the former, which is very likely because we're talking about black holes, there's absolutely no reason to think that energy associated with that radiation will have an expression resembling E = mc^{2}.

In general relativity "energy" is in no way as natural a concept as it is in classical or even quantum mechanics. Quantities associated with "time" translation symmetries are typically identified as energies, but inside a black hole even the roles of time and space become a bit twisted. For example, inside a Schwarzchild black hole, the simplest black hole solution known of Einstein's equation, time and space become so curved that any particle within that black hole eventually winds up at the singularity at its center, even if one exerts a force against it. Coordinates that represent time and "radial" distance outside the black hole flip roles inside of it. I am in no way certain how to even interpret an idea like "energy" inside a black hole, for a particle or anything else.

I hope you find all of this helpful. If you have any good ideas or relevant facts let me know!