OK, superceded, particularly as it applies to Hawking's discussions of black holes. Einsteinian black holes are absolute, singularities. The radiation that "comes out of" black holes via the proposed Hawking radiation requires Quantum effects that are not contemplated in Einstein's theories.
The singularity is at the center of the black hole. Hawking radiation comes from the event horizon. But in any case, these quantum effects are perfectly compatible with Einstein's field equations, which is to say that Hawking's quantum black hole is an exact solution to that set of equations, just as Schwarzschild's classical black hole was. The equations didn't have to be modified one jot or tittle.
The radiation that "comes out of" black holes via the proposed Hawking radiation requires Quantum effects that are not contemplated in Einstein's theories.
General relativity doesn't contemplate classical electromagnetic fields, either, but they fit right in. GR is only concerned with the shape of space.
[Geek alert: There is one way in which Einstein's theory of gravity seems to run afoul of quantum mechanics. That occurs when you try to quantize the gravitational field itself, into particles called gravitons (analogous with the photons of electromagnetism). When you try to calculate quantum gravitational interactions, you find that the quantities you calculate all become infinite. This is quite unlike any of the other forces, which all give finite, experimentally testable predictions for their interactions.]