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Does Hawking radiation assume that general relativity works perfectly, inside black holes, or on the event horizon?
Could someone please tell me the exact statements made in describing Hawking radiation? (Before we apply the math).
Would the concept of Hawking radiation be moot if general relativity somehow failed, inside a black hole, since at this point we only have a theoretical model for a black hole?
First let me direct you to An explanation of Hawking Radiation where I have attempted an accurate but relatively accessible explanation of the mechanism for Hawking radiation.
To address your specific points: Hawking's original calculation assumed that general relativity gives the correct description of the spacetime every inside and outside the black hole event horizon. He used quantum field theory in the curved spacetime predicted by GR to perform his calculations.
However the key feature that causes the radiation is the presence of a horizon, so alternatives to general relativity like Brans-Dicke theory also predict Hawking radiation as long as a horizon is present. General relativity would have to fail so badly that no event horizon exists for there to be no Hawking radiation.
No, we do not need to assume general relativity in order to predict Hawking radiation. In fact, the opposite is true. Hawking radiation is a prediction of semiclassical gravity, which is a method in which assumptions are made that violate general relativity. (And there is no other choice, since general relativity and quantum mechanics are incompatible.)
Actually, Unruh radiation is predicted to be emitted from any horizon. This includes event horizons in flat spacetime, which has no gravitational fields at all. Such a spacetime can be described using pure special relativity, with no need for general relativity.
Hawking radiation, which has not been observed, has been hypothesized as a resolution of the Black Hole Information Loss Paradox, whose resolution is necessary for that preservation of unitarity which is central to the current formalism of quantum mechanics.
Since the 2017 answer, a 2019 paper has appeared at https://arxiv.org/abs/1910.10819 , which makes a good case for resolution of the Black Hole Information Loss Paradox by passage of the information, within black holes formed by stellar collapse, to another local universe (possibly like our own, but presumably much smaller) "parented" within it, which will eventually contain black holes of its own, in a process that may continue ad infinitum. This passage of information, actually sketched in simpler terms by John Preskill at https://arxiv.org/abs/hep-th/9209058 in 1992, might preserve the unitarity of quantum physics without the extremely faint radiation conjectured by Hawking, which, at least without phenomenal magnification energy or one very long wait for some stellar collapses much closer to us than those already observed, is extremely likely to remain entirely hypothetical.
The "parenting" process involved may use GR in the formation of the black holes, but uses 1929's Einstein-Cartan Theory within them.
ECT, worked out thru conversations between Einstein and the great mathematician Elie Cartan, differs from GR in positing some TINY spatial extent for fermions (subatomic matter particles), which is essential for the torsion-based bounce effects involved in the formation of the LU's. After the discovery of particulate spin in the mid-1920's, the idealization of fermions as being absolutely point-like derived from Pauli, who realized that their rotation rate might otherwise exceed the speed of light. Although the reductions in that speed that occur when light rays enter material as common as water were known at the time, the possibility of smaller local universes whose average density might be much greater than our water's did not occur to Pauli, and Cartan laid it by that wayside where so many of physics' past "absolutes" remain. (Einstein had no objection to that, as his 1916 pop-sci book on 1915's GR mentioned just such "local" possibilities, but they hadn't been written into the formal theory, which antedated "multiverses" by several decades, and required observational or experimental proof.)
The 2019 paper I've linked to my answer requires, for its own observational proof, universal rotation, a concept often misunderstood because it may involve numerous axes. Astronomical results that might substantiate it have varied.
