Should (lone) black holes emit gravitational waves?
I understand this is a chain of dubious assumptions, but I'm not sure exactly where I go wrong with this line of thought.
Thanks to Hawking, we understand that black holes radiate particles. However, due to the low temperature, the particles emitted are mostly photons (as photons are massless).
Secondly, the quantum theory of gravitons isn't perfect, but is a good enough approximation in some regions. Now, if that applies to the event horizon, black holes should emit gravitons.
But now this is a purely gravitational interaction, and this should be completely solvable in Einstein's equations. (and remove the quantumness) So shouldn't black holes emit gravitational waves, and lose energy in the process? as far as I understand, in the GR model, they don't.
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To start with, black holes in the GR model don't radiate. At all. Even Hawking radiation isn't predicted by GR. Rather, it is predicted by doing quantum mechanics in curved spacetime. That is, you take the GR result as background to your quantum calculations (not unlike how you calculate the atom states in standard quantum mechanics by pretending a classical Coulomb potential, ignoring the fact that the electromagnetic field is actually a quantum field; if you take into account the quantum nature of the EM field, you get slight corrections known as Lamb shift).
In other words, you start with a classical GR description of the black hole, assuming that quantum effects are negligible for its description,. Then you do a quantum field theory calculation (in particular, quantum electrodynamics) to understand what quantum fields will do in such a background spacetime. Since you find that there will be radiation carrying away energy, you conclude that thanks to energy conservation, this should mean that the black hole becomes lighter.
So you have three components for evaporating black holes:

GR calculations not involving quantum effects (justified by the fact that black holes are massive, and quantum effects are negligible compared to them).

QED calculations not involving gravitational effects beyond the curved spacetime created by the black hole (justified by the fact that gravitation is weak enough that the gravitational effects of those electromagnetic processes will have a negligible effect, so the only gravitational effect to be considered is the effect of the black hole on spacetime curvature, which is modelled classically).

The general principle of energy conservation (which we expect to still hold even for processes that involve both gravitation and quantum mechanics).
Now to answer the question of gravitational Hawking radiation, we need to consider that we don't yet have a confirmed theory of quantum gravitation. We don't even know for sure that gravitation really is described by a quantum theory at all, and consequently don't know yet whether gravitons exist at all.
However we can make educated guesses on the assumption that gravitation actually follows quantum laws, and that at least for small energy perturbations, quantum gravitational effects can be described with a standard quantum field theory.
If those assumptions are true, then indeed black holes should emit gravitational waves. However, given that the gravitational force is many orders of magnitude lower than the electromagnetic force, the amount of gravitational radiation should be negligible to the amount of electromagnetic radiation. By the time that gravitational radiation would become significant, the black hole should be small enough that the approximations made for calculating Hawking radiation are no longer true.
So in short, we can't know for sure if black holes radiate gravitational waves, but we can say that almost certainly those gravitational waves will be negligible for all intents and purposes, except possibly for tiny black holes where quantum gravitational effects, if they exist, will be large enough that we can't say anything about how they will behave anyway.
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There are no gravitons in General Relativity, and black holes don't evaporate in General Relativity (i.e. there is no Hawking radiation). The consequence of Hawking radiation, or something similar, being empirically verified, would be an empirical refutation of General Relativity, precisely because it doesn't predict this. Specifically, General Relativity doesn't predict that black holes will "evaporate" at all.
There's a semiimplicit subquestion of whether Hawking radiation predicts that some of the energy radiated away would appear as gravitational waves. I don't know, but it seems plausible. Either way, General Relativity doesn't predict it.
If Hawking radiation occurs, models of a black hole in General Relativity would be like models of an iron beam in solid mechanics. Iron beams corrode but the solid mechanics model isn't going to predict that nor the change in the relevant mechanical properties, because it doesn't model chemistry.
Could we adapt General Relativity into a different, but still nonquantum, theory that did predict this? Probably, in the same way we could have the iron beam's mechanical properties change over time "just because" from the model's perspective. My impression is any such theory would be presented as and, even more, received as an approximation to an underlying quantum gravity theory and not as a new foundational theory of gravity.
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You're not far off, but there are a few key points to clarify. Hawking radiation is a result of quantum mechanics in curved spacetime and isn't predicted by general relativity alone. GR doesn't account for radiation from black holes; that's where quantum field theory comes in.
Regarding gravitational waves, if we assume a quantum theory of gravity, black holes might emit gravitons, resulting in gravitational waves. However, these emissions would be minuscule compared to electromagnetic radiation due to the weaker gravitational force. Thus, while theoretically possible, gravitational waves from such black holes would be negligible in practice.
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