Before quantum theory we knew accelerating electrons radiated electric fields. This is modelled classically (even though we know it is a quantum process emitting photons)
Similarly is there a possible classical theory of Hawking radiation? In this analogy the electron becomes the black hole radiating hawking radiation.
From a distance an astronomer may just observe a black hole as an object emitting classical electromagnetic waves and not feel the need to model this quantum mechanically.
The radiation coming out of a black hole would contribute to the stress energy tensor $T$ which is almost exclusively photons. Classically we could consider this just release of electromagnetic waves(?)
In GR we have:
$$G_{\mu\nu} =\kappa T_{\mu\nu}$$
which tells gravity how to evolve.
We have
$$\nabla_\mu T^{\mu\nu} = 0$$
which is the conservation of energy.
For any particular $T$ one could always solve it to find a $g$. But this $T$ might not correspond to any classically moving fields.
i.e. it may not satisfy:
$$\frac{\delta \int \sqrt{-g}T dx^4}{\delta \phi}=0$$
Of these 3 equations, which ones would we keep and which ones would we modify in order to model a semi-classical evaporating black hole. By semi-classical I mean modelling the radiation as smooth fields instead of particles (just as in the classical electron case).
Since we think of the electromagnetic radiation a "tunnelling" out of the black hole. Would this break one of the equations above?
Basically, if QM had not been discovered how would a classical physicists model an evaporating large black hole with classical field equations? (including the shrinking of the black hole as it evaporates)
Or perhaps this is not possibly classically due to the radiation being in a black-body spectrum which is only derivable using Plank's constant?