# Stability of the Hawking-Hartle vacuum in semiclassical gravity

Consider a free quantum field theory defined upon a static Lorentzian spacetime possessing a bifurcate Killing horizon, such as Schwarzschild spacetime.

These assumptions are sufficient to define a unique quantum state, the "Hawking-Hartle vacuum", which is stationary under Schrodinger evolution by the Killing observers, and which does not diverge should that Schrodinger evolution be continued through either event horizon. When decomposed into incoming and outgoing modes, one finds emission from and absorption by the black hole are exactly balanced.

This would seem to suggest that this state should exert no gravitational back-reaction (e.g. that the black hole should not evaporate). However, computations of the renormalized stress-energy tensor find a nonzero result.

So in the Hawking-Hartle vacuum, does the black hole evaporate or not? Rephrasing: in semiclassical gravity (e.g. the use of the expectation value of the renormalized stress-tensor as a source term to the Einstein equations), does the Schwarzschild-plus-Hawking-Hartle vacuum solution continue to be static?

P.S. I am aware that the evaporation of black holes is due to the Unruh, not the Hawking-Hartle, vacuum. This question is about the latter state specifically.

There are several issues to be considered here:

• First off, black hole could remain static even if quantum fields around it exert gravitational backreaction. Just like Reissner–Nordström solution is static while having nonzero stress-energy tensor, it is possible to have a static solution with non-zero renormalized stress-energy tensor. The requirement for that is the absence of net flux of energy. In Schwarzschild coordinates for Hartle–Hawking state $$\langle T^\mu_\nu\rangle_\text{HH}$$ is diagonal and thus $$\langle T^t_r\rangle_\text{HH}=0$$ and the black hole would be static even when backreaction is taken into account.

• Second issue, when taking into account the backreaction for HH state, is that stress-energy tensor has nonzero limit away from the black hole. This means that the solution no longer could be asymptotically flat, as the total mass of thermal radiation would be diverging with volume. To remedy that, one could consider thermal vacuum occupying only a finite volume around the black hole by imposing a suitable reflective boundary conditions at a finite distance from the horizon (placing a black hole in a box) or one could turn on the cosmological constant (by considering Schwarzschild—AdS solution).

• Third issue is that stability (as per question's title) of a solution is a larger question than it being a state of static equilibrium, since static solution could be unstable (under small perturbations). And for a black hole in a box (as a practical realization of HH state) the equilibrium could be either stable or unstable depending on the box's size and the mass of the black hole. If the total energy of radiation outside the black hole is more than the quarter of a black hole mass then equilibrium is unstable. If the black hole as a result of fluctuation absorbs larger than average amount of radiation in a given period of time, its mass would increase while its temperature would decrease so the next moment it would absorb even more while radiate away even less, and thus small perturbations would increase in time. On the other hand, if the total energy of radiation outside the black hole is small, then absorption of larger than average amount of radiation would deplete the reservoir so that even larger mass of a black hole would not lead to a larger absorption in the future, and so the equilibrium is stable.

No, it does not evaporate. The Hawking-Hartle vacuum describes a black hole in thermal equilibrium with its environment; the radiation emitted by the black hole is exactly balanced by radiation going in, as you pointed out.

The fact that local or averaged (semi-)classical energy conditions are violated doesn't contradict this directly, but it is an interesting technical point that needs to be addressed. See this paper and this one.

To put it in brief, those energy conditions are not the right ones to pick. There has been a lot of debate as to what the right energy conditions are. Note that the Universe (as we know it) violates the semiclassical energy conditions through e.g. dark energy.