On the stability of black holes

Vera Rubin just passed (RIP) and recent News reviving the theory that the dark matter might be due to (small?) black holes; the idea that smaller black holes may be more prevalent than thought (then wouldn't we see evidence in starlight distortion?).

So the interesting thing to me is the subject of black hole stability and I found a PSE post here that explains, at least with one answer, the stability in terms of a thermodynamic equilibrium; an inverse and unstable balance between mass and temperature; the Hawking temperature model.

I plan to research more, but would be pleased to hear from the experts here on the SE regarding their thoughts. I suppose it actually boils down to three questions:

1. How confident are we of the Hawking temperature model? Are there possibly other mechanisms or factors that might tend to stabilize smaller holes - that might balance the temperature model?

2. How confident are we of heat transfer/radiation in the outer galactic zones? Might the environment itself serve as a stabilizing factor? Or at least slow down the 'vaporization' of holes.

3. How possible might it be that there are other high energy processes that might be creating smaller black holes (other than collapsing stars) - a creation rate that at least balances their demise?

• This is a great question, but I expect the answer to be unsatisfying -- the Hawking radiation picture requires a semiclassical approximation, so you expect it to break down when the quantum radiation has an energy of order of magnitude equal to the black hole mass. There have been papers trying to deal with this (the cleverest I saw was one matching the Hawking radiation profile to the radiation in a Vaidya spacetime), but any answer is going to require full quantum gravity. – Jerry Schirmer Feb 10 '17 at 23:19
• Basically, no. The news article which I'm not sure anybody looked at, and not much lost anyway, is not aboout small black holes, but about maybe lots of stalwart size and above (like 10-40 solar masses) black holes. Nothing new there. As it says it'll take a decade to try to get some statistics on BHs, but there only skimpy support for them being enough for dark matter. 1)we feel very good on the temperature model. Small BHs would have evaporated, and a little small we'd be seeing more gamma rays. But still possible. 2)no, by now there's been most of it, and it's the size of the. -- see next – Bob Bee Feb 11 '17 at 18:54
• BHs now. 3)no, again it is not about small BHs. The article is about solar and up to maybe 100 solar masses. Definitely not enough real small ones, would be gone by now or we'd be seeing their gamma rays as they evaporate. – Bob Bee Feb 11 '17 at 18:56

This is not an easy question to address. I will outline what Hawking radiation is and how it works. I can then point to what the plausible difficulty with it lies. It is a semiclassical theory in that it treats the black hole as a classical system that emits quanta of radiation. The adjustment of the black hole to a smaller mass by a tiny increment is treated with a metric back reaction.

If you have a particle on an accelerated frame it is within what is called a Rindler wedge. Below is a diagram of spacetime for an accelerated frame.

We think of a particle in region I. The hyperbolic lines are regions of constant radius from the $45$ degree lines that are a particle horizon. An observer on an accelerated frame, with acceleration $g$, has the observer behind them at a distance $\rho~=~c^2/g$. The larger the acceleration the the closer to the horizon the observer is. It is also interesting that for two particles to remain a constant distance from each other they must have different accelerations.

One observer, call him Bob, in region $I$ is not able to ever observe anything in region $II$, say if there is an observer named Alice there, or communicate to region $II$, and Alice in region $II$ is not able to communicate to Bob.

The spacetime metric distances are parameterized as $$t~=~\rho sinh\omega,~x~=~\rho cosh\omega$$ the angle $\omega$ is a parametrized time. the metric in the Minkowksi form is then $$ds^2~=~-d\rho^2~~-~\rho^2 d\omega^2~-~dy^2~+~dz^2.$$ If we euclideanize this so that the metric is not Lorentzian we can then think of the unitary time development operator $U(t)~=~exp(-iHt)$ across region $I$ to region $II$. We do this to consider the evolution of a quantum fluctuation that encloses the origin of the diagram above. We then replace $i~\rightarrow~1$ and the time is evaluated for the entire loop, think of this as the perimeter of the loop, as $t~\rightarrow~\rho\omega|_0^{2\pi}$ $=~2\pi\rho$. We then have the operator $U(\omega)~=~exp(-2\pi\rho H)$.

Alice and Bob measure the quantum fluctuation, say a loop that encloses the origin, as a particle that emerges from the horizon and then approaches it again. The particle emerges from the past horizon slowly and then slowly approach the future horizon, for Bob in region $I$ can only observe in a redshifted and time dilated form. Alice in region $II$ observes the same. For this virtual loop we may think of Bob and Alice as witnessing different states $\phi(b,b')$ and $\chi(a,a')$, but which form an entangled state $\psi$ with density matrix $\rho_{AB}~=~\psi^*\psi$ $$\rho(a,a',b,b')~=~\chi^*(a,a')\phi^*(b,b')\phi(b,b')\chi(a,a'),$$ where Alice and Bob observe what can be found by tracing over Bob's and Alice's state variables $b,b'$ and $a,a'$.

The time evolution operator has become a thermal or Boltzmann operator. The temperature is then $\beta~=~2\pi\rho$ or $$T~=~\frac{1}{2\pi\rho k_B}.$$ This is the Unruh effect, explained in elementary terms. The Rindler wedge has no curvature. A black hole of course has curvature; the Riemann curvature has zero Ricci curvature and is all Weyl curvature for a sourceless region. We can however "map" the Unruh effect to the black hole case. This is done by considering the Unruh case as a case of an observer close to the horizon on an accelerated frame. The Hawking radiation emitted for large $\rho$, which persists because of spacetime curvature can be realized by the substitution $\omega~\rightarrow~tc^3/4GM$ with then gives the temperature for the black hole $$T~=~\frac{\hbar c^3}{8\pi k_B GM}.$$ This is a quick way to think of Hawking radiation.

In the Penrose diagram above there is the emergence of a particle EPR pair in region I and II as marked in red. There is also a blue hyperbolic curve in regions I and II. The event horizons marking regions I and II from the black hole regions III and IV are decoupled. The two black holes are less entangled, or in a sense no longer entangled. The blue horizons exist because the red particles act as a tiny Einstein lens that reduces the size of the horizon. This "jump" in a classical setting is put in by hand as a metric back reaction. However, if we understood quantum gravity more fully we would see this as a quantum superposed system. The event horizon would in a sense be a quantum system.

There are some reasons to think this would be the case. In holography all quanta or strings that compose a black hole are on the stretched horizon just a string or Planck length above the event horizon. The strings will form up in a long Ising-like chain or $1-d$ Toda lattice, which defines a sort of quantum membrane. This quantum membrane is due to a space filling process of this long chain of strings. Also holographic principle suggests the event horizon contains a quantum field theory that is equivalent to the gravitation in the larger spacetime outside. So this horizon should play a role in the quantum physics.

This would reflect where Hawking radiation as we understand it gives way to a more fundamental understanding of nature. As yet there is not a complete picture of this.