Black hole formation from radiation in a box

In a paper I am currently studying (here), it was assumed that a black hole can form from a pure state of radiation in a box.

Abstract:

If black hole formation and evaporation can be described by an S matrix, information would be expected to come out in black hole radiation. An estimate shows that it may come out initially so slowly, or else be so spread out, that it would never show up in an analysis perturbative in $M_{Pl}/M$, or in $1/N$ for two-dimensional dilatonic black holes with a large number $N$ of minimally coupled scalar fields.

What I want to understand is, what is the exact procedure behind it. Because when a star collapses due to gravity, there are various opposing forces (like degeneracy of fermions) that makes sure that only certain stars with huge masses can only form black holes. But for radiation(which is bosons) it is impossible for such pressures to exist.

Does that mean any radiation can form black hole? If there is a cut-off density, how is it defined?

By the way, looking at Einstein's equations it is obvious that $T_{\mu\nu}$ can be EM field and thus radiation is equally likely to form black holes as matter fields. I am just asking about the procedure.

• This would require checking my notes, but I do recall there's an amount of pure radiation that can cause black hole formation by virtue of its gravitational influence on itself alone. As you say putting only EM in $T_{\mu\nu}$ is a valid way to satisfy the EFEs. IIRC, that radiation density is so large that the practicality of it is near zero. The box is in the equation just to ensure that the radiation doesn't escape before forming the black hole, but it's a thought experiment only. There's no actual procedure other than "get a box with mirrored sides, put lots of EM in it, get black hole". – Jim Jul 31 '17 at 12:52
• arxiv.org/abs/0805.3880 – MBN Jul 31 '17 at 18:42

The answer seems simple enough. Old video games such as Pacman had a 2 dimensional torus, which is a sort of square with opposite edges identified with each other. A box with sides identified forms a $3$-torus. This appears to be the ideal box we can put a black hole in. If the black hole exists a face it just appears in the opposite face. There is a hitch. We are talking about relativity, and we can transform into a frame so the toroidal geometry in space assumes a time configuration. It is then possible to have closed timelike curves. This means time travel is possible in this space or spacetime! This may not be a feature one wants to admit in this model. It would mean that in one frame where the toroidal geometry is all spatial we may have an equilibrium of Hawking radiation and ingoing radiation, but in another frame some of the radiation will be on closed curves. In null coordinates closed spatial loops and closed timelike loops can as linear(like) combinations form closed null loops. It is not clear what is meant by equilibrium under these conditions.
Spacetimes that admit closed timelike curves have negative vacuum energy density. These are spacetimes that have some form of quantum field that has a negative vacuum energy not bounded below. This is in some ways pathological, for without a lowest energy state in the spectrum such spacetimes can in principle emit an endless stream of energy in the form of radiation. However, there is one form of this type of spacetime we can get some handle on. This is the anti-deSitter spacetime. On a conformal patch it is possible to have geodesic completeness. We might then put our black hole in a patch of and anti-de Sitter spacetime $AdS_4$ and avoid the issues of closed timelike geodesics. We now have the ideal box to put a black hole into.
We may now see there are some fascinating connections between black holes and anti-de Sitter spacetimes. An observer on an accelerated frame a constant distance from a black hole horizon is in a modified form of Rindler wedge. Given the Schwarzschild metric $$ds^2~=~\left(1~-~\frac{2m}{r}\right)dt^2~-~\left(1~-~\frac{2m}{r}\right)^{-1}dr^2~-~r^2d\Omega^2,$$ $m~=~GM/c^2$. We now consider the case of an observer close to the horizon $r~=~2m~+~\rho$, for $\rho~<<~2m$. This observer is on a highly accelerated reference frame. We then have that $$\left(1~-~\frac{2m}{r}\right)~=~\frac{\rho}{2m~+~\rho}~\simeq~\frac{\rho}{2m},$$ which means the metric is approximately $$ds^2~\simeq~\left(\frac{\rho}{2m}\right)dt^2~-~\left(\frac{2m}{\rho}\right)^{-1}dr^2~-~(2m)^2d\Omega^2.$$ This is $AdS_2\times\mathbb S^2$. In other words, the very near horizon condition for an accelerated frame is equivalent to a form of anti-de Sitter spacetime. The black hole and the $AdS$ “box" share the same spacetime physics! In fact, for a black hole in spacetime of $10$ dimensions this recovers $AdS_5\times\mathbb S^5$ and in $11$ dimensions $AdS_4\times\mathbb S^7$ or $AdS_7\times\mathbb S^4$ in the $AdS/CFT$ correspondence.