Let's say we have one black hole that formed through the collapse of hydrogen gas and another that formed through the collapse of anti-hydrogen gas. What happens when they collide? Do they (1) coalesce into a single black hole or do they (2) "annihilate" into radiation?
One would expect (1) to be the case if the No Hair Theorem were to hold. So I guess what I'm really asking for is a modern understanding of this theorem and its applicability given what we know today.
Best Answer
I suspect there is a bit of a difficulty here. A particle $p$ and its anti-particle $\bar p$ cancel each other, and if they have a mass $m$ this results in the production of massless bosons (photons). Quantum number which identify the particle $p$ are subtracted away by opposite quantum numbers of $\bar p$. These quantum numbers usually are the lepton or baryon numbers, electric charge, isospin and so forth. However, anti-matter does not have anti-mass.
Dirac’s original idea was that the Klein-Gordon equation had a square-root according to spinors. If the particle has a mass there are then two roots for the momentum, which define a surface in $\pm$ portions of the momentum light cone. The negative portion of the cone defines the so called Dirac sea, where the mass of particles is negative. All of these negative energy (mass) states are however completely filled up. This is why this is called a “sea,” for they define a ground state which has no dynamics. However, if you impart a package of energy to a state in the sea so its energy flips sign you can generate an anti-particle with positive mass-energy. However, all the other quantum numbers are reversed, including the charge.
One could then construct a black hole from a huge cloud of hydrogen and anti-hydrogen of equal mass. Based on the final state of the black hole it is not possible to determine whether it was formed by hydrogen or anti-hydrogen. So if you have two such black holes, one from $H$ and the other from $\bar H$ the two will coalesce into a larger black hole.
The Schwarzschild solution in its pure form has a past and future singularity, where the past singularity corresponds to a “white hole.” The white hole is about the closest thing there is to an “anti-black hole.” These do not exist in nature, or at least have not been identified astronomically. They may play some role in the very early universe, but nature is such that there is an asymmetry (asymmetry in their occurrence etc) between the black hole and white hole. However, the white hole does not have an “anti-mass.”