# Definition of black hole for the purposes of no-hair theorems?

1. Oct 31, 2010

### bcrowell

Staff Emeritus
definition of "black hole" for the purposes of no-hair theorems?

Living Reviews has a nice review article on no-hair theorems: http://www.livingreviews.org/lrr-1998-6 Their rough verbal statement of the no-hair theorem for GR coupled to E&M is: "all stationary black hole solutions to the EM equations (with non-degenerate horizon) are parametrized by their mass, angular momentum and electric charge."

I have a really elementary question, which is how "black hole" is defined in this context. Maybe I'm missing it, but I don't see anywhere in the Living Reviews article where they come out and say this plainly.

It seems to me that it can't be "a black hole is any electrovac solution with a singularity," because then the proof of the no-hair theorem would seem to be a proof of cosmic censorship in the case of GR+EM, which I assume has not been proved...? Is $\Lambda=0$ assumed? For the purposes of this theorem, does "black hole" include things like topological defects? Is it explicitly limited to things that have an event horizon? How about things that aren't asymptotically flat?

2. Oct 31, 2010

### atyy

Re: definition of "black hole" for the purposes of no-hair theorems?

Via the event horizon, at least in figure 1. They also give state asymptotically flat.

3. Oct 31, 2010

### bcrowell

Staff Emeritus
Re: definition of "black hole" for the purposes of no-hair theorems?

Thanks for the reply, atyy! So you think the definition is effectively given in the figure? I guess it makes sense that they define it as having an event horizon, since otherwise I guess you'd have to prove cosmic censorship in order to prove a no-hair theorem.

4. Nov 1, 2010

### George Jones

Staff Emeritus
Re: definition of "black hole" for the purposes of no-hair theorems?

Asymptotic flatness is used to define what "escape to infinity" means.

The standard definition of the black hole region of an asymptotically flat spacetime is the region of spacetime from which it is impossible to escape to future null infinity. An event horizon is the boundary of this region.

The review article Black Hole Boundaries by Ivan Booth,

http://arxiv.org/abs/gr-qc/0508107,

explores other characterizations of black holes, but I don't think thinks that it looks at "hair".