Photons Trapped on Expanding Event Horizons

[tom.stoer]

Suppose we have an event horizon H which is a light-like, closed 2-surface. Photons radiated outwards at the horizon along a light-like normal u of the horizon stay on the horizon (this is trivial b/c this statement is nothing else but the definition of the light-like 2-surface).

Now suppose we disturb the horizon, e.g. via infalling matter. This will result in a "dynamical expanding horizon". By definition this horizon is a light-like 2-surface, too.

Question: can one proof (or disprove) that photons trapped on the horizon will stay on the horizon even for disturbed, expanding horizons?

I think this question is equivalent to the question whether "the horizon expands along the normal u".

The answer is simple for Schwarzschild geometry and radially infalling shells of dust. But I don't see how to generalize this for horizons with arbitrary geometry and arbitrary perturbations.

 

[tom.stoer]

no idea?

 

[PeterDonis]

The only thing I can think of is to explicitly walk through the "simple" argument for the spherically symmetric case and see what, if any assumptions are required that actually depend on exact spherical symmetry.

Also (this would be the other only thing I can think of ), it might be worth consulting Hawking and Ellis; the theorems there about the behavior of null generators of horizons (which are the outgoing null geodesics that "expand along the normal u" in your terminology) are, IIRC, pretty general. In particular, I think there's a theorem to the effect that once a null generator has entered a horizon, it can never leave it, as long as the energy conditions hold. I can't remember exactly what other assumptions are needed to derive the theorem, but as I said, I think they're pretty general.

 

[George Jones]

Also (this would be the other only thing I can think of ), it might be worth consulting Hawking and Ellis; the theorems there about the behavior of null generators of horizons (which are the outgoing null geodesics that "expand along the normal u" in your terminology) are, IIRC, pretty general. In particular, I think there's a theorem to the effect that once a null generator has entered a horizon, it can never leave it, as long as the energy conditions hold. I can't remember exactly what other assumptions are needed to derive the theorem, but as I said, I think they're pretty general.

Yes, I too remember something like this, and I think Hawking and Ellis would be a good place to look (also the books by Wald, Penrose, Joshi).