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).

 

[sardar]

As a scientist, it is important to approach any question with a critical and analytical mindset. In this case, the statement and question presented are based on theoretical concepts and definitions within the field of general relativity. Therefore, any response should also be based on scientific principles and evidence.

Firstly, the concept of an event horizon is a fundamental aspect of general relativity, describing the boundary beyond which no information or matter can escape the gravitational pull of a black hole. This boundary is defined as a light-like, closed 2-surface, and any photons radiated outwards at the horizon will stay on the horizon due to the nature of light-like surfaces. This is a well-established concept and is not in question.

However, the idea of a "dynamical expanding horizon" is a more complex concept and is still an area of active research. It describes a situation where the event horizon is not static but is instead expanding due to the influence of infalling matter. This is a more realistic scenario as black holes in nature are constantly growing due to the accretion of matter.

The question then arises, can we prove or disprove that photons trapped on the horizon will stay on the horizon even for a dynamically expanding horizon? This is a challenging question as it requires a deeper understanding of the geometry and dynamics of the horizon, which is not fully understood at this point.

In the case of a Schwarzschild geometry and radially infalling shells of dust, it is possible to show that the horizon will expand along the normal u. However, this is a simplified scenario and may not hold true for horizons with arbitrary geometry and perturbations.

In conclusion, while it is tempting to draw conclusions based on simplified scenarios, it is important to acknowledge the complexity of the topic at hand and the need for further research and evidence before making any definitive statements. As a scientist, it is crucial to continue exploring and studying the dynamics of expanding event horizons to gain a deeper understanding of this phenomenon.