Regaining Causal Contact by passing Event Horizons

[craigi]

When an object approaches a particular event horizon, for a distant observer, in its own reference frame it is not destroyed. The distant observer sees the effect of the object's destruction at the event horizon. We resolve this paradox by the fact that the object and observer no longer have causal contact, at least from the perspective of the outside observer.

How do we resolve the case where the observer follows the object across the event horizon at a much higher speed?

 

[Chalnoth]

When an object approaches a particular event horizon, for a distant observer, in its own reference frame it is not destroyed. The distant observer sees the effect of the object's destruction at the event horizon. We resolve this paradox by the fact that the object and observer no longer have causal contact, at least from the perspective of the outside observer.

How do we resolve the case where the observer follows the object across the event horizon at a much higher speed?

Horizons are observer-dependent. We can never again interact with any object that has passed our horizon. But an object much closer to the object that crossed our horizon will have a different horizon, and may be able to interact with it.

 

[mathman]

Horizons are observer-dependent. We can never again interact with any object that has passed our horizon. But an object much closer to the object that crossed our horizon will have a different horizon, and may be able to interact with it.

Are you sure? My understanding is the event horizon is determined by the mass of the black hole and is not observer dependent.

Thr main difference as seen by the distant observer is that the object falling into the black hole never quite makes it (time dilation).

 

[craigi]

Are you sure? My understanding is the event horizon is determined by the mass of the black hole and is not observer dependent.

Thr main difference as seen by the distant observer is that the object falling into the black hole never quite makes it (time dilation).

I think I worked this out while I was waiting for a response.

Chalnoth is right, but is talking about the apparent horizon which is the surface from which nothing can reach the observer. This is distinct from the absolute horizon which is the surface from which nothing can escape. It is the apparent horizon that is relevant to causality, but the absolute horizon has the relations to other properties of a black hole.

If I'm right, this does answer the original question, but a second observer could observe the effect of the destruction of the object while the first observer follows the object to maintain causal contact, but doesn't cross the absolute event horizon. The two observers would still be in causal contact and would differ about the fate of the object.

I suspect that the answer to this variant of the original problem is that "by the time" the two observers have conferred then the object must have been destroyed by the singularity anyway. So that all they differ on is when and where and we end with up a 'one man's singularity is another man's event horizon' situation. Proving this isn't going to be easy though.

 

[Chalnoth]

Are you sure? My understanding is the event horizon is determined by the mass of the black hole and is not observer dependent.

Thr main difference as seen by the distant observer is that the object falling into the black hole never quite makes it (time dilation).

You're right, I was sloppy. It does depend upon which type of horizon you're talking about. There are apparent horizons and absolute horizons. The event horizon of a black hole is an absolute horizon (an invariant property of the space-time). The de Sitter horizon for a universe dominated by a cosmological constant is an apparent horizon. For some discussion of apparent horizons and absolute horizons, see here:
http://en.wikipedia.org/wiki/Apparent_horizon

Note that "event horizon" is not always synonymous with "absolute horizon", as some call the de Sitter horizon an event horizon as well.

 

[craigi]

Thr main difference as seen by the distant observer is that the object falling into the black hole never quite makes it (time dilation).

I don't think that's the full picture because it would preclude the growth of a black hole.

 

[bapowell]

I don't think that's the full picture because it would preclude the growth of a black hole.

Cf. Hawking's latest perspective.