Free falling into a black hole that evaporates by Hawking Radiation

[PeterDonis]

An absolute horizon must have no light from it escaping to future null infinity.

No; an absolute horizon is the *boundary* of the causal past of future null infinity. That means it is the null surface composed of outgoing light rays that just barely reach future null infinity.

Another way to see this is to work from the other end, so to speak: if the causal past of future null infinity is not the entire spacetime, then a "black hole" is present: the black hole is the region of spacetime that is not in the causal past of future null infinity. The absolute horizon is then the boundary of the black hole region. In other words, if there is a black hole present (which there clearly is in the Penrose diagram I referred to), there must be an absolute horizon. The only way there can be no absolute horizon is if there is no black hole at all--i.e., if the entire spacetime is in the causal past of future null infinity.

 

[George Jones]

Expanding on what Peter wrote:

Let B be a subset of topological space M. Define the interior of B, Int(B), as the union of all open subsets of M that are subsets of B. Define the closure of B, Cl(B), as the intersection of all closed subsets of M that contain B. The boundary of B is Cl(B) - Int(B). B might contain all (iff B is closed), some (iff B is neither open nore closed), or none (iff B is open) of its boundary.

Here B is the black hole region of spacetime M, i.e., B is set of all events that are not connected to future null infinity by a future-directed causal (lighlike or timelike) path. The absolute (event) horizon is the boundary of B.

The only way for an absolute horizon not to exist is if there isn't a black hole. This is what Hawking now argues, but doesn't prove.

 

[PAllen]

Expanding on what Peter wrote:

Let B be a subset of topological space M. Define the interior of B, Int(B), as the union of all open subsets of M that are subsets of B. Define the closure of B, Cl(B), as the intersection of all closed subsets of M that contain B. The boundary of B is Cl(B) - Int(B). B might contain all (iff B is closed), some (iff B is neither open nore closed), or none (iff B is open) of its boundary.

Here B is the black hole region of spacetime M, i.e., B is set of all events that are not connected to future null infinity by a future-directed causal (lighlike or timelike) path. The absolute (event) horizon is the boundary of B.

The only way for an absolute horizon not to exist is if there isn't a black hole. This is what Hawking now argues, but doesn't prove.

Ok, that's clear and sharp for me. The key is the interior world lines never escape (because they all reach a spacelike singularity; and the details of what evaporation means for the singularity is typically left out semi-classical treatment). However, then we must distinguish an evaporating horizon from an eternal horizon in that it no longer has the property that light from it never escapes. It is an absolute horizon, but no longer an eternally trapped surface! This would substantively change the behavior of an SC style time coordinate in the near horizon distant future.