I Evaporating black hole: First principles consideration

[Tomas Vencl]

TL;DR

Thinking about statement “In a evaporating spacetime it does not take an infinite amount of coordinate time to cross the horizon in Schwarzschild-like coordinates”

Here is first principles consideration:
Since it is a black hole there is an event horizon where timelike worldlines can enter but not exit, this is what defines a black hole instead of a white hole. When evaporation is finished there is no more event horizon, this is what defines the evaporation. Without loss of generality, assign the time of the evaporation event to be . Meaning, after there is no event horizon. All coordinate charts, by definition, are smooth and one-to-one, including the distant observer's Schwarzschild-like chart. So the crossing of the event horizon cannot be assigned a time coordinate . Therefore, indeed, it does not take an infinite amount of coordinate time to cross the horizon.

I was thinking about what it even means in the light of first principles to say that in Schwarzschild-like coordinates, an infalling worldline crosses the horizon of an evaporating black hole in finite coordinate time. Since we don’t know the metric, I don’t expect us to find an exact answer, but rather an attempt to see if we can infer anything more from these first principles.

Does this claim mean that we can map the geodesic of an infalling object even below the horizon in Schwarzschild coordinates? Isn’t one of the key properties of the horizon that it is uncrossable in Schwarzschild coordinates? If it loses this property, what other property defines it as a horizon?

One can think about some strange quantum effects on the infalling observer or on the metric, but that would imply that these effects must be dramatic already near the horizon, which is contrary to the generally accepted assertion that from the perspective of the falling observer, nothing dramatic happens during the passage through the horizon.

I am following up on a neighboring thread and for that reason, I don’t want to place the question in the ‘beyond the standard model’ section, but feel free to be more speculative and heuristic. I don’t expect a solution, rather your perspective on the matter. Thank you.

 

[PeterDonis]

All coordinate charts, by definition, are smooth and one-to-one

No, you cannot assume this. For a non-evaporating black hole, Schwarzschild coordinates are singular at the horizon. So you can't just assume that "Schwarzschild-like" coordinates for an evaporating black hole are not. You have to actually do the math and see. Which means you first need to define how the coordinates are assigned.

A correct statement would be that all coordinate charts, by definition, are smooth and one-to-one on some open region of the spacetime. But you cannot assume that that open region is the entire spacetime.

Isn’t one of the key properties of the horizon that it is uncrossable in Schwarzschild coordinates?

No.

what other property defines it as a horizon?

The property that it is the boundary of a region of spacetime that is not in the causal past of future null infinity. That is the standard definition of an "event horizon" (and the region itself is the "black hole"), which can be found in most GR textbooks (see, for example, Section 12.1 of Wald). This definition is also discussed in the Insights article I linked to in the other thread:

https://www.physicsforums.com/insights/black-holes-really-exist/

You marked this thread as "I" level, but this is a subject that really cannot be properly understood without an "A" level background in the subject. It is particularly not a subject in which you can rely on heuristic descriptions in ordinary language that you got from pop science sources.

One can think about some strange quantum effects on the infalling observer or on the metric, but that would imply that these effects must be dramatic already near the horizon

No, it doesn't. There are some proposed models (such as the "firewall" models) that have such dramatic effects near the horizon, but there are others that do not.

I am following up on a neighboring thread and for that reason, I don’t want to place the question in the ‘beyond the standard model’ section, but feel free to be more speculative and heuristic.

Nope, sorry, that is the exact opposite of what is needed for this subject. It requires a solid reference as a basis for discussion, not vague hand-waving. That is true of any discussion here at PF, but even more so for a discussion of a subject like this.

Thread closed.