Yet another "never crosses the horizon" question

[JoeMath]

I am still confused by the apparent contradictions between the world line of an immortal Observer at a distance (~100 M km) from a collapsing stellar core (-> Black Hole) and the reworking of coordinates to include the Schwarzschild radius. The argument that an in-falling astronaut would pass through the Schwarzschild radius (or event horizon) may be mathematically consistent with the assumptions of GR, but does not seem to take into account significant cosmological facts: 1) from the perspective of our distant Observer, the core collapse slows down as it approaches the Schwarzschild radius and never crosses. 2) After a brief time (Observer perspective) the surface of the core is close enough to the Schwarzschild radius that Hawking radiation is produced and could be detected by the Observer. 3) The Observer notes that the universe is expanding and the measured temperature of the Cosmic Background Radiation (CBR) is dropping, with a subsequent reduction in received energy.

At some point in the far future (one trillion years? 100 trillion years?) the Observer notes that the incoming CBR energy drops below the outgoing Hawking radiation energy. From that point on, the mass of the still collapsing core (Observer perspective) decreases. This results in a decrease of the Schwarzschild radius as well. Eventually the outgoing Hawking radiation becomes extreme and the core explodes in a massive shower of energy.

If this simple analysis is true, no in-falling astronaut or rock would ever cross the Schwarzschild radius (event horizon) irrespective of coordinate changes. Please help me understand if/why this argument is flawed.

 

[PeterDonis]

The argument that an in-falling astronaut would pass through the Schwarzschild radius (or event horizon) may be mathematically consistent with the assumptions of GR, but does not seem to take into account significant cosmological facts

Yes, it does.

Please help me understand if/why this argument is flawed.

It's flawed because you are assuming that there is a single, global, well-defined notion of "time" (and that it corresponds to your "Observer perspective"). There isn't.

Another way of putting it is that your "Observer perspective" is a particular system of coordinates, and coordinates have no direct physical meaning. The physics is in the invariants: the things that don't depend on the coordinates. What you should be doing is looking at the invariant properties of the spacetime, and seeing whether they say that a horizon forms, that things fall through it, etc. They do. In other words, the full spacetime geometry contains a horizon and a black hole region inside it, which eventually evaporates. It's just that, because of the way the geometry is curved, the Observer can't see anything inside the horizon, because light from that region can't escape it.

There have been other recent threads on a similar topic:

https://www.physicsforums.com/threa...s-event-horizon-crossing.804912/#post-5052804

https://www.physicsforums.com/threads/event-horizon-in-different-coordinate-systems.792112/

 

[harrylin]

[..] It's flawed because you are assuming that there is a single, global, well-defined notion of "time" (and that it corresponds to your "Observer perspective").[..]

I can find no such assumption in the OP; the assumption that I discern is that according to GR, the convenient reference system that was used for the analysis is valid. That assumption is similar to the assumption that according to classical mechanics an analysis that uses a center of mass frame is valid.

[Mentor's note - A second somewhat off-topic paragraph has been removed]

 

[PAllen]

I can find no such assumption in the OP; the assumption that I discern is that according to GR, the convenient reference system that was used for the analysis is valid. That assumption is similar to the assumption that according to classical mechanics an analysis that uses a center of mass frame is valid.

No it is not the same at all. The COM frame covers that whole region of interest, like any other frame. The issue in GR is that in a theoretical framework where coordinate patches are part of the fundamental mathematical structure (even the geometry of s 2-sphere cannot be fully described withtout them), some people insist on using a coordinates that don't cover the region of interest - the collapsing body itself. Even though GR makes unambiguous predictions about that, you can't explore those without using coordinates that actually cover the region of interest.

[Mentor's note: A few off-topic sentences have been removed from this post, at poster's suggestion]

 

[PAllen]

I should add something I described in one of the threads on this that Peter linked to. If a distance observer sets up coordinates using the radar method that is a generalization of Einstein's original recipe for SR, and applies these to the case of a BH that eventually evaporates, you find that unlike the eternal BH case:

1) A finite time coordinate is attached all of the following:

A) the last bit of the original body crossing the horizon

B) the time any later infalling matter crosses the horzion

C) the final evaporation

2) The time of A precedes B, which precedes C.

SC coordinates are inapplicable altogether because they describe the wrong external geometry - the distant geometry of an evaporating BH is a Vaidya metric not SC metric (which can only describe an eternal BH).

 

[PeterDonis]

I can find no such assumption in the OP

The assumption is in this statement:

If this simple analysis is true, no in-falling astronaut or rock would ever cross the Schwarzschild radius (event horizon) irrespective of coordinate changes.

 

[PeterDonis]

SC coordinates are inapplicable altogether because they describe the wrong external geometry - the distant geometry of an evaporating BH is a Vaidya metric not SC metric (which can only describe an eternal BH).

This is true in principle, but I think it's worth noting that, in practice, Hawking radiation is so faint for a black hole of stellar mass or larger that the Schwarzschild metric is an extremely good approximation for the external geometry. After all, the Hawking radiation energy density is much less than that of the CMBR, and we are willing to approximate a black hole in the real universe using the Schwarzschild geometry in the presence of the CMBR, rather than use the ingoing Vaidya metric, which would be a more exact description. If we wanted to make a better approximation to the entire spacetime, we would have a more complicated model with several spacetime regions:

- A region occupied by the matter that originally collapses to form the black hole.

- A region external to the matter but containing ingoing null radiation (the CMBR) described by the ingoing Vaidya metric, with the energy density of incoming radiation gradually decreasing (to account for the CMBR temperature decreasing as the universe expands); this corresponds to the mass $M$ in the ingoing Vaidya metric increasing more and more slowly with the "time" coordinate (which is actually a null coordinate labeling the ingoing null geodesics on which the radiation travels).

- A region external to the matter and described by the Schwarzschild geometry; this region would join the previous (ingoing Vaidya) region and the next (outgoing Vaidya) region. Physically, this region corresponds to the ingoing CMBR having become too faint to detect, and the outgoing Hawking radiation not yet having become detectable. This is an approximation, strictly speaking, but so are the Vaidya regions--the "energy density of radiation" in them is really the net ingoing/outgoing radiation obtained by subtracting the Hawking/CMBR density from the CMBR/Hawking density, respectively. An exact model would include both the ingoing and outgoing radiation, but we don't have an exact solution that describes that AFAIK.

- A region external to the matter but containing outgoing null radiation (Hawking radiation) described by the outgoing Vaidya metric, with the energy density of outgoing radiation gradually increasing (to account for the Hawking radiation temperature increasing as the hole evaporates); this corresponds to the mass $M$ in the outgoing Vaidya metric decreasing more and more quickly with the "time" coordinate (which here is a null coordinate labeling the outgoing null geodesics on which the radiation travels).

- A region to the future of the outgoing Vaidya region, which is flat; this corresponds to the spacetime after the hole has finally evaporated and the last flash of Hawking radiation has escaped to infinity. (Strictly speaking, it never gets to infinity in finite time, but at any finite radius $r$ there is a time when the flash passes that radius on its way out, and after that time spacetime is flat at that radius.)

 

[PAllen]

This is true in principle, but I think it's worth noting that, in practice, Hawking radiation is so faint for a black hole of stellar mass or larger that the Schwarzschild metric is an extremely good approximation for the external geometry. After all, the Hawking radiation energy density is much less than that of the CMBR, and we are willing to approximate a black hole in the real universe using the Schwarzschild geometry in the presence of the CMBR, rather than use the ingoing Vaidya metric, which would be a more exact description. If we wanted to make a better approximation to the entire spacetime, we would have a more complicated model with several spacetime regions:

- A region occupied by the matter that originally collapses to form the black hole.

- A region external to the matter but containing ingoing null radiation (the CMBR) described by the ingoing Vaidya metric, with the energy density of incoming radiation gradually decreasing (to account for the CMBR temperature decreasing as the universe expands); this corresponds to the mass $M$ in the ingoing Vaidya metric increasing more and more slowly with the "time" coordinate (which is actually a null coordinate labeling the ingoing null geodesics on which the radiation travels).

- A region external to the matter and described by the Schwarzschild geometry; this region would join the previous (ingoing Vaidya) region and the next (outgoing Vaidya) region. Physically, this region corresponds to the ingoing CMBR having become too faint to detect, and the outgoing Hawking radiation not yet having become detectable. This is an approximation, strictly speaking, but so are the Vaidya regions--the "energy density of radiation" in them is really the net ingoing/outgoing radiation obtained by subtracting the Hawking/CMBR density from the CMBR/Hawking density, respectively. An exact model would include both the ingoing and outgoing radiation, but we don't have an exact solution that describes that AFAIK.

- A region external to the matter but containing outgoing null radiation (Hawking radiation) described by the outgoing Vaidya metric, with the energy density of outgoing radiation gradually increasing (to account for the Hawking radiation temperature increasing as the hole evaporates); this corresponds to the mass $M$ in the outgoing Vaidya metric decreasing more and more quickly with the "time" coordinate (which here is a null coordinate labeling the outgoing null geodesics on which the radiation travels).

- A region to the future of the outgoing Vaidya region, which is flat; this corresponds to the spacetime after the hole has finally evaporated and the last flash of Hawking radiation has escaped to infinity. (Strictly speaking, it never gets to infinity in finite time, but at any finite radius $r$ there is a time when the flash passes that radius on its way out, and after that time spacetime is flat at that radius.)

True, but if you wait long enough for CMBR to be insignificant compared to Hawking radiation, you end up mostly just the outgoing Vaidya metric to describe the distant region in these 'late times' - but before final evaporation. Point is, the difference matters in late time as the hole's evaporation is significant.

 

[PeterDonis]

if you wait long enough for CMBR to be insignificant compared to Hawking radiation, you end up mostly just the outgoing Vaidya metric to describe the distant region in these 'late times' - but before final evaporation

Yes, that's the next to last region in my list.

the difference matters in late time as the hole's evaporation is significant.

Yes, I agree that's the key point; even if there is a region where the Schwarzschild metric is a good approximation to the external geometry, that region doesn't last; once the Hawking radiation is significant, the external geometry is no longer Schwarzschild to a good approximation (since that's what "the Hawking radiation is significant" means).

 

[m4r35n357]

If this simple analysis is true, no in-falling astronaut or rock would ever cross the Schwarzschild radius (event horizon) irrespective of coordinate changes. Please help me understand if/why this argument is flawed.

As you and others have pointed out, the Schwarzschild solution cannot describe the evolution of a black hole, particularly the case of infalling matter/astronauts. There is a pretty interesting but qualitative discussion here which might help answer some of your questions, and probably raise some more.

 

[Dale]

a collapsing stellar core (-> Black Hole) ... an in-falling astronaut ... Hawking radiation ... the Schwarzschild radius (event horizon)

Hmm. I don't think that you are using an appropriate model to describe everything that you are talking about. The Schwarzschild metric models an eternal black hole which never collapses or evaporates. To model an astronaut falling in during collapse you would need to use something like the Oppenheimer-Snyder metric. I don't know a metric for a black hole evaporating by Hawking radiation. It seems like you may be trying to apply a simplified static model to cover a dynamic process that it was never intended to cover.

Personally, I would pick a topic. Focus on either the collapse, or the infalling, or the evaporation. Don't try to do all of it at once.

 

[PeterDonis]

I don't know a metric for a black hole evaporating by Hawking radiation.

As PAllen noted, the outgoing Vaidya metric is a good classical approximation for this case.

 

[Dale]

As PAllen noted, the outgoing Vaidya metric is a good classical approximation for this case.

Excellent! That is good to know.