I Black hole formation and infinite redshift

[LightPhoton]

TL;DR Summary

Given that we now do know that black holes exist, how is this apparent paradox resolved?

In A short course in general relativity, Foster and Nightingale write:

If one assumes that the general features of a collapsing object are not too far removed from those that prevail in the spherically symmetric case, then one would expect the emergence of an event horizon which would shield the object in its collapsed state from view (see Fig. 4.14). An outside observer would see the object to be always outside the event horizon. However, it would effectively disappear from view because of the increasing redshift, and a black hole in space would be the result.¹⁸

¹⁸It would take an infinite time to disappear. If black holes do exist, then this is an argument that they must have been "put in" at the beginning.

So in modern astronomy, how is this apparent paradox resolved?

 

[Nugatory]

So in modern astronomy, how is this apparent paradox resolved?

The diagram itself shows the resolution of the paradox: The incoming matter does cross the horizon, rather quickly as measured by a clock that is part of that infalling matter. It does "take an infinite time" for light from that event to reach the outside observer, but that doesn't mean it didn't happen, it means that they don't see it happen. Lots of things happen that we don't see.

The "take an infinite time" language is actually quite sloppy; it is seldom used in serious textbooks and when it is it comes with a lot more context. It would be better to say that the outside observer is assigning time coordinates to events on the worldline of the infalling matter in such a way that they increase without bound as they approach the horizon. Phrased this way, it is easier to see that the infinity is an artifact of the way that we've chosen to assign these coordinates. Use coordinates that are suitable for describing the spacetime at the horizon (Kruskal coordinates would be a good choice) instead of Schwarzschild coordinates and the problem goes away.

 

[Dale]

I don’t think we have a paradox here. The second statement is just wrong.

 

[phinds]

So in modern astronomy, how is this apparent paradox resolved?

By noting that there IS no paradox. The distant observer's observation does not describe what is actually happening to the infaller, as he will know quite well if he understands physics, as @Nugatory explained.

 

[PeterDonis]

@LightPhoton please read this Insights article:

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

 

[Orodruin]

I never liked that spacetime diagram. It gives the inpression that the singularity is timelike rather than spacelike. What was wrong with using a Kruskal diagram?

 

[pervect]

TL;DR Summary: Given that we now do know that black holes exist, how is this apparent paradox resolved?

It wasn't clear to me what you thought was paradoxical about the quoted remarks. Could you point out what you thought was paradoxical?

Note that from context, we can interpret the word "see" quite literally - as in what a camera photographs. Basically, the diagram shows that light rays can only reach infinity from the region outside the event horizion - light rays , aka null geodesics, emitted inside the horizon can't escape, so we can't possibly "see" them (i.e no photograph will show such an escaping light ray).

I don't want to spend a lot of time belaboring some point that isn't relevant to your question, so I (and others) could use some more guidance on what it is that is puzzling you about the remarks you cite.

 

[PAllen]

In the OP, it appears a footnote apparently from the referenced text, is simply wrong. Only @Dale , in this thread seems to have caught this, so I want emphasize that the problem is an error the book referenced.

 

[PeterDonis]

the problem is an error the book referenced.

Yes, I agree.