Falling into a black hole "paradox"

[MigL]

2 hours ago, dimreepr said:

How will we ever know?

I know what you mean ...

I was thinking about it overnight, and almost ready to give in to Halc and md65536 PoV, thinking that maybe I had made some incorrect assumption. 
Then Markus posted, and now I'm not sure either.
It seems to be a scale problem, with differing scenarios, and gruesome calculations.

But I did enjoy th discussion 🙂 .

[md65536]

3 hours ago, Kino said:

There is no invariant sense in which you can say that "your toes have now crossed" until your head also crosses, so you cannot report this. In your local inertial frame your toes do cross first, and you are free to report that your toes have now crossed (but others are free to disagree).

That's an interesting point. Indeed, "Earth" receiving your message in the far distant future would say "that never happened." If your head could at that point, separate from the feet and escape the black hole, it could go from its local inertial frame where the feet crossed, and return to Earth where the feet never crossed. That seems weird, but it's no weirder than the Andromeda "paradox".

Everyone here agrees one way or another that the head while outside the EH never sees light that originated inside, so nothing contradictory is measured by anyone (no observation of an event that can later be said not to have happened).

The astronaut saying "my feet have now crossed" is a red herring that suggests there is a paradox when there is none. Without that, the astronaut simply sees images of her feet from outside the EH while outside the EH, and from inside the EH while inside. The details are the same, but may be easier to think about?

 

9 hours ago, Markus Hanke said:

I don’t see how the eyes of the astronaut could possibly “catch up” with the photon, while still preserving the usual local laws of SR. Perhaps the answer is obvious (lol), I just don’t see it right now.

I think it's obvious if you look at it in terms of intersecting world lines and light cones. I don't know the calculations, but it's easy to see using diagrams of lightcones made by people who've done the calculations. Also, it's easier to think about what information is available to an observer, and not about how it would look. If the object's world line and an event's future light cone intersect, that object "sees" the event. Those are invariant and can tell you what happens without needing to describe it in a given observer's local coordinates. Because the coordinate speed of light at a distance inside the BH is not c, there's going to be visual distortion at some point, and you will lose sight of your feet eventually as the light cones narrow as you approach the singularity (before, after, or during the inevitable spaghettification?, I don't know), but that doesn't always need to be figured out.

 

3 hours ago, Kino said:

Here's a Kruskal diagram, copied from Wikipedia, with toe and head worldlines drawn (badly hand-drawn black lines), and a few light signals exchanged (red from toe to head, green from head to toe).

Interesting! The red lines show light from toe to head going from a lower r to a higher r outside the EH (pink), constant r on the horizon, and from higher r to lower r inside the horizon. A 45 degree line toward +X anywhere in those regions would have those properties.

Edited March 14, 2021 by md65536

[md65536]

1 hour ago, md65536 said:

you will lose sight of your feet eventually as the light cones narrow as you approach the singularity

This is wrong, as Kino wrote and the diagram shows. It seems you never lose sight of your feet, even if you dropped them in ages before the rest of you went in, but you'd also never see them hit the singularity (of course). You just see older images from when they were above where you are now, and there'd be a last possible image of them which depends on how long ago they entered, so I suppose they'd have to appear increasingly red-shifted.

Edited March 14, 2021 by md65536

[beecee]

4 hours ago, MigL said:

I know what you mean ...

I was thinking about it overnight, and almost ready to give in to Halc and md65536 PoV, thinking that maybe I had made some incorrect assumption. 
Then Markus posted, and now I'm not sure either.
It seems to be a scale problem, with differing scenarios, and gruesome calculations.

But I did enjoy th discussion 🙂 .

As did I, despite not having much hair left.  Seriously though, thanks to all in this informative discussion. 

[md65536]

Looking for more details, I came across this:

Falling into the Schwarzschild black hole. Important details
S. Krasnikov

https://arxiv.org/abs/0804.3619

Quote

In Reissner-Nordström and Kerr black holes under their event
horizons (which are quite similar to Schwarzschild’s) there is another re-
markable surface — the Cauchy horizon. And that horizon does have the
property in discussion: an astronaut falling into the black hole reaches the
Cauchy horizon in a finite proper time and crosses it in a point p that con-
tains in its causal past the whole “external universe”. Such an astronaut,
indeed, will be able to see the death of stars and galaxies,

which is surprising, but is a reminder that the details describing a Schwarzschild BH can be vastly different for BHs in general. The Kruskal diagrams are of Schwarzschild spacetimes.

[Halc]

3 hours ago, md65536 said:

Looking for more details, I came across this:

Falling into the Schwarzschild black hole. Important details
S. Krasnikov

The author admits in the intro that an observer does not see the history of the universe. That question is discussed on stack exchange here which admittedly discusses the simplified case of a Schwarzschild black hole.

This image from there:

This clearly shows a finite history of some outside object (dotted worldline) seen by the blue worldline of our infalling observer, Cauchy horizon or not. Kino's Kruskal diagram shows the same sort of thing.

The Cauchy horizon for a charged or rotating BH seems to have special treatment in that the spacetime beyond it isn't 'smooth', but I still don't know how the history of the universe could be in the causal path of that region.

The wiki page on a charged black hole makes references to the 2nd horizon. No nice diagrams like above ☹️ and the ones in the paper don't seem to demonstrate the claim to my untrained understanding.

It seems that sufficient charge can prevent any mass from collapsing into a black hole (layman guess: because the EM repulsion is greater than the gravity attraction?), and perhaps below said Cauchy horizon, it prevents the singularity from forming, resulting in a chaotic fixed 'density' with no actual singularity?  Just guessing here, but such a construct could be maintained indefinitely. This does not seem to be what the author above is saying, since he talks of a point p where the observer crosses this horizon (an event) that has the whole universe in its causal history, not the collective set of events inside this horizon. I'm obviously out of my depth here. Still, Reissner-Nordström and Kerr black holes (and Schwarzschild) are not models of real black holes since none of those solutions include Hawking radiation.  Any BH has a finite life, and any events outside the past light cone of the final evaporation event will not be in the causal past of any event within the black hole.

Edited March 16, 2021 by Halc

[MigL]

2 hours ago, Halc said:

Still, Reissner-Nordström and Kerr black holes (and Schwarzschild) are not models of real black holes since none of those solutions include Hawking radiation. 

I would think that R-N and Schwarzschild BHs are highly unrealistic.
A R-N BH with any charge will be highly 'selective of the ionized plasma in its accretion disk, and quickly rid itself of that excess charge.
While any collapsing body, with even the smallest hint of rotation, will see it amplified tremendously by the 'ice-skater drawing in her arms' effect.
That makes Kerr BHs the only realistic solution.

Hawking radation is only a 'tacked-on' QM effect, due to having assigned entropy to the surface of the EH, and therefore, a 'temperature' to the BH.

Glad this conversation is ongoing.

[md65536]

2 hours ago, Halc said:

Still, Reissner-Nordström and Kerr black holes (and Schwarzschild) are not models of real black holes since none of those solutions include Hawking radiation.  Any BH has a finite life, and any events outside the past light cone of the final evaporation event will not be in the causal past of any event within the black hole.

Yes, exactly, so we can discuss what would happen or be seen in a particular model of a black hole, but must be careful not to make the same claim for just "a black hole" in general.

Having the universe's entire history in your causal past would describe a particle that is stuck on (or asymptotically approaching) some horizon, with a proper time that approaches infinitely slower than the rest of the clocks in the universe. That would be similar to an observer who could hover infinitely close to a Schwarzschild BH event horizon. So, EM repulsion exceeding gravity would make sense... it would be like the BH itself is causing you to hover (but this is inside the EH, so I don't think infinite energy would be needed?). I think the singularity still exists in those types of BHs, but with different geometry.

[beecee]

11 minutes ago, MigL said:

While any collapsing body, with even the smallest hint of rotation, will see it amplified tremendously by the 'ice-skater drawing in her arms' effect.
That makes Kerr BHs the only realistic solution.

While agreeing with that for the present time and the forseeable future, wouldn't even the angular momentum be very slowly negated over time? (friction with the accretion disk spiralling in for example) so that we end up with all Schwarzchild metric in the distant future. 

[MigL]

IIRC, that happens when a test mass is counter-rotatng wrt the BH, within the ergosphere, and by what is called the Penrose process, that mass can split in two, one half falling through the EH and gaining negative energy, while the other half has more energy than the original. This energy comes from the rotational energy of the BH, and slows its rotation down.

Usually, the accretion disk is rotating with the BH ( due to frame dragging effects ) and actually add to its rotational momentum.

[J.C.MacSwell]

22 hours ago, beecee said:

While agreeing with that for the present time and the forseeable future, wouldn't even the angular momentum be very slowly negated over time? (friction with the accretion disk spiralling in for example) so that we end up with all Schwarzchild metric in the distant future. 

Friction can degrade energy but it can't reduce the angular momentum of a closed system...so unless the accretion disc has angular momentum opposed to that of the BH there shouldn't be any negation over time.