The Black hole (BH) information paradox describes the apparent paradox of information being permanently lost in a BH, contradictory to QM. What I am asking myself is: According to General Relativity (GR), for an observer far away from the BH an object flying towards the BH will never arrive at the event horizon, but instead travel asymptotically slower towards it. Therefore, it will never dissapear in there, and therefore there would be no information loss. Can anyone explain me where this simple thought goes wrong and where the paradoxon appears then?
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1$\begingroup$ Your description of the paradox is incorrect. The information paradox is not that information is lost in a black hole. This part is fine, as your question suggests. The paradox arises when the black hole has completely evaporated. There is no black hole anymore, so where did the information go? Thus your question is based on an incorrect premise. $\endgroup$– safesphereCommented Sep 28, 2021 at 3:47
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According to General Relativity (GR), for an observer far away from the BH an object flying towards the BH will never arrive at the event horizon, but instead travel asymptotically slower towards it.
This is a common misconception.
In practice, the final photon is emitted at a finite time, but this may not satisfy you, as it would simply mean something is happening that we cannot see.
The oversimplified model of a black hole is the Schwarzschild metric, which assumes its mass is not changing and it is surrounded by vacuum. Clearly, this is not the Universe we live in: black holes are not lone entities surrounded by vacuum.
Clearly, we cannot use the Schwarzschild metric to describe a growing black hole, as that is circular logic: it assumes a black hole that does not grow, and then shows that it does not grow.
In detail, to model the situation of matter approaching a black hole, one would have to solve the time-dependent Einstein Field equations and evolve two masses. Because these equations are complex, every solution has its choice of simplifying assumptions.
Recently, due to numerical relativity, scientists can model time-dependent general relativity, which I hope I have convinced you is necessary to describe the time-dependent process of matter approaching a black hole.
The greatest success of numerical relativity is the simulation of binary black holes and the resulting gravitational wave signature as detected by LIGO. If you believe their simulations, then black holes indeed capture material in finite time and have time-dependent horizons during that process: see this LIGO video.
The simplest conceptual description of this time-dependent process is to imagine adding 1/10 the mass of a black hole to a black hole. Then, as the mass approaches the 1.1x the radius of the old event horizon, there will be enough mass within 1.1R to form a 1.1R black hole.
See here:
https://math.ucr.edu/home/baez/physics/Relativity/BlackHoles/fall_in.html and here:
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$\begingroup$ As a note, it's certainly fair to disagree with this description of finite time growth, but I hope it has at least been shown that the claim of infinite time growth is not proved, as it relies on the approximations and assumptions of the Schwarzschild metric. $\endgroup$– AlwinCommented Sep 27, 2021 at 20:53
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$\begingroup$ Just that I got your answer correctly: You say, since we cannot use the Schwarzschild metric (but rather Kerr metric for example), this is not true? Hypothetically, if there was a non rotating BH, just for fun, then my argument would be valid, wouldn't it? $\endgroup$ Commented Sep 28, 2021 at 2:14
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$\begingroup$ Thanks for checking. Actually, my argument applies to non-rotating BH. The point is that both Kerr and Schwarzschild assume that the mass is constant, so of course if you assume that they cannot grow, you will find they cannot grow. You need a two-body metric that changes with time to account for a black hole swallowing another body over time. The best examples of that are the numerical relativistic simulations. $\endgroup$– AlwinCommented Sep 28, 2021 at 2:36
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$\begingroup$ If you answer suggests that a growing horizon would swallow nearby objects, then it is incorrect. Due to the effect of linear frame dragging, things at the horizon move with the horizon and never cross it in the coordinate system of any external observer. The simplest example, a flying black hole hits an object. The horizon is moving, will it swallow the object? It will not, as is obvious by changing the coordinates to make the black hole stationary. The quote you took from the question is correct while your answer is misleading at best. $\endgroup$ Commented Sep 28, 2021 at 3:57
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$\begingroup$ Maybe you can explain further, as your statements circularly rely on the assumption of a stationary spacetime with an event horizon to prove that the black hole is stationary, whereas my statements refer to a dynamically evolving spacetime and dynamical horizons. See: arxiv.org/pdf/gr-qc/0604015.pdf $\endgroup$– AlwinCommented Sep 28, 2021 at 8:26