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?
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.