How can information ever get lost at the event horizon of a black hole?

Question

In the drawing, A and B are two entangled particles in Kruskal coordinates, A is falling into the black hole, B is remaining outside.

The lines going through the center are the time coordinates of the far-away observer : t = 0, t = 1, t = 2, limited by the event horizon where t = $\infty$. Until the end of time t, the worldline of B is remaining outside the event horizon, and until the end of time t, the worldline of A is on its travel to the event horizon without ever reaching it.

The question: As A will never reach the event horizon, according to the time coordinates of the far-away observer how could information ever get lost from the point of view of the reference frame of an outside observer? If information is remaining outside the event horizon until the end of time, I do not see how there can be any issue of loss of information in a black hole.

Note: I am aware of the fact that the result is very different from the point of view of the infalling observer A. According to the reference frame of A, A is entangled with B until it crosses the event horizon, and at this moment it is losing suddenly the entanglement.

Answer 1

$t$ is not really "the time of a faraway observer"; it's a time coordinate that anyone can use. But it's true that you can use $t$, or many other time coordinates with similar properties, to argue that there is no time at which information is unambiguously lost in a Schwarzschild black hole.

That argument doesn't work if the black hole evaporates, because the evaporation is unambiguously in your past. The information-loss paradox only dates back to the discovery of black hole evaporation, and as far as I know it has never been considered to apply to Schwarzschild and other eternal black hole solutions.

Answer 2

In Nikodem Poplawski's torsion-based cosmological model (described in numerous preprints, written between 2010 and 2021, that can be found by his name on the Arxiv website), information is not lost in an absolute sense, although "we" (meaning some majority of sentient beings generally) do lose sight of it for a phenomenally long time.

His model is based on 1929's Einstein-Cartan Theory (worked out by conversations between Einstein and the mathematician Cartan, a few years after the discovery of particulate spin), rather than 1915's General Relativity. In ECT, fermions have a specified spatial extent (a few orders of magnitude greater than the Planck length) in each causal patch, whereas in GR they don't (although they are commonly considered to have some minimum size).

Perhaps so that some of the astronomical evidence which might support it would be visible, Poplawski's model is based on the gravitational collapse of large rotating stars, after the star's expenditure of its nuclear fuel would leave it without radiation pressure sufficient to resist that collapse. (There has been evidence for at least 90 such collapses, evinced by the elliptical orbit still followed by the former binary partner: A substantial proportion of stars are in binary pairs.)

In the collapse, an event horizon propagates outward from the star's center, separating the fermions of many virtual particle/antiparticle pairs from each other through extreme tidal effects, with the outer one escaping and the inner one proceeding into contact with the vastly larger stellar fermions: That contact reverses and greatly acclerates the trajectories of the fermions newly-materialized by separation from their virtual partners, and they form a new "local universe" that subsequently expands indefinitely, within and beyond the spatial volume that the "parenting" star had occupied.

Poplawki's 2010 paper described his model as an "alternative" to cosmic inflation, although it's generally considered to be a version of inflation, without a need for the hypothetical scalar "inflaton" field required in the older model developed by Guth.

So, as Poplawski would have us finding ourselves in a local universe formed by the means described, how do we regain sight of the escaped particles? By waiting through some, or even all, of the phenomenally long "Poincaré recurrence time", which was confirmed mathematically by Cathéodory in 1919. There's even a slight possibility that we might have to wait through an unspecified number of recurrences, which occur in phase space and would probably have (meanwhile) been found to have some abstruse connection with quantum uncertainty.

Because the direction of passage through relativity's time dimension would be inherited by each "baby universe" from its parent, Poplawski's theory would be confirmed by a prevalent direction of motion in a rotating region perhaps far larger than our observable one (but still within our "local universe"). Although there have been many searches for evidence of such locally-universal rotation, formulated on different bases over many years, a recent one (by Lior Shamir) does appear compatible with Poplawski's model.

Nevertheless, getting a handle on GR's multiple equations is so difficult that any generalized replacement of it by ECT (or by ECSK or ECKS, as it's sometimes known, after modifications to it that were made several decades ago by Sciama and Kibble) may occur only after some appreciable piece of the recurrence time has passed. The heartening fact that it reduces to GR in vacuum is often overlooked.

Regarding Poincaré recurrence, I have to point out that it's not limited to Poplawski's model, which I'd chosen to describe because of its avoidance of a singularity: As the local speed limit for light (which certainly applies throughout our observable region) is based on the motion of potentially massive objects with respect to each other, it can be applied to all cosmological models taking account of that spatial expansion which was first noticed by Friedmann in the early 1920's, not just Poplawski's far more recent one. Spatial expansion is not relative motion, and, as noted by Davis of Lineweaver & Davis fame, it does not cause relative motion.