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I am not bothered that much by the fact that two observers describe the same phenomenon differently. Something similar , in principle ,  happens with simultaneity in special relativity,  and special relativity is a necessity based on experimental data that led to Maxwell's equations and the constancy of the speed of light in vacuum.

I think that the (mathematical ) existence of the singularity inside a black hole  is due to the inadequacy of the equations of GR (otherwise successful when quantum effects are negligible) related to what happens close to the center of a black hole, quantum gravity will eventually take care of that problem (hopefully ).

The fact that Alice merrily passes the EH and gets torn apart by gravitational tidal forces (and eventually by the "singularity"), while  Bob "thinks" (based on the information he receives far from the black hole ) that Alice's bits (or qubits ) of information content are spread over the EH of the black hole (after an eternity has passed for him ),  I don't see that as a big problem either ,  they just describe the universe from different perspectives.  ref. 1

The black hole information paradox is a more difficult problem, though. How do you reconcile unitarity in QM (information conservation,  and reversibility of physical laws ) with the fact that no information leaks from beyond  the EH of a black hole, this is the main problem, as far as I can see, right? .  

Hawking radiation (related to entangled virtual particles at the EH) , the firewall problem , these are several attempts to solve the problem. ref. 2

Every accelerating system generates gravitational waves, so in principle (emphasis on "in principle ") even a proton crossing the EH of a black hole generates gravitational waves (as faint and undetectable  as they may be ), that can encode information about the matter falling into a black hole. So even this more difficult problem might allow solutions , in principle. Am I missing something here?

Question: Could the persistent effects of gravitational waves solve the black hole information paradox?ref. 3

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Persistent effects of gravitational waves may not solve the problem, but they are at least being recognized (relatively recently) as an important part of understanding the problem. This is reviewed in

Since this is still an active area of research with unsettled questions, I'll just show this exerpt from that review (section 7.3), which expresses the state of the art according to an expert:

The no-hair theorem tells us that Einstein-Maxwell gravity has a three-parameter family of black hole solutions labeled by the mass $M$, charge $Q$, and total angular momentum $J$, up to diffeomorphisms and gauge transformations. It has long been accepted that a few diffeomorphisms (for example, boosts or translations) act nontrivially on the physical phase space and impart a sparse head of hair on black holes. We have shown that there are in fact an infinite number of diffeomorphisms... that act nontrivially and thereby impart a lush head of soft hair on black holes. The soft hairdo is classically measured by conserved superrotation charges at infinity. More hair is supplied by large gauge transformations. The hairdo is rearranged whenever anything is thrown into the black hole, and it thereby stores partial information about how the black hole is formed. Exactly how much information is stored, and how it is transmitted to the outgoing Hawking radiation, is presently unknown. At the very least, understanding black hole information requires understanding the properties of soft hair and more generally keeping careful track of how much information flows into or is stored in the deep infrared...

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    $\begingroup$ Wow, so it seems that the no-hair theorem has been quietly growing follicles! $\endgroup$ – Chappo Says Reinstate Monica May 16 at 11:28

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