This is a quite conceptual question, but I found it interesting. The holographic principle states that, roughly, any system containing gravity can be described on a theory on its surface. It applies to black holes, as far as I know. Well, if it is true, is the spacetime singularity INSIDE of the black hole described by something on the surface as well? If holography holds, it seems that even the spacetime singularity (if real, its physical existence as naked singularity is controversial), should be described on the surface or the spacetime singularity would disappear since the surface is not singular, unless the event horizon is also fake (as the firewall and black hole information paradox, unsolved, seems to suggest too). In summary, can holography erase the need of a spacetime singularity inside any black hole?
1 Answer
The firewall is a signature that we really do not know how to treat the interior of a black hole. Whether holography can inform us about the singularity is not clear. So far with the firewall it appears holography has not informed us. Of course this does not prevent us from thinking about this.
The firewall comes from the problem of entanglement. In some ways the firewall may be an obstruction in our understanding of entanglement as much as with Hawking radiation of black holes. If there is a black hole in the vacuum we know that the vacuum consists of virtual processes that are loops or equivalently the entanglement between a state and its complementary state, such as particle and antiparticle. A loop that straddles the horizon splits into a state entering the black hole and one escaping to ${\cal I}^+$. This means the Hawking radiated boson is entangled with the black hole. The can continue up to a certain point. After a time, called the Page time, the entanglement entropy of the black hole saturates the Bekenstein-Bousso limit. We may also think of this according to a thermal cavity, where around this time photons that are emitted must now be entangled to radiation previously emitted. This means we have something funny happening. The Hawking radiation that is emitted is now entangled with the black hole and previously emitted radiation. We now have a problem in that those previously emitted bosons that were in a bipartite entanglement with the black hole are now in a tripartite entanglement. Entanglement is unitary and the conversion of an entanglement of a certain symmetry, say bipartite, to another type of symmetry, say tripartite, is not a unitary process. This is the quantum monogamy problem, and to avoid this with black hole radiation the firewall is imposed in an ad hoc manner to prevent further violations of quantum unitarity. However, this violates the equivalence principle.
Susskind thinks however that the particle that enters the black hole is in fact identical to the previously emitted boson. This removes the problem of the conversion between entanglement types, but this ER = EPR hypothesis requires some quantum error correction code (QECC). A most general understanding of QECC with spacetime physics is with the Ryu-Takayanagi formulation. An overview of that can be seen in this paper https://arxiv.org/pdf/hep-th/0605073.pdf .
The QECC is something that works up to certain limits. Think of working in a library where you have to shelve books. Over time there are errors that creep into the shelving because of patrons who imprecisely return books and other errors. As you work you try to correct for these errors, which is not difficult if books are not too badly mis-shelved, and the library's catalog information is commensurate with the index, now on computers and once was the venerable card catalog. These small errors are an example of Hamming distances that are small enough to be corrected for by an algorithm. Suppose you go on vacation and upon returning you find the library has been horribly overturned with books terribly out of order. You can't so easily just do a bit of correcting, and if the jumble is bad enough you basically have to start over. This is a case where the Hamming distance may be too large to be simply corrected. The point where this happens turns out to be close to about the Page time.
What does this have to do with singularities? The entanglement of a Hawking boson with the black hole is connected to the horizon. However, if you decided to follow the ingoing boson into the black hole interior then in effect this entanglement is with the black hole singularity. The singularity is an odd thing, but we can see that it is a spatial slice that carries content that is also identified on ${\cal I}^+$. The singularity is then a hot region identified with all the Hawking radiation emitted. This in part is what Susskind identified. The particular slicing of spatial surfaces in the animation here illustrates this identification. This suggests in the BMS perspective of quantum hair that what quantum information is available on the horizon of the black hole is also available on the singularity. The horizon then from the perspective of a highly accelerated observer, where the horizon appears hot, contains the same information as what is on the singularity. The singularity is a very hot and unstable, and that a spatial surface in the BH interior is identified with a null infinity ${\cal I}^+$ might suggest this is an unstable vacuum, such as seen with the bosonic string or maybe a condensate of tachyons.
The firewall is in effect a sort of naked singularity, which is one reason many theorists hate this idea. It is either something physical or purely a gadget that illustrates an obstruction. At this time we really do not know. I suspect there is some conversion of quantum information into hidden forms with massive entanglements that construct spacetime. The singularity as a gas of bosonic strings with an unstable vacuum or tachyons is a sort of construction that might point to this. The condensate is mapped into these states and an entanglement structure for spacetime. Of course this is a bit speculative, but it might suggest something. The upshot at this time is that we really do not know the answer to this question with any great certainty.
