Dear Michel Planat,
I have been reading a paper by Maldacena and Susskind. This is a fairly bold paper that advances a pretty speculative idea. In keeping with my paper, which advances an associativity issue with quantum fields near the horizon , this seems to have a higher associator structure that is five fold. The most elementary is a three way associator (ab)c - a(bc) = [a,b,c], that defines a fundamental form for a quantum homotopy but a five fold system is a second homotopy group, The fluctuations across the inner and outer horizons of a black hole results in a 5-fold associative system. This is also a pentagonal system for the Kochen-Specker theorem.
The Susskind-Maldacena paper requires there to exist two black holes with the same BPS charge and angular momentum. Even if such black hole pairs exist it is not clear how they become EPR pairs. In most standard QM systems it requires some sort of mutual interaction to establish an EPR pair. To argue that a black hole is an EPR pair with another it means there exist in the multiverse some other black hole with an identical quantum configuration. This demands a multiverse landscape, for it is probably not likely this exists within the observable universe.
The idea is interesting though. The odd thing about this idea, which has been making the rounds these days, is it makes some physical sense of the interior of a black hole. Maldacena and Susskind work with a Schwarschild black hole. There idea is there are entanglements between black holes through wormhole. This sort of multiply connected topology does exist with black holes. A BPS or rotating black hole has two event horizons at r_{+/-} = m +/- sqrt{m^2 - Q^2}, where Q can be either a gauge charge or angular momentum parameter. There are then two event horizons and three regions; region I being where r > r_+, region II where r_+ > r > r_-, and region III where r < r_-. These regions are timelike, spacelike and timelike respectively. The region III has been regarded as suspect, since the r_- event horizon has a pile up of UV divergent radiation or quantum fields that implies the horizon is physically singular. This region has been regarded as a sort of mathematical fiction. However, maybe this region does play some sort of physical role.
The Kerr black hole appears in the first diagram is attach. The second attachment is a Penrose diagram of the Kerr black hole. It is evident that upon leaving region I (the normal timelike universe) the observer enters region II which is shared by another black hole. The horizon is split so the observer may enter two III type regions. The ring singularity in region III is where x^2 + y^2 = Q^2. In this III region the geodesics around the ring are similar to the flow of a hurricane around the eye. Also the singularity is repulsive; you can't reach it. In complex coordinates this singularity has a branch cut. If you make a complete orbit around the ring there is a branch cut which pops you into an identical copy of the III region as branch cuts link Riemann sheets of the complex plane. The interior region of a BH is naturally in a sense a sort of wormhole, and this approach might segue into this equivalency between wormhole multiple connectivity and entanglements. This interior region may physically play the role as an "entangler."
The argument for a firewall associated with a black hole concerns entanglement swaps between states in region I and II. These states are H_h ∊ I for Hawking radiation, H_s ∊ I for states on or near the stretched horizon associated with r_+ and interior states H_n ∊ II for states near the horizon and H_s ∊ II on the singularity. In the Kerr-Newman metric this singularity is identified with r_-. Physically r_- is a region with UV divergent quantum fields. However, we may remove this as a singularity if this divergence is regulated in some manner. That of course is an open question. The singularity states H_s are split into H_r- for states in the region II near r_- and those in the core region H_{III}. We now have a 5-fold system of states.
I have argued that three quantum states along a null ray are an associated quantum system. If the middle state is very near the horizon and the horizon has a quantum uncertainty the entanglement between the three states is an associator system [a,b,c] = (ab)c - a(bc). In quantum homotopy this is a fundamental group π^1. A 5 fold system is a set of states in a permuting structure that gives π^2, and there is a higher system that defines the Stasheff polytope π^3, and it goes on from there. The five fold system is equivalent to the pentagonal arrangement of states in the Kochen-Specker theorem in four dimensions. The associator system is then a form of the KS theorem. The KS theorem in 4 dimensions is a result of a three color graph with pentagonal symmetry.
If there is then this sort of black hole entanglement that is equivalent to a worm hole it may then be of this nature. Again the interior region, if it is physically real, is such that an orbit around the singularity pops the geodesic into an equivalent spacetime. It is a form of the multiple sheets of the complex plane connected by a branch cut. The argument for spacetime would be rather difficult, for this is not an elementary conformal argument in two dimension, but four dimensions.
Lawrence B. CrowellAttachment #1: kerr_bh.jpgAttachment #2: Penrose_diagram_for_Kerr.png
