I tried to post this last week and it would not let me. I try again.
I recently re-read the paper The Page curve of Hawking radiation from semiclassical geometry by Almheiri, Mahajan1, Maldacena, and Zhao arXiv:1908.10996v1. There is no paper of late that I have read so many times as this. This paper relies heavily upon the idea of quantum extremal surfaces and this paper Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime, Engelhardt and Wall arXiv:1408.3203v3 . This physics is the relationship between spacetime and quantum horizons. The metric g_{ab} = g0_{ab} + g^{Д§^ВЅ}_{ab} + g^Д§_{ab} + .... is expanded in units of the Planck length в„"_p = в€љGД§/c^3 в†' O(Д§^ВЅ). This is the logic of MTW with g = g0 + в„"_p/L + ... . This leads to a description of quantum states with an entanglement between exterior and interior states.
It occurred to me this could be seen in some simple ways. For the Kerr-Newman metric
ds^2 = (1 - 2m/r + Q^2/r^2)dt^2 - (1 - 2m/r + Q^2/r^2)^{-1}dr^2 - r^2dО©^2
The near horizon condition for an accelerated observer above the horizon is found to be
ds^2 ≈ (r/m)^2dt^2 - (m/r)^{-2}dr^2 - r^2dΩ^2
and curiously if one derives a similar metric for the near singularity condition r = 0 the metric is
ds^2 ≈ (Q/r)^2dt^2 - (Q/r)^{-2}dr^2 - r^2dΩ^2.
The two metrics are interestingly dual with respect to mass and the charge, or equivalently with angular momentum. Both metrics are AdS_2Г--S^2 or the anti-de Sitter spacetime in 2 dimensions, and in a BPS setting indicate a duality between the mass variable and quantum numbers associated with gauge fields. The extremal black hole with m = Q is the extremal black hole, and Carroll, Johnson and Randall showed this condition leads to the discontinuous map of the spacelike trapping region to AdS_2Г--S^2.
This dualism suggests that quantum states in the deep interior of the BH, on so called islands, near the singularity are dual or even entangled to states near the horizon or in the exterior. Conformal patches on the two AdS_2 spacetimes may correspond to each other, and the corresponding CFT_1, with AdS_2 ≈ CFT_1, are dual chains similar to a Haldane chain. The boundary of AdS_n is a timelike region, which means AdS space is not globally hyperbolic due to the existence of a timelike boundary at infinity. However, if there are boundary conditions at infinity there are then causal properties are still fine assuming that there are boundary conditions at infinity Hence a conformal patch in AdS_3 may have a boundary with CFT_2 that shares boundary conditions equivalent to the AdS_2. One way to think of this is that CFT_2 can define gauge-like gravity that has the same DoFs of bulk gravity in AdS_2. In this way the AMMZ argument for a higher dimensional space ties the two CFT_1 chains together in an entanglement.
The AdS_n black hole correspondence identifies holographic content of the event horizon with the CFT_{n-1} corresponding to the AdS_n the BH is embedding within. The AdS_3 corresponds to the BTZ black hole. The CFT_2 in a gauge-like gravity theory can describe quantum gravitation for the AdS_2. The area law S = /4в„"_p^2 + O(Д§) quantum corrections is then "constructed" through this correspondence between AdS_2 and AdS_3. The "area" in the AdS_2 case is a 0-dimensional, and in the AdS_3 as with the BTZ BH it is a circle. Hence for the AdS_2 the theory is a scalar field theory.
The construction is then a form of flag manifold. The AdS_n = O(n,2)/O(n,1), where O(n,2) is the isometry group of the space. The AdS spacetime is a Stiefel manifold and a form of flag manifold with
F^N_{d1,d2,...,dm}(в„‚) = SL(N, в„‚)/P^N_{d1,d2,...,dm}= U(N)/[U(k1)Г--U(k2)Г--...Г--U(kr)Г--U(kr+1)],
for k1 = d1, kr+1 = N, k_{n+1} = d_{n+1} - d_n. The real valued form of this flag manifold is
F^N_{d1,d2,...,dm}(в„ќ) = SL(N, в„ќ)/P^N_{d1,d2,...,dm}= O(N)/[O(k1)Г--O(k2)Г--...Г--O(kr)Г--O(kr+1)].
It is clear that AdS_n = F^{n,2}_{n,1}(в„ќ). The flag F^N_1(в„‚) is в„‚P^{N-1} = U(N)/[U(1)Г--U(N-1)]. Similarly, the real valued version is F^N_1(в„ќ) = в„ќP^{N-1} = O(N)/[O(1)Г--O(N-1)], where O(1) is a trivial group and в„ќP^{N-1} = O(N)/O(N-1) , where S^2 = O(3)/O(2). In the split form we then have
в„ќP^{N-1,2} = O(N,2)/O(N,1),
And AdS_n = в„ќP^{n-1,2}. The flag manifold F^4_1(в„‚) = в„‚P^3 = U(4)/[U(1)Г--U(3)] gives twistor space. In the split form
F^{2,2}_1(в„‚) = в„‚P^{2,1} = U(2,2)/[U(1)Г--U(2,1)],
this is according to the isometry group of AdS_5.
The flag manifold F^{2,2}_2(в„‚) = U(2,2)/[U(2)Г--U(2)Г--U(2)] is the Grassmanian space G_{4,2}(в„‚). This is identified as a spacetime, where dS and AdS spacetimes are defined according to a line element
A = t^2 В± u^2 - x^2 - y^2 - z^2,
with u^2 de Sitter and -a^2 anti-de Sitter. The light cone for A = 0 separates the AdS and dS. The flag F^{2,2}_{2,1} = U(2,2)/[U(1)Г--U(1)] with the bi-fibration
в„‚P^{2,1} в†ђ F^{2,2}_{2,1} в†' G_{4,2}(в„‚) вЉѓ dS_5 в€Є AdS_5.
These spaces are Kähler manifolds, where complex projective space is the Fubini-Study metric for projective Hilbert space. A simple entanglement geometry is of the form
в„‚^2вЉ--в„‚^2вЉ--... вЉ--в„‚^2/SL(2, в„‚)Г--SL(2, в„‚)Г--... Г--SL(2, в„‚).
where the flag manifold F^{2,2}_2(в„‚) = U(2,2)/[U(2)Г--U(2)] is similar to в„‚^2вЉ--в„‚^2/[SL(2, в„‚)Г--SL(2, в„‚)], but with a topological difference between в„‚^{вЉ--2} and SL(4, в„‚) ≈ U(2,2). U(2,2) has roots and weights with a Weyl chamber, while в„‚вЉ--в„‚ does not. The twistor bi-fibration is a projective map on a form of quantum entanglement.
This then concerns the emergence of space or spacetime from quantum entanglements. The geometry of spacetime, or G_{4,2}(в„‚) in 6 complexified dimensions is equivalent to a form of tripartite entanglement. Of course, we do not expect spacetime to be built up from just three states, but rather a spectrum of such states or a condensate of identical states. This returns us to the duality between the near singularity metric and the near horizon singularity. The near singularity metric is determined by gauge charges, while the near horizon condition is determined by mass. This duality between gauge charges and gravitation is a form of this equivalency between quantum mechanics and general relativity.