The singularity of a black hole represents a change in phase of a system.
The path integral defines states of the form
|ψ> = Z|0> = ∫D[φ]q e^{-iH(φ)t/ħ}|0>.
The partition function in statistical mechanics is of the form
Z = sum_n e^{-E_nβ}.
The primary differences are the partition function is Euclidean and
discrete, while path integrals are complex valued and for continuous fields
are themselves continuous. Let us consider the Ising model of spins with H
= κσ_iσ_j. The partition function may then be thought in the manner
Onsanger considered as
Z = sum_{ij} e^{-|i – j|ξ(β)},
where ξ(β) = (β - β_c)^n. Here β_c represents a critical temperature
1/kT_c. The system is discrete since the Hamiltonian operates for nearest
neighbor interactions. However for β = β_c the range expands “to infinity”
and the system is continuous in that limit. This occurs at a phase change.
We may then compare this to the hypothesis that spacetime is built up from
quantum entanglements. At the critical phase entanglements are entirely
nonlocal and the path integral, or in the Euclidean sense, is continuous.
The connection between the two is that ξ(β) = τ/ħ for τ = it a
Euclideanized time. At ξ(β) = 0 there is a quantum critical point and in
the setting of entanglements and spacetime what we think of as a continuous
spacetime is then defined.
It is better to consider the Reissnor-Nordstrom or Kerr-Newman black hole
with the outer and inner horizons
r_± = m ± sqrt{m^2 - a^2cos^2θ}.
The ring singularity occurs for r = 0 and θ = 0 or in Cartesian coordinates
a^2 = x^2 + y^2. The departure from spherical coordinates and Cartesian
coordinates is an oddity of spacetime being so twisted up in this region.
The outer ergosphere occurs at r = 2m, there is also an inner horizon that
occurs at r = a cosθ, This inner ergosphere is continuous with the ring
singularity at θ = 0. The region bounded by the inner ergosphere is where
timelike geodesics are forced into closed loops. These closed timelike
loops are then associated with a monodromy induced by a phase where
spacetime breaks down.
[image: Kerr-surfaces.png]
Dafermos and Luk found that within the inner horizon there is a breakdown
in uniqueness conditions for solution https://arxiv.org/abs/1710.01722 .
This is because within this region geodesics may be timelike, and within
the inner ergosphere they are constrained to be closed. I will confess to
have not as yet read their entire paper, as it is a long tome with rather
dense mathematics. However, the result appears at least commensurate with
the hypothesis that spacetime as understood in general relativity becomes
less defined.
Closed geodesics occur in anti-de Sitter spacetime as well. I have found a
homomorphism between black hole horizon states and states in AdS or
equivalently CFT states on the boundary. This interestingly is defined with
a form of the Riemann ζ-functions that give eigenvalues. The AdS_{n+1} has
in general topology S^1×R^n for S^1 timelike. Scott Aaronson have found
that closed timelike loops for quantum computers solve NP problems, and in
fact appear to cover all of P-SPACE https://arxiv.org/pdf/0808.2669.pdf .
The diagram illustrates this
[image: quantum computer with closed timelike curves.png]
There are then two registers of qubits R_{cr} that is causality respecting
and R_{ctc} for qubits on closed timelike paths. The closed timelike curves
in the path integral provide constructive and destructive interference of
the wave function that is NP. There is evidence the zeros of the Riemann
ζ-function is of a geometric complexity class that is NP
https://www.youtube.com/watch?v=Nn4B-9YspuI . The quantum eigenstates of
gravity are then “computed” by closed timelike paths in a path integral,
but where observers only have direct access to qubits in the R_{cr}.
LC
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