Abstract:We expand our recent work on the outer entropy, a holographic coarsegrained entropy defined by maximizing the boundary entropy while fixing the classical bulk data outside some surface. When the surface is marginally trapped and satisfies certain "minimar" conditions, we prove that the outer entropy is exactly equal to a quarter the area (while for other classes of surfaces, the area gives an upper or lower bound). We explicitly construct the entropy-maximizing interior of a minimar surface, and show that it s… Show more
“…In this section we review a classical geometric construction by Engelhardt and Wall (EW) [5,13]. In Sec.…”
Section: Classical Coarse-graining Of Black Hole Statesmentioning
confidence: 99%
“…We begin by fixing some notations and conventions; see Sec. 2 of [13] for details. Let σ be a Cauchy splitting surface, that is, σ is an achronal codimension two compact surface that divides a Cauchy surface Σ into two sides, Σ in and Σ out .…”
Section: A Classical Marginal Minimar and Stationary Surfacesmentioning
Behind certain marginally trapped surfaces one can construct a geometry containing an extremal surface of equal, but not larger area. This construction underlies the Engelhardt-Wall proposal for explaining Bekenstein-Hawking entropy as a coarse-grained entropy. The construction can be proven to exist classically but fails if the Null Energy Condition is violated.Here we extend the coarse-graining construction to semiclassical gravity. Its validity is conjectural, but we are able to extract an interesting nongravitational limit. Our proposal implies Wall's ant conjecture on the minimum energy of a completion of a quantum field theory state on a halfspace. It further constrains the properties of the minimum energy state; for example, the minimum completion energy must be localized as a shock at the cut. We verify that the predicted properties hold in a recent explicit construction of Ceyhan and Faulkner, which proves our conjecture in the nongravitational limit. * bousso@lbl.gov †
“…In this section we review a classical geometric construction by Engelhardt and Wall (EW) [5,13]. In Sec.…”
Section: Classical Coarse-graining Of Black Hole Statesmentioning
confidence: 99%
“…We begin by fixing some notations and conventions; see Sec. 2 of [13] for details. Let σ be a Cauchy splitting surface, that is, σ is an achronal codimension two compact surface that divides a Cauchy surface Σ into two sides, Σ in and Σ out .…”
Section: A Classical Marginal Minimar and Stationary Surfacesmentioning
Behind certain marginally trapped surfaces one can construct a geometry containing an extremal surface of equal, but not larger area. This construction underlies the Engelhardt-Wall proposal for explaining Bekenstein-Hawking entropy as a coarse-grained entropy. The construction can be proven to exist classically but fails if the Null Energy Condition is violated.Here we extend the coarse-graining construction to semiclassical gravity. Its validity is conjectural, but we are able to extract an interesting nongravitational limit. Our proposal implies Wall's ant conjecture on the minimum energy of a completion of a quantum field theory state on a halfspace. It further constrains the properties of the minimum energy state; for example, the minimum completion energy must be localized as a shock at the cut. We verify that the predicted properties hold in a recent explicit construction of Ceyhan and Faulkner, which proves our conjecture in the nongravitational limit. * bousso@lbl.gov †
“…We considered states |ψ 1 ; A , |ψ 2 ; A of fixed bulk HRT-area A that are also CPT-conjugate to each other in R, and argued that the bulk geometry dual to the sewn state |ψ 1 # R ψ 2 ; A can be obtained from the geometries g 1 (A), g 2 (A) dual to the original states |ψ 1 ; A , |ψ 2 ; A by extracting from g 1 (A), g 2 (A) the entanglement wedges of the regionsR 1 ,R 2 complementary to R and gluing these wedges together to define g(A) = g 1 (A)# R g 2 (A) as in the last line of figure 1. The work above assumed the bulk to be described by Einstein-Hilbert gravity, but using results from the forthcoming work [42] and assuming extensions of the matching conditions in [43] to the higher derivative context, analogous conclusions will continue to hold with arbitrary perturbative higher-derivative corrections.…”
Section: Discussionmentioning
confidence: 89%
“…Let us now discuss two further generalizations. First, one may note that the results of [43] allow two entanglement wedges with appropriately-compatible data on the HRTsurfaces to be directly sewn together without first embedding each in a larger geometry, and certainly without requiring the complementary wedges in that geometry to be CPTconjugate. It is thus natural to ask if there is a good CFT dual to this more general bulk gluing.…”
The CPT map allows two states of a quantum field theory to be sewn together over CPT-conjugate partial Cauchy surfaces R 1 , R 2 to make a state on a new spacetime. We study the holographic dual of this operation in the case where the original states are CPT-conjugate within R 1 , R 2 to leading order in the bulk Newton constant G, and where the bulk duals are dominated by classical bulk geometries g 1 , g 2 . For states of fixed area on the R 1 , R 2 HRT-surfaces, we argue that the bulk geometry g 1 #g 2 dual to the newly sewn state is given by deleting the entanglement wedges of R 1 , R 2 from g 1 , g 2 , gluing the remaining complementary entanglement wedges ofR 1 ,R 2 together across the HRT surface, and solving the equations of motion to the past and future. The argument uses the bulk path integral and assumes it to be dominated by a certain natural saddle. For states where the HRT area is not fixed, the same bulk cut-and-paste is dual to a modified sewing that produces a generalization of the canonical purification state √ ρ discussed recently by Dutta and Faulkner. Either form of the construction can be used to build CFT states dual to bulk geometries associated with multipartite reflected entropy. arXiv:1909.09330v1 [hep-th]
“…We can also define an entropy associated with a more general subset of observables that does not necessarily form a subalgebra. 9 Entropies of this type have been discussed previously in the context of holography in [39][40][41]. Given a set of (not necessarily commuting) operators A = {O α } and a global state ρ, we can look for a state ρ A that maximizes the von Neumann entropy subject to the constraint that…”
Section: Entropy Associated With a Subset Of Observablesmentioning
The BFSS matrix model provides an example of gauge-theory / gravity duality where the gauge theory is a model of ordinary quantum mechanics with no spatial subsystems. If there exists a general connection between areas and entropies in this model similar to the Ryu-Takayanagi formula, the entropies must be more general than the usual subsystem entanglement entropies. In this note, we first investigate the extremal surfaces in the geometries dual to the BFSS model at zero and finite temperature. We describe a method to associate regulated areas to these surfaces and calculate the areas explicitly for a family of surfaces preserving SO(8) symmetry, both at zero and finite temperature. We then discuss possible entropic quantities in the matrix model that could be dual to these regulated areas.
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