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On the interior geometry of a typical black hole microstate

Jan De Boer A, Rik van Breukelen, Sagar F. Lokhande, Kyriakos Papadodimas, Erik Verlinde

Published Jan 1, 2019• 22 citations
OpenAlex

Abstract

We argue that the region behind the horizon of a one-sided black hole can be probed by an analogue of the double-trace deformation protocol of Gao-Jafferis-Wall. This is achieved via a deformation of the CFT Hamiltonian by a term of the form O O ˜ $$ \\mathcal{O}\\tilde{\\mathcal{O}} $$ , where O ˜ $$ \\tilde{\\mathcal{O}} $$ denote the state-dependent “mirror operators”. We argue that this deformation creates negative energy shockwaves in the bulk, which allow particles inside the horizon to escape and to get directly detected in the CFT. This provides evidence for the smoothness of the horizon of black holes dual to typical states. We argue that the mirror operators allow us to perform an analogue of the Hayden-Preskill decoding protocol. Our claims rely on a technical conjecture about the chaotic behavior of out-of-time-order correlators on typical pure states at scrambling time.

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Abstract We argue that the region behind the horizon of a one-sided black hole can be probed by an analogue of the double-trace deformation protocol of Gao-Jafferis-Wall. This is achieved via a deformation of the CFT Hamiltonian by a term of the form $$ \mathcal{O}\tilde{\mathcal{O}} $$ O O ˜ , where $$ \tilde{\mathcal{O}} $$ O ˜ denote the state-dependent “mirror operators”. We argue that this deformation creates negative energy shockwaves in the bulk, which allow particles inside the horizon to escape and to get directly detected in the CFT. This provides evidence for the smoothness of the horizon of black holes dual to typical states. We argue that the mirror operators allow us to perform an analogue of the Hayden-Preskill decoding protocol. Our claims rely on a technical conjecture about the chaotic behavior of out-of-time-order correlators on typical pure states at scrambling time.

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We consider a class of non-equilibrium pure states, which are generally present in an isolated quantum statistical system. These are states of the form $|Ψ\rangle=e^{-{βH \over 2}} U e^{βH \over 2} |Ψ_0\rangle$, where $U$ is a unitary made out of simple operators and $|Ψ_0\rangle$ is a typical equilibrium pure state with sharply peaked energy. We argue that in a system with a holographic dual these states have a natural interpretation as an AdS black hole with transient excitations behind the horizon. We explore the interpretation of these states as pure states undergoing a time-dependent spontaneous fluctuation out of equilibrium. While these states are atypical and the microscopic phases of the wavefunction are correlated with the matrix elements of simple operators, the states are partly disguised as equilibrium states due to cancellations between contributions from different coarse-grained energy bins. These cancellations are guaranteed by the KMS condition of the underlying equilibrium state $|Ψ_0\rangle$. However, in correlators which include the Hamiltonian $H$ these cancellations are spoiled and the non-equilibrium nature of the state $|Ψ\rangle$ can be detected. We discuss connections with the proposal that local observables behind the horizon are realized as state-dependent operators. The states studied in this paper may be useful for implementing an analogue of the "traversable wormhole" protocol for a 1-sided black hole, which could potentially allow us to extract the excitation from behind the horizon. We include some pedagogical background material.

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In arXiv:2312.14108, we argued that a sliver at the black hole mass in the Hilbert space of a quantum field theory on an AdS black hole with a stretched horizon, has many desirable features of black hole microstates. A key observation is that the stretched horizon requires a finite Planck length, i.e., finite-$N$. Therefore it is best viewed as a ``quantum horizon" -- a proxy for the UV-complete bulk description, and not directly an element in the bulk EFT. It was shown in arXiv:2312.14108 that despite the manifest absence of the interior, the 2-point function in the sliver is indistinguishable from the smooth horizon correlator, up to the Page time. In this paper, instead of boundary correlators, we work directly in the bulk, and demonstrate the appearance of the bulk Hartle-Hawking correlator in the large-$N$ limit. This is instructive because it shows that the analytic continuation across the horizon is emergent. It is a $bulk$ transition to a Type III algebra and provides a structural distinction between black holes and weakly coupled thermal systems. We also identify a mechanism for universal code subspaces and interior tensor factors to appear via a quantum horizon version of thermal factorization. Our claims apply directly only within a ``Page window", so they are not in immediate tension with firewall arguments. During a Page window, typical heavy microstates probed by light single trace operators respond with effectively smooth horizons in low-point correlators. We work in 2+1 dimensions to be concrete, but expect our results to hold in all higher dimensions. We discuss some important differences between our approach and the conventional fuzzball program, and also argue that probe-notions (like infalling boundary conditions) must be distinguished from microstate-notions (like size of the Einstein-Rosen bridge) to make meaningful statements about post-Page smoothness.

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Holographic Quantum Error Correction and the Projected Black Hole Interior

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The ins and outs of black holes in AdS

Rik van Breukelen

This work starts by generalizing the traversable wormhole protocol from the thermofield double state black hole to time-shifted thermofield double state black holes and typical pure state black holes. The protocol is dependent on the time-shift and on the state respectively. The traversability of these states is evidence of the smoothness of the black hole horizons of these states. Inspired by these results, we show that the state-dependent construction of the interior of typical state black holes can be simplified using explicit time dependence. This reduces the state dependence to a single parameter.

•Jan 1, 2020•0 citations
We argue that the region behind the horizon of a one-sided black hole can be probed by an analogue of the double-trace deformation protocol of Gao-Jafferis-Wall. This is achieved via a deformation of the CFT Hamiltonian by a term of the form O O ˜ mathcalOtildemathcalO\\mathcal{O}\\tilde{\\mathcal{O}}mathcalOtildemathcalO , where O ˜ tildemathcalO\\tilde{\\mathcal{O}}tildemathcalO denote the state-dependent “mirror operators”. We argue that this deformation creates negative energy shockwaves in the bulk, which allow particles inside the horizon to escape and to get directly detected in the CFT. This provides evidence for the smoothness of the horizon of black holes dual to typical states. We argue that the mirror operators allow us to perform an analogue of the Hayden-Preskill decoding protocol. Our claims rely on a technical conjecture about the chaotic behavior of out-of-time-order correlators on typical pure states at scrambling time.
Abstract We argue that the region behind the horizon of a one-sided black hole can be probed by an analogue of the double-trace deformation protocol of Gao-Jafferis-Wall. This is achieved via a deformation of the CFT Hamiltonian by a term of the form OO~ \mathcal{O}\tilde{\mathcal{O}} OO~ O O ˜ , where O~ \tilde{\mathcal{O}} O~ O ˜ denote the state-dependent “mirror operators”. We argue that this deformation creates negative energy shockwaves in the bulk, which allow particles inside the horizon to escape and to get directly detected in the CFT. This provides evidence for the smoothness of the horizon of black holes dual to typical states. We argue that the mirror operators allow us to perform an analogue of the Hayden-Preskill decoding protocol. Our claims rely on a technical conjecture about the chaotic behavior of out-of-time-order correlators on typical pure states at scrambling time.
Abstract We investigate the possibility that the geometry dual to a typical AdS black hole microstate corresponds to the extended AdS-Schwarzschild geometry, including a region spacelike to the exterior. We argue that this region can be described by the mirror operators, a set of state-dependent operators in the dual CFT. We probe the geometry of a typical state by considering state-dependent deformations of the CFT Hamiltonian, which have an interpretation as a one-sided analogue of the Gao-Jafferis-Wall traversable wormhole protocol for typical states. We argue that the validity of the conjectured bulk geometry requires that out-of-time-order correlators of simple CFT operators on typical pure states must exhibit the same chaotic effects as thermal correlators at scrambling time. This condition is related to the question of whether the product of operators separated by scrambling time obey the Eigenstate Thermalization Hypothesis. We investigate some of these statements in the SYK model and discuss similarities with state-dependent perturba- tions of pure states in the SYK model previously considered by Kourkoulou and Maldacena. Finally, we discuss how the mirror operators can be used to implement an analogue of the Hayden-Preskill protocol.
We consider a class of non-equilibrium pure states, which are generally present in an isolated quantum statistical system. These are states of the form ∣Ψ⟩=e−βH2UeβH2∣Ψ0⟩|Ψ\rangle=e^{-{βH \over 2}} U e^{βH \over 2} |Ψ_0\rangle∣Ψ⟩=e−2βH​Ue2βH​∣Ψ0​⟩, where UUU is a unitary made out of simple operators and ∣Ψ0⟩|Ψ_0\rangle∣Ψ0​⟩ is a typical equilibrium pure state with sharply peaked energy. We argue that in a system with a holographic dual these states have a natural interpretation as an AdS black hole with transient excitations behind the horizon. We explore the interpretation of these states as pure states undergoing a time-dependent spontaneous fluctuation out of equilibrium. While these states are atypical and the microscopic phases of the wavefunction are correlated with the matrix elements of simple operators, the states are partly disguised as equilibrium states due to cancellations between contributions from different coarse-grained energy bins. These cancellations are guaranteed by the KMS condition of the underlying equilibrium state ∣Ψ0⟩|Ψ_0\rangle∣Ψ0​⟩. However, in correlators which include the Hamiltonian HHH these cancellations are spoiled and the non-equilibrium nature of the state ∣Ψ⟩|Ψ\rangle∣Ψ⟩ can be detected. We discuss connections with the proposal that local observables behind the horizon are realized as state-dependent operators. The states studied in this paper may be useful for implementing an analogue of the "traversable wormhole" protocol for a 1-sided black hole, which could potentially allow us to extract the excitation from behind the horizon. We include some pedagogical background material.
In arXiv:2312.14108, we argued that a sliver at the black hole mass in the Hilbert space of a quantum field theory on an AdS black hole with a stretched horizon, has many desirable features of black hole microstates. A key observation is that the stretched horizon requires a finite Planck length, i.e., finite-NNN. Therefore it is best viewed as a ``quantum horizon" -- a proxy for the UV-complete bulk description, and not directly an element in the bulk EFT. It was shown in arXiv:2312.14108 that despite the manifest absence of the interior, the 2-point function in the sliver is indistinguishable from the smooth horizon correlator, up to the Page time. In this paper, instead of boundary correlators, we work directly in the bulk, and demonstrate the appearance of the bulk Hartle-Hawking correlator in the large-NNN limit. This is instructive because it shows that the analytic continuation across the horizon is emergent. It is a bulkbulkbulk transition to a Type III algebra and provides a structural distinction between black holes and weakly coupled thermal systems. We also identify a mechanism for universal code subspaces and interior tensor factors to appear via a quantum horizon version of thermal factorization. Our claims apply directly only within a ``Page window", so they are not in immediate tension with firewall arguments. During a Page window, typical heavy microstates probed by light single trace operators respond with effectively smooth horizons in low-point correlators. We work in 2+1 dimensions to be concrete, but expect our results to hold in all higher dimensions. We discuss some important differences between our approach and the conventional fuzzball program, and also argue that probe-notions (like infalling boundary conditions) must be distinguished from microstate-notions (like size of the Einstein-Rosen bridge) to make meaningful statements about post-Page smoothness.
The quantum error correction interpretation of AdS/CFT establishes a sense of fluidity to the bulk/boundary dictionary. We show how this property can be utilized to construct a dictionary for operators behind horizons of pure black holes. We demonstrate this within the context of the SYK model with pure black hole microstates obtained via projecting out a single side of the thermofield double (and perturbed versions thereof). Assuming an erasure subsystem code for the duality between the eternal black hole and the thermofield double, this projection results in a rewiring of the dictionary so as to map the interior operators to the remaining boundary in a determinable way. We find this dictionary to be sensitive to the implemented projection in a manner reminiscent of previous state-dependent constructions of the black hole interior. We also comment on how the fluidity of the dictionary can be used to transfer information between two black holes connected by a wormhole, relating the ideas of entanglement wedge reconstruction and the Hayden-Preskill decoding criterion.
This work starts by generalizing the traversable wormhole protocol from the thermofield double state black hole to time-shifted thermofield double state black holes and typical pure state black holes. The protocol is dependent on the time-shift and on the state respectively. The traversability of these states is evidence of the smoothness of the black hole horizons of these states. Inspired by these results, we show that the state-dependent construction of the interior of typical state black holes can be simplified using explicit time dependence. This reduces the state dependence to a single parameter.