A class of non-equilibrium states and the black hole interior

TL;DR

The paper investigates a canonical class of non-equilibrium pure states $|\Psi\rangle = e^{-{\beta H \over 2}} U({\cal O}) e^{{\beta H \over 2}} |\Psi_0\rangle$ in holographic systems, proposing that they encode excitations behind the black hole horizon in AdS/CFT. By combining a small algebra ${\cal A}$, the Tomita–Takesaki construction, and a state-dependent interior operator framework, it shows these states appear equilibrium to simple probes but reveal non-equilibrium structure when the Hamiltonian is involved, signaling behind-horizon dynamics. The bulk interpretation is that the bulk geometry includes excitations localized behind the horizon, with the modular Hamiltonian $K$ and tilde-operators governing their boundary representation; this supports the possibility of interior reconstruction and connects to traversable-wormhole-like protocols in a one-sided context. The results illuminate how interior physics can be encoded in boundary data, while also outlining limitations and directions for extending these ideas to backreaction regimes and more general states.

Abstract

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.

Figures (5)

• Figure 1: a) Typical equilibrium state $|\Psi_{0}\rangle$ b) Non-equilibrium state of the form $U({\cal O})|\Psi_{0}\rangle$ with an excitation in free-fall c) Non-equilibrium state of the form $e^{-{\beta H \over 2}} U({\cal O}) e^{\beta H \over 2}|\Psi_{0}\rangle$ with a similar excitation. Notice that in these figures we show the conjectured Penrose diagram for a one-sided black hole and the dual CFT corresponds to the right asymptotic region. There is no dual CFT on the left side of the diagram and it is not clear until what distance the diagram can be continued into the left region.
• Figure 2: Left: a state of the form $e^{-{\beta H \over 2} } U({\cal O}) e^{{\beta H \over 2}} |\Psi_{0}\rangle$. Right: the same state seen at an earlier time.
• Figure 3: Left: a state of the form $U_R|0\rangle$, where $U_R$ is unitary localized in the right wedge. Right: the corresponding state $e^{-\pi M} U_R |0\rangle$, where $M$ is the Lorentz boost generator on the $t$-$x^1$ plane. The shaded regions show schematically the points where correlators differ from vacuum correlators. The state on the right represents an excitation which, from the point of view of an observer in $R$, remains behind the horizon for all time.
• Figure 4: The correlator \ref{['noneqcor']} keeping only the linear term in $\theta$ and setting $\theta=1, \beta=1, \Delta=2$. The correlator is time dependent around $t=t_0$ and decays exponentially for large $|t-t_0|$.
• Figure 5: Left: a non-equilibrium perturbation $e^{-{\beta H \over 2}}U({\cal O}) e^{{\beta H \over 2}}|\Psi_{0}\rangle$ of a typical state $|\Psi_{0}\rangle$. Right: a somewhat similar perturbation (purple particle) for a young black hole formed by the collapse of a shockwave emitted at $t=0$. The perturbation is created by dropping the particle before the shockwave.