Probing typical black hole microstates

Published for SISSA by Springer Received: August 22, 2019 Accepted: December 19, 2019 Published: January 13, 2020 Probing typical black hole microstates a Institute for Theoretical Physics and Delta Institute for Theoretical Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands b Theoretical Physics Department, CERN, CH-1211 Geneva 23, Switzerland c Geneva University, 24 quai Ernest-Ansermet, CH-1214 Geneva 4, Switzerland d International Centre for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy E-mail: , , , , 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 perturbations 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. Keywords: AdS-CFT Correspondence, Black Holes in String Theory ArXiv ePrint: 1901.08527 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP01(2020)062 JHEP01(2020)062 Jan de Boer,a Rik van Breukelen,b,c Sagar F. Lokhande,a Kyriakos Papadodimasd and Erik Verlindea Contents 1 2 On the interior geometry of a typical state 2.1 Typical black hole microstates and the “mirror region” 2.2 The mirror operators 2.2.1 Time dependence of mirror operators 2.3 On the boundary of the left region 2.4 Comments on the Hamiltonian 2.5 Perturbations of typical states 2.5.1 Autonomous excited states 2.5.2 Perturbations of the Hamiltonian 4 4 6 10 12 13 14 14 16 3 Traversable one-sided black holes 3.1 Double-trace perturbation of the two-sided black hole 3.2 State-dependent perturbations in a single CFT 3.2.1 Energy change after the perturbation 3.2.2 Shockwaves in one-sided black hole 3.2.3 Gravitational Wilson lines and the backreacted geometry 3.3 Probing the region behind the horizon 3.3.1 Thought experiment 1 3.3.2 Thought experiment 2 17 18 20 21 23 24 25 26 28 4 The SYK model as an example 4.1 Brief review of the SYK model 4.1.1 Equilibrium and non-equilibrium states in the SYK model 4.2 Mirror operators in the SYK model 4.3 Comments on the Kourkoulou-Maldacena states 4.4 Information behind the horizon in SYK 4.5 Numerical comparison 29 29 30 31 34 36 39 5 A conjecture about quantum chaos in pure and thermal states 5.1 General comments on the conjecture 5.1.1 Replacing typical pure states by microcanonical mixed state 5.1.2 Comments on comparing canonical to microcanonical ensembles 5.2 Evidence for the conjecture 5.2.1 Slow change with respect to energy 5.2.2 Connection to ETH 5.2.3 Time-order vs. out-of-time-order correlators 5.2.4 SYK numerics 39 40 41 44 45 46 47 49 49 –i– JHEP01(2020)062 1 Introduction 51 51 52 7 Discussion 56 A The exterior geometry of typical black hole microstates 57 B Time-dependence and choice of T of the mirror operators 58 C Spherical shells on an Einstein-Rosen bridge 61 D Numerics in the SYK model 63 1 Introduction The black hole information paradox is a long-standing open problem, which is related to the smoothness of the black hole horizon [1, 2]. The AdS/CFT correspondence provides an ideal setting to investigate the issue of smoothness. Large typical black holes in AdS are expected to be dual to typical high-energy pure states in the dual CFT. These typical black holes are approximately in equilibrium and hence do not evaporate. Even then, it is challenging to reconcile the smoothness of the horizon with unitarity of the dual CFT [3–5]. In this paper, we make some inroads into investigating the geometry of such a typical black hole microstate. Owing to robust arguments in the AdS/CFT framework, it is widely believed that at large N the geometry of a typical black hole microstate contains at least the region exterior to the black hole horizon, which is described by the AdS-Schwarzchild metric. The question then is: do there exist any other regions in the geometry dual to a typical black hole microstate? It seems reasonable that any proposed answer to this question needs to satisfy two constraints: (1) the geometry in the exterior should be that of the AdS-Schwarzschild black hole, (2) the geometry should manifest the approximate timetranslation-invariance of the typical pure state in the CFT, through the existence of an approximate timelike Killing isometry. We discuss the time-translation-invariance of typical pure states in section 2.1. In [2–4], it was suggested that the geometry of a typical black hole microstate contains only the exterior region, which gets terminated at the horizon by a firewall. However, for large typical black holes, the curvature near the horizon is low. Thus, this proposed solution demands a dramatic modification of general relativity and effective field theory in regions of low curvature. In this paper, we will explore the possibility that the bulk geometry of a typical pure AdS black hole microstate contains part of the extended AdS-Schwarzchild diagram, as shown in figure 1. Since the dual of this geometry is a typical pure state in a single CFT, the Penrose diagram cannot be extended arbitrarily to the left and there is no “left” –1– JHEP01(2020)062 6 Remarks on the mirror operators and the Hayden-Preskill protocol 6.1 The Hayden-Preskill protocol 6.2 Information recovery using the mirror operators Figure 1. A proposal for the geometry dual to a typical black hole microstate. Naively, the left region would be inaccessible from the CFT, at the level of effective field theory. However, starting with the work of Gao, Jafferis and Wall [6] and further work [7, 8], a new approach has been identified for probing the space-time beyond the horizon, including the left region. This new approach, which was formulated in the framework of the two-sided eternal black hole, is based on the observation of [6] that in the case of the two-sided eternal black hole there are perturbations of the boundary CFTs of the form δH = OL OR , which can create negative energy shock (...truncated)