Black hole state dependence as a single parameter

Published for SISSA by Springer Received: October 21, 2019 Revised: February 28, 2020 Accepted: April 10, 2020 Published: April 30, 2020 Rik van Breukelen Theoretical Physics Department, CERN, CH-1211 Geneva 23, Switzerland Département de Physique Théorique, Université de Genève, 24 quai Ernest-Ansermet, CH-1214 Geneva 4, Switzerland E-mail: Abstract: It has previously been proposed that the black hole interior of typical state large black holes in AdS can be described using state-dependent operators. We investigate the possibility that the interior can be described by explicit time dependence, which reduces the state-dependence of the interior operators to a single parameter. We also propose to use the natural cone, obtained from Tomita-Takesaki theory, as a candidate construction for the interior operators. Keywords: AdS-CFT Correspondence, Black Holes in String Theory ArXiv ePrint: 1910.00036 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP04(2020)210 JHEP04(2020)210 Black hole state dependence as a single parameter Contents 1 2 Explicit time dependence 2.1 Eternal black hole 2.2 Tomita-Takesaki theory 2.3 Typical black hole 1 2 3 4 3 Avoiding the paradoxes 3.1 The “Na 6= 0” paradox 3.1.1 The paradox 3.1.2 The resolution 3.2 Lack of left inverse 3.2.1 The paradox 3.2.2 The resolution 3.3 Other paradoxes 6 7 7 7 8 8 9 10 4 The natural cone 4.1 Basic properties 4.1.1 Restricting to the small algebra 4.1.2 Going from any state to P 4.1.3 Some subtleties 4.2 Application of the fixed mirror operators 4.2.1 How to apply 4.2.2 Time-dependence of the mirror operators 4.3 Consistency checks 4.3.1 Superpositions 4.3.2 Perturbations 11 12 13 14 16 17 17 20 21 21 22 5 Conclusions 22 A Thermal correlators 23 B Overlap of states 25 C Volume of self-dual cones C.1 The orthant cone C.2 The circular cone 26 27 27 –i– JHEP04(2020)210 1 Introduction 1 Introduction 2 Explicit time dependence Explicit time dependence is needed to get a consistent description of the black hole interior. In the following section, we will show this for the eternal black hole and typical pure state black holes. We will also discuss the Tomita-Takesaki construction used to describe the interior operators. –1– JHEP04(2020)210 The quantum mechanical behavior of black holes is an ongoing topic started by Hawking [1]. The Firewall paradoxes of AMPS [2–4] have formalized the problems surrounding the quantum mechanics of black holes. Moreover, these problems persist for large black holes in AdS. The AdS/CFT correspondence [5], therefore, provides a powerful tool to study this topic. The naive conclusion of these paradoxes is that there can be no operators in the CFT describing the interior of the black hole, and that, therefore, the horizon of the black hole is not smooth. It is, however, possible to construct interior operators that depend on the state of the black hole [6–8], and thus have a smooth horizon. The geometry described by these operators contains part of the extended AdS-Schwarzschild geometry, including part of the region beyond the interior [9, 10]. In this region it is obvious that time moves opposite to the Killing isometry. The interior operators must, therefore, be explicitly time dependent. In this paper, we will examine whether the state-dependence of the interior operators can be captured in a simple form by time dependence. State dependence can directly be rewritten as time evolution in the case of the thermofield double state and its time-shifted cousins [11, 12]. This is possible, more generally, for any ergodic system, because most states will become equal to the other states under time evolution for an ergotic system. State dependence is, therefore, equal to waiting the appropriate amount of time. We explore whether something similar can happen for typical pure state large black holes in AdS, i.e. whether explicit time dependence is enough to avoid the firewall paradoxes. We also investigate a candidate construction for the interior operators. TomitaTakesaki theory was used as a motivation for the construction of the state-dependent interior operators. We continue on this path by using the natural cone, described by Tomita-Takesaki theory, which has the elegant property that the interior operators are the same for all states in the natural cone. We investigate whether the natural cone together with explicit time dependence is enough to describe the interior operators of most typical states. This paper is organized as follows: In section 2, we will discuss explicit time dependence in the case of the thermofield double state, the basics of Tomita-Takesaki theory, and the construction of the state-dependent interior operators called the mirror operators. Next, in section 3, we will examine the various firewall paradoxes and see how explicit time dependence can avoid them. Finally, in section 4, the natural cone is proposed as a candidate construction. (a) (b) 2.1 Eternal black hole The eternal black hole is proposed [13] to be dual to two CFTs in a specific entangled state, the thermofield double state |ΨTFD i = X e−βEi /2 √ |Ei iL |Ei iR , Z i (2.1) where we denote the two CFTs with (L) left and (R) right, and β is the inverse temperature of the black hole. The geometry corresponding to this sate is depicted in figure 1a. The geometry has a Killing isometry corresponding to HR − HL , which flows up on the right part of the geometry and down on the left part of the geometry. This causes the left operator that is entangled with a right operator to move down when we move the right operator up, as depicted in figure 1b. To avoid closed timelike curves, we need to impose that time evolution is generated by HR + HL , i.e. up in both the left and right side of the geometry. This has consequences for left-right correlators, hOL (t1 )OR (t2 )i = f (t1 + t2 ), (2.2) where f is some function of t1 + t2 . This is different from the one-sided two-point function, which is a function of t1 − t2 . Left and right operators commute, even though the left-right two-point function is non-zero. This is because the two CFTs are causally disconnected. However, by considering a double trace perturbation [14, 15], of the form δH = gOL OR , a message can be send from one CFT to the other. This provides evidence for the smoothness of the black hole horizon in this state, as such a probe crosses the horizon. There is a class of states [11] that have the same entanglement structure called the time-shifted thermofield double states, |ΨT i = X e−βEi /2 √ eiEi T |Ei iL |Ei iR . Z i –2– (2.3) JHEP04(2020)210 Figure 1. a) The Penrose Diagram of the eternal black hole. b) The blue dots correspond to operators which are entangled. An operator at a later time, green dot, is entangled with an earlier operator on the other side. These states can also be made traversable [12], i.e. send a message from one side to other. However, the pro (...truncated)