Holographic entanglement and causal shadow in time-dependent Janus black hole
JHE
Holographic entanglement and causal shadow in
Yu¯ki Nakaguchi 0 1 3 4 6 7
Noriaki Ogawa 0 1 3 5 7
Tomonori Ugajin 0 1 2 3 7
0 RIKEN Nishina Center , Wako, Saitama 351-0198 , Japan
1 Bunkyo-ku, Tokyo 133-0022 , Japan
2 Kavli Institute for Theoretical Physics, University of California
3 Kashiwa , Chiba 277-8583 , Japan
4 Department of Physics, Faculty of Science, University of Tokyo
5 Quantum Hadron Physics Laboratory & Mathematical Physics Laboratory
6 Institute for the Physics and Mathematics of the Universe, University of Tokyo
7 Santa Barbara , CA 93106 , U.S.A
We holographically compute an inter-boundary entanglement entropy in a time-dependent two-sided black hole which was constructed in [1] by applying timedependent Janus deformation to BTZ black hole. The black hole contains “causal shadow region” which is causally disconnected from both the conformal boundaries. We find that the Janus deformation results in an earlier phase transition between the extremal surfaces and that the phase transition disappears when the causal shadow is sufficiently large.
Black Holes in String Theory; AdS-CFT Correspondence
1 Introduction 2 Properties of three-dimensional Janus Black Hole 2.1
The three-dimensional Janus metric
Time-dependent Janus deformation of BTZ metric
As a solution of Einstein-scalar theory
Main differences from BTZ black hole
Causal shadow region
The CFT interpretation of the Janus black hole
Calculation of holographic entanglement entropy
Covariant holographic entanglement entropy
Extremal areas in connected phase
How to calculate extremal areas in disconnected phase
Some limits of extremal surface areas in disconnected phase
Solving the equation of motion
Returning point (y∗, t∗)
Extremal surface area
Late time limit for large subsystem (t∞
r0−1)
Time evolution of entanglement entropy and phase transition
Introduction
The relation between entanglement and black hole interior has attracted much attention
recently [2–5]. For eternal AdS black holes, it was discussed that the time evolution of
holographic entanglement entropy [6–8] can capture some information about the black hole
interior, taking a particular time slicing with which the black hole looks time dependent.
For subsystems composed of two disjoint same intervals located in each of two CFT’s,
the original CFT and the thermofield doubled copy CFT [9], its holographic entanglement
entropy grows linearly in time for a while, in accordance with the growth of the wormhole
inside the black hole. At a certain critical time, the entropy becomes saturated at twice the
value of the black hole thermal entropy. In the dual CFT language, this time dependent
behavior of the entanglement entropy is interpreted in terms of global quench process [10].
For such Calabrese Cardy type of two dimensional quenches, a systematic construction of
their holographic duals was discussed in [11].
More general two-sided black holes can have even richer interior structures. For
example, similar inter-boundary entanglement entropies in charged or rotating black hole
geometries, which have vertically extended Penrose diagrams, were investigated in [12, 13].
In this paper, we focus on another interesting class of two-sided black holes with a so
called “causal shadow” region, which is a bulk region causally inaccessible from both the
boundaries. The implications of such a region for holographic entanglement entropy have
been discussed [14, 15]. For example, we can construct an asymptotically AdS black holes
with a causal shadow by sending shock waves from the boundaries of eternal AdS black
holes [16–18], and we can also discuss its dual CFT [19]. It is an interesting question how
the dual CFT encodes information on causal shadow regions.
To investigate this question further, we concentrate on another type of black hole with
a causal shadow called the three-dimensional time-dependent Janus black hole,1 which is a
one parameter deformation of the BTZ black hole and a solution of the Einstein-scalar
theory [1]. This black hole geometry has a nontrivial dilaton configuration, without which it
reduces to just the eternal BTZ black hole. From the viewpoint of the dual boundary theory,
this nontrivial dilaton configuration corresponds to the difference in the coupling constant
and so in Hamiltonian between the two CFT’s, the original CFT and the thermofield
doubled copy CFT [1]. Its corresponding CFT state was proposed [21] as a natural extension
of the usual eternal AdS black hole/thermofield double state correspondence [22, 23], and
this proposal was checked by computing a one point function both on the CFT side and
the gravity side [1, 21].
In this paper, we study the time evolution of an inter-boundary holographic
entanglement entropy in the Janus black hole geometry, expecting to capture some information on
its causal shadow. As in the BTZ black hole geometry, there are two extremal surfaces
for the subsystem we take, where the entanglement entropy (...truncated)