Holographic entanglement and causal shadow in time-dependent Janus black hole

Journal of High Energy Physics, Jul 2015

We holographically compute an inter-boundary entanglement entropy in a time-dependent two-sided black hole which was constructed in [1] by applying time-dependent 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.

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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)


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Yūki Nakaguchi, Noriaki Ogawa, Tomonori Ugajin. Holographic entanglement and causal shadow in time-dependent Janus black hole, Journal of High Energy Physics, 2015, pp. 80, Volume 2015, Issue 7, DOI: 10.1007/JHEP07(2015)080