Ⅱ-6. Particle Physics

RIKEN Accel. Prog. Rep. 48 (2015)

Holographic entanglement and causal shadow
in the time-dependent Janus black hole†
Y. Nakaguchi,∗1,∗2 N. Ogawa,∗3 and T. Ugajin∗4The relation between entanglement and the blackhole interior has been attracting attention recently.
For eternal AdS black holes, some interior information
is captured by the entanglement entropy of an intervalon the dual theory, i.e. a thermofield doubled CFT1) .
In some kinds of black holes, the interior appears
even more difficult to access since they have “causalshadow” regions inside. In this work, we consider
one such black hole, called the three-dimensional timedependent Janus black hole2) ,r0222+dθ),
ds2 = L2 dy 2 +(−dtg̃(y)2 cosh2 r0 t

2,
(1)
g̃(y) =1 + 1 − 2γ 2 cosh 2ywhere γ is a deformation parameter from the BTZ
black hole. This is a solution of the Einstein-scalar
theory and it corresponds to a pair of two entangled
2D CFTs, whose coupling constants are different fromeach other.
This geometry has a horizontally extended Penrose
diagram as shown in Fig. 1. The remarkable property of this geometry is the existence of a so-called“causal shadow region,” which is causally inaccessible
from both boundaries.

glement entropy in this system in a holographic manner, by using the Ryu-Takayanagi formula. As in theBTZ black hole, there are two candidate extremal surfaces in the bulk, whose area gives the entanglement
entropy. One of the extremal surfaces, which we callthe connected surface, connects two asymptotic boundaries and probes the black hole interior. The other extremal surface, which we call the disconnected surface,localizes near each of the asymptotic boundaries.
In the Janus black hole, when the deformation fromthe BTZ black hole is not so large, we found a similar behavior but with the critical time tc shorter thanthat of the BTZ black hole. Roughly speaking, this isbecause the deformation makes the wormhole region
longer and results in a longer connected surface. The
results for the time evolution of the entanglement entropy of each phase are plotted in Fig. 2.
In addition, we also found that with a sufficiently
large deformation, the disconnected surface is always
dominant, and that the holographic entanglement entropy is already proportional to the size of the subsystem from the initial time. This means that the entanglement entropy of this subsystem does not probe theblack hole interior.

At,Θ 5
11.0
10.5

t
+∞

10.0

EH

tu+∞

fu−∞

Γ 2  0.3
Adc Numerical
Early approx. tΘ
Late approx. Θt

9.5

reAH

9.0

ypaOΓ 2 Ac

st8.5

EH
8.0

−∞
Fig. 1. Penrose diagram of the 3D time-dependent Janus
black hole. The causal shadow is painted yellow. The
apparent horizons (green line) in the time slices τ =const. are located inside the future event horizon.

We studied the time-evolution of the same entan†

∗1
∗2
∗3
∗40

2

4

6

8tFig. 2. The time t dependence of the extremal surface area
A for a subsystem θ = 5, in the disconnected phase(black dotted line, numerically obtained) and in theconnected phase (gray line).

References

Condensed from the article arXiv:1412.8600 [hep-th].
Institute for the Physics and Mathematics of the Universe,University of Tokyo
Department of Physics, University of Tokyo
RIKEN Nishina Center
Kavli Institute for Theoretical Physics, University of California

1) T. Hartman and J. Maldacena, JHEP 1305 (2013)
014, [arXiv:1303.1080].
2) D. Bak, M. Gutperle, and S. Hirano, JHEP 0702
(2007) 068, [hep-th/0701108].
3) D. Bak, M. Gutperle, and A. Karch, JHEP 0712
(2007) 034, [arXiv:0708.3691].

- 153 -完全版2014_本文.indd 153

15/10/16 17:50