Islands in Schwarzschild black holes
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Springer
Received: April 21, 2020 Accepted: May 19, 2020 Published: June 12, 2020
Koji Hashimoto, Norihiro Iizuka and Yoshinori Matsuo Department of Physics, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka 560-0043, Japan
E-mail: [email protected], [email protected], [email protected] Abstract: We study the Page curve for asymptotically flat eternal Schwarzschild black holes in four and higher spacetime dimensions. Before the Page time, the entanglement entropy grows linearly in time. After the Page time, the entanglement entropy of a given region outside the black hole is largely modified by the emergence of an island, which extends to the outer vicinity of the event horizon. As a result, it remains a constant value which reproduces the Bekenstein-Hawking entropy, consistent with the finiteness of the von Neumann entropy for an eternal black hole. Keywords: Black Holes, Black Holes in String Theory ArXiv ePrint: 2004.05863
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP06(2020)085
JHEP06(2020)085
Islands in Schwarzschild black holes
Contents 1
2 No island, no entropy bound
6
3 Island saves the entropy bound 3.1 Close look at the black hole 3.2 View from a distance
7 7 8
4 Higher dimensions
10
5 Page time and scrambling time
12
A Early time growth of the entropy
15
B Geodesic distance and extremal volume
16
1
Introduction and our strategy
The information paradox [1] is the most fundamental problem in quantum gravity. The Hawking radiation behaves as thermal radiation [2], which implies that the entanglement entropy outside the black hole is monotonically increasing. On the other hand, quantum mechanics requires that the entanglement entropy goes to zero at the end of the evaporation since it must be the pure state. The time evolution of the entanglement entropy is described by the so-called Page curve [3, 4]. Thus, the information loss paradox is translated to the problem how the Page curve is reproduced for the entanglement entropy of the Hawking radiation. Recently it was proposed that the Page curve emerges from the effect of islands [5–8]. Regarding the state of the Hawking radiation as that in a region R outside the black hole, the density matrix of R is normally defined by taking the partial trace over the states in R, which is the complementary region of R. According to the prescription of the minimal quantum extremal surface [9–11], states in some regions in R, which are called islands I(⊂ R), should be excluded from the states to be traced out. Thus, the entanglement entropy of the Hawking radiation R is effectively given by that of states in R∪I. Explicitly, the entanglement entropy of the Hawking radiation is give by Area(∂I) S(R) = min ext + Smatter (R ∪ I) , (1.1) 4GN by using the prescription of the quantum extremal surface.
–1–
JHEP06(2020)085
1 Introduction and our strategy
I = Igravity + Imatter , Z Z √ √ 1 1 Igravity = d4 x −g R + d3 x −h K , 16πGN M 8πGN ∂M
(1.2) (1.3)
where GN is
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