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研究生：	孫柏鈞 Po-Chun Sun
論文名稱：	Time Evolution of the Holographic Entanglement Entropy from Black Hole Thermalization
指導教授：	陳江梅 Chiang-Mei Chen
口試委員:
學位類別：	碩士 Master
系所名稱：	理學院 - 物理學系 Department of Physics
論文出版年：	2021
畢業學年度：	109
語文別：	英文
論文頁數：	54
中文關鍵詞：	全像理論、反德西特/共形場論對偶、霍金粒子、糾纏熵、量子極面、黑洞資訊悖論
外文關鍵詞：	Holography, AdS/CFT, Hawking Radiation, Entanglement Entropy, Quantum Extremal Island, Information Paradox
相關次數：	點閱：21 下載：0
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我們藉由全像原理，考慮Hartle–Hawking 狀態下，(n+1)維度的永恆黑洞在反德西特空間(AdS)偶合到具有保角對稱的熱庫(CFT reservoir)，來研究霍金輻射的糾纏熵在(n+1)維的Kerr-Newman黑洞蒸發的過程隨時間演化。試想保角熱庫對偶到(n+2)維重力空間，而原本我們所考慮的(n+1)維度的反德西特空間黑洞被嵌入在高一維的流形，則此情況完全可以用Randall–Sundrum模型來描述。

根據島規則，糾纏熵在半古典重力可以分為來自量子效應與重力的貢獻。其中量子效應可以用Ryu–Takayanagi公式來得到，而重力部分等於量子極面(quantum extremal surface)除以四倍的牛頓常數。我們展示了在此全像系統演化晚期，糾纏熵成長是線性的。過了佩吉時間(Page time)，量子極面出現並且系統達到飽和。在這篇論文中，我們將強調在任意維度的時空，黑洞旋轉如何如何引響其糾纏熵。

We study the time evolution of the entanglement entropy of Hawking radiation in the (n+1)-dimensional Kerr-Newman black hole evaporation by the holographic approach that considering the (n+1)-dimensional AdS eternal black brane coupled to the auxiliary CFT reservoir is in the Hartle-Hawking state. The CFT reservoir itself has a holographic dual, the (n+2)-dimensional bulk geometry, and the original (n+1)-dimensional AdS-black brane is embedded into such bulk manifold, which is precisely Randall–Sundrum model.

According to the island rule, the entanglement entropy in semi-classical gravity can be divided into two parts, one is due to the quantum effects, which can be obtained by Ryu–Takayanagi conjecture. Another is the gravitational part, which is equal to the area of the quantum extremal surface divided by four times the Newton's constant. We show that the entanglement growth in our holographic system is linear in late times. After Page time, the system reaches saturation since the entanglement islands appear. In this thesis, we will emphasize how black hole rotation affects entanglement entropy in general dimensional spacetime.

Chinese Abstract                                                              ix
English Abstract                                                              xi
Acknowledgements                                                              xiii
List of Figures                                                               xvii
1 Introduction                                                                1
1.1 Holographic Entanglement Entropy . . . . . . . . . . . . . . . . . . . . .1
1.2 Entanglement Entropy of Hawking Radiation . . . . . . . . . . . . . . . . 3
1.3 Cylindrical Kerr-Newman Black Brane . . . . . . . . . . . . . . . . . . . 4
2 Holography Setup                                                            7
2.1 Review AdS/BCFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Black Brane Thermalization . . . . . . . . . . . . . . . . . . . . . . . .8
2.3 Induced Gravity on Planck Brane . . . . . . . . . . . . . . . . . . . . . 9
2.3.1 AdS on the Brane . . . . . . . . . . . . . . . . . . . . . . . . . . . .9
2.3.2 Black Hole on the Brane . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Time Evolution of Wormhole Area                                             13
3.1 Dynamics of Wormhole . . . . . . . . . . . . . . . . . . . . . . . . . . .13
3.2 Late Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15
4 Saturation and Quantum Extremal Island                                      17
5 Conclusions                                                                 19
5.1 Entanglement Growth during Thermalization . . . . . . . . . . . . . . . . 19
5.2 Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19
5.3 Quantum Information during Evaporation . . . . . . . . . . . . . . . . .  20
Appendices                                                                    21
A HEE in Cylindrical Kerr-Newman Black Brane                                  23
A.1 Integration form of Area and Boundary . . . . . . . . . . . . . . . . . . 23
A.2 Entanglement Entropy in the Small R Limit . . . . . . . . . . . . . . . . 25
A.2.1 Relation Between φR and η . . . . . . . . . . . . . . . . . . . . . . 25
A.2.2 Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26
A.3 Entanglement Entropy in the Large R Limit . . . . . . . . . . . . . . . . 28
References                                                                    31
                                

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