Evaporating Black Holes Coupled to a Thermal Bath Shan-Ming Ruan - PowerPoint PPT Presentation

Evaporating Black Holes Coupled to a Thermal Bath Shan-Ming Ruan Perimeter Institute & University of Waterloo Nov 2020 Strings and Fields 2020 arXiv:1911.03402,2007.11658 with Vincent Chen, Zachary Fisher, Juan Hernandez, and Robert C. Myers 1

Outline ❖ Motivations & Background ❖ Doubly Holographic Models ❖ Quantum Extremal Surfaces and Islands ❖ Entanglement Wedge Reconstruction Shan-Ming Ruan (PI) BH coupled to a Thermal Bath 2

̂ 01.Background- Page Curve Evaporating Black Hole S Hawking’s Result S radiation = − tr( ̂ ρ ln ρ ) S BH = A ( t ) 4 G N Fine Grained entropy Coarse Grained entropy Page Curve (unitarity) t t Page Shan-Ming Ruan (PI) BH coupled to a Thermal Bath 3

01. Generalized Entropy and QES Need a correct formula for entropy of BH/Radiations See 1905.08762, Almheiri, Engelhardt, Marolf, Maxfield 1905.08255, Penington Shan-Ming Ruan (PI) BH coupled to a Thermal Bath 4

� � � � � � � � � � � � 01. Generalized Entropy and QES Generalized entropy Cauchy Slice S gen [ X ] = A [ X ] + S vN ( Σ X ) 4 G N X A c A Σ X Quantum extremal surfaces X Minimizing generalized entropy Minimal surface Quantum Extremal Surface Ext X ( + S vN ( Σ X ) ) A [ X ] S gen = Min X 4 G N Shan-Ming Ruan (PI) BH coupled to a Thermal Bath 5

� � � � � � � � � � � � 01. Generalized Entropy and QES see: 1908.10096 A correct formula for 2006.06872 the fine grained entropy of Hawking radiation Ext X ( + S vN ( Σ radiation ∪ Σ Island ) ) A [ X ] S radiation = Min X Island Formula ： 4 G N t Radiations Island Radiations QES= Island ∂ t < t Page t ≥ t Page Shan-Ming Ruan (PI) BH coupled to a Thermal Bath 6

02. Doubly Holographic Models How to calculate in the semi-classical limit? S gen Shan-Ming Ruan (PI) BH coupled to a Thermal Bath 7

02. Doubly Holographic Models Two-dimensional CFT and JT gravity Shan-Ming Ruan (PI) BH coupled to a Thermal Bath 8

02. AEMM Model ( y + ) f ′ ( y − ) dy + dy − ds 2 = − 4 f ′ JT gravity + CFT Matter AdS 2 2 ( f ( y + ) − f ( y − ) ) AdS 2 t ∞ ∞ x + x − Vacuum solution: < T ++ > = 0 , < T −− > = 0 1 tanh ( π T 0 u ) f ( u ) = u π T 0 2 x + x − t u = 0 1 − ( π T 0 ) ϕ = 2 ¯ ϕ r ± 1 x + − x − π T 0 ¯ { f ( u ), u } = π ¯ ϕ r ϕ r T 2 E 0 ≡ 0 8 π G N 4 G N eternal black hole Shan-Ming Ruan (PI) BH coupled to a Thermal Bath 9

02. AEMM Model ( y + ) f ′ ( y − ) dy + dy − ds 2 = − 4 f ′ JT gravity + CFT Matter 2 ( f ( y + ) − f ( y − ) ) AdS 2 eternal black hole t ∞ ∞ x + x − u t u = 0 Evaporating black hole ± 1 π T 0 Put the black hole in a fridge! Shan-Ming Ruan (PI) BH coupled to a Thermal Bath 10

02. AEMM Model Thermal Bath on half line JT gravity + CFT Matter No gravity Joint quench T b = 0 AEMM model (1905.08762, Almheiri, Engelhardt, Marolf, Maxfield) It is a holographic model 1D dual system: 2D Bath QM L QM R Shan-Ming Ruan (PI) BH coupled to a Thermal Bath 11

02. Doubly Holographic Model JT gravity + CFT Matter Thermal Bath x + t ∞ QES x − 1 x ± = ± Shock wave QES π T 0 u = 0 New horizon T b = 0 QM R QM L Gravitational Region Bath Region Shan-Ming Ruan (PI) Complexity on the brane 12

02. Doubly Holographic Model log ( u Page ≈ 2 T 1 − T 0 + u HP k 5 3 k ) 8 π T 1 1 + ( T 1 − T 0 ) π + ⋯ u HP = 3 T 1 k 3 6 π T 1 2 π T 1 Page Curve Unitary evolution of an evaporating black hole More details:1905.0876,1911.03402 Shan-Ming Ruan (PI) BH coupled to a Thermal Bath 13

AEM 4 Z 02. Model JT gravity + Holographic CFT Holographic CFT on half line model AEM 4 Z (1908.10996, Almheiri, Mahajan, Maldacena, Zhao) It is a doubly holographic model! 3D dual system: ( AdS 3 / BCFT 2 ) Takayanagi, 1105.5165 See Dominik’s talk higher-dim construction Shan-Ming Ruan (PI) BH coupled to a Thermal Bath 14

02. Doubly Holographic Models- T b put a black hole in an oven Couple a black hole with a thermal bath T b ≠ 0 arXiv:2007.11658 with V.Chen, Z.Fisher, J.Hernandez, and R.Myers Shan-Ming Ruan (PI) BH coupled to a Thermal Bath 15

02. Doubly Holographic Models- T b Preheat the oven first! —Prepare a thermal state on cylinder QES x + t ∞ 1 QES Y = tanh( π T b y ) Shock wave x − π T b Bifurcation surface Ⅱ Ⅰ y − = ∞ ∞ 1 x ± = ± = + π T 0 y y ± = ˜ u ± ˜ ˜ σ Ⅲ Ⅳ u = 0 Half line Half cylinder y + − y − = 0 New horizon QM R QM L Purifying Region Gravitational Region Bath Region Shan-Ming Ruan (PI) BH coupled to a Thermal Bath 16

02. Doubly Holographic Models- T b Three equivalent descriptions 1D 2D 3D Shan-Ming Ruan (PI) BH coupled to a Thermal Bath 17

03. QES and Islands A Solvable Model a = 2 π T 2 1 − T 2 b k I ν ( a ) K ν ( ae − ku /2 ) − K ν ( a ) I ν ( ae − ku /2 ) f ( u , T b ) = 2 ν = 2 π T b Solutions ν ( a ) K ν ( ae − ku /2 ) − K ′ ν ( a ) I ν ( ae − ku /2 ) k ka I ′ k ≡ cG N ≪ 1 3 ¯ ϕ r : Temperature of thermal bath : Temperature of BH after joint T b T 1 ¯ ϕ r π b + ( T 2 T 2 T e ff ( u ; T b ) = T 2 1 − T 2 b ) e − ku E ( u ) = e ff ( u ) 4 G N Evaporating Equilibrium Thermalized T b < T 1 T b = T 1 T b > T 1 Shan-Ming Ruan (PI) BH coupled to a Thermal Bath 18

03. QES and Islands QES with Islands ϕ ( x ± ) = ϕ r ( f ′ ( y − ) ) 2 f ′ ( y − ) ′ ( y − ) f ′ ϕ x + − x − + S gen ( x + , x − ) = + S vN , 4 G N ∂ x ± S gen = 0 S vN = S Ω − 2 g = S UHP − c 6 ∑ log Ω ( x i ) x i ∈∂ Late time/Island Phase Γ e ff 8 π T e ff x + 4 − Γ e ff ( t ∞ − t ) x − ( t ∞ − t ) ; QES ≈ t ∞ + QES ≈ t ∞ − k ( 4 − Γ e ff ) 2 Γ e ff ( y − QES ) ≡ T b 1 − T e ff ( y − QES ) Shan-Ming Ruan (PI) BH coupled to a Thermal Bath 19

02. Doubly Holographic Models- T b Quantum Extremal Surface (QES) QES x + t ∞ QES Shock wave x − Bifurcation surface Ⅱ Ⅰ y − = ∞ 1 y + = ∞ x ± = ± π T 0 y ± = ˜ u ± ˜ ˜ σ Ⅲ Ⅳ u = 0 y + − y − = 0 New horizon QM R QM L Gravitational Region Bath Region Purifying Region Shan-Ming Ruan (PI) BH coupled to a Thermal Bath 20

03. QES and Islands 4 G N ( 1 − e ff ( u ) ) k π T e ff ( u ) ¯ T 2 dS gen,late ϕ r b Island Phase: ≈ − T 2 du 4 G N 4 π T b ku S gen ¯ ϕ r Thermalized T b > T 1 2 π T b T b > T 1 Equilibrium T b = T 1 T b = T 1 2 π T 1 T b < T 1 Evaporating T b < T 1 2 π T b 2 π ( T b + T 1 ) ku arXiv: 2007.11658 u Page 1 2 3 4 5 ku Shan-Ming Ruan (PI) BH coupled to a Thermal Bath 21

03. QES and Islands Equilibrium Status T 1 = T b 2 − 1 ) + k ( ( π T 1 t ) 1 ( ( π T 1 t ) 2 + 1 ) k 2 + π 2 T 2 x + QES ( t ) = 1 t 2 + 2 kt − 1 ) 1 ( π 2 T 2 π 2 T 2 2 − 1 ) + k ( ( π T 1 t ) 1 ( ( π T 1 t ) 2 + 1 ) k 2 + π 2 T 2 x − QES ( t ) = 1 t 2 + 2 kt + 1 ) 1 ( − π 2 T 2 π 2 T 2 1 − k log [ ϵ ( k + 2 G N ( 1 )]) ¯ ϕ k 2 + π 2 T 2 k 2 + π 2 T 2 S gen,late ( T 1 ) = Constant dx + QES ( t ) Island outside horizon x + QES ( t ) < t ∞ = x + ; QES ( t ∞ ) > 0 dt Shan-Ming Ruan (PI) BH coupled to a Thermal Bath 22

03. QES and Islands Island outside horizon 2 k 2 k T e ff ( y − T e ff ( y − QES ) QES ) T c 1 ( u ) ≈ 1 − T c 2 ( u ) ≈ 1 + π T 1 π T 1 Inside horizon T c 1 ( u ) < T b < T c 2 ( u ) On the horizon T b = T c 1 ( u ) or T b = T c 2 ( u ) Outside horizon T b < T c 1 ( u ) or T b > T c 2 ( u ) At very late time, QES always moves outside horizon 1 − T 2 π T 1 1 ku ≳ log T 2 8 k b Shan-Ming Ruan (PI) BH coupled to a Thermal Bath 23

03. QES and Islands A correct formula for Radiation Island formula Shan-Ming Ruan (PI) BH coupled to a Thermal Bath 24

03. Island Formula ρ out 2D Bath (with radiation) Pure State: QM L QM R ρ in Ext X ( + S vN ( Σ out ) ) A [ X ] S gen [ ρ out ] = Min X 4 G N Equivalent Ext χ ( + S vN ( Σ in ∪ Σ Island ) ) A [ χ ] S gen [ ρ in ] = Min χ 4 G N Island formula Shan-Ming Ruan (PI) BH coupled to a Thermal Bath 25

04. Entanglement Wedge Reconstruction 2D Bath (with radiation) QM L QM R Purification Entropy of QM L + bath (Hawking radiation) +Purification Entanglement Wedge of QM L + bath (Hawking radiation) +Purification Reconstruct the interior of BH after Page transition (Island Phase) Shan-Ming Ruan (PI) BH coupled to a Thermal Bath 26

04. EWR- a finite bath interval Who knows information of the BH interior? Only Hawking radiations is not enough How much Hawking radiation we need to reconstruct BH interior? Not all Hawking radiations are necessary What is the role of purification in EWR? Purification part may be also necessary Shan-Ming Ruan (PI) BH coupled to a Thermal Bath 27

04. EWR- a finite bath interval subsystem: QM L + part of the bath + Purification [ σ 1 , σ 2 ] Competing Channels S R = S gen S N = S gen QES − 1 + S 2 − IR ′ + S 1 − 2 + S 1 2 − line QES ′ Island Phase Reconstruct the interior of BH ! S R ≤ S N Shan-Ming Ruan (PI) BH coupled to a Thermal Bath 28