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.
Nov 2020 Strings and Fields 2020
arXiv:1911.03402,2007.11658 with Vincent Chen, Zachary Fisher, Juan Hernandez, and Robert C. Myers
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BH coupled to a Thermal Bath Shan-Ming Ruan (PI)
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BH coupled to a Thermal Bath Shan-Ming Ruan (PI)
Page Curve (unitarity)
Fine Grained entropy Coarse Grained entropy
Sradiation = − tr( ̂ ρ ln ̂ ρ)
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SBH = A(t) 4GN
Hawking’s Result
BH coupled to a Thermal Bath Shan-Ming Ruan (PI)
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See 1905.08762, Almheiri, Engelhardt, Marolf, Maxfield 1905.08255, Penington
BH coupled to a Thermal Bath Shan-Ming Ruan (PI)
Quantum extremal surfaces X Minimizing generalized entropy
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X Quantum Extremal Surface
BH coupled to a Thermal Bath Shan-Ming Ruan (PI)
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see: 1908.10096 2006.06872
t Island
Radiations
QES= Island
∂
BH coupled to a Thermal Bath Shan-Ming Ruan (PI)
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BH coupled to a Thermal Bath Shan-Ming Ruan (PI)
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BH coupled to a Thermal Bath Shan-Ming Ruan (PI)
x− x+
± 1 πT0
t∞
AdS2
t u
u = 0
∞
f(u) = 1 πT0 tanh (πT0u) eternal black hole
ds2 = − 4f′ (y+)f′ (y−)dy+dy− (f(y+) − f(y−))
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Vacuum solution: ϕ = 2 ¯ ϕr 1 − (πT0)
2 x+x−
x+ − x− < T++ > = 0 , < T−− > = 0 E0 ≡ ¯ ϕr 8πGN {f(u), u} = π ¯ ϕr 4GN T2
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AdS2
BH coupled to a Thermal Bath Shan-Ming Ruan (PI)
ds2 = − 4f′ (y+)f′ (y−)dy+dy− (f(y+) − f(y−))
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Put the black hole in a fridge!
x− x+
± 1 πT0
t∞
AdS2
t u
u = 0
∞
eternal black hole Evaporating black hole
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BH coupled to a Thermal Bath Shan-Ming Ruan (PI)
Joint quench
2D Bath QMR QML
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1D dual system:
No gravity
Complexity on the brane Shan-Ming Ruan (PI)
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t∞
x± = ± 1 πT0
u = 0
x+ x−
QES New horizon
Shock wave QES
Gravitational Region Bath Region
QML QMR
Tb = 0
BH coupled to a Thermal Bath Shan-Ming Ruan (PI)
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uPage ≈ 2 3 T1 − T0 T1k + uHP 3 + k 6πT1 5 (T1 − T0)π + ⋯ uHP = 1 2πT1 log ( 8πT1 3k ) More details:1905.0876,1911.03402
BH coupled to a Thermal Bath Shan-Ming Ruan (PI)
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3D dual system: ( ) AdS3/BCFT2
See Dominik’s talk higher-dim construction
Takayanagi, 1105.5165
BH coupled to a Thermal Bath Shan-Ming Ruan (PI)
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arXiv:2007.11658 with V.Chen, Z.Fisher, J.Hernandez, and R.Myers
BH coupled to a Thermal Bath Shan-Ming Ruan (PI)
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Half cylinder Half line
˜ y± = ˜ u ± ˜ σ t∞
x± = ± 1 πT0
Ⅰ Ⅱ Ⅲ Ⅳ
y− = ∞
y+ − y− = 0
y
+
= ∞ u = 0
x+ x−
QES New horizon
Shock wave
QES
Gravitational Region Bath Region Purifying Region
Bifurcation surface
QML QMR
BH coupled to a Thermal Bath Shan-Ming Ruan (PI)
Three equivalent descriptions
1D 2D 3D
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BH coupled to a Thermal Bath Shan-Ming Ruan (PI)
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Teff (u; Tb) = T2
b + (T2 1 − T2 b) e−ku
Tb = T1 Tb < T1 Solutions
a = 2π k T 2
1 − T 2 b
ν = 2πTb k
E(u) = ¯ ϕrπ 4GN T2
eff(u)
Tb > T1
f (u, Tb) = 2 ka Iν(a)Kν (ae−ku/2) − Kν(a)Iν (ae−ku/2) I′
ν(a)Kν (ae−ku/2) − K′ ν(a)Iν (ae−ku/2)
k ≡ cGN 3 ¯ ϕr ≪ 1
: Temperature of BH after joint
: Temperature of thermal bath
BH coupled to a Thermal Bath Shan-Ming Ruan (PI)
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Γeff (y−
QES) ≡
1 −
Tb Teff(y−
QES)
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ϕ (x±) = ϕr (
2f′ (y−) x+ − x− + f′ ′ (y−) f′ (y−) )
SvN = SΩ−2g = SUHP − c
6 ∑
xi∈∂
log Ω(xi) Late time/Island Phase
;
x+
QES ≈ t∞ +
Γeff 4 − Γeff (t∞ − t) x−
QES ≈ t∞ −
8πTeff k (4 − Γeff) (t∞ − t)
BH coupled to a Thermal Bath Shan-Ming Ruan (PI)
20 ˜ y± = ˜ u ± ˜ σ t∞
x± = ± 1 πT0
Ⅰ Ⅱ Ⅲ Ⅳ
y− = ∞
y+ − y− = 0
y+ = ∞ u = 0
x+ x−
QES New horizon
Shock wave
QES
Gravitational Region Bath Region Purifying Region
Bifurcation surface
QML QMR
BH coupled to a Thermal Bath Shan-Ming Ruan (PI)
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Tb > T1 Tb = T1 Tb < T1 arXiv: 2007.11658
4GN ¯ ϕr Sgen
ku
uPage 2πTb 1 2 2πT1 3 2πTb Tb < T1 Tb = T1 Tb > T1 4πTbku 4 5 2π(Tb + T1)ku
Island Phase:
dSgen,late du ≈ − ¯ ϕr 4GN (1 − T2
b
T2
eff(u)) kπTeff(u)
BH coupled to a Thermal Bath Shan-Ming Ruan (PI)
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T1 = Tb
x+
QES(t) =
k2 + π2T 2
1 ((πT1t) 2 − 1) + k ((πT1t) 2 + 1)
π2T 2
1 (π2T 2 1t2 + 2kt − 1)
x−
QES(t) =
k2 + π2T 2
1 ((πT1t) 2 − 1) + k ((πT1t) 2 + 1)
π2T 2
1 (−π2T 2 1t2 + 2kt + 1)
Sgen,late (T1) =
¯ ϕ 2GN (
k2 + π2T2
1 − k log [ϵ (k +
k2 + π2T2
1)])
Island outside horizon
;
x+
QES(t) < t∞ = x+ QES(t∞)
dx+
QES(t)
dt > 0
Constant
BH coupled to a Thermal Bath Shan-Ming Ruan (PI)
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Inside horizon
Tc1(u) < Tb < Tc2(u)
On the horizon
Tb = Tc1(u) or Tb = Tc2(u)
Outside horizon
Tb < Tc1(u) or Tb > Tc2(u)
Tc1(u) ≈ 1 − 2k πT1 Teff (y−
QES)
Tc2(u) ≈ 1 + 2k πT1 Teff (y−
QES)
ku ≳ log 1 − T2
1
T2
b
πT1 8k
At very late time, QES always moves outside horizon
BH coupled to a Thermal Bath Shan-Ming Ruan (PI)
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BH coupled to a Thermal Bath Shan-Ming Ruan (PI)
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2D Bath (with radiation) QMR QML
Pure State: Equivalent
BH coupled to a Thermal Bath Shan-Ming Ruan (PI)
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Entanglement Wedge of + bath (Hawking radiation) +Purification QML Entropy of + bath (Hawking radiation) +Purification QML
Reconstruct the interior of BH after Page transition (Island Phase) 2D Bath (with radiation) QMR QML Purification
BH coupled to a Thermal Bath Shan-Ming Ruan (PI)
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BH coupled to a Thermal Bath Shan-Ming Ruan (PI)
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SR = Sgen
QES−1 + S2−IR
SN = Sgen
QES′ ′ + S1−2 + S 1
2 −line
Competing Channels
Island Phase
BH coupled to a Thermal Bath Shan-Ming Ruan (PI)
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σ y+ u
uPage
y−
IR
s h
k
y2
˜ σ
Δσ(u)
˜ u
Purification
Tb > 0
Hawking radiation
Δσturn
σ y+ u
uPage
y−
IR
s h
k
y2
˜ σ
Δσ
˜ u
Purification
Tb = T1
Hawking radiation
σ2(u) ≳ T1 − T0 2k (T1 + Tb) + T1 4 (T1 + Tb) (u (1 − Tb T1 )
2
+ uHP (1 − T 2
b
T 2
1 )) +
log (
6Es cT1 )
2π (T1 + Tb) + ⋯
BH coupled to a Thermal Bath Shan-Ming Ruan (PI)
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SR = Sgen
QES−1 + S2
SN = Sgen
QES′ ′ + S1−2
Reconstruct the interior of BH !
SR ≤ SN
Only If :
Tb ≲ Tp ≈ T1 + T0 2 + k 2π log ( 6Es cT1 )
BH coupled to a Thermal Bath Shan-Ming Ruan (PI)
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BH coupled to a Thermal Bath Shan-Ming Ruan (PI)
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