Islands Beyond Ads

Islands beyond AdS

Edgar Shaghoulian University of Pennsylvania

Island Hopping 2020: from Wormholes to Averages November 17, 2020

[Hartman, Shaghoulian, Strominger 2004.13857]

[Hartman, Jiang, Shaghoulian 2008.01022] I These approaches may be the same via Euclidean gravity = holography [Harlow, ES] I Euclidean gravity approach is generalizable to other spacetimes; let’s consider ﬂat/cosmological spacetimes (−→ holography?)

Introduction

I Page curve for Hawking radiation from holography [Penington; Almheiri, Engelhardt, Marolf, Maxfield] and Euclidean gravity path integral [Penington,

Shenker, Stanford, Yang; Almheiri, Hartman, Maldacena, ES, Tajdini] I Euclidean gravity approach is generalizable to other spacetimes; let’s consider ﬂat/cosmological spacetimes (−→ holography?)

Introduction

I Page curve for Hawking radiation from holography [Penington; Almheiri, Engelhardt, Marolf, Maxfield] and Euclidean gravity path integral [Penington,

Shenker, Stanford, Yang; Almheiri, Hartman, Maldacena, ES, Tajdini] I These approaches may be the same via Euclidean gravity = holography [Harlow, ES] Introduction

I Page curve for Hawking radiation from holography [Penington; Almheiri, Engelhardt, Marolf, Maxfield] and Euclidean gravity path integral [Penington,

Shenker, Stanford, Yang; Almheiri, Hartman, Maldacena, ES, Tajdini] I These approaches may be the same via Euclidean gravity = holography [Harlow, ES] I Euclidean gravity approach is generalizable to other spacetimes; let’s consider flat/cosmological spacetimes (−→ holography?) Quantum effective action exact at large c. Equivalent to flat-space JT Z 2 √ I = d x −g [ΦR + 1] + ICFT[g]

Conformal gauge ds2 = −e2ρdx+dx−

U(1) current ∂µ(ρ − φ) equivalent to Φ shift symmetry.

Flat space

CGHS/RST model: Z √ h c i I = d2x −g e−2φ(R + 4(∇φ)2 + 4) − φR + I [g] cl 24 CFT | {z } RST Conformal anomaly leads to c Z √ I = − d2x −g R −1R anom 96π Flat space

CGHS/RST model: Z √ h c i I = d2x −g e−2φ(R + 4(∇φ)2 + 4) − φR + I [g] cl 24 CFT | {z } RST Conformal anomaly leads to c Z √ I = − d2x −g R −1R anom 96π Quantum eﬀective action exact at large c. Equivalent to ﬂat-space JT Z 2 √ I = d x −g [ΦR + 1] + ICFT[g]

Conformal gauge ds2 = −e2ρdx+dx−

U(1) current ∂µ(ρ − φ) equivalent to Φ shift symmetry. Linear dilaton vacuum: 1 dx+dx− Ω = −x+x− − log(−4x+x−) , ds2 = = −dσ+dσ− 4 x+x− ﬂat 2 2 Tx±x± = Tx±x± − ((∂+ρ) − ∂+ρ) =⇒ Tx±x± = Tσ±σ± = 0

Solutions: linear dilaton vacuum

Field redeﬁnitions and further gauge-ﬁxing:

2 ﬂat ∂+∂−Ω = −1 , ∂±Ω = −Tx±x± Restrict to Ω ≥ 1/4. Solutions: linear dilaton vacuum

Field redeﬁnitions and further gauge-ﬁxing:

2 ﬂat ∂+∂−Ω = −1 , ∂±Ω = −Tx±x± Restrict to Ω ≥ 1/4.

Linear dilaton vacuum: 1 dx+dx− Ω = −x+x− − log(−4x+x−) , ds2 = = −dσ+dσ− 4 x+x− ﬂat 2 2 Tx±x± = Tx±x± − ((∂+ρ) − ∂+ρ) =⇒ Tx±x± = Tσ±σ± = 0 + − ds2 ≈ −dσ+dσ˜− near I+ with x+ = eσ , x− + M = −e−σ˜ , so observer sees 1 1 − T − − ≈ 1 − , σ˜ ∈ (−∞, 4M) σ˜ σ˜ 4 1 + Meσ˜−

Evaporating black hole Shock wave impinges on linear dilaton vacuum 1 Ω = −x+x−− log(−4x+x−) − M(x+ − 1)θ(x+ − 1) 4 + Tσ+σ+ = Mδ(x − 1) ,Tσ−σ− = 0 Evaporating black hole Shock wave impinges on linear dilaton vacuum 1 Ω = −x+x−− log(−4x+x−) − M(x+ − 1)θ(x+ − 1) 4 + Tσ+σ+ = Mδ(x − 1) ,Tσ−σ− = 0

+ − ds2 ≈ −dσ+dσ˜− near I+ with x+ = eσ , x− + M = −e−σ˜ , so observer sees 1 1 − T − − ≈ 1 − , σ˜ ∈ (−∞, 4M) σ˜ σ˜ 4 1 + Meσ˜− Evaporating black hole

c − S (R) = S (R) = log(1 + Meσ˜ ) + UV/IR divergences QG QFT 12 [Fiola, Preskill, Strominger, Trivedi]

Remnants! Proposed modiﬁcation: sum over topologies to get Area(∂I) S (R) = min ext S (I ∪R) ,S (I ∪R) = +S (I ∪R) . QG I gen gen 4 QFT

I+ non-dynamical.

Let’s revisit the evaporating black hole! cf [Gautason, Schneiderbauer, Sybesma,

Thorlacius; Anegawa, Iizuka]

Island rule entropy [Hartman, ES, Strominger]

CGHS/RST model are deﬁned on ﬁxed topology: no replica wormholes. I+ non-dynamical.

Let’s revisit the evaporating black hole! cf [Gautason, Schneiderbauer, Sybesma,

Thorlacius; Anegawa, Iizuka]

Island rule entropy [Hartman, ES, Strominger]

CGHS/RST model are deﬁned on ﬁxed topology: no replica wormholes.

Proposed modiﬁcation: sum over topologies to get Area(∂I) S (R) = min ext S (I ∪R) ,S (I ∪R) = +S (I ∪R) . QG I gen gen 4 QFT Let’s revisit the evaporating black hole! cf [Gautason, Schneiderbauer, Sybesma,

Thorlacius; Anegawa, Iizuka]

Island rule entropy [Hartman, ES, Strominger]

CGHS/RST model are deﬁned on ﬁxed topology: no replica wormholes.

Proposed modiﬁcation: sum over topologies to get Area(∂I) S (R) = min ext S (I ∪R) ,S (I ∪R) = +S (I ∪R) . QG I gen gen 4 QFT

I+ non-dynamical. Island rule entropy [Hartman, ES, Strominger]

CGHS/RST model are deﬁned on ﬁxed topology: no replica wormholes.

Proposed modiﬁcation: sum over topologies to get Area(∂I) S (R) = min ext S (I ∪R) ,S (I ∪R) = +S (I ∪R) . QG I gen gen 4 QFT

I+ non-dynamical.

Let’s revisit the evaporating black hole! cf [Gautason, Schneiderbauer, Sybesma,

Thorlacius; Anegawa, Iizuka] Evaporating black hole

σ˜− ∼ O(M) with M 1:

+ c − − c σ SQG(R) = min extI Sgen(I ∪ R) = min {2˜σ , 4M − σ˜ } + log 24 6 uv

tPage = 4M/3 = tevap/3: −δSBH = 2δSradiation ds2 = e2ρdzdz¯, place defect (represents ∂I) at z = 0 and take z ∼ 0, n ∼ 1:

flat flat flat ρ → ρ + (n − 1)δρ, φ → φ + (n − 1)δφ, Tµν → Tµν + (n − 1)δTµν

1/n 1 Smoothness on replica manifold w = z implies δρ ∼ − 2 log(zz¯).

Conformal ward identity gives

c/12 ∂Sﬂat 2πδT ﬂat = − CFT + reg , barred zz z2 z Expanding dilaton in series respecting replica symmetry around z = 0 and solving singular terms of constraint equations (gzz and gz¯z¯ equations) gives

Area(∂I) Area(∂I) ∂ + S = ∂¯ + S = 0 . 4 CFT 4 CFT

Replica wormholes

Consider general dilaton gravity Z I = (φR + U(φ)) + ICFT Σ2 Conformal ward identity gives

Area(∂I) Area(∂I) ∂ + S = ∂¯ + S = 0 . 4 CFT 4 CFT

Replica wormholes

Consider general dilaton gravity Z I = (φR + U(φ)) + ICFT Σ2 ds2 = e2ρdzdz¯, place defect (represents ∂I) at z = 0 and take z ∼ 0, n ∼ 1:

flat flat flat ρ → ρ + (n − 1)δρ, φ → φ + (n − 1)δφ, Tµν → Tµν + (n − 1)δTµν

1/n 1 Smoothness on replica manifold w = z implies δρ ∼ − 2 log(zz¯). Replica wormholes

Consider general dilaton gravity Z I = (φR + U(φ)) + ICFT Σ2 ds2 = e2ρdzdz¯, place defect (represents ∂I) at z = 0 and take z ∼ 0, n ∼ 1:

flat flat flat ρ → ρ + (n − 1)δρ, φ → φ + (n − 1)δφ, Tµν → Tµν + (n − 1)δTµν

1/n 1 Smoothness on replica manifold w = z implies δρ ∼ − 2 log(zz¯).

Conformal ward identity gives

Area(∂I) Area(∂I) ∂ + S = ∂¯ + S = 0 . 4 CFT 4 CFT Necessary condition 1 [Almheiri, Engelhardt, Marolf, Maxfield]: d Sgen(I ∪ R) = 0 =⇒ dλ+ d Area(∂I) d SQFT(I) + + [SQFT(I ∪ R) − SQFT(I) − SQFT(R)] = 0 dλ+ 4 dλ+ for outgoing null derivative. Monotonicity of mutual information gives d d Sgen(I) = IQFT(I,R) ≥ 0 dλ+ dλ+ For inward null derivative d d Sgen(I) = IQFT(I,R) ≤ 0 dλ− dλ− So ∂I is in “quantum normal region”:

d ± Sgen(I) ≥ 0 dλ±

Looking for islands [Hartman, Jiang, ES] How do we guide our search for islands in general dimensions/spacetimes? Looking for islands [Hartman, Jiang, ES] How do we guide our search for islands in general dimensions/spacetimes?

Necessary condition 1 [Almheiri, Engelhardt, Marolf, Maxfield]: d Sgen(I ∪ R) = 0 =⇒ dλ+ d Area(∂I) d SQFT(I) + + [SQFT(I ∪ R) − SQFT(I) − SQFT(R)] = 0 dλ+ 4 dλ+ for outgoing null derivative. Monotonicity of mutual information gives d d Sgen(I) = IQFT(I,R) ≥ 0 dλ+ dλ+ For inward null derivative d d Sgen(I) = IQFT(I,R) ≤ 0 dλ− dλ− So ∂I is in “quantum normal region”:

d ± Sgen(I) ≥ 0 dλ± Rearrange to get Area(∂I) S (I) − > S (I) + S (I ∪ R) − S (R) QFT 4 QFT QFT QFT Araki-Lieb S(I ∪ R) ≥ |S(I) − S(R)| gives

Area(∂I) SbQFT(I) & 4

“Phenomenology”: in our past lightcone, for thermal matter entropy, bound violated at T ∼ 1 TeV.

Looking for islands

Necessary condition 2: Area(∂I) + S (I ∪ R) < S (R) 4 QFT QFT “Phenomenology”: in our past lightcone, for thermal matter entropy, bound violated at T ∼ 1 TeV.

Looking for islands

Necessary condition 2: Area(∂I) + S (I ∪ R) < S (R) 4 QFT QFT Rearrange to get Area(∂I) S (I) − > S (I) + S (I ∪ R) − S (R) QFT 4 QFT QFT QFT Araki-Lieb S(I ∪ R) ≥ |S(I) − S(R)| gives

Area(∂I) SbQFT(I) & 4 Looking for islands

Area(∂I) SbQFT(I) & 4

“Phenomenology”: in our past lightcone, for thermal matter entropy, bound violated at T ∼ 1 TeV. I I

Looking for islands in the sky