2D Is Fire Free

2D is Fire Free

Ahmed Almheiri UCSB

w/ James Sully; 2013

Black Holes in String Theory Workshop UMich, 10/16/13 Movaon

• Need a controllable seng to aempt to answer long-standing QG quesons: – Informaon Loss? – remnants? – Firewall? • 2D provides such a seng: It has completely solvable models, – CGHS [Callan, Giddings, Harvey, Strominger] – RST [Russo, Susskind, Thorlacius] Movaon

• Ashtekar et al simulate CGHS in the Mean Field Approximaon (Large-N semiclassical analysis) and ﬁnd: – Unitary Evaporaon* – No Firewall [Ashtekar, Pretorius, Ramazanola, Taveras, Varadarajan]

• This looks like it contradicts the Firewall proposal. Outline

• The Model – Classical CGHS – CGHS in MFA • Entanglement Entropy of 1+1 CFTs and Moving Mirrors

• CGHS Remnants

• Upli

• Discussion The Model – Classical CGHS

• The CGHS model is given by the acon

1 2 2φ a 2 S = d x√ge− R +4 φ φ +4κ G G ∇ ∇a coupled to N scalar ﬁelds via 1 S = d2x√g af i f i M −2 ∇ ∇a [Callan et al] • It describes the near horizon physics of magnecally charged extremal dilatonic black holes in 4 and 5 dimensions [Giddings, Strominger] The Model – Classical CGHS

• Soluon ﬁxed by scalar ﬁeld

+ κ(z z−) Θ(z±)=e − + + z z i 2 + + G + κz + κz ∂f+ Φ(z±)=Θ(z±) dz e dz e− − 2 ∂z + i −∞ −∞ ab 1 ab 2φ Where g = ΦΘ− η , Φ = e− • Form a black hole by sending in a scalar shockwave

1 i 2 MADM + (∂ + f ) = δ(z ) 2 z + N The Model – Classical CGHS

• Singularity is at Φ =0

+ • and are R L− Icomplete I w.r.t. the metric gab

+ • Past of covers only IR part of − IL y− = ln(z− z−) − h − = Informaon Loss! ⇒ • Next: Include backreacon of Hawking Radiaon The Model – CGHS in MFA

• Do so by working in a Mean Field Approximaon: – Replace geometric ﬁelds by expectaon values – Include scalar ﬁeld quantum contribuons

– Requires large N [callan et al, Ashtekar et al] [Christensen, Fullings] • Einstein’s equaons are sourced by the one-loop trace anomaly G = T µν µν The Model – CGHS in MFA

+ • The MFA equaons give a balance law at from R which one deﬁnes a Bondi Mass and Flux: I

dM ATV Bondi = F ATV dy− −

2 NG d dy− where F ATV = ln 48 dy dz − − + and are the aﬃne parameters on and z− y− − IL IR respecvely. The metric is ﬂat in these coordinates. The Model – CGHS in MFA

• Comments on the model: NG – Quantum corrected singularity Φ = 12 – Scaling Symmetry: ATV ATV ATV ATV (M ADM,F ,M ,N) λ(MADM,F ,M ,N) Bondi → Bondi • Physical parameters are raos, eg: MADM M ∗ = N – A black hole is considered Planckian if it evaporates in a Planck me when assuming the standard hawking ﬂux M ∗ GMPl The Model – CGHS in MFA

• Again, form a black hole by sending in a coherent î state of scalar fields, , peaked at the classical f+ o profile f+ s.t.:

1 o 2 MADM + (∂ + f ) = δ(z ) 2 z + N

• The right movers, , are inially in their ground fˆi state, so the classical proﬁles are − f i =0 −

The Model – Results The Model – Results

0.9

0.85

0.8 * m

0.75

0.7 w=0 fitted curve

0.65 [Ashtekar et al] 0 5 10 15 M* • Universal small Bondi mass at the last ray

m∗ = M ∗ Last ray 0.86GMPl Bondi| ∼ The Model – Results

• Small Bondi mass at the last ray, m∗ The Model – Results

6 0.1

0 5 ï0.1

) ï0.2 4 6 ) ï ï0.3 /dz

3 ))/d(ln ï

ï ï0.4

/dz ï0.5 ( dy 2 ï 10 ï0.6 log 1 ï0.7 d(ln(dy

ï0.8 0 ï0.9

ï1 ï1 [Ashtekar et al] ï10 ï5 0 ï10 ï5 0 log log 106 106 • is ﬁnite at the last ray, and goes as y− y− = kz− dy 1 • Intermediate region: (like classical case)− = dz −κ(z− z ) − h − − + • is not complete w.r.t. gab IR The Model – Results

• Small Bondi mass at the last ray, m∗

+ • is not complete IR w.r.t. gab y− = kz− The Model – Results

8 ï5 0 5

ï8 ï10 4

ï7 ï10

ï sing 0 1/5 z R ï

ï ï2 z ï6 ï10 ï4

1eï5 ï6 1eï6 ï5 ï10 1eï8 ï8 2 4 8 16 8 12 16 20 [Ashtekar et al] z+ z+ • No thunderbolt singularity. The last ray is regular. • No singularity in the horizon region.

The Model – Results

• Small Bondi mass at the last ray, m∗

+ • is not complete IR w.r.t. gab y− = kz−

• No Thunderbolt Singularity

• Horizon region is regular = No Firewalls ⇒ The Model – Results

• Comments: + – might be extendable s.t. it becomes unitarily IR equivalent to L− – Assuming the singularity is alleviated by quantum I correcons. • So we have unitary black hole evaporaon with semiclassical physics in the horizon region with no drama---!....Relax • Will show that this is a Remnant scenario. The small bondi mass object is entangled with the hawking radiaon. The Model – Analysis

• Compute entropy of an interval containing the Hawking radiaon

• The radiaon is due to having two diﬀerent asymptoc killing vectors with which to deﬁne posive energy

• This is now exactly like compung entanglement entropy of radiaon from moving mirrors [Holzhey, Larsen, Wilczek] The Model – Analysis

• We obtain from f(z−) Ashtekar et al’s plots. The Model – Analysis

6 0.1

0 5 ï0.1

) ï0.2 4 6 ) ï ï0.3 /dz

3 ))/d(ln ï

ï ï0.4

/dz ï0.5 ( dy 2 ï 10 ï0.6 log 1 ï0.7 d(ln(dy

ï0.8 0 ï0.9

ï1 ï1 [Ashtekar et al] ï10 ï5 0 ï10 ï5 0 log log 106 106 • Mirror Trajectory: – Inial y− = z− dy 1 – Intermediate − = dz −κ(z− z ) − h − − – Near the last ray and past y− = kz− EE in 1+1 CFT

• We resort to 1+1 CFT EE techniques. • EE of an interval in the vacuum is given by

c +¯c Σ Sent = ln 6 √ 1 2

ε2 Σ ε1

[Callan, Wilczek, Srednicki] EE in 1+1 CFT

• The mirror transforms the vacuum state to an excited state

• EE for excited states: – These states relate to the ground state by a conformal transformaon f(x) – This transformaon preserves the noon of inside and outside And so, Exc c +¯c f(x2) f(x1) Sent = ln − 6 f (x2)2f (x1)1

[Holzhey, Larsen, Wilczek] EE in 1+1 CFT

• We are interested in compung the excess entropy on top of the vacuum. – This gives the entropy of the new excitaons produced by the mirror (or black hole). • This deﬁnes a renormalized EE, c +¯c (f(x ) f(x ))2 SRen = ln 2 − 1 ent 12 (x x )2f (x )f (x ) 2 − 1 2 1 – By subtracng oﬀ the vacuum piece we got rid of the

dependence on the UV regulator. [Holzhey, Larsen, Wilczek] EE in 1+1 CFT and Moving Mirrors

• Let’s apply the formula for the mirror trajectory that models the black hole evaporaon 1 • Mirror path given by y − = f − (z−) 1 – Far past: f − (z−)=z− 1 – Near the last ray and beyond: f − (z−)=kz− • We wish to compute the entropy of an interval with y in the far past, and near the last ray. Then: 1− y2−

f(y2−) f(y1−) f (y1−) 1 − 1 → y− y− → 2 − 1 as y− 1 →−∞ EE in 1+1 CFT and Moving Mirrors

• The Ren. Entropy simpliﬁes to Ren c S = ln(f (y−)) where f (y−)=1/k ent −12 2 2

• The entropy is independent of and thus all the excess y1− entanglement is with modes past the last ray • This excess comes from pulling modes below the cut-oﬀ EE in 1+1 CFT and CGHS Remnants

• We can do beer than just plugging in . We can k Ren obtain the scaling of with SEnt MADM • From the formulas of the Bondi Mass and Flux we have ATV 2 dMBondi ATV NG d = F = ln(f (y)) − dy 48 −dy − − NG d κ ln(f (y)) ∼ 48 −dy − dy− 1 Used the intermediate region approx: = dz− −κ(zh− z−) • Now integrate from to the last ray: − −∞ ATV NG M + M = κ ln(f (y−)) + ln(f (y−)) − Bondi|last ray ADM 48 − 2 1 EE in 1+1 CFT and CGHS Remnants

• To leading order we ﬁnd

Ren 4 SEnt = MADM κG • We found very large entanglement between the Hawking radiaon and the region past the last ray, where the Bondi Mass is very small (Planckian). This is a Remnant!

• Long lived. The evaporaon is energecally constrained to be 2 slower than MADM Upli

• The soluon can be uplied to describe the process of throwing in excitaons inside an extremal dilatonic black holes in 4d or 5d • Before the pulse we had an inﬁnite throat with a linear dilaton • The infalling pulse pushes the black hole away from extremality • The black hole then proceeds to evaporate away to extremality Upli

• This suggests an old version of the remnant scenario where the remnants are given by these inﬁnite throats – cornucopions [Banks, Giddings, Loughlin, Strominger] • These are ruled out in higher dimensions by the usual arguments: AdS/CFT and inﬁnite pair producon [Giddings] • But didn’t we just show that remnants are indeed what you get? • No: Reducon and Quanzaon do not commute. Discussion

• Why didn’t we get anything new? • Gravity in 2d is a local renormalizable ﬁeld theory • We do not expect a complementary descripon to emerge for such theories.