[PPT] - Holographic Entanglement in Gauss-Bonnet gravity: time and shadows E PowerPoint Presentation

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Holographic Entanglement in Gauss-Bonnet gravity: time and shadows

ELENA CÁCERES

Facultad de Ciencias Universidad de Colima, Mexico Theory Group University of Texas at Austin

(work in progress with M. Sanchez and J. Virrueta)

March 25, 2015

E. Cáceres (UCol/UT-Austin)

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Outline

1. Motivation
2. Gauss Bonnet gravity
3. HEE in time dependent GB gravity
4. HEE shadows in GB gravity
E. Cáceres (UCol/UT-Austin)

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MOTIVATION

Bulk reconstruction. Emergence of spacetime.

1. String corrections, finite but large tHooft

I Generic form of higher derivatives corrections is not known I Effective five-dimensional gravity theory

S = 1 16⇡GN Z d5xp−g(R−2⇤+L2(↵1R2 +↵2Rµ⌫Rµ⌫ +↵3Rµ⌫⇢Rµ⌫⇢))

where ⇤ = −6/L2 and we assume ↵i << 1.

2. Higher derivative theories can lead to interesting physics: i.e.

η s

bound violation for ↵3 > 0 ⌘ s = 1 4⇡(1 − 8↵3) + O(↵2

i )

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3. Higher derivative theories have c 6= a

I Conformal anomaly of 4 dimensional CFT

hTµ

µ i =

c 16⇡2 I4 − a 16⇡2 E4 where I4 = CabcdCabcd = RabcdRabcd − 2RabRab + 1 3R2 E4 == RabcdRabcd − 4RabRab + R2

I Holographically,

hTµ

µ i = something

16⇡2 I4 − something0 16⇡2 E4

I comparing both expressions we get

↵3 ∼ c − a 8c

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TWO QUESTIONS

1) How deep behind the horizon does the HEE probe in time dependent GB theories?

1 2 3 4 5 6 7 0.2 0.4 0.6 0.8 1.0 1.2 vHxL zHxL-1

2) In global AdS 9 regions not probed by minimal surfaces, "shadows". Effect of GB on shadows? Do CFT dual to higher derivative theories "know" more about the bulk?

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GAUSS-BONNET GRAVITY

Sgrav = 1 16⇡GN Z d5xp−g ✓ R + 12 L2 + L2 2 L(2) ◆ , L(2) = Rµ⌫⇢Rµ⌫⇢ − 4Rµ⌫Rµ⌫ + R2

I Exact solutions are known

Black hole solution, ds2 = −L2 z2 f(z) f0 dv2 + L2 z2 ✓ − 2 pf0 dzdv + d¯ x2 ◆ , f(z) = 1 2[1 − q 1 − 4(1 − mz4)].

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dv = dt − dz f(z) f0 = 1 2 ⇣ 1 − p 1 − 4 ⌘ . (1) In Poincarè coordinates ds2 = −L2 z2 f(z) f0 dt2 + L2 z2 d¯ x2 + L2 z2 dz2 f(z). (2)

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Note that:

I Causality bounds:

− 7/36   9/100, (3)

I Central charges:

c = ⇡2 L3 lp (1 − 2 f0), a = ⇡2 L3 l3

p

(1 − 6 f0) (4)

I Singularity at finite z for < 0,

zsing = 1 p 2m(v)1/4 (−1/ + 4)1/4

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HEE IN TIME DEPENDENT GAUSS-BONNET

S = Sgrav + Sext where the external source is unespecified ds2 = −L2 z2 f(z, v) f0 dv2 + L2 z2 ✓ − 2 pf0 dzdv + d¯ x2 ◆ , where f0 = 1 2(1 − p 1 − 4), f(z, v) = 1 2  1 − q 1 − 4(1 − m(v)z4)

I m(v) is arbitrary

I Sext yields the following energy-momentum tensor

(16⇡GN)T ext

µ⌫ = 3

2z3 dm dv µv⌫v.

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Previous work focused on thermalizarion time ( Li, Wu and Yang 2013)

I Apparent horizon zAH = m(v)1/4 I Event horizon: z0 EH(v) = − 1 2pf0 f(zEH, v)

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Covariant prescription

Hubeny, Rangamani, Takayanagi 07 d A ∂A

γ

A AdS d+1

SA = Areaextrm(A) Gd+1

N

Codimension 2 surface Homology condition A ∼ A 9 bulk region r s.t. r = A [ A

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Entanglement entropy in GB

(Hung, Myers, Solkin, 2011)

SEE = 1 4GN Z

⌃

d3⇠p(1 + L2R⌃) + 1 2GN Z

@⌃

d2⇠ p h K

I R⌃ : Ricci scalar for intrinsic geometri on ⌃ I K : trace of extrinsic curvature on @⌃

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I Study “rectangular strip" for the time-dependent case. I z(x), v(x) I Induced metric on the co-dimension two surface is

ds2 = L2 z2 (dx2

2 + dx2 3) + L2

z2 ✓ 1 − f f0 v02 − 2 pf0 v0z0 ◆ dx2, (5)

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Thus, p = L3 pf0 1 z3 ⇣ f0 − fv02 − 2 p f0v0z0⌘1/2 , (6) L2pR⌃ = (2L3 p f0) z02 z3 f0 − fv02 − 2pf0v0z01/2 + dF dz , (7) where, F(x) = (4L3 p f0) z0 z2 f0 − fv02 − 2pf0v0z01/2 (8)

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Finally, action to be extremized is, Seff = L3 4GN pf0 Z dz z3 ⇣ f0 − fv02 − 2 p f0v0z0⌘1/2 + 2f0z02

f0 − fv02 − 2pf0v0z01/2

#

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I Time dependent, minimal surfaces penetrate the horizon,

but do not reach singularity

I How does this change with I in other words , the region accesible to the holographic

probes increases or decreases with ?

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Results

tb=0.8 tb=1 tb=1.2 Schw AdS

0.0 0.5 1.0 1.5 2.0 2.5 3.0

1.0
0.5

0.0 0.5 1.0 l S l=-0.05

tb=0.8 tb=1 tb=1.2 Schw AdS

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

1.0
0.5

0.0 0.5 1.0 l S l=0.05

1.0
0.5

0.0 0.5 1.0 0.0 0.5 1.0 1.5 v r l=-0.05

1.0
0.5

0.0 0.5 1.0 0.0 0.5 1.0 1.5 v r l=0.05

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Figure : illustration of rmin vs `/2 (numerics in progress)

Theories with < 0 probe deeper behind the horizon than Einstein gravity . Theories with > 0 explore less For < 0 and large ` the entanglement probes can reach arbitrarily close to the singularity

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HEE SHADOWS IN GB

Global, static. Holographic Entanglement entropy

Ryu, Tagayanagi 06 d A ∂A

γ

A AdS d+1

SA = Areamin(A) Gd+1

N

Codimension 2 surface Homology condition A ∼ A 9 bulk region r s.t. r = A [ A

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Minimal surfaces in global BTZ SA(#) = 8 < :

c 3 log

⇣

2 r∞ r+ sinh(r+#/2)

⌘ , #  #X

c 3 ⇡ r+ + c 3 log

⇣

2 r∞ r+ sinh(r+(2⇡ − #)/2)

⌘ , # #X #X (r+) = 2 r+ coth−1 (2 coth(⇡ r+) − 1) .

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SHADOWS

I Entanglement shadow: regions of the bulk not reached by

any HEE probe i.e. maximum depth among all boundary regions.Balasubramanian et. al. 2014

I Behavior associated with phase transition

1 2 3 4 5 6 J

0.5

0.5 1.0 1.5 2.0 2.5 3.0 S

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Entanglement Shadow

Freivogel et. al 14.12.5175

= r⇤ − rh = 2rHe−⇡rH sinh(⇡rH)

0.5 1.0 1.5 2.0 2.5 3.0 rh 0.1 0.2 0.3 0.4 0.5 0.6 rss

I ∼ #rH + .....

for rH << `AdS

I ∼ r2 He−#rH + ....

for rH >> `AdS Similar limiting behaviour in AdS5.

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Entanglement Shadows in Gauss-Bonnett

I Black hole in global AdS5, small I Assume the 3-dimensional boundary region of interest is O(3)

symmetric, r(✓) ds2 = −f(r) f∞ dt2 + dr2 f(r) + r2(d✓2 + sin2(✓)d⌦2) f(r) = 1 + r2 2(1 − q 1 + 4((rh2 + rh4 + ))/r4 − 4])

where f∞ is a convenient normalization factor, f∞ = 1−p

1−4λ 2λ

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HEE, prescription for higher derivatives. Action to minimize, L = s r0(✓)2 f(r) + r(✓)2 r(✓)2 sin(✓)2 + 2

+

2r(✓)2 cos(✓)2 + sin(2✓)r(✓)r0(✓) + sin(✓)2r0(✓)2 q

r0(✓)2 f(r) + r(✓)2

Study shadows numerically –in progress.

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Following Frievogel et al 14125175, approximate solution near the horizon

I expand eom close to the horizon I assume r0(✓) is small

r00(✓) + 2 cot(✓)r0(✓) + r(✓)H(rh, ) + ˜ H(rh, ) Can be solved with r(0) = r⇤, r0(0) ∼ 0. For small , r(✓) = 1 k3rh2 csc(✓)(k3rh3 sin(✓) + (rh − rs)(6k(1 + 2rh2)✓ cosh(k✓) − (6 + rh2(k2 + 12)) sinh(k✓))

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where k = p 5 + rh2 Shadow size: r⇤ − rh ⌘ ,

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I Large black holes, similar behaviour as = 0

∼ rh2e−#rh

I Small black holes

∼ #rh + p(rh) where p(rh) > 0 ! For < 0 shadow is smaller

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