Holographic Entanglement in Gauss-Bonnet gravity: time and shadows E LENA C ÁCERES 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) 1
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) 2
M OTIVATION 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 Z d 5 x p − g ( R − 2 ⇤ + L 2 ( ↵ 1 R 2 + ↵ 2 R µ ⌫ R µ ⌫ + ↵ 3 R µ ⌫⇢� R µ ⌫⇢� )) 1 S = 16 ⇡ G N where ⇤ = − 6 / L 2 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 ) E. Cáceres (UCol/UT-Austin) 3
3. Higher derivative theories have c 6 = a I Conformal anomaly of 4 dimensional CFT c a h T µ µ i = 16 ⇡ 2 I 4 − 16 ⇡ 2 E 4 where I 4 = C abcd C abcd = R abcd R abcd − 2R ab R ab + 1 3R 2 E 4 == R abcd R abcd − 4R ab R ab + R 2 I Holographically, I 4 − something 0 µ i = something h T µ E 4 16 ⇡ 2 16 ⇡ 2 I comparing both expressions we get ↵ 3 ∼ c − a 8c E. Cáceres (UCol/UT-Austin) 4
T WO QUESTIONS 1) How deep behind the horizon does the HEE probe in time dependent GB theories? 1.2 1.0 0.8 z H x L - 1 0.6 0.4 0.2 0 1 2 3 4 5 6 7 v H x L 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? E. Cáceres (UCol/UT-Austin) 5
G AUSS -B ONNET GRAVITY Z d 5 x p − g L 2 + � L 2 ✓ ◆ 1 R + 12 S grav = 2 L ( 2 ) , 16 ⇡ G N L ( 2 ) = R µ ⌫⇢� R µ ⌫⇢� − 4R µ ⌫ R µ ⌫ + R 2 I Exact solutions are known Black hole solution, ds 2 = − L 2 dv 2 + L 2 f ( z ) ✓ ◆ − 2 x 2 p f 0 dzdv + d ¯ , z 2 z 2 f 0 f ( z ) = 1 q 1 − 4 � ( 1 − mz 4 )] . 2 � [ 1 − E. Cáceres (UCol/UT-Austin) 6
dv = dt − dz f ( z ) p f 0 = 1 ⇣ ⌘ (1) 1 − 1 − 4 � . 2 � In Poincarè coordinates ds 2 = − L 2 dt 2 + L 2 x 2 + L 2 dz 2 f ( z ) (2) z 2 d ¯ f ( z ) . z 2 z 2 f 0 E. Cáceres (UCol/UT-Austin) 7
Note that: I Causality bounds: (3) − 7 / 36 � 9 / 100 , I Central charges: c = ⇡ 2 L 3 a = ⇡ 2 L 3 (4) ( 1 − 2 � f 0 ) , ( 1 − 6 � f 0 ) l 3 l p p I Singularity at finite z for � < 0 , 1 2m ( v ) 1 / 4 ( − 1 / � + 4 ) 1 / 4 p z sing = E. Cáceres (UCol/UT-Austin) 8
HEE IN TIME DEPENDENT G AUSS -B ONNET S = S grav + S ext where the external source is unespecified ds 2 = − L 2 dv 2 + L 2 f ( z , v ) ✓ − 2 ◆ x 2 p f 0 dzdv + d ¯ , z 2 z 2 f 0 p f 0 = 1 where 2 � ( 1 − 1 − 4 � ) , � f ( z , v ) = 1 q 1 − 4 � ( 1 − m ( v ) z 4 ) 1 − 2 � I m ( v ) is arbitrary I S ext yields the following energy-momentum tensor µ ⌫ = 3 2z 3 dm ( 16 ⇡ G N ) T ext dv � µv � ⌫ v . E. Cáceres (UCol/UT-Austin) 9
Previous work focused on thermalizarion time ( Li, Wu and Yang 2013) I Apparent horizon z AH = m ( v ) 1 / 4 I Event horizon: z 0 1 EH ( v ) = − 2 p f 0 f ( z EH , v ) E. Cáceres (UCol/UT-Austin) 10
Covariant prescription Hubeny, Rangamani, Takayanagi 07 S A = Area extrm ( � A ) AdS d+1 G d + 1 d N ∂ A Codimension 2 surface A Homology condition � A ∼ A γ A bulk region r s . t . � r = � A [ A 9 E. Cáceres (UCol/UT-Austin) 11
Entanglement entropy in GB (Hung, Myers, Solkin, 2011) Z Z p d 3 ⇠ p � ( 1 + � L 2 R ⌃ ) + 1 1 d 2 ⇠ S EE = h � K 4G N 2G N ⌃ @⌃ I R ⌃ : Ricci scalar for intrinsic geometri on ⌃ I K : trace of extrinsic curvature on @⌃ E. Cáceres (UCol/UT-Austin) 12
I Study “rectangular strip" for the time-dependent case. I z ( x ) , v ( x ) I Induced metric on the co-dimension two surface is ds 2 = L 2 3 ) + L 2 ✓ ◆ 1 − f 2 v 0 2 − v 0 z 0 (5) z 2 ( dx 2 2 + dx 2 dx 2 , p f 0 z 2 f 0 E. Cáceres (UCol/UT-Austin) 13
Thus, p � = L 3 p 1 f 0 v 0 z 0 ⌘ 1 / 2 ⇣ f 0 − fv 0 2 − 2 (6) p f 0 , z 3 p z 0 2 � L 2 p � R ⌃ = ( 2L 3 � f 0 − fv 0 2 − 2 p f 0 v 0 z 0 � 1 / 2 + dF (7) f 0 ) dz , z 3 � where, p z 0 F ( x ) = ( 4L 3 � (8) f 0 ) f 0 − fv 0 2 − 2 p f 0 v 0 z 0 � 1 / 2 z 2 � E. Cáceres (UCol/UT-Austin) 14
Finally, action to be extremized is, Z dz L 3 p ⇣ f 0 v 0 z 0 ⌘ 1 / 2 f 0 − fv 0 2 − 2 S eff = p f 0 + z 3 4G N # 2 � f 0 z 0 2 f 0 − fv 0 2 − 2 p f 0 v 0 z 0 � 1 / 2 � E. Cáceres (UCol/UT-Austin) 15
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 � ? E. Cáceres (UCol/UT-Austin) 16
Results l= 0.05 l=- 0.05 1.0 1.0 t b = 0.8 t b = 0.8 t b = 1 t b = 1 t b = 1.2 t b = 1.2 0.5 0.5 Schw Schw AdS AdS S S 0.0 0.0 - 0.5 - 0.5 - 1.0 - 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 l l l=- 0.05 l= 0.05 1.5 1.5 1.0 1.0 r r 0.5 0.5 0.0 0.0 - 1.0 - 0.5 0.0 0.5 1.0 - 1.0 - 0.5 0.0 0.5 1.0 v v E. Cáceres (UCol/UT-Austin) 17
Figure : illustration of r min 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 E. Cáceres (UCol/UT-Austin) 18
HEE S HADOWS IN GB Global, static. Holographic Entanglement entropy Ryu, Tagayanagi 06 S A = Area min ( � A ) AdS d+1 G d + 1 d N ∂ A A Codimension 2 surface Homology condition � A ∼ A γ A bulk region r s . t . � r = � A [ A 9 E. Cáceres (UCol/UT-Austin) 19
Minimal surfaces in global BTZ 8 ⇣ ⌘ c 2 r ∞ # # X < 3 log r + sinh ( r + # / 2 ) , S A ( # ) = ⇣ ⌘ 2 r ∞ c 3 ⇡ r + + c # � # X : 3 log r + sinh ( r + ( 2 ⇡ − # ) / 2 ) , # X ( r + ) = 2 coth − 1 ( 2 coth ( ⇡ r + ) − 1 ) . r + E. Cáceres (UCol/UT-Austin) 20
E. Cáceres (UCol/UT-Austin) 21
S HADOWS 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 S 3.0 2.5 2.0 1.5 1.0 0.5 J 1 2 3 4 5 6 - 0.5 E. Cáceres (UCol/UT-Austin) 22
Entanglement Shadow Freivogel et. al 14.12.5175 � = r ⇤ − r h = 2r H e − ⇡ r H sinh ( ⇡ r H ) r ss 0.6 0.5 0.4 0.3 0.2 0.1 rh 0.5 1.0 1.5 2.0 2.5 3.0 for r H << ` AdS I � ∼ # r H + ..... H e − # r H + .... for r H >> ` AdS I � ∼ r 2 Similar limiting behaviour in AdS 5 . E. Cáceres (UCol/UT-Austin) 23
Entanglement Shadows in Gauss-Bonnett I Black hole in global AdS 5 , small � I Assume the 3-dimensional boundary region of interest is O ( 3 ) symmetric, r ( ✓ ) dt 2 + dr 2 ds 2 = − f ( r ) f ( r ) + r 2 ( d ✓ 2 + sin 2 ( ✓ ) d ⌦ 2 ) f ∞ f ( r ) = 1 + r 2 q 1 + 4 � (( rh 2 + rh 4 + � )) / r 4 − 4 � ]) 2 � ( 1 − where f ∞ is a convenient normalization factor, f ∞ = 1 − p 1 − 4 λ 2 λ E. Cáceres (UCol/UT-Austin) 24
HEE, prescription for higher derivatives. Action to minimize, s r 0 ( ✓ ) 2 r ( ✓ ) 2 sin ( ✓ ) 2 + 2 � f ( r ) + r ( ✓ ) 2 � � L = + 2 � r ( ✓ ) 2 cos ( ✓ ) 2 + sin ( 2 ✓ ) r ( ✓ ) r 0 ( ✓ ) + sin ( ✓ ) 2 r 0 ( ✓ ) 2 q r 0 ( ✓ ) 2 f ( r ) + r ( ✓ ) 2 Study shadows numerically –in progress. E. Cáceres (UCol/UT-Austin) 25
Following Frievogel et al 14125175, approximate solution near the horizon I expand eom close to the horizon I assume r 0 ( ✓ ) is small r 00 ( ✓ ) + 2 cot ( ✓ ) r 0 ( ✓ ) + r ( ✓ ) H ( rh , � ) + ˜ H ( rh , � ) Can be solved with r ( 0 ) = r ⇤ , r 0 ( 0 ) ∼ 0 . For small � , 1 k 3 rh 2 csc ( ✓ )( k 3 rh 3 sin ( ✓ ) + ( rh − r s )( 6k ( 1 + 2rh 2 ) ✓� cosh ( k ✓ ) r ( ✓ ) = − ( 6 � + rh 2 ( k 2 + 12 � )) sinh ( k ✓ )) E. Cáceres (UCol/UT-Austin) 26
p where k = 5 + rh 2 Shadow size: r ⇤ − r h ⌘ � , E. Cáceres (UCol/UT-Austin) 27
I Large black holes, similar behaviour as � = 0 � ∼ rh 2 e − # rh I Small black holes � ∼ # rh + � p ( rh ) where p ( rh ) > 0 ! For � < 0 shadow is smaller E. Cáceres (UCol/UT-Austin) 28
Conclusions: I In time dependent case, EE can explore arbitrarily close to the singularity. I In static global case, shadow size is smaller for � GB < 0 Theories with � GB < 0 "know more" of the bulk than � � 0 . I Bulk reconstruction. What CFT observables access regions in entanglement shadow? i.e what is the right probe? I How generic is the entanglement shadow region? I Nonlocality? E. Cáceres (UCol/UT-Austin) 29
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