CAUSAL WEDGES in AdS/CFT

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1 CUSL WEDGES in ds/cft Veronika Hubeny Durham University Gauge/Gravity Duality 2013 Max Planck Institute for Physics, 29 July 2013 to 2 ugust 2013 Based on: VH & M.Rangamani: , VH, M.Rangamani, E. Tonni: , VH, H.Maxfield, M.Rangamani, E. Tonni:

2 ds/cft correspondence String theory ( gravity) gauge theory (CFT) in bulk asymp. ds K on boundary Invaluable tool to: Use gravity on ds to learn about strongly coupled field theory (as successfully implemented in e.g. ds/qcd & ds/cmt programs) Use the gauge theory to define & study quantum gravity in ds Pre-requisite: Understand the ds/cft dictionary... esp. how does spacetime (gravity) emerge?

3 ds/cft correspondence String theory ( gravity) gauge theory (CFT) in bulk asymp. ds K on boundary Invaluable tool to: Use gravity on ds to learn about strongly coupled field theory (as successfully implemented in e.g. ds/qcd & ds/cmt programs) Use the gauge theory to define & study quantum gravity in ds Pre-requisite: Understand the ds/cft dictionary... esp. how does spacetime (gravity) emerge? One pproach: Consider natural (geometrical) bulk constructs & try to identify their field theory duals. (We can then use these CFT `observables to reconstruct part of bulk geometry.)

4 Natural bulk constructs Specify a spatial region on the ds boundary. (Natural to assume that CFT observer has access to physics in, e.g. the reduced density matrix ) ρ boundary bulk previously considered in different contexts by e.g. [Marolf, Bousso et.al., Kabat et.al.,...]

5 Natural bulk constructs Specify a spatial region 1) Extremal surface E (bulk co-dimension 2, anchored on ) Entanglement entropy of region is S = Tr ρ log ρ on the ds boundary. (Natural to assume that CFT observer has access to physics in, e.g. the reduced density matrix ) ρ Proposal of [Ryu & Takayanagi], covariantized by [VH, Rangamani, Takayanagi] : S In the bulk is captured by area of extremal co-dimension 2 bulk surface E (anchored on ): S = rea(e ) 4 G N bulk boundary E previously considered in different contexts by e.g. [Marolf, Bousso et.al., Kabat et.al.,...]

6 Natural bulk constructs Specify a spatial region 1) Extremal surface E (bulk co-dimension 2, anchored on ) Entanglement entropy of region is S = Tr ρ log ρ on the ds boundary. (Natural to assume that CFT observer has access to physics in, e.g. the reduced density matrix ) ρ Proposal of [Ryu & Takayanagi], covariantized by [VH, Rangamani, Takayanagi] : S In the bulk is captured by area of extremal co-dimension 2 bulk surface E (anchored on ): S = rea(e ) 4 G N bulk boundary E 2) Causal wedge (= even more basic construct, since only depends on conformal structure) previously considered in different contexts by e.g. [Marolf, Bousso et.al., Kabat et.al.,...]

7 Causal Wedge construction Consider a bdy region bulk boundary t z x sketch for planar ds: ds 2 = dt2 + dx 2 + dz 2 z 2 (boundary at z =0)

8 Causal Wedge construction Consider a bdy region t Construct the bdy domain of dependence of, denoted (observables in the entire region can be determined solely from the initial conditions specified on ) z x

9 Causal Wedge construction Bulk causal wedge t J [ ] J + [ ] = { bulk causal curves which begin and end on } Causal information surface Ξ z Ξ x Ξ + ( ) ( ) Causal holographic information χ χ rea(ξ ) 4 G N

10 Causal Wedge construction Bulk causal wedge t J [ ] J + [ ] = { bulk causal curves which begin and end on } Causal information surface Ξ z Ξ x Ξ + ( ) ( ) Causal holographic information χ χ rea(ξ ) 4 G N What is their interpretation within the dual CFT? To gather hints, study their properties.

11 Properties of causal wedge: Quasi-teleological (on light-crossing timescale) i.e. location of is sensitive to bulk geometry even in future of

12 Toy model for dynamics: Vaidya-dS spacetime, describing a null shell in ds: ds 2 = f(r, v) dv 2 +2dv dr + r 2 dω 2 where f(r, v) =r 2 +1 ϑ(v) m(r) r+ 2 +1, in ds 3 with m(r) = r 2 + r (r ), in ds 5 0, for v<0 pure ds and ϑ(v) = 1, for v 0 Schw-dS (or BTZ) we can think of this as δ 0 limit of smooth shell with thickness δ : ϑ(v) = 1 tanh v 2 δ +1

13 Causal wedge profile in Vaidya: There are two separate regimes when intersects the shell: For t < tshell, E is undeformed but Ξ is deformed For t > tshell, both E and Ξ are deformed

14 Causal nature of χ : Hence naively, since is quasi-teleological while is causal, one might have expected that correspondingly quasi-teleological (while i.e. we d expect χ Ξ χ would likewise be S E is causal). to start increasing before t = t shell 0 However this does not happen! In thin shell Vaidya-dS, fully causal. This is due to special property of causal wedges in ds: Θ =0 χ is Hence difference χ S is only non-zero for 0 <t < ϕ χ S t χ In fact finding is easier than finding in this regime! S

15 Properties of causal wedge: Quasi-teleological (on light-crossing timescale) Curiously, in thin shell Vaidya-dS, χ remains causal. Topologically non-trivial (Even for simply-connected, can have holes)

16 Causal wedge can have holes In 5 dim ST: Projection of Ξ to Poincare disk for varying size of : BH Ξ Ξ

17 Causal wedge can have holes Even if is simply connected region, the causal wedge can be topologically complicated. e.g. in Schw-dSd with d>3, for sufficiently large region (and fixed BH size), the causal wedge `wraps around the BH. conversely, for fixed region > hemisphere, small enough BH s.t. the causal wedge has a hole Ξ transitions from single to two disconnected pieces. Ξ Ξ Size of 3 Π 4 Π d=4, 5, 6,... Π 2 Π 4 0 Π 4 Size of BH Π 2

18 Remarks on holes We do not need to be large in order for to have holes (e.g. for sufficiently boosted BH, can be arbitrarily small.) For multiple black holes in ds, can have multiple holes. Don t need causally non-trivial geometry (e.g. for sufficiently compact star, is the same as for a BH.) For small deformations of ds, For 3-d bulk (BTZ) black hole, does not have holes. does not have holes. In general, must be simply connected [Ribeiro].

19 No holes in 3 dimesions BTZ black hole is never effectively small due to low dim. BH Ξ

20 Properties of causal wedge: Quasi-teleological (on light-crossing timescale) Curiously, in thin shell Vaidya-dS, χ remains causal. Topologically non-trivial (Even for simply-connected, can have holes in >3 ST dims) Expels extremal surfaces ( penetrates deeper into the bulk than ) E Ξ

21 CW expels extremal surfaces: Ξ In general does not penetrate as far into the bulk as the bulk extremal surface associated with E Justification for Ξ and E lying on a single spacelike surface -- based on expansion of null generators: By construction, Θ Ξ 0 while Θ E =0 E Proof by contradiction: suppose lay closer to bdy than Ξ. Then tangent to E, there is a surface Ξà for some smaller region Ã. But for such configuration, Θ Ξà < 0, which is a contradiction. bdy à bdy Θ α < Θ β S α Ξà S β p E p Ξ

22 CW expels extremal surfaces: Ξ In general does not penetrate as far into the bulk as the bulk extremal surface associated with E Justification for Ξ and E lying on a single spacelike surface -- based on expansion of null generators: By construction, Θ Ξ 0 while Θ E =0 E Proof by contradiction: suppose lay closer to bdy than Ξ. Then tangent to E, there is a surface Ξà for some smaller region Ã. But for such configuration, Θ Ξà < 0, which is a contradiction. bdy à bdy Θ α < Θ β S α Ξà S β p E p Ξ Holds in full generality assuming can be foliated by α (true for static ST)

23 Implication for entanglement entropy Important implication: whenever is large enough for to have two disconnected pieces, there cannot exist a single connected extremal (minimal) surface homologous to! However, the homology constraint is required for EE, i.e. part of must reach around the BH. E So must likewise have two disconnected pieces, one on the horizon and one homologous to c (=complement of ) Hence we have the universal formula for the entanglement entropy, whenever is large enough: S = S + S c BH utomatically saturates the raki-lieb inequality = entanglement plateau [VH, Maxfield, Rangamani, Tonni] So we can extract BH (thermal) entropy from entanglement entropy [cf. zeyanagi, Nishioka, Takayanagi] E E Ξ

24 Summary: In general bulk extremal surface does not penetrate as far into the bulk as the associated with The causal wedge construction is mildly teleological (but only on light-crossing timescale) Causal Holographic Information (and coincides in special cases) However in general bounds EE can violate strong subadditivity -- hence it cannot be a von Neumann entropy... Ξ Ξ E While cannot penetrate black hole event horizons, can penetrate dynamical black hole horizons (by a limited amount). Even if is simply connected region, the causal wedge can be topologically complicated. Conjecture: χ χ E S can admit holes if the geometry admits a null circular orbit.

25 Future directions Most important questions still remain: What is the direct boundary interpretation/construction of Ξ χ the causal holographic surface and information? What is the bulk dual of the reduced density matrix? Given a bulk location, how do we extract the geometry there from the CFT? How does the CFT encode bulk locality and causality? ρ