CFT PROBES OF BULK GEOMETRY

Transcription

1 CFT PROBES OF BULK GEOMETRY Veronika Hubeny Durham University May 24, 2012 Based on: VH: VH & M.Rangamani:

2 OUTLINE Motivation & Background Features of Extremal Surfaces Probing Horizons Causal Holographic Information Summary & Future directions

3 AdS/CFT correspondence String theory ( gravity) gauge theory (CFT) in bulk asymp. AdS K on boundary Key aspects: Gravitational theory maps to non-gravitational one! Holographic: gauge theory lives in fewer dimensions. Strong/weak coupling duality. Invaluable tool to: Use gravity on AdS to learn about strongly coupled field theory Use the gauge theory to define & study quantum gravity in AdS

4 AdS/CFT correspondence String theory ( gravity) gauge theory (CFT) in bulk asymp. AdS K on boundary Key aspects: Gravitational theory maps to non-gravitational one! Holographic: gauge theory lives in fewer dimensions. Strong/weak coupling duality. Invaluable tool to: Use gravity on AdS to learn about strongly coupled field theory Use the gauge theory to define & study quantum gravity in AdS Pre-requisite: Understand the AdS/CFT dictionary...

5 Motivation To understand the AdS/CFT dictionary; esp. how does spacetime (gravity) emerge? Most QG questions rest on bulk locality (& its breakdown)... Given a specific bulk location, what quantities in the CFT should we examine in order to learn about the physics at that location? How deep into the bulk can various CFT probes see? esp.: can convenient CFT probes see into a black hole? Given full knowledge of physics ( region A, A ) in a certain boundary in what region of the bulk does it determine the bulk geometry? in what region of the bulk is it sensitive to the bulk geometry?

6 Probes of bulk geometry The bulk metric can be extracted using various CFT probes (which are described by geometrical quantities in the bulk): Examples: CFT probe expectation values of local gauge-invariant operators correlation functions of local gauge-invariant operators Wilson loop exp. vals. entanglement entropy bulk quantity asymptotic fall-off of corresponding conjugate field in WKB approx., proper length of corresponding geodesic area of string worldsheet vol of extremal co-dim.2 surface

7 Correlators see inside horizons Consider correlators of high-dimension operators in CFT 3-dim BH: r = 0 2 asymptotic regions 2 CFTs r = 8 r = 8 correlators of high-dim operators: CFT 2 CFT 1 x 2 x 1 r = 0 h (x 1 ) (x 2 )i e m L 12 where L 12 = regularised proper distance along spacelike geodesic between x1 and x2. Correlators between operators in CFT1 and CFT2 give access to full spacetime. However, in 3-dim, these are insensitive to the BH singularity. [Kraus, Ooguri, Shenker]

8 Probing black hole singularity Signatures of bulk black hole singularity are encoded in analytically continued correlators of high-dimension operators in CFT 5-dim BH: [Fidkowski, VH, Kleban, Shenker] spacelike geodesics repelled by singularity t=0 h i(t) 1 (t t c ) 2m divergence in CFT correlation fn. as t! t c t c subtlety: requires analytic continuation Hence CFT correlators are sensitive to geometry deep in bulk, even when causally disconnected from the boundary.

9 Bulk-cone singularities More direct CFT probes (not reliant on analytic continuation) = bulk-cone singularities: [VH, Liu, Rangamani] Green s functions on curved bulk spacetime are singular at nullseparated points Boundary correlation functions h (x) (y)i inherit these singularities Hence h (x) (y)i!1 when x and y are null-separated (either along boundary or through the bulk) The set of bulk-cone singularities in the CFT directly give the endpoints of bulk null geodesics. One can use this information to learn about the bulk geometry

10 Bulk-cone singularities Consider asymp.(global) AdS bulk, and study projection of null geodesics in (r, t) and (r, ') geodesic endpoints clearly distinguish between different bulk geometries: ' t r Null geodesics in AdS: cf. Null geodesics in AdS star geometry: const. t (r, t) plane const. t (r, t) plane

11 Holographic entanglement entropy Proposal [Ryu & Takayanagi] for static configurations: Entanglement entropy of region is S A = Tr A log A A In the bulk this is captured by area of minimal co-dimension 2 bulk surface S anchored bulk boundary A S In time-dependent situations, prescription must be covariantised: minimal surface extremal surface equivalently, S is the surface with zero null expansions; cf. light sheet construction [Bousso] [VH, Rangamani, Takayanagi]

12 Holographic entanglement entropy Entanglement entropy growth during thermalisation: Bulk geometry = collapsing black hole (in 3-d): behaviour of extremal surfaces at times v0 during collapse corresponding entanglement entropy: v0 2 v0 1 v0 0 L v v0 0.1 v0 0.5 v [VH, Rangamani, Takayanagi]

13 OUTLINE Motivation & Background Features of Extremal Surfaces Probing Horizons Causal Holographic Information Summary & Future directions

14 Probe geodesics For simplicity, focus on static, spher. sym., asymp.(global)ads ds 2 = f(r) dt 2 + h(r) dr 2 + r 2 d 2 Probe geodesics = bulk geodesics with both endpoints anchored on the (same) AdS bdy Can only be spacelike or null (timelike geods don t reach bdy) Consider how deep into the bulk these can probe (= rmin) and the regularized proper length (= Lreg ), for a given ' angular ( ) and temporal ( ) separation of the endpoints: What part of the bulk is accessible to probe geodesics? Which are optimal geodesics for probing the bulk? t

15 Probe geodesics geodesics w/ energy E and ang.mom. L have radial potential ṙ 2 + V e (r) =0, V e (r) = 1 h(r) for turning point rmin, the endpoints are separated by t 2 where Z 1 r min E f(r) g(r) s g(r) dr and ' 2 h(r) apple + E2 f(r) = L 2 r 2 apple apple Z 1 r min 1 p Ve (r) = 1 ṙ(r) E 2 f(r) + L2 r 2 L g(r) dr, r2 with proper length L R =2 Z R r min g(r) dr

16 Probe geodesics in AdS Distinct E in distinct panes, denoted by color-coding Distinct L are distinct curves in each pane const. t ' (r, t) plane t

17 Results for probe geodesics In causally trivial spacetime, all of the bulk is accessible to spacelike as well as null geodesics In presence of a black hole, probe geodesics cannot penetrate the horizon (cf. part II) Spacelike geodesics probe deeper than null ones for fixed parameters E & L The `optimal geodesics for probing bulk are the E=0 spacelike ones, as these minimize rmin at fixed (However, they have larger Lreg) '

18 Extremal surfaces Simplified context: Focus on extremal surfaces S anchored on bdy n-dim region R in static planar asymp. (Poincare) AdSd+1 Parameters we can dial: bulk geometry (specified by 2 fns of 1 variable): ds 2 = 1 z 2 g(z) dt 2 + k(z) dx i dx i + dz 2 shape (1 fn of d variables) & extent X (1 real number) of bdy region R dimensionality n (1 integer) of the surface S Key feature of S: Bulk depth reached z (Note that z is geometrically well-defined.)

19 Results for extremal surfaces Preview: Higher-dimensional surfaces probe deeper i.e. z increases with n for fixed extent X(R) Surfaces anchored on R = ball reach deepest compared to differently-shaped R with same extent or area Surfaces in pure AdS reach deeper for fixed R, compared asymp.ads geometry

20 Results for extremal surfaces Higher-dimensional surfaces probe deeper: Consider R = n-dim l strip on bdy of pure AdS -z * z * x z * z n=5 n=2 cross-sections of S n=1 z * êdx n We should compare z for fixed extent X(R): For R = n-ball, z = X/2 For R = n-ball in deformed AdS, at fixed, X decreases with n at fixed X, z z increases with n n (i.e. all S hemispherical) in pure AdS z r z +

21 Results for extremal surfaces Consider R with fixed dimensionality n and `area A(R) What shape of R maximizes z, i.e. when does S reach deepest? bdy R bulk S Fig. 9: Various extremal surfaces, anchored on a boundary region R (bounded by the thick red curve).

22 Results for extremal surfaces Surfaces anchored on R=ball reach deepest: Linearize around the hemisphere To 2nd order, At fixed ds 2 = 1 2 cos 2 ( =0, )= 0, the area A(R) = 2 0 increases as R is perturbed from ball 2 (, )= 0 4 dt 2 + d d 2 +sin 2 d 2 + in pure AdS: Hence R=round ball S has greatest reach for fixed A(R) Confirmed numerically at non-linear level as well. Xd 1 j=3 dỹ 2 j (, )= (, )+ 2 2 (, )+O( 3 ) 1 (, ) = tan`( /2) (1 + ` cos ) cos `. 2 (, )= tan 2`( /2) {(1 + ` cos ) 2 + µ (1 + 2 ` cos )+`2 cos 2 cos 2` } O( 4 )

23 OUTLINE Motivation & Background Features of Extremal Surfaces Probing Horizons Causal Holographic Information Summary & Future directions

24 Can extremal surfaces probe horizon? Preview: Causal probes (e.g. null geods) can never penetrate event horizon No such causal obstruction for spacelike curves/surfaces However, for static spacetime, extremal surfaces cannot penetrate event horizon But since event horizon is defined teleologically, extremal surfaces can penetrate event horizon for sufficient time-dep.

25 Probe geods can t penetrate horizons in static bulk Consider a geod. crossing the horizon; what can happen? HaL HbL HcL HdL HeL? (b) & (c) are allowed, but don t correspond to probe geods (d) is disallowed by energy conservation: Ingoing coords: ds 2 = f(r) dv 2 +2 p f(r) h(r) dv dr + r 2 d 2 p conserved energy along geods: E = f(r) v f(r) h(r)ṙ (e) is disallowed by assumption of reaching horizon from bdy

26 Probe geods can t penetrate horizons in static bulk Eg: BTZ Schw-AdS5 RN-AdS5 spacelike geodesics can penetrate arb. close to horizon however as, and limiting ' =2, for r + =0.5 r min! r + '!1 L reg!1 r min r + r + = 8 < : for BTZ for SAdS for RNAdS 5 null geodesics can ever reach a finite distance from horizon

27 Extremal surfaces can t penetrate horizons in static bulk Consider extent X(R) for n-strip in Schw-AdS5 at fixed z bdy z x n=2 n=1 cross-sections of S z n=4 bulk horizon z + We see extremal surfaces are repelled by the horizon

28 Extremal surfaces can t penetrate horizons in static bulk Cf. extent X(R) for n-strip in extremal RN-AdS5 at fixed z bdy z é x cross-sections of S in standard coords z n=2 bulk horizon x n=1 n=4 cross-sections of S in FG coords z Again, extremal surfaces are repelled by the horizon

29 Extremal surfaces can t penetrate horizons in static bulk Completely general proof, for any n, R, & bulk geom: Consider extremal surface S param. by in bulk spacetime At horizon, Lagrangian: EOM: ds 2 = 1 z 2 f! 0 h! ±1 and Around the turning point, z = z : z(x 1,...,x n ) f(z) dt 2 + dx i dx i + h(z) dz 2 L(z,z,1,...,z,n ; x 1,...,x n )= p G = X i z,ii X j z 2,j 1 A h(z) X i,j z,ij z,i z,j + X i q 1+h(z) z 2, z2,n z n n z + h0 (z) + n 2 h(z) zh(z) =0 In order for z to be maximum, z,ii (z ) < 0, which forces z 2,i X z,ii + i n zh(z) =0 h(z ) > 0 Hence turning point z must be outside the horizon.

30 Extremal surfaces can penetrate horizons in time-evolving bulk Gedanken-experiment to demonstrate that causality does not pose a fundamental obstacle to extracting information via CFT: [VH] uses the teleological nature of event horizon & non-local nature of AdS/CFT: Measure bulk event by spacelike CFT probe (precursor), e.g. geodesic g Afterwards, collapse a shell s, such that the resulting event horizon H encompasses the measured event. Then g is a probe geodesic which penetrates the event horizon seen explicitly for geods in Vaidya-AdS

31 OUTLINE Motivation & Background Features of Extremal Surfaces Probing Horizons Causal Holographic Information Summary & Future directions

32 Bulk dual to a bdy region? A What is the most natural bulk region associated to a given region A on the bdy? natural : try to be minimalistic, use only bulk causality Take A to be d-1 dimensional spatial region on bdy of asymp. AdSd+1 bulk spacetime. The unique minimal construction gives a bulk causal wedge associated with A, and a corresponding d-1 dimensional bulk surface A Using geometrical information, we can associate a number to A, corresponding to area of A What is the CFT interpretation of and A? A A

33 Causal construction domain of dependence or be influenced by events in domain of influence I ± [A] be influenced by events in D ± [A] A = region which must influence = region which can influence or A J + [A] D + [A] D [A] A A A D + [A] [ D [A] J [A] Given A, we can determine observables in the entire A

34 Causal construction Bulk causal wedge A J + [A] t A J [ A ] \ J + [ A ] Causal information surface A + ( A ) ( A ) z A * A A x Causal holographic information A Area( A) 4 G N A J [A]

35 General properties of : A Causal information surface bulk surface which: is anchored lies within (on boundary of) is a d-1 dimensional spacelike reaches deepest into the bulk from among surfaces in is a minimal-area surface among surfaces A ) in general does not penetrate as far into the bulk as the bulk extremal surface E A A associated with A A A c A A However, in all cases where one is able to compute entanglement entropy in QFT from first principles independently of coupling, the surfaces and agree! E A A A c E A A

36 Cases when and coincide: A E A When EE can be related to thermal entropy cf. [Myers et.al.] Case CFT density matrix Holographic duals (a). CFT vacuum global AdS 3 (3.13) (b). Thermal density matrix static BTZ geometry (3.14) (c). Grand canonical density matrix rotating BTZ spacetime (3.15) (a) (b) (c)

37 General properties of A : The Causal Holographic Information in special* cases, coincides with Entanglement entropy A Area( A) S 4 G A Tr ( A log A )= Area(E A) = N 4 G N in general diverges more strongly than entanglement entropy hence provides a bound on entanglement entropy unlike entanglement entropy, does NOT satisfy strong subadditivity A S A1 + S A2 S A1 [A 2 + S A1 \A 2 S A1 + S A2 S A1 \A 2 + S A2 \A 1 S A Geometric proof in static bulk and support in time-dep bulk [Headrick et.al.] But counter-examples for A.

38 Conjectured meaning of A : We conjecture that A characterizes the amount of information contained in A which can be used to reconstruct the bulk geometry (entirely in A but possibly further)... What bulk region can we fully reconstruct? (& What is the most efficient reconstruction method?) What is the bulk dual of the reduced density matrix A? Cf. several recent discussions: [Bousso, Leichenauer, Rosenhaus] and [Czech, Karczmarek, Nogueira, Van Raamsdonk]

39 OUTLINE Motivation & Background Features of Extremal Surfaces Probing Horizons Causal Holographic Information Summary & Future directions

40 Summary Spacelike geodesics reach deeper than null geodesics (at fixed spatial separation of endpoints). Higher-dimensional extremal surfaces reach deeper (at fixed extent of bounding region). Extremal surfaces anchored on sphere reach deepest (at fixed extent or volume of bounding region). Extremal surfaces of any dimension, anchored on any region, in any static planar black hole spacetime, cannot penetrate the horizon. Extremal surfaces can penetrate horizon of dynamically evolving black hole. Natural bulk region corresponding to a bdy region A = causal wedge Associated causal holographic information... z + z r

41 Future directions Most important questions still remain: What characterizes the bulk region accessible to extremal surfaces in general time-dependent geometry? How close to curvature singularity (e.g. in collapse situation) can various probes reach? Given a bulk location, how do we extract the geometry there from the CFT? Given the CFT reduced density matrix in a certain boundary region, what part of the bulk geometry can we fully reconstruct (and how)?