Positive gravitational subsystem energies from CFT cone relative entropies

Journal of High Energy Physics, Jun 2018

Abstract The positivity of relative entropy for spatial subsystems in a holographic CFT implies the positivity of certain quantities in the dual gravitational theory. In this note, we consider CFT subsystems whose boundaries lie on the lightcone of a point p. We show that the positive gravitational quantity which corresponds to the relative entropy for such a subsystem A is a novel notion of energy associated with a gravitational subsystem bounded by the minimal area extremal surface à associated with A and by the AdS boundary region  corresponding to the part of the lightcone from p bounded by ∂A. This generalizes the results of arXiv:1605.01075 for ball-shaped regions by making use of the recent results in arXiv:1703.10656 for the vacuum modular Hamiltonian of regions bounded on lightcones. As part of our analysis, we give an analytic expression for the extremal surface in pure AdS associated with any such region A. We note that its form immediately implies the Markov property of the CFT vacuum (saturation of strong subadditivity) for regions bounded on the same lightcone. This gives a holographic proof of the result proven for general CFTs in arXiv:1703.10656. A similar holographic proof shows the Markov property for regions bounded on a lightsheet for non-conformal holographic theories defined by relevant perturbations of a CFT.

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Positive gravitational subsystem energies from CFT cone relative entropies

Accepted: May Positive gravitational subsystem energies from CFT cone relative entropies Dominik Neuenfeld 0 1 2 3 4 Krishan Saraswat 0 1 2 3 4 Mark Van Raamsdonk 0 1 2 3 4 0 tum Gravity , Conformal Field Theory 1 6224 Agricultural Road, Vancouver , B.C. V6T 1W9 , Canada 2 Department of Physics and Astronomy, University of British Columbia 3 The surface @A is then described by 4 is constant. On the light cone , only the @ The positivity of relative entropy for spatial subsystems in a holographic CFT implies the positivity of certain quantities in the dual gravitational theory. In this note, we consider CFT subsystems whose boundaries lie on the lightcone of a point p. We show that the positive gravitational quantity which corresponds to the relative entropy for such a subsystem A is a novel notion of energy associated with a gravitational subsystem bounded by the minimal area extremal surface A~ associated with A and by the AdS boundary region A^ corresponding to the part of the lightcone from p bounded by @A. This generalizes the results of arXiv:1605.01075 for ball-shaped regions by making use of the recent results in arXiv:1703.10656 for the vacuum modular Hamiltonian of regions bounded on lightcones. As part of our analysis, we give an analytic expression for the extremal surface in pure AdS associated with any such region A. We note that its form immediately implies the Markov property of the CFT vacuum (saturation of strong subadditivity) for regions bounded on the same lightcone. This gives a holographic proof of the result proven for general CFTs in arXiv:1703.10656. A similar holographic proof shows the Markov property for regions bounded on a lightsheet for non-conformal holographic theories de ned by relevant perturbations of a CFT. AdS-CFT Correspondence; Gauge-gravity correspondence; Models of Quan- - 1 Introduction 2 Background 2.1 2.2 4.1 4.2 4.3 4.4 lightcone Relative entropy in conformal eld theories Gravity background Light cone coordinates for AdS HRRT surface in pure AdS The bulk vector eld Perturbative formulae for H 4.4.1 4.4.2 Vanishing of the rst order expression Relative entropy at second order 3 Bulk interpretation of relative entropy for general regions bounded on a 4 Perturbative expansion of the holographic dual to relative entropy 5 Holographic proof of the Markov property of the vacuum state 5.1 The Markov property for states on the null-plane 5.2 The Markov property for states on the lightcone 6 Discussion A Equivalence of H on the boundary and the modular Hamiltonian B The HRRT surface ending on the null-plane C Calculation of the binormal D Hollands-Wald gauge condition 1 Introduction Via the AdS/CFT correspondence, it is believed that any consistent quantum theory of gravity de ned for asymptotically AdS spacetimes with some xed boundary geometry B corresponds to a dual conformal eld theory de ned on B . Recently, it has been understood that many natural quantum information theoretic quantities in the CFT correspond to natural gravitational observables (see, for example [1], or [ 2, 3 ] for a review). Through this correspondence, properties which hold true for the quantum information theoretic { 1 { quantities can be translated to statements about gravitational physics. In this way, we can obtain a alternative/deeper understanding of some known properties gravitational systems, but also discover novel properties that must hold in consistent theories of gravity. A particularly interesting quantum information theoretic quantity to consider is relative entropy [4]. For a general state j i of the CFT, we can associate a reduced density matrix A to a spatial region A by tracing out the degrees of freedom outside of A. Relative entropy S( Ajj 0A), which we review in section 2, quanti es how di erent this state is from the vacuum density matrix 0A reduced on the same region. Relative entropy is typically UV- nite, always positive, and has the property that it increases as we increase the size of the region A (known as the monotonicity property). According to the AdS/CFT correspondence, this should correspond to some quantity in the gravitational theory which also obeys these positivity and monotonicity properties. This has previously been explored in [5{11]. As we review in section 2, by making use of the holographic formula relating CFT entanglement entropies to bulk extremal surface areas (the \HRRT formula" [1, 12]), it is possible to explicitly write down the gravitational quantity corresponding to relative entropy as long as the vacuum modular Hamiltonian (HA0 = log 0A) for the region A is \local", that is, it can be written as a linear combination of local operators in the CFT. Until recently, such a local form was only known for the modular Hamiltonian of ballshaped regions [13]. For these regions, relative entropy has been shown to correspond to an energy that can be associated with the bulk entanglement wedge corresponding to this ball [8, 10].1 The positivity of relative entropy then implies an in nite family of positive energy constraints (reviewed below) [11]. Ball-shaped regions (of Minkowski space) have the property that their boundary lies on the past lightcone of a point p and the future lightcone of some other point q. In the recent work [14], it has been shown that the vacuum modular Hamiltonian for a region A has a local expression so long as the boundary @A of A lies on the past lightcone of a point p or the future lightcone of a point q.2 Thus, we have a much more general class of regions for which the relative entropy and its properties can be interpreted gravitationally. The main goal of the present paper is to explain this interpretation. In the general case, we denote by A^ the region of the lightcone bounded by @A, as shown in gure 1. The modular Hamiltonian can then be written as HA0 = Z ^ A(x)T (x) ; A (1.1) where T is the CFT stress-energy tensor, is a volume form de ned in section 2, and A(x) is a vector eld on A^ directed towards the tip of the cone and vanishing at the tip To describe the gravitational interpretation of the relative entropy for region A, we consider any codimension one spacelike surface in the dual geometry such that inter1The entanglement wedge is a region de ned by the union of spacelike surfaces with one boundary on the HRRT surface from the ball and the other boundary on the domain of dependence of the ball at the AdS boundary. 2The existence of such a region depends on the relativistic nature of the theory under consideration, which guarantees the existence of a codimension-0 domain of dependence. { 2 { A ^ A sects the AdS boundary at A^ and is bounded in the bulk by the HRRT surface A~ (the minimal area extremal surface homologous to A). This is illustrated in gure 2. Next, we de ne a timelike vector eld in a neighborhood of with the properties that approaches A at the AdS boundary and behaves near the extremal surface A~ like a Killing vector associated with the local Rindler horizon at A~. The timelike vector eld a particular choice of time on the surface and we can de ne an energy H represents associated with this. While generally there are many choices for the surface and the vector eld , we can show that all of them lead to the same value for the energy H . It is this quantity that corresponds to the CFT relative entropy S( Ajj 0A).3 The independence of H on the surface used to de ne it can be understood as a bulk conservation law for this notion of energy. In the case of a ball-shaped region [11], 3In this paper, we focus on the leading contribution to the CFT relative entropy at large N and make can be de ned on the full entanglement wedge for A, i.e. the union of spacelike surfaces ending on A~ and on any A0 in DA, so we can think of the energy H as being associated with the entire entanglement wedge. In the more general case considered in this paper, the collection of allowed surfaces generally still de ne a codimension zero region WA of the bulk spacetime (equivalent to the bulk domain of dependence of any particular ), but this region intersects the boundary only on the lightlike surface A^ rather than the whole domain of dependence region DA. In section 4, we consider the limit where the geometry is a small deformation away from pure AdS. For pure AdS, we show that the extremal surface A~ associated with a region A whose boundary lies on the lightcone of p always lies on the bulk lightcone of p. Thus, in a limit where perturbations to AdS become small, the wedge WA collapses to the portion A^bulk of this lightcone between p and A~. We present an analytic expression for the extremal surface A~ and a canonical choice for the vector eld on A^bulk. In terms of these, we can write an explicit expression for the leading perturbative contribution to the energy H , which takes the form of an integral over A^bulk quadratic in the bulk eld perturbations. In section 5, we point out that the explicit form of the extremal surface A~ in the pure AdS case (in particular, the fact that it lies on the bulk lightcone) leads immediately to a holographic proof of the Markov property for subregions of a CFT in its vacuum state, namely that for two regions A and B the strong subadditivity inequality S(A) + S(B) S(A \ B) S(A [ B) 0; (1.2) is saturated if their boundaries lie on the past or future lightcone of the same point p. This was shown for general CFTs in [14], so it had to hold in this holographic case. The holographic proof extends easily to cases where the eld theory is Lorentz-invariant but non-conformal, for example a CFT deformed by a relevant perturbation. In this case, the statement holds for subregions A, B whose boundaries lie on a null-plane. We conclude in section 6 with a discussion of some possible future directions. 2 2.1 Background Relative entropy in conformal eld theories For a general quantum system or subsystem described by a density matrix , the relative entropy quanti es the di erence between and a reference state . It is de ned as S( jj ) = tr( log ) tr( log ) ; which can be shown to be nonnegative as well as vanishing if and only if = . Relative entropy also obeys a monotonicity property: when A is a subsystem of the original system, the relative entropy satis es S( Ajj A) S( jj ) ; { 4 { (2.1) (2.2) where A and A are the reduced density matrices for the subsystem de ned from and respectively. entropy as [4] Using the de nition S( ) = tr( log ) of entanglement entropy and H = log( ) of the modular Hamiltonian associated with , we can rewrite the expression for relative S( k ) = hH i S where indicates a quantity calculated in the state minus the same quantity calculated in the reference state . For a conformal eld theory in the vacuum state, the modular Hamiltonian of a ballshaped region takes a simple form [13]. For a ball B of radius R centered at x0 in the spatial slice perpendicular to the unit timelike vector u , the modular Hamiltonian is ^ dx d 1 is a volume form and B is the conformal where Killing vector = (d 11)! B = R [R2 (x x0)2]u + [2u (x x0) ](x x0) ; with some four-velocity u . The modular Hamiltonian is the same for any surface B0 with the same domain of dependence as B. Using the expression (2.4) in (2.3), the relative entropy for a state compared with the vacuum state may be expressed entirely in terms of the entanglement entropy and the stress tensor expectation value. For a holographic theory in a state with a classical gravity dual, these quantities can be translated into gravitational language using the HRRT formula (which also implies the usual holographic relation between the CFT stress-energy tensor expectation value and the asymptotic bulk metric [15]). Thus, the CFT relative entropy for a ball-shaped region corresponds to some geometrical quantity in the gravitational theory with positivity and monotonicity properties. In [10] and [11], this quantity was shown to have the interpretation of an energy associated with the gravitational subsystem associated with the interior of the entanglement wedge associated with the ball. Recently, Casini, Teste, and Torroba have provided an explicit expression for the vacuum modular Hamiltonian of any spatial region A whose boundary lies on the lightcone of a point [14]. To describe this, consider the case where @A lies on the past lightcone of a point p and let A^ be the region on the lightcone that forms the future boundary of the domain of dependence of A. For x 2 A^, de ne a function f (x) that represents what fraction of the way x is along the lightlike geodesic from p through x to @A (so that f (p) = 0 and f (x) = 1 for x 2 @A). Now, de ne a lightlike vector eld on A^ by (2.3) (2.4) (2.5) (2.6) (2.7) In general, we cannot extend the vector eld away from the surface A^ such that the expression (2.7) remains valid when integrated over an arbitrary surface A0 with @A0 = @A. In equation (4.3) we give an explicit expression for HA in a convenient coordinate frame. Using this expression in (2.3), we can express the relative entropy for the region A in a form that can be translated to a geometrical quantity using the HRRT formula. We would again like to understand the gravitational interpretation for this positive quantity. 2.2 Gravity background We now focus on states in a holographic CFT dual to some asymptotically AdS spacetime with a good classical description. For any spatial subsystem A of the CFT, there is a corresponding region on the boundary of the dual spacetime (which we will also call A). The HRRT formula asserts that the CFT entanglement entropy for the spatial subsystem A in a state j i, at leading order in the 1=N expansion, is equal to 1=(4GN ) times the area of the minimal area extremal surface A~ in the dual spacetime which is homologous to the region A on the boundary. For pure AdS, when the CFT region is a ball B, the spatial region between B and B~ forms a natural \subsystem" of the gravitational system, in that there exists a timelike Killing vector B de ned on the domain of dependence D of and vanishing on B~. At the boundary of AdS, this reduces to the vector B appearing in the modular Hamiltonian (2.4) for B. The vector B gives a notion of time evolution which is con ned to D . From the CFT point of view, this time evolution corresponds to evolution by the modular Hamiltonian (2.4) within the domain of dependence of B, which by a conformal transformation can be mapped to hyperbolic space times time. For states which are small perturbations to the CFT vacuum state, it was shown in [10] that the relative entropy for a ball B at second order in perturbations to the vacuum state corresponds to the perturbative bulk energy associated with the timelike Killing vector B in D (known as the \canonical energy" associated with this vector). This result was extended to general states in [11]. While there are no Killing vectors for general asymptotically AdS geometries, it is always possible to de ne a vector eld B that behaves near the AdS boundary and near the extremal surface in a similar way to the behavior of the Killing vector B in pure AdS. Speci cally, we impose conditions a jB = a B ; r [a b]jB~ = 2 nab; jB~ = 0 ; g ! g + L g : { 6 { where nab is the binormal to the codimension two extremal surface B~. Given any such vector eld, we can de ne a di eomorphism This represents a symmetry of the gravitational theory, so we can de ne a corresponding conserved current and Noether charge. The resulting charge H turns out to be the same for any vector eld satisfying the conditions (2.8){(2.10). It can be interpreted as an (2.8) (2.9) (2.10) (2.11) the ow (2.11) in the phase space formulation of gravity. The main result of [11] is that the CFT relative entropy for a state j i comparing the reduced density matrix vacuum counterpart (vac) is equal to the di erence of this gravitational energy between B to its S( Bjj B (vac)) = H (M ) H (AdS) : where L is the Lagrangian density and is de ned by J = (L g) L ; L(g) = d ( g) + E(g) g : We will review the derivation of this identity in the next section when we generalize it to our case. To write H explicitly, we start with the Noether current (expressed as a d-form) HJEP06(218)5 Thus, for a background satisfying the gravitational equations, we have (2.12) (2.13) (2.14) (2.15) (2.16) (2.17) (2.18) (2.19) Here, is any spacelike surface bounded by the HRRT surface B~ and by a spacelike surface @M on the AdS boundary with the same domain of dependence as B. For a ball-shaped region B, the quantity H is independent of both the bulk surface (as a consequence of di eomorphism invariance) and also the spacelike surface @M at the boundary of AdS (as a consequence of the fact that B de nes an asymptotic symmetry). The quantity K in the boundary term is de ned so that ( K) = As explained in [11], this ensures that the di erence (2.12) does not depend on the regularization procedure used to calculated the energies and perform the subtraction. We can rewrite H completely as a boundary term using the fact that on-shell, J can be expressed as an exact form [11] Here, E(g) are the equations of motion obtained in the usual way by varying the action. The Noether current is conserved o -shell for Killing vector elds and on-shell for any vector eld , Then, up to a boundary term, the energy H is de ned in the usual way as the integral of the Noether charge over a spatial surface: dJ = E(g) L g: Z K: J = dQ : { 7 { This shows that the de nition of H is independent of the details of the vector eld in the interior of . In our derivations below, it will be useful to have a di erential version of this expression that gives the change in H under on-shell variation of the metric. By combining (2.19) with (2.17), we obtain The interpretation of H as a Hamiltonian for the phase space transformation associated with (2.11) can be understood by recalling that the symplectic form on this phase space is de ned by HJEP06(218)5 Z H = ( Q ) where the d-form ! is de ned in terms of In terms of ! we have that for an arbitrary on-shell metric perturbation !(g; 1g; 2g) = 1 (g; 2g) 2 (g; 1g) : To begin, we choose a bulk vector eld satisfying a jA^ = a A ; r [a b]jA~ = 2 nab; jA~ = 0: { 8 { (2.20) (2.21) (2.22) (2.23) (3.1) (3.2) (3.3) (3.4) This amounts to the usual relation dH = vH between a Hamiltonian (in this case H ) and its corresponding vector eld (in this case L g) via the symplectic form . 3 Bulk interpretation of relative entropy for general regions bounded on a lightcone Consider now a more general spacelike CFT subsystem A whose boundary lies on some lightcone. In this case | unless the boundary is a sphere | there is no longer a conformal Killing vector de ned on the domain of dependence region DA and we cannot write the boundary modular Hamiltonian as in (2.4) where the result is independent of the surface B^. Nevertheless, we have a similar expression (2.7) for the modular Hamiltonian as a weighted integral of the CFT stress tensor over the lightcone region A^ (shown in gure 1). Thus, making use of the formula (2.3) for relative entropy, together with the holographic entanglement entropy formula and the holographic dictionary for the stress-energy tensor, we can translate the CFT relative entropy to a gravitational quantity. In this section, we show that this can again be interpreted as an energy di erence, S( Ajj vAac) = H (M ) H (AdS) for an energy H extremal surface A~. associated with a bulk spatial region bounded by A^ and the bulk We will now evaluate H for this vector eld starting from (2.20) and nd that it matches with a holographic expression for the change in relative entropy. First, we evaluate the part at the AdS boundary. Explicit calculations in the FG gauge, which are done in appendix A, show that where was de ned in the previous section and Q jz!0 = d Using the standard holographic relation between the asymptotic metric and the CFT stress tensor expectation value, we obtain Z ^ A Here, HA^ is the boundary modular Hamiltonian for the region A, so this term represents the variation in the modular Hamiltonian term in the expression (2.3) for relative entropy. Next, we look at the part of (2.20) coming from the other boundary of , at the extremal surface. By condition (3.4) we have that 1 integral over Q . Q can be brought into the form 16 r a b we obtain the entanglement entropy using the HRRT conjecture, vanishes on A~ and we are left with the ^ab [16] and by virtue of (3.3) The argument that the latter two conditions can be satis ed is the same as in [11], making use of the fact that we can de ne Gaussian null coordinates near the surface A~. To enforce the rst condition, we will make use of Fe erman-Graham (FG) coordinates for which the near-boundary metric takes the form 1 ds2 = z2 (dz2 + dx dx + zd (d)dx dx + O(zd+1)) and choose a vector eld expressed in these coordinates as = A + z 1 + z2 2 + : : : z = z 1z + z2 2z + : : : : (3.5) (3.6) (3.7) (3.8) (3.10) (3.11) (3.12) Combining both contributions to (2.20), we have that Z ~ A 1 4GN Z ~ A Q = = S: where the variation corresponds to an in nitesimal variation of the CFT state. Integrating this from the CFT vacuum state up to the state j i, we have that Thus, we have established that for a boundary region A with @A on a lightcone, the CFT relative entropy is interpreted in the dual gravity theory as an energy associated with the timelike vector eld . The energy H is naturally associated with a certain spacetime region of the bulk, foliated by spatial surfaces bounded by the boundary lightcone region A^ and the bulk extremal surface A~. That such spatial surfaces exist is a consequence of the fact that the extremal surface A~ always lies outside the causal wedge of the region A (the intersection of the causal past and the causal future of the domain of dependence of A) [17]. 4 Perturbative expansion of the holographic dual to relative entropy (4.2) (4.3) In this section, we consider the expression for H in the case where the CFT state is a small perturbation of the vacuum state so that the density matrix can be written perturbatively as A = vac + A with metric g ( ) = g(0) + g(1) + 2g(2) + : : : . 1 + 2 2 + : : : . In this case, the CFT state will be dual to a spacetime We recall that relative entropy vanishes up to second order in perturbations; making use of the expression (2.23), we will check that the gravitational expression for relative entropy also vanishes up to second order for general regions A bounded on a light cone. We then further make use of (2.23) to derive a gravitational expression dual to the rst non-vanishing contribution to relative entropy, expressing it as a quadratic form in the rst order metric perturbation. 4.1 Light cone coordinates for AdS It will be convenient to introduce coordinates for AdSd+1 tailored to the light cone on which the boundary of A lies. Starting from standard Poincare coordinates with metric ds2 = 1 z2 dz2 dt2 + d~x2 ; (t; ; ) = (t; ; 1; : : : ; d 2) centered at ~x = 0 and de ne = t we assume that the point p whose light cone contains @A is at ~x = z = 0 and t = where 0+ is an arbitrary constant. On the AdS boundary, we introduce polar coordinates + = 0+ and some function = ( i). With these coordinates, the vector eld (2.6) de ning the boundary modular ow takes the form and the modular Hamiltonian (2.7) may be written explicitly as HA = 4 d d Z Z 0 + ( i) d 1 + 0 ( i) ( i) T : For the choice ( i) = 0+ the region A is a ball of radius 0+ centered at the origin on the t = 0 slice and the expression reduces to the usual expression for a modular Hamiltonian of such a ball-shaped region. jA^ = 0 + 0 )( 2 In the bulk, we similarly de ne polar coordinates (t; r; ; 1; : : : ; d 2) where ( ; z) = r(cos ; sin ) and de ne r t r so that the bulk light cone of the point p is r+ = We will see below that for pure AdS, the extremal surface A~ lies on this bulk light cone + 0 on a surface that we will parameterize as r The AdSd+1 line element in these coordinates reads ( ; i), where ( = 0; i) is the function ds2 = 1 sin2 4dr+dr (r+ r )2 + d 2 + cos2 gij d id j ; where gij is the metric on the unit d In this section, we derive an analytic expression for the extremal surface A~ in pure AdS whose boundary is the region @A on the lightcone of p. This will be useful in giving more explicit expressions for the relative entropy at leading order in perturbations. We choose static gauge, parameterizing the surface using the spacetime coordinates and i and describing its pro le in the other directions by ( ; i). The equations which determine its location are 8p ab sin2 (r+ ! : (4.4) HJEP06(218)5 (4.5) (4.6) + 0 (4.7) The solution which corresponds to ball-shaped entangling surfaces is well known to be located at 2 + z2 = const. In order to obtain the solution for entangling surfaces of arbitrary shape (but still on a lightcone) we substitute the expression for the induced metric and separate the equation for r into 1 1 f ( ; i) = 1 pg : (4.8) the lightcone r+ = surface is 0+ and r Let us make the ansatz that even away from the boundary the extremal surface lives on = ( ; i). The induced metric of this codimension two ab = 1 sin2 a b + cos2 gij ai bi ; where a; b 2 f ; 1; : : : ; d 2g and i; j 2 f g 1; : : : ; d 2 . This metric is independent of r ; we will see in section 5 that this is related to the Markov property of CFT subregions with boundary on a lightcone. Since the induced metric is independent of r , the left hand side of the equations of motion (4.5) vanishes and we can see from the right hand side that the ansatz r+ = solves the equations. The remaining equation for f ( ; i) + 0 ( ; i) reads p sin2 ab 1 ! : To x the constants in (4.10) it helps to use intuition from the solutions in the case where the boundary of a subregion is located on a null-plane instead of a lightcone (see appendix B). In that case it is clear that e ects from perturbations away from a constant entangling surface on the extremal surface die o as z ! 1. Under a transformation which maps the Rindler result to a ball-shaped region, the distant part of the extremal surface corresponds to = =2. Consequently, we require that hn( =2) ! 0 for n 1 and hn( =2) = 1 for n = 0. At the same time, for ! 0 we need that hn( ) is constant and di erent from zero. These constraints are easily solved with c1 = 0; c2 = 1. Introducing a normalization factor to ensure that hn(0) = 1, we are left with hn( ) = cosn ( d+2n ) ( d 1+n ) 2 ( 2 d 1 + n) ( ) 2 d 2F1 n 2 1 n d ; ; 2 1 + 2n 2 ; cos2 : r = ( ; i) with In conclusion this shows that extremal surfaces in the bulk are located at r+ = 0+ and + ( ; i) = 0 1 C0 + P1 n=1 Pl Cn;lhn( ) ln( i) : Here, n runs over spherical harmonics in d 2 dimensions and l over their respective degeneracy. They intersect the boundary at Here, we followed our conventions and used indices i; j for the angular coordinates i. If we write f1 = h( ) ( i) we nd that the left hand side can be solved if ( i) is a spherical harmonic. In d 2 dimensions, the eigenvalues of the Laplacian on Sd 2 are given by n(3 d n) for the n-th harmonic. Every level n has a corresponding set of degenerate eigenfunctions ln with l = 1; : : : ; 2n+d 3 n+d 4 [18]. The left hand side reads n n 1 This di erential equation can be solved in terms of hypergeometric functions, h( ) = c1 cos3 d n 2F1 + c2 cosn 2F1 n 2 2 d 2 1 n d ; ; 2 Thus, the constants Cn;l are determined in terms of the function parameterizing the boundary surface by performing the spherical harmonic expansion As a simple example, one choice of surface involving only the n = 1 harmonics for the AdS4 case takes the form + ( ) = 0 + ; ( ) = 0 + 2 0+p1 1 + cos 2 ; and correspond to ball-shaped regions in a reference frame boosted relative to the original one by velocity in the x-direction. + ( i) = 0 C0 + P1 n=1 1 Pl Cn;l ln( i) : + 0 1 ( i) 1 n=1 l = C0 + X X Cn;l ln( i) : 4.3 Our next step is to provide an explicit expression for the vector eld on the extremal surface which obeys equations (3.2){(3.4), such that the quantity H is dual to relative entropy. Using (4.2), the explicit form of equation (3.2) is jA^ = = n n Equation (3.3) requires knowledge of the unit binormal but thanks to the knowledge about the expression for the extremal surface which we found in the preceding section it is possible to calculate it explicitly. Here, n1;2 denote two orthogonal normal vectors to the RT surface. The calculation is delegated to appendix C. The non-zero components of the unit binormal read n + = g + ; n a = where a again runs over coordinates ( ; i). One possible choice of a vector eld satisfying the boundary conditions given by equations (3.3) is:4 (4.16) (4.17) (4.18) (4.19) = 2 ( 0+ vector in the case when component (along the lightcone) is nonzero, and this has the same qualitative behavior as the vector on the boundary lightcone. It is immediately clear that conditions (3.2) and (3.4) are satis ed. It is also straightforward to verify the condition involving the unit binormal using the fact that for a torsion free connection we have r r Calculating the Lie derivative of the metric with respect to this vector eld gives zero on the light cone r+ = 0+ but not away from the light cone. This is in contrast to the case of a ball-shaped region, where the Lie derivative vanished everywhere inside the entanglement wedge. 4Upon expanding the sums in equation (4.19) it looks like the i components of the vector eld diverge as ! 2 and for d > 3 as i ! 0; due to the metric on the S d 2 sphere. However, these divergences can be shown to be mere coordinate singularities: from equation (4.11) we see that @i cos . Hence the i components of the vector eld go only as cos 1 . This happens as a consequence of the coordinate singularity at = =2 in polar coordinates which can be removed by going into Poincare coordinates (t; z; ~x). Similar arguments also hold for singularities due to the S d 2 metric. Coordinate independent quantities like the norm of the spatial part of the vector eld remain nite as can be seen from inspecting the metric. 4.4 To write an explicit perturbative expression for H , we begin with the on-shell result 1 2 1 2 1 2 = g g g g g g g g g g g g + g g g : (4.22) Z H = Here, the symplectic d-form ! is explicitly given by d d At second order, we have d 2 d 2 S(g( )jjg0)j =0 = Z d d d d and ^ depend on the metric they will have a series expansions in when we express the metric as a series. Also in this case we will use sub- or superscripts in parenthesis to indicate the order of the term in . Here and in the following we will use to denote covariant derivatives with respect to g (0) = g(0). It will be convenient for us to choose a gauge for the metric perturbations such that the extremal surface stays at the same coordinate location for any variation of the metric. It was shown in [16] that this is always possible. In this case, we have at rst order H(1) = Z d d We will see in the next section that this vanishes, in accord with the general vanishing of relative entropy at rst order (also known as the rst law of entanglement). 1 2 =0 : (4.20) (4.23) (4.24) (4.25) (4.26) We will calculate this more explicitly in section (4.4.2). 4.4.1 Vanishing of the rst order expression In this section, we demonstrate that our gravitational expression for the relative entropy vanishes for rst order perturbations as required. Expanding the rst order expression (4.23) for ! yields Z d d where repeated lower case letters a; b imply summation over angular coordinates ( ; i). Using the de nition of the Lie derivative and the fact that since gab is independent of r all Christo el symbols of the form vanish at leading order, the problem reduces to a problem of only the angular coordinates. Substituting the general form of a from equation (4.19) and using that ga(0b) = a(0b) we end 1 0 ! g(0) and g(1) are the bulk metric and its perturbation and a(0b), a(1b) are the induced metric and the induced metric perturbation, respectively. This expression is proportional to the equation for an extremal surface, equation (4.7), and therefore vanishes. If we drop the assumption that the Einstein equations are satis ed, one can show that the rst law of entanglement entropy implies that the Einstein equations hold at rst order around pure AdS. This was done in [15] where only ball-shaped CFT subregions were considered. Utilizing more general subregions bounded by a lightcone does not yield new (in-)equalities at rst order. 4.4.2 Relative entropy at second order We will now provide a more explicit expression for the leading perturbative contribution to relative entropy, which appears at second order in the perturbations. Starting from (4.24) and using our explicit expression for !, we obtain four potentially contributing terms, We obtain up with d 2 The rst and third terms vanish because of our rst order results of section 4.4.1. The last term is reminiscent of the standard canonical energy associated with the interior of the entanglement wedge, except that is no longer a Killing vector. The non zero contributions take the form (2)H = Z Here, a; b; c; d run over angular coordinates, ; run over all coordinates. Note that although we are calculating relative entropy at second order, the expression only depends on rst order metric perturbations. Due to the fact that is no longer a Killing vector eld, =0 ab (4.27) (4.28) (4.29) (4.30) HJEP06(218)5 we appear to have a contribution in addition to the rst term which appears for the case of ball-shaped regions. However, we have not yet imposed the Hollands-Wald gauge condition on the rst order metric perturbations, for which the coordinate location of the extremal surface is the same as in the case of pure AdS. We have additional gauge freedom on top of this, and it may be that for a suitable gauge choice, the nal term in the expression above can be eliminated. We have checked that this is the case for a planar black hole in AdS4. We discuss this, as well as the procedure of choosing the Hollands-Wald gauge condition in more detail in appendix D. 5 Holographic proof of the Markov property of the vacuum state In [14] it was pointed out that the vacuum states of subregions of a CFT bounded by curves = A and = B on the lightcone + = 0+ saturate strong subadditivity, i.e. HJEP06(218)5 SA + SB SA\B SA[B = 0: This is also known as the Markov property. Moreover, even for CFTs deformed by relevant perturbations, the reduced density matrices for regions A and B describe Markov states if A and B have their boundary on a null-plane. In its most general form the proof used that the modular Hamiltonians for such regions obey HA + HB HA\B HA[B = 0; which can be proven using methods of algebraic QFT. In this section we will give a holographic proof of the Markov property which uses the Ryu-Takayanagi proposal for entanglement entropy. We will start with the proof for a subregion of a deformed CFT with boundary on a null-plane and after that also show the property for subregions of CFTs with boundary on a lightcone. 5.1 The Markov property for states on the null-plane The vacuum state of a deformed CFT is dual to a geometry of the form ds2 = f (z)dz2 + g(z)( 2dx+dx + dx?dx? ): and x+ = x+(~x?; z) simpli es the equation to An undeformed CFT corresponds to the special case f (z) = g(z) = z12 . The entanglement entropy of a subregion A can then be calculated using the RT prescription, following the same steps as in section B. We assume that the boundary @A is described by x = const and x+ = x+(~x?). To describe the corresponding extremal surface we go to static gauge, where z and x? are our coordinates and x (z; x?) is the embedding. The ansatz x = const (5.1) (5.2) (5.3) (5.4) The relevant solution to this equation in the case of pure AdS is discussed in appendix B and is given by Here, Kd=2 is the modi ed Bessel function of the second kind and the coe cients aki are given in terms of the entangling surface x+(0; xi ) as More generally, the induced metric on the extremal surface in the bulk is ? ak = Z d d 2x ds2 = f (z)dz2 + g(z)(dx?dx? ) (5.5) (5.6) (5.7) and independent of the embedding x+(~x?; z). Thus, it is clear that the areas of all extremal surfaces ending on x = const are the same, potentially up to terms which depend on how the area of the extremal surface is regularized as we approach the boundary. The standard prescription given by cutting o z at some distance away from the boundary gives a universal cuto term for all such extremal surfaces and therefore the entanglement entropies for all regions with boundary on x are identical and strong subadditivity is saturated. Our argument is an explicit version of very similar arguments which have been used to show the saturation of the Quantum Null Energy condition [19].5 5.2 The Markov property for states on the lightcone If we consider an arbitrary region on the lightcone we expect the Markov property to hold for undeformed CFTs, since the lightcone is conformally equivalent to the null-plane. The solution for an extremal surface in pure AdS ending on a lightcone at the boundary was already discussed in section 4.2. Consider the case where we have two di erent entangling surfaces given by = A( i) and = B( i). We have seen before that the metric on the extremal surface is in fact r independent. However, again the dependence on the entangling surface can enter through regularization of the integral and would show up in the cuto -dependent term. In the coordinates of our choice ; i the divergent term in the area comes from the integral over . Following the standard way of regulating the surface integral we introduce a cuto z = , which translates into cutting o the integral at = r . From this is follows that if we choose the canonical way of regulating the entropy, the integral runs from ( 0+2 ) to =2. The entropy which is proportional to the area term can now be calculated using the explicit form of the induced metric, equation (4.6), and is given by Z p = Z d Z =2 d cosd 2 sind 1 : (5.8) 5We thank Adam Levine for pointing this out to us. (5.9) A (5.10) HJEP06(218)5 The only way the shape of the entangling surface appears is through the cuto , i.e. the surface area can be expanded as A = 0 X where the coe cients cn are the same for all entangling surfaces. In the light of equation (5.8) saturation of strong subadditivity for two regions on a lightcone de ned by and B is guaranteed if Z d ( 0+ A( i)) + ( 0+ max( 0+ A( i); 0+ B( i)) B( i)) min( 0+ A( i); 0+ B( i)) = 0; which is trivially pointwise true. This again shows that strong subadditivity is saturated, or in other words, reduced density matrices for regions on the lightcone describe Markovian states. For more details on the form of the coe cients cn in the expansion, see [20]. The authors of [14] also speculated about the possibility of introducing a cuto to regulate the area of extremal surfaces such that the area of the extremal surfaces of subregions on the lightcone are all exactly equal. The previous discussion explicitly shows that choosing to introduce a cuto = instead of z = realizes such a regularization procedure in which all entanglement entropies for regions on the lightcone are in fact the same. 6 Discussion The results of this paper imply that for any classical asymptotically AdS spacetime arising in a consistent theory of quantum gravity, the energy H must be positive and must not decrease as we increase the size of region A. It would be interesting to understand if it is possible to prove this result directly in general relativity, by requiring that the matter stress-energy tensor satisfy some standard energy condition. It may be useful to point out that there is a di erential quantity whose positivity implies all the other positivity and monotonicity results considered here. If we consider a deformation of the region A by an in nitesimal amount v( ), where v is some vector eld on @A pointing along the lightcone away from p, the change in relative entropy to rst order must take the form S( Ajj vAac) = Z v( )SA( ) (6.1) The monotonicity property implies that the quantity SA( ) must be positive for all A and all .6 It would be interesting to make use of our results to come up with a more explicit expression for the gravitational analogue of the quantity SA( ). One approach to providing a GR proof of the subsystem energy theorems would be to prove positivity of this. 6A special case of this positivity result was utilized in the proof of the averaged null energy condition in [21]. The Markov property discussed in section 5 suggests that it should be interesting to consider (for general states) the gravitational dual of the combination S(A) + S(B) B) S(A \ B) of entanglement entropies for regions A and B on a lightcone. Since strong subadditivity is saturated for the vacuum state, this gravitational quantity will vanish for pure AdS, but must be positive for any nearby physical asymptotically AdS spacetime according to strong subadditivity. Thus, strong subadditivity for these regions on a light cone will lead to a constraint on gravitational physics that appears even when considering small perturbations away from AdS. For two-dimensional CFTs, this quantity was already considered previously in [5, 8]; the analysis there suggests that this gravitational constraint takes the form of a spatially integrated null-energy condition. See [2] for some additional discussion of gravitational constraints from strong subadditivity. Acknowledgments We would like to thank Alex May for helpful discussions. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada and by the Simons Foundation. DN is supported by a UBC Four Year Doctoral Fellowship. A Equivalence of H on the boundary and the modular Hamiltonian In this appendix we will show that H reduces to the modular Hamiltonian on the boundary, even in the case of a deformed entangling surface. We take the in nitesimal di erence between pure AdS and another spacetime that satis es the linearized Einstein's equations around pure AdS, i.e. we want to calculate Q boundary. We can nd in the appendix of [11] that on a constant z slice near the = 1 16 GN ^ab 1 2 gac rc b gcr c a b + c b ga c r b rc gca + b a gcc : (A.1) r The next step is to expand the sum over a and b. As we approach the boundary we consider volume elements on constant z slices and thus the term involving the volume element ^ vanishes. In Fe erman-Graham gauge ( gzc = 0) we nd given in equation (3.6). tion (A.2) are Now all we need to do is nd the leading order behaviour near z = 0. To this e ect we assume that the vector elds have a asymptotic expansion near the conformal boundary We also take gab = z d 2 (adb) + zd 1 (d+1) + : : :. The leading order terms of equa ab Q Q (A.2) g : (A.3) = 1 16 GN + 1 16 GN ^ z 1 2 g r z ^ z g r z 1 2 r z g g r c r z + gcz + r z g rc gcz z r g + z r d 16 GN ^ z (d) zd+1 + : : : = O(1); The HRRT surface ending on the null-plane In order to derive the HRRT surface which ends on a curve located on a boundary nullplane, we split the coordinates into x = t x (here x is the spatial direction parallel to the null-plane), boundary directions xi orthogonal to the null-plane, and the bulk coordinate z. The metric on the Poincare patch in these coordinates is (A.4) (A.5) (B.1) (B.2) (B.3) (B.4) (B.5) 1 ; We choose static gauge for the coordinates on our extremal surface, such that x = x (z; xi?). The entangling surface on the boundary is then given by x The equations which determine the embeddings x (z; xi?) are given by = x (0; xi?). where the induced metric is denoted by ab. Having the extremal surface ending on a boundary null-plane means that either x+ or x are constant. Without loss of generality, we choose x = x 0 = const. This reduces the two equations (B.2) to a single equation for x+(z; xi?). Making the ansatz x+(z; xi?) = hk(z)gk(xi?) we can separate the equation into where we use the fact that for a CFT traceless stress-energy tensor implies that and ^ z = O z (d+1) . Finally, employing the relation between the metric perturbation in FG coordinates and the stress-energy tensor, and the de nition of given in section 2 we arrive at h T i = d 16 GN (d) z=0 Q = h T i + O(z): ?gk(xi?): The general solutions for the functions hk(z) and gk(xi?) are given by gk(x?) = akieikixi?; hk(z) = ckzd=2Id=2(zk) + dkzd=2Kd=2(zk); xi. I and K where k = jkij and xi?ki denotes the Euclidean inner product between the vectors ki and denote the modi ed Bessel functions of rst and second kind, respectively. We also de ne h0 = limz!0 hk(z). It turns out that we do not want the full solution for hk. Intuitively, it is clear that the e ect of deformations of the entangling surface on the boundary should die o as z ! 1. At the same time we also require that the shape of the extremal surface is uniquely determined by boundary conditions. The asymptotic behavior of hk as z ! 1 and z ! 0 is z!1 lim hk(z) = ck zli!m0 hk(z) = dk2 2 1 2 k r We can only ful ll above requirements if we set ck = 0. Hence any extremal surface ending The normalization is chosen such that determines ak in terms of the entangling surface x+(0; x?). C Calculation of the binormal The binormal n is de ned as ( ; i)) = 0. A convenient set of tangent vectors is given by where n1 and n2 are orthogonal 1 normalized normal vectors to the extremal surface. To calculate them start by calculating the d 1 tangent vectors to the surface which will g + 0 ) = 0 and and so on for all i. It is easy to see that these vectors form an orthonormal basis on the Ryu-Takayanagi surface. Requiring that n1 and n2 are orthogonal to all tangent vectors, g n1;2ta = 0. This requirement is ful lled by choosing n1+;2 = g + ; a n1;2 = where a stands again for all angular components. The condition that n1 and n2 be orthogonal and normalized to +1 and 1, respectively, is obeyed provided we choose 1 2 n 1 = (1 n 2 = 1 2 One can check that the only non-zero components of the binormal are given by: n + = g + ; n a = (B.6) (B.7) (B.8) (B.9) (C.1) (C.2) (C.3) (C.4) (C.5) (C.6) (C.7) (C.8) 2 ~ A (0) ab ~ A : As a warm-up consider a ball-shaped entangling surface with a corresponding extremal surface at r+ = 0+; r = 0+ placed in a planar black hole background, Hollands-Wald gauge condition as a perturbation of pure AdS, we can choose a gauge where g(1a)j = 0 = g(1) j In this appendix, we argue that for the example of a planar black hole in AdS4, considered which at the same time is compatible with Hollands-Wald gauge. In this case, the nal term in our second order expression (4.30) for the relative entropy vanishes. Hollands-Wald gauge is determined by requiring that the extremal surface in the deformed spacetime sits at the same coordinate location than the extremal surface in the undeformed spacetime. In particular this means that ( ; ); r : The requirement that also after a perturbation of the metric the extremal surface A~ sits at its old coordinate location translates into (D.1) (D.2) (D.3) (D.4) (D.5) (D.6) (D.7) (D.8) (D.9) (D.10) (1 zd)dt2 + dz2 (1 zd) + dx2 ; ab (1) (0) ab ~ A at leading order in . The equations for the extremal surface now become at rst order We can use the symmetry of the perturbation under time translations and regularity at the boundary to nd a vector eld v that generates a di eomorphism g ! Lvg which locates the extremal surface in the perturbed geometry at the same coordinate location as the extremal surface in the unperturbed geometry. This di eomorphism brings the metric perturbation into the form ds2 = 8 r )3 cos3 cot d 2: v+ = v = v = v = 0: 64 64 64 r ) 1 + sin2 sin sin (1 + sin2 )(r+ r )2; sin (1 + sin2 )(r+ r )2; The only non-vanishing components of the metric in the new coordinates are g+ ; g g . In particular, we have that g(1a) = 0 = g(1) . The main bene t of these coordinates and is that equation (D.5) holds automatically. Hence at least for a ball-shaped entangling surface we are in Hollands-Wald gauge and the extremal surface is located at r = It can be see from the metric that lines of constant r are lightlike and therefore we know that the new entangling surface still is on the bulk lightcone of a point p at the boundary. + 0 . From this we can conclude that the entanglement wedge associated to any region bounded by a lightcone does not contain any point outside the causal wedge. As we have seen this is true for ball-shaped regions. A deformation of the entangling surface cannot change this, since the boundary domain of dependence is smaller than that of some ballshaped region. At the same time, the extremal surface cannot lie within the causal domain of dependence and therefore we must conclude that the extremal surface also lies on the lightcone. This means that the transformations (D.6){(D.9) bring the RT surface to its correction r+ location. The only additional adjustment we need to make to the coordinate system is to reparameterize r around the extremal surface, e.g. by rescaling the r coordinate in an angle-dependent way. To nd a solution to the general Hollands-Wald gauge condition, equation (D.3), we alter the plus-component of the vector eld, v+ ! v+ + v~+( ; ), around the extremal surface such that it shifts the extremal surface into its new correct location on the lightcone. This vector eld can be chosen such that at the extremal surface A~ it remains constant along r and r+ and thus depends only on and . It should be clear that such a solution exists, since at the boundary the correction v~+( ; ) vanishes and is smooth everywhere else. More formally, in this case equation (D.3) reduces to 16 16 ( ; ))2) + ( ; ))2 1 cos ~ A ~ A (D.11) For small deformations of the ball shaped entangling surface we can write ( ; ) as a series expansion in the deformations. At rst order, this gives us a linear PDE which can be solved. Higher orders become inherently non-linear and thus this equation is in general very hard to solve. An interesting observation one can make for small n = 1 deformations of the entangling surface is that the linear order correction is zero. Open Access. 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Dominik Neuenfeld, Krishan Saraswat, Mark Van Raamsdonk. Positive gravitational subsystem energies from CFT cone relative entropies, Journal of High Energy Physics, 2018, 50, DOI: 10.1007/JHEP06(2018)050