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Constraints on geometry from causal holographic information and relative entropy Saraswat, Krishan 2017

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Constraints on Geometry from Causal Holographic Informationand Relative EntropybyKrishan SaraswatB.Sc., University of Alberta, 2015a thesis submitted in partial fulfillmentof the requirements for the degree ofMaster of Scienceinthe faculty of graduate and postdoctoral studies(Physics)The University of British Columbia(Vancouver)August 2017c© Krishan Saraswat, 2017AbstractIn this thesis we find constraints to asymptotically anti de-Sitter space dual to holographic con-formal field theory states using the holographic duality. A recent conjecture involving the causalholographic information surface propsed that for smooth asymptotically anti de-Sitter spacetimesthat obey the null energy conditions, the area of the Ryu-Takayanagi surface will always be lessthan or equal to the area of the causal holographic information surface. This conjecture is exploredin three dimensional spacetimes that are dual to translation invariant states on the boundary con-formal field theory in two dimensions. A series expansion of the Ryu-Takayanagi surface and causalholographic information surface is derived, and is used to translate the constraint between the ar-eas of the two surfaces into a constraint on the asymptotic structure of such geometries near theconformal boundary. The translated constraints are compared to the constraints given by the nullenergy condition - and it is found that the first two leading order constraints are the same. Wethen outline some preliminary results of an ongoing project whose goal is to understand the dualof relative entropy of holographic states defined on null cone regions on the conformal boundary.We derive the modular Hamiltonian for vacuum states defined on null cone regions in a conformalfield theory using known results for modular Hamiltonians on null planes. We also derive the Ryu-Takayanagi surface associated with such null cone regions. Using these results, it is argued that,for null cones whose base is cut by a constant time cut, will not give new constraints beyond whatis already known for ball shaped regions.iiLay SummaryA quantum field theory describes particles as excitations of an underlying quantum field. Thesetheories usually exhibit some kind of symmetries. In particular, a conformal field theory is a specialtype of quantum field theory that has a large number of symmetries, which include conformaland Poincare symmetries. The anti de-Sitter conformal field theory correspondence tells us thatspecial states of a conformal field theory, called holographic states, are related to a special classof spacetime geometries called asymptotically anti de-Sitter geometries. In this thesis we usethis relation between holographic states and asymptotically anti de-Sitter spacetimes to translateconstraints on the field theory to constraints on geometry.iiiPrefaceThe questions the author explores in this thesis were posed by the author’s supervisor Dr. MarkVan Raamsdonk. The calculations done in chapter 2 were done independently and guided bysuggestions from the author’s supervisor. The materials in chapter 3 are a result of a collaborativeeffort with Dominik Neuenfeld under the guidance of the author’s supervisor. The calculations inappendix A.4, A.6, and A.7 were originally done by Dominik. The calculations in section 3.3 and3.4 were done jointly by both Dominik and the author and then checked for consistency.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Historical Overview of the AdS/CFT Correspondence and Holography . . . . . . . . 11.2 Basics of Entanglement Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Holographic Entanglement Entropy and the Ryu-Takayanagi Conjecture . . . . . . . 41.4 Boundary Stress Energy Tensor from Asymptotic Behaviour in Bulk . . . . . . . . . 72 Constraints From Causal Holographic Information Surface . . . . . . . . . . . . 102.1 Defining the Causal Holographic Information Surface . . . . . . . . . . . . . . . . . . 102.2 Series Expansion for CHI Curve for AAdS3 Spacetimes . . . . . . . . . . . . . . . . . 112.3 Series Expansion for the Area of the CHI Curve . . . . . . . . . . . . . . . . . . . . . 162.4 Ryu-Takayanagi Curve for AAdS3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.5 Series Expansion for Area of the Ryu-Takayanagi Surface . . . . . . . . . . . . . . . 202.6 Constraints on AAdS3 Spacetimes from CHI Inequality . . . . . . . . . . . . . . . . 223 Constraints from Relative Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . 263.1 Basic Properties of Relative Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2 Relative Entropy for Ball Shaped Regions in Terms of Bulk Quantities . . . . . . . . 273.3 Modular Hamiltonian on the Cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.4 Ryu-Takayanagi Surface Anchored to Light Cone on CFTd Boundary . . . . . . . . 333.5 Relative Entropy as Quasi-Local Bulk Energy . . . . . . . . . . . . . . . . . . . . . . 373.6 Writing Modular Hamiltonian in Covariant Form . . . . . . . . . . . . . . . . . . . . 40v3.7 Extending Boundary Vector Field into Bulk . . . . . . . . . . . . . . . . . . . . . . . 414 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46A Supplementary Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48A.1 Co-Dimension 2 Extremal Surface in d+ 1 Dimensional Spacetime . . . . . . . . . . 48A.2 Quadratic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49A.3 Mapping Half Space to a Ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50A.4 Calculating Jacobian for SCT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52A.5 Coordinates on Null Plane to Coordinates on Null Cone . . . . . . . . . . . . . . . . 53A.6 Conformal Transformation of the Stress Energy Tensor of a CFTd . . . . . . . . . . 54A.7 Unit Binormal to RT surface Anchored to Cone Regions . . . . . . . . . . . . . . . . 56viAcknowledgmentsFirstly, I would like to thank my supervisor, Dr. Mark Van Raamsdonk for giving me be theopportunity to do research with him and for all his patience and guidance throughout the time Iwas supervised by him. Without his guidance this thesis would not have come to fruition. I wouldalso like to thank Dominik Neuenfeld for his contributions to the materials presented in chapter 3and the appendices and also for the many helpful conversions during our collaboration. I wouldalso like to thank Ali Izadi Rad and Bin Guo for helpful discussions. Last but not least, I wouldlike to thank my parents for their unconditional support throughout my Masters degree program.viiDedicationTo my parents, who always encourage and support me.viiiChapter 1Introduction1.1 Historical Overview of the AdS/CFT Correspondence andHolographyThe Anti-de Sitter/Conformal Field Theory (AdSd+1/CFTd) correspondence was first proposedby Juan Maldacena in 1997 in the context of string theory [1]. The conjecture states that acertain class of conformal field theories, which are sometimes called holographic, defined on a d-dimensional Minkowski background, are equivalent to theories of quantum gravity on a d + 1-dimensional asymptotically Anti-de Sitter (AAdSd+1) background [2]. Since its initial conceptionit has proved to be a powerful tool to do calculations in strongly coupled quantum field theories,[3], as well as being a promising approach to formulating a consistent theory of quantum gravity[4]. In particular one important development that has come out of studying the duality is theRyu-Takayanagi (RT) conjecture. The conjecture states that the entanglement entropy of somesub-region of a CFTd is proportional to the area of a co-dimension 2 extremal surface in thedual AAdSd+1 geometry [5]. Since the area of an extremal surface is a geometric quantity whichdepends on the metric, the conjecture provides a direct relation between the quantum informationquantity of entanglement entropy and the geometry of AAdSd+1 spacetimes. Since there are certainconstraints for the quantum information quantities on CFTd’s one can use the RT conjecture totranslate these quantum information constraints to constraints on the dual spacetimes [4, 6, 7]. Byunderstanding such constraints one can understand what types of geometries and energy conditionsare physically allowed in any consistent theory of quantum gravity.1.2 Basics of Entanglement EntropyIn the context of holography entanglement entropy usually refers to the entanglement entropy ofsome sub-region of spacetime over which the state of a quantum field theory is defined. Beforeunderstanding this notion of entanglement entropy, we start by introducing the density matrixformalism of quantum mechanics. In this formalism the central object that describes the state ofa quantum system is called the density matrix which is a non-negative hermitian operator with11.2. Basics of Entanglement Entropyunit trace. More explicitly, given a complete set of orthonormal quantum states, {|ψ1〉 , ..., |ψn〉},for an n-dimensional Hilbert space, along with a set of “classical” probabilities that add to one,{p1, ..., pn}, the density matrix of the system can be written as:ρ =n∑i=1pi |ψi〉 〈ψi| (1.2.1)The operator defined above is a hermitian operator with non-negative eigenvalues {pi}ni=1 andclearly the trace of the operator is equal to 1 since∑ni=1 pi = 1. The expectation values of anoperator O with respect to the density matrix ρ can be defined by using the trace:〈O〉ρ = Tr(ρO) (1.2.2)Density matrices in which only one eigenvalue is one and the rest are zero are referred to aspure states. In particular one can show that a density matrix defines a pure state iff ρ = ρ2 oralternatively iff Tr(ρ2) = 1. For a pure state defined in terms of a state vector, σ = |ψ〉 〈ψ|, thedefinition of the expectation value with respect to σ simplifies to the usual expression using thestate vector |ψ〉. This can be seen by calculating the trace using a complete set of basis states |ψi〉:〈O〉σ =n∑i=1〈ψi|σO |ψi〉 =n∑i=1〈ψi|ψ〉 〈ψ| O |ψi〉 =n∑i=1〈ψ|ψi〉 〈ψi| O |ψ〉 = 〈ψ| O |ψ〉 (1.2.3)We can also define density matrices for composite systems. Suppose we have two quantum sys-tems A and B each with its own complete set of states {|e1〉 , |e2〉 , ..., |en〉} and {|e˜1〉 , |e˜2〉 , ..., |e˜m〉}respectively. The composite system is in a mn-dimensional Hilbert space that is spanned by thefollowing basis vectors {|ei〉⊗ |e˜j〉}, where i ∈ {1, ..., n} and j ∈ {1, ...,m}. Using this basis we canwrite a general state vector |Ψ〉 in the composite system as:|Ψ〉 =n∑i=1m∑j=1ψij |ei〉⊗|e˜j〉 (1.2.4)As before, given a state vector, we can define a corresponding density matrix for the compositesystem:ρAB = |Ψ〉 〈Ψ| =n∑i,l=1m∑j,k=1ψijψ∗kl(|ei〉⊗|e˜j〉)(〈e˜k|⊗〈el|)(1.2.5)Given a density matrix for a composite system, such as the one defined above, one can definethe reduced density matrix for the subsystem A denoted ρA by taking a partial trace with respect21.2. Basics of Entanglement Entropyto subsystem B. Explicitly we find:ρA = TrB(ρAB) =n∑i,l=1m∑p,j,k=1ψijψ∗kl |ei〉 〈el| δjpδkp =n∑i,l=1m∑j=1ψijψ∗jl |ei〉 〈el| (1.2.6)The reduced density matrix can then be used to quantify the amount of entanglement betweenthe sub-systems A and B. This is done by calculating the Von Neumann entropy of the reduceddensity matrix. The Von Neumann entropy for the density matrix ρ which will be denoted as S(ρ)is given by the following equation:S(ρ) = −Tr(ρln(ρ)) = −n∑i=1piln(pi) (1.2.7)Where pi are the eigenvalues of the density matrix ρ. Most of the time we will be interested indensity matrices for composite systems in pure states. One special property of pure states is thatthe entanglement entropy of a sub-system is equal to the entanglement entropy of its complement.This can be seen by using an important theorem in quantum information called the Schmidtdecomposition theorem. It states that if |Ψ〉 is a state vector for a composite system AB then thereexists an orthonormal bases {|Ai〉}ni=1 and {|Bi〉}mi=1 for subsystems A and B respectively such thatthe state can be written as:|Ψ〉 =min(n,m)∑i=1√pi |Ai〉⊗|Bi〉 (1.2.8)Using this we can construct a density matrix which is given as:ρAB = |Ψ〉 〈Ψ| =min(n,m)∑i,j=1√pipj |Ai〉 〈Aj |⊗|Bi〉 〈Bj | (1.2.9)It is important to note that if we are to think of the operator above as a matrix it will be anm-dimensional square matrix however the non-zero information will be contained in an min(n,m)dimensional square sub-block. Now, we can compute the reduced density matrices for the subsys-tems by taking a partial trace:ρA = TrB(ρAB) =m∑k=1min(n,m)∑i,j=1√pipj |Ai〉 〈Aj |⊗〈Bk|Bi〉 〈Bj |Bk〉 =min(n,m)∑i,j,k=1√pipjδkiδjk |Ai〉 〈Aj |=min(n,m)∑k=1pk |Ak〉 〈Ak|(1.2.10)31.3. Holographic Entanglement Entropy and the Ryu-Takayanagi ConjectureρB = TrA(ρAB) =n∑k=1min(n,m)∑i,j=1√pipj 〈Ak|Ai〉 〈Aj |Ak〉⊗|Bi〉 〈Bj | =min(n,m)∑i,j,k=1√pipjδkiδjk |Bi〉 〈Bj |=min(n,m)∑k=1pk |Bk〉 〈Bk|(1.2.11)As one can see, the reduced density matrices will have the exact same non-zero eigenvalueswhich implies that the entanglement entropy of the sub-system A will be equal to the entanglemententropy of subsystem B. Also note that we did not make any assumptions on the sizes of theHilbert spaces of the two subsystems. This means that for pure states, entanglement entropy doesnot scale with the volume of the Hilbert space of the subsystems. The formalism discussed abovecan be applied to any quantum system whose state can be summarized in terms a density matrix.Now we want to define entanglement entropy of a subregion of a CFTd. To start one chooses somed− 1 dimensional Cauchy slice of the background spacetime. On this slice we define a state usingan Euclidean path integral. This defines a global state over the whole Cauchy slice which is oftencalled the wave functional denoted, |Ψ[Φ(x)]〉. The wave functional is defined by the fields, Φ(x),which depend on the coordinates on the slice. Using this wave functional one can define an assoici-ated density matrix ρ = |Ψ〉 〈Ψ| over the entire slice. After doing this, one can restrict themselvesto some subregion on the slice- call it A, it will have a boundary, ∂A, which is sometimes called theentangling surface. This splits the slice into two regions A and its complement Ac. Naturally, onecan now define the reduced density matrix, ρA, on A by integrating out the field configurations inthe complement. The Von-Neumann entropy of the reduced density matrix can now be computedand this is defined to be the entanglement entropy of the subregion A of the CFTd. Unsurprisinglywhen one calculates the entanglement entropy it diverges due to the fact that one is dealing with acontinuous system. This is why the entanglement entropy is usually given in terms of some latticespacing cutoff which regulates the UV divergence. Typically ground states of CFTd’s obey what isknown as an area law of entanglement, which states that the entanglement entropy has a leadingorder UV divergence that scales like the area of the entangling surface ∂A for a fixed lattice spacing.For a more complete discussion of how to calculate entanglement entropy using the ideas discussedabove one should refer to [8].1.3 Holographic Entanglement Entropy and the Ryu-TakayanagiConjectureSo far our discussions of Entanglement entropy had nothing to do with the AdSd+1/CFTd corre-spondence. The AdSd+1/CFTd correspondence comes into the picture when we consider a specialsub-class of CFTd states which are called holographic. For such CFTd states it can be shown [5, 8],41.3. Holographic Entanglement Entropy and the Ryu-Takayanagi Conjectureusing the prescription outlined in the previous section, that the result for calculating the entangle-ment entropy of some subregion A will give the same leading order divergent term as the area ofa co-dimension 2 surface in the bulk that is anchored to the entangling surface on the conformalboundary of AdSd+1. This observation leads to the more general conjecture first formulated by Ryuand Takayanagi called the Ryu-Takayanagi conjecture. This is summarized by the Ryu-Takayanagiformula [5, 8]:SEE =Area(γA)4GN(1.3.1)Where γA is a co-dimension 2 extremal surface in the bulk which extremizes the area functional.This surface obeys the following boundary condition ∂A = ∂γA, which is to say that the extremalsurface in the bulk ends on the conformal boundary on the entangling surface of the sub-regionA. Sometimes there can be more than one extremal surface that satisfies this boundary conditionin such a case one would choose the surface that minimizes the area. This formula will be usedthroughout this thesis and is sometimes referred to as the holographic entanglement entropy sinceit computes the entanglement entropy of holographic CFTd states.To make the definition more concrete and illustrate how such calculations will be done, we willdo the calculation for the RT surface in pure AdSd+1 anchored to the boundary of a ball shapedregion ∂B on a constant time slice t = 0, on the conformal boundary. The first thing that we mustdo is write down the line element for pure AdSd+1. A convenient coordinate system to write it inis called Poincare coordinates given by the line element:ds2 = gµνdxµdxν =1z2[dz2 − dt2 + (dx1)2 + ...+ (dxd−1)2](1.3.2)In these coordinates z is a space-like coordinate that goes into the bulk and the rest are boundarycoordinates. The conformal boundary exists at z = 0 and the metric on the conformal boundary isgiven by z2gµν |z=0. In this case we have the flat Minkowski metric on the conformal boundary. Wewill change the space-like boundary coordinates to hyper-spherical coordinates (ρ, φ1, ..., φd−2) dueto the fact we want to consider ball shaped regions on the boundary. The line element becomes:ds2 =1z2(dz2 − dt2 + dρ2 + ρ2gΩijdφidφj)(1.3.3)Where gΩij is the metric on the unit d − 2 sphere. Now we need to write the area functional forsurfaces that are anchored to the entangling surface of the ball on the conformal boundary on theconstant time slice, t = 0. To do this we apply the formalism discussed in appendix A.1 by settingtwo of the coordinates equal to functions of the other d− 1 coordinates. In particular we define:Xt = t = 0 (1.3.4)51.3. Holographic Entanglement Entropy and the Ryu-Takayanagi ConjectureXz = f(ρ) (1.3.5)Xρ = ρ = σρ (1.3.6)Xφi= φi = σi (1.3.7)The first embedding equation is simple; since we are considering a static slice of the boundarywe know the surface will be on the same static slice in the bulk. The second embedding equationis some function of the radial boundary coordinate; of course, a more general anzatz would be toinclude φi. However, since the boundary entangling surface has no φi dependence we can eliminatesuch dependence from our anzatz. The coordinates on the surface will be the remaining coordinatesof the background space, σa = (σρ = ρ, σi = φi). Now we can write the induced metric, γab, on thesurface which is given by:γab = gµν∂Xµ∂σa∂Xν∂σb= gzz∂aXz∂bXz+gρρδρaδρb+gijδiaδjb =1f2(ρ)[δρaδρb (1 + ∂ρf(ρ)∂ρf(ρ)) + ρ2gΩijδiaδjb](1.3.8)Using this we define the area functional:A =∫ √γdd−1σ (1.3.9)The integral goes over the boundary coordinates within the ball shaped region. To find theextremal surface we must extremize the Area functional defined above. In appendix A.1 we derivedthe equation that needs to be satisfied which is given by:δAδXB=12√γγab∂aXµ∂bXν∂Bgµν − ∂a(√γγab∂bXµgµB)(1.3.10)Where B is given by the two coordinates that we used to define the co-dimension 2 surface. Inour case we have B = t, z. We see that t = 0 trivially satisfies the equation, so all we need to do issolve the equation when B = z. We use the fact that ∂zgµν = −2zgµν this simplifies the first term.We can also simplify the second term by noting that Xz only depends on ρ. This implies we haveonly one non-zero term in the sum thus, we find that:d− 1f(ρ)√γ + ∂ρ(√γγρρ∂ρf(ρ)1f2(ρ))= 0 (1.3.11)Since the induced metric is diagonal it, is easy to see√γ =ρd−2√gΩ√1+(∂ρf)2fd−1(ρ) and γρρ = f2(ρ)1+(∂ρf)2.Plugging everything in, one can check that f(ρ) =√R2 − ρ2 solves the equation where R is theradius of the ball on the boundary. This reproduces the well known result that the Ryu-Takayanagi61.4. Boundary Stress Energy Tensor from Asymptotic Behaviour in Bulksurface for pure AdSd+1 anchored to the entangling surface of a ball on a constant time slice is ad− 2-dimensional hemisphere in the bulk.One should keep in mind that the calculation we just did for the Ryu-Takayanagi surfacewas relatively simple for a number of reasons. The first is that the background geometry was amaximally symmetric space called pure AdSd+1. This resulted in the metric having only explicitdependence on ρ and z. We also utilized the fact that the metric is diagonal. In more generalAAdSd+1 geometries these facts will no longer hold true, which will lead to more complicatedequations. The second reason is that the entangling surface on which the extremal surface isanchored to has a very simple coordinate description for our choice of coordinates. In fact, thisallowed us to justify the anzatz that f was only a function of ρ and not φi. For more generalentangling surfaces on the boundary, we obviously cannot assume this making the equations moredifficult to solve. If one plugs in the solution for the extremal surface back into the area functionalone will find that the integral will diverge. Therefore one must introduce a cutoff in the bulk nearthe boundary this corresponds to the UV lattice cutoff we described in the previous section whenone does the calculation on the CFTd side. For holographic states of a CFTd one will find thatthe leading order divergent term in the area for the Ryu-Takayanagi surface will coincide withthe leading order divergent term in the entanglement entropy of the CFTd. In particular for theextremal surface we calculated here its area divided by 4GN would correspond to the entanglemententropy of a vacuum state of a dual CFTd.1.4 Boundary Stress Energy Tensor from Asymptotic Behaviourin BulkIn this section we will do a brief review of the Einstein vacuum equations in d + 1 dimensionalspacetimes1 for negative cosmological constant. We will give a formal definition of what it meansfor a space to be AAdSd+1 and how exactly the asymptotics of the geometry of such spaces give usinformation about the boundary CFTd stress energy tensor [9, 10]. The Einstein vacuum equationswith non-zero cosmological constant, Λ, for d+ 1 dimensional spacetime is given by [11]:Rµν − 12Rgµν + Λgµν = 0 (1.4.1)In particular when Λ < 0, there exists a maximally symmetric spacetime that solves the equa-tions known as pure AdSd+1. This space can be written in Poincare coordinates and the lineelement will read:ds2 = gµνdxµdxν =l2z2[dz2 − (dx0)2 + (dx1)2 + ...+ (dxd−1)2]=l2z2[dz2 + ηijdxidxj](1.4.2)1Throughout this thesis we adopt the following convention for the spacetime signature (-,+,...,+) where the minussign comes for time-like coordinates71.4. Boundary Stress Energy Tensor from Asymptotic Behaviour in BulkWhere the constant l is related to the cosmological constant through the following relation,Λ = −d(d−1)2l2, which can be obtained by plugging the metric into the vacuum equations. Throughoutthis thesis we will simply set l = 1 unless otherwise stated. A general AAdSd+1 space can be writtenin the form:ds2 =1z2[dz2 + gij(x, z)dxidxj]= Gµν(z, x)dxµdxν (1.4.3)Where z is called the defining function. The defining function satisfies the following two con-ditions. The first is that z ≥ 0 and only vanishes on the conformal boundary. The second is thatgµν∂µz∂νz 6= 0 on the boundary where gµν = z2Gµν . It can be shown that there is always a preferreddefining function in a small neighbourhood of the conformal boundary such that gµν∂µz∂νz = 1. Bychoosing this preferred defining function, it was shown in that gij(x, z) has the following asymptoticexpansion near the boundary at z = 0 for pure Einstein gravity (i.e no matter fields) [9, 10]:gij(x, z) = g(0)ij (x) + z2g(2)ij + ...+ zdg(d)ij + ... (1.4.4)Moreover it was shown that once one fixes g(0)ij (x) the only non-zero higher order terms occurin even powers of z, which are can be determined order by order using the Einstein equations up toorder zd. Terms beyond order zd are undetermined by the Einstein equations near the boundary.In this thesis we will mainly be interested in the case where the AAdSd+1 space is conformally flat,g(0)ij (x) = ηij . In this case it was shown that the asymptotic expansion of the AAdSd+1 space isgiven by [10]:ds2 =1z2[z2 + ηijdxidxj + zdΓ(d)ij (x) + zd+1Γ(d+1)ij (x) + ...](1.4.5)From such an expansion it was shown using the AdSd+1/CFTd correspondence that the coeffi-cient in the asymptotic expansion, Γ(d)ij , was fixed by the expectation value of the boundary CFTdstress energy tensor through the following relation [10]:〈Tij(x)〉 = d16piGNΓ(d)ij (x) (1.4.6)The reader should keep in mind that the result above is only true if we are dealing with aconformally flat AAdSd+1 spacetime. In general an extra term will be added that reflects conformalanomalies which occur due to lower order terms in the expansion [10]. Now we are ready to discussthe correspondence between states of a CFTd on a Minkowski background and the correspondingAAdSd+1 dual geometry. Start by noting that pure AdSd+1 occurs when all the terms Γ(n≥d)ij (x) = 0.In particular the pure AdSd+1 geometry corresponds to a dual state on the CFTd whose stressenergy tensor expectation value is zero. This leads us to the correspondence that Pure AdSd+1 isdual to the vacuum state of a CFTd and vice-versa. More generally when Γ(n>d)ij (x) 6= 0 then thiswill correspond to some state deformed away from the vacuum. Furthermore it is assumed thatsmall perturbations away from the vacuum state of the CFTd corresponds to small perturbations81.4. Boundary Stress Energy Tensor from Asymptotic Behaviour in Bulkaway from pure AdSd+1 dual geometry. Using this correspondence between the bulk AAdSd+1geometry and states of a CFTd as a starting point one can begin to understand how constraintson states of a CFTd translate to constraints in the bulk geometry.9Chapter 2Constraints From Causal HolographicInformation Surface2.1 Defining the Causal Holographic Information SurfaceBefore understanding the causal holographic information (CHI) surface we need to understandhow to construct a geometric quantity in the bulk called the causal wedge. The causal wedgeconstruction in the bulk was motived by a need to understand how bulk geometry of AAdSd+1spacetimes emerged from the the dual CFTd. In particular, it was argued that if one was given thedensity matrix on the CFTd of some closed bounded region on the boundary, B, with boundary∂B. Then the bulk geometry of the dual AAdSd+1 spacetime could be reconstructed within aregion known as the causal wedge of B [12]. We will give a quick summary of the constructionof the wedge as given in [12]. We start with a Cauchy slice Σ of the spacetime the CFTd resideson, then define a closed and bounded d − 1 dimensional region on the slice and call it B. Thisregion has an associated d dimensional future domain of dependence, denoted D+[B] and a pastdomain of dependence, denoted D−[B]. More intuitively we can say that a point p− ∈ D−[B] if allfuture oriented null geodesics originating from p− intersect with B. Similarly we can say a pointp+ ∈ D+[B] if all past oriented null geodesics originating from p+ intersect with B. Another way ofsaying this is that the set of points in D+[B] and D−[B] are determined by doing future and pasttime evolution of some prescribed data on the region B. Together the union of the past and futuredomain of dependence is known simply as the domain of dependence of the region B and is denotedas D[B]. Now that we have defined the domain of dependence of our region B we define the causalwedge of B to be the intersection of the future and past domains of influence of D[B]. This makesa wedge that extends into the bulk, the wedge at the conformal boundary coincides with D[B].The boundary of the wedge is made of two null surfaces whose intersection defines a co-dimension2 (d − 1 -dimensional) space-like surface in the bulk which is called the causal holographic (CHI)surface which is denoted as ΞB. It is anchored to ∂B on the conformal boundary. Using this102.2. Series Expansion for CHI Curve for AAdS3 Spacetimesco-dimension 2 surface in the bulk one defines the CHI associated with the boundary region B as:χB =Area(ΞB)4GN(2.1.1)In this thesis we will be interested in the following property that Ξ is conjectured to obey forsmooth spacetimes satisfying the null energy condition [13]:χB − SB = 14GN(Area(ΞB)−Area(RT )) ≥ 0 (2.1.2)Where Area(RT ) is the area of the Ryu-Takayanagi surface. We want to see what how thisconjecture constraints the asymptotic geometry of highly symmetric and static AAdS3 spacetimes.2.2 Series Expansion for CHI Curve for AAdS3 SpacetimesWe start with a general AAdS3 metric given in Poincare coordinates (t, x, z) that exhibits transla-tion invariance in the boundary coordinates x and y. The line element is given as:ds2 =l2z2(dz2 − g(z)dt2 + f(z)dx2) (2.2.1)Where g(z) and f(z) have the following asymptotic expansions near the boundary conformalboundary situated at z = 0:f(z) = 1 +z2f22!+z3f33!+z4f44!+ ... (2.2.2)g(z) = 1− z2f22!+z3g33!+z4g44!+ ... (2.2.3)From this point, throughout the rest of chapter 2 when we say AAdS3 spacetime, we meana spacetime that has the line element given by (2.2.1). The quadratic order coefficient in bothexpansions differ by a sign in order to have a traceless holographic stress energy tensor due to thefact that the spacetime is dual to a CFT2 state. The goal now is to translate the constraint givenby the inequality (2.1.2) into constraints on the coefficients in the asymptotic expansions givenabove. We will consider tilted intervals on the conformal boundary to be our region B from whichwe will construct the domain of dependence of the interval D[B]. We will in use D[B] to define theCHI surface that extends into the bulk. To define the interval we start by looking at the inducedline element on a constant z-slice:ds2z=z0 =l2z20(−g(z0)dt2 + f(z0)dx2) (2.2.4)Define an interval on the slice which is a straight line connecting the space-like separated pointsP1 = (−δt,−δx, z0) and P2 = (δt, δx, z0) where δx − δt > 0. Now we want to find the domain ofdependence for this interval. To construct the domain of dependence we emit null geodesics from112.2. Series Expansion for CHI Curve for AAdS3 Spacetimesthe end points of the interval. It is not difficult to see that the geodesics will intersect at two points.One will be a point to the past of the interval and another to the future which we will denote P−and P+ respectively. We can explicitly calculate these points by computing the null geodesics. Wefind that null geodesics emitted from the point P1 are given by the equation:t1±(x) = ±√f(z0)g(z0)(x+ δx)− δt (2.2.5)For null geodesics emitted at P2:t2±(x) = ±√f(z0)g(z0)(x− δx) + δt (2.2.6)We get P+ by setting t1+ = t2− and P− by setting t1− = t2+ we find that P+ =(√f(z0)g(z0)δx,√g(z0)f(z0)δt, z0)and P− =(−√f(z0)g(z0)δx,−√g(z0)f(z0)δt, z0). The points P1, P+, P2, and P− are the vertices of a dia-mond shape which is the domain of dependence for the interval, sometimes it is called the causaldiamond. We can get the domain of dependence on the conformal boundary by letting z0 → 0 then√f(z0)g(z0)→ 1.Now we will find a series expansion for the CHI surface associated with the space-like intervalwe defined. To do this we will emanate a family of null geodesics from the from the point (t =δx, x = δt, z = 0) towards the past of the point, and another family of null geodesics from thepoint (t = −δx, x = −δt, z = 0) towards the future of the point into the bulk geometry. The set ofpoints where the geodesics intersect will form a curve and this will be the CHI surface. We will beinterested in finding a series expansion for it. To begin, we will start by simplifying the problemof a tilted interval to a more simpler case where the interval is on a constant time slice. To dothis we make use of a Lorentz transformation. Since we have a space-like interval on the conformalboundary that connects two points separated by ∆t = 2δt and ∆x = 2δx. We want to choose aboost parameter v such that:δt′ = γ(δt− vδx) = 0 (2.2.7)γ =1√1− v2 (2.2.8)Here we see that we require that v = δtδx , in this new frame we have that:L := δx′ = γ(δx− vδt) = δxγ(2.2.9)Note that L is the proper length of the interval which is a Lorentz invariant quantity. Since wetransformed the interval we must also transform our original metric as well using the infinitesimal122.2. Series Expansion for CHI Curve for AAdS3 Spacetimesversions of the Lorentz transformations:dt′ = γ(dt− vdx)⇒ dt = γ(dt′ + vdx′) (2.2.10)dx′ = γ(dx− vdt)⇒ dx = γ(dx′ + vdt′) (2.2.11)dz′ = dz (2.2.12)Substituting into our original metric we obtain the transformed line element which will read:ds′2 =l2z′2[dz′2 + γ2[−dt′2(g(z′)− v2f(z′)) + dx′2(f(z′)− v2g(z′)) + 2vdt′dx′(f(z′)− g(z′))]](2.2.13)Hence we have transformed our problem of finding the CHI curve associated with a tilted intervalon the boundary with the bulk metric defined by equation (2.2.1) to an equivalent problem with aconstant time interval on the boundary and a bulk metric defined by equation (2.2.13). The boostparameter, v, gives a way to go from one description to the other. Now we must compute nullgeodesics using the transformed metric. We will use the Lagrangian approach. Define the followingLagrangian:L′ = lz′√z˙′2 + γ2[−t˙′2(g(z′)− v2f(z′)) + x˙′2(f(z′)− v2g(z′)) + 2vt˙′x˙′(f(z′)− g(z′))] (2.2.14)Where we use the dot notation to represent the derivative of each coordinate with respect tosome parameter along the geodesic which we will eventually take to be the coordinate time. Weget geodesics by solving the Euler Lagrange equations. In particular since there is no explicitdependence on on either t′ or x′ we know that:γ2l2z′2L′ [x˙′(f(z′)− v2g(z′)) + vt˙′(f(z′)− g(z′))] = c′x (2.2.15)γ2l2z′2L′ [vx˙′(f(z′)− g(z′))− t˙′(g(z′)− v2f(z′))] = c′t (2.2.16)Where c′x and c′t are constants. Dividing one equation by the other and rearranging we obtainthe following:dx′dt′=vf(z′)(1 + cv)− g(z′)(c+ v)vg(z′)(c+ v)− f(z′)(1 + cv) (2.2.17)Where c = −c′x/c′t is some constant (the negative sign is simply a convention we adopt). Morephysically, the parameter c will label the different geodesics, and different values of c will corrspondto geodesics being emitted along different directions from a given point in spacetime. Now we take132.2. Series Expansion for CHI Curve for AAdS3 Spacetimesthe parameter along the geodesics in the Lagrangian as t′ and substitute our above expression fordx′dt′ into the Lagrangian (2.2.14). We then set the Lagrangian to zero (null condition) and solve fordz′dt′ . We obtain:dz′dt′= γ√g(z′)[vdx′dt′+ 1]2− f(z′)[v +dx′dt′]2(2.2.18)Using the equations (2.2.17) and (2.2.18) we get the following two integral equations:∫dt′ =∫ √1− v2dz′√g(z′)[v dx′dt′ + 1]2 − f(z′) [v + dx′dt′ ]2 (2.2.19)∫dx′ =∫vf(z′)(1 + cv)− g(z′)(c+ v)vg(z′)(c+ v)− f(z′)(1 + cv)√1− v2dz′√g(z′)[v dx′dt′ + 1]2 − f(z′) [v + dx′dt′ ]2 (2.2.20)Using the two integral equations above, we can describe how geodesics will evolve from a pointof interest on the conformal boundary into the bulk geometry. In particular, for a past directed nullgeodesic starting at the future tip of the causal diamond on the boundary, (t′ = L, x′ = 0, z = 0),we have that:t+(z, c+) = L−∫ z0√1− v2dz′√g(z′)[v dx′dt′ + 1]2 − f(z′) [v + dx′dt′ ]2 (2.2.21)For future directed null geodesics starting at the past tip of the causal diamond on the boundary,(t′ = −L, x′ = 0, z = 0), we get:t−(z, c−) = −L+∫ z0√1− v2dz′√g(z′)[v dx′dt′ + 1]2 − f(z′) [v + dx′dt′ ]2 (2.2.22)We do a power series expansion of the right hand side of the equation at z = 0, and find powerseries of the form:t+(z, c+) = L−∞∑k=1Ak(c+)zk (2.2.23)t−(z, c−) = −L+∞∑k=1Ak(c−)zk (2.2.24)We can also find similar integral expressions for the x coordinate along the past and futuredirected geodesics and expand in a power series in z with coefficients depending on c±:x+(z, c+) =∞∑k=1Bk(c+)zk (2.2.25)142.2. Series Expansion for CHI Curve for AAdS3 Spacetimesx−(z, c−) =∞∑k=1Bk(c−)zk (2.2.26)Now fix a particular c+, this will correspond to a particular geodesic starting at the point(t = L, x = 0, z = 0). This should intersect with some other geodesic characterized by c− startingat the point (t = −L, x = 0, z = 0). The intersection point between the two geodesics willcorrespond to one point on the CHI curve. To find this point we set t+ = t− and x+ = x−. Weobtain:2L =∞∑k=1[Ak(c+) +Ak(c−)]z′kint (2.2.27)0 =∞∑k=1[Bk(c+)−Bk(c−)]z′kint ⇒ Bk(c+)−Bk(c−) = 0, ∀k (2.2.28)Where zint is the z coordinate in the bulk where the geodesics intersect. Equation (2.2.28) willbe satisfied if we set c+ = c− = c. By substituting this relation between c+ and c− into (2.2.27) wewill find:L =∞∑k=1Ak(c)z′intk (2.2.29)It states that the intersection point in the bulk of the two geodesics is controlled by the properlength of the interval on the conformal boundary. This implies that if the proper length of theinterval on the boundary is sufficiently small, then the z-coordinate of the intersection point willalso be small. To make this statement more precise we can invert the series given by equation(2.2.29) to write z′int, as a power series in L with c and v dependent coefficients:z′int(c) =∞∑k=1ηk(c)Lk (2.2.30)From this point we must assume that L << 1 in order for the series to converge. Since we knowwhat zint is we can use equations (2.2.23)−(2.2.26) to obtain t′int = t±(z′int, c) and x′int = x±(z′int, c):x′int(c) =∞∑k=1Bk(c)z′int(c)k =∞∑k=1ζk(c)Lk (2.2.31)t′int(c) = 0 (2.2.32)The coefficients in the series expansions are well defined when −1 ≤ c ≤ 1. We now have the setof intersection points of geodesics that generate the null boundary of the causal wedge in the bulk.This gives us the CHI curve parameterized by c ∈ [−1, 1]. The curve is connected to the endpointsof the interval of interest z′int(±1) = 0 and x′int(±1) = ±L as expected. We can revert back to the152.3. Series Expansion for the Area of the CHI Curveoriginal coordinates where the interval is tilted by applying the inverse Lorentz transformation tothe CHI curve coordinates. We get:zint(c) = z′int(c) =∞∑k=1ηk(c)Lk (2.2.33)xint(c) = γ(x′int + vt′int) =x′int(c)√1− v2 =1√1− v2∞∑k=1ζk(c)Lk (2.2.34)tint(c) = γ(t′int + vx′int) =vx′int(c)√1− v2 =v√1− v2∞∑k=1ζk(c)Lk (2.2.35)The three equations above give a complete perturbative expansion of the CHI curve associatedwith a tilted interval of proper length L on the boundary for AAdS3 spaces whose metric can bewritten in the form given by equation (2.2.1). The coefficients in the expansions can be written interms of the coefficients Ak and Bk defined in equations (2.2.23)− (2.2.26) through the procedureoutlined above.2.3 Series Expansion for the Area of the CHI CurveNow that we have a series expansion for the CHI curve we can find a series expansion for its length.To do this we will begin by defining a new parameter along the CHI curve, λ =√1− c2, where0 ≤ λ ≤ 1. Then we can relate λ to c by using a piecewise definition, c± = ±√1− λ2 where,−1 ≤ c− ≤ 0 and 0 ≤ c+ ≤ 1. The main reason we choose to define this different parameter isbecause we will need to regulate the integrals involving the length of the CHI curve by introducinga cutoff near the conformal boundary. Doing this will simply amount to setting λ to a smallparameter. This will make it easier to split the expressions into a finite part and a divergent part.Since we are going to use λ, we need to deal with the two halves of the CHI curve separately, inparticular we define (tL, xL, zL) to be points on the left half of the CHI curve and (tR, xR, zR) tobe the points on the right half of the CHI curve. We can define these points very easily usingequations (2.2.33)− (2.2.35):(tL, xL, zL) = (tint(c−), xint(c−), zint(c−)) (2.3.1)(tR, xR, zR) = (tint(c+), xint(c+), zint(c+)) (2.3.2)162.3. Series Expansion for the Area of the CHI CurveNow we can write the integral for the total length of the curve CHI curve as:ACHI =∫ 101zL(λ)√(dzLdλ)2+ f(zL(λ))(dxLdλ)2− g(zL(λ))(dtLdλ)2dλ+∫ 101zR(λ)√(dzRdλ)2+ f(zR(λ))(dxRdλ)2− g(zR(λ))(dtRdλ)2dλ(2.3.3)Consider the first integral involving the length of the left half of the CHI curve and analyze theintegrand near the boundary, λ = 0, as this is where any divergence in the integral will occur. Todo this we need only to understand the asymptotic expansions of (tL, xL, zL). They will have theform:zL = λ+O(λ2) (2.3.4)xL = −L+O(λ2) (2.3.5)tL = vxL = −vL+O(λ2) (2.3.6)Where for equation (2.3.6) we used the fact that tint/xint = v. We can also give the asymptoticexpansions for f and g in terms of λ:f(zL(λ)) = 1 +O(λ2) (2.3.7)g(zL(λ)) = 1 +O(λ2) (2.3.8)This implies that near the boundary the integrand has the following asymptotic expansion:1zL(λ)√(dzLdλ)2+ f(zL(λ))(dxLdλ)2− g(zL(λ))(dtLdλ)2=1λ+O(1) (2.3.9)This means the integral will diverge. The same will also hold true for the integrand involvingthe right side of the CHI curve. This is an expected result, what we can do now is rewrite it intoa finite part and divergent part. This amounts to subtracting off the divergence and doing theintegral from 0 to 1 and adding on the same divergence with cutoffs ˆL and ˆR going to zero this172.4. Ryu-Takayanagi Curve for AAdS3gives the following expression:ACHI =∫ 10 dλzL(λ)√(dzLdλ)2+ f(zL(λ))(dxLdλ)2− g(zL(λ))(dtLdλ)2− 1λ dλ+∫ 10 1zR(λ)√(dzRdλ)2+ f(zR(λ))(dxRdλ)2− g(zR(λ))(dtRdλ)2− 1λ dλ− limˆR→0ln(ˆR)− limˆL→0ln(ˆL)(2.3.10)Here the first integrals will converge and all the divergence is contained in the last two loga-rithmic terms as ˆL and ˆR go to zero. We can expand the finite part as a power series in L andexplicitly do the integrals order by order. We find that:ACHI =2ln(2) +132pi(−g3v2 + f3)1− v2 L3 +1903f22 v2 − 2g4v2 − 3f22 + 2f41− v2 L4 +O(L5)− limˆR→0ln(ˆR)− limˆL→0ln(ˆL)(2.3.11)This gives us the area of the CHI curve as a power series in the proper length of the interval onthe boundary with coefficients that depend on the asymptotic structure of the geometry and thetilt of the interval characterized by v.2.4 Ryu-Takayanagi Curve for AAdS3Here we reproduce the expressions given in [4] that the Ryu-Takayanagi surface will obey. Startwith the metric expressed in terms of a line element:ds2 =1z2(dz2 + f(z)dx2 − g(z)dt2) (2.4.1)The Ryu-Takayanagi surface for this space will be a curve. We will parameterize the curve interms the bulk coordinate z. The length functional will be:L =∫ z00dzz√1 + f(z)(dxdz)2− g(z)(dtdz)2(2.4.2)The functions x(z) and t(z) for the RT-curve will extremize the length functional. We candefine the associated Lagrangian as:L(x, x˙, t, t˙, z) = 1z√1 + f(z)x˙2 − g(z)t˙2 (2.4.3)Where we use Newton’s dot notation for a derivative with respect to z. The associated Euler182.4. Ryu-Takayanagi Curve for AAdS3Lagrange equations for x(z) and t(z) and given by:ddz∂L∂x˙= 0⇒ ddz[1zf(z)x˙√1 + f(z)x˙2 − g(z)t˙2]= 0 (2.4.4)ddz∂L∂t˙= 0⇒ ddz[1zg(z)t˙√1 + f(z)x˙2 − g(z)t˙2]= 0 (2.4.5)These tell us that along the curve parameterized by z the quantities in the square bracketsare constants. In particular we can choose to evaluate the expression in the square brackets atthe end point z0 which is the maximal depth that the curve goes into the bulk. This implies that[x˙(z0)]−1 = [t˙(z0)]−1 = 0. Then it follows that the terms in the expressions in the square bracketsevaluated at the point z0 are:[1zf(z)x˙√1 + f(z)x˙2 − g(z)t˙2]z=z0=f0z0√1− β20(2.4.6)Where we defined β0 =√g(z)f(z)dtdx∣∣∣∣z=z0, f0 = f(z0), and g0 = g(z0). Using the expression givenin equation (2.4.4) we can write:1zf(z)x˙√1 + f(z)x˙2 − g(z)t˙2=f0z0√1− β20(2.4.7)Then we can write t˙ in terms of x˙ by noting that:1zg(z)t˙√1+f(z)x˙2−g(z)t˙21zf(z)x˙√1+f(z)x˙2−g(z)t˙2=g(z)f(z)t˙x˙=√g0f0β0 ⇒ t˙ = f(z)g(z)√g0f0β0x˙ (2.4.8)Plugging this into equation (2.4.7) and rearranging for x˙2 gives us:x˙2 =z2f0f2z201[1− z2f0z20f]− β20[1− z2g0z20g] (2.4.9)Doing a similar calculation also shows that:t˙2 = β20z2g0z20g21[1− z2f0z20f]− β20[1− z2g0z20g] (2.4.10)This gives us equations that the Ryu-Takayanagi curve should obey.192.5. Series Expansion for Area of the Ryu-Takayanagi Surface2.5 Series Expansion for Area of the Ryu-Takayanagi SurfaceNow we want to get a similar series expansion for the area of the RT curve. From the previoussection we know that the RT curve satisfies the following differential equations:dxdz=z√f0z0f1√1− z2f0z20f− β20[1− z2g0z20g]dtdz= β0z√g0z0g1√1− z2f0z20f− β20[1− z2g0z20g] (2.5.1)Where z0 is the maximal value of z reached by the RT curve and we defined β0 =√g(z)f(z)dtdx∣∣∣∣z=z0,f0 = f(z0), g0 = g(z0), g = g(z), and f = f(z). This is the only information we need to find a seriesexpansion of the length of the RT curve using the metric we are given. However, this expansionwill have coefficients that depend on β0 and z0 which are not the parameters we used to express theCHI curve. This problem can be resolved by expressing the parameters β0 and z0 in terms of theparameters v and L used in the CHI expansion. To do this we start by integrating the expressionsabove to obtain:∆x =∫ z00z√f0z0fdz√1− z2f0z20f− β20[1− z2g0z20g] (2.5.2)∆t =∫ z00β0z√g0z0gdz√1− z2f0z20f− β20[1− z2g0z20g] (2.5.3)Note that these integrals only give half of the RT curve since we choose to parameterize in termsof the bulk coordinate z. However, due to the high degree of symmetry of the metric in the boundarycoordinates, the total change in x and t are double the integrals we have above. We can now relatethe proper length of the interval on the boundary to the integrals above L = δxγ =√(∆x)2 − (∆t)2.We do a series expansion for the integrands at ∆x and ∆t at z = 0. We integrate the series termby term from z = 0 to z = z0 and obtain a series expansion for δx and δt in terms of z0. We canthen plug in the series into the equation for δxγ and get:L(z0, β0) =δxγ=∞∑k=1bk(β0)zk0 (2.5.4)Then we can do a series reversion and find z0 as a power series inδxγ = L:z0(L, β0) =∞∑k=1ck(β0)Lk (2.5.5)202.5. Series Expansion for Area of the Ryu-Takayanagi SurfaceOnce again we find that the maximum depth the curve goes into the bulk is controlled by theproper length of the boundary interval. We can also do a series expansion of v = δtδx =∆t∆x at z = 0and integrate term by term to get v as a power series in z0 with β0 dependent coefficients:v(z0, β0) = β0 +∞∑k=2vk(β0)zk0 (2.5.6)Since we know that z0 is related to L by the series expansion given by equation (2.5.5), we canrewrite v as a power series in L by substituting z0(L, β0) into the right had side of equation (2.5.6).Then we expand in powers of L, this will give a series expansion for v of the form:v(L, β0) = β0 +∞∑k=2Vk(β0)Lk (2.5.7)We can invert the series above and write β0 as a power series in L with v dependent coefficients.To accomplish this we define the following series in L:β0(L, v) = v +∞∑m=2Km(v)Lm (2.5.8)To find the coefficients Km(v), we substitute the series for β0(L, v) into the right hand side ofequation (2.5.7) and expand as a power series in L and calculate up to whatever order in L we want.This will give us β0(L, v). Now we want to find z0(L, v). Going back to the series expansion givenby equation (2.5.5) we note that the coefficients ck(β0) are dependent on β0. If we plug in our seriesexpansion β0(L, v) into the coefficients ck(β0(L, v)) and expand, we will get a series expansion ofz0(L, v). This will enable us to express the series expansion of the length of the RT curve in termsof the parameters L and v. The length of the curve parameterized in terms of z is given by:ART =∫ z002z√1 + f(z)(dxdz)2− g(z)(dtdz)2dz (2.5.9)Just like for the CHI curve, we know that the length of the RT curve will diverge. We will splitthe area into a finite and divergent part:ART =∫ z002z√1 + f(z)(dxdz)2− g(z)(dtdz)2− 2z dz + lim→02ln(z0)=∫ 102u√√√√ 1− β201− u2 f(z0)f(z0u) − β20[1− u2 g(z0)g(z0u)] − 2u du+ lim→02ln(z0) (2.5.10)The divergence is contained in the logarithmic terms expressed in terms of the cutoff . We canthen substitute the series expansions of z0(L, v) and β0(L, v) and expand the finite part of the area212.6. Constraints on AAdS3 Spacetimes from CHI Inequalityas a power series in L. We find that:ART =2ln(2) +13f2(1 + v2)1− v2 L2 +132pi(−g3v2 + f3)1− v2 L3− 1452f22 v4 − g4v4 + f4v2 + g4v2 + 2f22 − f4(1− v2)2 L4 +O(L5) + lim→02ln(L) (2.5.11)Which gives us the series expansion for the area of the RT curve in terms of the parametersdefined for the CHI curve.2.6 Constraints on AAdS3 Spacetimes from CHI InequalityNow that we have a series expansion for both the CHI and RT curve associated with the intervalon the conformal boundary, we will see what kind of constraint we can get by using the conjecturethat ACHI − ART ≥ 0. Before doing this, we must recall that the cutoff was described differentlyfor the RT and CHI curve. We must first relate the cutoff ˆ for the CHI curve to the cutoff  forthe RT curve. In particular we must satisfy: = zL(λ = ˆL) = ΩL(L, v)ˆL +O(ˆ2L) (2.6.1) = zR(λ = ˆR) = ΩR(L, v)ˆR +O(ˆ2R) (2.6.2)We expand the right hand side of the two equations as power series in ˆL and ˆR with v andL dependent coefficients. We will only need to retain terms up to first order since we are taking alimit to zero in the end. Then:ˆL =ΩL(L, v)(2.6.3)ˆR =ΩR(L, v)(2.6.4)We will then substitute these expressions into our cutoff for the CHI curve and and expand thecutoff term in a series in L when we do this we obtain the following expansion with a cutoff thatis identical to the RT curve cutoff:ACHI =2ln(2) +13f2(1 + v2)1− v2 L2 +132pi(−g3v2 + f3)1− v2 L3− 1452f22 v4 − g4v4 + f4v2 + g4v2 + 2f22 − f4(1− v2)2 L4 +O(L5) + lim→02ln(L) (2.6.5)Now we can compute the difference in area between the CHI and RT which will be finite. We222.6. Constraints on AAdS3 Spacetimes from CHI Inequalityget:ACHI −ART = 1192pif2v2(g3 − f3)(1− v2)2 L5+12764800(1− v2)2 [(1440f23 + 8640g3f3 + (6075pi2 − 44640)g23)v4+ ((2700pi2 − 41280)(f23 + g23) + (−17550pi2 + 151680)g3f3 + 16384f2(g4 − f4))v2+ (6075pi2 − 44640)f23 + 8640f3g3 + 1440g23]L6 +O(L7)(2.6.6)As we can see, the area of RT curve and CHI curve area are identical up to fourth order in L.In particular when v = 0, the leading order term is of order L6 given by:161440(135pi2 − 992)f23 +1320g3f3 +11920g23 (2.6.7)We claim that the leading order term for v = 0 is always non-negative for any AAdS3 geometrygiven. To see this fix g3 to any arbitrary value then the sixith order coefficient is a quadratic in f3.We want to start by understanding the zeros of the quadratic by solving for f3 in terms of g3 wefind:f3 =4g3(−24± i√270pi2 − 2560)135pi2 − 992 (2.6.8)From this we see that f3 is complex and can only be real if g3 = 0 which, in turn, impliesthat f3 = 0. Which makes the entire 6th order term equal to zero. This means that if g3 6= 0 theparabola in f3 will not cross the f3 axis. If g3 = 0, then there is a doubly degenerate zero at theorigin and the parabola will not cross the f3 axis. Hence to check positivity it suffices to chooseany value for f3 and g3 and see that it is greater than zero. In particular let g3 = 0, and let f3be arbitrary. Then the sixth order term is positive because 135pi2 − 992 > 0. This proves thatfor v = 0 the leading order term is always non-negative. This makes sense because we know thatfor a constant time slice the RT curve is a minimal length curve that is anchored to the boundaryinterval. Now we consider the case in which v is non-zero. The leading order term is is 5th orderin L. We require that the term be non-negative for v ∈ (0, 1). This gives the constraint:f2(g3 − f3) ≥ 0 (2.6.9)We should assume that f2 > 0 because f2 is related to the expectation value of the stressenergy tensor of the CFT2. In particular, f2 is proportional to the energy density which should benon-negative. Using this fact we have that:f3 ≤ g3 (2.6.10)232.6. Constraints on AAdS3 Spacetimes from CHI InequalityNow suppose that g3 = f3, then the leading order term will be 6th order in L. Requiringnon-negativity implies that:Ax2 − 2Bx+A ≥ 0A =(pi2 − 25645)f23 ≥ 0B = A+81926075f2(f4 − g4)x = v2 ∈ (0, 1)(2.6.11)Start with the case when f3 = 0 then we are dealing with a linear equation in x. Clearly inorder for the expression to be non-negative we require that:B ≤ 0⇒ f2(f4 − g4) ≤ 0⇒ f4 ≤ g4 (2.6.12)Now, we deal with the case that f3 6= 0 start by rewriting the inequality as follows:x2 − 2BAx+ 1 ≥ 0 (2.6.13)Using results from appendix A we show that the quadratic satisfies the inequality in the intervalx ∈ (0, 1) when:f2(f4 − g4) ≤ 0⇒ f4 ≤ g4 (2.6.14)We can compare these leading order constraints to the constraints derived in [4] which we willquickly review. It was shown that for AAdS3 spacetimes we are considering the non-vanishingcomponents of the stress energy tensor are given as:Tzz = − 12zg′g− 12zf ′f+14f ′fg′g(2.6.15)Ttt =g4z(2f ′f+ zf ′2f2− 2z f′′f)(2.6.16)Txx = − f4z(2g′g+ zg′2g2− 2z g′′g)(2.6.17)We can do an asymptotic expansion of these expressions to obtain:Ttt = −14f3z − 16f4z2 +16f2f3z3 +O(z4) (2.6.18)Txx =14g3z +16g4z2 +16f2g3z3 +O(z4) (2.6.19)242.6. Constraints on AAdS3 Spacetimes from CHI InequalityWe will apply the null energy condition (NEC) which states Tµνuµuν ≥ 0. We take u to bea null vector in the boundary directions, u = ux∂x + ut∂t this gives Tttf(z)g(z) + Txx ≥ 0. We cancalculate in terms of the asymptotic expansion we get:Tttf(z)g(z)+ Txx = −14(f3 − g3)z − 16(f4 − g4)z2 +O(z3) ≥ 0 (2.6.20)The leading order term in z gives us f3 ≤ g3, which is exactly the same constraint we got in(2.6.10). When we assume f3 = g3 then the leading order term is of order z2 and the NEC givesf4 ≤ g4, which we got in (2.6.14). In conclusion, we find that the conjecture ACHI − ART ≥ 0 atleading orders does not give any tighter constraints to the asymptotic structure of AAdS3 space-times than what we obtain using the NEC in the boundary field theory directions. In fact, wefound that to the first two leading orders the constraints are identical.A natural question to ask in light of these leading order results is whether this is also true forhigher order terms. That is, do constraints from the series expansion of ACHI −ART ≥ 0 in L givethe same constraints as the NEC for null vectors in the boundary directions at higher than thefirst two leading orders? A good way to start answering this question is to simply calculate andcompare a few more terms in the series expansions. If we find that the higher order terms match,then this might indicate a more deeper relation between the NEC in AAdS3 spacetimes and theconstraint that ACHI−ART ≥ 0. If the higher order constraints between the two conditions do notmatch, then one could ask why the leading order terms match and whether the results we obtainedare a result of choosing AAdS3 metrics that are translation invariant in the boundary coordinates.In either case, there are still some open questions that one could try to answer that would give abetter insight of what causal holographic information can tell us about bulk spacetimes.25Chapter 3Constraints from Relative Entropy3.1 Basic Properties of Relative EntropyRelative entropy is yet another quantum information quantity that can be defined for states onsome subregion of a CFTd. This means that we can translate constraints on relative entropy toconstraints in bulk geometry. Such constraints have been extensively studied in [6, 7, 14] for ballshaped regions. We will review some of these results and use them as a basis for understanding thedual of relative entropy for holographic states defined on null cone sub-regions. Start by definingrelative entropy. Suppose we are given two states of a quantum system in terms of the densitymatrices ρ and σ. We can define a quantity called relative entropy in terms of these two states:S(ρ||σ) = Tr(ρlnρ)− Tr(ρlnσ) (3.1.1)Here, σ is often called the reference state. Relative entropy is always greater than or equal tozero and will be equal to zero iff ρ = σ. This property is often referred to as the positivity ofrelative entropy. Furthermore, for reduced density matrices ρA and σA obtained by a partial traceoperation from ρ and σ, one can show:S(ρA||σA) ≤ S(ρ||σ) (3.1.2)This is called the monotonicity of relative entropy. If we consider the case where the densitymatrices ρ and σ describe states in a subregion B of a CFT . We can view ρA and σA as the samestates defined in a subregion A such that A ⊂ B.Now that we have introduced the notion of relative entropy we will move on and reformulateit in terms of a quantity called the modular Hamiltonian. The modular Hamiltonian for a state σis defined by the formula Hσ = −lnσ. Using this definition we can recast the equation of relative263.2. Relative Entropy for Ball Shaped Regions in Terms of Bulk Quantitiesentropy as follows:S(ρ||σ) = Tr(ρlnρ)− Tr(ρlnσ) + Tr(σlnσ)− Tr(σlnσ)= [Tr(ρHσ)− Tr(σHσ)]− [−Tr(ρlnρ) + Tr(σlnσ)]=[〈Hσ〉ρ − 〈Hσ〉σ]− [S(ρ)− S(σ)]= ∆ 〈Hσ〉 −∆S(3.1.3)Where ∆ 〈Hσ〉 is the difference in the expectation value of the modular Hamiltonian Hσ withrespect to the states ρ and σ and ∆S is the difference between the Von Neumann entropies of thestates ρ and σ. Before explaining how this formula will be used in the setting of holography, wewant to prove the so called first law of entanglement entropy for states close to the reference stateσ. Let ρ = σ + X where 0 <  << 1 and X is a traceless hermitian matrix. We then substitutethis into the equation for relative entropy given at the beginning of the section and do a seriesexpansion in . We find that:S(σ + X||σ) = Tr[(σ + X)(lnσ + Xσ−1 − 122X2σ−2 +O(3))]− Tr [(σ + X)lnσ]= Tr(X) +122Tr(Xσ−1X) +O(3) = 122Tr(Xσ−1X) +O(3)(3.1.4)Where we used the cyclic property of trace and the fact that X is traceless. The leading ordernon-zero term is of order 2 and is called quantum Fisher information. Furthermore, since the firstorder term in  vanishes, this tells us that the first order variation of relative entropy for statesnear the reference state σ vanish. We can use this result to see that the first order variation of themodular hamiltonian equals to the first order variation in the Von Neumann entropy:δS = δ 〈Hσ〉 (3.1.5)The equation above is often called the first law of entanglement entropy. It was shown in [14]that when one considers the bulk dual of the first law for ball shaped regions on the boundaryCFTd, one finds the linearized Einstein equations in the bulk.3.2 Relative Entropy for Ball Shaped Regions in Terms of BulkQuantitiesHere we will review relative entropy in the context of AdSd+1/CFTd. We start with the formulafor relative entropy in terms of the modular Hamiltonian.S(ρB||σB) = ∆ 〈HB〉 −∆S (3.2.1)273.2. Relative Entropy for Ball Shaped Regions in Terms of Bulk QuantitiesHere we let σ = σB be the vacuum state of a CFTd on a ball shaped region B for a constanttime slice. Let ρ = ρB be some other excited state on B. In this case, the modular Hamiltonianfor the vacuum state σB takes a simple form given by the following integral expression [6, 7, 14]:HB = 2pi∫ballR2 − |~x− ~xc|22RTtt(tc, ~x)dd−1x (3.2.2)Where R is the radius of the d− 1 ball centred at the point (tc, ~xc). Ttt(tc, ~x) is the time-timecomponent of the stress energy tensor on the constant time slice t = tc. This can be generalized toa more covariant version given by the equation below [6, 7, 14]:HζB =∫B′∈D[B]ζµBTµν ˆν (3.2.3)Where the integral is over a d−1 space-like surface B′ that has the same domain of dependenceas the ball of radius R centred at (tc, ~xc). The vector field ζB is a conformal killing vector fielddefined as [6, 7, 14]:ζµB =piR[R2 − (t− tc)2 − |~x− ~xc|2]∂t − 2piR(t− tc)d−1∑i=1(x− xc)i∂i (3.2.4)The vector field defines what is known as the modular flow associated with the domain ofdependence D[B]. The stress energy tensor Tµν is now on B′ and ν is a d− 1 form defined usingthe d dimensional background Minkowski metric ηµν :ˆν =√η(d− 1)!ν−1dxa1 ∧ ... ∧ dxad−1 (3.2.5)It has a property such that by contracting the form with a normal vector nν to the surface B′,we get the volume form for the d − 1 dimensional surface B′. One can check that this covariantversion reproduces the older result for the ball on the constant time slice t = tc given by equation(3.2.2). Using the covariant formula along with the fact that 〈Tµν〉 = d16piGN Γ(d)µν , we can write thequantity ∆ 〈HζB 〉 as follows 2:∆ 〈HζB 〉 =∫B′ζµB∆ 〈Tµν〉 ˆν =d16piGN∫B′ζµBΓ(d)µν ˆν (3.2.6)This gives the change in the modular Hamiltonian in terms of metric quantities of the dualAAdSd+1 spacetimes. We can address the term ∆S using the RT formula which will tell us that∆S is the difference in the areas of the RT surfaces in the different backgrounds:∆S =∆A4GN(3.2.7)2We will assume from this point onwards that σ is the vacuum state, we know that its dual geometry is pureAdSd+1. We also know that the excited state ρ will be dual to some other AAdSd+1 spacetime which will have thefollowing asymptotic expansion in Poincare coordinates ds2 = 1z2[dz2 + ηµνdxµdxν + zdΓ(d)µν dxµdxν +O(zd+1)].283.3. Modular Hamiltonian on the ConeCombining the two results together gives us relative entropy expressed in terms of the bulkquantities:S(ρB||σB) = d16piGN∫B′ζµBΓ(d)µν ν − ∆A4GN(3.2.8)The quantity above is called holographic relative entropy. We defined it for states on d − 1dimensional sub-regions within the domain of dependence of a ball. We see that to calculate thisquantity we need two things; the first is the modular Hamiltonian of the sub-region on the bound-ary, and the second is the area of the RT surfaces. For ball shaped regions it has been shown thatholographic relative entropy can be interpreted as a quasi-local bulk energy [6]. At first order thevanishing of the variation of this quantity leads to the linearized Einstein equations. At secondorder, it was shown that quantum Fisher information was dual to a canonical energy defined in thebulk [6].One should keep in mind that the results discussed above apply to ball shaped regions. Thisis because for more arbitrary shaped sub-regions, the modular Hamiltonian is unknown and isassumed to take on a non-local form. Furthermore, calculating RT surfaces anchored to arbitraryentangling surfaces can be a difficult problem as we already discussed in the introduction. However,recent results by Casini and collaborators showed that if we restrict ourselves to the future horizonof some cut on a null plane, then one we can write the modular Hamiltonian as a simple integral.We will review this recent result and apply a conformal transformation to get the correspondingmodular Hamiltonian on a past light-cone whose base can be defined by an arbitrary cut.3.3 Modular Hamiltonian on the ConeIn this section, we want to consider the modular Hamiltonian of regions on a CFTd bounded byan entangling surface that lies on a light cone. Our starting point will be a result derived byCasini, Teste, and Gonzalo [15]. To start we consider the CFTd on a flat Minkowski backgroundwith coordinates xµ = (x0, x1, ..., xd−1). In these coordinates the Minkowski metric is diagonal,diag(−1, 1, ..., 1). Now we change coordinates to what we will call null plane coordinates definedas x− = x0 − x1 and x+ = x0 + x1 leaving the other transverse coordinates the same. The lineelement becomes:ds2 = −dx+dx− +d−1∑i=2(dxi)2 (3.3.1)In particular, if we set x− = 0, then this defines a null plane hyper-surface. Now we considersome cut along the null plane defined by setting the null coordinate x+ equal to some function,γ(x⊥), of the transverse coordinates x⊥ = (x2, .., xd−1). Then the modular Hamiltonian on the293.3. Modular Hamiltonian on the Conefuture horizon to the cut, γ, is given by the following integral expression [15]:Hγ = 2pi∫dx2...dxd−1∫ ∞γ(x⊥)(x+ − γ(x⊥))T++(x− = 0)dx+ (3.3.2)We know from the results in appendix A.3 that there exists a special conformal transformation(SCT) that maps this null sheet to a null cone. This means that by doing a conformal transformationof the integral expression above, we can obtain the corresponding modular Hamiltonian on the lightcone. To start, we want to see how the integration measure dx+dx2...dxd−1 on the plane transformsafter applying the SCT. The first step will be to do a transformation of the transverse coordinates(x+, x2, .., xd−1) → (x+, y2, ..., yd−1) where the coordinates on y are obtained after applying theSCT map given in appendix A.3:yi =xiΩ(x)Ω(x) =−(x+ + 2R)(x− − 2R) + (x⊥)24R2⇒ Ω(x− = 0) = 1 + x+2R+(x⊥2R)2(x⊥)2 =d−1∑i=2(xi)2(3.3.3)Using the map above one can calculate the elements of the Jacobian matrix associated withchanging coordinates. The result is:∂yi∂xk= Ω−1[δik −xixk2R2Ω](3.3.4)Where i, k ∈ {2, 3, ..., d − 1}. The determinant of the matrix can be found by finding theeigenvalues of the matrix which is outlined in appendix A.4, the result is:J⊥ = det(∂yi∂xk)= Ω2−d(1− (x⊥)22R2Ω)(3.3.5)If we restrict ourselves to the null plane x− = 0 then:J⊥∣∣x−=0 =2 + x+R − ΩΩd−1∣∣∣∣x−=0(3.3.6)This allows us to make the following statement:dx+dy2dy3...dyd−1 = J⊥dx+dx2dx3...dxd−1 ⇒ dx+dx2...dxd−1∣∣x−=0 =Ωd−12 + x+R − Ω∣∣∣∣x−=0dx+dy2...dyd−1(3.3.7)303.3. Modular Hamiltonian on the ConeNow we use the fact that on the null plane x− = 0 and this implies x+ = 2x1, thus:dx+dx2...dxd−1∣∣x−=0 =2Ωd−11 + 2x1R − Ω∣∣∣∣x−=0dx1dy2...dyd−1 =2Ωd−12 + 2x1R − Ω∂x1∂y1∣∣∣∣x−=0dy1dy2...dyd−1(3.3.8)Since we know that points on x− = 0 get mapped to points on y0 + |~y| = R we will calculatethe partial derivative under this restriction and find:∂x1∂y1∣∣∣∣y0+|~y|=R=1ωR− |~y| − y1|~y|∣∣∣∣y0+|~y|=R(3.3.9)Combining everything and using that Ω = 1/ω we find that:dx+dx2...dxd−1∣∣∣∣x−=0=2R|~y|ωd−1dy1...dyd−1∣∣∣∣y0+|~y|=R(3.3.10)We can change to hyper-spherical coordinates with ρ as the radial coordinate and φ1, ..., φd−2the angular coordinates. This gives:dx+dx2...dxd−1∣∣∣∣x−=0=2R√gΩωd−1ρd−3dρdφ1...dφd−2∣∣∣∣y0+ρ=R(3.3.11)Where gΩ is the determinant of the metric on a unit d− 2 sphere. Finally of define radial nullcoordinates by letting ρ± = y0 ± ρ it follows that:dx+dx2...dxd−1∣∣∣∣x−=0= −R√gΩωd−1(R− ρ−2)d−3dρ−dφ1...dφd−2∣∣∣∣ρ+=R(3.3.12)This tells us how the area measure on the plane changes when to change coordinates using theSCT that maps a half plane to a ball.We want to understand exactly how the cut x+ = γ(x⊥) on the null plane is mapped to thenull cone. We need to understand this due to the fact that the integral involving x+ is not over allspace, but rather only to the future of the cut and also because γ(x⊥) shows up in the integrand.To do this we use the result from appendix A.5 that, ρ− on the cone is related to x+ on the planevia:ρ− =x+1 + x+2R +(x⊥2R)2 −R (3.3.13)Now we will write the cut on the plane in the following form:x+(x⊥) = 2R(1g(x⊥)− 1)(1 +(x⊥2R)2)(3.3.14)313.3. Modular Hamiltonian on the ConeWhere g(x⊥) is some arbitrary function whose properties we can understand by substitutingthis expression for x+ into the equation (3.3.13). We find that:ρ− = R(1− 2g(x⊥)) (3.3.15)This tells us that if g(x⊥) = 0 ⇒ x+ = ∞. This we are sitting on the tip of the cone. Ifg(x⊥) = 1 ⇒ x+ = 0, then we are on the boundary of the ball on the zero time slice. This tellsus any cut that we make on the cone that is between the tip and ball on the zero time slice willbe specified by some function that obeys the following inequality 0 ≤ g(x⊥) ≤ 1. For example,if the function is a constant, then this will correspond to a constant cut of the light cone. Onemay actually be concerned because the function we are specifying is not a function of the angularcoordinates. However, one can check that any function of the transverse coordinates on the planewill give some function of the angular coordinates on the cone. This means that in principle wecould pick some cut on the cone that we want g(φ) and then use the relations in the appendix toexpress all the φ dependence in terms of x⊥ and vice-versa. Using the integration limit given byequation (3.3.14) we can write:Hγ = 2pi∫ ∫ ∞2R(1g(x⊥)−1)(1+(x⊥2R)2)[x+ − 2R(1g(x⊥)− 1)(1 +(x⊥2R)2)]T++dx+dx2...dxd−1(3.3.16)When we change coordinates to the light cone we can write the terms in the large square bracketas:x+ − 2R(1g(x⊥)− 1)(1 +(x⊥2R)2)=ρ− −R+ 2Rg(φ)ωg(φ)(3.3.17)Where we use the fact that:1 +(x⊥2R)2=1ω(1− R+ ρ−2R)(3.3.18)Using the result from appendix A.6, for doing a conformal transformation of the stress energytensor in a CFTd allows us to write:T++ =ωdR2(R− ρ−2)2T˜−− (3.3.19)The upper integration limit becomes R and the lower integration limit becomes R(1− 2g(φ)).Combining everything gives us the result for the modular Hamiltonian on a cone:Hcone = 2pi∫ ∫ RR(1−2g(φ))√gΩ(R− ρ−2)d−1 [R(1− 2g(φ))− ρ−Rg(φ)]T˜−−dρ−dφ1...dφd−2 (3.3.20)323.4. Ryu-Takayanagi Surface Anchored to Light Cone on CFTd BoundaryWe define ρ−0 (φ) = R(1− 2g(φ)) and rewrite the result as:Hcone = 4pi∫ ∫ Rρ−0 (φ)√gΩ(R− ρ−2)d−1 [ρ−0 (φ)− ρ−R− ρ−0 (φ)]T˜−−dρ−dφ1...dφd−2 (3.3.21)Where −R ≤ ρ−0 (φ) ≤ R. We can do one more change in the integration variable to makethe integral start at zero. Introducing the new integration parameter u = R−ρ−2 and definingγ(φ) =R−ρ−02 will give3:Hcone = 2pi∫ ∫ γ(φ)0√gΩud−1(uγ(φ)− 1)T¯uududφ1...dφd−2 (3.3.22)Where 0 < γ(φ) < R.Now that we have an expression for the modular Hamiltonian on a light cone cone with a cutbase we know we can calculate the modular Hamiltonian term in the relative entropy formula. Westill need a description of the corresponding RT surface anchored to this cone on the boundary.This will be address this in the following section in the case of a pure AdSd+1 background.3.4 Ryu-Takayanagi Surface Anchored to Light Cone on CFTdBoundaryIn this section we want to derive the Ryu-Takayanagi surface for pure AdSd+1 anchored to someregion on the boundary light cone. To do this start by writing the pure AdSd+1 metric in Poincarecoordinates and rewrite the boundary coordinates in hyper-spherical coordinates, the line elementreads:ds2 =1z2(−dt2 + dz2 + dρ2 + ρ2gΩijdφidφj) (3.4.1)where ρ is the radial distance on the boundary, i, j ∈ {1, 2, ..., d− 2}, and gΩij are components ofthe metric on the unit d−2 sphere with angular coordinates (φ1, ..., φd−2). Then we define anotherchange of coordinates by defining:z = rsin(θ)ρ = rcos(θ)θ ∈ [0, pi/2]r ∈ (0,∞)(3.4.2)3At the time of writing this thesis we determined that the result in the paper [15] for the light cone modularHamiltonian is not correct. We verified this with the authors and used a conformal transformation outlined in thissection to get the correct result.333.4. Ryu-Takayanagi Surface Anchored to Light Cone on CFTd BoundaryThe line element will now read:ds2 =1r2sin2(θ)(−dt2 + dr2 + r2dθ2 + r2cos2(θ)gΩijdφidφj) (3.4.3)In these coordinates, r defines a radial coordinate for the bulk geometry which will simplifyto radial coordinates to ρ on the boundary situated at θ = 0. We do one final transformation bydefining r± as follows:r± = t± r (3.4.4)This gives us the final form of the line element we will need expressed in the coordinates(r+, r−, θ, φ1, .., φd−2):ds2 =1sin2(θ)[−4dr+dr−(r+ − r−)2 + dθ2 + cos2(θ)gΩijdφidφj](3.4.5)We will refer to these coordinates as bulk radial null coordinates. In these coordinates we willdefine the following co-dimension 2 surface through the following two embedding equations:r+ = R (3.4.6)r− = f(θ, φi) (3.4.7)Where R is a constant and f(θ, φi) is some function that will be fixed by solving some PDEswhich we will write down shortly. More intuitively equation (3.4.6) specifies a past bulk null conewhose tip is at (t = R, z = 0, xbdry = 0). Equation (3.4.7) will specify a cut at the base of thebulk cone. The induced metric on this co-dimension 2 hyper-surface is given by the following d− 1dimensional metric:Gab =δθaδθb + cos2(θ)gΩijδiaδjbsin2(θ)(3.4.8)Where the indices a, b ∈ (θ, φ1, .., φd−2). The hyper-surface will be extremal if it satisfies thefollowing two PDE equations which we derived in appendix A.1:∂a[ √GGab∂br∓(r+ − r−)2sin2(θ)]= 0 (3.4.9)Clearly r+ = R will satisfy the PDE above. This leaves us with the following equation for thefunction f(θ, φi):∂a[ √GGab∂bf(θ, φi)(r+0 − f(θ, φi))2sin2(θ)]= ∂a[√GGabsin2(θ)∂b(1f˜(θ, φi))]= 0 (3.4.10)343.4. Ryu-Takayanagi Surface Anchored to Light Cone on CFTd BoundaryWhere we defined f˜(θ, φ) = R− f(θ, φi). It is not difficult to see that:√G =cotd−2(θ)sin(θ)√gΩ (3.4.11)Gab = sin2(θ)(δθaδθb +gΩijδaiδbjcos2(θ))(3.4.12)Substituting these expressions into equation (3.4.10) and expanding the sum we find that:tand−2(θ)sin(θ)cos2(θ)∂θ[cotd−2(θ)sin(θ)∂θ(1f˜)]+1√gΩ∂i[√gΩ(gΩ)ij∂j(1f˜)]= 0 (3.4.13)Now we will apply separation of variables between the boundary angular coordinates φi andthe bulk angle θ. Define 1f˜= h(θ)Φ(φi). Substituting this into the equation above gives:tand−2(θ)sin(θ)cos2(θ)1h(θ)ddθ[cotd−2(θ)sin(θ)dhdθ]+1Φ(φi)1√gΩ∂∂φi[√gΩ(gΩ)ij∂Φ∂φj]= 0 (3.4.14)Now define the constant of separation to be α. Then we know:1√gΩ∂i[√gΩ(gΩ)ij∂jΦ]= −αΦ(φi) (3.4.15)The PDE given by equation (3.35) is the Laplace-Beltrami operator acting on the function Φon the unit d− 2 sphere. The solutions to the PDE are well known and are called hyper-sphericalharmonics, Φn(φi) with eigenvalues −α = n(3− d− n), where n ∈ {0, 1, 2, ...} 4. This means thatthe ODE involving θ will be:−sin(θ)cos2(θ)d2hdθ2+ cos(θ)(cos2(θ) + d− 2) dhdθ− n(3− d− n)sin(θ)h(θ) = 0 (3.4.16)The general solution is given by hypergeometric functions:hn(θ) = C1cosn(θ)2F1(n2,n− 12;2n+ d− 12, cos2(θ))+ C2cos3−d−n(θ)2F1(2− d− n2,3− d− n2;5− d2− n, cos2(θ)) (3.4.17)4Note that we decide to be sloppy in labeling the hyper-spherical harmonic functions. It should be noted that inhigh dimensions there is more than one function that can give the same eigenvalues these degenerate functions areorthogonal and have their own labels but we choose to suppress these labels and only write the label that tells us theeigenvalues. For a more complete treatment one can look at [16].353.4. Ryu-Takayanagi Surface Anchored to Light Cone on CFTd BoundaryWe throw out the second term due to the fact it is ill defined for certain values of n and d andalso generally vanishes on the boundary which are not the type of solutions we are looking for.This means:hn(θ) = C1cosn(θ)2F1(n2,n− 12;2n+ d− 12, cos2(θ))(3.4.18)This solution satisfies the following conditions:h0(θ) = 1 (3.4.19)limθ→0hn>0(θ) =Γ(2n+d−12 )Γ(d2)Γ(n+d−12 )Γ(n+d2 )6= 0 (3.4.20)limθ→pi/2hn>0 = 0 (3.4.21)In summary, we find that the Ryu-Takayanagi surface can be written defined by the embeddingequations:r+ = R (3.4.22)r− = f(θ, φi) = R− 1C0 +∑∞n=1Cnhn(θ)Φn(φi)(3.4.23)Where we define Cn =cnhn(0)as an arbitrary constant normalized by the value of hn(θ = 0). Bydoing this we can see that on the boundary we have that:r+ = R = ρ+ (3.4.24)r−(θ = 0, φi) = R− 1C0 +∑∞n=1 cnΦn(φi)= ρ−(φi) (3.4.25)Where we used the fact:ρ±(θ, φi) = R− 1∓ cos(θ)2(C0 +∑∞n=1Cnhn(θ)Φn(φi))(3.4.26)We see that, on the extremal surface ends on some boundary light cone whose base is cut by afunction written as a series in hyper-spherical harmonics. To get an intuitive sense of the equationslets go back to hyper-spherical coordinates in the bulk the equations tell us the extremal surface isdescribed by the set of points satisfying:r =12C0 + 2∑∞n=1Cnhn(θ)Φn(φi)(3.4.27)363.5. Relative Entropy as Quasi-Local Bulk Energyt = R− 12C0 + 2∑∞n=1Cnhn(θ)Φn(φi)(3.4.28)If we set the higher order terms to zero then C0 → ∞ places us at the tip of the cone wherer = 0 and t = R. If we set C0 = 1/2R then we have a constant time slice cut at t = 0 of the conewhich is a hyper-sphere in the bulk of radius R. If we restrict or cut to be between the coordinatetime 0 ≤ t < R, then we have to require that 12R ≤ C0 +∑∞n=1 hn(θ)Φn(φi) < ∞. Hence we seethat the higher order terms can be thought of as perturbing away from the constant time cut to amore general cut which can be expressed in terms of hyper-spherical harmonics.3.5 Relative Entropy as Quasi-Local Bulk EnergyUsing the formula for holographic relative entropy discussed in the previous section, one could tryto directly calculate the quantities. Generally this will be quite difficult due to the fact that it ishard to solve the equations describing the RT surface in an arbitrary AAdSd+1 spacetime. To workaround this issue we will review the formalism discussed in [6]. This allows us to write relativeentropy in a form that is more naturally suited to handle arbitrary perturbations in the bulk. Tostart we define a d+ 1 form related to the Lagrangian density L of our system:L(g) = Lˆ (3.5.1)Where g is a shorthand for the fields in the Lagrangian. In our case since we are interested inpure gravity the Lagrangian density for our system will be the usual Einstein-Hilbert density givenby:L = 116piGNR− Λ (3.5.2)The d+1 form, ˆ, is defined in terms of the determinant of d+1 dimensional background metric,gab:ˆ =√−g(d+ 1)! ∧ dxa2 ∧ ... ∧ dxad+1 (3.5.3)We define (d+ 1− n)- dimensional forms ˆ , where n < d+ 1 as follows:ˆ =√−g(d+ 1− n)! ∧ ... ∧ dxd+1 (3.5.4)Where√−g is still the determinant of the d+1 dimensional metric. It is useful for describingvolume forms on co-dimension n surfaces embedded in the d+ 1 dimensional background.As a quick example to see how these forms operate we will look at the case where gab is themetric in pure AdSd+1 in Poincare coordinates. In this case the indices we sum over will take373.5. Relative Entropy as Quasi-Local Bulk Energyvalues, a1, a2, .., ad+1 ∈ {t, z, x1, .., xd−1}. It is easy to see that the form  will be given as:ˆ =1zd+1tx1..xd−1zdxt ∧ dx1 ∧ .... ∧ dxd−1 ∧ dxz = 1zd+1dtdx1...dxd−1dz (3.5.5)Which is what we would expect for the pure AdSd+1 metric in Poincare coordinates. Nowsuppose that we want to embed a co-dimension 1 surface. For simplicity let it be a constant z slice.It is clear that the normalized unit normal vector n = 1√gzz ∂z. If we contract this with the d + 1form ˆ with the unit normal we will get:ˆ · nz∂z = ˆznz = 1√gzzzd+1tx1..xd−1dxt ∧ dx1 ∧ .... ∧ dxd−1 = 1zddtdx1...dxd−1 (3.5.6)Which gives the correct form on the slice. For co-dimension 2 surfaces, we would have to finda unit binormal to the surface to define the d− 1 volume form.Having defined the forms we can continue and take a variation of the d + 1 form given byequation (3.5.1). One will get the following results:δL(g) = (−Eg)δgˆ+ dΘ(g, δg) (3.5.7)The first term is the equations of motion associated with the Lagrangian density L. For usthey will be the Einstein vacuum equations with the cosmological constant. The second term isboundary term which is defined in terms of a d-form, Θ(g, δg) and dΘ is the exterior derivative ofthe d-form. This will be used to define another d-form called the symplectic d-form which will bedefined as:ω(δ1g, δ2g) = δ1Θ(g, δ2g)− δ2Θ(g, δ1g) (3.5.8)Note that it is defined in terms of metric perturbations δ1g and δ2g. Now we want to use this inthe context of holography. We consider the subregion given by the light-cone regions, Aˆ, for whichthe modular Hamiltonian Hcone is known for vacuum states. This subregion, Aˆ, has an associatedco-dimension 2 extremal surface that extends into the bulk A˜ which we found in section 3.4 forpure AdSd+1. One then defines a d dimensional surface in the bulk that is bounded by the extremalsurface, A˜, in the bulk and Aˆ on the boundary which we denote as Σ. In our case, for pure AdSd+1bulk geometry, we know that this surface, Σ, will be the bulk light cone r+ = R, whose base iscut by a function described by r− = f(θ, φi). On Σ, we define a vector field ξc whose purpose willbe to generate diffeomorphisms. Finally, we can define δHξc (not to be confused with the modularHamiltonian) in terms of the d-form ω and the vector field ξc that exists on Σ:δHξc =∫Σω(δg,Lξcg) =∫Σ[δΘ(g,Lξcg)− LξcΘ(g, δg)] (3.5.9)383.5. Relative Entropy as Quasi-Local Bulk EnergyWe can expand the Lie derivative of the d-form Θ along the vector field ξc using the identityLξcΘ = ξc · dΘ + d(ξc ·Θ). Where ξc ·Θ is notation that tells us to contract Θ with the vector fieldξc. Using the identity we get:δHξc =∫Σ[δΘ− ξc · dΘ− d(ξc ·Θ)] =∫Σ[δΘ− ξc · dΘ]−∫∂Σξc ·Θ (3.5.10)Where ∂Σ = A˜ ∪ Aˆ. We define the Noether current d-form associated with the diffeomorphismgenerated by ξc to be, Jξc = Θ − ξc · L. Assuming that the equations of motion are satisfied weknow from equation (3.5.7) that dΘ(δg) = δL(g). We can show that the exterior derivative of theNoether current form vanishes through the following calculations:dJξc = dΘ(Lξcg)− d(ξc · L(Lξcg)) = δL(Lξcg)− d(ξc · L(Lξcg))= LξcL(g) + ξc · dL(g)− LξcL(g) = ξc · dL(g) = 0(3.5.11)Where for the last equality we used that fact that L is a d+1-form and dL = 0. This means thatwhen the equation of motion is satisfied then we can find a d− 1 form, Qξc , such that Jξ = dQξc .Now consider the variation of Noether current form as follows:δJξc = δΘ− ξc · δL = δΘ− ξc · dΘ (3.5.12)This allows us to write:δHξc =∫ΣδJξc −∫∂Σξc ·Θ (3.5.13)Now one needs to find a d-form K on the boundary, ∂Σ, that has the following property:δ(ξc ·K)|∂Σ = ξc ·Θ|∂Σ (3.5.14)It turns out that such a K exists if the following condition holds true:∫∂Σξc · ω(δ1g, δ2g) = 0,∀δ1g, δ2g (3.5.15)Using this K along with the fact that Jξc = dQξc , we can write:Hξc =∫ΣJξc −∫∂Σξc ·K =∫∂Σ[Qξc − ξc ·K] (3.5.16)Here Hξc is referred to as the quasi-local energy associated with the vector field ξc on Σ. Theimportant fact to note here is that Hξc can be written completely in terms of a oriented integralover ∂Σ = A˜ ∪ Aˆ. Due to this fact it has been shown in case for ball shaped subregions5 that onecan choose a particular ξB such that the integral reproduces the holographic relative entropy given5The arguments made to this point also apply to ball shaped regions simply replace the cone region on boundarywith ball regions along with the extremal surface anchored to the ball region393.6. Writing Modular Hamiltonian in Covariant Formby equation (3.2.8). This tells us that for ball shaped regions, the relative entropy has a bulk dualin the form of the quasi-local energy we defined. What we want to do is use similar arguments butadapt them to regions on light-cones to reproduce the formulas we have for holographic relativeentropy for light-cone subregions and their associated extremal surfaces. What we have outlinedhere is a starting point to doing this. In the following sections we will give a sketch as to how someof the arguments will be used in the case of null cone subregions on the boundary.3.6 Writing Modular Hamiltonian in Covariant FormIn this section we want to gain insight as to the form of ξc on the conformal boundary. We willdo this by assuming that modular Hamiltonians on cone shaped regions on the boundary take thefollowing form:Hcone =∫coneζµc Tµν ˆν (3.6.1)Where ζc will be a vector field such that when it is restricted to the surface of the cone we areintegrating over will, it reproduce the modular Hamiltonian given by equation (3.3.21). We defineˆ is a d-form and ˆν is a d− 1 form such that when it is contracted with a unit normal vector nν itgives the d− 1 dimensional volume form on the perpendicular subspace:ˆν =√−g(d− 1)!νa2a3...addxa2 ∧ ... ∧ dxad (3.6.2)We will work in boundary radial null coordinates (ρ+, ρ−, φ1, ..., φd−2) and adopt the convention+−φ1..φd−2 = 1. For the past light cone volume element we are interested in a normal vector to thesurface ρ+ = R. We calculate it as follows:nµ = gµν∂ν(−ρ+ +R) = −gµ+ ⇒ n = nµ∂µ = −g+−∂− (3.6.3)We contract this normal vector with the d-form ˆ to get the d− 1 form on the cone:ˆ·n =[−g+−√−gd!a1...addxa1 ∧ ... ∧ dxad]·∂− = −[√gΩd!(R− ρ−2)d−2a1...addxa1 ∧ ... ∧ dxad]·∂−(3.6.4)We use the fact that dx+ · ∂− = −1 and we find the following d− 1 form:ˆ+ =√gΩ(d− 1)!(R− ρ−2)d−2+a2...addxa2 ∧ ...∧dxad =√gΩ(R− ρ−2)d−2dρ−dφ1...dφd−2 (3.6.5)Which gives us the correct volume form for the past light-cone. Finally, we use the fact that403.7. Extending Boundary Vector Field into Bulkˆ− = gν−ˆν = g+−ˆ+, this gives:ˆ− = −2√gΩ(R− ρ−2)d−2dρ−dφ1...dφd−2 (3.6.6)We also require that on the cone ρ+ = R, the vector field obeys ζc|ρ+=R = ζ−c ∂−. Which gives:Hcone =∫coneζ−c T−−ˆ− = −∫cone2ζ−c T−−(R− ρ−2)d−2√gΩdρ−dφ1...dφd−2 (3.6.7)Comparing this expression to equation (3.3.21), we find that we require the vector field:ζ−c = −4pi(R− ρ−)[R− ρ−R− ρ−0− 1]= 4pi(R− ρ−)(−ρ−0 + ρ−)R− ρ−0(3.6.8)By choosing this vector field, we will reproduce the modular Hamiltonian given by equation(3.3.21). Hence we have found the following boundary condition that the bulk vector field ξc mustsatisfy on the conformal boundary:ξc|Aˆ =4pi(R− ρ−)(ρ− − ρ−0 )R− ρ−0∂− (3.6.9)3.7 Extending Boundary Vector Field into BulkNow we want to extend this vector field on the boundary cone to a bulk vector field on the surfaceΣ. An obvious and simple way of doing this in pure AdSd+1 is to simply replace the boundaryradial null coordinates ρ− and ρ+ with bulk radial null coordinates r+ and r−. In these coordinatesΣ is the bulk cone r+ = R with a base that is cut by some function of θ and φi. Explicitly we havethat:ξc|Σ = 4pi(R− r−)(r− − f(θ, φi))R− f(θ, φi) ∂− (3.7.1)Where f(θ = 0, φi) = ρ−0 (φi). One can check that this trivial extension of the boundary vectorfield will satisfy the following conditions on the RT surface, A˜, given by r+ = R and r− = f(θ, φi):ξc|A˜ =4pi(R− r−)(r− − f(θ, φi))R− f(θ, φi) ∂−∣∣∣∣A˜= 0 (3.7.2)∇aξbc −∇bξac |A˜ = 4pinab (3.7.3)Where we define nab as the unit binormal tensor to the RT surface in pure AdSd+1, which wederive in appendix A.7. Note that the boundary conditions we have defined by equations (3.6.9),(3.6.11), and (3.6.12) are analogous to the boundary conditions given in [6] for the bulk vector fieldξB, corresponding to ball shaped regions. Due to this, we expect that many of the same arguments413.7. Extending Boundary Vector Field into Bulkused in [6] for ball shaped regions will also apply to our light-cone regions. For example, we canconsider how the quasi-local energy integral gives us the area term in the holographic relativeentropy formula. We use the result from [6], which states Qξc =−116piGN∇aξbc ˆab where ˆab is a d− 1form defined in our discussion of forms in section 3.5. We can use the boundary conditions to get:∫A˜[Qξc − ξc ·K] =∫A˜Qξc = −116piGN∫A˜12ˆab[∇aξbc −∇bξac]= − 18GN∫A˜ˆabnab= − 14GN∫A˜+− = −Area(A˜)4GN(3.7.4)Which gives the area term in the holographic relative entropy formula in the pure AdSd+16.So far, we have argued that in the light-cone case relative entropy is dual to a bulk quasi-localenergy on Σ, and many of the same arguments used to show this in the ball case also apply to cutlight-cone regions. These statements were made by only knowing how the vector field ξc behaveson ∂Σ. Now we want to consider extending this vector field away from Σ. As a first step we canconsider this bulk vector field, ξc, defined on Σ in pure AdSd+1 and compare it to the bulk ξB givenin [14]. The vector field, ξB, plays the role of ξc for ball shaped regions on the boundary and isgiven as:ξB =piR[R2 − z2 − t2 − |~x|2] ∂t − 2piRt(d−1∑i=1xi∂i + z∂z)(3.7.5)Where the ball shaped region on the boundary is centred at tc = 0, ~xc = 0, and simplifies to theconformal Killing vector field given by equation (3.2.4) on the boundary z = 0. We can rewrite thisin the coordinate basis given by the coordinates (t, r, θ, φi) by noting that r∂r = z∂z +∑d−1i=1 xi∂i.This allows us to write:ξB =piR[R2 − t2 − r2] ∂t − 2piRtr∂r (3.7.6)Using this we can easily write down the non-zero components of the vector fields in bulk radialnull coordinates (r+, r−, θ, φi):ξ+B =∂r+∂tξtB +∂r+∂rξrB =piR[R2 − (r+)2] (3.7.7)ξ−B =∂r−∂tξtB +∂r−∂rξrB =piR[R2 − (r−)2] (3.7.8)Now we want to compare this vector field for the ball shaped region to the vector field we defined6This argument is exactly the same as for ball shaped sub-regions on the boundary we also expect that ourderivation of the term that will reproduces the modular Hamiltonian term from the quasi-local energy will be identicalto the derivation given in [6] for ball shaped regions.423.7. Extending Boundary Vector Field into Bulkin equation (3.7.1). This time we we take the cut of the cone defined by the function f(θ, φi) = −R.We do this because a cut r− = f(θ, φi) = −R corresponds to a constant time slice cut of the coneat t = 0. By doing this, the null boundary of the causal wedge for the ball and Σ will coincide inPure AdSd+1 between the coordinate times t ∈ [0, R]. We find that:ξB|Σ = piR[R2 − (r−)2] ∂− (3.7.9)ξc|Σ = 2piR[R2 − (r−)2] ∂− (3.7.10)The vector fields are identical up to a factor of two. We could then extend the vector field ξcaway from the surface Σ by simply defining its extension away from the surface to be the sameas in ξB by a factor of two. By doing this we know that, up to a constant, ξc will have all thesame properties as ξB and the arguments used for ball shaped regions in [6] should also apply toconstant cut cones. This suggests that for constant time slices of the cone we do not expect to getany new constraints from relative entropy inequalities. For more arbitrary cut cones, we know thevector field ξc will not coincide with the vector field ξB on Σ. In this case it is not as obvious howto extend the vector field away from Σ. We hope that new constraints will arise by understandingξc away from Σ for arbitrary cuts and using it in the formalism described in section 3.5.43Chapter 4ConclusionIn this thesis we have tried to obtain constraints on AAdS spacetimes using information theoreticquantities for holographic CFT states that are dual to such geometries. In chapter 2, we trans-lated the constraint that ACHI −ART ≥ 0 to statements about the asymptotic structure of AAdS3spacetimes that have translation invariance in the boundary coordinates. This was done by findingseries expansions in the proper length of the boundary intervals for both the area of the CHI andRT curves. These series expansions were used to construct the series expansion for the quantityACHI − ART . The constraints on the asymptotic geometry from this conjecture was obtained byrequiring that the leading order term be positive. We found that the first two leading order con-straints provided no new information about the possible asymptotic structure of AAdS3 spacetimes.However, when the results were compared to the constraints obtained from the series expansionin small z of the null energy condition Tµνuµuν ≥ 0 for null vectors parallel to the boundary; wefound that the first two leading order terms gave the exact same constraints as what we got bysimply considering ACHI − ART ≥ 0. We proposed that this observation may be a result of someinteresting connection between the constraint ACHI − ART ≥ 0 and the null energy condition inthe bulk. In chapter 3 we reviewed the progress of a ongoing research project whose goal is tounderstand the bulk dual of relative entropy constraints for holographic states defined on cut nullcone regions on the boundary CFTd. We derived the modular Hamiltonian cone regions whosebase is cut. Our strategy was to start with the result for the null plane given in [15], and usinga conformal transformation to get the result on the cone. We then derived the RT surface in thebulk for pure AdSd+1 spacetime anchored to the cut cone region on the boundary. Using theseresults we argued that for sub-regions on the boundary that are on null cones, one could still usethe machinery developed in [6] for ball shaped regions. We gave a rough sketch as to how onecan begin to prove that relative entropy between states on cone subregions is dual to quasi-localenergy in the bulk. We argued that for cone regions with constant time cuts, the constraints wouldbe identical to the constraints from ordinary ball shaped regions. For future work we stated thatwe need to understand the quasi-local energy for more generally cut cone sub-regions, and whatit has to say about the constraints that relative entropy imposes. The work in chapter 3 can bethought of as a starting point for understanding relative entropy duals for regions on the boundary44that are deformed away from the ball. By carefully studying these quantities we hope to sharpenour understanding of the role that relative entropy plays in the reconstruction and dynamics ofAAdSd+1 spacetimes.45Bibliography[1] J. Maldacena, “The large n limit of superconformal field theories and supergravity,” Int. J.Theor. Phys., vol. 38, pp. 1113–1133, 1999. → pages 1[2] M. V. Raamsdonk, “Lectures on gravity and entanglement,” in Proceedings, TheoreticalAdvanced Study Institute in Elementary Particle Physics: New Frontiers in Fields andStrings, 2017, pp. 297–351. → pages 1[3] H. Nastase, “Introduction to ads-cft,” arXiv:0712.0689 [hep-th], 2007. → pages 1[4] N. Lashkari, C. Rabideau, P. Sabella-Garnier, and M. V. Raamsdonk, “Inviolable energyconditions from entanglement inequalities,” JHEP, p. 067, 2015. → pages 1, 18, 24[5] S. Ryu and T. Takayanagi, “Holographic derivation of entanglement entropy from ads/cft,”Phys. Rev. Lett., vol. 96, p. 181602, 2006. → pages 1, 4, 5[6] N. Laskari, J. Lin, H. Ooguri, B. Stocia, and M. V. Raamsdonk, “Gravitational positiveenergy theorems from information inequalities,” PTEP, vol. 2016, May 2016. → pages 1, 26,28, 29, 37, 41, 42, 43, 44[7] N. Lashkari and M. V. Raamsdonk, “Canonical energy is quantum fisher information,”JHEP, vol. 04, p. 153, 2016. → pages 1, 26, 28[8] M. Rangamani and T. Takayanagi, “Holographic entanglement entropy,” Lect. Notes Phys.,vol. 931, pp. 1–246, 2017. → pages 4, 5[9] K. Skenderis, “Lecture notes on holographic renormalization,” Class. Quant. Grav, vol. 19,pp. 5849–5876, 2002. → pages 7, 8[10] S. de Haro, K. Skenderis, and S. Solodukhin, “Holographic reconstruction of spacetime andrenormalization in the ads/cft correspondence,” Commun. Math. Phys., vol. 217, pp.595–622, 2001. → pages 7, 8[11] S. Carroll, Spacetime and Geometry An Introduction to General Relativity. Addison Wesley,2004. → pages 7[12] V. Hubeny and M. Rangamani, “Causal holographic information,” JHEP, vol. 06, p. 114,2012. → pages 10[13] W. Kelly and A. Wall, “Coarse-grained entropy and causal holographic information inads/cft,” JHEP, vol. 03, p. 118, 2014. → pages 11[14] T. Faulkner, M. Guica, T. Hartman, R. Myers, and M. V. Raamsdonk, “Gravitation fromentanglement in holographic cfts,” JHEP, vol. 03, p. 051, 2014. → pages 26, 27, 28, 4246Bibliography[15] H. Casini, E. Teste, and G. Torroba, “Modular hamiltonians on the null plane and themarkov property of the vacuum state,” J. Phys., vol. A50, no. 364001, 2017. → pages 29, 30,33, 44[16] C. Frye and C. Efthimiou, “Spherical harmonics in p dimensions,” arXiv:1205.3548[math.CA], 2012. → pages 3547Appendix ASupplementary MaterialA.1 Co-Dimension 2 Extremal Surface in d+ 1 DimensionalSpacetimeIn this section, we will go over how one can define a Co-dimension 2 surface in a d+ 1 dimensionalspacetime as well as the equations the surface must obey to be extremal. Start with a metric for thed+ 1 dimensional spacetime gµν(X). The metric is a function of the coordinates of the spacetimeXµ. To define a co-dimension two surface in the spacetime we write two of the coordinates whichlabel with capital letter indices XB1 and XB2 . The remaining coordinates of the d− 1 coordinateswhich we label by lower case latin letters Xa will serve as the coordinates on the co-dimension 2surface we collectively label these coordinates as σa = Xa. Now one can define the d−1 dimensionalinduced metric, γab,on the surface as follows:γab(σ) = gµν(X(σ))∂Xµ∂σa∂Xν∂σb(A.1.1)Using the induced metric one can define an area functional for the surface in terms of thedeterminant of the induced metric γ:A =∫ √γdd−1σ (A.1.2)Now we can consider fixing the background metric gµν and doing a variation to the surface.We want to know when the variation of the area functional vanishes. This amounts to havingXB1 → XB1 + δXB1 and XB2 → XB2 + δXB2 and calculating the difference to first order in δX:δA =∫ √det[gµν(X + δX)∂(Xµ + δXµ)∂σa∂(Xν + δXν)∂σb]−√det[gµν(X)∂Xµ∂σa∂Xν∂σb]=∫ √det [γab + gµν (∂aδXµ∂bXν + ∂aXµ∂bδXν) + ∂aXµ∂bXν∂ρgµνδXρ + ...]−√γ(A.1.3)48A.2. Quadratic AnalysisNote that we have the functional in the form δA =∫ (√det(γab + δγab + ...)−√γ)dd−1σ,this is easily expanded to first order using the formula δA =∫12√γγabδγabdd−1σ where δγab =gµν (∂aδXµ∂bXν + ∂aXµ∂bδXν) + ∂aXµ∂bXν∂ρgµνδXρ. Plugging into our formula we find:δA =∫12√γγab (2∂aδXρ∂bXµgµρ + ∂aXµ∂bXν∂ρgµνδXρ) dd−1σ (A.1.4)After integrating the first term by parts and using the fact the variation should vanish at theboundary we are left with the result:δA =∫ [12√γγab∂aXµ∂bXν∂ρgµν − ∂a(√γγab∂bXµgµρ)]δXρ (A.1.5)This gives us the condition for the co-dimension 2 surface to be extremal:δAδXB=12√γγab∂aXµ∂bXν∂Bgµν − ∂a(√γγab∂bXµgµB)= 0 (A.1.6)Where B = B1, B2.A.2 Quadratic AnalysisWe want to understand for what values of C the following quadratic will be greater than zero onthe interval x ∈ (0, 1):x2 + Cx+ 1 ≥ 0 (A.2.1)Start by noting that if C ≥ 0 then the inequality holds trivially on our interval. The onlypossible way it could be less than zero is for some set of values C ≤ 0. Start by calculating theroots which will be given by:x =−C ±√C2 − 42=−C ±√(C + 2)(C − 2)2(A.2.2)We assume that C ≤ 0 then C−2 ≤ 0 in order for the root to be real we require that C+2 ≤ 0.If C = −2, then x = 1. Now consider C = −2− ,  > 0. It follows that the real roots are:x± = 1 +2±√(+ 4)2(A.2.3)Considering the minus root we have that:x− = 1 +2−√2 + 42≤ 1 + 2−√22= 1⇒ x ≤ 1 (A.2.4)Hence if C < −2 there will always be a root in the interval x ∈ (0, 1). Now consider the casewhen −2 ≤ C ≤ 0. These two conditions imply that C2 − 4 ≤ 0. These automatically tell us thatthere are no real roots and since the quadratic has a y-intercept of 1, then the quadratic is positive.Hence we find that the inequality (A.1) is satisfied in the interval (0, 1) if C ≥ −2.49A.3. Mapping Half Space to a BallA.3 Mapping Half Space to a BallIn this section we will go over the special conformal transformation (SCT) that will map the halfspace on a constant time slice to a ball shaped region on a Minkowski background with signature(−1, 1, ..., 1). We start by introducing coordinates to the half space given as xµ = (x0, x1, ..., xd−1)then define the following change of coordinates:yµ(x) =xµ − (x · x)cµ1− 2(c · x) + (c · c)(x · x) + 2R2cµ (A.3.1)Where cµ = (0,−1/(2R), 0, ..., 0) and x ·x = ηµνxµxν . It is straight forward to check that thesechange of coordinates will change the flat Minkowski metric by a local scale factor, which impliesthat this is a conformal change of coordinates. In particular one can show:ηµν∂yµ∂xα∂yν∂xβ=1Ω2(x)ηαβΩ(x) = 1− 2(c · x) + (c · c)(x · x)(A.3.2)This implies that:ds2 = ηµνdyµdyν =1Ω2(x)ηαβdxαdxβ (A.3.3)Now we will show that the half space described by the points {xµ : x1 > 0, x0 = 0} are mappedto a ball. To do this we start by calculating y · y = ηµνyµyν in terms of the coordinates x. We findthat:y · y = Ω(x)R2 − 2x1RΩ(x)⇒ x1 = Ω(x)2R(R2 − y · y) = Ω(x)2R(R2 + (y0)2 − |~y|2) (A.3.4)By using the SCT one can verify that the set of points on the constant time slice x0 = 0are mapped to points on the constant time slice y0 = 0. On this time slice one can verify thatΩ(x0 = 0) ≥ 0. Using this information one can see that points in the region {xµ : x1 > 0, x0 = 0}are mapped to points in the region {yµ : |~y| ≤ R, y0 = 0}. This proves the statement that the halfspace is mapped to a ball shaped region on a constant time slice. We can also calculate the inverseof the SCT transformation given by equation (B.1). This will amount to finding xµ(y) with theproperty that xµ(yµ(x¯)) = x¯µ. To do this we split the SCT given by (B.1) into two parts given by:yµ(x) = y′µ(x) + 2R2cµ = y′µ(x) +cµ2c2(A.3.5)Where we defined:y′µ(x) =xµ − (x · x)cµ1− 2(c · x) + c2(x · x) (A.3.6)50A.3. Mapping Half Space to a BallWe make the claim that the inverse of y′µ(x) is given by the following:xµ(y′µ) =y′µ + (y′ · y′)cµ1 + 2(c · y′) + c2(y′ · y′) (A.3.7)To verify this claim start by noting that:y′(x¯) · y′(x¯) = x¯ · x¯Ω(x¯)(A.3.8)Using equation (B.6) we find that:y′µ(x¯) + y′(x¯) · y′(x¯)cµ = x¯µΩ(x¯)(A.3.9)Using this, calculate xµ(y′µ(x¯)) and find:xµ(y′µ(x¯)) =x¯ω(y′µ(x¯))Ω(x¯)(A.3.10)Where we defined ω(y)as :ω(y) = 1 + 2(c · y) + c2(y · y) = 14− y12R+y · y4R2(A.3.11)One can check that ω(y′µ(x¯))Ω(x¯) = 1 this proves our claim. Now we can use the result (B.7)and substitute for the argument y′µ = yµ − cµ2c2this will give us the inverse of the SCT given by(B.1) we find that:xµ(y) =yµ − cµ2c2+ (yν − cν2c2)(yν − cν2c2 )cµ1 + 2cν(yν − cν2c2 ) + c2(yν − cν2c2)(yν − cν2c2 )=yµ + 2(y · y)cµ14 + c · y + c2(y · y)− cµc2(A.3.12)We can also give an interpretation of ω(y) by the following argument. Using (B.12) we can seethat:x1(y) =R2 − y · y2Rω(y)(A.3.13)Comparing this with (B.4) tells us that:Ω =1ω(y)(A.3.14)Hence ω will be the scale local scale factor in particular we can see by rearranging (B.2) that:ηαβ∂xα∂yµ∂xβ∂yν=1ω2(y)ηµν (A.3.15)Now we want to show that the co-dimension 1 null surface given by setting x− := x0 − x1 = 0gets mapped to the past null cone of the point (y0 = R, 0, ..., 0). To see this we begin by calculating51A.4. Calculating Jacobian for SCT|~y|2 in terms of y we find that:|~y|2 = (x0 −RΩ)2 + 2RΩx−Ω2(A.3.16)Now we set x− = 0 and use the fact y0 = x0/Ω this gives us:|~y|2 = (y0 −R)2 ⇒ |~y| = R− y0 (A.3.17)Which defines the surface of a past null cone with its tip at (y0 = R, 0, ..., 0).A.4 Calculating Jacobian for SCTHere we derive the equation for the elements of the Jacobian matrix as well as its determinant. Forthe transverse coordinates we know that:yi =xiΩ(x)Ω(x) =−(x+ + 2R)(x− − 2R) + (x⊥)24R2(x⊥)2 =d−1∑i,j=2δijxixj(A.4.1)Now we compute the elements of the Jacobian associated with the mapping above:∂yi∂xk=∂xi∂xkΩ− xi ∂Ω∂xkΩ2(A.4.2)We can calculate ∂Ω∂xkas follows:∂Ω∂xk=∂∂xk[∑d−1i,j=2 δijxixj4R2]=xk2R2(A.4.3)Combining everything gives the result:[J⊥]ik =∂yi∂xk= Ω−1[δik −xkxi2R2Ω](A.4.4)We compute the determinant of the matrix by finding its eigenvalues. To find the eigenvalueswe need eigenvectors. We can write the eigenvectors in a basis where the first eigenvector is givenas v2 =∑d−1k=2 xk∂k. If we apply this vector to the Jacobian we will find:d−1∑k=2[J⊥]ikvk2 = Ω−1[1− (x⊥)22R2Ω]xi = Ω−1[1− (x⊥)22R2Ω]vi2 (A.4.5)It has an eigenvalue of Ω−1[1− (x⊥)22R2Ω]. We can choose the other d − 3 eigenvectors to be52A.5. Coordinates on Null Plane to Coordinates on Null Coneorthogonal to v2, this implies that∑d−1k=2 xkvkb = 0, b ∈ {3, 4, ..., d− 1}. Hence we see that:d−1∑k=2[J⊥]ikvkb = Ω−1vkb (A.4.6)Which states that we have d − 3 eigenvalues of Ω−1. The determinant of the matrix is theproduct of eigenvalues this gives the result:det([J⊥]ik) = J⊥ = Ω2−d[1− (x⊥)22R2Ω](A.4.7)A.5 Coordinates on Null Plane to Coordinates on Null ConeIn appendix A.3 we defined a change of coordinates which was a SCT that maps a null sheet to anull cone. The mapping was done between cartesian coordinates on the plane (x0, x1, x2, ..., xd−1)and cartesian coordinates on the cone (y0, y1, y2, ..., yd−1). We had a complete understanding ofthe maps that go from one coordinate to the other. Here we want to write the coordinates on theplane in terms of cartesian null coordinates (x+, x−, x2, ..., xd−1) and use the SCT to go to radialnull coordinates on the cone (ρ+, ρ−, φ1, ..., φd−2). Recall that from equation (A.3.12):xµ(y) =yµ + 2(y · y)cµ14 + c · y + c2(y · y)− cµc2=yµ + 2(y · y)cµω− cµc2(A.5.1)Now we compute x± = x0 ± x1 which is given by:x+ =y0ω+R2 − y · y2Rω=(R+ y0)2 − |~y|22Rω=(R+ ρ−)(R+ ρ+)2Rω(A.5.2)x− =y0ω− R2 − y · y2Rω= −(R− y0)2 − |~y|22Rω= −(R− ρ+)(R− ρ−)2Rω(A.5.3)xi =yiω(A.5.4)where ρ± = y0 ± |~y| and ω = 14 − y12R +y·y4R2. Notice from these coordinates that it is clear thatif ρ+ = R then x− = 0, applying these restrictions, one can easily relate the null coordinate on theplane which is x+ and the null coordinate on the cone which is ρ−:x+ =R+ ρ−ω|ρ+=R(A.5.5)53A.6. Conformal Transformation of the Stress Energy Tensor of a CFTdWe can explicitly calculate ω|ρ+=R as follows:ω =14− y12R+y · y4R2=R2 − 2Ry1 − (y0)2 + |~y|24R2= −ρ+ρ− +R(2y1 −R)4R2⇒ ω|ρ+=R = −ρ− + 2y1 −R4R(A.5.6)We can also write ρ− in terms of x+ by rearranging C.5 and using Ω = 1/ω:ρ− =x+Ω|x−=0−R (A.5.7)Where we explicitly can compute Ω|x−=0 as follows:Ω = 1 +x1R+x · x4R2=−(x0)2 + (x1 + 2R)2 + (x⊥)24R2=−(x− − 2R)(x+ + 2R) + (x⊥)24R2⇒ Ω|x−=0 = 1 +x+2R+(x⊥2R) (A.5.8)This gives us the equation used in equation (3.12) :ρ− =x+1 + x+2R +(x⊥2R)2 −R (A.5.9)A.6 Conformal Transformation of the Stress Energy Tensor of aCFTdHere we go over the calculations for applying a conformal transformation to the stress energy tensorcomponent when we apply the SCT defined in appendix A.3. The conformal transformation of thestress energy tensor associated with the SCT can be implemented through a standard change ofcoordinates to the tensor along with Weyl rescaling to make the background metric flat again. Westart by calculating the general elements of the Jacobian matrix associated with the SCT:∂yµ∂xν=∂∂xν[xµ − (x · x)cµΩ]=δµν − 2xνcµΩ−∂Ω∂xν (xµ − (x · x)cµ)Ω2=1Ω[δµν − 2xνcµ −∂Ω∂xν(yµ − cµ2c2)]=1Ω[δµν − 2xνcµ −(−2cν + 2c2xν)(yµ − cµ2c2)]=1Ω[δµν − xνcµ + 2cνyµ − 2c2xνyµ −cνcµc2]=1Ω[δµν − δµ1 δ1ν −yµRδ1ν +xν2R(δµ1 −yµR)](A.6.1)54A.6. Conformal Transformation of the Stress Energy Tensor of a CFTdBy similar calculations one can show:∂xµ∂yν= Ω[δµν − xµcν + 2cµyν − 2c2xµyν −cνcµc2](A.6.2)We want to use these to calculate how the stress energy tensor will change from the change incoordinates:T++ =∂yµ∂x+∂yν∂x+Tµν (A.6.3)We can use the formula (A.6.1) to calculate the partial derivative:∂yµ∂x+=12(∂yµ∂x0+∂yµ∂x1)=12Ω[δµ0 −yµR+x+2R(δµ1 −yµR)](A.6.4)From this point forward throughout this section we will be restricted to the null plane. Weknow that x+ = −x− = 0. This means that:T++∣∣x−=0 =14Ω2(δµ0 −yµR)(δν0 −yνR)Tµν∣∣ρ+=y0+|~y|=R (A.6.5)Now we make the following claim:δµ0 −yµR=2|~y|R∂yµ∂ρ−(A.6.6)To see this is true, we start by calculating the following quantity:|~y|R∂ρ−∂yµ+ 2δ0µ(1− |~y|R)=|~y|Rδ0µ −yµR(1− δ0µ) + 2δ0µ(y0R)= δ0µ(y0 + |~y|R)− yµR= δ0µ −yµR(A.6.7)Where we used the fact that y0 + |~y| = R. Now we can compute the following quantity:(δ0ν −yνR)(δν0 −yνR)=R2 + y · yR2=2|~y|R(A.6.8)At the same time we also can compute:[ |~y|R∂ρ−∂yµ+ 2δ0µ(1− |~y|R)][2|~y|R∂yµ∂ρ−]=2|~y|2R2+4|~y|R(1− |~y|R)∂y0∂ρ−=2|~y|R(A.6.9)Where we used the fact that y0 = 12(ρ+ + ρ−). This means that we know the left hand sides of(A.6.8) and (A.6.9) are equal. Furthermore, using equation (A.6.7), we can deduce that that the55A.7. Unit Binormal to RT surface Anchored to Cone Regionsclaim given in equation (A.6.6) is true. Using the result gives:T++∣∣x−=0 =1R2Ω2(R− ρ−2)2T˜−−∣∣ρ+=R(A.6.10)This takes care of the coordinate transformation. In order to complete the conformal trans-formation, we need to do a Weyl rescaling of the stress energy tensor to figure out what powerof the conformal factor we need we apply the following argument. Suppose we have a confor-mal change of coordinates such as in the SCT. Then we know that when we go from coordinates(x0, x1, ..., xd−1) → (y0, y1, ..., yd−1) the new metric will be rescaled ηµν → Ω2ηµν . Now considerthe term in the action where the stress energy tensor couples to the metric:∫ddy√det(Ω2ηµν)ηµνΩ2T˜µν =∫ddxΩd−2ηµν T˜µν =∫ddxηµνTµν (A.6.11)In order to cancel the conformal factor we need to rescale the stress energy tensor by Ω2−d.Using this the conformally transformed stress energy tensor is now:T++∣∣x−=0 =1R2Ωd(R− ρ−2)2T˜−−∣∣ρ+=R=ωdR2(R− ρ−2)2T˜−−∣∣ρ+=R(A.6.12)Which gives the result in equation used in chapter 3 equation (3.3.19).A.7 Unit Binormal to RT surface Anchored to Cone RegionsIn this section we want to derive the unit binormal on the extremal surface we derived in the previoussection. To calculate the unit binormal to our extremal surface start by calculating the d−1 tangentvectors to the surface which will be labeled with the index a as Ta = T µa ∂µ, a ∈ {1, 2, ..., d − 1}.We can also write in component form Ta = (T +a , T −a , T θa , T ia ) in the coordinate basis. These vectorswill have components that satisfy the following equations:T µa ∂µ(r+ −R) = T +a = 0 (A.7.1)T µa ∂µ(r− − Λ(θ, φi)) = T −a − T θa ∂θΛ− T ia∂iΛ = 0 (A.7.2)The first equation tells us that the tangent vectors will have no components in the direction ∂+.Now we will introduce a slightly abusive labeling of the tangent vectors. We let labelling indices tobe coordinate indices a ∈ {θ, φj}, then we can define the following vectors:Ta = (∂aΛ)∂− + δθa∂θ + δia∂i (A.7.3)56A.7. Unit Binormal to RT surface Anchored to Cone RegionsWe can clearly see that these satisfy the second equation. We can check for orthogonality:gµνT µa T νb = gθθT θa T θb +d−2∑i=1giiT iaT ib = gθθδθaδθb +d−2∑i=1giiδiaδib = gaaδab (A.7.4)Where we used the fact that the metric on the unit d − 2 sphere is diagonal and g−− = 0,normalization can be trivially done by dividing by the metric components. These vectors forman orthogonal basis d − 1 basis on the Ryu-Takayanagi surface. Now we need to find two morevectors n1 and n2 that are orthogonal to each other and the tangent vectors we defined. Start withthe most general form for the normal vectors with no constraints on the components n1 = nµ1∂µand n2 = nµ2∂µ. Now we write the condition that the vectors should be orthogonal to the tangentvectors starting with n1:gµνnµ1T νa = g+−n+1 ∂aΛ + gθθnθ1δθa +d−2∑i=1giini1δia = 0 (A.7.5)This equation constrains the components d− 1 components na1 = −g+−n+1 ∂aΛgaa. The exact sameargument holds true for n2 hence we have that the following two vectors with be normal to alltangent vectors:n1 = n+1 ∂+ + n−1 ∂− + na1∂a (A.7.6)n2 = n+2 ∂+ + n−2 ∂− + na2∂a (A.7.7)To simplify calculations we let n+1 = n+2 = g+−. This implies that na1 = na2 and the orthogonalitycondition between the two vectors will be:gµνnµ1nν2 = (n−1 + n−2 ) +∑a=θ,i(∂aΛ)2gaa= 0 (A.7.8)We also want gµνnµ1nν1 = 1 this means that:n−1 =1−∑a=θ,i (∂aΛ)2gaa2(A.7.9)Finally, we can use the orthogonality condition to get the component n−2 :n−2 = −1 +∑a=θ,i(∂aΛ)2gaa2(A.7.10)We can verify that gµνnµ2nν2 = −1. In summary we found the normalized normal vectors to the57A.7. Unit Binormal to RT surface Anchored to Cone RegionsRyu-Takayanagi surface to be:n1 = g+−∂+ +1− Z2∂− −∑a=θ,i∂aΛgaa∂a (A.7.11)n2 = g+−∂+ − 1 + Z2∂− −∑a=θ,i∂aΛgaa∂a (A.7.12)Z =∑a=θ,i(∂aΛ)2gaa(A.7.13)Now we can define the unit binormal components using the normal vector components:nab = na2nb1 − nb2na1 (A.7.14)One can check that the only non-zero components of the binormal are given by:n+− =1g+−nθ− = −∂θΛgθθni− = −∂iΛgii(A.7.15)58


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