On the (A)dS/CFT correspondence and the Hartle-Hawking-Maldacena proposal

Transcription

1 On the (A)dS/CFT correspondence and the Hartle-Hawking-Maldacena proposal Milad Nafeli, Vrije Universiteit Amsterdam Supervisor: dr. Sebastian de Haro, Second Examiner: prof. dr. Piet Mulders Abstract This thesis focuses on the Anti-de Sitter / Conformal Field Theory (AdS/CFT) correspondence and notably the de Sitter / Conformal Field Theory (ds/cft) proposed conjecture by way of the HHM-proposal. This proposal states that on the gravity side one can analytically continue physics from Euclidean AdS (EAdS) to ds via a wick rotation and that this ds should have a hypothetical dual CFT. Furthermore, we introduced asymptotic metrics in conformal cylindrical coordinates to describe EAdS and ds near the boundary. Using these metrics we have performed new calculations on the holographic stress-energy tensor in d = asymptotically EAdS, which pertains to the Weyl anomaly. We compared it to the Weyl Anomaly in plain EAdS and using the HHM-proposal we predicted the ds holographic stress-energy tensor by analytically continuing the asymptotically EAdS finding. We confirmed this finding by calculating the ds holographic stress-energy tensor independently.

2 Contents Introduction. Conformal field theories The Bulk Anti-de Sitter spacetime De Sitter spacetime Asymptotic (A)dS spaces AdS/CFT Hawking-Hartle-Maldacena proposal 4. Calculations in EAdS Calculations in ds Further remarks on the structure of the space-time Einstein equations in the bulk 8 3. Euclidean Anti-de Sitter spacetimes Poincaré coordinates Conformal cyndrical coordinates De Sitter spacetime in conformal cyndrical coordinates The Holographic Stress-Energy Tensor 3 4. Euclidean Anti-de Sitter in conformal cyndrical coordinates De Sitter in conformal cyndrical coordinates Conclusion 8 Introduction Quantum Field Theory (QFT) and the Standard Model are the most successful theories in the history of physics to date, since it reconciled three of the four fundamental forces: weak, strong and electromagnetic. The forth, being gravity, is described by General Relativity (GR) on large scales. QFT is a theory that combines quantum mechanics (QM) with special relativity. While in classical mechanics one describes a particle by its position as a function of time (i.e x(t)), in quantum mechanics one describes a particle by its wave function as a function of position and time (i.e ψ( x, t)). Moving on to QFT, the function ψ( x, t) attains a different meaning. It describes a field that encompasses the whole space-time, and particles are localized excitations of that field. This change of perspective has not only united QM and special relativity, but also paved the way to predicting a plethora of particles. This zoo of particles has been experimentally confirmed and is known as the Standard Model. However, QFT breaks down when applying it to gravity and thus when trying to reconcile it with GR. Both theories are the standard in their domains, but when a phenomenon requires both theories they do not mix. Moreover, QFT has problems of its own due to the localized nature of particles as excitations of the field. Localization assumes weak coupling, but if the coupling is strong the wave functions start to overlap and the QFT mathematical framework breaks down. The best candidate known at present for such a quantum gravity theory is string theory, which uses non-local operators and has produced numerous exotic and interesting predictions.

3 One such theory is the gauge/gravity duality, most notably the famous AdS/CFT duality proposed by Maldacena []. AdS/CFT stands for Anti-de Sitter spacetime / Conformal Field Theory. The former describes a theory of gravity on a space-time with a negative cosmological constant Λ. The latter describes a special set of quantum fields that have extra symmetries, namely scale invariance and special conformal symmetries. The duality states that objects in AdS and some corresponding CFT are equivalent and a natural phenomena in on theory can also be calculated in the other theory. Moreover, AdS/CFT has the following key features []: There is a mapping between a quantum theory of gravity (namely string theory) and an ordinary (non-gravitational) quantum field theory. Furthermore, the theory is argued to be background independent [3]. This duality could give insight in to quantum gravity by studying the dual quantum field theory. AdS/CFT is a strong/weak coupling duality: when the gauge theory is strongly-coupled (and hence all perturbative techniques fail), we can study it using the weakly-coupled string theory context. The AdS/CFT mapping is holographic. A (d + )-dimensional AdS gravity theory (the bulk) is equivalent to a d-dimensional CFT on the boundary of the bulk. AdS/CFT has undergone numerous tests (most notably [4]) and has also been applied in strongly coupled condensed matter physics [5 7]. In light of this profound theory it is important to note that the cosmological constant in our observable universe is in fact positive and not negative as in the anti-de Sitter space. Therefore, there is a natural motivation from a cosmological viewpoint to find a de Sitter / CFT correspondence. If such a correspondence exists it would have a great impact on our understanding of the universe. It proves to be a non-trivial endeavour to find such a ds/cft correspondence. The AdS/CFT construction greatly benefits from explicit string solutions in the bulk. These solutions have not been found yet for a dual to ds, however, promising progress has been made in [8]. Nonetheless, several attempts have been made to make a consistent ds/cft [9 ], with the Hawking-Hartle- Maldacena (HHM) proposal [3] being the simplest, which is the focus of this thesis. Briefly, the HHM-proposal entails that the wave function of a universe can be computed in terms of a conformal field theory Ψ HH [g] = Z[g], (.) where Ψ[g] is the Hartle - Hawking (HH) wave function (described in [4]) and the right hand side is the partition function of some dual conformal field theory. Then the HH state can be obtained by calculating the corresponding renormalized on-shell action on Euclidean AdS 4 (EAdS 4 ) and analytically continuing it to ds 4 via l AdS il ds, (.) where l AdS and l ds are the radius of curvature of their respective spacetimes. This thesis has two parts, namely a general literature thesis on AdS/CFT in the following subsections of section and the HHM-proposal in section, where most notably the papers [5], [6] and [7] are studied. As well as a research thesis in section 3 on setting up the Einstein equations for a general holographic metric in (d + )-dimensions. Using this framework we introduce in subsection.3 new asymptotic metrics in conformal cylindrical coordinates fitting to the HHM-proposal. Finally, in section 4 we use these metrics and the HHM-proposal to 3

4 perform new calculations on the holographic stress-energy tensor T ij in EAdS, and analytically continuing it to ds. This finding will then be compared to an independent calculation of the holographic stress-energy tensor in ds.. Conformal field theories One of the cornerstones of the duality is that both sides are conformal. A conformal coordinate transformation is a local coordinate transformation such that if we take x µ x µ (x) the line element rescales by dx µ dx µ = Ω (x) dx µ dx µ e φ(x) x µ dx µ or in other words, the metric rescales as g µν = e φ(x) g µν. The special case that Ω(x) is constant is for dilatations. In general, conformal coordinate transformation rescales the length while preserving the angles between vectors. To probe the structure of Ω(x), consider it at the infinitesimal level, so Ω(x) = φ(x) + O(φ ) and x µ = x µ + v µ (x). Using this we can derive the conformal Killing equation µ v ν + ν v µ = φ(x)g µν. (.3) In particular, let us look at d-dimensional space-time and consider the flat metric (i.e. g µν = η µν ). Taking the trace to give us µ v µ = φ(x)d results in the equation µ v ν + ν v µ d ( ρ v ρ ) η µν = 0, (.4) and if we take the derivative of (.4) we get the equation ( ηµν + (d ) µ ν ) ρ v ρ = 0. (.5) We see immediately from equation (.5) that there is the special case d = such that there are infinite solutions for the Killing vectors. Equation (.5) is second order will then simply become the Cauchy-Riemann differential equation. For d > equation (.5) yields η µν ρ v ρ = 0, which has at most second order solutions for v ρ. This leads us to use the Ansatz v µ (x) = a µ + c µν x ν + b µνσx µ x σ, (.6) where a µ, c µν, b µνσ are some constants. Subjecting this to the conformal Killing equation (.4) results in c µν = ω µν + λη µν, where ω µν is an anti-symmetric tensor and λ some scalar, and we have b µνσ = b σ η µν + b ν η µσ b µ η σν, where b µ is some constant vector contracted from b µνσ. In summary this leads to the most general solution ν v µ (x) = a µ + ω µνx + λx µ + b µ x (b.x)x µ, (.7) }{{}}{{}}{{}}{{} P µ J µν D K µ where P µ the translation generator, J µν the Lorentz generator, D the dilatation generator and K µ is the special conformal transformation generator. From the K µ term we see that the special conformal transformation produces d independent components. In summary, the total independent components are d + d(d ) + + d = (d+)(d+). Motivated by this result we define a generator J ab where the indices a and b run over {, 0,,..., (d )}, such that J µν = J µν, J,µ = (P µ K m ), J,0 = D, J 0,µ (P µ + K µ ). (.8) The conformal Killing equation is a specific realisation of the normal Killing vector which is µv ν + νv µ = 0. 4

5 Then this generator obeys the algebra [J ab, J cd ] = i(g ad J bc + g bc J ad g ac J bd + g bd J ac ). (.9) For Euclidean space it is isomorphic to SO(d +, ), in Lorentzian space it becomes isomorphic to SO(d, ). Killing vectors are useful as they generate conserved quantities. To see this consider a geodesic γ(λ) with tangent vector t µ, then we can set up t µ µ (v ν t ν ) = t µ t ν µ v ν + v ν t µ µ t ν (.0) The second term vanishes due to the geodesic equation dt dλ + Γν αβ tα t β = t µ µ t ν = 0. The first term can be rewritten using the fact that t µ t ν is symmetric t µ t ν µ v ν = tµ t ν ( µ v ν + ν v µ ). (.) This would simply vanish for normal killing vectors and v ν t ν would be conserved for any geodesic. However, for conformal killing vectors this is more subtle, using equation (.4) gives t µ µ (v ν t ν ) = t µ t µ ρ v ρ. (.) Now v ν t ν is only conserved if t µ t µ = 0 which corresponds to null geodesics. Referring to normal killing vectors, Wald [8] argues that space-time motions are represented by timelike geodesics and light rays by null geodesics, such that these killing vectors give rise to a conserved quantity for particles and light rays. However, in our case this means that conformal killing vectors only generate conserved quantities for massless objects. This subtle difference in conformal killing vectors also constraints the trace of the stress-energy tensor. We will show this in two different ways. The first method is to look at the conserved current j µ = T µν v µ, where T µν is the symmetric stress-energy tensor. Taking the derivative of this current results in µ (T µν v ν ) = µ T µν v ν + T µν µ v ν = T µν ( µ v ν + ν v µ ) = d ρ v ρ T µ µ. (.3) So to require that j µ is conserved the trace of the stress-energy tensor is constrained to be zero (i.e T µ µ = 0). The second method is to manipulate the action of a CFT. We add a source term g µν, which is in fact the metric, to the standard action of a conformal theory δs = δs 0 + d d x T µν δg µν. (.4) The generating functional in this case is W [g] ln Z[g]. Using this T µν can be found by doing a functional derivative of W [g] with respect to the source g T µν = g δw [g] δg µν. (.5) To have conformal invariance, we need to have that under a variation of a conformal coordinate transformation δ c the generating functional is equal to zero. We know that under variation of conformal coordinate transformation δ c, the metric behaves like δ c g µν = Ω(x) g µν, thus δ c W [g] = d d x 5 δw [g] δg µν δ c g µν = 0. (.6)

6 Substituting in (.5) gives d d x g T µ µ Ω(x) = 0, (.7) which implies T µ µ = 0 for any arbitrary function Ω(x). So summarizing, for a CFT the stress energy tensor is not only conserved but also traceless µ T µν = 0, T µ µ = 0. (.8) Generally, in quantum field theories like QCD and QED, the conformal invariance is broken by having a stress-energy tensor with a trace that is non-zero. In Yang-Mills theory for example the trace of the stress-energy tensor is T µ µ β(g)f. The theory becomes scale invariant if the β-function, which describes the dependence of th coupling g to the energy scale µ, becomes zero. The beta function can vanish in two cases: when the theory is finite, in the sense that there are no divergences in the theory, or when g runs over a curve that has minima or maxima with respect to µ, such that β(g) = µ dg = 0. These are called fixed points. In this case the dµ Renormalization Group equation becomes simply the ward identity for translations. In essence the β-function introduces a scale at the quantum level and it is due to the effects of interaction. For a general massless quantum theory the Lagrangian is scale invariant, however, the Green s functions controlling the interaction are not if the β function is non-zero. With scale invariance it makes no sense to define an interaction in local coordinates as is done in QFT. This is because every state with a given energy level could be scaled to any arbitrary energy level from zero to infinity with dilatations. To circumvent this problem, CFT interactions are calculated using correlation functions. Specifically we are interested with guage theories with correlation functions that have good dilatation properties. The conformal symmetry is so strong in such a CFT, that it determines the form of -point functions and 3-point function. The -point function is zero due to scale invariance, while the two-point function is of the form O i (x)o j (y) = Aδ ij x y i, (.9) where the constant A is determined by the field properties and is the scaling dimension. The scaling dimension stems from the fact that under a dilatation x λx any given operator picks up a factor λ, which in the case of a CFT helps to constraint the form of the operator. For a 3-point function we have O i (x)o j (y)o k (z) = λ ijk x y i+ j l y x j + k i z x k+ i j, (.0) where again the constant λ ijk is determined by the field properties and generally higher point functions are fixed by the scaling dimension and the three-point coefficients λ ijk. Fields that are annihilated by the lowering operator K µ are called primary operators and have a scaling dimension and quantum number attributed to them.. The Bulk The primary focus of this thesis concerns maximally symmetric spacetimes in the vacuum bulk (T µν = 0), which are spacetimes that have a maximum numbers of Killing vectors: D(D + )/ for a D-dimensional spacetime. This is because a maximally symmetric space is both homogenous and isotropic. Homogeneity implies that at any point in spacetime there is a Killing vector 6

7 in the direction of any other point, thus there are D Killing vectors. In an isotropic spacetime a point in spacetime remains unchanged under the coordinate transformation x µ = x µ + ξ µ (x), thus ξ µ (x) = 0, however its derivative is non-zero. In this case we are dealing with normal Killing vectors ξ µ such that they satisfy µ ξ ν + ν ξ µ = 0, (.) thus µ ξ ν represents an antisymmetric tensor, which gives D(D )/ Killing vectors. Combining all vectors indeed gives D(D + )/ Killing vectors and we have the constraints ξ µ = 0, ν ξ µ = µ ξ ν (.) The fact that we have a maximum number of Killing vectors implies that we have a constant curvature, in other words the Riemann tensor should be expressed by the Ricci scalar. To see this start of with the definition the Riemann tensor for a vector [ µ, ν ] = R σ µνρξ σ, (.3) which holds for any vector and thus also for the Killing vector ξ µ. We can use the Killing equation and the permutation identity of the Riemann tensor Rµνρ σ + Rρµν σ + Rνρµ σ = 0 to rewrite equation (.) to µ ν ξ ρ = Rνρµξ σ σ. (.4) We can use the definition of the Riemann tensor again applied to µ ξ ρ, which is now a tensor [ σ, ν ] µ ξ ρ = R λ ρσν µ ξ λ R λ µσν ρ ξ λ. (.5) This equation constraints equation (.). Using this constraint and the Killing equation one can set up the following equation ( ν R λ σρµ σ R λ νρµ)ξ λ = ( R λ ρσνδ κ µ + R λ µσνδ κ ρ R λ σρµδ κ ν + R λ νρµδ κ σ) κ ξ λ. (.6) Using the constraints in (.0), the left-hand side of equation (.4) results in being equal to zero. We can use the antisymmetric constraint to set up R λ ρσνδ κ µ + R λ µσνδ κ ρ R λ σρµδ κ ν + R λ νρµδ κ σ λ κ = 0, (.7) Contracting this with g κµ and using the permutation identity of the Riemann tensor gives (D )R α ρσν = R νρ δ α σ R σρ δ α ν. (.8) From this we can calculate the Ricci tensor by contracting with g ρν, which gives DR α σ = Rδ α σ and thus R µν = D g µνr. (.9) So now we can finally construct the unique Riemann tensor for maximally symmetric spacetimes (.6) in terms of the Ricci scalar and the metric R µνσρ = This is a generalization of the the definition of the Riemann tensor [8]. R D(D ) (g νρg σµ g σν g µρ ), (.30) 7

8 and indeed the spacetime has constant curvature. Using the bianchi identity it can be shown that this holds for everywhere in space so we can also write R µνσρ = ε l (g νρg σµ g σν g µρ ), (.3) where l is the radius of curvature and ε is a signature constant which will be linked to the cosmological constant. The Einstein equations in the vacuum are R µν RG µν + ΛG µν = 0, (.3) where Λ is the cosmological constant. It is convenient from now on to use D = d+ dimensions. This equation can be simplified by taking the trace and solving for R and inserting that in to (.3). R = d + Λ. (.33) d Looking at equation (.30) and (.3) we see that ε becomes (d+)d. We can now solve for Λ which d(d ) Λ = ε l. (.34) Now the einstein equation can be written in the convenient form l = R R µν = ε d l G µν. (.35) The solutions of these space times for (d+)-dimensions may be represented by the hypersurface d ηx0 + εxd + Xi = εl, (.36) where η = ± corresponds to the metric signature and ε = ± the signature of the cosmological constant Λ. This equation describes four different spacetimes, namely: de Sitter (ds d+ ), a d+ sphere (S d+ ), anti-de Sitter (AdS d+ ), and Euclidean anti-de Sitter (EAdS d+ ). The parameters η and ε corresponding to these spacetimes are presented in table. It is interesting that ds and EAdS in (d + )-dimension both have the isometry group SO(d +, ). In the CFT side we have seen that a conformal theory in d-dimension in euclidean space has the isometry group SO(d +, ). Thus, a CFT in a lower dimension has equivalent algebra. Which is the basis principle of the HHM-proposal. Indeed when choosing EAdS (η, ε) = (, ) and performing the transformations X 0EAdS ix 0dS and l EAdS il ds as prescribed by the proposal one gets (η, ε) = (, ), which is ds space time. i= space η ε isometry group ds d+ - SO(d +, ) S d+ SO(d + ) AdS d+ - - SO(d, ) EAdS d+ - SO(d +, ) Table : The four spacetimes and their corresponding parameters as described in equation (.36). 8

9 .3 Anti-de Sitter spacetime AdS is the maximally symmetric solution with negative cosomological constant. As such it can not be applied to our universe, however, its role in AdS/CFT makes it an important spacetime to study. The quadratic equation is d X0 + Xd + Xi = l. (.37) A parametrization of AdS can be given in the following coordinates X 0 = l cosh ρ cos τ X d = l cosh ρ sin τ i= X i = l sinh ρ Ω i (i =,..., d ). (.38) These are global coordinates that describe the entire hyperboloid if we restrict ρ R + and ρ = is the boundary at spacelike infinity. The coordinate τ is restricted to τ [0, π), due to the periodicity of the metric, but encompasses the whole space. The unit vector Ω i parametrizes the (d )-sphere, satisfying Σ (d ) i= Ω i =. For example if we have d + = 5 we get the following parametrization X 0 = l cosh ρ cos τ X 4 = l cosh ρ sin τ X = l sinh ρ cos θ X = l sinh ρ sin θ cos φ X 3 = l sinh ρ sin θ sin φ. (.39) From this the AdS line element can be constructed is found to be ds = l ( cosh ρ dτ + dρ + sinh ρdω d ). (.40) The most often spacetime used in AdS/CFT is EAdS. The quadratic equation is then d X0 + Xd + Xi = l. (.4) The EAdS coordinates can be found by simply wick rotating the AdS parametrization, which we will do by using the transformation τ iτ. The parametrization then becomes i= X 0 = l cosh ρ cosh τ X d = l cosh ρ sinh τ From which we can again construct the line element X i = l sinh ρ Ω i (i =,..., d ). (.4) ds = l (cosh ρ dτ + dρ + sinh ρdω d ). (.43) 9

10 We can conveniently transform the metric to make it conformal to R S d via tan θ := sinh ρ such that the line element becomes ds = l cos θ ( dτ + dθ + sin θ dω d ). (.44) We see that θ has only the range [0, π ) as θ = π, which is the equator, produces a singularity. This means that this metric describes only half of the spacetime and this case the northern hemisphere. The singularity that arises from the conformal factor poses no problem as the equator was not part of the spacetime and spacetime is non-compact. The rest of the line element is completely regular. The boundary at which the conformal factor produces a singularity is timelike and corresponds to the region ρ, where all coordinates in (.4) become divergent. In other words under a conformal mapping a asymptotic infinity can be compactified to a finite parameter. Since the boundary of AdS are timelike and light rays reach the boundary in finite time, we must fix the boundary conditions at spacelike infinity to have a well-defined Cauchy problem. In comparison, Minowski space has lightlike boundaries and the fields only need to be specified on a spatial slice. The null rays then connect this slice causally to the entire spacetime and the values of the fields determine the time evolution. Another parametrization are the Poincaré patch coordinates. These coordinates, as we will show, will make the metric conformally flat. This will be convenient as the metric of the CFT is induced on the boundary up to a conformal factor. The coordinates are X 0 = r (r + l + g ij x i x j ) X d = r (r l + g ij x i x j ) X i = l r x i, (i =,..., d ), (.45) where g ij is a (d )-dimensional flat metric. For AdS g ij is Minkowski metric and for EAdS it is the euclidean metric. As we will see in subsection.5 the coordinate r will play an important role. As such we invert the above expression to get r = l X 0 X d. (.46) These coordinates are not global and are only conformal to half of the metric on R d+ and will only cover r > 0. The line element is ds = l r (dr + g ij dx i dx j ). (.47) The metric is conformal to half of the flat metric on R d. The boundary is located at r 0 and the boundary metric g ij is the flat metric on R d..4 De Sitter spacetime De Sitter space is the maximally symmetric solution with positive cosmology constant, which according to current observations, is a valid description of our universe. The quadratic equation is d X0 + Xd + Xi = l. (.48) 0 i=

11 Analogous to AdS we will be using the spherical parametrization which results in X 0 = l sinh(t/l) X i = l cosh(t/l) Ω i (i =,..., d), (.49) where now we have that the unit vector Ω represents a (d )-dimensional sphere instead (d ) this can be easily seen from the quadratic equation which looks like a (d )-sphere except for one timelike coordinate. The line element corresponding with this parametrization is ds = dt + l cosh (t/l)dω d, (.50) these are global coordinates and describe the full quadratic equation. This can be considered a cylindrical spacetime where t parametrizes the symmetry axis. The cosh (t/l) factor expands and contracts the spatial term of the metric as a function of t. As such the spatial volume of the universe contracts from t to t = 0. For t the universe expands indefinitely. At t = 0 the universe is just a (d )-sphere with radius l, moreover at any constant t the universe is a sphere with the radius depending on the value of t and l. The metric in Poincaré patchnotes for ds are ds = l η ( dη + g ij dx i dx j ), (.5) where the background metric g ij is Euclidean. This metric has only one boundary (η 0) and this tells us that it only describes half of the ds space-time..5 Asymptotic (A)dS spaces We can now work towards asymptotic spacetimes. We follow the steps by Skenderis made in [9]. Let us for convenience start with the EAdS d+ metric (i.e. (η,ε) = (, )) ds = l cos θ ( dt + dθ + sin θdω d ), (.5) Consider a function r(x), which is positive inside the bulk and has a first order pole at the boundary. Since it has a conformal structure at the boundary we can multiply the metric by r g (0) = r G θ= π, (.53) where g (0) is the metric at the boundary. A simple finite metric would be if we choose r = cos(θ). Then we can state that if r is a good defining function then so is r e w, where w is a smooth function with no zeros. Or in other words the metric is indeed conformal at the boundary, which we want. On to generalizing the metric we need to define conformally compact manifolds [0]. Skenderis then goes on to a more general approach by defining X to be the interior of a manifold with boundary X, and M = X be its boundary. The bulk metric G is called conformally compact has a second order pole at M, but there exists a defining function that has general properties r(m) = 0, dr(m) 0 and r(x) > 0, such that g = r G, (.54) smoothly extends to X, g M = g (0) and is non-degenerate. An asymptotically AdS space is approximately Anti-de Sitter at the boundary. To see this consider the transformation G = r ḡ = e φ ḡ. Now if the Ricci tensor is conformally transformed R µν [G] = R µν [ḡ] (d ) µ ν φ ḡ µν φ + (d ) µ φ ν φ (d )ḡ µν ( φ ), (.55)

12 which becomes R µν [G] = R µν [ḡ] r d(ḡρσ ρ r σ )ḡ µν + ( g r ρσ ( ρ σ r Γ λ ρσ λ r) g µν + (d )( µ ν r Γ ) λ µν λ r) (.56) So as r 0. The second term, which comes from the φ term, becomes the dominant term and as Graham [] defined dr ḡ = ḡ µν µ r ν r, we have R µν [G] = d r dr ḡ ḡ µν + O(r ). (.57) Then we can consider this to be a maximally symmetric space near the boundary if it satisfies the Einstein equation in equation (.33) for AdS, which is in this case R µν = d ḡ l r µν. Thus we constraint dr ḡ = near the boundary, which means that r may not depend on any other l coordinate x. The same calculations can be shown to hold for de Sitter space, which will lead in general that dr ḡ = ε. Notice how the metric is not defined explicitely as such we can consider a l general background metric g ij in Guassian coordinates. The radial coordinates is taken to be the affine parameter of the geodesic with tangent r. The radial coordinate becomes the defining function. This leads to the Fefferman-Graham metric. For AdS in Poincaré patch coordinates this is given as 3 [] ds = r (dr + g ij (x, r)dx i dx j ). (.58) We can generalize this further to the formalism we used in subsection. to get ds = G µν dx µ dx ν = e φ ( ηdr + g ij dx i dx j), (.59) where g ij is called the boundary metric, which lives in d-dimensions and is part of the (d + )- dimensional bulk metric G µν and e φ is the conformal scaling factor. In the case of ε = the g ij is Euclidean and for the case ε = it is Lorentzian. As this is a conformally compact manifold it has by construction that g ij (r, x) has a smooth limit as r, such that we can write the expansion g ij = g (0)ij + rg ()ij + r g ()ij + r 3 g (3)ij +... (.60) The coefficients g (n)i j can then be solved for n > 0 order by order. Furthermore, we will adopt a synchronous guage, as is done in [3], which entails that G ri = 0 and G rr can be set to a special value..6 AdS/CFT The isometries of AdS and CFT may be equivalent, but the dimensions where the theories live in is different. The missing link to these apparently incompatible theories is the holographic principle. There have been hints of holography, namely the maximal entropy of a black hole is determined by its area not its volume. What is more interesting is that generally the degrees of freedom of a quantum gravity theory grows with the area not the volume. Furthermore, t Hooft showed that for large N expansion 4 Yang-Mills theory the CFT started to show stringy characteristics. Motivated by this, and other more specific arguments coming from string theory (specifically: the emission and absorption rates of black holes made of D-branes), Maldacena formulated AdS/CFT in his paper []. 3 From now on l = will be taken. It can be easily reinstated by dimensional analysis.

13 In AdS/CFT the holographical principle is used wherein the CFT is said to live on the boundary of the bulk where the gravity theory lives. Which is similar to Green s theorem where a surface can be described by a closed line and vice versa. Recall that we can add a source term to the CFT action S b = S b0 + d d x h(x)o, (.6) where h is the source and O is some operator. The spectrum of the CFT is specified by primary operators. The AdS/CFT correspondence states that the source h corresponds to a field in AdS and is associated with a operator O in a CFT. Then in terms of the generating functional W [h] = ln Z[h]. The n-point function is then O...O c = δn W [g] δh n h=0. (.6) The h(x) source is a boundary of a higher dimension field in AdS that is subjected to the equations of motion of the bulk described by the effective action S bulk of AdS. In other words there is a field h(x) in AdS with a boundary at h(x, x b ), where x b at the position of the boundary. Now we can formulate the fundemental statement of AdS/CFT: e W [h] = e ho = e S AdS d+ (.63) CFT d In summary, on the left hand side we have a d-dimensional CFT and on the right hand side we have a (d + )-dimensional gravity theory. To inspect this in further detail we use the Fefferman-Graham metric and the expansion in equations (.58) and (.60) and consider AdS. So we have (η, ε) = (, ), at r 0 the metric becomes d-dimensional and the conformal metric at the boundary is g (0)ij := g ij (0, x). With this we can solve the Einstein equations using g ij (r, x) and some initial conditions. The Einstein equations are second order, thus, two initial conditions will be needed: g (0) and g (d). Solving the Einstein equations will determine the coefficients g (n) as an expression of g (0) and g (d), which are not determined by the Einstein equations and are unrelated to each other. Now if we look at equation (.63) in the semi-classical limit we have the semi-classical correspondence: W CFTd [g (0) ] S AdSd+ [g (0) ]. (.64) For a CFT the expectation value of the stress energy tensor can be found by using (.5). Thanks to the semi-classical correspondence we can relate this to the renormalized Brown-York tensor [4] Π ij (x) = δs AdSd+ [g (0) ], (.65) g(0) δg ij (0) such that we can define the semi-classical correspondence as Π ij (x) = T ij (x) CFT. (.66) Thus, we can use this equation to calculate the boundary stress energy that corresponds with the gravity theory in the bulk. This will be presented in section 4, where we will show how to derive the Brown-Yorke tensor. It will be shown that as well as in EAdS and ds, and thus the boundary theory, the Brown-York stress-energy tensor will be dependent on g (d), for the specific case d =. Which we can use to conclude that g (d) can be interpreted as the stress-energy tensor up to a factor. 3

14 Hawking-Hartle-Maldacena proposal Following some fruitful years of AdS/CFT, Maldacena finds a prescription to relate correlators in EAdS to ds [3]. As we have seen in subsection. the isometry group of ds d + is equivalent tot that of EAdS d+. Maldacena starts by computing corrections to the Guassian primordial fluctuations in the early universe by single field inflationary models. This is done by calculating 3-point correlator functions of scalar and graviton fields with variables ζ and γ s respectively. The ζ variable is associated with the trace of the stress-energy tensor while γ s is associated with the traceless part of the stress-energy tensor. This is coupled in the usual source form as discussed in subsection. dk 3 [ (π) 3 ζ k Ti i ( k) + γ s T s ( ] k). (.) k He then argues that a single scalar field with a negative potential, which would be AdS, has the same mathematical structure as this inflationary model. Furthermore, it is reported that in approximately de Sitter space the wave function of gravity is suppose to have properties of a conformal field theory [9] [5]. Now starting from the results of a 3-point function of an inflationary model, Maldacena relates this to a hypothetical dual CFT for a ds/cft theory. He notes that if we were to use the wave function of a ds universe as described in [4], this Hartle-Hawking (HH) wave function is conjectured to be dual to the partition function of a CFT: Ψ HH [g] = Z[g]. (.) There is a subtlety that must be addressed for ds space times, it namely has two boundaries. As such it is difficult to define a ground state for a globally compact space in the usual sense, that it is the state with the lowest energy. Hartle and Hawking argue that in a certain sense the total energy of the universe is zero. Thereby they propose to extend the ds space-time with a Euclidean compact regime such that the ground state wave function is given by Ψ 0HH [h ij ] = Dg e SE[g], (.3) where S E is the Euclidean action for compact gravity g and h ij is the induced metric. The de Sitter space is thought in this process to be a path of fat slices h ij and that all fields start in the Bunch-Davies vacuum. The space-time C that results from the summation of these slices is given as [6] C C = { g(η, x) g is compact, η η, g(η, x) = h ij (x) }, (.4) In other words they would like to glue a Euclidean space-time which would represent the past such that there are is only a boundary at future times. Anninos et al. also argued in [7] in favour of having the boundary at future times, since if we impose Dirichlet boundary conditions the structures of ds become very similar to that of AdS. This framework would enable the possibility to define a ground state for the HHM-proposal these are called Bunch-Davies vacuum states and are studied in [8]. There is no preferred vacuum in a curved space-time. However, if a spacetime has isometries with timelike Killing vectors then that would lead to a natural way of ordering negative and positive frequencies. With the help of creation and annihilation operators a vacuum state can then be defined. The Bunch- Davies vacuum state is invariant under the isometries of de Sitter spacetime and is therefore a good candidate. 4

15 Moreover, the Bunch-Davies state has the feature that at past infinity the wavelengths of a given mode are much shorter than the Hubble length, such that there is no distinction between de Sitter and Minkowski space in this state [9]. The de Sitter spacetime must be considered in the region where the timelike Killing vectors are valid. However, for global de Sitter spacetime one must restrict to the region of the timelike Killing vectors. Maldacena argues that it is valid in a small patch in a future where it can be approximated by a Poincaré patch. This construction was adapted due to (.) being valid in EAdS space too, which is described in [30 3].. Calculations in EAdS 4 To show the relation between EAdS 4 and ds 4 in terms of expectation values we can start with the action of a canonically normalized scalar field in EAdS 4, which Maldacena did in [3]: dz S = lads z [( zφ) + ( φ) ]. (.5) The partition function Z e S can be in this case approximated by simply the classical action shown in equation (.5). We can find the solution in momentum space φ k by imposing the Euler-Lagrange equation and Fourier transforming it ( ) L µ L ( µ φ) φ = z φ k k φ k z zφ k = 0. (.6) The solution will be restricted to classical constraints, namely the solution must go to zero for large z and it must have the appropriate boundary properties at the boundary z = z c, such that in momentum space φ k = φ 0 k ( + kz)e kz ( + kz c )e kzc, k = k. (.7) If z becomes large the solution goes to zero and at boundary z = z c the solution becomes φ k = φ 0 k, which will be the boundary condition. Inserting this back in to the action in momentum space and integrating by parts, Maldacena finds: S = d 3 k (π) 3 l AdS φ0 k zc dφ k dz = z=zc d 3 k k (π) 3 l AdS φ0 k φ0 k z c ( + kz c ). (.8) k We can expand this in z c such that z = ( k c(+kz c) z c + k ). The action then becomes d 3 k S = (π) 3 l AdS φ0 k φ0 k ( k + k ) (.9) z c The divergent term k z c and can be subtracted by a local counterterm as it is in local in position space since it is proportional to z c dx 3 ( φ0 ). The term k 3 is non-local and thus a -point correlator function can be set up where φ 0 and φ 0 k are the sources and Z is the partition in k Euclidean CFT (i.e. Z e S ) O( k)o( k ) = EAdS δ Z δφ 0 k δφ 0 k φ 0 =0 (π)3 δ( k + k )l AdS k3. (.0) 5

16 . Calculations in ds 4 In this case the partition function Z e is can not be trivially approximated by the classical action. In [7] these calculations have been done and we will show it here. We start with the wave function Ψ HH [φ k, η c ] = Dφ k e is[φ ] k, (.) where φ k respresents the field in de Sitter in momentum space and η c is conformal time coordinated evaluated at the cut-off. Here we integrate over fields that satisfy φ k e ikη in the limit kη and φ k (η c ) = φ k. This wave-function is then subjected to the Schrödinger equation. When considering a massless scalar field the solution to the Schrödinger equation is exact and is given by Anninos et al. to be Ψ HH [φ k, η c ] = k ( k 3 π ) 4 exp [ il ( k R d k η c ( ikη c ) ) φ k φ k ] e ikηc/ ( ikηc ). (.) To find the action we can take the logarithm of this wave function, this results in log Ψ HH [φ k, η c ] = is ds. Evaluating it for small negative η c, which corresponds with late times, the exponential e factor is evaluated to be ikηc/, and we can make the expansion i k k ( ikηc) η c( ikη c) = (i η c k ). The logarithm of the wave function then reads log Ψ HH [φ k, η c ] = d 3 k (π) 3 l ds φ k φ k (i k k ). (.3) η c Again the divergent term can be subtracted by adding local counterterms. -point function gives O( k)o( k ) = EAdS δ Z δφ 0 k δφ 0 k Note that if we analytically continue equations (.9 -.0) with Computing the φ 0 =0 (π)3 δ( k + k )l ds k3. (.4) z iη, l AdS il ds, (.5) we get equations (.3-.4). This is a deep statement and also points towards a direction of a possible ds/cft via analytical continuation from AdS/CFT. However, an explicit wave function and is still treated as hypothetical. In the paper [33] it is report that it might not be possible to devise a ds/cft system. They argument is from a quantum mechanical black hole point of view, where the local proper temperature goes to infinity at the horizon. In de Sitter space there is an even horizon as well and it surrounds the observer. This becomes a problem if we adopt the formalism for ds/cft by Strominger [34], where it is needed to have a finite entropy at the boundary horizon. Furthermore, Dyson et al. state that it may be possible that there is a CFT that matches the boundary conditions needed, but it would be an unfamiliar theory..3 Further remarks on the structure of the space-time In [3] Maldacena shows that we can use S 4 for the Euclidean action S E. He starts with the partition function of EAdS (Z EAdS ) and transforms this first to the S 4 of which we will use 6

17 the southern hemisphere, since we want it to be at the past boundary. And then analytically continuing it to ds 4. This is done with the transformations l EAdS il d s ρ ρ + i π, (.6) using the Poincaré coordinates stated in his paper, where ρ is the radial conformal coordinate. Ultimately this results in log Z log Ψ HH. (.7) This was also shown more explicitely in [6]. Moreover, this means that going from EAdS to ds the boundary shifts from 0 to π. Since we interpret this to be a late times boundary we must compactify our conformal time coordinate such that it has a boundary at π. Using the asymptotic formalism discussed in subsection.5 and notably the metric in equation (.59) we can use the conformal factor e φ(r) as well as η and ɛ to specify a space-time - we will discuss this in more detail in section 3. For now, let us introduce the metrics we will be using for the HHM-proposal. We can define an asymptotic space-time in conformal cylindrical coordinates for ds: ds = cos r ( dr + g ij dx i dx j ). (.8) This metric has two boundaries and describes the full asymptotic ds space-time but compactified to r ( π, π ), which is what we want. We will focus on the future boundary r = π. To get the Euclidean space-time we will be using for the proposal we only need to Wick rotate r ir, the space-time then becomes the hypersphere S. The metric in cylindrical conformal coordinates for a S space-time is ds = cosh r (dr + g ij dx i dx j ). (.9) In this paper we will glue the two spaces at r = 0, which will have the same isometry for constant r. The actual metric will then be ds = { cosh r (dr + g ij dx i dx j ) r 0 cos r ( dr + g ij dx i dx j ) 0 < r < π, (.0) where S will be the southern hemisphere that runs from < r 0 and we have ds in 0 < r < π. In figure we have plotted the conformal factors of equation (.0) and we see that it smoothly extends from S to ds. In [3] and [7] arguments have been made that this glueing should be done at late times, which means close to the boundary π. Ultimately, for the calculations which we will do in section 4, it does not matter where we glue S, because we will only do calculations precisely at the boundary of ds. Since we will use conformal cylindrical coordinates for ds we must also use it for EAdS. The corresponding asymptotic metric is ds = sinh r (dr + g ij dx i dx j ). (.) For this metric to describe EAdS we restrict < r 0, where we indeed only get a space-like infinity at r = 0. Normally, it is the convention to use positive r as is done for example in [5]. De Haro et al. used curvature conventions such that the curvature of AdS comes out positive. 7

18 5 e φ 4 3 S e φ = cosh (r) ds e φ = cos (r) r Figure : A plot of the conformal factors e φ seen in equation (.0). This in turn results in using positive r (i.e. 0 r < ). In this thesis it is more natural to use negative r as we want to relate this to the southern hemisphere S via analytical continuation. This in turn defines a more natural future times boundary for ds, which is r = π as we have discussed. 3 Einstein equations in the bulk It is now important to see how the Einstein equations look like in the bulk for asymptotic spaces. To ease the calculations we will first start with the general metric given in equation (.59), but without the conformal factor e φ : ds = Ḡµνdx µ dx ν = ηdr + g ij dx i dx j. (3.) We can easily go back the general metric in equation (.59) by taking the conformal transformation Ḡµν e φ Ḡ µν = G µν. We will use this conformal factor to specify which space-time we are considering and as discussed in subsection.5, we want the conformal factor to only depend on r. Using the definition of the Christoffel symbol Γ α µν = Gαβ ( µ G αν + ν G µα β G µν ), we find the non-zero Christoffel symbols to be Γ r ij[ḡ] = η g ij Γ i rj[ḡ] = gik g jk Γ i jk [Ḡ] = Γi jk [g], (3.) where the r index represents the radial coordinate and the latin indices represent the spatial coordinates which in our case is d dimensional, since our bulk has (d + ) dimensions. It is 8

19 interesting to note that if we run the indices over the spatial coordinates the Christoffel symbol simply becomes that of the boundary metric g ij as can be seen in (3.). With these Christoffel symbols calculated we can now derive the Riemann tensor, we will use the definition R µ ναβ = αγ µ νβ βγ µ να + Γ µ ασγ σ νβ Γµ βσ Γσ να. (3.3) We find the components to be Rjkl i [Ḡ] = Ri jkl [g] η ( 4 gim g jl g km ) g jk g ml (3.4) R irj[ḡ] r = η g ij + η 4 (g g g ) ij (3.5) Rijk r [Ḡ] = η ( j g ki kg ij ). (3.6) The last step is to calculate the Ricci tensor and set up the Einstein equations. The ricci tensor can be derived using R µν = R α µαν = G αβ R µανβ and thus R ij [Ḡ] = R ij[g] + η( g ij 4 Tr(g g )g ij + (g g g ) ij ) (3.7) R rr [Ḡ] = Tr(g g ) + 4 Tr(g g ) (3.8) R ir [Ḡ] = ( j g ij i Tr(g g )). (3.9) We have now calculated the Ricci tensor for this simple generic metric. We will now conformally transform the metric G µν = e φ Ḡ µν such that we get the metric mentioned in (.59). With this transformation the Ricci tensor will transform as follows: R µν [G] = R µν [Ḡ] (d ) µ ν φ Ḡµν φ + (d ) µ φ ( ) ν φ (d )Ḡµν φ. (3.0) We can use the fact that φ only depends on rr this leads to the conformal transformation R µν [G] = R µν [Ḡ] (d ) ( r φδµδ r ν r Γ r rr r φδµδ r ν r Γ r ij r φδµδ i ν j ) ) Ḡ r ( ḠḠ rr r φ + (d )( r φ) δµδ r ν r (d )ḠµνḠrr ( r φ). (3.) Ḡµν ( ḠḠ ) Inserting Ḡ r rr r φ = ηφ + η Tr(g g )φ, we arrive at the conformally transformed components of the Ricci tensor R ij [G] = R ij [Ḡ] η ( (d )g ijφ + g ij φ + g ijtr(g g )φ + (d )g ij (φ ) ) (3.) R rr [G] = R rr [Ḡ] dφ Tr(g g )φ (3.3) R ir [G] = R ir [Ḡ]. (3.4) Using the equations ( ) we can set up Einstein s equations R µν [G] = ε d l eφ Ḡ µν, (3.5) 9

20 which will then be the three equations ε d ( l eφ g ij = R ij [g] + η g ij 4 Tr(g g )g ij + (g g g ) ij (d )g ijφ g ij φ g ijtr(g g )φ (d )g ij (φ ) ) (3.6) ε d l eφ η = Tr(g g ) + 4 Tr(g g ) dφ Tr(g g )φ (3.7) 0 = j g ij i Tr(g g ). (3.8) We will only have to deal with these three Einstein equations. The equations (3.6), (3.7) and (3.8) will be refered to the first, second and third Einstein equation respectively. These equations can be solved perturbatively by deriving it and solving it order by order using the metric expansion (.60). Note that the third Einstein equation is conformally invariant. We can conveniently rewrite it as 0 = j (g ij g ij Tr(g g )), (3.9) since the total derivative of g ij g ijtr(g g ) is zero we can then write g ij = g ij Tr(g g ) + t ij, (3.0) with t ij a symmetric conserved tensor (i.e. j t ij = 0) that arrives as an integration constant. 3. Euclidean Anti-de Sitter spacetimes 3.. Poincaré coordinates Generally, in literature Poincaré coordinates are used for asymptotically AdS spaces. The metric is given as ds = r (dr + g ij (r, x)dx i dx j ), (3.) as such e φ = and (η, ε) = (, ) have been used in the metric (.59). In this spacetime r r has an interval of < r 0 where r = 0 is the boundary of the bulk. The defining function has the right properties for an asymptotic EAdS space, it extends smoothly to the boundary r and has a second order pole at the boundary. We can now easily set up the Einstein equations. The φ function in this case is φ = log r + C, where C is some constant, this results in φ = r, φ = r. (3.) Using this we set up the first and second Einstein equation 4 0 = R ij [g] g 4 Tr(g g )g + g g g + r ((d )g + gtr(g g )) (3.3) 0 = Tr(g g ) + 4 Tr((g g ) ) + r Tr(g g ). (3.4) 4 We are going to suppress trivial indices for the metric, unless noted specifically otherwise. 0

21 It is important to note that if we use r = ρ we indeed get the established Einstein equations as described in [5]. Solving the first equation up to the zeroth order results at first hand that g () = 0. With this knowledge the solution of the zeroth order becomes R ij [g (0) ] + (d )g () + g (0) Trg () = 0, (3.5) where we used Tr g () as a notation for Tr(g (0) g ()), which we will use from now on. The second Einstein equation does not provide any information as g () is not determined by it. However, the third equation produces g () = g (0) Tr g () + t ij, (3.6) such that the solution becomes R ij [g (0) ] + (d )g () t ij = 0. (3.7) Let us look at d = as it is the special dimension of CFTs where there is an infinitedimensional algebra of local conformal transformations. This simplifies the conformal theory heavily and is also studied often with respect to string theory. In d = (.30) results in R µανβ = (g µνg αβ g µβ g να )R, R µν = g αβ R µανβ = g µνr, (3.8) from which we could calculate R ij [g (0) ] = g (0)R, so equation (3.8) for d = becomes Taking the trace of equation (3.7) results in as expected. 3.. Conformal cyndrical coordinates g () = (g (0)R t ij ). (3.9) Trt = R, (3.30) As discussed in subsection.3, the HHM-proposal is done by glueing the southern hemisphere of S (d+) to ds d+. As such we use the asymptotic conformal cylindrical metric for EAdS: ds = sinh r (dr + g ij dx i dx j ), (3.3) again our conformal radial coordinate is restricted to r < 0 and the conformal boundary is at r = 0. We have: φ = log sinh r, φ = tanh r, φ = Constructing the Einstein equations leads to 0 = R ij [g] g 4 Tr(g g )g + g g g + sinh r. (3.3) ( (d )g + gtr(g g ) ) (d )g tanh r 0 = Tr(g g ) + 4 Tr(g g ) + Tr(g g ). (3.33)

22 Solving the first Einstein equation up to first zeroth order results in 0 = R ij [g (0) ] g () + (d )g () + g (0) Trg () + (d )g (0). (3.34) Again, the second equation does not determine g () so it can be ignored. Using (3.6), (3.8) and inserting d = gives the expression for g (), which is g () = (g (0)(R + ) + t ij ). (3.35) Using this solution and taking the trace of (3.34) we can calculate the trace of t ij, which becomes Tr t = R + (3.36) 3. De Sitter spacetime in conformal cyndrical coordinates The asymptotic conformal cylindrical metric for this space-time is ds = cos r ( dr + g ij dx i dx j ), (3.37) we now have e φ = and (η, ε) = (, ). With the conformal coordinate being restricted to cos r π < r < π. As discussed in subsection.3, we restrict the conformal coordinate to 0 < r π and the boundary will be at r = π. In this spacetime the φ function and its derivatives are φ = log cos r, φ = tan r, φ = cos (3.38) r Using this prescription we can set up the Einstein equations 0 = R ij [g] + tan r ( (d )g + gtr(g g ) ) + 4 Tr(g g )g (d )g ( g g g ) + g 0 = Tr(g g ) Tr(g g ) + Tr(g g ) tan r. (3.39) To solve this order by order we need to evaluate the expansion of tan r at r = π. To do this it is convenient to introduce a translated coordinate τ = r π. Due to this shift the boundary will be conveniently at τ = 0 and now the conformal coordinate will be restricted to π < τ 0. We also have to shift the coordinate in the southern hemisphere S d+ so that the coordinate there runs from < τ π. Now we can write tan r = tan(τ + π ) = τ + O(τ ) (3.40) Again when solving the first equation up to order τ 0 we find g () = 0 as it should. Then solving it again at the zeroth order gives R ij [g (0) ] (d )g () g (0) Tr(g () ) (d )g (0) = 0. (3.4) The second equation again does not describe g () and the third equation produces the same solution. So combining the first and third equation results in the solution R ij [g (0) ] (d )g () (d )g (0) + t ij = 0. (3.4) For d = we can again use identity (3.8) to simplify the solution, which becomes Calculating the trace of t ij now results in g () = (g (0)(R ) + t ij ). (3.43) Tr t = (R ). (3.44)