hep-th/0312039 BOUNDARY RIGIDITY AND HOLOGRAPHY 4 0 M. Porrati a and R. Rabadan b 0 2 n a Department of Physics, New York University a J 4 Washington Pl., New York NY 10012, USA 2 2 b School of Natural Sciences, Institute for Advanced Studies Olden Lane, Princeton NJ 08540, USA 2 v 9 3 0 2 Abstract 1 3 0 We review boundary rigidity theorems assessing that, under appropriate conditions, Rieman- / h nian manifolds with the same spectrum of boundary geodesics are isometric. We show how t - p to apply these theorems to the problem of reconstructing a d+1 dimensional, negative curva- e ture space-time from boundary data associated to two-point functions of high-dimension local h : operators in a conformal field theory. We also show simple, physically relevant examples of v i negative-curvature spaces that fail to satisfy in a subtle way some of the assumptions of rigidity X r theorems. Inthoseexamples, weexplicitly showthat thespectrumof boundarygeodesics is not a sufficient to reconstruct the metric in the bulk. We also survey other reconstruction procedures and comment on their possible implementation in the context of the holographic AdS/CFT duality. e-mail: [email protected], [email protected] Contents 1 Introduction 1 2 Green’s Functions, Geodesics, and Boundary Rigidity Theorems 3 2.1 From Green’s Functions to Geodesics . . . . . . . . . . . . . . . . . . . . 3 2.2 Boundary Rigidity Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 Examples and “Counterexamples” 6 3.1 Point Particle in AdS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 3.1.1 Constant Time Section . . . . . . . . . . . . . . . . . . . . . . . . 6 3.1.2 Finite and Zero Temperature AdS with a Point Particle . . . . . 9 3 3.2 Lorentzian BTZ Black Hole . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2.1 Description of the Space . . . . . . . . . . . . . . . . . . . . . . . 10 3.2.2 Boundary Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2.3 Geodesics Outside the Horizon . . . . . . . . . . . . . . . . . . . . 12 3.2.4 Geodesics Crossing the Horizon . . . . . . . . . . . . . . . . . . . 14 3.3 Euclidean BTZ Black Hole . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.4 The RP2 Geon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.5 Euclidean RP2 Geon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.6 Higher-Dimensional Finite-Temperature AdS Black Holes . . . . . . . . . 18 4 Other Bulk Reconstruction Procedures 18 4.1 Dirichlet-to-Neumann Map . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.2 Scattering Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.3 Bulk to Boundary Functions . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.4 Spectral Boundary Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5 Summary, Conclusions, Speculations 23 1 Introduction The holographic principle [1] is a potentially revolutionary new paradigm in quantum gravity, since it gives up the idea that a fundamental description of physics is local. In place of locality, the principle states that the fundamental degrees of freedom that de- scribe quantum gravity in a region of d + 1-dimensional space-time, called “the bulk” hereafter, arelocatedonanappropriated-dimensionalsubspace, a“screen” locatedsome- where in that region. A proper definition of such holographic screen can be given also in cases where the bulk has no boundary [2]. What is generally unknown, instead, is the 1 physics of the degrees of freedom that live on that screen. In the case that the back- ground bulk space-time is Anti de Sitter (AdS) space, much more can be said. In that case it has been conjectured that quantum gravity –or better string theory– on AdS d+1 space (times some compact manifold of dimension 9 d) has a dual description in terms − of a d-dimensional (local) conformal field theory (CFT) defined on the boundary M of d AdS [3]. A comprehensive review of the evidence in support of that conjecture can be d+1 found in [4]. The relation between quantum gravity in AdS and the CFT on M is a duality, d+1 d because when one description is perturbative, the other is strongly coupled. So, for instance, in the canonical case when the duality is between the Type IIB superstring on AdS S and N=4, SU(N ) super Yang-Mills in four dimensions, one can trust the 5 5 c × low-energy supergravity approximation to the superstring in the large N limit, and only c when the ’t Hooft coupling constant of the N=4 theory, g2 N is large. YM The fact that the two dual descriptions are never simultaneously weakly coupled makes it difficult to establish an explicit “dictionary” associating states to the quantum gravity inAdStostates ofthedualconformalfield theory. Consider inparticular thecase where the quantum gravity wave function is peaked around a given classical geometry. A natural question one can ask is how to reconstruct this geometry from CFT boundary data only. This question does not have as yet a complete answer, even though much progress has been made in the last few years. For instance, proposals exist for the CFT description of precursors [5], and for how to detect, through CFT correlators, the region behind the horizon of an AdS black hole [6]. In this paper, we continue the program of “holographic” reconstruction of space-time by looking at a special class of CFT observables, namely the two-point correlators of local operators with high conformal dimension. We will investigate to what extent they can determine the geometry of the bulk space-time. The Green’s functions we select are particularly simple because they are directly related to the geodesic distance of two boundary points in the (regularized) bulk space-time. The reconstruction of the bulk space-time from boundary data reduces, in this ap- proximation, to a classical problem in mathematics: the boundary rigidity problem. Its precise definition will be given in Section 2, here we can formulate it as follows: under what conditions are two spaces with the same spectrum of geodesics, whose endpoints lie on the boundary, isometric? In Section 2, we will review the argument of ref. [7] connecting Green’s functions to geodesic distance, and we will summarize existing theorems about boundary rigidity, paying particular attention to the assumption necessary to prove them. We will use some of these known results to show, for instance, that a small deformation of (Euclidean) AdS space is boundary rigid in any dimension. 2 In Section 3, we will examine some specific examples of bulk space-times: point particles in AdS , their equal-time sections, theBTZ black hole [8], the RP2 geon[9], and 3 their Euclidean continuation. We will show that some of those spaces are not boundary rigid. The reason for that failure will be traced back to the violation of some of the most subtle assumptions needed in proving general boundary rigidity theorems. The examples of Section 3, the AdS point particle in particular, are in some sense the flip side of the 3 findings in ref. [7]. That reference used Green’s functions of operators with high conformal weight as holographic probes. Among other things, it showed that they can detect the formation ofAdS black holesinthecollision oftwo pointparticles. So, thosesimple observables are 3 nevertheless able to detect physics behind the black-hole horizon. In Section 3, instead, we find that there exist situations where the bulk space-time has no horizons, yet its metric cannot be reconstructed from the spectrum of its boundary geodesics. In Section 4, we survey, without any pretense of completeness, other holographic reconstruction procedures, and we discuss which of them could be implemented using the AdS/CFT duality, i.e. from knowledge of CFT data only. Section 5 contains our conclusions, together with a conjecture about a possible ex- tension of boundary rigidity theorems, and its relation to the holographic duality. 2 Green’s Functions, Geodesics, and Boundary Rigid- ity Theorems 2.1 From Green’s Functions to Geodesics This subsection, included here for completeness, follows closely ref. [7]. Near the boundary, the metric of an asymptotically Anti de Sitter space is L2 ds2 = [dz2 +g (z,x)dxµdxν], µ,ν = 1,.,4, g (z,x) = g0 (x)+O(z2). (1) z2 µν µν µν All non-light-like geodesics ending on the boundary z = 0 have infinite length, so the space must be regularized by cutting off a small region near the boundary, specifically, by restricting z ǫ. The length ǫ has a holographic counterpart in the boundary CFT: it ≥ is the UV cutoff one needs to regularize the theory [10, 11, 12]. Let us denote the cutoff d + 1 dimensional bulk with ǫ . In ǫ , geodesics have finite length. Moreover, Md+1 Md+1 in this space, the boundary-to-boundary Green’s function of a free scalar field field of mass m is well defined. This Green’s function, G(x,y), with x,y ∂ ǫ , is interpreted ∈ Md+1 as the (regularized) two-point function of some scalar composite operator in the dual CFT. The conformal dimension of the operator, ∆, is (generically) the largest root of the 3 equation L2m2 = ∆(∆ d). (2) − For large mass mL 1, ∆ = mL+d/2+O(1/mL) mL. The Green’s function of a ≫ ≈ free scalar field in AdS can be also represented as a functional integral G(x,y) = [dX(t)]exp( ∆D[X]/L), (3) − Z where D[X] is the length of the path X(t) joining x to y. When mL 1, the path ≫ integral is dominated by its saddle point, i.e. the boundary-to-boundary geodesic joining x to y: G(x,y) = const 1+O[L/∆D (x,y)] exp[ ∆D (x,y)/L]. (4) min min { } − Notice that in Eq. (4) we are ignoring inverse powers of the distance, so, even when more than one geodesic can be drawn between the points x,y, we should only take into account the contribution of the shortest one. To include the others would be inconsistent with our approximation 1. In summary, we have found that the holographic correspondence and known results about the semi-classical approximation to free-field Green’s functions relate, by Eq. (4), a CFT quantity (the two-point function of an operator of dimension ∆ 1) to a geometrical quantity: the minimal geodesic distance between the two points. ≫ 2.2 Boundary Rigidity Theorems Assume that a direct problem is well behaved, i.e. that its solution exists, is unique, stable etc. The inverse problem is to extract some properties of the original object or system from the solution of the direct problem. These problems in general are ill-posed (in the sense of Hadamard): there may be no solution, or the solution may be non- unique, or unstable (small changes in the input data may result in large changes in the solution). Examples of inverse problems include inverse scattering (how to reconstruct the shape of a target, or a potential from the scattered field at large distances), the inverse gravimetry problem, tomography, inverse conductivity problems, inverse seismic problems, many problems in inverse spectral geometry & c. ConsiderinparticularaRiemannianmanifold( ,g)withaboundary. LetD (x,y) min M be the geodesic distance between two points at the boundary x,y ∂ 2. The function ∈ M D (x,y) is called the hodograph (a term borrowed from geophysics). The inverse min problem is to find to what extent the Riemannian manifold is determined by the lengths ofthegeodesicsbetweenpointsattheboundary. Equivalently, thequestionis: uptowhat extent do the two-point functions in the conformal theory determine the bulk metric? 1Attempts to go beyond this limitation will be discussed in Sections 4 and 5. 2See reference [14] for a survey. 4 Solutions to this problem come into sets, related by diffeomorphisms that reduce to the identity at the boundary. That, of course, changes the metric in the interior, while keeping the same geodesic spectrum. A manifold is said boundary rigid if there exists only one such set of solutions. Boundary rigidity theorems analyze the uniqueness of the solution. They give the conditions that a Riemannian manifold must satisfy to be boundary rigid, i.e. to be completely determined by the hodograph. If we take a manifold where there exist in- terior points that cannot be reached by any geodesic, then one can always change the metric close to this point without changing the length spectrum. So, general Riemannian manifolds are not boundary rigid. What are the conditions that a manifold should satisfy to be boundary rigid? One of the most natural conditions is that the manifold is simple; namely, its boundary is strictly convex, and every two points at the boundary are joined by a unique geodesic. Such a manifold is diffeomorphic to a ball. R. Michel conjectured in 1981 [15] that every simple manifold is rigid. Another natural condition considered by Croke [16] is that the manifold is strongly geodesic minimizing. This means that every segment of a geodesic that lies on the interior of the manifold is strongly minimizing, i.e. it is the unique path. The length spectrum determines the volume of the manifold for both simple and strongly geodesic minimizing manifolds. The problem is not solved in general, but there are some partial results that will be useful to us. Simple Riemannian manifolds with negative curvature (like AdS) are deformation boundary rigid [17], i.e. we cannot deform the metric keeping the boundary distance fixed. This result was generalized [18] and further in [20] for compact dissipative Riemannian manifolds (convex boundary plus a condition on the maximal geodesics) satisfying some inequality concerning the curvature. There is a semi-global result in [19] when one of the metrics is close to the Euclidean and the other satisfies a bound on the curvature. For general metrics, not just deformations, a theorem exists in two dimensions [22]: every strong geodesic minimizing manifold with non-positive curvature is boundary rigid. This theorem has been recently generalized to subdomains of simple manifolds in two dimensions [21]. Any compact sub-domain with smooth boundary of any dimension in a constant curvature space (Euclidean space, hyperbolic space or the open hemisphere of a round sphere) is boundary rigid [15, 23, 24]. Apart from that spaces and sub-domains of negatively curved symmetric spaces and some products of spaces3 there are no other boundary rigid examples. The Lorentzian case has not been analyzed very much. The two dimensional case is analyzed in ref. [25], which tries to extend the result of Croke [22] to the Lorentzian case. The condition analogous to being strong geodesically maximizing is not enough to 3See the survey [14]. 5 guarantee that the manifold is boundary rigid. 3 Examples and “Counterexamples” 3.1 Point Particle in AdS 3 The metric for a point particle in AdS is locally the same as AdS , but with different 3 3 global identifications. It reads 1 ds2 = dr2 (r2 +γ2)dt2 +r2dφ2, r 0, 0 φ < 2π. (5) r2 +γ2 − ≥ ≤ By redefining r = γrˆ, tˆ= γt, φˆ= γφ, this metric can be recast in a standard AdS form, 3 ˆ ˆ but with a different periodicity for the φ coordinate: 0 φ < 2πγ. This implies of course ≤ 0 γ2 < 1. Negative γ2 gives the non-rotating BTZ black hole metric. ≤ To correctly analyze the geodesic between any two points at the boundary one has to consider the Euclidean version of the problem. In the Lorentzian version the problem is ill defined. The problems associated with Lorentzian signature (there are no geodesics between some points at the boundary, or an infinite number of them with the same length & c) can already be found in simple examples as AdS. In this section we will be mainly concerned with Euclidean metrics. Next we will study Euclidean AdS with a 3 point particle in three cases: at infinite temperature, where one studies its constant time section (which is the same as inthe Lorentzian problem), at zero temperature, andfinally at finite, nonzero temperature. We will find that, in some cases, non-rigidity appears. 3.1.1 Constant Time Section Consider now thet = 0section of thismetric. Its geodesics canbeeasily found, e.g. using the Hamilton-Jacobi method. A standard calculation gives the angular distance between the boundary endpoints of the geodesic, ∆φ, as a function of its minimum distance from the center, r¯: 1 r¯2 γ2 ∆φ = θ, cotθ = − . (6) γ 2γr¯ (Here we chose γ > 0). By definition, 0 θ < π. In this range, Eq. (6) is one-to-one. ≤ This does not mean that there is only one geodesic joining any two boundary points! Indeed, when the angular distance ∆φ is in the range π < ∆φ < π/γ, we have a second geodesic joining the same two boundary points, with ∆φ′ = 2π ∆φ < ∆φ. Since Eq. (6) − is one-to-one, this means that the minimum radii of the two geodesics are different, hence the geodesics are distinct. 6 So, even if our space is “almost” AdS , and its sectional curvature is negative, this 3 space may not be boundary rigid, since it fails to satisfy the simplicity condition. More- over, it is singular at r = 0; removing the point r = 0 makes the space non-simply connected, so, again, non-simple. If we were given the lengths of all geodesics between boundary points, it would be still far from obvious that the point-particle space could be deformed without changing some geodesic lengths. In our case, though, more than one geodesics can be drawn between the sametwo points, sowehavetobecarefulabouttheidentificationofphysically meaningful holographic data. As we mentioned in Subsection 2.1, the physical quantities one is given in the bound- arytheoryarethetwo-point functionofcompositeoperators. Geodesicareusedtoobtain a saddle point approximation of these functions. Since the saddle point approximation neglects inverse powers of the geodesic distance [see Eq. (4)], one should also neglect contributions from sub-dominant saddle points. So, the physical data are the lengths of minimal geodesics in between boundary points. Generically speaking, the minimal geodesic spectrum is not enough to reconstruct the bulk metric from boundary data. In our case, one can be more specific, and prove that there exist deformations of the metric that do not change the spectrum of minimal length geodesics. So, not only the conditions for boundary rigidity are not met in our simple example, but we can explicitly show that the bulk metric can be changed without affecting boundary data. To see this, notice that the shortest geodesic is that for which ∆φ < π. This means that no minimal-length geodesic can come closer to the center than 1 cos(γπ/2) r = min r¯= γ − . (7) min 0≤θ≤γπ vu1+cos(γπ/2) u t So, any change of the metric confined to the region r < r is undetectable, within our min approximation. Now, let us ask whether it is possible to smooth out the singularity at r = 0 without changing the spectrum of minimum-length geodesics. This would mean that hologra- phy could be blind to qualitative features of the bulk space geometry, such as the very existence of singularities. It is convenient to change coordinates in Eq. (5) by setting r = γsinhρ, and write the metric at t = 0 as ds2 = dρ2 +γ2sinh2ρdφ2, ρ > 0. (8) Eq. (7) implies that the minimum distance ρ probed by minimal-length geodesics min obeys γsinhρ < 1. Now the question is, can we smooth out the metric by changing min only the region ρ < ρ , while preserving some basic characteristics of the metric, for min instance, that the curvature is negative? The answer is no. To see this, consider the 7 change γsinhρ F(ρ) 0. The new range of the coordinate ρ is from ρ , the point 0 → ≥ where F vanishes, to + . Smoothness atρ requires (dF/dρ) = 1. To leave the metric ∞ 0 |ρ0 outside ρ unchanged, we must also require (dF/dρ) = γcoshρ min |ρmin min To keep the curvature negative, we must have d2F/dρ2 > 0, whence the inequality ρmin d2F (dF/dρ) (dF/dρ) = γcoshρ 1 = dρ > 0. (9) |ρmin − |ρ0 min − Zρ0 dρ2 By using the value of r = γsinhρ given in Eq. (7), we finally find that, in order to min min smooth out the singularity without changing the geodesic spectrum, we must have 2 γcoshρ = γ > 1. (10) min s1+cos(γπ/2) This equation is never satisfied in the range 0 < γ < 1. So we have seen that there is no metric preserving rotational invariance that coincide with the point particle metric in the region accessible by geodesics and that has negative curvature. That means that all the metrics with this hodograph have positive curvature in some region so the theorems about dispersive manifolds with negative curvature (ref. [17] e.g.) do not apply. We can extend this proof for general deformations of the metric by considering the integral 1 k(Σ) = , (11) 4π R ZΣ where Σ is a region in the interior of the Euclidean section of the space. On a com- pact manifold without boundary, k is the Euler number. In two dimensions the scalar- curvature density is a total derivative. In our case, the curvature has two contributions: one from the point particle (a delta function at its position) and one from the AdS space itself. The first contribution to the number k can be expressed in terms of the deficit angle δ = 2π(1 γ) − 1 k (Σ) = δ. (12) part 2π The AdS space has constant negative curvature = 2/R2. So, the total contribution R − in a region that contains the point particle is: 1 1 k(Σ) = δ Vol . (13) 2π − 2πR2 Σ The metric Eq. (8) has R = 1, so the critical volume when k = 0 is: Vol = R2δ = 2π(1 γ). (14) c − 8 By Eq. (10), the volume of the ball B of radius r –i.e. the region not probed by min minimal-length geodesics– is 2 V = 2πγ 1 . (15) B s1+cos(γπ/2) − ! Can we change the metric within a region Σ B to a smooth, negative-curvature one, 0 ⊂ without touching the outside metric? Again, the answer is no, since if this were possible, then, for that metric, k(Σ ) < 0. On the other hand, for the AdS point-particle metric: 0 1 2 k(Σ ) = (1 γ) Vol > (1 γ) γ 1 > 0. (16) 0 − − 2πR2 Σ − − s1+cos(γπ/2) − ! So we need k(Σ ) to bepositive fora metric, andnegative for another. This is impossible, 0 because for any open region Σ, k(Σ) is invariant under any change of the metric inside Σ, that reduces to the identity on its boundary, since the scalar curvature is a total derivative. 3.1.2 Finite and Zero Temperature AdS with a Point Particle 3 Now let us consider the whole Euclidean AdS with a point particle in it. It is easy to see 3 that the shortest geodesic joining points separated by a very long Euclidean time can get arbitrarily close to the origin, as the time interval gets larger. Let us consider geodesics with only a time separation (∆φ = 0). The trajectory satisfy the equation: dr (γ2 +r2) = m2(γ2 +r2) E2, (17) dt E − q that can be integrated to: γ γ m2(γ2 +r2) E2 +Er t = log − , (18) 2 γqm2(γ2 +r2) E2 Er − − q where E and m are the energy and the mass of the particle. Starting at the boundary there is a family of solutions with E2 > m2 γ2 that do | | | | not touch the origin (see figure 1). The geodesics start from the boundary and go back at ∆φ = 0. The time to come back is: E +mγ ∆T = γlog . (19) E mγ − Notice that for E2 m2 γ2 the time interval diverges. That means that they can | | → | | be arbitrarily long, and joining any two points on the boundary. The turning point is at 9