IISCOs in AdS/CFT
David Berenstein † , Ziyi Li † , Joan Sim´on ‡ † Department of Physics, University of California at Santa Barbara, CA 93106 ‡ School of Mathematics and Maxwell Institute for Mathematical Sciences,University of Edinburgh, Edinburgh EH9 3FD, UK
We study stable circular orbits in spherically symmetric AdS black holes in various dimensionsand their limiting innermost stable circular orbits (ISCOs). We provide analytic expressions fortheir size, angular velocity and angular momentum in a large black hole mass regime. The dualinterpretation is in terms of meta-stable states not thermalising in typical thermal scales and whoseexistence is due to non-perturbative effects on the spatial curvature. Our calculations reproduce thebinding energy known in the literature, but also include a binding energy in the radial fluctuationscorresponding to near circular trajectories. We also describe how particles are placed on these orbitsfrom integrated operators on the boundary: they tunnel inside in a way that can be computed fromboth complex geodesics in the black hole background and from the WKB approximation of the waveequation. We explain how these two computations are related. a r X i v : . [ h e p - t h ] O c t I. INTRODUCTION
The AdS/CFT correspondence [1] is an equivalence between two seemingly distinct theories : a gauge field theoryin d dimensions and a gravity theory, usually realized as a string theory, on an asymptotically AdS d +1 spacetime.The nature of the duality involves an equivalence of Hilbert spaces and a dictionary of observables from one theoryto the other.The nature of the dictionary maps any phenomenon on one side of the duality to a corresponding realization on theother side. A simple example is that states in both Hilbert spaces fit into representations of the conformal group, thelatter acting as the isometry group on the gravity side. A more striking one is unitarity in the quantum field theorypredicting that quantum gravity in AdS d +1 must be unitary.The purpose of this paper is to understand what are the consequences for the dual theory that follow from theexistence of stable geodesic orbits in spherically symmetric AdS d +1 black holes for d ≥ d = 2, recovering the result in [2],whereas for d ≥
3, their angular momentum (cid:96) must be above the threshold set by (cid:96) isco , the angular momentum of theISCO. Below this threshold, orbits plunge into the black hole.In the field theory side, plunging orbits are associated with thermalization dynamics [3–7]. Hence, stable circularorbits must be associated to physics that does not thermalize on a thermal time scale. However, the existence ofquantum processes such as radiation or tunneling, even if subleading, can make these states to thermalize eventually.Thus, the field theory states mapped to the gravity excitations must be metastable.From the point of view of wave equations on AdS, that metastability can be analyzed using a WKB approximation,an analysis that was started in the work of Festuccia and Liu [8]. Our work is complementary to theirs, spending a lotmore effort understanding the classical dynamics of these orbits and finding limits where the ISCO can be computed.We deepen the understanding of the dual CFT physics that may be responsible for these phenomena. We also providea way to place particles in these trajectories from the boundary.The kinematic conditions determining the size ( r isco ) and the angular momentum ( (cid:96) isco ) of stable orbits are hard tosolve analytically. However, working in a large mass limit, we determine their precise scaling with the temperature ofthe black hole in any dimension. Such scaling was also observed in [8].We argue that the origin of these meta-stable states in the dual theory are non-perturbative effects in the spatialcurvature where the field theory is defined. One observation supporting this claim is the fact that at very hightemperatures, the physics of the dual theory should not distinguish between flat space and a sphere, up to finitevolume effects. However, the scaling with T of the ISCO energy and (cid:96) isco , which sets the window for these statesto exist, is different than the standard linear in T thermal excitations. This necessarily requires the existence of afurther scale, the spatial curvature. Moreover, lack of thermalization for these excitations and that this failure tothermalize stops abruptly at (cid:96) isco can not be explained through any perturbative argument, i.e using any polynomialin momentum. It is important to stress that finite volume is not enough. Indeed, in 2d CFTs on a finite circle, thesestates do not exist, matching the bulk result [2] and our argument due to the absence of curvature.A further outcome of our analysis is the calculation of the binding energy of the stable orbits in terms of conservedquantities that can be measured at the boundary. The latter decreases as we increase the angular momentum. Whenthis binding energy is small, we can think of it as giving small contributions to the dimension of a composite operator[12]. We further compute the binding energy in the radial fluctuations by including the excitations in near circularorbits. The latter is also small at large angular momentum. Both these corrections are absent in free field theorieswhere all conformal dimensions are half-integer. Any universality that one might want to ascribe to these effects needsto be tempered with usual caveats about field theories with a gap in anomalous dimensions and where one can arguethat the contributions from the gravity sector in the bulk, the stress tensor multiplet in the boundary, can dominatethe result [9–12]. Results from the bootstrap at large angular momentum can show consistency with these ideas,at least to the extent that it shows an expansion in inverse powers of the angular momentum [13–15], although theanalysis is usually done with four point functions of identical operators of small dimension, and the leading inversepower in the angular momentum is different: it is controlled by the twist of the stress energy tensor multiplet.We also provide an operational way to prepare the bulk particles in these orbits by the insertion of operators inthe boundary theory, extending the previous work [21] to the black hole case. This mechanism is best understoodas a tunneling process starting from the boundary. We show how the particle tunneling from the boundary on anEuclidean geodesic and the WKB calculation are related to each other using separation of variables in the Hamilton-Jacobi formulation of the geodesics. When we have angular momentum in the problem, these are complex geodesics.The paper is organized as follows. In section II, we discuss gravitational physics. An exact analysis on existence,stability, angular velocity and energy of circular orbits is provided in section II A. The large mass scaling is studiedin section II B and the calculation of the radial frequency for near-circular orbits is given in section II C. The dualinterpretation is given in section III. This contains our arguments for the relevance of spatial curvature, the rangeof quantum numbers for which these states exist in section III A and also on the relation between our work and thebootstrap approach. A discussion on how to operationally define these states in the boundary theory by the insertionof boundary operators together with its bulk interpretation in terms of tunneling is addressed in section III B. Weconclude with some summary of results and outlook of future directions in IV. Appendix A contains a more detaileddiscussion on some of the arguments given in section III B. II. CIRCULAR TIMELIKE GEODESICS FOR ADS SCHWARZSCHILD BLACK HOLES
The (d+1)-dimensional global AdS Schwarzschild black hole metric ds = − H ( r ) dt + H − ( r ) dr + r d Ω (1)involves a redshift factor H ( r ) = 1 + r L − Mr d − (2)depending on the radius of AdS L and the parameter M , related to the mass of the black hole M BH by M BH = ( d − πG ω d − (2 M ) . (3)Here ω d − stands for the volume of the (d-1)-sphere in (1). The black hole outer horizon r h satisfies H ( r h ) = 0 andis related to the black hole temperature ( T ) by T = d L r h + d − r h . (4)The case d = 2, corresponding to a BTZ black hole [22] if M > /
2, is special because it allows an analytic solutionto H ( r h ) = 0. This can also be achieved, for any d , in the large mass limit, keeping L fixed, when the redshift factorcan be approximated by H ( r ) ≈ r L − Mr d − , (5)allowing to find an explicit analytic expression for r h and its relation to the temperature r h L ≈ (cid:18) ML d − (cid:19) /d , T L ≈ d r h L (6)The case d = 3 is a black hole in AdS . The generic structure of geodesics in this spacetime has been analyzed in [23].We generalize these to higher dimensions and pay special attention to the large mass limit where additional analyticresults can be found. A. Classical stability and ISCOs
To study the classical stability of timelike circular orbits in these black holes, consider the geodesic quadratic action S = 12 (cid:90) ds (cid:16) − H ˙ t + H − ˙ r + r ˙ φ (cid:17) , (7)together with the constraint equation − − H ˙ t + H − ˙ r + r ˙ φ = g (cid:18) dds , dds (cid:19) . (8)The latter guarantees the action is proportional to the proper time s . Notice we already used spherical symmetry toreduce the dynamics to an effective 3d problem (7). Indeed, spherical symmetry guarantees any geodesic takes placein a plane of rotation where φ describes the evolution of the angular variable in such plane.The black hole metric (1) is invariant under time translations and rotations. These give rise to two conservedquantities (on top of the constraint) e = H ( r ) ˙ t, (cid:96) = r ˙ φ , (9)corresponding to the energy and angular momentum per unit mass, respectively. This is because our action (7) hasno dependence on the mass m of the particle. Plugging this back into (8) − − e + ˙ r ) H − ( r ) + (cid:96) r , (10)the latter can be arranged into e = ˙ r + V ( r ) (11)where V ( r ) is an effective potential given by V ( r ) = (cid:18) (cid:96) r (cid:19) H ( r ) = 1 + (cid:96) L + (cid:96) r + r L − M (cid:96) r d − Mr d − . (12)Circular orbits occur when ˙ r = 0. They sit at critical points r o of V , V (cid:48) ( r o ) = 0. Classical stability requires V (cid:48)(cid:48) ( r o ) > e can be compensated by a small change δr , while still finding aturning point in the constraint equation (11). The notion of innermost stable circular orbit (ISCO) r isco correspondsto the marginal stable orbits satisfying V (cid:48)(cid:48) ( r isco ) = 0. Due to the change in sign V (cid:48)(cid:48) ( r p ) < r p < r isco , all suchorbits would necessarily plunge into the black hole. a. Global AdS. It is convenient to analyze circular orbits in global AdS ( M = 0) first. This is because, at fixedmass M , large circular orbits should approach the ones existing in global AdS. In this case, critical points V (cid:48) ( r o ) = 0occur when − (cid:96) r o + 2 r o L = 0 ⇒ r o = √ (cid:96) L , e = (cid:96)L + 1 (13)where we used (11) and (12) for M = 0. Since V (cid:48)(cid:48) ( r o ) > (cid:96) , all circular orbitsare classically stable. Hence, there is no ISCO in global AdS. The particle angular velocity in global time equalsΩ = dφdt = (cid:96)r o r o /L + 1 e = 1 L . (14)This equals ω = ded(cid:96) and corresponds to propagation at the speed of light in the boundary of AdS. To sum up, thereexists a single circular orbit for any value of the angular momentum (cid:96) related to the size of the orbit r o by the scalingrelation (13). b. Schwarzschild AdS black holes. Let us consider stable circular orbits in the black holes (1). Criticality imposesthe condition V (cid:48) ( r o ) = − (cid:96) r o + 2 r o L + d M (cid:96) r o d +1 + ( d −
2) 2 Mr d − o = 0 . (15)Solving for (cid:96) ( r o ), we get (cid:96) = r o L ( r d o + ( d − M L ) r d o − dM r o . (16)As expected, when r o → ∞ and d >
2, at fixed M , we recover the circular orbits of global AdS (13), where r o (cid:39) √ (cid:96) L .In fact, for d > (cid:96) L > r o . This means circular orbits get pushed in relative to global AdS. Equivalently, the radiusof the orbit at fixed angular momentum shrinks. Even though the current discussion is coordinate dependent, thisconclusion reflects the attractive nature of gravity. The d = 2 BTZ black hole behaves differently. Notice (cid:96) is alwaysnegative whenever M > /
2. Thus, there are no circular geodesic orbits in BTZ, reproducing a well known statement[2].Circular orbits stop existing at the radius r min , defined by r d min − dM r min = 0 ⇒ r d − min = dM . (17)This is because (cid:96) is negative for r o < r min , whereas (cid:96) is large for r o (cid:38) r min . A better understanding of this geometriclocus can be gained in terms of a limiting set of light-like geodesics. Indeed, the constraint equation for light-likegeodesics ( − e + ˙ r ) H − ( r ) + (cid:96) r = 0 (18)gives rise to the effective potential V l ( r ) = (cid:96) H ( r ) r = (cid:96) (cid:18) L + 1 r − Mr d (cid:19) . (19)Circular light-like geodesics correspond to V (cid:48) l = 0. These are independent of (cid:96) and occur when V (cid:48) l ( r ) = − r + d Mr d +1 = 0 . (20)or equivalently, at r d − dM r = 0. Thus, the locus r min corresponds to the light-ring of the black hole. Notice there isno light-ring for BTZ black holes since V (cid:48) l ( r ) > d = 2 and 2 M >
1, in agreement with our previous observation.To sum up, all circular timelike orbits, stable and unstable, must occur beyond the light-ring radius, i.e. r o > r min . c. Stability. Stability requires the second derivative of the effective potential (12) to be positive r o V (cid:48)(cid:48) ( r o ) = 6 (cid:96) r o + 2 r o L − d ( d + 1) 2 M (cid:96) r d +1 o − ( d − d −
1) 2 Mr d − o > , (21)together with the critical condition (15). We discuss the notion of marginal stability, i.e. the one algebraicallycharacterised by V (cid:48)(cid:48) ( r isco ) = 0, first.Solving (15) and r isco V (cid:48)(cid:48) ( r isco ) = 0 for r isco /L and equating the corresponding expressions, we derive the constraint8 (cid:96) isco r d − isco = d ( d + 2) 2 M (cid:96) isco + d ( d −
2) 2
M r isco (22)This relates the locus of the orbit r isco with its angular momentum (cid:96) isco . When the latter holds, one must still solve asecond constraint, say V (cid:48) ( r isco ) = 0.When plotting or numerically studying these two conditions, one finds that classical stability requires (cid:96) > (cid:96) isco , with (cid:96) isco satisfying a non-trivial algebraic condition. This statement can be made explicit for d = 4. In this particularcase, (22) becomes a quadratic equation that is solved by r isco = 3 2 M (cid:96) isco (cid:96) isco − M . (23)Obviously, this requires (cid:96) isco > M . Furthermore, plugging this back into (15), we obtain a second condition for (cid:96) isco L ( (cid:96) isco − M ) = 27 (2 M (cid:96) isco ) . (24)This condition was already derived in [8]. The lesson of this discussion is that for d > (cid:96) isco , the one corresponding to the marginal orbit r isco , such that for (cid:96) > (cid:96) isco , there exist classical stablecircular orbits in the AdS black holes (55). d. Angular velocity and energy. The angular velocity Ω of the stable orbits equalsΩ = ˙ φ ˙ t = (cid:96)r o H ( r o ) e . (25)This is equivalent to the definition ω = ded(cid:96) , (26)because for circular orbits the angular momentum depends on the size of the orbit through (16). Hence, the constraintequation (11) becomes e = V ( (cid:96), r o ) = (cid:18) (cid:96) r o + 1 (cid:19) H ( r o ) . (27)Taking the total derivative with respect to r , we find that2 e dedr = ∂ (cid:96) V d(cid:96)dr + ∂ r V = ∂ (cid:96) V d(cid:96)dr (28)From this expression, we derive the equivalence ω = (cid:18) dedr (cid:19) (cid:18) d(cid:96)dr (cid:19) − = ded(cid:96) = ∂ (cid:96) V e = (cid:96)r o H ( r o ) e = Ω (29)Using the identity 1 + (cid:96)r o = H − dM/r d − o (30)in (27) and plugging the corresponding energy in (29), we obtain an exact expression for the angular velocity ofcircular stable orbits ω L = (cid:115) d − M L r d o . (31)At fixed M and large orbit size, ω L → ded(cid:96) (cid:39) L (cid:115) d − M L ( L(cid:96) ) d/ (cid:39) L (cid:18) d − M L ( L(cid:96) ) d/ (cid:19) , (32)where we used r o ≈ ( (cid:96) L ) / , allow us to determine the binding energy of this large size orbits by integrating withrespect to (cid:96) e ( (cid:96) ) = (cid:90) (cid:96) ded(cid:96) (cid:39) (cid:96)L + 1 − M ( L(cid:96) ) d/ − . (33)Notice the arbitrary constant was fixed by matching the global AdS result (13). The negative term provides a gaugeinvariant characterisation of the binding energy of the circular orbit, i.e. independent of the choice of radial coordinate r . That the binding energy is negative indicates the attractive nature of gravity . Our result is consistent with thebinding energy found in [12] (see also [24]). Additional corrections will arise from both expanding the square rootand remembering that r ( (cid:96) ) < √ (cid:96) L receives corrections from large (cid:96) .Once more, the d = 2 case is special. The angular velocity ω L = 1 everywhere, and not just close to the boundary.Hence, the binding energy must be independent of the orbit size. Indeed, solving (16) for r o /L and plugging this intothe energy (27), we find e ( (cid:96) ) = √ − M + (cid:96)L , M < . (34)This is the energy of the orbit in an AdS conical defect. As soon as M > /
2, the energy becomes imaginary,indicating the corresponding circular orbits become unstable, as we already established.Observe that some analogue of Kepler’s third law, i.e. a simple scaling relation between the size of the orbit andthe period, would exist if the second term in (31) would be dominant. Since stable orbits have size r o > r min , thisrequires a small M and r o limit, i.e. L (cid:29) r o , while satisfying r d − o > dM . In this limit, we can approximate (16) by (cid:96) r o ≈ ( d − Mr d − o − dM/r d − o . (35)Plugging this back into (21) and after some algebra, we find r o V (cid:48)(cid:48) ( r o ) ≈ d − M/r d − o − dM/r d − o (cid:18) − d + 2 (cid:18) − d Mr d − o (cid:19)(cid:19) + 2 r o L (36) The correct binding energy requires to multiply our correction by the particle mass m due to our choice of classical action (7). We learn these orbits are not stable for d ≥
4, but they are stable for d = 3 if 3 M/r o < /
2, a situation where thefrequency is approximately ω ≈ √ M r − / o , (37)the same scaling in Kepler’s law. In this case r o (cid:28) L , so these orbits do not really feel the curvature of AdS andcan be considered as circular orbits in flat space Schwarzschild. These are known to satisfy Kepler’s third law in thestandard Schwarzschild coordinate system where the radius of the sphere d Ω determines the radial coordinate. Thisis exactly our choice in equation (1). B. Large mass limit
We established all circular orbits satisfy r o > r min , whereas the classically stable ones occur for r o > r isco , with themarginal orbit r isco determined by solving the kinematic constraints (22) and (15).These constraints can not be solved analytically, unless we consider a large black hole mass limit. Since r isco > r min and r min ∼ M / ( d − for any mass, it is natural to consider a large M limit keeping the locus of the black hole light-ring M/r d − fixed. Notice this scaling is different from the one determining the horizon in (6).To further appreciate the relevance of this physical scaling, introduce the dimensionless coordinate ˆ r = r/L andparameter ˆ M = M/L d − and further plug (16) into (21). After some algebra, this can be written as V (cid:48)(cid:48) = 1 L − d ˆ M ˆ r d − (cid:34) − M ˆ r d − (cid:32) d ( d + 2) + d − r (cid:32) d − d M ˆ r d − (cid:33)(cid:33)(cid:35) . (38)When we consider the large M limit keeping M/r d − fixed, the value of ˆ r is large and we can approximate the answerby V (cid:48)(cid:48) ≈ L − d ˆ M ˆ r d − (cid:34) − ˆ M ˆ r d − ( d ( d + 2)) (cid:35) . (39)The marginal orbits ( V (cid:48)(cid:48) ( r isco ) = 0) are located at r isco ≈ (cid:18) d ( d + 2) M (cid:19) / ( d − (40)and have angular momentum (cid:96) isco L ≈ (cid:114) d + 2 d − r isco . (41)This large mass limit also allows us to compare the ISCO size with the horizon size (6). Indeed, r isco L ≈ (cid:18) d ( d + 2)8 (cid:19) / ( d − (cid:16) r h L (cid:17) / ( d − . (42)Using (38), classical stability reduces to r > r isco . This is because whenever the numerator in (38) is positive, i.e. r > r isco , the denominator factor is also positive for d >
2. Furthermore, as stressed earlier, the angular momentumof stable orbits must indeed be above the marginal (cid:96) isco . Indeed, the difference (cid:96) L − (cid:96) isco L = ˆ r − d +2 a (cid:20) − d + 2 d − a + 4 d − a d +2 (cid:21) with a ≡ r isco r (43)is a monotonically decreasing function ∀ a ∈ (0 , e ( (cid:96) ) since this requires to invert (16) in order to write r o ( (cid:96) ).This is because the powers of M/r d − o appearing in (16) are kept fixed in the large mass limit, making the analyticinversion not possible.What we can determine is the scaling with the black hole temperature of both the energy and angular momentumof these orbits. According to (6), the temperature of the black hole in the large mass limit is T L ≈ d M ) /d . (44)Hence, the energy and angular momentum of the classically stable circular orbits scale with the temperature as e , (cid:96)L (cid:39) ( T L ) d/ ( d − . (45)Notice that since 2 d/ ( d − >
1, the power of the temperature in these expressions indicates a scaling that increasesfaster than T for the stable orbits. C. Near-circular orbits
We can extend our analysis to near-circular orbits by including radial excitations at fixed (cid:96) . The dynamics of theseorbits can be understood by expanding the constraint equation (11) around a circular orbit using r = r ◦ + δr and e = e ◦ + δe . The constraint becomes 2 e ◦ δe = (cid:18) dδrdt (cid:19) ˙ t + 12 V (cid:48)(cid:48) ( r ◦ ) δr (46)This is identical to the energy of a harmonic oscillator with mass m = ˙ t /e ◦ and spring constant k = V (cid:48)(cid:48) ( r ◦ ) / (2 e ◦ ).It follows the angular frequency ω r of these radial oscillations is given by ω r = km = 12 V (cid:48)(cid:48) ( r ◦ ) (cid:18) − d Mr d − ◦ (cid:19) = 1 L (cid:34) − ˆ M ˆ r d − ◦ (cid:32) d ( d + 2) + d − r ◦ (cid:32) d − d M ˆ r d − ◦ (cid:33)(cid:33)(cid:35) , (47)where in the second equality, we used ˙ t = e ◦ H = 1 (cid:113) − d Mr d − ◦ , (48)and in the third we simplified using (38). The frequency (47) is exact and valid for any stable orbit ( r ◦ > r isco ).For both large orbits at fixed mass M and the large mass limit discussed in section II B, the frequency of the radialoscillations reduces to ( ω r L ) ≈ − d ( d + 2) Mr d − ◦ (49)For large orbits at fixed mass M , specifically, r ◦ (cid:39) √ (cid:96) L , we can rewrite this frequency as ω r L (cid:39) − d ( d + 2)4 M ( (cid:96) L ) d/ − ( d ≥ . (50)The case d=2 is, once more, special. From the exact expression (47), we learn ω r L = 2 √ − M . (51)This is (cid:96) independent and it vanishes at the threshold of black hole formation, i.e. M = 1 /
2. The frequency becomesimaginary for
M > /
2, in agreement with these orbits being unstable, and it decreases from ω r L = 2, for globalAdS , to zero, as the conical defect mass M increases. III. THE CFT DUAL INTERPRETATION
To discuss the dual field theory interpretation of the bulk results presented in the previous section, we first usethe basic holographic dictionary mapping a bulk particle of mass m to a single trace operator of dimension ∆ (cid:39) m L + d/ d/ e and angularmomentum (cid:96) per unit mass of the bulk particle excitation translate into an energy E and angular momentum J inthe CFT given by E = ∆ e , J = ∆ (cid:96)L . (52)What is the possible meaning of the ISCO in the dual field theory? In the bulk spacetime, the ISCO defines an orbitsize separating plunging orbits, i.e. those falling directly into the black hole, from classically stable circular orbits.Once we include semiclassical corrections, these orbits become meta-stable and bulk particles can decay semiclassicallyby tunneling past the potential barrier and plunging into the black hole geometry. These effects are suppressed bya tunneling amplitude of the form A tun (cid:39) exp( − ∆ S ), with S the tunneling action for a point particle of unit massto overcome the potential barrier . The semiclassical tunneling of wave solutions (which can be computed using aWKB approximation) has also been studied to discuss black hole stability [29] and is an important ingredient in theanalysis.There is plenty of evidence in the AdS/CFT literature indicating that the classical plunging should be interpretedas the dynamics of thermalization for typical excitations of the field theory. This is supported by the scramblingproperties of black holes [3], the time evolution of holographic entanglement entropy [4–6] and two-point functions [5]after CFT quench perturbations, and the butterfly velocity spread of quantum information under small perturbationsnear the horizon of a black hole [7]. Applying the same logic, meta-stable orbits should describe field theory long-livedexcitations that do not thermalise like typical excitations . We are interested in understanding when these excitationsappear and how their parameters scale with the temperature of the black hole. Using the AdS/CFT correspondence,the bulk kinematics presented in the previous section will provide answers to these questions.However, before computing the dual CFT charges E and J for these excitations, we would like to interpret theexistence of these orbits as a non-perturbative curvature effect on the boundary field theory. In our case, the curvatureof the (d-1)-sphere.To gain some intuition, let us compare our AdS black holes (1) with planar AdS black holes having a flat cross section.These have infinite mass, but finite mass density. Their dual involves a CFT on flat space at finite temperature. Theirdual infinite energy is due to the infinite volume of flat space. Since this field theory has no scale on its own, the onlyenergy scale in the thermal CFT is the temperature itself.One way to derive these black holes from their global AdS versions in (1) (see [32] for example), is to combine amass rescaling M → λ d M with a conformal transformation ( M, L ) → ( λ − M, λ L ), in the limit λ → ∞ so that( M, L ) → ( λ d − M , λ L ) λ → ∞ (53)Combining this transformation with the coordinate rescaling r = λ ( d +1) /d ρ , the metric (1) becomes ds = − dt λ /d ( ρ /L − M ρ − d ) + dρ ( ρ /L − M ρ − d ) − + λ d +1) /d ρ d Ω . (54)To take the limit λ → ∞ requires the time coordinate rescaling ˜ t = λ /d t and approximating the sphere metric byits tangent plane at a point. To see the latter, expand the sphere metric around the north pole in a flat coordinatesystem, d Ω = d(cid:126)θ + a (cid:126)θ d(cid:126)θ + . . . and define a new rescaled angular variable (cid:126)x ⊥ = λ ( d +1) /d (cid:126)θ . The sphere metricthen becomes d Ω = λ − d +1) /d ( d (cid:126)x ⊥ + O ( aλ − d +1) /d )). Altogether, the dominant limiting λ → ∞ metric becomes ds = − d ˜ t ( ρ /L − M ρ − d ) + dρ ( ρ − M ρ − d ) − + ρ d(cid:126)x ⊥ . (55)Besides time translation symmetry in ˜ t , the resulting metric is invariant under the full euclidean group acting on (cid:126)x ⊥ , in agreement with the flat cross-section of the AdS black hole that resulted from zooming into a small angularregion around the north pole of the coordinate system. The location of the horizon ρ d h = 2 M L ≡ γ , determinesthe scale of the energy (˜ e ) and linear momentum ( p ⊥ ) for particles in these planar AdS black holes to be ˜ e ∼ γ − /d and p ⊥ ∼ γ − /d . Both scale with the temperature since T ∼ γ − /d . Thus, the double scaling limit preserves CFTexcitations scaling like E (cid:39) ∆ T and J (cid:39) ∆ T , the latter being converted to linear momentum in the limit.From the CFT point of view, black holes are dual to a finite temperature configuration. On the sphere, there arenow two scales : the radius of the sphere and the thermal length scale. However, at very high temperature, the Gravitational radiation can also cause the orbit to lose energy, but this effect is expected to be small [12]. The existence of such long-lived meta-stable states was already stressed in [8] when analytically estimating the frequency of the AdSblack hole quasi-normal modes in a WKB approximation.
E, p (cid:39) T , exactly like the double scaling limit of the dual black hole tells us.By contrast, the energies and angular momentum of the stable particles in ISCO trajectories around global AdSblack holes (1) scale like (45) e , (cid:96)L (cid:39) ˆ M / ( d − (cid:39) ( T L ) dd − . (56)This is a different scaling from the flat λ → ∞ double scaling limit, i.e. the global ISCO energies and angular momentagrow faster than linearly in the temperature. The same scaling has been observed in [8]. This means the global ISCOtrajectories get pushed out of the flat rescaled coordinate system, i.e. they are too far in the ρ direction to be capturedby the flat black hole cross section. Hence, the existence of global ISCO trajectories must be a property of the globalAdS geometry (1). They disappear in the planar limit. The one distinction between these two in the boundary fieldtheory is that the boundary geometry S d − has an additional scale: the radius of curvature, which is absent in R d − .This is precisely the scale L appearing on the right hand side of (56) to restore the units.The natural interpretation of this observation is that the metastability of the states associated to the stable orbitsis due to a curvature effect in the field theory that competes with the temperature. The effect is absent in 3 d blackholes (BTZ black holes), even if there is still a finite volume on the boundary. As we have seen, all geodesics plungein this case. Finite volume alone is not enough. We need the change in the curvature of the boundary. The effectdoes not exist either if we take a flat space black hole with periodic spatial identifications: all geodesics plunge inthat case as well.Moreover, the width of the wave functions for these states becomes exponentially small, rather than power law,in the temperature. This exponentially small behaviour is the semiclassical result of the gravity computation, whichwe assume gives the correct CFT description of the metastability of the perturbations. Since perturbative Feynmandiagrams can only give power laws in E and J, we must conclude that the dynamics responsible for this effect is non-perturbative in the curvature of the ( d − A. Quantum numbers
Having clarified the origin for the metastability of these states in the dual CFT, we return to the discussion of theirquantum numbers E and J . For fixed mass and large orbits, using the dictionary (52) in (33), stable circular orbitsaround the black hole should correspond to CFT excitations with energy E = ∆ + J − ∆ ML d − (cid:18) ∆ J (cid:19) d/ − . (57)Before including the radial fluctuations due to near circular orbits, we discuss the relation between our result (57)and the existing literature. First, notice that if we use M = πG N ( d − ω d − M BH , which follows from (3), we reproduce thedominant contribution to equation (2.18) in [12] up to a missing 1 / M . This observable in [12] was computed assuming that one particle (the black hole in ourcase) is fixed, while the other orbits with large (cid:96) , precisely the bulk situation we discussed in the previous section.Next, we compare with the Bootstrap programme computing the anomalous dimensions of composite operators[ O O ] (cid:96) in the CFT. Bootstrap calculations require one to have operators with dimension ∆ BH + ∆ + J + O (1 /J τ )where τ is the twist of the stress energy tensor, τ = d − O (1 /J τ/ ) . These are not in contradiction to each other. Theblack hole is also expected to have internal excitations with energies and spin very close to the semiclassical calculationabove. These would correspond to black holes with angular momentum J and energy ∆ BH + ∆ + J + O (1 /J τ ). Theyare not these states. The states we are discussing are only expected to be metastable and do not represent exactdimensions of conformal primaries. The metastable states should eventually decay to these black hole excitations, In the special case d = 2 we get the correct answer with twist zero. (cid:96) of O with the total angularmomentum (cid:96) , requires the latter to be much larger than the sum of the dimension of the two primaries, i.e. (cid:96) (cid:29) ∆ +∆ .In such situation, one would indeed expect the black hole to back react and move. Notice this is not the regime whereour approximations hold, where the black hole is expected to be heavy and to have ∆ BH (cid:29) J .Finally, let us compare with the bootstrap analysis of 4-pt functions involving two heavy and two light operators,as studied in [16–20], and references therein. This approach is based on an analytic continuation to the Regge limit,so that in the bulk one focuses on geodesic motion along null geodesics in the presence of a black hole. This providesa computation of the phase shift. The latter can be dominated by an exchange in the stress tensor multiplet in the t -channel (in the particle physics language). To understand the dimensions of composite operators in this situation,one needs to transform the t -channel expansion to the s -channel. To do so, the literature typically assumes that onecan treat the system as a “double trace” operator near a generalized free field limit. Under that assumption, onecan recover the same correction O (1 /J τ/ ) in the binding energy as in (57). However, there is no decay observedin these discussions since, ab initio, due to the generalised free field approximation, such possibility does not exist.A complete calculation would deal with operators whose conformal dimensions are near the bound state energy. Toobserve the decay would require to work with a very fine-grained density of states that captures the asymptotics ofOPE coefficients. Our understanding is the approximations used in this body of work are able to estimate an averageof these OPE coefficients convoluted with the density of states with the right intermediate quantum numbers. A fullcomputation would then be able to give a prediction of the ‘lifetime’ of the orbits. We would like to stress that ourresults seem to get the exact correct answers for the case of conical singularities in AdS . In these cases, since thereare no black holes in the bulk, the density of states is reduced (there is very little entropy) and the double traceapproximation is probably better justified, especially at large values of the angular momentum.We can now add the radial fluctuations associated to nearly circular orbits by treating these excitations as aharmonic oscillator with frequency ω r , derived in equation (50). These provide an additional contribution to theenergy δE = k ω r L , with k the occupation number , so that the total CFT energy equals E = ∆ + J + 2 k − ∆ (cid:18) k ∆ d ( d + 2)4 (cid:19) ML d − (cid:18) ∆ J (cid:19) d/ − ( d ≥
3) (58)Again, the δω r < k (cid:39) ∆, that is, when the radial excitationenergy is comparable to the mass of the particle.The analogous energy for 2d CFTs equals E = ∆ √ − M (cid:18) k ∆ (cid:19) + J ( d = 2) (59)For M = 0, we recover the usual descent relation for energies (dimensions) in the representations of the conformalgroup, where E = ∆ + J + 2 k counts descendants of the operator O ∆ of the type ∂ Jµ (cid:3) k O ∆ 6 . When we deal with ablack hole where M (cid:54) = 0, these can not be thought of as descendants. Descendants have a difference of energy with aprimary that is exactly an integer. The fact that we see small corrections in the energy means that we need to lookfor the descendants of these states elsewhere. The descendants will be generated by the center of mass motion of theblack hole instead. If we think of the black hole as an object that leads to a spontaneous breaking of the conformalsymmetry, the associated “goldstone bosons” would be associated to collective motion of the black hole itself. Theother Virasoro descendants will still be generated by boundary gravitons.In a sense, the fact that in (58) we get corrections in both J and k from the conformal group representationtheory values suggest that we should interpret these states as different (approximate ) primaries with dimension∆ tot (cid:39) E BH + E , The fact that the corrections in the energy are small but not zero at large J means that theseresults do not apply to free field theories, where all dimensions of operators are integers or half integers. One can inprinciple expect that these answers might be universal in some sense for theories at strong coupling that have a gapin the spectrum of dimensions of operators. To be more precise, the bulk analysis for near circular orbits (46) gives rise to a quantum harmonic oscillator with (cid:126) = 1 /m . Hence, δe = k (cid:126) ω r = ( k/ ∆) ω r L , with k the occupation number. Using our AdS/CFT dictionary (52), the CFT energy δE = ∆ δe = k ω r L . In [12] a different argument is used to suggest a similar result. These states are only metastable, so they do not describe exact dimensions of operators. B. Comments on operators generating these metastable states
We have translated the increase in energy and angular momentum
E, J in the boundary theory due to the bulkexcitations on classical stable orbits. However, we have neither specified how to generate these states nor whichcorrelation functions do not thermalize in a typical thermal time in the dual CFT.Two of the authors of this paper recently addressed how to get bulk excitations placing particles in specific globalAdS geodesics [21]. The main idea is that to create particles in the bulk with angular momentum J , associated to anoperator of dimension ∆, one considers boundary operator insertions of O (cid:15),J (cid:39) (cid:90) d Ω Y Jm ( θ ) exp( − (cid:15)H ) O ( θ, t ) exp( (cid:15)H ) , (60)acting on a state representing the background, which in the original paper was the ground state of the conformalfield theory. The integrals over the spherical harmonics Y Jm project onto the correct angular momentum state andthe presence of (cid:15)H regularizes the operator, so that the insertion is normalizable [30]. Because of the Euclideantime evolution on the boundary, the operators in question are not local at an instant of time. The natural way tointerpret the preparation of the bulk particle is in terms of tunneling from the boundary. The amplitude to producethe bulk particles can be computed from the Euclidean action of a point particle evaluated on a minimising trajectoryconnecting the AdS boundary to the turning point (aphelion, r (cid:63) ) of the classical lorentzian orbit where we wantto create the bulk excitation. In this way, the parameter (cid:15) becomes the Euclidean time that it takes the euclideangeodesic to connect both points and it can be determined by the properties of the euclidean geodesic solving thematching to the lorentzian orbit. We will be a bit more precise below figure 1.We want to apply this construction to the current discussion, where the state dual to the black hole backgroundis the thermofield double state [31]. The main idea is that this field theory mechanism based on boundary operatorinsertions allows us to place particles in the classical bulk orbits discussed in the previous section. Moreover, correlationfunctions of these operators will not only be sensitive to the dynamics of these trajectories, but attuned to them.That is, the physics of these trajectories can be turned to a problem of correlation functions on the boundary. Thus,if we include real time evolution and imaginary time preparation, we at least answer in principle an operational wayto access this dynamics.To further analyze the interpretation of placing particles of mass m (cid:39) ∆ in the classical stable geodesics identifiedin section II, we need to perform a semiclassical tunneling calculation in Euclidean AdS. Here, we want to clarify howthe result of this euclidean bulk calculation relates to the WKB approximation for the wave function (field) of theparticle in the bulk, as discussed in [8].Since our bulk particle has no spin, consider the wave equation for a scalar field Φ of mass m in the black holebackground (1), with angular momentum J = m(cid:96) , and energy E = me . Schematically (see appendix A for a moreprecise derivation), the wave function can be decomposed asΦ (cid:39) f ( r, (cid:96), e ) exp( − iEt ) exp( iJφ ) . (61)In a geometric optics approximation, the radial ODE equation for f ( r, (cid:96), e ) (cid:39) exp( − m ˜ W ( r )) can be written as − g rr ( ∂ r ˜ W ) + (cid:96) r − e H ( r ) + 1 = 0 . (62)The separation of variables in the quantum theory can usually be related to the separation of variables in the Hamilton-Jacobi theory for the classical motion of a particle via the WKB approximation. In the Hamilton-Jacobi theory, ˜ W is a generating function of a canonical transformation, where the conjugate momentum is p r = ∂ r ˜ W . We can indeedestablish this relation by comparing (62) to the constraint equation (11). The match is identical, up to a sign, if weset g rr p r = ˙ r , or equivalently, p r = g rr ˙ r . This sign corresponds to taking V ( r ) → − V ( r ) , e → − e and it is standardin the analytic continuation to an Euclidean geometry, as we discuss below and in more technical detail in appendixA 1.The euclidean point particle Lagrangian equals L = 12 g rr ˙ r + 12 g ττ ˙ τ + 12 g φφ ˙ φ , (63)where τ is euclidean time and g ττ is the corresponding euclidean metric component. To match the WKB expression forthe effective potential, the angular momentum needs to be analytically continued (cid:96) → i(cid:96) , as we analytically continuethe Euclidean time. The change of sign of g ττ relative to g tt takes care of the change e → − e , while the change of signin the extra term in the potential associated with the constraint equation arises from requiring that the Hamiltonian3constraint has solutions for regular Euclidean geodesics. All in all, the result changes e − V ( r ) → − ( e − V ( r )) as isstandard in tunneling calculations for a regular Schr¨odinger equation (see appendix A 1 for an explicit derivation ofthe matching between (62) and (11)).If we were to consider a more general generating function W ( r, θ, τ ) = ˜ W ( r ) − eτ + i(cid:96)φ , (64)this would correspond to the Hamilton-Jacobi theory generating function W for all the variables, not just the radialdirection. This is identical to W = (cid:90) p r dr − (cid:90) edτ + (cid:90) ˜ (cid:96)dφ = (cid:90) p r ˙ rds − (cid:90) e ˙ τ ds + (cid:90) ˜ (cid:96) ˙ φds = (cid:90) ds (65)where the integral over the worldline (cid:82) ds arises when the constraint equation (62) is applied in the expression.Since (cid:96) is imaginary, the relevant geodesic for the tunneling problem must be complex. In particular, φ must beimaginary so that W remains real. Thus, according to (65), evaluation of W computes the length of the geodesic.Hence, exp( − mW ) is the probability of tunneling, while exp( − mW ) is the tunneling amplitude itself for the complextrajectory. Notice the mass m of the particle in AdS units plays the role of 1 / (cid:126) , as discussed when computing thefrequency of the radial oscillations for nearly circular stable orbits. By contrast, the tunneling amplitude exp( − m ˜ W )is the one to be computed when we want to create a particle in one of our classical stable orbits at fixed angularmomentum. This is because the integration over the Y (cid:96)m in (60) fixes the angular momentum, on top of the classicalenergy. Notice we go from one tunneling expression to the other by a Legendre transform. This modification is alsonecessary to have a proper variational problem.Having clarified the relation between the WKB approximation to the bulk wave function and the euclidean actionin the inverted potential − V , let us have a qualitative discussion on how to implement the ideas from [21] in thecurrent black hole background situation. The typical inverted potential is shown in figure 1, where the yellow linestands for the fixed value of the classical energy. r - V FIG. 1. Effective potential for Euclidean action problem. The horizontal orange line indicates a fixed energy for the motion ofthe particle.
The prescription in [21] requires us to evaluate the Euclidean geodesic at fixed (cid:96) (as determined from Y (cid:96)m ) andbetween times τ = ± (cid:15) . The values of (cid:15) are obtained by solving for the geodesic once we know the turning point of thetrajectory in the inverted potential at fixed energy. In principle, one then inverts the problem to solve for the valueof the energy based on (cid:15) . The geodesic will be time symmetric with respect to τ = 0, which forces us a to the turningpoint of the dynamics in the radial direction at τ = 0, namely, to the point where ˙ r = 0. The geodesic will then needto go to the boundary. The point where the orbit folds back is exactly the aphelion of the regular orbit. Once it isplaced at that point, it will oscillate in the radial variable in real time, starting at that initial condition.Given a fixed angular momentum (cid:96) , there are three different tunneling problems one may consider depending onthe value of the energy e ◦ . The first tunneling problem corresponds to the situation where the energy crosses thepotential function just once, below the minimum of − V . The second corresponds to three crossing points (as the lineindicated in the figure). The third corresponds to a single crossing above the maximum of − V .The prescription in [21] determines the euclidean time (cid:15) as a function of the turning point r (cid:63) once the energy isfixed. To invert this relation gives rise to a multi-valued function r (cid:63) ( (cid:15) ) indicating there is more than one euclideantrajectory solving the problem. However, it is only the trajectory that minimises the euclidean action globally thatwill dominate the tunneling amplitude.4Remember the maximum of − V corresponds to the circular trajectory in the Lorentzian problem. When the energyis below the minimum of − V , the tunneling is to a position of large r , i.e. a high energy state in the Lorentzianproblem. The corresponding Lorentzian geodesic, starting from r (cid:63) plunges through the horizon and thermalizes. Thistunneling problem is very similar to the one discussed in global AdS [21]. Hence, it should dominate at small (cid:15) . Forthe second type of energy, the tunneling is to a stable orbit with some radial motion, starting from the outermost r (cid:63) .As the energy approaches the local maximum of − V , the proper Euclidean time diverges since the trajectoryspends a very long time near the top of the potential. This means that exactly for the circular orbit we have that (cid:15) ( e ◦ , (cid:96) ) → ∞ . Beyond this energy, the time coordinate (cid:15) needs to come back down again and be finite. There wouldalso be a discontinuous jump in r (cid:63) to a small value. These geodesics would have the same (cid:15) as other geodesics thathave high energy. That is, if we invert the problem to find e ( (cid:15), (cid:96) ), the energy e is multivalued. Basically, for eachsuch value of (cid:15) there is more than one Euclidean geodesic that can contribute. The one that dominates is the onethat ends at the largest value of r , just like in vacuum AdS, because it has the shortest length. This should be theone with high energy, which is most similar to vacuum AdS. Effectively, our prescription with (cid:15) will not let us probethe small r region with plunging geodesics that start closer to the origin than the circular orbit: they correspond to asubdominant saddle in the path integral. Many of these will have e < (cid:96) . This is very similar to results in [37], whichargue that certain smearing functions can not be constructed for situations where the energy is small relative to theangular momentum e (cid:28) (cid:96) .In our setup, the full Euclidean calculation should not use just the naive analytic continuation of the time variable,but should also include the temperature of the black hole, with a “cigar-like” geometry. The time variable shouldbe periodic. That is, we usually need to impose regularity at the horizon. The periodicity of the circle at infinityis the temperature of the black hole. This would force an upper bound of (cid:15) < β/
4, because we need to identify τ → τ + nβ . The physics of this is as follows: if we have too long of a wait in Euclidean time, we will be sensitiveto thermal fluctuations and this will take the particle away from the trajectory we desire. These thermal fluctuationsare calculated from the geodesics with the different values of τ → τ + nβ . The bound guarantees the geodesic wefocus on is the dominant one. One should be careful of this analysis in the case of microstate geometries, rather thanin the canonical ensemble. There the periodicity in Euclidean time is not guaranteed and this can make its presenceknown in Euclidean correlators [35].To summarize, there can be many tunneling Euclidean geodesics subtended from fixed points in the boundary.Only one such geodesic will be dominant and these dominant geodesics miss regions of the interior where in principleexcitations can be placed in the bulk.For completeness, we can also consider the analysis of the oscillating region within a WKB approximation. Thesewill give the real part of the exact quantized energies of the particles in the stable orbits. The orbit in the radialdirection would be periodic in time, and one can semiclassically quantize the energy by requiring that (cid:73) p r dr = (cid:18) n + 12 (cid:19) π (cid:126) (66)where the contour integral covers the physical region twice. Here one can take (cid:126) = 1 /m , or take (cid:126) = 1, but use a p r that includes the factor of m derived from the correct normalization of the action. This is the type of analysisthat was performed in [8], see also the earlier work [36]. For small oscillations near the circular orbit, this is alreadyimplicit in our analysis (58). That analysis quantized a harmonic oscillator with the frequency of oscillations of theradial variable near the circular orbit. A more general analysis would compute the contour integrals and impose thecondition (66) to evaluate the energies of the orbits. IV. OUTLOOK
In this paper we took some steps to understand the dual interpretation to the physics of stable geodesics in blackhole backgrounds in AdS/CFT. We argued their existence indicates that there is some physics that does not thermalizeon a thermal time scale and we interpreted the bulk excitations as metastable states in the dual field theory. Westudied the size, angular velocity, energy and angular momentum of these trajectories. Using the standard AdS/CFTdictionary, the last two provide the quantum numbers of these states. The least energetic marginally stable trajectory,the ISCO, provides a sharp way to identify the window of quantum numbers where this type of dynamics exists.We were careful to compute the spectrum of these excitations near the circular stable geodesics, including the radialexcitation quantum numbers. The results indicate a binding energy for both the radial fluctuations and for the circulartrajectories, which usually vanishes at very large angular momentum. Such a small correction is not compatible witha free field theory analysis, where the spectrum of excitations has an integer spacing between them. We do notyet understand under what conditions these results are universal. However, using holographic considerations would5suggest the dual field theory to have a gap in the spectrum and to be sparse so that the gravity dual might be a goodapproximation to the physics [9, 10].In the large temperature limit, the CFT predictions on a sphere or on the plane should be equivalent, up to finitevolume effects. Since the energy of the bulk excitations scales differently with the temperature than linearly, it requiresa second scale in the problem. We argued this scale is the curvature of the boundary field theory since these orbitsdo not exist when the boundary is flat, like in BTZ black holes, nor in flat AdS black holes.The positive curvature of the sphere in the boundary is suggestive. Light rays emanating from a point focus andmight reconstitute objects when they do. Since we don’t have a CFT calculation to perform, we can guess that thisfocusing is an important ingredient in the corresponding calculation. Such an idea suggests that for negatively curvedboundaries the corresponding black holes do not have such stable trajectories, as the null geodesics defocus. Thesealso have an important role to play in the understanding of entanglement entropy [33]. This analysis is beyond thescope of the present paper.At least in principle, our analysis provides a probe/test of which strongly coupled theories are holographic: theyneed to have these long lived excitations. Because these only occur when the boundary has spatial curvature, we needa 2d material sample in the shape of a sphere to use as a probe. The quantum numbers of these excitations dependon an anomalous scaling with the temperature and occur at non-zero angular momentum. Alternatively, we can fixthe temperature and change the size of the sample: their energies would depend non-trivially on the radius of thesphere. Another probe that depends on the geodesic structure is the light ring (Einstein ring). This has been arguedto be probed by CFT correlators that see gravitational lensing in [34]. In our results, the light ring radius also hasan anomalous scaling at high temperatures that is different than the horizon radius.Our work focused on spherically symmetric AdS black holes. It would obviously be interesting to extend thesecalculations to other black holes in AdS, like Reissner-Nordstrom or Kerr, whose structure will necessarily be richer.We also discussed a CFT mechanism to prepare the bulk particles on these trajectories. This involves tunnelingfrom the boundary using euclidean geodesics. In general, these tunneling amplitudes are hard to compute for theseblack holes, as they require solving complicated integrals. It would be interesting to find cases where some of thesecan be performed analytically. Since these tunneling geodesics to the boundary have infinite action, to understandtheir physics better one needs to find a regularization scheme that can be used to compare directly with the physicsof empty AdS. Having explicit analytical examples would provide a guide to do so. We also analyzed in more detailthe relation between this tunneling computation and the WKB approximation for the radial part of the scalar waveequations for massive particles in AdS.Another important observation we have is that there can be more than one tunneling geodesic related to fixed endpoints and angular momentum on the boundary. This is related to the Euclidean time being periodic in the Euclideanblack hole background. The bulk geodesics attached to these points can have different winding numbers. These makeit hard to prepare states in certain regions of the black hole geometry with our prescription: near the horizon at largeangular momentum, as they correspond to a subdominant contribution in the tunneling trajectory. One can speculatethat these obstructions are related to the absence of smearing functions for modes very close to the black hole withlarge angular momentum [37], which seem somewhat similar in character.
ACKNOWLEDGMENTS
D. B. would like to thank D. Harlow, J. Maldacena, J. Santos for discussions. The work of D.B. is supported inpart by the Department of Energy under grant DE-SC 0011702.
Appendix A: Geometric optics approximation
On general grounds, one expects our geodesic analysis in section II to capture the geometric optics limit of a massivescalar field propagating in (1). We make this connection explicit below, explaining why the temperature scaling in(56) was observed in [8] when studying the temperature dependence on the quasi-normal modes propagating in (1).This discussion will also allow us to connect with our euclidean tunneling observations in the main text.Consider a massive bulk scalar field propagating in (1) with action S = − (cid:90) d d +1 x √− g (cid:0) ( ∇ Φ) + m Φ (cid:1) (A1)Decomposing the scalar wave function as Φ = e − iωt Y lm ( (cid:126)θ ) r − ( d − / ψ ( r ), its Klein-Gordon equation of motion reduces6to the radial equation [8] (cid:0) ∂ z + V Φ − ω (cid:1) ψ = 0 , V Φ = H ( r ) (cid:18) (2 l + d − + 14 r + ν −
14 + 2 M ( d − r d (cid:19) . (A2)The parameter ν = ( mL ) + d /
4, i.e. the conformal dimension of the boundary operator O is ∆ = ν + d/
2, and theradial coordinate z is defined through dzdr = − H ( r ) . (A3)Since the conserved charges (9) were per unit mass, the relation between these and the Fourier modes of the scalarfield must be ω = mL e l = mL (cid:96) (A4)The geometric optics limit requires to work with a large mass, so that ν ≈ mL . Notice the last term in the potential V Φ is negligible both for large orbits at finite M and in the large M limit keeping M/r d − fixed. Both situationsinvolve large angular momentum, so that we can approximate the scalar potential by V Φ ≈ ν H ( r ) (cid:18) (cid:96) r (cid:19) = ν V (cid:96) ( r ) . (A5)Looking for a radial wave function of the form ψ ( r ) = e νS and only keeping the dominant contribution to the waveequation (A2), i.e. keeping terms not suppressed by negative powers of ν , the wave equation reduces to − ( ∂ z S ) + V (cid:96) = e . (A6)This equation reproduces the geodesic constraint (11) if S = i ˜ W and we use the Hamilton-Jacobi equation p r = ∂ r ˜ W .Indeed, − ( ∂ z S ) = − ( H ( r )) ( ∂ r S ) = H ( ∂ r ˜ W ) = p r H = H ( g rr ˙ r ) = ˙ r (A7)We learn the radial wave function of the scalar field equals ψ ( r ) = e iν ˜ W in the geometric optics limit, where ˜ W standsfor the lorentzian particle action in the Hamilton-Jacobi formulation, as claimed in the main text.
1. Euclidean continuation
The discussion above related the radial wave equation of the massive scalar field with the geodesic constraintequation (11). Here, we discuss the euclidean continuation of the lorentzian geodesic action in order to compute theamplitude for tunneling of a bulk particle as the interpretation of the boundary operator insertion (60).To avoid any confusions regarding our starting action (7) and our conventions in the main text, remember thestandard lorentzian relativistic point particle lagrangia L lor = − m (cid:112) − ˙ x (A8)is classically equivalent to L lor = m (cid:18) ˙ x em − em (cid:19) = m ˆ L lor + constant (A9)Fixing the gauge em = 1 gives rise to our starting lorentzian action (7) together with the constraint (8). This laststep, where we dropped the constant piece, justifies why the conserved charges (9) were defined per unit of mass.One possible euclidean continuation of the Lorentzian lagrangianˆ L lor = 12 (cid:16) g tt ˙ t + g rr ˙ r + g φφ ˙ φ (cid:17) (A10)is given by t → iτ and s → − is , where s is proper time. The change in the measure ds together with an overall signchange in the lagrangian density allows to transform the amplitude in the lorentzian path integral e iS lor into e − mS euc with corresponding euclidean lagrangian given by L euc = 12 (cid:16) − g tt ˙ τ + g rr ˙ r + g φφ ˙ φ (cid:17) ≡ (cid:16) g ττ ˙ τ + g rr ˙ r + g φφ ˙ φ (cid:17) (A11)7where g ττ = − g tt stands for the standard euclidean time component of the metric and L euc is based on the euclideanmetric, as it should .Let us check what happens to the constraint equation (8) under this euclidean continuation − e g tt − g rr ˙ r − g φφ ˙ φ e (A12)Since both t and s are Wick rotated, e does not change, but the kinetic terms in the spatial directions flip sign.Writing this in terms of the euclidean metric components, this is equivalent to e g ττ = − g rr ˙ r + 1 − g φφ ˙ φ e (A13)Finally, ˙ φ = − i ˙ φ e . 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