Quantum Fluctuations and the Unruh Effect in Strongly-Coupled Conformal Field Theories
Elena Caceres, Mariano Chernicoff, Alberto Guijosa, Juan F. Pedraza
aa r X i v : . [ h e p - t h ] A p r UTTG-03-10
Quantum Fluctuations and the Unruh Effectin Strongly-Coupled Conformal Field Theories
Elena C´aceres † , Mariano Chernicoff ⋆ , Alberto G¨uijosa ♮ and Juan F. Pedraza ♮ † Facultad de Ciencias,Universidad de Colima,Bernal D´ıaz del Castillo 340, Col. Villas San Sebasti´an,Colima, Colima 28045, M´exico ⋆ Departamento de F´ısica,Facultad de Ciencias,Universidad Nacional Aut´onoma de M´exico,M´exico D.F. 04510, M´exico and
Departament de F´ısica Fonamental,Universitat de Barcelona,Marti i Franqu`es 1, E-08028 Barcelona, Spain ♮ Departamento de F´ısica de Altas Energ´ıas,Instituto de Ciencias Nucleares,Universidad Nacional Aut´onoma de M´exico,Apdo. Postal 70-543, M´exico D.F. 04510, M´exico
Abstract:
Through the AdS/CFT correspondence, we study a uniformly accel-erated quark in the vacuum of strongly-coupled conformal field theories in variousdimensions, and determine the resulting stochastic fluctuations of the quark trajec-tory. From the perspective of an inertial observer, these are quantum fluctuationsinduced by the gluonic radiation emitted by the accelerated quark. From the pointof view of the quark itself, they originate from the thermal medium predicted bythe Unruh effect. We scrutinize the relation between these two descriptions in thegravity side of the correspondence, and show in particular that upon transformingthe conformal field theory from Rindler space to the open Einstein universe, the ac-celeration horizon disappears from the boundary theory but is preserved in the bulk.This transformation allows us to directly connect our calculation of radiation-inducedfluctuations in vacuum with the analysis by de Boer et al. of the Brownian motionof a quark that is on average static within a thermal medium. Combining this samebulk transformation with previous results of Emparan, we are also able to computethe stress-energy tensor of the Unruh thermal medium. [email protected] mchernicoff@ub.edu [email protected] [email protected] Introduction and Summary
The radiation emitted by an accelerated charge inevitably backreacts on the charge.One effect, present already at the classical level, is a reaction force on the charge, thattends to damp its motion. But if the system is quantized, one additionally expectsthe emission of radiation to induce stochastic fluctuations of the charge’s trajectory.In the context of a quantum Abelian gauge theory, the first effect has been exploredin [1, 2, 3, 4], and the second, in [4, 5]. The discovery of the gauge/gravity duality[6, 7, 8, 9] has opened the possibility of extending the exploration to the previouslyuncharted terrain of strongly-coupled non-Abelian gauge theories.The first step in this direction was taken recently in [10, 11], where it was shownthat the duality allows a simple derivation of an equation of motion for a ‘composite’or ‘dressed’ quark that correctly incorporates the effects of radiation damping. Theresult is a non-linear generalization of the classic (Abraham-)Lorentz-Dirac equation[12] that is physically sensible and (unlike Lorentz-Dirac) has no self-accelerating orpre-accelerating solutions. The damping effect follows directly from the fact that, inthe context of this duality, the quark corresponds to the endpoint of a string, whosebody codifies the profile of the non-Abelian (near and radiation) fields sourced by thequark, and can thus act as an energy sink. In other words, the quark has a tail, andit is this tail that is responsible for the damping force.This mechanism had been previously established in the computations of the dragforce exerted on the quark by a thermal plasma, which is described in dual languagein terms of a string living on a black hole geometry [13, 14]. The analysis of [10, 11]makes it clear that the damping effect is equally present in the gauge theory vacuum,i.e., irrespective of whether or not there is a black hole in the dual geometry (although,of course, the detailed form of the damping force is different). On the other hand,energy loss via the string does turn out to be closely associated with the appearanceof a worldsheet horizon, as noticed initially in [15, 16] at finite temperature andemphasized in [17] for the zero temperature case. This association has been furtherstudied in [18, 19, 20, 21].As is customary in the gauge/gravity setting, the calculation in [10, 11] treatedthe quark as a heavy particle coupled to the fully quantized gluonic (+ other gaugetheory) field(s). In this paper we go beyond the classical description of the particleand study the quantum fluctuations of the quark trajectory induced by its couplingto the gluonic field.While we expect the physics of interest to us to be present under rather generalcircumstances, for simplicity we will restrict attention to the anti-de Sitter (AdS) /conformal field theory (CFT) subcases of the gauge/gravity duality, with the CFTdefined on Minkowski spacetime of arbitrary dimension d . We begin in Section 2.1by recalling the basic entries of the AdS/CFT dictionary that are of interest to us,as well as the results (12)-(15) of [10, 11] on radiation damping for a quark thataccelerates in the CFT vacuum. In Section 2.2 we specialize to the case of uniformacceleration, deriving the relevant classical string embedding (21) as a particular1nstance of the general solution obtained in [22], and verifying that the induced metricon the worldsheet contains a black hole (as had been established previously in [19, 23]),a fact that plays a central role in our investigation. Such a black hole would in factbe present for any accelerated quark trajectory [17], but we are able to carry out thecalculations of interest to us only in the case where the worldsheet geometry is static,which corresponds to uniform acceleration.When we go beyond the classical description of the string, two new effects arefound, both of which are suppressed by a factor of the string length divided by theAdS curvature radius, or, equivalently (via (6)), by an inverse factor of the CFTcoupling. On the one hand, we pick up the usual quantum fluctuations arising fromthe determinant of the path integral over string embeddings. These are present evenfor a static string (see, e.g., [24, 25]), and lead to spontaneous deviations from theaverage endpoint/quark trajectory of the type studied, e.g., in [26]. On the otherhand, the worldsheet black hole emits Hawking radiation, which populates the variousmodes of oscillation of the string. In what follows we will concentrate solely on thissecond effect, which is present only for an accelerated trajectory and is thus associatedwith the quantum fluctuations induced by the gluonic radiation emitted by the quark.Needless to say, in the future it would be interesting to determine the way in whichthe quark behavior is modified when both effects are combined.As always, small perturbations about the average string embedding (21) are de-scribed by free scalar fields propagating on the corresponding induced worldsheetgeometry. Our task is then to quantize these modes and establish the way in whichthe excitations generated in them by the presence of the worldsheet horizon make thestring endpoint fluctuate. This analysis is in complete parallel with [27, 28], where thesame question was studied for a static string on the (planar) Schwarzschild-AdS d +1 geometry, which is dual to a static quark in a thermal bath of the CFT. It was shownin those works that worldsheet Hawking radiation indeed gives rise to the expectedBrownian motion of the endpoint/quark, whose detailed form is captured by a gen-eralized Langevin equation. The authors of [27] reached this conclusion in arbitrarydimension by assuming (following [29, 30]) that the state of the quantized embeddingfields is the usual Hartle-Hawking (or Kruskal) vacuum, which describes the blackhole in equilibrium with its own thermal radiation. The authors of [28] focused onthe case d = 4 and followed a different but equivalent route, employing the dualrelation between the Kruskal extension of the Schwarzschild-AdS geometry and theCFT Schwinger-Keldysh formalism [31, 32, 33], together with the known connectionbetween the latter and the generalized Langevin equation. These calculations werelater generalized and elaborated on in [34, 35, 36, 37].When we attempt to run through either of these procedures for the case of a quarkundergoing uniform acceleration at zero temperature, the analysis is complicated bythe explicit time dependence present in the worldsheet geometry (22), which is onlyto be expected, given that the velocity and rate of radiation of the quark vary astime marches on. It is then natural to expect the problem to simplify if instead ofworking in the coordinates appropriate for an inertial observer we transform to a2indler coordinate system adapted to an observer sitting on the quark. We thereforepostpone the study of the string fluctuations until Section 4, and dedicate Section 3to a close scrutiny of the relevant transformations and their physical consequences.This exercise turns out to be rather interesting in itself, and sheds light on the AdSimplementation of the celebrated Unruh effect [38] (for reviews, see, e.g., [39]). Earlieranalyses of this implementation can be found in [19, 23], as well as in the very recentwork [40], which appeared while the present paper was in preparation.In Section 3.1 we present the bulk diffeomorphism that implements the transitionfrom Minkowski to Rindler coordinates (which we denote with primes), equation (29).This transformation gives rise to an acceleration horizon both in the boundary andbulk descriptions. As a result, a state that is pure from the inertial perspective willgenerally be mixed from the point of view of the Rindler observers, because the fielddegrees of freedom accessible to the latter will be entangled with degrees of freedomin the region beyond their horizon, which they must trace over. In particular, thepure AdS geometry expressed in Rindler coordinates, equation (30), which is dual tothe Minkowski vacuum of the CFT (as evidenced by the vanishing of the expectationvalue of the stress-energy tensor), is interpreted as a thermal bath at the expectedUnruh temperature (27). In Rindler coordinates, the string embedding takes the form(33), which as expected is static and bends towards the Rindler horizon.In Section 3.2 we observe that the Rindler horizon of the CFT can be removedvia a Weyl transformation. The resulting geometry (37) (which we label with doubleprimes) is that of the open Einstein universe. Following [41], we identify the corre-sponding bulk transformation, equation (39). Unlike the diffeomorphism discussedin Section 3.1, it drastically alters the radial foliation of the AdS geometry. Eventhough, by construction, in this new conformal frame the acceleration horizon is nolonger visible in the boundary description, we show that it is still present in the bulk,but lies at the fixed radial position that according to the AdS/CFT dictionary corre-sponds to the Unruh temperature (27). In other words, after the transformation (39),the thermal character of the CFT state arises not from entanglement with degrees offreedom that lie beyond a spacetime horizon, but from the direct identification of thespecific energy scale (27) as the temperature of the CFT, in exact parallel with thedual interpretation of the Schwarzschild-AdS geometry.From the doubly primed AdS metric (40), we can extract the expectation valueof the stress-energy tensor in the conformal Minkowski vacuum of the CFT on theopen Einstein geometry. The result, given in (41), is in complete agreement with [42],where this same quantity was computed in the context of a more general investigationof hyperbolic black holes in AdS/CFT. It was elucidated long ago [43, 44, 45] thatfor even d this vev is shifted to a non-zero value as a result of the Weyl anomaly,which implies that the transformation that takes us from Rindler to open Einsteinspacetime is not a true symmetry of the CFT. The AdS counterpart of this statementis also well understood [46]. Using this information and the results obtained in [42]for the conformal Rindler vacuum of the CFT on the open Einstein universe, we cantranslate back to Rindler spacetime to determine the energy-momentum tensor of the3nruh thermal medium, equation (49).We close Section 3.2 by noting that in the doubly primed coordinate system, thestring embedding (50) is static and completely vertical. This encourages us to carryout our study of small perturbations about the average string trajectory precisely inthis frame, and in Section 4 we finally proceed to do so. Interestingly, both the basestring embedding (50) and the background geometry (40) at the location of the stringare found to coincide exactly with the d = 2 thermal setup analyzed in [27], whichallows us to obtain the information we are after simply by translating to our languagethe results of that work. This close relation between the quantum fluctuations of theuniformly accelerated quark on Minkowski spacetime and the thermal fluctuations ofa static quark in a thermal medium is evidently a direct consequence of the Unruheffect, but the reader should be aware that, for d >
2, the detailed properties ofthis thermal medium are found to differ from those of the familiar homogeneous andisotropic thermal ensemble dual to the Schwarzschild-AdS d +1 geometry.In Section 4.1, we carry over from [27] the generalized Langevin equation (55),describing the way in which the thermal medium on the open Einstein geometry makesour quark fluctuate, and deduce its local approximation (58), which is valid when thefluctuations are examined over time scales that are large compared to the quarkCompton wavelength. In Section 4.2, we then translate all relevant quantities backto the original, Minkowski (unprimed) frame, thereby concluding that the radiationemitted by the quark induces quantum fluctuations of its trajectory that obey theequations (64)-(67), which constitute our main result. Within an appropriate rangeof temporal resolutions, these equations simplify to the local form (68)-(71).The most prominent feature of these equations (in either nonlocal or local form) istheir manifest time dependence, which is in marked contrast with the static nature ofthe open Einstein thermal medium, but was of course entirely expected, given that, aswe emphasized above, the uniformly accelerated quark plus gluonic radiation systemis certainly not in equilibrium. Another salient property is the anisotropy between thelongitudinal and transverse directions, which is again inherent to the definition of thesystem in the inertial frame. In this second respect, our equations of motion for thefluctuations of a quark that undergoes uniform acceleration in the CFT vacuum aresomewhat akin to those of [34, 36], which considered thermal fluctuations of a quarkploughing at constant (and possibly relativistic) velocity through a thermal plasma.In our case, the anisotropy goes so far as to result in a longitudinal equation of motionthat, unlike the transverse equation, contains a term linear in the fluctuation and isconsequently not of generalized Langevin form. As discussed below (71), anothercurious feature is found in the signs of this and the frictional term, which in thelongitudinal case turn out to be counter-intuitive within a certain period of time.Curiosity aside, the direct connection with the generalized Langevin equation in theopen Einstein frame of course ensures that the novel longitudinal equation leads tophysically sensible evolution. The fact that we have been able to get our hands onthis strongly coupled physics constitutes yet another illustration of the usefulness ofthe AdS/CFT correspondence. 4 AdS/CFT and Uniformly Accelerated Quarks
According to the AdS/CFT correspondence [6, 7, 8], string theory (or M-theory) ona background that asymptotically approaches the ( d + 1)-dimensional anti-de Sitter(AdS) spacetime ds = G MN dx M dx N = L z (cid:0) − dt + d~x + dz (cid:1) , (1)where ~x ≡ ( x , . . . , x d − ), is dual to a d -dimensional conformal field theory (CFT).The directions x µ ≡ ( t, ~x ) are parallel to the AdS boundary z = 0, and are directlyidentified with the CFT directions. The radial direction z is mapped holographicallyinto a variable length/energy scale in the CFT, in such a way that z = 0 and z → ∞ are respectively dual to the ultraviolet (UV) and infrared (IR) limits of the CFT[47, 48].When AdS is radially foliated with Poincar´e (or horospheric) coordinates as in (1),the dual CFT lives on Minkowski spacetime, ds = η µν dx µ dx ν , but by choosingdifferent foliations we can obtain gravity descriptions of the same CFT on other back-ground geometries. Any asymptotically AdS metric can be written in the Fefferman-Graham [49] form ds = L z (cid:0) g µν ( z, x ) dx µ dx ν + dz (cid:1) , (2)from which the dual CFT metric ds = g µν ( x ) dx µ dx ν can be directly read offas g µν ( x ) = g µν (0 , x ). The full function g µν ( z, x ) is uniquely determined (via theEinstein equations in the bulk) from this boundary value together with data dual tothe expectation value of the CFT stress-energy tensor T µν ( x ). More specifically, interms of the near-boundary expansion g µν ( z, x ) = g µν ( x ) + z g (2) µν ( x ) + . . . + z d g ( d ) µν ( x ) + z d log( z ) h ( d ) µν ( x ) + . . . , (3)the standard GKPW recipe for correlation functions [7, 8] leads after appropriateholographic renormalization to [50, 51] (see also [52, 53, 54]) h T µν ( x ) i = d L d − πG ( d +1)N (cid:0) g ( d ) µν ( x ) + X ( d ) µν ( x ) (cid:1) , (4)where X ( d ) µν = 0 ∀ odd d , X (2) µν = − g µν g (2) αα , (5) X (4) µν = − g µν h(cid:0) g (2) αα (cid:1) − g (2) βα g (2) αβ i − g (2) αµ g (2) αν + 14 g (2) µν g (2) αα , Cross a compact space, which will be mentioned briefly around (6) but will otherwise play norole in our discussion. X (6) µν is given by a similar but more longwinded expression that we will nottranscribe here. In (5) it is understood that the indices of the tensors g ( n ) µν ( x ) areraised with the inverse boundary metric g µν ( x ).Known examples of this duality include the cases d = 2 , , ,
6, which involve thenear-horizon geometries and low-energy worldvolume theories associated with systemsof multiple D1/D5-, M2-, D3- and M5-branes, respectively [9]. For particle theoryapplications we are mainly interested in the case d = 4, where the best understoodexample equates Type IIB string theory on AdS × S (with a constant dilaton and N c units of Ramond-Ramond five-form flux through the five-sphere) to N = 4 SU ( N c )super-Yang-Mills (SYM) with ’t Hooft coupling λ ≡ g Y M N c = L l s . (6)Replacing the five-sphere with other compact geometries one obtains gravity duals ofCFTs with fewer supersymmetries. For all d , we will find it notationally convenientto use the rightmost expression as a definition of λ , and to refer to the CFT fields as‘gluonic’.It follows trivially from (4) that the string theory state described by the unper-turbed metric (1) is dual to a CFT state where h T µν ( x ) i = 0, i.e., the Minkowskivacuum. The closed string sector describing (small or large) fluctuations on top ofthis bulk geometry fully captures the gluonic physics. An additional, ‘quark’ sectorcan be introduced into the CFT by appropriately adding to the bulk a set of N f ‘fla-vor’ D-branes, whose excitations are described by open strings [55]. In this context,an isolated quark is dual to an open string that extends radially from the AdS horizonat z → ∞ to a position z = z m where it ends on the flavor branes. In particular,a static, purely radial string corresponds to a static quark, and by computing theenergy of the former one finds that z m is related to the quark mass m through z m = √ λ πm . (7)We will work in the regime where the geometry is weakly curved in units of the stringlength, L ≫ l s , in which the string (or M-) theory in the bulk essentially reduces tosupergravity, and the dual CFT is strongly coupled. We also assume that N f ≪ N c and consequently neglect the backreaction of the flavor branes on the geometry; inthe field theory this corresponds to working in a ‘quenched’ approximation whichdisregards quark loops.In more detail, it is really the z = z m endpoint of the string that is dual to thequark, while the body of the string encodes the (near and radiation) gluonic fieldprofile set up by the quark. It should be borne in mind that the quark so describedis automatically not ‘bare’ but ‘composite’ or ‘dressed’ [56, 11], and is surrounded bya gluonic cloud with characteristic thickness (Compton wavelength) z m .The string dynamics follows as usual from the Nambu-Goto action S NG = − πα ′ Z d σ p − det h ab ≡ Z d σ L NG , (8)6here h ab ≡ ∂ a X M ∂ b X N G MN ( X ) ( a, b = 0 ,
1) denotes the induced metric on theworldsheet. The quark worldline is identified with the trajectory of the string end-point, x µ ( τ ) = X µ ( τ, z m ). We can exert an external force ~F on the endpoint/quarkby turning on an electric field F i = F i on the flavor branes. This amounts to addingto the Nambu-Goto action the usual minimal coupling S F = Z dτ A µ ( x ( τ )) ∂ τ x µ ( τ ) . (9)The string here is being described, as usual, in first-quantized language, and, aslong as it is sufficiently heavy, we are allowed to treat it semiclassically. In CFTlanguage, then, we are at this point coupling a first-quantized quark to the strongly-coupled gluonic fields, and then carrying out the full path integral over the latter (theresult of which is codified by the AdS spacetime), but treating the path integral overthe quark trajectory x µ ( τ ) in a saddle-point approximation.Variation of the string action S NG + S F implies the standard Nambu-Goto equationof motion for all interior points of the string, plus the usual boundary condition [57]Π zµ ( τ ) | z = z m = F µ ( τ ) ∀ τ , (10)where Π zµ ≡ ∂ L NG ∂ ( ∂ z X µ ) = √ λ π ( ∂ τ X ) ∂ z X µ − ( ∂ τ X · ∂ z X ) ∂ τ X µ z p ( ∂ τ X · ∂ z X ) − ( ∂ τ X ) (1 + ( ∂ z X ) ) ! (11)is the worldsheet Noether current associated with spacetime momentum, and we havedefined F µ ≡ − F νµ ∂ τ x ν . The latter coincides with the Lorentz d -force ( − γ ~F · ~v, γ ~F )if the parameter τ is chosen to be the quark proper time, which will be understoodhenceforth unless otherwise stated.When the quark accelerates, it will radiate, and we generally expect this radia-tion to exert a damping force on the quark already at the classical level. A givenquark/endpoint semiclassical trajectory x µ ( τ ) = X µ ( τ, z m ) is thus determined onlythrough the combined effect of the applied external force and the concomitant back-reaction of the gluonic fields. In [10, 11] it was shown, building upon the results of[22, 17], that the standard string dynamics, and in particular the boundary condition(10), imply that the quark obeys the equation of motion ddτ m dx µ dτ − √ λ πm F µ q − λ π m F = F µ − √ λ πm F dx µ dτ − λ π m F . (12)This is a generalized, non-linear version of the Lorentz-Dirac equation, whose physicalcontent is most clearly displayed when we rewrite it in the form dP µ dτ ≡ dp µq dτ + dP µ rad dτ = F µ , (13)7here p µq = m dx µ dτ − √ λ πm F µ q − λ π m F (14)is the intrisic d -momentum of the quark (including the near-field contribution, andsatisfying the on-shell relation p q = − m ), and dP µ rad dτ = √ λ F πm dx µ dτ − √ λ πm F µ − λ π m F ! (15)is the rate at which d -momentum is carried away from the quark by gluonic radiation(which in the limit of infinite mass reduces to the standard Lienard formula fromclassical electrodynamics [22]). As announced in the Introduction, in the present paper we would like to go one stepbeyond the classical approximation for the quark, and study quantum fluctuations δx µ ( τ ) induced on the quark trajectory by its coupling to the gluonic field. For this,in the gravity side of the AdS/CFT correspondence we need to identify the stringconfiguration dual to the average quark trajectory of interest, and analyze smallperturbations about it.Of course, specifying an endpoint trajectory x µ ( τ ) = X µ ( τ, z m ) does not uniquelydetermine the full evolution of the string, just like specifying the quark trajectory doesnot uniquely determine the gluonic field profile. Both on the AdS and CFT sides, forany given endpoint/quark worldline there exist an infinite number of configurations,which differ in the boundary condition on the string/gluonic waves at infinity (orequivalently, in the corresponding initial conditions). As in [17, 10, 11], we will focussolely on solutions that are retarded , in the sense that the string/gluonic configurationat any given time depends only on the behavior of the string endpoint at earlier times.These are the solutions that capture the physics of present interest, with influencespropagating outward from the quark to infinity.Remarkably, the retarded solution to the Nambu-Goto equation of motion on AdSis known for an arbitrary timelike trajectory of the endpoint/quark [22]. In terms ofthe coordinates used in (1), and directly parametrized in terms of the endpoint/quarkworldline x µ ( τ ) and the external d -force F µ ( τ ) (related to x µ ( τ ) through (12)), it reads[17, 11] X µ ( τ, z ) = z − z m p − z m / F ! (cid:18) dx µ dτ − z m / F µ (cid:19) + x µ ( τ ) . (16)Generically, a black hole is found to develop on the string worldsheet described bythis solution [17], even though no spacetime black hole is present in the background The paper [22] considered a string on AdS , but the resulting solution can be readily generalizedto AdS d +1 with other values of d . x ≡ x , and results from the application of a constant external force ~F =( F, , . . . ), with F >
0. In this case the equation of motion (12) implies that the quarkmoves with constant proper acceleration [11] A ≡ r d x µ dτ d x µ dτ = ddt ( γv ) = F q m − λF π m , (17)and so follows the hyperbolic trajectory x ( t ) = √ A − + t , (18)where for convenience we have made a particular choice of the spacetime origin. Theproper time of the quark is then τ = A − arcsinh( At ) . (19)Using these data as input, and defining t ret ≡ t ( τ, z m ), the general solution (16) takesthe form t ( t ret , z ) = ( z − z m ) γ ( t ret ) hp z m A − v ( t ret ) z m A i + t ret , (20) X ( t ret , z ) = ( z − z m ) γ ( t ret ) h v ( t ret ) p z m A − z m A i + x ( t ret ) , which upon eliminating t ret , reduces to X ( t, z ) = p A − + t − z + z m . (21) Note that when we turn on the constant electric field needed to accelerate the quark, theembedding of the flavor branes in AdS changes [58], implying in turn a modification of the relationbetween the radial location z m of the string endpoint and the Lagrangian mass of the quark. Asa result, m in (7) and (17) must be interpreted as the effective quark mass in the presence of theelectric field. In the case where the quark is subjected to an arbitrary time-dependent external force,it would be much more complicated to work out the detailed relation between the correspondingmasses, but the same general idea still applies. We are grateful to Arnab Kundu for a discussion ofthis point. We thank Eric Pulido for help with this simplification.
9s a (rather trivial) consistency check, notice that the endpoint trajectory X ( t, z m )indeed coincides with (18). This solution was also found in [19, 23], albeit continuedall the way down to the AdS boundary z = 0 and parametrized in terms of theproper acceleration of the corresponding string endpoint, which would be dual to aninfinitely massive quark undergoing uniform acceleration. Having here obtained (21)as a special case of (16), we are assured that it is the unique retarded embedding thatcodifies the physics of interest to us.Combining (1) and (21), the induced metric on the worldsheet is found to be h tt = − L z (cid:18) A − − z + z m A − + t − z + z m (cid:19) , (22) h tz = − L z (cid:18) tzA − + t − z + z m (cid:19) ,h zz = L z (cid:18) A − + t + z m A − + t − z + z m (cid:19) . We see here that h tt vanishes (indicating that the downward-pointing lightconesbecome horizontal) at z = p A − + z m ≡ z h , and so z h marks the location of aworldsheet horizon. As promised, then, we find that the string embedding dual to auniformly accelerated quark contains a worldsheet black hole. The quark/endpointfluctuations δx ( t ) that we intend to analyze are therefore causally connected (alongthe worldsheet) only with the z m ≤ z ≤ z h portion of the string. The fact that thegeometry is static will become manifest in the next section.It is interesting to note that the solution (22) penetrates into the bulk only up to z = p A − + t + z m ≡ z c ( t ). The full string of interest to us must of course extendbeyond this radial position, all the way up to z → ∞ , in order to be dual to anisolated quark, but has an inflection point at z = z c ( t ) (where ∂ z X ( t, z c ( t )) → ∞ ).It should be possible to derive the form of the z > z c ( t ) portion of the embedding[21], but we will not need it here, because it lies beyond the worldsheet horizon andtherefore cannot influence the string endpoint. Having determined in the previous section the string embedding dual to a quark thaton average follows the uniformly accelerated trajectory (18), we would next like to The endpoint can of course also be causally connected to the rest of the string along spacetimetrajectories outside the worldsheet, but emission/absorption of the closed string modes that couldcarry information along such trajectories is suppressed in the large N c limit. Another possibility is to complete the embedding (21) with its reflection across x = 0, in whichcase the full string lies at z ≤ z c ( t ) and is dual to a quark-antiquark pair, with the particles uniformlyaccelerating back-to-back. Indeed, this is the presentation in which (21) was found in [19]. δ~x ( t ) induced by the coupling to the gluonic field. Forthis purpose, following [27, 28] we must determine the way in which the Hawkingradiation emanating from the worldsheet horizon at z = z h populates the variousmodes of oscillation of the string, thereby making the endpoint/quark jitter. It willturn out to be easier to address this calculation in coordinates different from thoseseen in (22), so in this subsection and the next we will present the relevant bulkdiffeomorphisms and explain their CFT interpretation. Along the way, we will learnsome lessons about the implementation of the Unruh effect in the context of theAdS/CFT correspondence.For starters, it is natural to expect our problem to simplify if we switch to acoordinate system adapted to an observer sitting on the quark. To this end, werewrite the CFT in terms of the Rindler coordinates [39, 45] t = A − exp( Ax ′ ) sinh( At ′ ) , x = A − exp( Ax ′ ) cosh( At ′ ) , ~x ⊥ = ~x ′⊥ (23)( x ′ here is the frequently used tortoise longitudinal coordinate, related to Rindler’soriginal choice through ˇ x ′ ≡ A − exp( Ax ′ )). As ( t ′ , x ′ , ~x ′⊥ ) range over the interval( −∞ , ∞ ), they cover the right Rindler wedge ( x ≥ | t | ) of the original p -dimensionalMinkowski spacetime. The CFT line element takes the Rindler form ds = e Ax ′ (cid:16) − dt ′ + dx ′ (cid:17) + d~x ′⊥ . (24)From the point of view of an inertial observer, the worldlines with constant ( x ′ , ~x ′⊥ ) de-scribe a family of uniformly accelerated observers with proper acceleration A exp( − Ax ′ )and proper time t ′ exp( Ax ′ ), so as desired, in the new coordinate system our quark liesat the fixed position x ′ = 0, ~x ′⊥ = 0 and has proper time t ′ . In accord with the equiv-alence principle, objects in this frame are immersed in a gravitational field analogousto that of an infinite flat Earth located at the x ′ → −∞ (ˇ x ′ = 0) plane, and so tendto fall to the left. This effect manifests itself in various ways, and leads in particularto an x ′ -dependence in the local temperature that we will shortly determine.The form of the relation between t and t ′ implies that inertial and acceleratedobservers will disagree in their identification of positive frequency modes, and con-sequently, in their definition of the CFT vacuum. We will denote the correspondingvacua by | Ω i (Minkowski) and | Ω ′ i (Rindler). For the accelerated observers, whofollow orbits of the timelike Killing vector ξ = ∂ t ′ , there is a horizon at the edge ofthe Rindler wedge, x = | t | , or equivalently, x ′ → −∞ , with surface gravity k h ≡ −
12 ( ∇ µ ξ ν ∇ µ ξ ν ) horizon = A . (25)As a result, a state that is pure from the inertial perspective will generally be mixedfrom the point of view of the Rindler observers, because the field degrees of freedomin the Rindler wedge will be entangled with degrees of freedom in the region beyondthe horizon, which are traced over. In particular, the Rindler observers will interpretthe Minkowski vacuum | Ω i as a thermal bath [38, 39, 45] with local temperature T ( x ′ ) = k h π √− ξ · ξ = T U exp( − Ax ′ ) , (26)11here T U = A π (27)denotes the Unruh temperature, which directly gives the temperature of the bath atthe (average) location of our quark.More specifically, given a set of local operators O i ( x ′ µ ) evaluated in the rightRindler wedge, the Unruh effect essentially amounts to the statement that h Ω |O ( x ′ µ ) · · · O n ( x ′ µn ) | Ω i = Tr (cid:16) e − H ′ /T U O ( x ′ µ ) · · · O n ( x ′ µn ) (cid:17) , (28)where H ′ ≡ − P t ′ = − A ( xP t + tP x ) denotes the Rindler Hamiltonian (the generatorof translations in t ′ ), and the trace runs over all Rindler states. This equivalence isoften discussed explicitly in terms of free scalar fields, but has been proven to holdfor an arbitrary interacting field theory on flat space [59].As noted below (1), when AdS is radially foliated in terms of Poincar´e coordinates,the bulk point x µ of the leaf at each radial position z is directly identified with thepoint x µ of the CFT, so the bulk transformation dual to (23) is simply t = A − exp( Ax ′ ) sinh( At ′ ) , (29) x = A − exp( Ax ′ ) cosh( At ′ ) ,~x ⊥ = ~x ′⊥ ,z = z ′ , which reexpresses the metric (1) in the form ds = L z ′ h e Ax ′ (cid:16) − dt ′ + dx ′ (cid:17) + d~x ′⊥ + dz ′ i . (30)From (4) it follows trivially that the spacetime (30) describes the CFT state where h T ′ µν ( x ′ ) i = 0 , (31)namely, the Minkowski vacuum | Ω i . This assignment might seem to conflict with thestatement (28), but the stress-energy of the expected thermal medium does manifestitself in the difference h T ′ µν ( x ′ ) i medium ≡ h Ω | T ′ µν ( x ′ ) | Ω i − h Ω ′ | T ′ µν ( x ′ ) | Ω ′ i = −h Ω ′ | T ′ µν ( x ′ ) | Ω ′ i . (32)In other words, the Rindler vacuum | Ω ′ i is naturally assigned a negative energy den-sity, reflecting the absence of the thermal medium [60]. We will return to this point inthe next subsection, where a further bulk transformation will enable us to determine(32). Strictly speaking, the trace on the right-hand side is not well-defined in the field-theoreticcontext, so the reinterpretation of the left-hand side as a thermal state must be stated in terms ofa KMS condition.
12n (30) we find an acceleration horizon at x ′ → −∞ just like in (24), extendingin the radial direction as a reflection of the fact that the CFT horizon equally affectsall gluonic modes, from the extreme UV ( z = 0) to the deep IR ( z → ∞ ). Theassociated Unruh temperature is again (27).In the primed coordinates, the string embedding (21) translates into X ′ ( t ′ , z ′ ) = 12 A ln (cid:16) − A ( z ′ − z m ) (cid:17) , (33)which as expected is at rest. The fact that the string is not vertical but bends towards − x ′ is a reflection of the fact that a purely radial embedding would not have minimalarea: just like in the CFT, trajectories at fixed x ′ are non-geodesic, and objects in thisframe tend to fall towards the left (aside from being attracted upward, toward largervalues of the radial coordinate, which they were already in the unprimed frame).Since our string lives on the Rindler-AdS spacetime (30), we expect it to beexposed to a thermal medium. To see this in more detail, notice that the point ( t, z )on the string embedding (21) follows the hyperbolic trajectory x − t = A − − z + z m ,which from the CFT perspective corresponds to the d -acceleration (defined in (17)) A CFT ( z ) = 1 / p A − − z + z m , and from the bulk perspective corresponds to the( d + 1)-acceleration A AdS ( z ) ≡ p ( U M D M U N )( U P D P U N ) = 1 L q A ( z ) z , (34)with D M the bulk covariant derivative and U M the proper ( d + 1)-velocity. The AdSversion of the Unruh effect predicts that an observer undergoing such motion will feelimmersed in a thermal medium with local temperature [61, 23]T( z ) = 12 π q A ( z ) − L − = zA CFT ( z )2 πL . (35)It is easy to check that, up to the factor of z/L arising from the difference in propertimes for the CFT and AdS observers, this is precisely the expected CFT temperature(26) at the location x ′ assigned to the point z ′ = z of the string by the embedding(33).The induced worldsheet metric reads h t ′ t ′ = − L z ′ (cid:16) − A ( z ′ − z m ) (cid:17) , (36) h t ′ z ′ = 0 ,h z ′ z ′ = L z ′ (cid:18) A z m − A ( z ′ − z m ) (cid:19) , corresponding to a manifestly static black hole with horizon at z ′ = z h , and associatedUnruh/Hawking temperature again given by (27).13 .2 Removing the Rindler horizon via a change of conformalframe The presence of a Rindler horizon in the CFT arises from the exp(2 Ax ′ ) factor in theline element (24), so the fact that we are dealing with a conformal theory presentsus with an interesting possibility: we can remove this factor from the ( t ′ , x ′ ) portionof the metric via a change of conformal frame. To be more precise, among theinfinite number of different conformal frames available to an observer sitting on thequark, we can choose the one related to (24) via the specific Weyl transformation ds → exp( − Ax ′ ) ds , to be left with ds = − dt ′′ + dx ′′ + e − Ax ′′ d~x ′′⊥ . (37)By definition, Weyl transformations locally rescale the metric while leaving the co-ordinates untouched; nevertheless, we have relabeled the coordinates with doubleprimes instead of primes because this will shortly prove convenient. The line ele-ment (37) is that of the open Einstein universe, R × H d − . Our parametrizationof the hyperbolic space is related to the standard Poincar´e (or horospheric) coordi-nates through ˇ x ′′ ≡ A − exp( Ax ′′ ), and to the perhaps more familiar ( k = −
1) staticRobertson-Walker form of the line element through [44]ˇ x ′′ = A − cosh χ ′′ − sinh χ ′′ cos θ ′′ ,x ′′ = A − sinh χ ′′ sin θ ′′ cos θ ′′ cosh χ ′′ − sinh χ ′′ cos θ ′′ , (38)... x ′′ d − = A − sinh χ ′′ sin θ ′′ · · · sin θ ′′ d − cosh χ ′′ − sinh χ ′′ cos θ ′′ . Naively, the possibility of removing the horizon might make it seem like the in-terpretation of the Minkowski vacuum as a thermal state depends on the choice ofconformal frame, in which case the Unruh effect would somehow not be fully presentin a CFT. This issue was explored in the free field context a couple of decades ago For maximal clarity, we note that, in the GR literature, these mappings are directly calledconformal transformations, but it is frequent in the CFT and string theory literature to reservethe latter denomination for mappings that leave the metric untouched while pushing the points ofthe manifold around in a way that preserves angles. In either presentation, these transformationsgenerally induce position-dependent rescalings of proper lengths, and are therefore distinct fromconformal diffeomorphisms (or conformal isometries), which transform both the metric and thecoordinates leaving proper lengths invariant. A conformal transformation, in the second sense of thephrase, can always be pictured as a conformal diffeomorphism composed with a Weyl transformationchosen to bring the metric back to its original form. It follows then that, in any theory invariantunder diffeomorphisms, Weyl invariance implies conformal invariance, but the converse holds onlyin 2 dimensions, because for d > SO ( d,
2) for conformally flat metrics). g µν ( x ) → exp(2 ω ( x )) g µν ( x ) corresponds to a bulktransformation involving z → exp( ω ( x )) z and a compensating change in x µ thatprevents the appearance of a z - µ cross-term in the bulk metric. (So, taking thisfamily of bulk diffeomorphisms into account, a given bulk metric is understood toinduce not a specific boundary metric, but a specific boundary conformal structure[8].) In particular, the Weyl transformation leading from (24) to (37) is dual to abulk diffeomorphism that changes the radial foliation according to t ′ = t ′′ , (39) x ′ = x ′′ + A − ln (cid:16)p A z m p − A z ′′ (cid:17) ,~x ′⊥ = p A z m ~x ′′⊥ ,z ′ = p A z m z ′′ exp( Ax ′′ ) . This converts the metric (30) into ds = L z ′′ (cid:20) − (cid:16) − A z ′′ (cid:17) dt ′′ + dx ′′ + e − Ax ′′ d~x ′′⊥ + dz ′′ − A z ′′ (cid:21) , (40)which indeed gives rise to (37) at the boundary. It is clear from this line element that the acceleration horizon is still present inthe bulk, and still has an associated Unruh temperature (27), but now lies on thefixed radial plane z ′′ = A − = 2 π/T U , which explains why it is no longer visible asa horizon from the CFT perspective. In other words, in this new conformal frame,the thermal character of the CFT state arises not from entanglement with degrees offreedom that lie beyond a spacetime horizon, but from the direct identification of thespecific energy scale T U as the temperature of the CFT.For the case d = 2 (where we are dealing with AdS ), and under an appropriateperiodic identification of the x ′′ coordinate, the metric (40) is that of the BTZ blackhole [62, 63]. In our case, however, x ′′ is non-compact, and so what we have, for allvalues of d , is naturally not a black hole but just a portion of AdS d +1 .Using (4), we can deduce from (40) the expectation value of the stress-energy For later convenience, we also include here a rigid rescaling of the transverse directions. And can be brought back to Fefferman-Graham form through the trivial bulk diffeomorphism z ′′ = ˜ z/ (1 + A ˜ z / h T ′′ µ ν i = d odd ,LA πG (3)N diag( − , d = 2 ,L A πG (5)N diag( − , , , d = 4 ,L A πG (7)N diag( − , , , , , d = 6 . (41)This is in complete agreement with the results of [42], where these stress tensors wereobtained in the context of a more general investigation of hyperbolic black holes inAdS/CFT (whose conclusions will be of further use to us below). We notice from (41) that, in even dimensions, the expectation value is nonva-nishing in spite of the fact that we are in the state that is conformally related tothe Minkowski vacuum. The reason is of course well-understood: our CFT is classi-cally Weyl invariant, but at the quantum level this symmetry is anomalous preciselyfor even d [64] (for reviews see, e.g., [45, 65]). This is evidenced by the fact that,when the theory is defined on a curved background with metric g µν , the trace of theenergy-momentum tensor is generically non-zero, h T µµ i g = c π R d = 2 , π (cid:20) α (cid:18) R µνλρ R µνλρ − R µν R µν + 13 R (cid:19) + β (cid:18) R µνλρ R µνλρ − R µν R µν + R (cid:19)(cid:21) d = 4 , (42)(and similarly for d = 6), where c, α, β are numerical constants that depend on thefield content.One of the most impressive pieces of evidence supporting the AdS/CFT correspon-dence is the fact that the Weyl anomaly can be reproduced from the dual classicalgravity setup, as was first demonstrated in [46] (following a suggestion of [8]). Indeed,the energy-momentum tensor (4), derived by functional differentiation of the appro-priately renormalized gravity action (i.e., Einstein-Hilbert plus counterterms chosento eliminate the IR divergences), has a trace that agrees with (42), with c = 3 L G (3)N , α = πL G (5)N = − β . (43)These coefficients can be shown to coincide with those expected at weak coupling(and in the large N c limit) for the field content of the CFT, in every case where the The case d = 4 was also considered very recently in [40], which appeared while the present paperwas in preparation, but the result of that work differs from ours by an overall factor of exp( − Ax ′′ ) / N = 4 SYM, α = − β = N c /
4) [46]. On the gravity side, theWeyl anomaly translates into the statement that, generically, the diffeomorphismsdual to Weyl transformations of the CFT [41] are not true symmetries of the theory.It is easy to check that, for the Rindler metric (24) and open Einstein metric (37),the geometric expression for the Weyl anomaly vanishes (the first case is flat, andhence trivial). This is consistent with the fact that the corresponding stress tensors(31) and (41) are traceless, and implies that, starting with either of these backgrounds, infinitesimal
Weyl transformations are true symmetries. But in going from the primedto the doubly primed frame we have made a finite
Weyl transformation, which is not a true symmetry because the anomaly happens to be non-zero for all metrics ‘inbetween’. This is the origin of the anomalous shift in the energy-momentum tensor.In more detail, for a theory that is classically Weyl-invariant, the transformation¯ g → g ≡ exp(2 ω )¯ g , with ¯ g a flat metric, induces a change in the energy-momentumtensor that follows directly from integration of the Weyl anomaly. E.g., for d = 4[66, 45], h T µν i g = r ¯ gg h T µν i ¯ g + 116 π (cid:20) αH (1) µ ν + 2 βH (3) µ ν (cid:21) , (44)where H (1) µν ≡ − ∇ µ ∇ ν R − g µν ∇ R + 12 g µν R − RR µν ,H (3) µν ≡ − R µρ R ρν + 23 RR µν + 12 R λρ R λρ g µν − R g µν . (45)Since AdS/CFT correctly reproduces the Weyl anomaly, the energy-momentum ten-sor (4) should automatically transform in this manner. This was verified in [50] forinfinitesimal Weyl transformations about an arbitrary metric. In our setting, giventhe fact that h T ′ µν i = 0 in the Minkowski vacuum, and the form of the metrics (24)and (37), it is straightforward to verify that the doubly primed tensor (41) indeedhas the form expected from the finite Weyl transformation, and in particular satisfies(44) when d = 4.The shift in the energy-momentum tensor induced by the Weyl anomaly is purelygeometric, and therefore independent of the state in which the expectation valueis computed. In particular, (44) holds equally for the Minkowski vacuum | Ω i andthe Rindler vacuum | Ω ′ i . This implies that the stress tensor of the Unruh thermalmedium, defined in (32) as the difference between the Minkowski and Rindler expec-tation values, is related to the corresponding difference in the open Einstein universesimply through h T ′ µ ν i medium = s g ′′ g ′ (cid:18) h Ω | T ′′ µν | Ω i − h Ω ′ | T ′′ µν | Ω ′ i (cid:19) . (46)The conformal Minkowski vev in the right-hand side of the preceding expression isalready available to us in (41). Direct field-theoretic analysis shows that the conformal17indler vev vanishes for d = 4 [43]. The corresponding result for arbitrary d wasdeduced in [42] via AdS/CFT. That paper considered an infinite family of hyperbolicblack holes in AdS d +1 parametrized by their temperature. The translation of thesesolutions to our language involves the usual inversion of the radial coordinate, r → z = L /r , followed by the bulk diffeomorphism x µ /R → Ax µ , z/R → Az (which isdual to a conformal transformation in the CFT), to be left with ds − BH = L z ′′ h − (cid:16) − A z ′′ − µL d − A d z ′′ d (cid:17) dt ′′ + dx ′′ + e − Ax ′′ d~x ′′⊥ + dz ′′ − A z ′′ − µL d − A d z ′′ d , (47)with µ a mass-density parameter that controls the black hole temperature T BH . Thecase µ = 0 clearly coincides with our pure AdS metric (40), and was indeed identifiedin [42] as dual to the conformal Minkowski vacuum of the CFT on the open Einsteinuniverse. The zero temperature, extremal, solution has µ/L d − = − d − d/ − /d d/ ,and was argued in [42] to be dual to the conformal Rindler vacuum. The correspond-ing stress tensor is h Ω ′ | T ′′ µν | Ω ′ i = L d − A d πG ( d +1)N ǫ d ) d − − d − d/ − d d/ ! diag( d − , , . . . , , (48)with ǫ d ) = 0 for odd d , and ǫ d ) = ( d − /d ! for even d .Using (41) and (48) in (46) we can finally deduce the stress-energy of the Unruhthermal medium detected in the Minkowski vacuum by the Rindler CFT observer, h T ′ µ ν i medium = ( d − d/ − L d − A d πd d/ G ( d +1)N e − dAx ′′ diag( d − , , . . . , . (49)We notice here the expected divergence ( ∝ ˇ x ′′− d ) of the local energy density as theRindler horizon is approached [60].Now that we understand the precise meaning of our coordinate transformation,let us shift our focus back to the behavior of the quark/string. In the doubly primedcoordinates, the string embedding (33) is simply X ′′ ( t ′′ , z ′′ ) = 0 , (50)which is not only static but also vertical. This reflects the fact that in the newconformal frame there is no longer a gravitational potential pulling objects to theleft. Notice that the spacetime geometry (40) at the location of the string, which isall that is relevant for the small string perturbations that we will compute in the next More generally, the geometry (47) is dual to the state of the open Einstein CFT conformal to athermal ensemble of the Rindler observer at temperature T BH . This result was also implicit in [42], but the Unruh effect was not a central concern in that work. x ′′ and ~x ′′⊥ , so the dual quark isimmersed in a thermal medium that (to first order) is isotropic. The string endpointis at z ′′ = z m / p A z m ≡ z ′′ m . As explained in [56, 11], in the CFT the length z ′′ m gives the characteristic size of the ‘gluonic cloud’ surrounding the quark, or in otherwords, the quark Compton wavelength.The induced worldsheet metric now directly coincides with the ( t ′′ , z ′′ ) block ofthe spacetime metric, h t ′′ t ′′ = − L z ′′ (cid:16) − A z ′′ (cid:17) , (51) h t ′′ z ′′ = 0 ,h z ′′ z ′′ = L z ′′ (cid:16) − A z ′′ (cid:17) − , and so displays the same horizon at z ′′ = A − ≡ z ′′ h . Notice, however, that, becausethe worldsheet is only 2-dimensional, the line z ′′ = A − (presented either in theunprimed, primed or doubly primed coordinates) is an event horizon, and the regionbehind it is a true black hole. The small fluctuations δ ~X ′′ ( t ′′ , z ′′ ) that are of interestto us are free massless scalar fields that propagate on this geometry, and, just like in[27, 28], get excited by the Hawking radiation emanating from the horizon.Combining (29) and (39), the relation between the unprimed and doubly primedcoordinates is seen to be t = A − p A z m p − A z ′′ exp( Ax ′′ ) sinh( At ′′ ) , (52) x = A − p A z m p − A z ′′ exp( Ax ′′ ) cosh( At ′′ ) ,~x ⊥ = p A z m ~x ′′⊥ ,z = p A z m z ′′ exp( Ax ′′ ) . This transformation was written down already in [63], and considered in the AdS/CFTcontext in [19] (as well as in the very recent work [40], which appeared while thepresent paper was in preparation), but its precise CFT interpretation had not beenpreviously elucidated.
We are now ready to determine the form of the quantum fluctuations of the stringendpoint, by examining how small perturbations of the string embedding get excitedby Hawking radiation emanating from the worldsheet black hole. Clearly it will beeasier to perform this calculation in either the primed or doubly primed coordinatesystems defined in the previous section, where the black hole geometry is manifestlystatic. From the analysis of the previous section we know that the doubly primed19etup should be directly dual to a static quark in a thermal medium that to zerothorder looks isotropic at the location of the quark, so it is natural to expect thecomputation in this system to run in complete parallel with the analyses of thermalfluctuations in [27, 28].As a matter of fact, we can notice that, for the case d = 2, the string embedding(50) and the spacetime metric (40) at the location of the string coincide exactly withthe corresponding metric and embedding considered in Section 2.2 of [27], under theidentifications t ′′ = t there , x ′′ = x there , z ′′ = ℓ r , L = ℓ , A = r H ℓ , z ′′ m = ℓ r c . (53)The embedding of the flavor branes is different: whereas in [27] they cover the regionof AdS between the boundary and the z ′′ = z ′′ m plane, in our case they extend up to z ′′ = z ′′ m e − Ax ′′ , (54)which is the image of the unprimed locus z = z m . This implies that at the averageposition of its endpoint, x ′′ = 0, our string should satisfy a Neumann boundarycondition along the line z ′′ = z ′′ m (1 − Ax ′′ ), instead of simply along z ′′ . This distinction,however, is itself of first order in the perturbation about the string embedding (50),and consequently negligible at our level of approximation. For d = 2, we can thereforecarry over to our setting the worldsheet perturbation analysis of [27] and the resultingLangevin equation describing the evolution of thermal fluctuations of the quark alongthe single spatial direction x ′′ .For d >
2, our doubly primed metric (40) differs from the corresponding metric in[27], because the latter involves exponents that depend on d . But the only novelty inour calculation is the presence of the transverse fluctuations δ ~X ′′⊥ ( t ′′ , z ′′ ), which, giventhe isotropy of the spacetime metric (40) at the location of the string, must evolve inexactly the same way as the longitudinal fluctuations δX ′′ ( t ′′ , z ′′ ), whose behavior isalready known to us via the contact with the d = 2 case of [27].For arbitrary dimension, then, we find that as a result of the thermal mediumpresent in the doubly primed system, the position of the quark fluctuates in such away that both longitudinal and transverse perturbations obey the generalized non-relativistic Langevin equation m ′′ d δx ′′ i dt ′′ ( t ′′ ) + Z t ′′ −∞ ds ′′ η ′′ ( t ′′ − s ′′ ) dδx ′′ i ds ′′ ( s ′′ ) = f ′′ i ( t ′′ ) , (55)where m ′′ ≡ √ λ/ πz ′′ m = p A z m m and f ′′ i is a random force with statisticalaverages h f ′′ i ( t ′′ ) i = 0 , h f ′′ i ( t ′′ ) f ′′ j ( s ′′ ) i = δ ij κ ′′ ( t ′′ − s ′′ ) . (56)The friction kernel η ′′ ( t ′′ ) and the stochastic force correlator κ ′′ ( t ′′ ) (respectively de-20oted mγ ( t ) and κ ( t ) in [27]) can be specified in terms of their Fourier transforms η ′′ ( ω ) ≡ Z ∞ dt ′′ η ′′ ( t ′′ ) e iωt ′′ = im ′′ ω + √ λ π π T − i π √ λ m ′′ ω − i √ λω πm ′′ ! , (57) κ ′′ ( ω ) ≡ Z ∞−∞ dt ′′ κ ′′ ( t ′′ ) e iωt ′′ = √ λπ π T + ω λω π m ′′ ! | ω | exp( | ω | T U ) − . These two functions are connected through the fluctuation-dissipation relation [27, 28]2Re η ′′ ( ω ) = [(exp( | ω | /T U ) − / | ω | ] κ ′′ ( ω ). If we study the fluctuations over time scalesmuch larger than the quark Compton wavelength, ∆ t ′′ ≫ z ′′ m , we can approximate (57)by its low-frequency form ( ω ≪ πm ′′ / √ λ ), and then (55) reduces to the standardlocal Langevin equation with white noise, m ′′ th d δx ′′ i dt ′′ ( t ′′ ) + η ′′ dδx ′′ i dt ′′ ( t ′′ ) = f ′′ i ( t ′′ ) , h f ′′ i ( t ′′ ) f ′′ j ( s ′′ ) i = δ ij κ ′′ δ ( t ′′ − s ′′ ) , (58)where η ′′ ≡ η ′′ ( ω = 0) = 2 π √ λT , κ ′′ ≡ κ ′′ ( ω = 0) = 4 π √ λT and m ′′ th ≡ m ′′ (1 − λT /m ′′ ). The force correlation strength κ ′′ obtained here coincides with the resultderived previously in [19], in the context of a momentum broadening computation. Having understood the way in which the thermal (Unruh) medium in the doublyprimed frame makes our quark fluctuate, we can now transform back to the unprimed,inertial frame, to obtain the radiation-induced quantum fluctuations which are ourmain interest. For this purpose we need to relate the corresponding fluctuating stringembeddings X i ( t, z ) = p A − + t + z m − z δ i + δX i ( t, z ) ←→ X ′′ i ( t ′′ , z ′′ ) = 0 + δX ′′ i ( t ′′ , z ′′ ) , a task which requires two separate steps. First, using (52) evaluated at the locationof the string, x ′′ = 0, we recognize that the coordinates on the worldsheet transform Notice that, by causality, η ′′ ( t ′′ ) is understood to vanish for negative argument. This last definition is a bit of a puzzle to us. We expected the linear term in the low frequencyapproximation to η ( ω ) to contribute to the mass term in (58) with the known thermal correction,as was verified in [28] for the case d = 4, but that would have led to m ′′ th = m ′′ (1 − z ′′ m /z ′′ h ) = m ′′ (1 − √ λT U /m ′′ ) . There is presumably some slight error either in the results of [27] or in our interpretation of them, but,regrettably, we have not been able to find it. We should also note that the questionable quadratictemperature-dependence seen in the thermal mass shift deduced by us has a common origin withthe quadratic dependence in the friction coefficient η ′′ , which is definitely correct. We are grateful to Bo-Wen Xiao for pointing out that the numerical disagreement reported inv1 of this paper on the arXiv is actually nonexistent. t ′′ = A − arcsinh At p − A ( z − z m ) ! , z ′′ = z p A z m . (59)When evaluated at the string endpoint, the first relation tells us that t ′′ agrees withthe quark proper time (19), as it should. So the time derivatives in (55) or (58) areconnected to their inertial counterparts via d/dt ′′ = √ A t d/dt . Next, at anygiven point ( t, z ) ↔ ( t ′′ , z ′′ ) on the string worldsheet, we can perturb (52) to concludethat δX = (1 + A z m ) p A ( t + z m − z )1 − A ( z − z m ) δX ′′ , δ ~X ⊥ = p A z m δ ~X ′′⊥ . (60)Evaluating at the string endpoint, this tells us how to relate the quark fluctuations δx i to δx ′′ i .The only other element needed to complete the translation of (55) into the un-primed frame is the transformation rule for the forces, which can be obtained asfollows. As we have done all along, let ~v and γ denote the velocity and Lorentz factorof the quark undergoing the hyperbolic motion (18), and ~F the corresponding external( d − ~f that we are now trying to determine,and as a result acquires a perturbed velocity ~v tot ≡ ~v + δ~v and corresponding Lorentzfactor γ tot . We will denote the associated d -force by F CFT µ = γ tot ( − ~F tot · ~v tot , ~F tot ).Knowing that F AdS M = (cid:18) dτ CFT dτ AdS F CFT µ , (cid:19) (61)(where dτ CFT and dτ AdS = (
L/z ) dτ CFT respectively denote the CFT and AdS propertimes) transforms as a ( d + 1)-vector, we can deduce that F ′′ CFT µ = ∂x ν ∂x ′′ µ F CFT ν . (62)Taylor-expanding this relation to first order in the perturbation, we find that f ′′ = f , ~f ′′⊥ = p A z m √ A t ~f ⊥ . (63)Putting all of this together, we finally conclude that, in the original inertial frame,the radiation emitted by the quark induces quantum longitudinal and transversefluctuations respectively obeying the differential equations m d δxdt ( t ) + Z t −∞ ds (cid:18) η ( t, s ) dδxds ( s ) − ζ ( t, s ) δx ( s ) (cid:19) = q λA π m f ( t ) √ A t (64)and m d δ~x ⊥ dt ( t ) + Z t −∞ ds η ⊥ ( t, s ) dδ~x ⊥ ds ( s ) = q λA π m ~f ⊥ ( t ) √ A t , (65)22ith h f i ( t ) i = 0 , h f i ( t ) f j ( s ) i = δ ij κ i ( t, s ) , (66)where we have defined η ( t, s ) ≡ η ′′ ( A − arcsinh( At ) − A − arcsinh( As )) q λA π m √ A t √ A s − mA t (1 + A t ) δ ( s − t ) ,ζ ( t, s ) ≡ A s η ′′ ( A − arcsinh( At ) − A − arcsinh( As )) q λA π m √ A t (1 + A s ) / + mA (1 − A t )(1 + A t ) δ ( s − t ) ,η ⊥ ( t, s ) ≡ η ′′ ( A − arcsinh( At ) − A − arcsinh( As )) q λA π m (1 + A t ) + mA t (1 + A t ) δ ( s − t ) , (67) κ ( t, s ) ≡ κ ′′ (cid:0) A − arcsinh( At ) − A − arcsinh( As ) (cid:1) ,κ ⊥ ( t, s ) ≡ κ ′′ ( A − arcsinh( At ) − A − arcsinh( As )) (cid:0) λA π m (cid:1) √ A t √ A s . Notice that, in contrast with the transverse fluctuations, the longitudinal one doesnot evolve according to a generalized Langevin equation: there is in (64) an additionaldissipative force that depends (nonlocally) on δx ( t ) itself. This distinction reflectsthe inherent anisotropy of the system in the inertial frame.When we analyze the trajectory at quark proper times larger than the effectivequark Compton wavelength, ∆ τ ≫ z m / p A z m , but over time intervals ∆ t smallerthan the characteristic time A − set by the acceleration, the low-frequency approxi-mation of the previous subsection becomes appropriate, and the equations of motionagain reduce to a local form, m th d δxdt ( t ) + η ( t ) dδxdt ( t ) − ζ ( t ) δx ( t ) = q λA π m f ( t ) √ A t , (68)and m th d δ~x ⊥ dt ( t ) + η ⊥ ( t ) dδ~x ⊥ dt ( t ) = q λA π m ~f ⊥ ( t ) √ A t , (69)with m th ≡ m m ′′ th /m ′′ = m ′′ th / p λA / π m , h f i ( t ) i = 0 , h f i ( t ) f j ( s ) i = δ ij κ i ( t ) δ ( t − s ) , (70)23here we have defined the time-dependent coefficients η ( t ) ≡ A (cid:16) √ λ π √ A t − mt (cid:17) (1 + A t ) q λA π m ,ζ ( t ) ≡ A (cid:16) √ λ π √ A t A t + m (1 − A t ) (cid:17) (1 + A t ) q λA π m ,κ ( t ) ≡ √ λ π A √ A t , (71) η ⊥ ( t ) ≡ A (cid:18) √ λ π (1 + A t ) + q λA π m mt (cid:19) (1 + A t ) / q λA π m ,κ ⊥ ( t ) ≡ √ λ π A (cid:0) λA π m (cid:1) √ A t . These equations (as well as their nonlocal progenitors) reveal interesting structure inthe fluctuation/dissipation setup induced by gluonic radiation in our strongly-coupledCFT. The main feature is the prominent time dependence of the problem, which isof course expected in the unprimed frame but in stark contrast with the situation inthe doubly primed frame, where the quark is exposed to a static thermal medium.For the inertial observer, the effects produced by the emitted gluonic radiation doevolve with time, and the system can at best be regarded as quasi-stationary if it isexamined in the regime ∆ t ≪ A − .In more detail, the velocity and the rate at which energy and longitudinal mo-mentum are radiated by the quark follow from (18), (15) and (17) as v = At √ A t ,dE rad dt = A − A t q π m λ + A √ A t , (72) dp rad dt = A At √ A t − A q π m λ + A , (73)which clearly approach a constant only at very late (or very early) times, | t | ≫ A − .In this limit, all of the dynamical coefficients (71) are seen to vanish, except for thelongitudinal force correlation coefficient κ ( t ). The latter diverges, but does so ata rate that is too slow to give a finite contribution to the equation of motion (68),which therefore becomes free. 24t can be seen from (17) that, as the external force on the quark increases fromzero to its maximal value F crit = m / π √ λ (which would be strong enough to nu-cleate quark-antiquark pairs from the vacuum [16, 17, 11]), the proper acceleration A covers the full range [0 , ∞ ). Nevertheless, the situation of main physical interestis √ λA/ πm <
1, corresponding to a heavy (and therefore close to pointlike) quarkthat is not too violently accelerated (i.e., a quark that does not change velocity sig-nificantly in a time period smaller than its Compton wavelength z m ). Inspection of(71) shows that under these circumstances ζ always starts out positive around t = 0,becomes negative at a finite time and then asymptotically approaches zero from belowas t → ∞ . It is therefore particularly curious that the sign in front of the term linear δx ( t ) in (68) (and (64)) turns out to be negative, because, when ζ > ζ > η (as well as η ) generically becomes negative after acertain amount of time, implying that the effect of the emitted radiation is to speedup the longitudinal fluctuations of the quark instead of slowing them down. Of course,given that the equations of motion (64)-(71) have been obtained simply by translating(55)-(58) to the inertial frame, the combined effect of all terms appearing in themcannot possibly lead to runaway behavior.We can similarly read off directly from [27] the expression for the displacementsquared in the open Einstein frame and translate it to the inertial frame. We willrefrain from writing out the results here, since they are not particularly illuminating.It is easy to extract from them the ballistic behavior expected at small times, but thediffusive regime of the doubly primed frame is not accessible to the inertial observerwithin the quasi-stationary regime.
Acknowledgements
This work is dedicated to the memory of Blanca G¨uijosa. We are grateful to AlejandroCorichi, Roberto Emparan, Bartomeu Fiol, Antonio Garc´ıa, Gast´on Giribet, Dami´anHern´andez, Mart´ın Kruczenski, David Mateos, ´Angel Paredes and Daniel Sudarskyfor useful discussions. EC thanks the Theory Group at the University of Texas atAustin for hospitality, and the Aspen Center for Physics for hospitality and partialsupport. When this paper was begun, MCh was at the Facultad de Ciencias, UNAM,and his work was supported by a DGAPA-UNAM postdoctoral fellowship. For thefinal stage of this paper MCh is at the Departament de Fisica Fonamental, Universitatde Barcelona, and his work is supported by a postdoctoral fellowship from Mexico’sNational Council of Science and Technology (CONACyT). The work of the remainingauthors was partially supported by CONACyT grants 50-155I, CB-2008-01-104649and 50760, as well as DGAPA-UNAM grant IN116408 and by the National Science In the process one encounters an infrared divergence that happened to cancel out in the finalresult of [27], and could be regularized as in [37].
References [1] E. J. Moniz and D. H. Sharp, “Absence of runaways and divergent self-mass innonrelativistic quantum electrodynamics,” Phys. Rev. D (1974) 1133;E. J. Moniz and D. H. Sharp, “Radiation Reaction In Nonrelativistic QuantumElectrodynamics,” Phys. Rev. D (1977) 2850.[2] G. D. R. Martin, “Classical and Quantum Radiation Reaction,” arXiv:0805.0666[gr-qc];A. Higuchi and G. D. R. Martin, “Quantum Radiation Reaction and the Green’sFunction Decomposition,” Phys. Rev. D (2006) 125002 [arXiv:gr-qc/0608028];A. Higuchi and G. D. R. Martin, “Radiation reaction on charged particles inthree-dimensional motion in classical and quantum electrodynamics,” Phys. Rev.D (2006) 025019 [arXiv:quant-ph/0510043];A. Higuchi and G. D. R. Martin, “The Lorentz-Dirac force from QED for linearacceleration,” Phys. Rev. D (2004) 081701 [arXiv:quant-ph/0407162];A. Higuchi, “Radiation reaction in quantum field theory,” Phys. Rev. D (2002)105004 [Erratum-ibid. D (2004) 129903] [arXiv:quant-ph/0208017].[3] R. Rosenfelder and A. W. Schreiber, “An Abraham-Lorentz-like equation for theelectron from the worldline variational approach to QED,” Eur. Phys. J. C ,161 (2004) [arXiv:hep-th/0406062].[4] P. R. Johnson and B. L. Hu, “Uniformly accelerated charge in a quantum field:From radiation reaction to Unruh effect,” Found. Phys. (2005) 1117 [arXiv:gr-qc/0501029];“Stochastic theory of relativistic particles moving in a quantum field. II: ScalarAbraham-Lorentz-Dirac-Langevin equation, radiation reaction and vacuum fluc-tuations,” Phys. Rev. D (2002) 065015 [arXiv:quant-ph/0101001];“Stochastic theory of relativistic particles moving in a quantum field. I: Influencefunctional and Langevin equation,” arXiv:quant-ph/0012137;“Worldline influence functional: Abraham-Lorentz-Dirac-Langevin equationfrom QED,” arXiv:quant-ph/0012135.[5] R. Parentani, “The Recoils of the accelerated detector and the decoherence ofits fluxes,” Nucl. Phys. B (1995) 227 [arXiv:gr-qc/9502030].[6] J. M. Maldacena, “The large N limit of superconformal field theories and super-gravity,” Adv. Theor. Math. Phys. , 231 (1998) [Int. J. Theor. Phys. , 1113(1999)] [arXiv:hep-th/9711200]. 267] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, “Gauge theory correlatorsfrom non-critical string theory,” Phys. Lett. B , 105 (1998) [arXiv:hep-th/9802109].[8] E. Witten, “Anti-de Sitter space and holography,” Adv. Theor. Math. Phys. ,253 (1998) [arXiv:hep-th/9802150].[9] O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri and Y. Oz, “Large N field theories, string theory and gravity,” Phys. Rept. , 183 (2000) [arXiv:hep-th/9905111].[10] M. Chernicoff, J. A. Garc´ıa and A. G¨uijosa, “Generalized Lorentz-Dirac Equationfor a Strongly-Coupled Gauge Theory,” Phys. Rev. Lett. (2009) 241601[arXiv:0903.2047 [hep-th]].[11] M. Chernicoff, J. A. Garc´ıa and A. G¨uijosa, “A Tail of a Quark in N = 4 SYM,”JHEP (2009) 080 [arXiv:0906.1592 [hep-th]].[12] P. A. M. Dirac, “Classical theory of radiating electrons,” Proc. Roy. Soc. Lond.A (1938) 148.[13] C. P. Herzog, A. Karch, P. Kovtun, C. Kozcaz and L. G. Yaffe, “Energy loss of aheavy quark moving through N = 4 supersymmetric Yang-Mills plasma,” JHEP (2006) 013 [arXiv:hep-th/0605158].[14] S. S. Gubser, “Drag force in AdS/CFT,” Phys. Rev. D (2006) 126005[arXiv:hep-th/0605182].[15] S. S. Gubser, “Momentum fluctuations of heavy quarks in the gauge-string du-ality,” Nucl. Phys. B (2008) 175 [arXiv:hep-th/0612143].[16] J. Casalderrey-Solana and D. Teaney, “Transverse momentum broadening of afast quark in a N = 4 Yang Mills plasma,” JHEP (2007) 039 [arXiv:hep-th/0701123].[17] M. Chernicoff and A. G¨uijosa, “Acceleration, Energy Loss and Screening inStrongly-Coupled Gauge Theories,” JHEP , 005 (2008) [arXiv:0803.3070[hep-th]].[18] F. Dom´ınguez, C. Marquet, A. H. Mueller, B. Wu and B. W. Xiao, “Comparingenergy loss and p ⊥ -broadening in perturbative QCD with strong coupling N = 4SYM theory,” Nucl. Phys. A (2008) 197 [arXiv:0803.3234 [nucl-th]].[19] B. W. Xiao, “On the exact solution of the accelerating string in AdS space,”Phys. Lett. B (2008) 173 [arXiv:0804.1343 [hep-th]].[20] G. Beuf, C. Marquet and B. W. Xiao, “Heavy-quark energy loss and thermaliza-tion in a strongly coupled SYM plasma,” arXiv:0812.1051 [hep-ph].2721] A. G¨uijosa and E. J. Pulido, in preparation.[22] A. Mikhailov, “Nonlinear waves in AdS/CFT correspondence,” arXiv:hep-th/0305196.[23] A. Paredes, K. Peeters and M. Zamaklar, “Temperature versus acceleration: theUnruh effect for holographic models,” JHEP (2009) 015 [arXiv:0812.0981[hep-th]].[24] H. Dorn and H. J. Otto, “Q anti-Q potential from AdS-CFT relation at T ≥ (1998) 021 [arXiv:hep-th/9807093].[25] J. Greensite and P. Olesen, “Worldsheet fluctuations and the heavy quark poten-tial in the AdS/CFT approach,” JHEP (1999) 001 [arXiv:hep-th/9901057].[26] P. Johnson, “Relativistic Particle Trajectories from Worldline Path IntegralQuantization,” In the Proceedings of IEEE Particle Accelerator Conference (PAC2001), Chicago, Illinois, 18-22 Jun 2001, pp 1781-1783 .[27] J. de Boer, V. E. Hubeny, M. Rangamani and M. Shigemori, “Brownian motionin AdS/CFT,” JHEP , 094 (2009) [arXiv:0812.5112 [hep-th]].[28] D. T. Son and D. Teaney, “Thermal Noise and Stochastic Strings in AdS/CFT,”JHEP , 021 (2009) [arXiv:0901.2338 [hep-th]].[29] A. E. Lawrence and E. J. Martinec, “Black Hole Evaporation Along MacroscopicStrings,” Phys. Rev. D (1994) 2680 [arXiv:hep-th/9312127].[30] V. P. Frolov and D. Fursaev, “Mining energy from a black hole by strings,” Phys.Rev. D , 124010 (2001) [arXiv:hep-th/0012260].[31] J. M. Maldacena, “Eternal black holes in Anti-de-Sitter,” JHEP , 021 (2003)[arXiv:hep-th/0106112].[32] C. P. Herzog and D. T. Son, “Schwinger-Keldysh propagators from AdS/CFTcorrespondence,” JHEP , 046 (2003) [arXiv:hep-th/0212072].[33] J. Casalderrey-Solana and D. Teaney, “Heavy quark diffusion in strongly coupled N = 4 Yang Mills,” Phys. Rev. D (2006) 085012 [arXiv:hep-ph/0605199].[34] G. C. Giecold, E. Iancu and A. H. Mueller, “Stochastic trailing string andLangevin dynamics from AdS/CFT,” JHEP , 033 (2009) [arXiv:0903.1840[hep-th]].[35] G. C. Giecold, “Heavy quark in an expanding plasma in AdS/CFT,” JHEP (2009) 002 [arXiv:0904.1874 [hep-th]].2836] J. Casalderrey-Solana, K. Y. Kim and D. Teaney, “Stochastic String Mo-tion Above and Below the World Sheet Horizon,” JHEP , 066 (2009)[arXiv:0908.1470 [hep-th]].[37] A. N. Atmaja, J. de Boer and M. Shigemori, “Holographic Brownian Motion andTime Scales in Strongly Coupled Plasmas,” arXiv:1002.2429 [hep-th].[38] W. G. Unruh, “Notes on black hole evaporation,” Phys. Rev. D (1976) 870;P. C. W. Davies, “Scalar particle production in Schwarzschild and Rindler met-rics,” J. Phys. A , 609 (1975).[39] L. C. B. Crispino, A. Higuchi and G. E. A. Matsas, “The Unruh effect and itsapplications,” Rev. Mod. Phys. , 787 (2008) [arXiv:0710.5373 [gr-qc]];S. Takagi, “Vacuum Noise And Stress Induced By Uniform Acceleration:Hawking-Unruh Effect In Rindler Manifold Of Arbitrary Dimensions,” Prog.Theor. Phys. Suppl. , 1 (1986).[40] T. Hirayama, P. W. Kao, S. Kawamoto and F. L. Lin, “Unruh effect and Holog-raphy,” arXiv:1001.1289 [hep-th].[41] C. Imbimbo, A. Schwimmer, S. Theisen and S. Yankielowicz, “Diffeomorphismsand holographic anomalies,” Class. Quant. Grav. (2000) 1129 [arXiv:hep-th/9910267].[42] R. Emparan, “AdS/CFT duals of topological black holes and the entropy ofzero-energy states,” JHEP , 036 (1999) [arXiv:hep-th/9906040].[43] T. S. Bunch, “Stress Tensor Of Massless Conformal Quantum Fields In Hyper-bolic Universes,” Phys. Rev. D , 1844 (1978).[44] P. Candelas and J. S. Dowker, “Field Theories On Conformally Related Space-Times: Some Global Considerations,” Phys. Rev. D , 2902 (1979).[45] N. D. Birrell and P. C. W. Davies, “Quantum Fields In Curved Space,” Cam-bridge Univ. Pr., UK (1982) 340p ;R. M. Wald, “Quantum field theory in curved space-time and black hole ther-modynamics,”
Chicago Univ. Pr., USA (1994) 205 p .[46] M. Henningson and K. Skenderis, “The holographic Weyl anomaly,” JHEP (1998) 023 [arXiv:hep-th/9806087].[47] L. Susskind and E. Witten, “The Holographic Bound In Anti-De Sitter Space,”arXiv:hep-th/9805114;A. W. Peet and J. Polchinski, “UV/IR relations in AdS dynamics,” Phys. Rev.D (1999) 065011 [arXiv:hep-th/9809022].2948] V. Balasubramanian, P. Kraus, A. E. Lawrence and S. P. Trivedi, “Holographicprobes of anti-de Sitter space-times,” Phys. Rev. D (1999) 104021 [arXiv:hep-th/9808017].[49] C. Fefferman, C. R. Graham, Conformal invariants, in ´Elie Cartan et lesMath´ematiques d’Aujourd’hui , (Ast´erisque, 1985), 95.[50] S. de Haro, S. N. Solodukhin and K. Skenderis, “Holographic reconstructionof spacetime and renormalization in the AdS/CFT correspondence,” Commun.Math. Phys. (2001) 595 [arXiv:hep-th/0002230];K. Skenderis, “Asymptotically anti-de Sitter spacetimes and their stress energytensor,” Int. J. Mod. Phys. A , 740 (2001) [arXiv:hep-th/0010138].[51] K. Skenderis, “Lecture notes on holographic renormalization,” Class. Quant.Grav. (2002) 5849 [arXiv:hep-th/0209067].[52] V. Balasubramanian and P. Kraus, “A stress tensor for anti-de Sitter gravity,”Commun. Math. Phys. (1999) 413 [arXiv:hep-th/9902121].[53] R. C. Myers, “Stress tensors and Casimir energies in the AdS/CFT correspon-dence,” Phys. Rev. D (1999) 046002 [arXiv:hep-th/9903203].[54] R. Emparan, C. V. Johnson and R. C. Myers, “Surface terms as countertermsin the AdS/CFT correspondence,” Phys. Rev. D , 104001 (1999) [arXiv:hep-th/9903238].[55] A. Karch and E. Katz, “Adding flavor to AdS/CFT,” JHEP (2002) 043[arXiv:hep-th/0205236].[56] J. L. Hovdebo, M. Kruczenski, D. Mateos, R. C. Myers and D. J. Winters,“Holographic mesons: Adding flavor to the AdS/CFT duality,” Int. J. Mod.Phys. A (2005) 3428.[57] A. Abouelsaood, C. G. Callan, C. R. Nappi and S. A. Yost, “Open Strings InBackground Gauge Fields,” Nucl. Phys. B (1987) 599.[58] T. Albash, V. G. Filev, C. V. Johnson and A. Kundu, “Quarks in an ExternalElectric Field in Finite Temperature Large N Gauge Theory,” JHEP (2008)092 [arXiv:0709.1554 [hep-th]].[59] J. J. Bisognano and E. H. Wichmann, “On The Duality Condition For A Her-mitian Scalar Field,” J. Math. Phys. (1975) 985;J. J. Bisognano and E. H. Wichmann, “On The Duality Condition For Quan-tum Fields,” J. Math. Phys. (1976) 303. G. L. Sewell, “Quantum Fields onManifolds: PCT and Gravitationally Induced Thermal States,” Ann. Phys. (1982) 201. 3060] P. Candelas and D. Deutsch, “On The Vacuum Stress Induced By Uniform Ac-celeration Or Supporting The Ether,” Proc. Roy. Soc. Lond. A (1977) 79;“Fermion Fields In Accelerated States,” Proc. Roy. Soc. Lond. A (1978) 251.[61] S. Deser and O. Levin, “Accelerated detectors and temperature in (anti) de Sit-ter spaces,” Class. Quant. Grav. , L163 (1997) [arXiv:gr-qc/9706018];S. Deser and O. Levin, “Equivalence of Hawking and Unruh temperaturesthrough flat space embeddings,” Class. Quant. Grav. , L85 (1998) [arXiv:hep-th/9806223];S. Deser and O. Levin, “Mapping Hawking into Unruh thermal properties,” Phys.Rev. D , 064004 (1999) [arXiv:hep-th/9809159].[62] M. Ba˜nados, C. Teitelboim and J. Zanelli, “The Black hole in three-dimensionalspace-time,” Phys. Rev. Lett. (1992) 1849 [arXiv:hep-th/9204099].[63] M. Ba˜nados, M. Henneaux, C. Teitelboim and J. Zanelli, “Geometry of the (2+1)black hole,” Phys. Rev. D (1993) 1506 [arXiv:gr-qc/9302012].[64] D. M. Capper and M. J. Duff, “Trace anomalies in dimensional regularization,”Nuovo Cim. A (1974) 173;S. Deser, M. J. Duff and C. J. Isham, “Nonlocal Conformal Anomalies,” Nucl.Phys. B (1976) 45;S. Deser and A. Schwimmer, “Geometric classification of conformal anomalies inarbitrary dimensions,” Phys. Lett. B (1993) 279 [arXiv:hep-th/9302047].[65] M. J. Duff, “Twenty years of the Weyl anomaly,” Class. Quant. Grav. , 1387(1994) [arXiv:hep-th/9308075].[66] L. S. Brown and J. P. Cassidy, “Stress Tensors And Their Trace Anomalies InConformally Flat Space-Times,” Phys. Rev. D (1977) 1712;T. S. Bunch, “On Renormalization Of The Quantum Stress Tensor In CurvedSpace-Time By Dimensional Regularization,” J. Phys. A , 517 (1979);A. Cappelli and A. Coste, “On the stress tensor of conformal field theories inhigher dimensions,” Nucl. Phys. B314