JHEP10(2018)129 Springer , the Holo- May 9, 2018 3 : October 14, 2018 October 19, 2018 : : September 23, 2018 : Eives a Rindler Hori- Received

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JHEP10(2018)129 Springer , the holo- May 9, 2018 3 : October 14, 2018 October 19, 2018 : : September 23, 2018 : eives a Rindler hori- Received . For AdS 1 nt of view, there are no − Accepted a source has fallen across Published Revised d ler-AdS space directly from re from the two-point func- H pacetime, and ﬁnd from the ncepts in Science, × iscuss an alternate foliation of uding numerical factors, using R -gravitational aspects of Rindler Published for SISSA by https://doi.org/10.1007/JHEP10(2018)129 b are holographically dual to an entangled conformal +1 [email protected] d , AdS . 3 1211.7370 and Prasant Samantray The Authors. a c AdS-CFT Correspondence, Black Holes in String Theory

In anti-de Sitter space a highly accelerating observer perc , [email protected] Tempe, AZ 85287, U.S.A. Department of Physics, ArizonaTempe, State AZ University, 85287, U.S.A. E-mail: Department of Physics andArizona Beyond: State Center University, for Fundamental Co b a local signifiers of theRindler-AdS presence which of is the dual horizon. to a Finally, we CFT d living inKeywords: de Sitter space. ArXiv ePrint: the boundary conformal field theory.tion We and derive obtain the the temperatu the Rindler Cardy formula. entropy density We alsobehavior precisely, probe of incl the the causal one-point structure functionthe of the that Rindler s the horizon. CFT “knows” This when is so even though, from the bulk poi graphic duality is especiallyhorizons tractable, to be allowing probed. quantum We recover the thermodynamics of Rind field theory that lives on two boundaries with geometry Open Access Article funded by SCOAP Abstract: zon. The two Rindler wedges in Maulik Parikh Rindler-AdS/CFT JHEP10(2018)129 7 8 1 2 5 6 10 11 15 18 22 23 to then use that corre- graphic duality between bserver undergoing con- avity. This has not been s to consider accelerating gravity in asymptotically de Sitter space does exist: in anti-de Sitter space with mple of a spacetime with an udied. Nevertheless, most of ngth scale) have acceleration chniques. Instead one would t many of the most interesting itter space that such observers acetime with a horizon. As the f Rindler-AdS space. It is worth g the techniques of quantum field – 1 – Fortunately, a tractable theory of quantum gravity in anti- 4.3 Relation between4.4 Rindler-AdS space and The AdS omniscient4.5 black CFT holes Signatures of across-horizon physics 4.1 Temperature and4.2 two-point correlators Entropy like to be abledone to for study the Rindler space simpleflat in reason space a that does theory a not of presently tractable quantum exist. theory gr of quantum near-horizon limit of allobserver-dependent nonextremal horizon, black Rindler holes space andthe has an literature been on exa the much subject st hastheory treated in Rindler curved space spacetime, usin whereasproblems it is of now horizon recognized tha physics are not accessible with those te 1 Introduction Rindler space, the portionstant acceleration of can Minkowski interact, space is with perhaps which the an simplest sp o 6 Summary and discussion A Rindler and Eddington-Finkelstein coordinates for AdS 5 De Sitter space as the boundary of Rindler-AdS 3 Thermodynamics of Rindler-AdS 4 The boundary theory Contents 1 Introduction 2 The geometry of Rindler-AdS space suitably high proper accelerationhorizons; (compared Rindler-AdS with space the is AdS thuscan le the interact portion with. of anti-deRindler-AdS The S purpose space of and this aspondence paper to boundary is investigate conformal quantum-gravitational to aspects field set o theory, up and a holo it is defined by theobservers not AdS/CFT in correspondence. Minkowski space This but motivates in u AdS space. Observers JHEP10(2018)129 4 of (2.1) it is known to be . Rindler-AdS/CFT. 5 , we show that the 2 3 L ribed and studied pre- tter space. We briefly − tools necessary to study Bekenstein-Hawking en- brought to bear when the = t the Rindler horizon even k holes. Next, we turn to . The results of the paper s a boost generator. It was ce that falls freely into the 2 ensuing advantages. In par- e of AdS to AdS rdinates natural to an accel- ts that mark the presence of in AdS, Rindler-AdS has no nd the two-point correlation llen across the horizon. This ons: boundary operator, we show g entropy density, including Rindler-AdS space. We then t Rindler space, the existence cetime. Remarkably, we find e boundary theory becomes a umerical factors. Even more tageous spacetime for studying +1 . We summarize and conclude in d ly and the Rindler entropy density X , we present the classical geometry of − 2 2 ) isometry group is manifest. In the em- d , d X (2 , we consider an alternate foliation of Rindler- O 5 – 2 – + ... + 2 and the 1 L , we quickly review Rindler-AdS thermodynamics. Section X 3 + can conveniently be described using embedding coordinates 2 0 +1 d X − with some remarks about directions and puzzles suggested by 6 This paper is organized as follows. In section While Rindler-AdS space in general dimensions has been desc = 4 super Yang-Mills theory. Thus in principle one has all the + 2-dimensional Minkowski space with two time-like directi bedding space, a Rindler observer is one whose Hamiltonian i tropy density of theinterestingly, one Rindler can horizon probe precisely,that the the including causal boundary n theory structure “knows”though, of when from the a a source spa bulk has pointthe fallen of event pas horizon. view, there are no local invarian viously, the real power ofbulk the spacetime AdS/CFT dimension correspondence is cantwo-dimensional three. be CFT For living that in special Minkowskiticular, case, space, the th with two-point function all can the can be calculated be explicit derived from the Cardy formula. The result matches the Here the AdS curvature scale is N that a “boundary theorist” canis tell the whether main the result source of has the fa paper. In section Cardy formula preciselythe reproduces numerical the coefficient, Bekenstein-Hawkin bothdiscuss for the nonrotating relation and rotating betweenperhaps Rindler-AdS our space most and interestingRindler AdS derivation. horizon. blac By We calculating consider the a one-point sour function of a event horizons in a theory of quantum gravity. d 2 The geometry ofWe Rindler-AdS would space like to covererating anti-de Sitter observer. space AdS in the Rindler coo AdS in which thediscuss some boundary subtleties of conformal this variant fieldsection of Rindler-AdS/CFT theory lives in de Si Rindler-AdS space. In section describes the boundary theoryare as and follows. contains our Wefunction calculate main of the results bulk-boundary operators propagator in a the boundary theory. Specializing the quantum gravity ofsingularities horizons. where Unlike bulk physics eternal breaksof black down. holes a And dual unlike conformal fla field theory is assured; indeed, in the cas emphasizing that Rindler-AdS space is a particularly advan JHEP10(2018)129 , e 2 in χ (2.2) (2.5) (2.3) (2.4) (2.6) +1 → d ) . χ/L 2 ( ) is satisfied, hypersurfaces 2 − 2 d . ξ 2.1 Ω ) d ) sinh ) 2 . L ]. χ/L χ/L ( 3 1 χ/L ( 2 ( 2 − 0. The above metric has , 2 2 d . 2 Ω indler-AdS coordinates as > ace [ (as one does in flat Rindler 0 and awking or Unruh tempera- ) − d cosh sinh 2 d 2 sinh g 2 ng equation ( 2 se coordinates cover the part ) an be obtained directly from +1 ρ → 2 Ω L d s useful to consider AdS t is L t/L er) can be regarded as Rindler d e Wightman function which in turn L ξ + L ter 2 + + ,X χ χ/L 2 + ( 1 ) In particular, Rindler observers in ) 2 to rescale the time coordinate such ) + 1-dimensional Rindler space. To 2 + cosh( X d 2 1 2 ξ L dχ ρ/L ]. t/L t/L -vectors) that can just as well be computed in ( dρ 2 ( cosh = dχ 2 2 2 1 ) ) 2 + 0 and 1 + ( X 2 L > direction: + ξ/L dξ + χ/L 2 ) 1 2 2 – 3 – ) + sinh + ) + sinh +2-dimensional Minkowski space (with two time ξ 0 ) X dτ 2 d t/L ) 1 + ( X sinh( 2 dt χ/L ) t/L χ/L 2 ( 2 ( ) − + 2 2 L 2 cosh( 2 ρ/L is unphysical, there is no loss of generality. Choosing the ) 1 + ξ ξ/L sinh( parameterizes the worldline of an accelerating observer in g ( cosh 2 cosh ξ X 2 ξ t − ] that both acceleration and “true” horizons in an Einstein ξ/L 2 2 = 1 dξ p ξ (1 + ( ξ = 0 − 2 = = X ; since 1 + ( . These are the hypersurfaces on which the boundary CFT will b = ds 1 ) = 2 ) with ψ ] in various other contexts. Note that the constant- ρ − /L + 2 d 9 2 2.1 – ds τ/L H tan 4 ( dt 2 2 × ) R cos ξ/L ( − = is replaced by 1 2 g Consider then a Rindler observer in ds This is because the response of Unruh detectors depends on th 1 rest of the coordinates (see appendix) such that the embeddi that space), we have used the existence of the AdS scale understand the global properties of Rindler-AdS space, it i Here, instead of choosing an arbitrary acceleration parame The global coordinates can then be expressed in terms of the R global coordinates (see appendix) for which the line elemen This line element describesof AdS the in hypersurface Rindler ( coordinates. The which is just the line element of standard (i.e. flat) AdS. Indeed, as the AdS curvature scale diverges, so that directions) uniformly accelerating in the defined. The coordinate time the Rindler-AdS metric becomes: space (such as sayhorizons Schwarzschild, de in Sitter, a or higher-dimensionalture anti-de flat detected Sitt embedding by space. observersaccelerating The in H trajectories the in lower-dimensional the spaceAdS embedding are c also space Rindler [ observers in the embedding Minkowski sp we recover are of the form depends only on geometricthe invariants embedding (constructed space. out of bi been discussed in [ shown in an elegant paper [ JHEP10(2018)129 . . 2 r 1 and (2.8) (2.9) (2.7) dχ γξ = 0. At 2 β τ → − hypersurface. ξ metry of the . Our Rindler 1 1 − hypersurface is d 2 ) H ξ 2 at the boundary of × → ±∞ ξ/L R π/ t tant- ± . as ns, let us briefly consider is a = 1 + ( 2 2 π ξ τ = 0 each point in the interior r space is the existence of a . dχ + ρ = 0 corresponds to 2 2 → ± ) t 2 2 . This is illustrated in figure ξ τ L e dχ 2 , the transformation . One copy of the CFT lives on the ξ/L , whose orbits are a Rindler observer’s + dξ t ∂ 1 + 2 dt → ∞ 1 + ( + − is ξ + 2 2 3 r 2 ξ 2 L 2 are the same in both coordinate systems. In ξ L dχ dξ – 4 – 2 r + − 1 + are tilted with respect to the constant-time slices d 2 t S βdt + 2 , which we have explicitly separated from the angles dξ 2 space. A surface of constant 2 1 ξ 2 − − dt L d +1 ]: 2 2 r d 2 S ξ L → dt 10 on the − 2 2 , φ ds β = i θ 2 − ds 2 β . The Rindler-AdS region extends only up to 2 − − 1 d ) is the usual scale-radius duality, and is manifestly an iso S 2 ) χ, t ( . The angles 1 2 − ξ/L − ( γ d . Geometry of Rindler-AdS are the time and radius in global coordinates; except at is the polar angle on the S − ρ → ψ Since many of our calculations will be done in three dimensio Another feature unique to three-dimensional anti-de Sitte ) = . Furthermore, with the other coordinates held fixed, τ 2 and χ, t of other times, the constant-time slices of particular, the last equation indicates that the time-slic boundary within the region shown in red. Here worldline; the arrow is reversed for the antipodal observer on the corresponds to a τ AdS. The arrow on the right points in the direction of We see that, unlike in higher dimensions, the metric on a cons ds conformal to Minkowski space.( Moreover, as that special case. The metric for Rindler-AdS Its asymptotic behavior near the AdS boundary is given by Figure 1 coordinates therefore cover a finite interval of global time asymptotic metric. kind of rotating Rindler space [ JHEP10(2018)129 - β , has (3.1) (3.2) (3.3) τ (2.10) ∂ ) before = 0) in de /L embedding a = 1 c -vacuum annihi- a 1. Both rotating β ≤ and local temperature β a ies (they are in different , ti-de Sitter space just as ≤ amiltonians for nonrotat- sic observer ( ler space is classically the globally the portion of the onding vacuum states (“ 1 hat covered by nonrotating by an observer moving with ruh temperature. icles. Interestingly, rotating t event horizon but even an orizons. From the embedding ime appears to the nonrotat- cceleration ( − portional to the proper accel- easure a temperature. Such an r embed eration t have horizons. For example, an a ]. That is, the βt acceleration see zero-temperature . The proper acceleration of such π ]. , moving in the direction of 1 ξ 10 2 − ρ respond to spacelike trajectories in 10 r [ = χ . 2 . 2 2 β a 1 → L 1 πξ − + χ 2 1 + 1) 2 = p 1 ξ – 5 – − r β/ 2Λ d = local ( = 2 βχ d T a − s r ξ/L π t 1 2 → = is a rotation parameter with t β local T ) we get and 3.1 ] is ∞ 3 < r is the proper acceleration of the Rindler observer in the flat ) into ( from [ < χ 3.2 +1 d embed a dS −∞ ) Consider then a Rindler-AdS observer at constant A where Inserting ( 3 Thermodynamics of Rindler-AdS Contrary to the situationconstant in acceleration flat space, in the curvederation. temperature spacetime seen is Rather, not the general always pro formula relating proper accel vacua”) of scalar field theory are particle-inequivalent [ maps one spacetime tosame the spacetime other. as In nonrotatinging that Rindler and sense, space. rotating rotating Rindlerconjugacy Rind However, classes space the of are H the not AdS related isometry by group) AdS and isometr the corresp and nonrotating Rindler-AdS spaceflat Rindler are space of is coursespacetime a a piece covered piece of by Minkowski of theRindler space. an coordinates coordinates. In above The fact, diffeomorphism is even identical to t Here in ( lated by the Hamiltonian thating generates Rindler a observer rotating as RindlerRindler-AdS an t space excited possesses state not populatedobserver-dependent ergosphere only with at part an observer-dependen an observer is space. This agrees for example with the fact that even a geode Sitter space sees a temperature. In AdS, there is a critical a the observer detects thermality. Observersextremal at horizons. the critical Observers with lower accelerationobserver do at no a constant nonzero global radial coordinate a constant nonzero acceleration butobserver nevertheless moves does vertically up not the m Penrose diagrampoint and has of no view, h sub-criticalthe acceleration higher-dimensional trajectories space cor and therefore do not give an Un JHEP10(2018)129 ], ary 17 (3.6) (3.4) (4.1) (3.5) , an be 16 , 11 = 0 and the ξ ), in a single- ) contribution z ). Later, we will 2.1 . In the Euclidean 3.3 S/CFT [ , niquely determines the )] global time coordinate, ∂AdS must have an imaginary )-invariant vacuum state , z,x ading (in ( r t , d φ ropagator. Thus bulk fields ies and the precise form of dχ . [ subtleties in the Lorentzian ich defines quantum gravity ) dχ ical supergravity action: space) is the state annihilated s simpler to study than eternal ime, indicating that an Unruh rature. Finally, the proper time sugra 2 ction in the boundary theory. χ/L β iS ( e 2 − )-invariant vacuum, when expressed − 1 d , d , giving precisely ( i ≈ = 0. As in flat Rindler space, the area ) tt . x g ξ Z ( ), the temperature and entropy are sinh − O 3 ) on the boundary ) +1 ∞ 2.9 1 x d x G √ 0 1 ( 1. The event horizon is still at ( 4 0 Z G 0 φ 4 2 ≤ φ – 6 – = − β d = S ∂AdS L s ≤ R i ] because of the existence of normalizable modes in ∼ 1 e h ) 13 − 2 , = +1 β d , is finite and obeys the universal relation: 12 πL − s 2 1 CFT )] (AdS = x ) takes the value ) then implies that the Rindler time H ( and large ’t Hooft coupling, the string partition function c 0 T A φ 2.2 z, x [ ( N Z φ can be assigned to each point on the entire space, ( ) acts as a source term in the CFT, and specification of the bound τ x ( 0 . Thus the Green’s function of the SO(2 φ πL ) (along with the assumption of regularity in the interior) u is the rotation parameter, x ( β 0 φ Next consider the entropy. The horizon is at . Being global, We are now interested inin Rindler-AdS the space. holographically As dual emphasizedAdS theory, earlier, wh black Rindler-AdS holes. i the Rindler-AdS boundary space CFT does is not known have singularit in certain cases. Now, as usual in Ad entropy is of course infinite. 4 The boundary theory However, the entropy density, where where the bulk field formulation, For three-dimensional rotating Rindler space ( the bulk. These are bulk excitations that do not change the le approximated at saddle point by the exponential of the class in the limit of large derive this temperature from the two-point correlation fun valued manner. But ( of the horizon in Rindler-AdS space is infinite: in Rindler coordinates isdetector carried similarly by periodic the Rindler in observerof will imaginary the record t a Rindler tempe observer has an extra factor of period of 2 (analogous to the vacuum in Poincaré-invariant Minkowski by the modesτ that have positive frequency with respect to the This can also be seen directly from the coordinates. The SO(2 field bulk field, which can beare determined dual using to the bulk-boundary boundaryversion p sources. of However, the there correspondence are [ additional JHEP10(2018)129 . 0 e, t bulk (4.4) (4.3) (4.2) (4.5) = t ) is the uation in and ξ, χ, t 0 χ . ive scalar field, . 2 = ∆ m χ 1+ 0 t √ . L − t ll perform a calculation k-boundary propagator 0 1+ n global AdS spacetime. operators are taken with . dt re the contribution of the y correlation functions in horizon. Remarkably, the 0 2 2 peraters are inserted on the t be given by a direct product igher dimensions. Below we i ce and sink happen to be on L − n dχ cosh 1 t -boundary propagators defined ) E ξ L 0 , t × | − available to the CFT about events 0 1 1 χ i 0 cosh 1 ( for computational convenience; most n χ 0 L − − E φ 3 | ) χ 2 0 2 / χ n , t ]: ], the two-point function between conformal 0 L − 1 βE ). In order to compute correlation functions χ 15 17 χ cosh ; − , , – 7 – e 2 2 ). The normalizable modes are dual to states 4.2 ξ L x n 14 16 ( X , 0 ξ, χ, t ) cosh 9 φ ( 1 + β . We also take the complete state to be an entangled ( 2 K 1 q = Z Z i ) becomes a delta function supported at p ⊗ H 2 1 ) = ) = , t = K 0 2 H i , ) of this entangled state is unique and related to the AdS scal χ , t ( Ψ = 0 | /β ξ, χ, t χ O is the conformal dimension of the boundary operator dual to a ( ) ; H → ∞ 1 2 φ ) acts as a source on the boundary while the bulk point ( ξ 0 , t m 1 , t χ ξ, χ, t 0 . The bulk-boundary propagator satisfies the massive wave eq ( ( χ 1 + m ) defined by K hO √ , the bulk-boundary propagator for a minimally coupled mass 0 3 , t 0 χ ; ξ, χ, t ( Here ∆ = 1+ Using the standard rules for AdS/CFT [ Here the point ( operators inserted on the same boundary is in the boundaryK theory, one needs the explicit form of the bul This state corresponds toAlso, the the temperature vacuum (1 of the boundary theory i upto normalization, is state of the two CFTs, as studied in [ as we willrespect see to later. the entangled All state expectation given values by of ( the conformal to the boundary value of the field, sink. In AdS in the boundary theory.normalizable modes For and our just present analyticallyin purpose, continue Euclidean the we bulk signature will igno Lorentzian in signature. order to We study will the also various focus boundar on AdS We take the complete Hilbertof space of two Hilbert conformal operators spaces, to will first recover thethat thermodynamics from indicates the how CFT. thecalculation indicates Then boundary that we theorist at wi leastthat could partial are perceive information across is the the Rindler horizon. 4.1 Temperature and two-point correlators of the results can be extended without loss of generality to h same boundary. As Rindler-AdS coordinates and is validthe as same long side as of both the the Rindler sour horizon i.e. when the conformal o scalar of mass JHEP10(2018)129 (4.8) (4.6) (4.7) (4.9) (4.10) ). The ), with ature. A.1 4.2 . 2 . m 2 m 1+ e Rindler horizon √ 1+ , the Rindler horizon the other side of the √ 1+ 5 1+ 0 angled state ( S/CFT we arrive at the t lt the operators on opposite L 2 − t t om the dual theory: as the L tropy. The free field entropy − al AdS coordinates and then dinates. dχ , 1 ). The number of degrees of t ) can be calculated in standard o different from what happens undaries N s are always spacelike separated. . ing to AdS , as can be seen from ( cosh 1 χ/L ξ ( L 2 3 iπL CFT 1 + T + cosh − . sinh t 0 0 2 χ 3 5 χ in imaginary time; evidently the boundary CFT χ L 0 − → V L − G scales the boundary theory. Specifically, for χ Z 2 1 t πL 2 χ πL ξ 2 5 N 2 = – 8 – G πL π 2 cosh 3 2 N cosh 2 2 ξ L = →∞ 0 χ = i 1 + = lim CFT ) 2 S q , t = 4 SYM theory, with a gauge field, four Weyl spinors and 2 χ ( Rindler N ) = 2 S 0 O ) , t 1 0 . The size of the gauge group is related to the AdS radius by 2 χ , t ; . Hence the boundary theory gives the correct horizon temper 1 N χ 1 πL ( 2 1 ξ, χ, t = ( hO K H T . This is in agreement with the fact that the temperature of th πL , the dual theory is 5 In general, correlation functions can be calculated in glob To evaluate the bulk-boundary propagator when the sink is on = 2 Using the above bulk-boundarytwo-point function propagator of and operators the inserted rules on the of opposite Ad bo AdS The two-point function is nonsingular becauseThe the operator reason the expectationboundaries value commute does is not that vanish the even CFTs though are entangled. six conformally coupled scalars, all in the adjoint of SU( freedom is thus 15 bulk-boundary propagator then becomes β A priori, there areentropy now of two a ways gas of ofcomputation calculating thermal for free the a fields, entropy thermal and fr as CFT entanglement is en done using the standard resu is indeed The two-point functions has a periodicity of 2 horizon, we analytically continue the time as has entropy CFT is thermal in nature, as mentioned previously for the ent which diverges as expected. The coordinate in flat Rindler spaceMinkowski for coordinates which and (bulk) then correlation transformed functions to Rindler coor 4.2 Entropy First consider the entropy in higher dimensions. Specializ transformed to Rindler-AdS coordinates. This is of course n JHEP10(2018)129 y ), 2.7 (4.13) (4.11) (4.17) (4.12) (4.14) (4.18) (4.15) (4.16) n and . For ( t it is very e entropy of 3 . Since by the πL . = 2 , ndler-AdS BH β S . BH S = do not yet understand how eld CFT. In the exact case 2 2 r = Ω dχ . e of the fact that we have as- d dχ ) reement. dχ oundary data into the above ex- ), we see that the free field CFT Z , , χ L Z χ/L 2 . 2 0 ( . β 2 2 1 3 4.10 1 πξ dχ πL and the physical temperature at the πL dχ − 2 2 G 2 + 4 1 R 3 sinh 1 = 0 2 L πL 3 G χ 2 Rindler 3 L = + sinh 2 0 G tt dτ 3 S 2 g Z 3 2 3 π 1 6 2 H 3 0 ) is given by = = 3 − π G A T L – 9 – dχ L ), albeit with 4 πξ = √ 2.7 H , we can now use the Cardy formula to calculate 4 T + = 4.8 = 2 2 CFT CFT →∞ L BH dt S 0 Z 2 boundary S χ CFT − Volume = ds T = Volume = c = lim 2 0 β ξ , the boundary metric is π 3 cT AdS L = , and the last equality follows from the fact that the boundar 3 CFT π Z = V 2 ≫ ) and inserting them into ( = πL ds 0 ) is periodic in imaginary time with period ξ CFT + 2 4.13 S = 4.5 CFT τ limit and large ’t Hooft coupling. In this approximation, th S ξ ), ( = is the central charge of the unitary CFT as calculated by Brow N β 3 ]. Of course the entropy of the Rindler horizon is infinite, bu L 4.12 G 3 + 2 26 τ is given by ), ( = ∼ 4.9 c τ CFT V So far this is all mostly familiar. We can do much better for Ri We can also use the Cardy formula for the rotating CFT: Using ( the boundary theory ishowever, computed the using CFT the could results beto for a calculate a fully the free interacting entropy fi field for theory; such we a theory directly. The horizon temperature is given by where and This familiar numerical disagreementsumed is the presumably large becaus boundary is interesting that the entropy densities are now in precise ag the Bekenstein-Hawking entropy is given by entropy scales in the same manner as ( Henneaux [ AdS/CFT correspondence where The Euclideanized boundary metric for ( pression. At fixed Evaluating this “holographically” implies substituting b the entropy of the CFT: two-point function ( JHEP10(2018)129 , o- 18 ) to R ), the (4.22) (4.21) (4.19) (4.20) , 2 m 2.10 1+ . SL(2 ) both have √ 2 × 1+ . ) r 4.18 i at the boundary. R dt 2 , t t 2 ∂ β ction is lost; neither r L − 1 t Rindler − i ) and ( SL(2 ) r bserver-dependent. Put 2 near-horizon limit of all ]. The black hole solution ng entropy density are in e recalls the rotating BTZ 8 , t ndler-AdS space and black dχ 4.17 s of Rindler-AdS two-point rators inserted on opposite – 2 1 4 time in the study of horizons. χ dS space and the BTZ black dicuar to ( 2 + cosh he diffeomorphism ( e, is flat Rindler space. nt to clarify that Rindler-AdS e [ 2 r r-AdS is, as we shall see below, O der has a preferred frame, singled + o the rotating BTZ black hole [ ) πn 1 2 +2 dr 2 L ) χ πn, t 2 2 − 1 β π . 1 β χ − + 2 − , and therefore ( − 2 1 + 2 2 r 1 r χ 2 r dχ ( ) r gives the two-dimensional boundary Minkowski χ 1 2 s – 10 – 1)( cosh π β h ∼ hO = − r − ∞ ∞ +2 ξ 2 χ r −∞ −∞ r (1 ( χ X X = =+ = =+ . n n n n → + ∼ G 4 2 r ∼ ∼ r ]. Hence there is a preferred direction of time. / χ dχ dt 21 ) 2 , BTZ β i 20 ) 2 − is the universal cover of the BTZ black hole also means that tw , t 2 3 2 r 2 χ r ( 1)( 2 O − ) 2 1 r ( , t 1 − χ U(1). Consequently, the freedom of picking out the time dire ( 1 × = ) 2 hO R , ds That Rindler-AdS The existence of an ergosphere in rotating Rindler-AdS spac ] via The relative simplicity of theanother one two-point of function the advantages in of Rindle Rindler-AdS as a model space another way, the identification the event horizon nor the ergosphere of the BTZ black hole is o Rindler-AdS is thus the universal cover for the BTZ black hol SL(2 However, there is an importanthole. difference between The Rindler-A identification breaks the symmetry group down from is obtained by making an identification in a direction perpen volume element transforms as the universal entropy density 1 space a cylinder topology. Butout special relativity by on the a identification cylin [ A change of coordinates puts the metric in the familiar BTZ form: 19 precise agreement, including the numerical factor. Under t Once again the CFT entropy density and the Bekenstein-Hawki functions summed over allboundaries, image the points. BTZ For two-point example, correlator for is ope point functions in the CFT for BTZ black holes are infinite sum Let us pause here toholes comment briefly in on anti-de the Sitter relation space.space between Ri is From the not outset,non-extremal the it black is near-horizon holes, importa limit including black of holes black in AdS holes spac black in hole. AdS; Indeed, rotating the Rindler-AdS space is related t 4.3 Relation between Rindler-AdS space and AdS black holes JHEP10(2018)129 fields. These couple FT can tell whether an r that is dual to the bulk n. Here we will consider lack hole formation and urrent configuration. For le is far from clear. Here able literature on how to calar field. The boundary er the case where the bulk dler horizon, before being en has partial information ng because, from the local ht-cone or Eddington-type he field itself; this would be tant in the boundary CFT. n or off even after it crosses nitary theory. Nevertheless, he bulk field here would be imelike trajectory, for which trast, gauge/gravity duality n from any earlier moment. rdinate and we would not be ending in an electromagnetic es of our infalling source, we as thrown in. The alternative l field theory dual to anti-de per time. The source couples hat one expects to be able to ce would have its wave-packet bed by the bulk matter energy- ng source. Such a construction ght like to send in a source that consider a point-like source, but y. Therefore, by considering the ular momentum) such as, say, an nd evaporation of AdS black holes he horizon. In principle, we could source) through smearing functions – 11 – Our goal then is to study the response of the boundary operato There are of course several different ways to probe the horizo would depend on the naturefield is of the the metric. source. Thenmomentum For the example, tensor, infalling consid source which would in beto turn descri the would CFT be through madecreate localized their up bulk of boundary fields (that bulk values; constitute matter there the infalling is a consider field, as the source fallsthis into the is horizon. a good For simplicity approximationblue-shifted we since and even increasingly more localized a asalso realistic it explicitly sour approaches construct t a CFT operator dual to the infalli able to distinguish theThe moment advantage of the sending packet in crossed athe source the ingoing is horizo null that coordinate it can timesignatures travel varies of on along a the the t “switching trajector will on” see and that “switching the off” CFTthe can process Rindler tell horizon. whether the source is switched o electric dipole, to test whether theto CFT throwing can in determine a what source w analogous would be to to probing send our in Reissner-Nordstromwave some black excitation which hole of will by t s propagatecoordinates, the on wave null would have trajectories. a constant ingoing Hence null in coo lig value of the bulk fieldConsider, in as turn an plays analogy, thethe a role electromagnetic Reissner-Nordstrom of field a black and coupling hole.the a cons purpose source T of would understanding be informationcarries retrieval, any no one charge coarse-grained mi hair or (i.e. c no mass, charge, or ang “switching on” a“switched point off” source after which the passageto freely of a falls some bulk into finite field interval the which, of pro Rin for simplicity, we will take to be a free s evade the paradoxes of black holes. we will take ainfalling step source in has that crossedabout direction the by events horizon. that showing that happenbulk In the across point fact, dual the of C the horizon. view,is CFT the This nonlocal horizon ev and is is it promisi a is nondescript precisely place; in by a con theory with nonlocality t It is nowevaporation widely is believed, unitary. if TheSitter not space existence lends proven, of support to a that thisis belief, unitary the as presumably conforma the a process formation process a of thata b has detailed a account dual of description how within a information u emerges from a black ho 4.4 The omniscient CFT JHEP10(2018)129 is he the v u (4.24) (4.25) (4.27) (4.26) (4.23) e time =const so χ as an infinite Rindler coordinates . .   1, we have that n, we consider timelike 1 , with the region outside 2 ≡ − + 1 ∞ dχ , r r 1, we find that the source’s ndary one-point functions of 1. Then setting mensions. L L on is somewhat irrelevant to 2 2 ing source) yield information m − # − r L 2 2 g = 0. To that end, we transition to 1 + 1 + f the global time coordinate. The < v < be , v < v q q τ 1 + + 1 u L 2 u 2 is conserved along geodesics. Since 2   . ) −∞ τ 0 v L dχ ln + p X + 2 = 1 L ( and r 2 L 2 2 2 − + r 1 + sin 2 cos ∞ L dv τ t 4 2 L ) 2 √ − dvdr 1 1 + = + – 12 – . Since the metric is invariant under translations 2 2 and using 3 X 2 " r r ) L L 2 2 q L/ r < r < ln cos u 2 + = ( 2 2 L L 1 + 2 b 2 dr L/ q dv ds (0) = − r ) = ) = r r L 2 L 2 τ τ ( ( metric in EF coordinates becomes . In particular, these coordinates are perfectly smooth at t − J J is Z 3 r v ∞ r ]. Therefore, in principle there is no problem in describing = 2 L + ξ 2 2 t 28 is the velocity vector), and setting , ds ≡ ≡ a 7 < r < u r v , 6 = 0. These coordinates span one patch of the Rindler-AdS spac r ). In the Penrose diagram, the entire space time can be viewed (where ∞ v coordinate, the momentum component is the proper time. = 0 (which corresponds to radial infall) we have v τ mu χ < r < u In order to describe a source falling into the Rindler horizo In order to describe an infalling source, we need to define the = 2 L v − radially ingoing geodesics in Rindler-AdS of the p conserved. For simplicity, let the conserved value of that The ranges of the coordinates is beyond the horizon i.e. into the region ( geodesic trajectory is given by which is conformally flat, an advantage of working in three di where Choosing the initial condition With this, the Rindler-AdS infalling bulk source inthe the question CFT. we are However, trying suchoperators to a consider dual constructi here to i.e. the howon bulk do across-horizon the field physics? bou (that is sourced by the infall boundary metric at large concatenation of such identical patches, in the direction o ingoing Eddington-Finkelstein (EF) coordinates by definin at the boundary [ the horizon being 0 future horizon ( JHEP10(2018)129 , J 2). √ (4.32) (4.31) (4.33) (4.30) (4.28) (4.29) 0, then ≥ ln(1+ i . Using the 2 for which τ nt since the L − prescription is ) we have Lπ/ = , iǫ = ′ v 4.29 to evaluate the bulk , dv ′ max )) τ dτ . τ from the boundary per- dχ ′ 2 and ( ral by its saddle-point )) . in path integrals, such as J at elike separation, can yield τ dr he Feynman propagator to L/ χ ( ith proper time J ′ elation functions with respect ln 2 = − , , v , v 2 r ′ L χ ) ( τ , δ − ( , χ ′ ] CFT J )) r cl = E ) = 0 for simplicity. In addition, we τ φ [ ( , χ τ h O ) ( J 0 J iI τ J φ v e ( ′ R . The AdS/CFT dictionary mandates the χ J e − , v φ ∼ r , v ′ D ] v ; = 0, we have ( φ [ δ = , χ τ ′ iI ] , sourced by a freely falling localized source, r )) – 13 – = 0 ), and φ φ ; [ τ , we can solve for the bulk scalar field as r, χ, v φ e ( h iI ( J J D r 4.27 G φ e f , r r, χ, v − τ i D bulk τ r Z Z ( ) before getting “switched off” or terminated at a later G 4 π bulk . Given ), are Feynman propagators; Feynman’s L Z Z cl ) = dτ δ 4.27 φ = f τ 4.30 i ) = h τ τ Z r, χ, v ) is the bulk-bulk propagator. For our source ( ′ ( ) are given by ( = cl , v τ r, χ, v ′ φ ( J ( J cl , χ v ′ φ r ), which is the AdS analog of the Hartle-Hawking vacuum state . ; f τ 4.2 in order to be consistent with our setup. This is very importa ] is the action for the classical field configuration. In order ) and cl τ cl is the boundary value of the bulk field r, χ, v φ ( ( φ 0 [ J r G I φ = 0. In particular, the source crosses the Rindler horizon at In order to describe the infall of the source into the horizon The conditions are chosen such that, at We now consider a bulk scalar field, max where where traverse the geodesic path ( action, we need to first find to the state ( equivalence of the bulk and boundary vacua. We evaluate corr require the source to get “switched on” at a certain instant w spective, we use the basic AdS/CFT tool on the left-hand side of ( An important point to note is that the propagators that arise where SUGRA approximation, we can approximate the bulk path integ proper time, The source exits the patch covered by Eddington coordinates v Feynman propagator, which cruciallysignatures does about across-horizon not physics. vanish at spac where necessary for path integralsevaluate to converge. Hence we must use t which we model as JHEP10(2018)129 , r ]. 27 et for (4.35) (4.38) (4.41) (4.37) (4.39) (4.40) (4.36) (4.34) → ∞ r . 1 χ L − ) to get 2 χ , dτ . 4.33 )) τ 2 ( cosh J , γ 2 , v ∆ r L ) 2 , e boundary, i.e. at τ and large ’t Hooft coupling, 2 1 ( , ndix), we find that dχdvdr . J , ) N + 1 + ν , ivalent to the global vacuum [ , χ ) is calculated using normalizable ) ). Let us first evaluate the action. Σ 2 ∆ cl v, χ τ d Jφ 1 ( , δI φ ( r L 0 cl µ 2 J 4.34 2 + ∆ 4.11 , r ∂ φ δφ 2 ; µ rµ φ ∂ h ab 1 1 + g 2 η cl 1 F − b g h 2 2 m 2 √ q − − r, χ, v L 2 1 δφ X ( ∆ a 1 2 √ √ coordinate and the variation of the bulk term − µν γ − G L is the Minkowski metric in the embedding space X – 14 – g →∞ 2 f v r ∼ ) ) τ i ab 1 = τ →∞ Z lim ) η r r = lim Z ( 2 ∂φ γ f ), the one-point function is ∼ ( i µ , v ) ), of the scalar field can be obtained by taking ] ∂ 1 2 )+ 2 cl 2 rµ r − 4.37 φ L ( , χ [ v, χ g χ, v hOi ∼ f 2 ( ( , where g h r − 0 δI a 2 ; 1 − − φ Z hO 1 v X − √ √ , v is 2 ). The explicit form for the bulk-bulk Feynman propagator fo 1 v ] = ] and is given by φ . The bulk-bulk propagator ( φ and 2 [ , χ →∞ lim χ, v 1 27 r I a 1 ) by parts, and separating the bulk and the surface terms, we g ( m r 0 ( X cosh φ is pulled down by differentiation. We now plug in ( G 2 4.38 1 + r φ 2 1 L √ hOi ∼ r ) = 4 + is the surface normal to the q ν r, χ, v Σ + ( is related to the AdS invariant geodesic distance, is the determinant for the boundary metric ( d cl was derived in [ γ h = φ Integrating ( According to the AdS/CFT correspondence, at large The boundary value, 3 γ →∞ lim (i.e. with two time directions). In EF coordinates (see appe for any two vectors AdS Here r using the above expression and ( where the variation of the action The action for the field modes in the Poincarécoordinates; is Poincarévacuum equ Here as one power of where ∆ = 1 + vanishes on-shell. Since we wish to evauate this action at th the one-point function is given by JHEP10(2018)129 . . ) r h 2 ∞ 1 r ) = 0, 4.36 < τ (4.43) (4.44) (4.47) (4.45) (4.46) (4.42) ) J τ/L 6 χ τ/L , Lπ/ 2 2 . cos(2 . 1+sin(2 i = . ∂r 2 ∂γ √ ) f i 6 τ τ ) ives. Therefore in ( ) √ v/L g τ e 2 ( v/L , ... g − − − 2 v L ) + − 5 . The above integral can ∂r 2 v L ∂γ = 0 and 4 γ O i τ/L 3 e is both switched on and 3 τ 1 cosh ) + sinh( √ ... . The appearance of the − i cosh ator 1 + τ + 2 τ L 2 L 2 v/L τ 2 + L 2 τ √ 2 L 2 2 2 γ γ 3 1 + sin (2 cos ln cos ln ∆ = 2 for ease of calculation, 1+sin dτ p 2 2 h L cos L q 3 cosh( ) contributes. Noting that in the large 4 ∂ 1 + ⇒ ∂γ dτ ln v/L q √ − = = 1 ), we obtain − 2 1 1 ) i f − 4.42 − e − χ = 0 ) poles at →∞ →∞ 6 = 4.45 − r r χ √ m – 15 – χ # = lim = lim 1 cosh( four ) − +1 , v ) ) cosh( τ 2 τ τ L in terms of Pochhammer symbols. We then get ( ( 2 L L ) cosh J J τ 6 v/L , γ r r γ L ) into the above expression. Next, we notice from ( 2 2 2 2 √ , , we have for the one-point function τ/L 2 1+ 1+ / 4.36 2 . We can therefore perform a power series expansion of the 2 ∞ cosh( q q 3 r L 4 , 5 + 2 " 1 + cos − 1 , v 2 ln 1 + cos q → → ∞ 1 h 2 √ 3 L q F r ), and ( 1 + cos(2 rr h 2 ∞ √ − r ∞ 2 p 2 ∞ ) r 2 + 1 f 2 γ , g r 4.34 τ τ i 2 + ), we have √ τ ( limit, only the first term in ( r ∞ Z = J ∞ L r ), ( ∂ i ∼ v r 2 ∞ f ln 2 ln τ r 4.36 i 0) τ 2 i ∼ ∂r L L → ∂γ →∞ 4.27 ) Z lim = v, r ( ) = = 0 and performing the integral ( h = = r τ is the infrared cutoff that marks the surface on which the CFT l i∼ i ( − χ v, χ f →∞ hO ) lim goes to zero as g ( u r u ∞ √ r γ hO v, χ ( the large where Here Setting be further simplified to yield hO hypergeometric function for small factor is consistent with the scaling dimensions of the oper limit, that switched off outside the horizon. For instance, we could take 4.5 Signatures ofFirst, across-horizon physics let us consider the one-point function when the sourc Notice that the one-point function has where, from ( Finally, we assume a massless scalar field and insert ( JHEP10(2018)129 . . f f τ τ ≤ (4.49) (4.48) τ icate the ≤ ). We have i is related to τ us of poles is u 4.23 . es; 3 1 as to wait till future − izon, there is no pole ction is related to the ) since at these points . 2 r it crosses the horizon. f ) ation about the location √ nderstand. We considered rizon. In our radial infall he horizon, the boundary , τ indler horizon; if there are i τ the source do not reach the orizon-crossing. But in order v/L ln l of proper time, For example, a source that is ry perspective. There are also es towards the AdS boundary r crossing the horizon. If there poles) as well as the advanced L ted in figure n will contribute a future-light- g the Feynman propagator, the u = sinh( , allows the boundary theorist to when the source switches off at i ) u . hOi 2 poles. They are at v/L 1 . Evaluating the integral, we find coordinate and the location of a switching h ) is given by the log term in ( χ cosh( coordinates. If the source switches off before three r ( > τ v − f – 16 – 2 2 coordinates for reasons that will be clear soon. √ u and Lπ/ , v u ∞ = r hypersurface where the CFT lives with the past/future f ), where r τ r ( i ∼ 2 + 1 f 2 0) ). The poles therefore come from taking the derivative of a √ − is discontinuous at the endpoints ( v, ( v ln . 4.41 cl = 0 and φ hO L = i τ u = 0 = poles) of the propagator. In a certain sense, these poles ind i f u v v (spatial) direction on the boundary. That is because the loc are ingoing and outgoing Eddington-Finkelstein coordinat χ v and ) pole in the near-infinite future. So the boundary theorist h u u -coordinate by In fact, the boundary theorist even acquires partial inform The appearance of poles in the one-point function is easy to u Now, consider the case where the source switches off only afte Evidently, the poles of the one-point function, Now the crucial point is that, once the source crosses the hor v As a result, the field light cone emanating from the endpoint. See figure the intersection of the constant of the switching offscenario, event, we even have if effectively that suppressed event the was across the ho an idealized source which is nonzero only for a finite interva derivative of the field ( we abruptly switch the source on and off. But the one-point fun infinity to determine whether there are three poles or four. In this case the one-point function has only are four poles, theonly source three switched poles, it off meansto before that it determine the reached source whether switched the thetheorist off R after has source h to switches observe offswitched the before off one-point just or function infinitesimally for aftercone before all t ( crossing time. the horizo determine whether the source was annihilated before or afte creation and annihilation of thepoles source from in a the boundary theo component (the For example, choose discontinuous field. The discontinuityalong in light-like the trajectories. field propagat propagation Moreover, of since these we signals are occur usin via the retarded (the expressed two of the poles in terms of Here the on/off event is characterized by its This is because oncesurface past where the the horizon, CFT lives. retarded signals This is from schematically illustra corresponding to the outgoing Eddington coordinate JHEP10(2018)129 Χ ing from one endpoint 4 certain time period outside the es. b) The right figure shows a lation of the source respectively. dary of the Eddington-Finkelstein y, and therefore the CFT perceives CFT lives. indicate signals propagating towards ded signal from the annihilation (or the case where the source switches off 2 poles coming from the intersection of the past v v L 1 2 3 4 - - - - – 17 – 2 - 4 - . The locus of points on the boundary where there are poles com . a) The left figure illustrates when the source is active for a source that crosses theswitching off) horizon. of the Itjust source is no three evident longer poles reaches that as thecoordinates. the CFT shown. retar boundar The dashed lines indicate the boun Figure 3 horizon in the right Rindlerthe wedge (R). AdS The boundary red and whichThe blue correspond four lines poles to are the indicated creation on and the annihi boundary where the CFT liv Figure 2 of the source trajectory.precisely The on specific the values horizon, plotted for are which for there are only light cone of the endpoint with the hypersurface on which the JHEP10(2018)129 ” v (5.1) values of v direction: 1 X + 2-dimensional d and the ) we can see that the u 3 for the one-sided CFT nnot single-handedly set le signature of the across- ding in the other Rindler if the source switches off . cit in our set-up since we dge (F) also intersects the ation about events beyond erver in ) the third pole (i.e. the “ rossing the horizon — even e structure of both CFTs is rist can read off the results he boundary theorist would h the e time. The key difference is . However, to actually create he horizon in finite time, and ). The missing fourth pole is ndler-AdS with a potentially value of the event. So partial tive to beyond-horizon metric s the event horizon. Contrast ion if the source were to get undary correlation function is erating in the . This is precisely the spirit of ion of a CFT whose relation to t/L horizon, the boundary theorist re AdS behind the horizon, the v e figure ts in the upper Rindler wedge. cosh( r = ˜ 1 X ], one needs both the right as well as the left – 18 – 28 ) t/L , the boundary theorist has to make certain assumptions sinh( v r = ˜ 0 X In order to exactly read off We have used a boundary correlation function to detect a simp Minkowski space (with two time directions) uniformly accel In this section, we touchwide upon spectrum an of alternate formulation applications. of Ri Consider again a Rindler obs therefore necessary to fully reconstruct switching off even 5 De Sitter space as the boundary of Rindler-AdS if such a theorist mayour not calculation. recognize it More aspast precisely, an from light infalling cone our source of construction aantipodal (se switching CFT off (associated event with inactually a the in hypersurface upper the in Rindler antipodal region we CFT; (L) complete knowledge of the pol CFTs. Hence a boundaryup theorist with the access experiment. to onlyof Nevertheless, one CFT the a ca one-sided experiment boundarydepending — on theo there whether the are source distinct switches off and before measurable or effects after c the event so thatafter it its crosses precise the location horizon, the can CFT be still knows identified. about the Even it traverses the horizon, the CFT pole structure records bot wedge). Nevertheless, even if therequalitative were result deviations from regarding pu thestill number hold. of That poles is,would irrespective “seen” still of by the perceive t geometry onlyswitched behind three off the behind poles the in horizon. the one-point Only the funct precise location of about the geometry behindhave assumed the that Rindler the horizon. geometry is This pure is AdS impli everywhere (inclu that the boundary theorist hasthe access bulk to the is one-point non-local funct the — horizon. and which can therefore encode inform information is obtained even aboutthis events with that a happen bulk observer acros therefore who would does also not not see anything see fall any into information t come out in finit coordinate of the pole inperturbations. the boundary theory) would be sensi the bulk source in the boundary theory [ horizon physics of anaccessible infalling to bulk a source. boundary theorist Notably, with the access bo to only one CFT JHEP10(2018)129 2 (5.3) (5.6) (5.4) (5.2) (5.7) (5.5) χ < π/ . ≤ in the region 2 π . − χ 2 2 d Ω , we finally obtain d sin , χ is reversed for this 2 φ < L r t a Rindler observer at +1 ≤ sin ∂ 2 , d + = 0 L 2 2 r ) 2 dX − = − and d ions). Rindler observers at 2 d − 2 r φ θ by φ . For the region 0 0 r/L e conformal to static de Sitter Ω 2 d . π d ( dr 1 X 2 − H cos sin cos ∂ χd ≤ − ) d dX 2 2 3 2 i 1 S θ − − − + d d d r/L 1 + θ θ θ ≤ ( is given by 2 ... + sin 0 . Defining t ) − . The geometry of Rindler-AdS space is ) sin sin sin dt 2 χ + ). The first two coordinates are of the 2 1 2 2 2 dχ ...... t/L ) between t/L /Z ) 2.1 + cos 1 1 1 1 d . θ 2 2 dX θ θ θ R 5.1 ) B 2 R r/L + – 19 – R ( sinh( cosh( sin sin cos sin 2 r < L + covers the static patch of the antipodal observer. . We foliate AdS as ˜ χ χ r + χ χ χ χ − R/L 2 ˜ r d ( ≤ π ) 2 1 / ]. The ranges of the coordinates are 0 L sin sin cos cos sin sin + s ≤ 2 2 − 30 1 p R R R R R R R R/L now being , i.e. the hyperbolic space dt dR 2 2 = r d ) ======), the spatial part of AdS that corresponds to a Rindler 29 < χ ... [ ∞ a 1 0 1 2 2 d ˜ 2 1 + ( r L , the relation ( L +1 − − AdS X X X X R/L χ . The proper acceleration of such an observer is d d d π/ = r whose topology is − X X X 2 r < L + ( 2 < t < = ds 2 ≤ ) /Z 2 = cos d . is related to the polar angle on the ds 4 2 − ∞ r H r R/L dR 2. The range and constant 1 + ( R have proper acceleration 1 R r ) with what we called ˜ = ≤ 2 0 5.1 ds χ < π/ Incidentally, the spatial geometry at constant To compute the temperature of the Rindler horizon, consider ≤ This satisfies the AdS embedding equation ( which is locally Euclidean depicted in figure We see that Rindler-AdS canspace also be with foliated de in Sitter slices radius that ar (corresponding to 0 constant 0 observer is really observer. Note that, since ˜ form ( which, indeed, isconstant Rindler ˜ space (albeit with two time direct This turns the flat space line element into The coordinate JHEP10(2018)129 2 ) (5.9) (5.8) (5.10) (5.11) (5.12) R/L , which ( = 0 each R ˆ t/ ρ = t elerating observer at are the time and radius in 2 at the boundary of AdS. . ρ e, we define 2 π/ − ± 2 d and , whose orbits are a Rindler/de t Ω , = τ e. ∂ d τ 2 , 2 f ) r 2 to 2 ) , ) . r/L 2 ntipodal observer. Except at ervers. ( 1 r/L 1 l ( − πR R/L local 2 1 − T + ( 1 tt = p g ) 2 ) r s − ( 1 H πR dr π f – 20 – r √ 2 1 4 ( is πR ′ 2 + = . f r = 2 2 ˆ t space. The shaded region is a surface of constant H − = = d d T ) S +1 T r d ( local f T boundary − T = part of the metric in the form 2 R ds ) we get 3.1 ) into ( 5.7 . Geometry of Rindler-AdS in static de Sitter space. To obtain the de Sitter temperatur r From the boundary point of view the Rindler observer is an acc global coordinates. The Rindler-AdS region extends only up Inserting ( The arrow in the right shaded region points in the direction o Then the de Sitter temperature is and the local temperature at constant fixed Sitter observer’s worldline; the arrow is reversed for the a which is again the physically-measured Rindler temperatur which puts the constant point in the interior corresponds to a covers the static patches of a pair of antipodal de Sitter obs Figure 4 and the horizon temperature is JHEP10(2018)129 . , ). 0 D +1 R d 4.9 (5.15) (5.14) (5.13) = ] using , R 35 ) d s foliation, using ( /L scaling of the 2 0 2 0 R N ( computation for 2 tor. We see that R 1 ) ) has the required retation of Rindler − /L 0 dary is intriguing. It 5.15 sinh R in the reverse way: by L e finite entropy, however, andard area formula. The d calculation, quite apart ace. The , − n de Sitter space [ d an extensive entropy that coupled field theories in de alar field in Rindler-AdS o the horizons of the static 2 s in the static de Sitter patch ∆ ) ) ions is to Wick-rotate the time /L D 0 1 R cos . . The two-point function of the dual − d 2 0 S 5 1 + ( (1 ]; the de Sitter horizon acts as a surface G , and yields the correct Rindler tempera- p ). We observe that ( 2 34 0 = 2 πLR ψ – θ i R ) ≈ 31 2 2 5 cos dimensions, which in two dimensions becomes – 21 – , ψ , the Rindler horizon has entropy 1 G 2 5 θ d πL 2 θ : ( Rindler 2 0 = O S AdS R ) 2 1 ) ) + cos , ψ 2 1 ψ θ R/L dR ( 2 − , is the conformal dimension of the dual operator, and 1 2 R hO ψ 1 + ( m ; the CFT then lives on an p 0 cos ( 1 + R 2 0 . Specializing to iLψ scales the boundary theory in this parameterization. At fixe √ θ Z λ 5 R ]. There are some subtleties, however. Unlike in our previou → π sin = though oddly the entropy in this case is precisely ( scaling strongly suggests that the correct boundary interp G t ), seems to indicate, perhaps surprisingly, that a free field 1 34 r 2 0 3 0 the entropy scales like θ = R R 0 is a cut-off radius which acts in the bulk as an infrared regula 5.14 R ). 0 R = (sin 5.9 Rindler To calculate the two-point correlator consider a massive sc That a certain foliation of AdS has de Sitter space as its boun The entropy of the Rindler horizon is calculated using the st S D operator can now be easily calculated as The easiest way to calculatecoordinate the as boundary correlation funct entropy could be as entanglement entropy [ therefore, the theory is a UV cut-off CFT in static de Sitter sp for large where ∆ = 1 + The coordinate cos using gravity in Rindler-AdSSitter space space [ to learnthe about boundary itself strongly- nowdiamond of contains de a Sitter horizon,since space. there corresponding The is t boundary no horizon gravity does in the not boundary hav theory. periodicity in the imaginary time coordinate, would be interesting tothis try setup. to understand It the may vacuum allow states us i to use the AdS/CFT correspondence The actual across which the conformal fieldsof are the entangled antipodal with observer. the field horizon is at where is the de Sitter invariant distance in ture ( entropy, ( scales like a thermal CFT willfrom not being give off the by numerical right factors, result would either. be expected A to free yiel fiel JHEP10(2018)129 tion about the ]. It might also 38 , 36 two-point and higher server complementar- ill quantum field theory direction to learn about Hagedorn transition. In d study the quantization tersection surface cannot picture to holographically the pole structure in both this work but it would be m gravity of horizons and, χ or over a fixed background calculations using Rindler- able to take a step beyond oundary theory by calculat- ntzian version of AdS/CFT s. ulk from the boundary value cannot be attributed unam- celeration horizons. The key by both CFTs. In particular, r intuition about information rizon thermodynamics includ- horizon [ on-crossing source. Evidently, subject. . We also showed that physics observer, it might be necessary tor insertions on the boundary 3 an in flat space, so as to be able ween the bulk and the boundary arly related, but in string theory ng to set up a problem in which aphy of AdS-Schwarzschild black ther more realistic scenarios that ] or to map the antipodal CFT to some 37 ) and map them to the boundary theory. 2.3 – 22 – ]. For example, in our scenario we know that complete informa 37 , More speculatively, we could try to implement some kind of ob It should be noted that what we have done was, in some sense, st Still more speculatively, there might be connections to the Also, as mentioned earlier, there are subtleties in the Lore Among the obvious directions for future study are to work out 36 ity [ of the bulk field whichgeometry. in turn By was considering determined graviton usingQFT a fluctuations, in propagat curved we spacetime. might be in curved spacetime. The boundary theory learned about the b biguously to either Rindlerof wedge. global Correspondingly, opera AdSobviously at be precisely thought the of as points insertions where in either it ofquantum touches field the that theory, two acceleration CFT and in temperature are line switching off event in theCFTs. upper In Rindler order wedge for was allto provided this information perform by to some be kind available of toother one antipodal surface identification in [ the originalbe, wedge, however, such that as complete at informationthe the is points stretched not where provided even the two antipodal Rindler wedges intersect One can also determineconditions. the This spectrum will throw of more light normalizabledescriptions on in modes the Rindler-AdS/CFT. dictionary an bet because of the presenceinteresting of to normalizable work modes. out We mode ignored solutions in for ( In this paper, weidea have was presented to a consider holographic acceleration horizons dualityto in exploit for AdS, the ac rather AdS/CFT th correspondence.probe We properties then of used the the Rindler dual ing horizon. the We precise recovered the entropy ho density for the case of Rindler-AdS 6 Summary and discussion AdS/CFT and then finally make a global identification in the correlation functions for infalling sourcesmight and probe to the look horizon. at Itinformation o would fell be into particularly the interesti return Rindler is horizon, borne to out. see whether Another ou obvious direction is to perform beyond the horizon can be probeding from the the response perspective of of theRindler-AdS/CFT the boundary b holds theory much to promise an formoreover, infalling it studying horiz is the considerably quantu more tractableholes; than the we hologr have surely only scratched the surface of this rich the holography of BTZ black holes. JHEP10(2018)129 (A.1) (A.3) (A.2) π . 2 φ < 2 − ≤ d θ φ 0 φ cos cos sin 2 FT correspondence might n part by John Templeton 3 2 2 − ing its early stages, as well d Yao Ji and Thomas Jacques π − − − n de Boer, Paul Davies, and θ φ mperature reaches the Hage- d d d φ θ θ θ ≤ i sin cos cos θ sin sin sin 2 3 2 − − − ≤ . d d d ...... ) θ θ θ 0 1 1 1 1 ) θ θ θ θ sin sin sin τ/L . τ/L ). The ranges of the coordinates are ) ...... ) cos ) sin ) sin ) sin ∞ 1 1 1 1 cos( sin( 2.1 θ θ θ θ 2 2 χ/L χ/L χ/L χ/L χ/L – 23 – R ρ sin sin cos sin + + To view Rindler observers as part of AdS, define ψ ψ ψ ψ ψ < χ < 2 2 Global coordinates are related to embedding coordi- ) sinh( sinh( sinh( sinh( cosh( ) L L . 2 2 2 2 2 sin sin sin sin cos . ξ ξ ξ ξ ξ p ρ ρ ρ ρ ρ p t/L t/L +1 + + + + + d +1 d 2 2 2 2 2 ======... L L L L L 2 1 2 1 0 d sinh( cosh( +1 − − p p p p ξ p ξ X X X X d d d ∞ − ∞ X X X ======... 2 2 1 0 1 d − +1 − < t < X X X X d d d X X X − ∞ < ξ 0 This satisfies the AdS embedding equation ( A Rindler and Eddington-Finkelstein coordinatesRindler for coordinates AdS for AdS We thank Erik Verlindeas for for collaborating subsequent on discussions. thisDavid We Lowe would project for also dur helpful like discussions. tofor thank P. their S. Ja generous would help likeFoundation in to grant making thank 60253. figures. M. P. is supported i dorn temperature. Perhapsprovide the a new existence angle of from a which to Rindler-AdS/C examine this old issue. Acknowledgments it is possible that something nontrivial happens when the te nates via Global coordinates for AdS JHEP10(2018)129 , ] (A.4) (A.5) (A.6) , pace ommons       spaces 1 1 − − , ]. ero energy states ]. + 1 + 1 hep-th/0506118 [ r/L r/L r/L r/L Ingoing Eddington- SPIRE SPIRE ]. IN IN 1 + 2 1 + 2 . ][ 3 ][ 1 + 2 1 + 2 p p ]. redited. p p , SPIRE ]. inates through IN rL rL 2 2 (2006) 086003 ][ SPIRE Local bulk operators in AdS/CFT: A Holographic representation of local v u u t v u u t IN SPIRE ][ IN v/L v/L D 73 ][ − − ]. e e hep-th/9806223 hep-th/0606141 [ [ − + ). These coordinates are nonsingular at the SPIRE 4.23 IN 1 1 – 24 – + 1 + 1 Phys. Rev. arXiv:0805.3488 ][ [ , − − gr-qc/9706018 [ (1998) L85 r/L r/L )) )) , r r r/L r/L ( ( 15 arXiv:1006.1263 (2006) 066009 Accelerating Branes and Brane Temperature [ f f 1 + 2 1 + 2 χ L χ L ), which permits any use, distribution and reproduction in − − 1 + 2 1 + 2 as given by ( p p v v ( Seeing a c-theorem with holography D 74 ( # p p 1 1 L 1 (2008) 175017 (1997) L163 L Equivalence of Hawking and Unruh temperatures through flat s Accelerated detectors and temperature in (anti)-de Sitter sinh cosh − +1 rL rL hep-th/9906040 r r 2 2 [ L L 2 2 25 14 rL rL v u u t v u u t 1+ 1+ CC-BY 4.0 (2010) 046006 cosh sinh q q + 2 + 2 v/L v/L Phys. Rev. " 2 2 This article is distributed under the terms of the Creative C = 0. e e , rL rL AdS/CFT duals of topological black holes and the entropy of z L L r       2 2 Class. Quant. Grav. ln D 82 , 1 1 2 2 p √ p √ 2 (1999) 036 L ]. ======06 ) = 2 1 3 0 r ( X X X X SPIRE f IN embeddings bulk operators Boundary view of horizons[ and locality Class. 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