Near-Horizon Geometry and Warped Conformal Symmetry

We provide boundary conditions for three-dimensional gravity including boosted Rindler spacetimes, representing the near-horizon geometry of non-extremal black holes or flat space cosmologies. These boundary conditions force us to make some unusual choices, like integrating the canonical boundary currents over retarded time and periodically identifying the latter. The asymptotic symmetry algebra turns out to be a Witt algebra plus a twisted u(1) current algebra with vanishing level, corresponding to a twisted warped CFT that is qualitatively different from the ones studied so far in the literature. We show that this symmetry algebra is related to BMS by a twisted Sugawara construction and exhibit relevant features of our theory, including matching micro- and macroscopic calculations of the entropy of zero-mode solutions. We confirm this match in a generalization to boosted Rindler-AdS. Finally, we show how Rindler entropy emerges in a suitable limit.


Introduction
Rindler space arises generically as the near horizon approximation of non-extremal black holes or cosmological spacetimes. Thus, if one could establish Rindler holography one may expect it to apply universally. In particular, obtaining a microscopic understanding of Rindler entropy could pave the way towards some of the unresolved puzzles in microscopic state counting, like a detailed understanding of the entropy of the Schwarzschild black hole, which classically is one of the simplest black holes we know, but quantum-mechanically seems to be among the most complicated ones.
Our paper is motivated by this line of thought, but as we shall see the assumptions we are going to impose turn out to have a life of their own and will take us in somewhat unexpected directions. This is why we will label what we do in this work as "quasi-Rindler holography" instead of "Rindler holography".

Three-dimensional gravity
For technical reasons we consider Einstein gravity in three spacetime dimensions [1]. While simpler than its higher-dimensional relatives, it is still complex enough to exhibit many of the interesting features of gravity: black holes [2], cosmological spacetimes [3] and boundary gravitons [4].
In the presence of a negative cosmological constant the seminal paper by Brown and Henneaux [4] established one of the precursors of the AdS/CFT correspondence. The key ingredient to their discovery that AdS 3 Einstein gravity must be dual to a CFT 2 was the imposition of precise (asymptotically AdS) boundary conditions. This led to the realization that some of the bulk first class constraints become second class at the boundary, so that boundary states emerge and the physical Hilbert space forms a representation of the 2dimensional conformal algebra with specific values for the central charges determined by the gravitational theory.
As it turned out, the Brown-Henneaux boundary conditions can be modified, in the presence of matter [5], in the presence of higher derivative interactions [6] and even in pure Einstein gravity [7]. In the present work we shall be concerned with a new type of boundary conditions for 3-dimensional Einstein gravity without cosmological constant. Let us therefore review some features of flat space Einstein gravity.
In the absence of a cosmological constant Barnich and Compère pioneered a Brown-Henneaux type of analysis of asymptotically flat boundary conditions [8] and found a specific central extension of the BMS 3 algebra [9], which is isomorphic to the Galilean conformal algebra [10][11][12]. Based on this, there were numerous developments in the past few years, like the flat space chiral gravity proposal [13], the counting of flat space cosmology microstates [14], the existence of phase transitions between flat space cosmologies and hot flat space [15,16], higher spin generalizations of flat space gravity [17], new insights into representations and characters of the BMS 3 group [18,19] with applications to the 1-loop partition function [20], flat space holographic entanglement entropy [21] and numerous other aspects of flat space holography [22][23][24].

Quasi-Rindler gravity
As we shall show in this paper, the flat space Einstein equations, R µν = 0, (1.1) do allow for solutions that do not obey the Barnich-Compère boundary conditions but instead exhibit asymptotically Rindler behavior, The main goal of the present work is to set up these boundary conditions, to prove their consistency and to discuss some of the most relevant consequences for holography, in particular the asymptotic symmetry algebra and a comparison between macroscopic and microscopic entropies. Some previous literature on asymptotically Rindler spacetimes is provided by Refs. [25][26][27][28][29][30][31][32]. Before we delve into the relevant technicalities we address one conceptual issue that may appear to stop any attempt to Rindler holography in its track. For extremal black holes the usual near-horizon story works due to their infinite throat, which implies that one can consistently replace the region near the horizon by the near-horizon geometry and apply holography to the latter. By contrast, non-extremal black holes do not have an infinite throat. Therefore, the asymptotic region of Rindler space in general has nothing to do with the near-horizon region of the original theory. So even if one were to find some dual field theory in some asymptotic Rindler space, it may not be clear why the corresponding physical states should be associated with the original black hole or cosmological spacetime. However, as we shall see explicitly, the notion of a "dual theory living at the boundary" is misleading; one could equally say that the "field theory lives near the horizon", since (in three dimensions) the canonical charges responsible for the emergence of physical boundary states are independent of the radial coordinate. While we are going to encounter a couple of obstacles to apply Rindler holography the way we originally intended to do, we do not consider the finiteness of the throat of non-extremal black holes as one of them.
Starting with the assumption (1.2) we are led to several consequences that we did not anticipate. The first one is that, on-shell, the functions specifying the spacetime metric depend on retarded time u instead of the spatial coordinate x, as do the components of the asymptotic Killing vector fields. As a consequence, canonical charges written as integrals over x all vanish and the asymptotic symmetries are all pure gauge. However, upon writing surface charges as integrals over u and taking time to be periodic, the asymptotic symmetry algebra turns out to describe a warped CFT of a type not encountered before: there is no Virasoro central charge nor a u(1)-level; instead, there is a non-trivial cocycle in the mixed commutator. Based on these results we determine the entropy microscopically and find that it does not coincide with the naive Rindler entropy, as a consequence of the different roles that u and x play in quasi-Rindler, versus Rindler, holography. In the quasi-Rindler setting, we show that our microscopic result for entropy does match the macroscopic one. As we shall see, the same matching occurs in our generalization to Rindler-AdS. In summary, in this paper we describe a novel type of theory with interesting symmetries inspired by, but slightly different from, naive expectations of Rindler holography.
This work is organized as follows. In section 2 we state boosted Rindler boundary conditions and provide a few consistency checks. In section 3 we derive the asymptotic symmetry algebra with its central extension and discuss some implications for the putative dual theory. Then, in section 4, we address quasi-Rindler thermodynamics, calculate free energy and compare macroscopic with microscopic results for entropy, finding exact agreement between the two. Section 5 is devoted to the generalization of the discussion to quasi-Rindler AdS. Finally, we conclude in section 6 with some of the unresolved issues. Along the way, we encounter novel aspects of warped conformal field theories, which we explore in appendix A. Questions related to standard Rindler thermodynamics are relegated to appendix B.

Boosted Rindler boundary conditions
which, up to boundary terms, is equivalent to the Chern-Simons action [33] with an isl(2) connection A and corresponding invariant bilinear form · · · . Explicitly, the isl(2) generators span the Poincaré algebra in three dimensions (n, m = 0, ±1), and their pairing with respect to the bilinear form · · · is (all bilinears not mentioned here vanish). The connection can be decomposed in compo- In terms of these components the line-element reads Thus, from a geometric perspective the components A n M correspond to the dreibein and the components A n L to the (dualized) spin-connection [34,35].

Boundary conditions
In the metric formulation, boosted Rindler boundary conditions at null infinity r → +∞ are given by where a(u, x) and η(u, x) are arbitrary, fluctuating O(1) functions. The equations of motion (1.1) imply homogeneity of the Rindler acceleration: The function η(u, x) and subleading terms are also constrained by the equations of motion. These constraints are solved by functions η that depend on u only 1 : For simplicity, from now on we always implement the asymptotic on-shell conditions (2.8), (2.9) together with the boundary conditions (2.7), i.e., we assume both a and η depend on the retarded time u only.
Since the x-independence of the functions a and η has important consequences we stress that the conditions (2.8)-(2.9) are forced upon us by the Einstein equations and our choice of boundary conditions (2.7). In fact, similar boundary conditions were proposed already in four dimensions [30], but no attempt to identify the dual theory was made in that paper. 1 The most general solution of these constraints (up to gauge transformations) is actually η(u, x) = η(u)+k(x) exp [− u a(u ) du ]. We considered this more general case but did not find relevant new features when k(x) is non-zero. In particular, the function k(x) does not contribute to the canonical surface charges, so we set it to zero with no loss of generality.
For many applications it is useful to recast these boundary conditions in first order form in terms of an isl(2) connection with the ISL(2) group element b = exp r 2 M −1 (2.11) and the auxiliary connection where ± refers to L ±1 and M ±1 . Explicitly, where the second term comes from b −1 db and the linear term in r from applying the Baker-Campbell- Using (2.6), this Chern-Simons connection is equivalent to the metric (2.1).

Variational principle
For a well-defined variational principle the first variation of the full action Γ must vanish for all variations that preserve our boundary conditions (2.7). Since we shall later employ the Euclidean action to determine the free energy we use Euclidean signature here too.
We make the ansatz 2 where α is some real parameter. If α = 1, we recover the Gibbons-Hawking-York action [36]. If α = 1 2 , we recover an action that is consistent in flat space holography [16]. We check now which value of α -if any -is consistent for our boosted Rindler boundary conditions (2.7).
Dropping corner terms, the first variation of the action (2.14) reads on-shell [16] In the Lorentzian theory the boundary is time-like (space-like) if a is positive (negative). To accommodate both signs one should replace K by |K|. To reduce clutter we assume positive a and moreover restrict to zero mode solutions, a = const., η = const. It can be shown that the variational principle is satisfied for non-zero mode solutions if it is satisfied for zero mode solutions, as long as ∂xgrx = O(1/r 2 ). We thank Friedrich Schöller for discussions on the variational principle and for correcting a factor two.
We introduce now a cut-off at r = r c and place the boundary at this cut-off, with the idea of letting r c → ∞ at the end of our calculations. Some useful asymptotic expressions are Inserting these expressions into (2.15) establishes Therefore, picking α = 1 we have a well-defined variational principle. Demanding a well-defined variational principle for the first order action (2.3) with the boundary conditions (2.10)-(2.12) also requires the addition of a boundary term of the form where the sign depends on the conventions for the boundary volume form, ux = ∓1. This result agrees with the general expression found in three-dimensional gravity in flat space [34,37].
In conclusion, the full Euclidean second order action suitable for quasi-Rindler holography is given by where the boundary contribution is the Gibbons-Hawking-York boundary term. This action is the basis for Rindler thermodynamics discussed in section 4.

Asymptotic symmetry transformations
The allowed diffeomorphisms preserving the boundary conditions (2.7) are generated by vector fields ξ whose components are where t(u) and p(u) are arbitrary real functions. Infinitesimally, the corresponding diffeomorphisms take the form on the spacetime boundary at infinity. In other words, the gravitational system defined by the boundary conditions (2.7) is invariant under time reparametrizations generated by t(u)∂ u − rt (u)∂ r and under time-dependent translations of x generated by p(u)∂ x . These symmetries are reminiscent of those of two-dimensional Galilean conformal field theories [11]. The Lie bracket algebra of allowed diffeomorphisms (2.21) is the semi-direct sum of a Witt algebra and a u(1) current algebra. This can be seen, for instance, by thinking of u as a complex coordinate (which will indeed be appropriate for thermodynamical applications) and expanding t(u) and p(u) in Laurent modes. Another way to obtain the same result is to take u periodic, say u ∼ u + 2π L (2.23) (where L is some length scale), and to expand the functions t(u) and p(u) in Fourier modes. Introducing the generators t n ≡ ξ| t(u)=L e inu/L , p(u)=0 and p n ≡ ξ| t(u)=0, p(u)=L e inu/L (2.24) then yields the Lie brackets up to subleading corrections that vanish in the limit r → +∞. Thus, quasi-Rindler boundary conditions differ qualitatively from usual AdS holography, which relies on conformal symmetry, and from flat space holography [4,38], which relies on BMS symmetry [8,12]. Instead, if there exists a dual theory for quasi-Rindler boundary conditions, it should be a warped CFT [39] whose conformal symmetry is replaced by time-reparametrization invariance. 3 We will return to the interpretation of this symmetry in section 3. The allowed diffeomorphisms (2.21) can also be obtained from the Chern-Simons formulation: upon looking for isl(2) gauge parameters ε that obey where δA denotes the fluctuations allowed by the boundary conditions (2.10)-(2.12), and writing [in terms of the group element (2.11)], one finds Here the functions t(u) and p(u) are those of (2.21), while Υ(u) solves the differential equation Upon imposing periodicity (2.23), this solution is unique. Using (2.26)-(2.28) and the on-shell connection (2.12), we find that the functions a and η transform as follows under allowed diffeomorphisms: (Prime denotes differentiation with respect to u.) Note in particular that translations by p leave a invariant; note also the inhomogeneous term t in the infinitesimal transformation law of a under conformal transformations, which hints that the asymptotic symmetry algebra has a central extension. We are now going to verify this.

Asymptotic symmetry group
This section is devoted to the surface charges associated with the asymptotic symmetries (2.21). First we show that the conventional approach leads to a trivial theory where all asymptotic symmetries are gauge transformations, as any on-shell metric is gaugeequivalent to Minkowski space. We then opt in subsection 3.2 for a non-standard definition of surface charges, providing us with a centrally extended asymptotic symmetry algebra. In subsection 3.3 we work out the finite transformations of the dual energy-momentum tensor.

An empty theory
We saw in the previous section that asymptotic symmetries include time reparametrizations. This is a somewhat ambiguous situation: on the one hand, asymptotic symmetries are generally interpreted as global symmetries, but on the other hand, time reparametrizations are usually seen as gauge symmetries. In this subsection we show how the standard approach to surface charges in gravity selects the latter interpretation.
In the Chern-Simons formulation [34,35,37,46,47], the variation of the canonical current j associated with an asymptotic symmetry generated by ε reads where A and a are related by (2.10) and · · · denotes the invariant bilinear form (2.5).
The integral of that expression along a line or a circle at infinity gives the variation of the corresponding surface charge [48]. In the present case, the region r → +∞ is spanned by the coordinates u and x, but only the latter is space-like. 4 Accordingly, the natural surface charges are integrals over x; unfortunately the boundary conditions (2.12) [and the ensuing asymptotic symmetries (2.28)] set to zero the x-component of the variation (3.1), so that these charges all vanish. It follows that, from this viewpoint, all asymptotic symmetries are in fact gauge symmetries; there are no global symmetries whatsoever, and the theory is empty. While this conclusion is somewhat disappointing, it does not prevent us from studying the group of gauge symmetries in its own right, and these considerations will in fact be useful once we turn to an alternative interpretation. Upon integrating the infinitesimal transformations (2.22), one obtains finite diffeomorphisms of the plane R 2 (spanned by the coordinates u and x) given by where f is an orientation-preserving diffeomorphism of the real line (so that f (u) > 0 for all u), and p is an arbitrary function. Such pairs (f, p) span a group where the vector space C ∞ (R) is seen as an Abelian group with respect to pointwise addition. Diffeomorphisms act on functions according to so G is a semi-direct product with a group operation It is a centerless version of the symmetry group of warped conformal field theories. One may then ask how finite gauge transformations affect the on-shell metrics (2.1), given that the infinitesimal transformations are (2.30). We show in appendix A.1 that, under the action of a gauge transformation (f, p), the functions η(u) and a(u) are mapped on new functionsη andã given bỹ It is easily verified that these transformations reduce to (2.30) upon taking f (u) = u+ t(u), replacing p(u) by p(u) and expanding to first order in . This formula shows explicitly that the phase space of the theory is empty, as any diffeomorphism f such that maps a(u) onã = 0. When combining this map with a suitable translation p(u), the whole metric (2.1) is mapped on that of Minkoswki space, so that indeed any solution is pure gauge. Note that the inhomogeneous term proportional to f /f in the transformation law of a is crucial in order for the latter statement to be true. In principle one can impose suitable fall-off conditions on the functions a(u) and η(u) at future and past infinity, and study the subgroup of (3.3) that preserves these conditions. For example, a(u) ∼ const.+O(1/|u|) would include Rindler spacetime, potentially leading to interesting asymptotic symmetries at |u| → +∞. We will not follow this approach here; instead, we will try to make the theory non-trivial by using an unconventional prescription for the definition of surface charges.

Quasi-Rindler currents and charges
In the previous subsection we interpreted asymptotic symmetries as gauge symmetries, in accordance with the fact that all surface charges written as integrals over a space-like slice at infinity vanish. However, another interpretation is available: instead of integrating (3.1) over x, we may decide to integrate it over retarded time u. Despite clashing with the usual Hamiltonian formalism, this approach is indeed the most natural one suggested by the u-dependent asymptotic Killing vector fields (2.21) and the solutions (2.1).
For convenience we will also assume that the coordinate u is periodic as in (2.23). This condition introduces closed time-like curves and breaks Poincaré symmetry (even when a = 0 !); it sets off our departure from the world of Rindler to that of quasi-Rindler holography. While it seems unnatural from a gravitational/spacetime perspective, this choice is naturally suggested by our asymptotic symmetries and our phase space. In the remainder of this paper we explore its consequences, assuming in particular that the functions t(u), p(u), η(u) and a(u) are 2πL-periodic. This will in fact lead us to study new aspects of warped conformal field theories, which we believe are interesting in their own right.
In the quasi-Rindler case, the variation of the surface charge associated with the symmetry transformation (t, p), evaluated on the metric (a, η), reads (3.8) Using (3.1) and inserting expressions (2.12) and (2.28) yields where so that the charges are finite and integrable: This expression shows in particular that the space of solutions (a, η) is dual to the asymptotic symmetry algebra 5 [18,[49][50][51]. More precisely, the pair of functions (T (u), P (u)) transforms under the coadjoint representation of the asymptotic symmetry group, with T (u) dual to time reparametrizations and P (u) dual to translations of x. This observation will be crucial when determining the transformation law of T (u) and P (u) under finite asymptotic symmetry transformations, which in turn will lead to a Cardy-like entropy formula. From the variations (2.30) of the functions (a, η), we deduce corresponding variations of the functions T and P in (3.10): This result contains all the information about the surface charge algebra, including its central extensions. On account of 2πL-periodicity in the retarded time u, we can introduce the Fourier-mode generators with κ = k. As it must be for a consistent theory, this algebra coincides with the Lie bracket algebra (2.25) of allowed diffeomorphisms, up to central extensions. Note that T 0 /L, being the charge that generates time translations u → u + const, should be interpreted as the Hamiltonian, while P 0 is the momentum operator (it generates translations x → x + const). The only central extension in (3.14) is a twist term in the mixed commutator; it is a non-trivial 2-cocycle [52]. In particular, it cannot be removed by redefinitions of generators since the u(1) current algebra has a vanishing level.

Warped Virasoro group and coadjoint representation
As a preliminary step towards the computation of quasi-Rindler entropy, we now work out the finite transformation laws of the functions T and P . For the sake of generality, we will display the result for arbitrary central extensions of the warped Virasoro group, including a Virasoro central charge and a u(1) level. We will use a 2πL-periodic coordinate u, but our end result (3.17)-(3.18) actually holds independently of that assumption.

Finite transformations of the stress tensor
The asymptotic symmetry group for quasi-Rindler holography is (3.3) with R replaced by S 1 , and it consists of pairs (f, p), where p is an arbitrary function on the circle and f is a diffeomorphism of the circle; in particular, (For instance, the diffeomorphisms defined by (3.7) are generally forbidden once that condition is imposed.) However, in order to accommodate for inhomogeneous terms such as those appearing in the infinitesimal transformations (2.30), we actually need to study the central extension of this group. We will call this central extension the warped Virasoro group, and we will denote it byĜ. Its Lie algebra reads and is thus an extension of (3.14) with a Virasoro central charge c and a u(1) level K. Note that, when K = 0, the central term in the mixed commutator [T, P ] can be removed by defining L n ≡ T n + iκ K nP n . In terms of generators L n and P n , the algebra takes the form (3.16) without central term in the mixed bracket, and with a new Virasoro central charge c = c − 12κ 2 /K. [We will see an illustration of this in eq. (5.11), in the context of quasi-Rindler gravity in AdS 3 .] But when K = 0 as in (3.14), there is no such redefinition.
We relegate to appendix A.1 the exact definition of the warped Virasoro groupĜ, together with computations related to its adjoint and coadjoint representations. Here we simply state the result that is important for us, namely the finite transformation law of the stress tensor T and the u(1) current P . By construction, these transformations coincide with the coadjoint representation ofĜ, written in eqs. (A.21)-(A.22); thus, under a finite transformation (3.2), the pair (T, P ) is mapped to a new pair (T ,P ) with where {f ; u} is the Schwarzian derivative (A.10) of f . These transformations extend those of a standard warped CFT [39], which are recovered for κ = 0. In the case of quasi-Rindler spacetimes, we have c = K = 0 and κ is non-zero, leading tõ which (for κ = k) actually follows from (3.6) and the definition (3.10). Note that these formulas are valid regardless of whether u is periodic or not! In the latter case, f (u) is a diffeomorphism of the real line.

Modified Sugawara construction
Before going further, we note the following: since P (u) is a Kac-Moody current, one expects that a (possibly modified) Sugawara construction might convert some quadratic combination of P 's into a CFT stress tensor. This expectation is compatible with the fact that P (u) du is a one-form, so that P (u) du ⊗ P (u) du is a quadratic density. Let us therefore define and ask how it transforms under the action of (f, p), given that the transformation law of P (u) is (3.20). Writing this transformation as M −→ M, the result is which is the transformation law of a CFT stress tensor with central charge Once more, this observation is independent of whether the coordinate u is periodic or not. This construction is at the core of a simple relation between the quasi-Rindler symmetry algebra (3.14) and the BMS 3 algebra. Indeed, by quadratically recombining the generators P n thanks to a twisted Sugawara construction the brackets (3.14) reproduce the centrally extended BMS 3 algebra [8]: This apparent coincidence implies that the representations of the warped Virasoro group (with vanishing Kac-Moody level) are related to those of the BMS 3 group, but one should keep in mind that the similarity of group structures does not imply similarity of the physics involved; in particular, the Hamiltonian operator in (3.25) is (proportional to) T 0 , while the standard BMS 3 Hamiltonian is M 0 . Nevertheless, unitary representations of the warped Virasoro group with vanishing Kac-Moody level K can indeed be studied and classified along the same lines as for BMS 3 [18]. As we do not use these results in the present work we relegate this discussion to appendix A.2.

Quasi-Rindler thermodynamics
In this section we study quasi-Rindler thermodynamics, both microscopically and macroscopically, assuming throughout that surface charges are defined as integrals over time and that the coordinate u is 2πL-periodic. We start in subsection 4.1 with a microscopic, Cardy-inspired derivation of the entropy of zero-mode solutions. Section 4.2 is devoted to certain geometric aspects of boosted Rindler spacetimes, for instance their global Killing vectors, which has consequences for our analytic continuation to Euclidean signature in subsection 4.3. In subsection 4.4 we evaluate the on-shell action to determine the free energy, from which we then derive other thermodynamic quantities of interest, such as macroscopic quasi-Rindler entropy. In particular, we exhibit the matching between the Cardy-based computation and the purely gravitational one.

Modular invariance and microscopic entropy
Here, following [39], we switch on chemical potentials (temperature and velocity) and compute the partition function in the high-temperature limit, assuming the validity of a suitable version of modular invariance. Because the u(1) level vanishes in the present case, the modular transformations will not be anomalous, in contrast to standard behaviour in warped CFT [39]. (This will no longer be true in AdS 3 -see section 5.) The grand canonical partition function of a theory at temperature 1/β and velocity η is where H and P are the Hamiltonian and momentum operators (respectively), and the trace is taken over the Hilbert space of the system. In the present case, the Hamiltonian is the (quantization of the) charge (3.11) associated with t(u) = 1 and p(u) = 0, i.e. the zero-mode of T (u) up to normalization: As for the momentum operator, it is the zero-mode of P (u) (again, up to normalization). If we denote by I the Euclidean action of the system, the partition function (4.1) can be computed as the integral of e −I+η dτ P over paths in phase space that are periodic in Euclidean time τ with period β. Equivalently, if we assume that the phase space contains one Lagrange variable at each point of space (i.e. that we are dealing with a field theory), the partition function may be seen as a path integral of e −I over fields φ that satisfy φ(τ + β, x) = φ(τ, x + iβη) since P is the generator of translations along x. Note that both approaches require the combination H − ηP to be bounded from below; in typical cases (such as AdS, where P is really an angular momentum operator and η is an angular velocity), this is a restriction on the allowed velocities of the system. Now, our goal is to find an asymptotic expression for the partition function at high temperature. To do this, we will devise a notion of modular invariance (actually only S-invariance), recalling that the symmetries of our theory are transformations of (the complexification of) S 1 × R of the form (3.2). Seeing the partition function (4.1) as a path integral, the variables that are integrated out live on a plane R 2 spanned by coordinates u and x subject to the identifications While the transformations (4.4) do not belong to the group of finite asymptotic symmetry transformations (3.2), their analogues in the case of CFT's, BMS 3 -invariant theories and warped CFT's [14,39] apparently lead to the correct entropy formulas. We shall assume that the same is true here, which implies that the partition function Z(β, η) satisfies a property analogous to self-reciprocity, This, in turn, gives the asymptotic formula where H vac and P vac are respectively the energy and momentum of the vacuum state. These values can be obtained from the finite transformations (3.19) [with T = P = p = 0] and (3.20) [with P = 0] by considering the map f (u) = Le inu/L with some integer n and declaring that the vacuum value of the functions T (u) and P (u) is zero, exactly like for the map between the plane and the cylinder in a CFT. Accordingly, the vacuum values of these functions "on the cylinder", sayT vac andP vac , arẽ The asymptotic expression (4.7) of the partition function thus becomes where the sign of the dominant vacuum value in (4.9) is determined by the sign of η. (More precisely, the vacuum ±iκ is selected when sign(η) = ∓1.) The free energy F ≡ −T log Z is given by F ≈ −2πκL |η|T (4.11) at high temperature, and the corresponding entropy is In subsection 4.4 we will see that this result exactly matches that of a macroscopic (i.e. purely gravitational) computation. Before doing so, we study Euclidean quasi-Rindler spacetimes and elucidate the origin of the vacuum configuration (4.8).

Boosted Rindler spacetimes and their Killing vectors
Any solution of 3-dimensional Einstein gravity is locally flat and therefore locally has six Killing vector fields. However, these vector fields may not exist globally. We now discuss global properties of the Killing vectors of the geometry defined by the line-element (2.1) with the identification (2.23). For simplicity, we present our results only for zero-mode solutions, a, η = const. The six local Killing vector fields are This algebra is isomorphic to isl(2), as displayed in (2.4), with the identifications M 0 = ξ 2 , In terms of the generators t n , p n of the asymptotic Lie bracket algebra (2.25) we have the identifications t 0 ∼ ξ 1 , p 0 ∼ ξ 2 , t 1 ∼ ξ 4 and p −1 ∼ ξ 5 ; the vector field ξ 3 generates trivial symmetries, while the Killing vector ξ 6 is not an asymptotic Killing vector, as it is incompatible with the asymptotic behavior (2.21). This shows in particular that the boundary conditions of quasi-Rindler gravity actually break Poincaré symmetry even when the coordinate u is not periodic. Interestingly, the four generators t 0 , t 1 , p −1 , p 0 obey the harmonic oscillator algebra where the Hamiltonian is formally given by t 0 = H = α † α+const., the annihilation/creation operators formally by t 1 = α, p −1 = α † and p 0 = z commutes with the other three generators. The algebra is written here in terms of Poisson brackets, but becomes the standard harmonic oscillator algebra after quantization. Note, however, that t 1 and p −1 are not generally adjoint to each other. 6 In the canonical realization (3.14) the first commutator acquires an important contribution from the central extension If we wish to identify the vacuum as the most symmetric solution then our vacuum spacetime takes the form with some non-zero integer n. We shall demonstrate in subsection 4.3 that |n| = 1 is the only choice consistent with the u-periodicity (2.23). Thus, we have uncovered yet another unusual feature of quasi-Rindler holography: the vacuum metric (4.18) with |n| = 1 is complex. Our vacuum is neither flat spacetime (as one might have guessed naively) nor a specific Rindler spacetime, but instead it is a Rindler-like spacetime with a specific imaginary Rindler "acceleration", the value of which depends on the choice of the periodicity L in (2.23). Another way to see that the solution a = η = 0 is not maximally symmetric for finite L is to consider the six local Killing vectors ξ (0) i in the limit a, η → 0: Only four of them can be defined globally, because ξ 4 and ξ 5 have a linear dependence on u that is incompatible with finite 2πL-periodicity (2.23).
While the vacuum metric (4.18) (with n = 1) is preserved by all six Killing vectors (4.13c), at most three of the associated generators of the asymptotic symmetry algebra can annihilate the vacuum; since we expect from the discussion in subsection 4.1 that P 0 is non-zero we pick the vacuum by demanding that T 0 , T 1 and P −1 annihilate it. The generator P 0 then acquires a non-zero eigenvalue due to the central term in (4.17), while the remaining two Killing vectors are excluded for reasons stated above (one acts trivially, while the other violates the boundary conditions for asymptotic Killing vectors). In fact, upon performing a shift P 0 → P 0 − iκ, the symmetry algebra (3.14) becomes so that the vacuum is now manifestly invariant under T 0 , T 1 , P 0 and P −1 . These considerations reproduce the result (4.9) and thereby provide a consistency check. The fact that the eigenvalue of P 0 is imaginary indicates that the warped Virasoro group is represented in a non-unitary way if the assumption of self-reciprocity (4.6) holds.

Euclidean boosted Rindler
In order to prepare the ground for thermodynamics, we now study the Euclidean version of the metrics (2.1). We consider only zero-mode solutions for simplicity. Then, defining new coordinates For non-zero Rindler acceleration, a = 0, there is a Killing horizon at ρ h = 0, or equivalently at The patch ρ > 0 coincides with the usual Rindler patch for positive Rindler acceleration a. For negative a the patches ρ > 0 and ρ < 0 switch their roles. We assume positive a in this work so that τ is a timelike coordinate in the limit r → ∞. The Euclidean section of the metric (4.23) is obtained by defining which yields the line-element Demanding the absence of a conical singularity at ρ = 0 and compatibility with (2.23) leads to the periodicities Given the periodicities (4.27), we can now ask which values of n in (4.14) give rise to a regular spacetime with metric (4.18). Consider the Euclidean line-element (4.26) and define another analytic continuation, which yields with the periodicities (τ, y) ∼ (τ + iβ, y − iβη) ∼ (τ + 2πL, y) . The point now is that the Euclidean line-element (4.30) with the periodicities (4.31) has a conical singularity atρ = 0 unless |n| = 1. Thus, we conclude that the vacuum spacetime (in the sense of being singularity-free and maximally symmetric) is given by (4.18) with |n| = 1, confirming our discussion in sections 4.1 and 4.2.

Macroscopic free energy and entropy
The saddle point approximation of the Euclidean path integral leads to the Euclidean partition function, which in turn yields the free energy. The latter is given by temperature times the Euclidean action (2.20) evaluated on-shell: where we have inserted the periodicities from subsection 4.3 and used √ γK = a + O(1/ρ).
The absolute value for η was introduced in order to ensure a positive volume form 7 for positive L and β. From this result we extract the entropy which coincides with the Cardy-based result (4.12). As a cross-check, we derive the same expression in the Chern-Simons formulation by analogy to the flat space results [23]: We show in the next section that the same matching occurs in Rindler-AdS.

Boosted Rindler-AdS
In this section we generalize the discussion of the previous pages to the case of Rindler-AdS spacetimes. In subsection 5.1 we establish quasi-Rindler-AdS boundary conditions and show that the asymptotic symmetry algebra can be untwisted to yield a standard warped CFT algebra, with a u(1) level that vanishes in the limit of infinite AdS radius. Then, in subsection 5.2 we derive the entropy microscopically, and we show in subsection 5.3 that the same result can be obtained macroscopically.
(5.1) 7 One pragmatic way to get the correct factors of i is to insert the Euclidean periodicities in the ranges of the integrals and to demand again positive volume when integrating the function 1. In the flat space calculation this implies integrating u from 0 to β, while here it implies integrating u from 0 to −i2πL.
Starting from this ansatz, we now adapt our earlier discussion to the case of a nonvanishing cosmological constant. Since the computations are very similar to those of the quasi-Rindler case, we will simply point out the changes that arise due to the finite AdS radius. When we do not mention a result explicitly we imply that it is the same as for the flat configuration; in particular we assume again 2πL-periodicity in u.
The Chern-Simons formulation is based on the deformation of the isl(2) algebra to so(2, 2), where the translation generators no longer commute so that the last bracket in (2.4) is replaced by The on-shell connection (2.10)-(2.12) and the asymptotic symmetry generators (2.27)-(2.29) are modified as The connection A changes correspondingly as compared to (2.13), In particular, the limit → ∞ reproduces the algebra (3.14). At finite the presence of the non-vanishing level in (5.10) enables us to remove the central extension of the mixed bracket of (3.14b) thanks to a twist L n = T n − i κ 2k nP n (5.11) in terms of which the asymptotic symmetry algebra reads with the expected Brown-Henneaux central charge 8 [4]

Microscopic quasi-Rindler-AdS entropy
As in the case of a vanishing cosmological constant, it is possible to derive a Cardylike entropy formula that can be applied to zero-mode solutions. The only difference with respect to subsection 4.1 is the non-vanishing u(1) level K = −2k/ that leads to a slightly different form of modular invariance. Namely, according to [39], the self-reciprocity of the partition function, eq. (4.6), now becomes Z(β,η) = e βKLη 2 /2 Z (2πL) 2 β , iβη 2πL (5.14) leading to the high-temperature free energy (We have renamed the chemical potential conjugate to P 0 asη for reasons that will become clear in subsection 5.3.) This is the same as in (4.7), up to a temperature-independent constant proportional to the u(1) level. The vacuum values of the Hamiltonian and the momentum operator are once more given by the arguments above (4.8); in particular, the u(1) level plays no role for these values. Accordingly, the free energy at high temperature T = β −1 1 boils down to F ≈ −2πLk|η|T + kLη 2 / (5. 16) and the corresponding entropy is again given by (4.12): Notably, this is independent of the AdS radius. The same result can be obtained by absorbing the twist central charge through the redefinition (5.11) and then using the Cardy-like entropy formula derived in [39]. In the next subsection we show that this result coincides with the gravitational entropy, as in the flat quasi-Rindler case discussed previously.

Macroscopic quasi-Rindler-AdS entropy
Generalizing the macroscopic calculations from section 4 for zero-mode solutions (5.3) with constant a and η we find that the outermost Killing horizon is located at and has a smooth limit to the quasi-Rindler result (4.24) for infinite AdS radius → ∞. We assume η > −a /4 so that r h is real and surface-gravity is non-zero. The vacuum spacetime reads where again we defined "vacuum" as the unique spacetime compatible with our boundary conditions, regularity and maximal symmetry.
Making a similar analytic continuation as in subsection 4.3 we obtain the line-element with the Killing norm K(r) = 4r 2 2 + 4ηr + 2ar + η 2 (5.21) and the periodic identifications where inverse temperature β = T −1 and boost parameterη are given by In particular, the chemical potentialη no longer coincides with the parameter η appearing in the metric. Note that the limit of infinite AdS radius is smooth and leads to the expressions in subsection 4.3 for line-element, periodicities, temperature and boost parameter. Converting the zero-mode solution (5.20)-(5.23) into the Chern-Simons formulation and using formula (4.34) then yields the entropy where independence of the AdS radius follows from the fact that the connection given by (5.4a) has no non-zero component along any of the M n 's. This agrees with the microscopic result (5.17). The macroscopic free energy compatible with the first law dF = −S dT − P 0 dη is given by where H is given by the zero mode charge T = aη through (4.2), H = kLaη = −F , and P 0 is given by the zero mode charge P = a through the right eq. (3.13), P 0 = kLa. The result (5.25) also coincides with the microscopic one (5.16).

Rindlerwahnsinn?
In this final section 9 we highlight some of the unusual features that we unraveled in our quest for near horizon holography. We add some comments, explanations and possible resolutions of the open issues.

Retarded choices?
Let us summarize and discuss aspects of the dependence of Rindler acceleration on retarded time. (Note that our whole paper can easily be sign-flipped to advanced time v, which may be useful in some applications.) Rindler acceleration depends on retarded time. We started with the ansatz (1.2) since we wanted a state-dependent Rindler acceleration to accommodate a state-dependent temperature.
We left it open whether Rindler acceleration a was a function of retarded time u, spatial coordinate x or both. The Einstein equations forced us to conclude that Rindler acceleration can depend on retarded time only. We give now a physical reason why this should be expected. Namely, if the zeroth law of black hole mechanics holds then surface gravity (and thus Rindler acceleration) must be constant along the horizon. In particular, it cannot depend on x. If the horizon changes, e.g. due to emission of Hawking quanta or absorption of matter, then Rindler acceleration can change, which makes the dependence on u natural, much like the corresponding dependence of Bondi mass on the lightlike time.
Retarded time is periodic. While many of our results are actually independent of the choice (2.23), it was still a useful assumption for several purposes, e.g. the introduction of Fourier modes. For some physical observables it is possible to remove this assumption by taking the limit L → ∞. We shall provide an important example in section 6.2 below. 9 The play on words in this section title is evident for German speaking physicists; for the remaining ones we point out that "Rinderwahnsinn" means "mad cow disease" and "Wahnsinn" means "madness".
Boundary currents are integrated over retarded time. If we wanted our theory to be non-empty we could not use the standard definition of canonical charges integrated over space, but instead had to consider boundary currents integrated over retarded time. We have no further comments on this issue, except for pointing out that in four dimensions, a bilinear in the Bondi news is integrated over retarded time in order to yield the ADM mass. Thus, despite of the clash with the usual Hamilton formulation we believe that we have made here the most natural choice, given our starting point.

Rindler entropy?
Let us finally take a step back and try to connect with our original aim of setting up Rindler holography and microscopically calculating the Rindler entropy [28]. We summarize in appendix B results for Rindler thermodynamics and Rindler entropy, which match the near horizon results of BTZ black holes and flat space cosmologies. We consider a limiting procedure starting with our result for entropy (4.33). We are interested in a limit where simultaneously the compactification length L in (2.23) tends to infinity, the boost parameter η tends to zero, the length of the spatial cycle x appears in the entropy and all unusual factors of i are multiplied by something infinitesimal. In other words, we try to construct a limit towards Rindler entropy (B.12).
Consider the identifications (4.3) with a complexified β → β 0 + 2πiL and split them into real and imaginary parts: The rationale behind this shift is that the real part of the periodicities untwists. As in appendix B we call the (real) length of the x-cycleL and thus have the relatioñ L = 2πLη . (6.2) Therefore, taking the decompactification limit for retarded time, L → ∞, while keeping fixedL simultaneously achieves the desired η → 0, so that the periodicities (6.1) in this limit simplify to which, if interpreted as independent periodicities, are standard relations for non-rotating horizons at inverse temperature β 0 and with a length of the spatial cycle given byL. Apart from taking limits our only manipulation was to shift the inverse temperature β in the complex plane. Thus, any observable that is independent from temperature should remain unaffected by such a shift; moreover, the "compactification" of β along the imaginary axis is then undone by taking the decompactification limit L → ∞. We conclude from this that entropy S from (4.33) should have a smooth limit under all the manipulations above and hopefully yield the Rindler result (B.12). This is indeed the case: Thus, we recover the usual Bekenstein-Hawking entropy law as expected from Rindler holography. In this work we have also provided a microscopic, Cardy-like derivation of this result. A different singular limit was considered in [53], where the Rindler entropy (6.4) was derived from a Cardy formula for holographic hyperscaling theories.

Other approaches
Relation to BMS/ultrarelativistic CFT. Using our quadratic map (3.24) of warped CFT generators to BMS 3 generators together with the result (3.23) for the central charge, we now check what microstate counting would be given by an ultrarelativistic CFT (or equivalently a Galilean CFT) [14]. Using the "angular momentum" h L = aη, the "mass" h M = a 2 /(2k), the central charge c M = 12k with k = 1/(4G N ), and introducing an extra factor of L to accommodate our periodicity (2.23), the ultrarelativistic Cardy formula gives This entropy thus coincides with the warped CFT entropy (4.12), and matches the gravity result (4.33).
Other Rindler-type boundary conditions. While finishing this work the paper [54] appeared which proposes alternative Rindler boundary conditions, motivated partly by Hawking's claim that the information loss paradox can be resolved by considering the supertranslation of the horizon caused by ingoing particles [55]. 10 (See also [57,58].) In [54] the state-dependent functions depend on the spatial coordinate and thus allow for standard canonical charges. The corresponding Rindler acceleration (and thus temperature) is state-independent and the asymptotic symmetry algebra has no central extension. We checked that the Rindler acceleration of that paper can be made state-dependent, but in accordance with our discussion in section 6.1 it cannot depend on the spatial coordinate; only dependence on retarded/advanced time is possible. Thus, we believe that if one wants to allow for a state-dependent temperature in Rindler holography the path described in the present work is unavoidable.
Generalizations. We finish by mentioning a couple of interesting generalizations that should allow in several cases straightforward applications of our results, like generalizations to higher derivative theories of (massive or partially massless) gravity, theories that include matter and theories in higher dimensions. In particular, it would be interesting to generalize our discussion to topologically massive gravity [59] in order to see how the entropy computation would be affected.

A On representations of the warped Virasoro group
This appendix is devoted to certain mathematical considerations regarding the warped Virasoro group. They are motivated by the questions encountered in section 3 of this paper, although they are also interesting in their own right. First, in subsection A.1 we study the coadjoint representation of the warped Virasoro group, which is needed to derive formulas (3.17)-(3.18) for the transformation law of the stress tensor. Then, in subsection A.2 we classify all irreducible, unitary representations of this group with vanishing u(1) level using the method of induced representations. We assume throughout that the coordinate u is 2πL-periodic.

A.1 Coadjoint representation
We call warped Virasoro group, denotedĜ, the general central extension of the group where the notation is the same as in (3.3) up to the replacement of R by S 1 [so that in particular f satisfies property (3.15)]. In this subsection we display an explicit definition of G and work out its adjoint and coadjoint representations, using a 2πL-periodic coordinate u to parametrize the circle. We refer to [52,60] for more details on the Virasoro group and its cohomology.
Since the differentiable, real-valued second cohomology space of G is three-dimensional [52,61], there are exactly three central extensions to be taken into account when defininĝ G; in other words,Ĝ ∼ = G × R 3 as manifolds. Accordingly, elements ofĜ are pairs (f, p) belonging to G, supplemented by triples of real numbers (λ, µ, ν). The group operation in G is where σ is the action (3.4) while B, C and D are non-trivial 2-cocycles on G given explicitly by In particular, B is the standard Bott-Thurston cocycle [60] defining the Virasoro group.

Adjoint representation and Lie brackets
To write down an explicit formula for the coadjoint representation of the warped Virasoro group, we first need to work out the adjoint representation, which acts on the Lie algebrâ g ofĜ. As follows from the definition ofĜ, that algebra consists of 5-tuples (t, p; λ, µ, ν) where t = t(u) ∂ ∂u is a vector field on the circle, p = p(u) is a function on the circle, and λ, µ, ν are real numbers. The adjoint representation ofĜ, which we will denote as Ad, is then defined as where e t is to be understood as an infinitesimal diffeomorphism e t (u) = u+ t(u)+O( 2 ). Given the group operation (A.2), it is easy to verify that the central terms λ 1 , µ 1 and ν 1 play a passive role, so we may simply set them to zero and write Ad (f,p;λ,µ,ν) ≡ Ad (f,p) . Using multiple Taylor expansions and integrations by parts, the right-hand side of (A.6) yields and the central charges the Lie brackets defined by (A.14) turn out to read Here we recognize the centrally extended algebra (3.16), up to the fact that the central charges Z i are written as operators; eventually they will be multiples of the identity, with coefficients c, κ and K corresponding to Z 1 , Z 2 and Z 3 respectively.

Coadjoint representation
The coadjoint representation ofĜ is the dual of the adjoint representation, and coincides with the finite transformation laws of the functions T and P introduced in where Ad * (f,p) T and Ad * (f,p) P are a quadratic density and a one-form on the circle (respectively) whose components, evaluated at f (u), are and Ad These are the transformation laws displayed in (3.17)-(3.18), with Ad * (f,p) T ≡T and Ad * (f,p) P ≡P . They reduce to the transformations of a standard warped CFT [39] for κ = 0. In the Rindler case, however, c = K = 0 and κ = k is non-zero.
The transformation law of the function η(u) in (3.6) under finite asymptotic symmetry transformations can be worked out in a much simpler way. Indeed, it is easily verifed that the left formula in (3.6) reproduces (2.30) for infinitesimal transformations, and is compatible with the group operation (3.5) of the warped Virasoro group. One can also check that the transformation laws (3.19) follow from (3.6) and the definition (3.10).

A.2 Induced representations
As indicated by the imaginary vacuum values (4.9) (which are actually fairly common in the world of warped CFT's [39]), the asymptotic symmetry group is not represented in a unitary way in quasi-Rindler holography. Nevertheless, since the standard interpretation of symmetries in quantum mechanics requires unitarity [63], it is illuminating to study unitary representations of the warped Virasoro group. Here we classify such representations for the case of vanishing Kac-Moody level K, but non-vanishing twist κ. As in the case of the Euclidean, Poincaré or BMS groups, the semi-direct product structure (A.1) [or similarly (3.9)] is crucial; indeed, all irreducible unitary representations of such a group are inducedà la Wigner [64,65]. We refer to [66] for more details on induced representations and we mostly use the notations of [18,19].

A lightning review of induced representations
The construction of induced representations of the warped Virasoro groupĜ with vanishing u(1) level follows the same steps as for the Poincaré group [65,67] or the BMS 3 group [18]. One begins by identifying the dual space of the Abelian group C ∞ (S 1 ), which in the present case consists of currents P (u) du. [The elements of this dual space are typically called "momenta", and our notation P (u) is consistent with that terminology.] One then defines the orbit O P and the little group G P of P = P (u) du as and G P ≡ f ∈ Diff + (S 1 ) | f · P = P , (A. 23) where the action of f on P is given by (3.20). Then, given an orbit O P and an irreducible, unitary representation R of its little group G P in some Hilbert space E, the corresponding induced representation T acts on square-integrable wave functions Ψ : O P → E : q → Ψ(q) according to [66] ( where (f, p) belongs to Diff + (S 1 ) C ∞ (S 1 ). Let us briefly explain the terms of this equation: • The real, positive function ρ f on O P denotes the Radon-Nikodym derivative of the measure used to define the scalar product of wavefunctions. It is an "anomaly" that takes the value ρ f = 1 for all f when the measure is invariant, but otherwise depends on f and on the point q at which it is evaluated. In simple cases (e.g. the Poincaré group), the measure is invariant and ρ f (q) = 1 for all f and all q ∈ O P .
• The operator R g −1 q f g f −1 ·q is a "Wigner rotation": it is the transformation corresponding to f in the space of spin degrees of freedom of the representation T . We denote by g q the "standard boost" associated with q, that is, a group element such that g q · P = q. For scalar representations, R is trivial and one may simply forget about the Wigner rotation.

Induced representations ofĜ
Our goal now is to classify all irreducible, unitary representations of the warped Virasoro group with vanishing Kac-Moody level, under the assumption that there exists a quasi-invariant measure on all the orbits (see [68] for the construction of such measures). According to the lightning review just displayed, this amounts to the classification of orbits, as defined in (A.23). We start with two preliminary observations: 1. For any constant current P (u) = P 0 = const, the little group G P consists of rigid time translations f (u) = u + u 0 .

The charge Q[P ] defined as
is constant along any coadjoint orbit of the warped Virasoro group, regardless of the values of the central charges c, κ and K. In other words, for any current P , any (orientation-preserving) diffeomorphism f of the circle and any function p, the zero-mode of P is left invariant by the coadjoint action: where Ad * (f,p) P is given by (3.18). This result holds, in particular, in the case c = K = 0 that we wish to study, and corresponds physically to the fact that the average value of acceleration is invariant under asymptotic symmetries.
The proof of both results is straightforward, as they can be verified by brute force. In fact, they follow from a stronger statement: it turns out that the orbits O P foliate the space of currents into hyperplanes of constant Q[P ], so that any current P (u) can be brought to a constant by acting with a diffeomorphism. To prove this, note that constancy of Q[P ] implies that that constant, if it exists, coincides with the zero-mode P 0 of P (u). The question thus boils down to whether or not there exists an orientation-preserving diffeomorphism f such that This condition is equivalent to an inhomogeneous first-order differential equation for 1/f , whose solution is where A is a real parameter. Since u is assumed to be 2πL-periodic, this function must be 2πL-periodic as well. This selects a unique solution (and such a solution exists for any value of P 0 ), meaning that, for any P (u), there always exists a diffeomorphism f of the circle such that f · P be a constant 12 ; furthermore, that diffeomorphism is uniquely specified by P (u) up to a rigid time translation. This proves that all orbits O P are hyperplanes specified by the value of the charge (A.25), in accordance with the fact that P 0 is a Casimir operator in (3.16). One can then apply the usual machinery of induced representations to the warped Virasoro group, and compute, for instance, the associated characters along the lines of [19]; however, as the interpretation of these characters in the present context is unclear, we refrain from displaying them. 12 An alternative way to prove the same result is to recall the modified Sugawara construction (3.21), by which a coadjoint orbit of the Virasoro group with non-negative energy is associated with each orbit OP . Since the only Virasoro orbits with non-negative energy are orbits of constants, we know that there always exists a diffeomorphism f that brings a given Sugawara stress tensor (3.21) into a constant, which in turn brings the corresponding current P to a constant.

Physical properties of unitary representations
We have just seen that unitary representations of the warped Virasoro group [with vanishing u(1) level] can be classified according to the possible values of the Casimir operator P 0 in (3.16). Accordingly, the orbits (A.23) are affine hyperplanes with constant zero-modes embedded in the space of currents P (u). In particular, each orbit contains exactly one constant representative. The physical meaning of this statement is that any (generally time-dependent) acceleration a(u) can be brought to a constant, a 0 , by using a suitable reparametrization of time that preserves 2πL-periodicity. Furthermore, a 0 coincides with the Fourier zero-mode of a(u). Note that, with the requirement of 2πL-periodicity, it is no longer true that any time-dependent acceleration a(u) can be mapped onã = 0 because the diffeomorphisms defined by (3.7) generally do not preserve that requirement.
Having classified the orbits, we know, in principle, the irreducible unitary representations of the warped Virasoro group at vanishing Kac-Moody level. In three spacetime dimensions, these representations describe the Hilbert space of metric fluctuations around the background specified by the orbit O P and the representation R [18,20,49,69], as follows from the fact that the phase space coincides with the coadjoint representation of the asymptotic symmetry group. For instance, an induced representation ofĜ specified by an orbit O P and the trivial representation of the little group gives the Hilbert space of metric fluctuations around the background Here u is still understood as a 2πL-periodic coordinate; in particular, the solution at a = 0 is not Minkowski spacetime because of that identification. Note that, in any unitary representation of the type just described, the eigenvalues of the Hamiltonian T 0 are unbounded from below (and from above). There is thus a trade-off between unitarity and boundedness of energy: if we insist that the representation be unitary, then it is a (direct integral of) induced representation(s), and energy is unbounded both from below and from above; conversely, if we insist that energy be bounded from below, then the asymptotic symmetry group cannot act unitarily on the Hilbert space of the putative dual theory. This property has actually been observed in representations of the Galilean Conformal Algebra in two dimensions, gca 2 , and its higher-spin extensions [17,70]. Indeed, when T 0 is interpreted as the Hamiltonian, demanding that energy be bounded from below amounts to considering highest-weight representations of the symmetry algebras (3.14) or (3.25), the highest weight being the lowest eigenvalue of T 0 in the space of the representation. This representation is easily seen to be non-unitary when the central charge of the mixed commutator is non-zero. We stress, however, that in the more common interpretation of the warped Virasoro group [39] where P 0 plays the role of the Hamiltonian, there is no such conflict between unitarity and boundedness of energy.

B Rindler thermodynamics
We have not found a holographic setup that leads to Rindler thermodynamics, but it is still of interest in its own right to consider it. In this appendix we describe Rindler thermodynamics in a non-holographic context and show that it recovers the near horizon thermodynamics of BTZ black holes and flat space cosmologies, in the sense that temperatures and entropies agree with each other. If a consistent version of Rindler holography exists, it should reproduce the results of this appendix (see also [28]).
The main change as compared to the main text is that the periodicities in the Euclidean coordinates are no longer given by (4.27), but instead by (t E , y) ∼ (t E + β, y − iβη) ∼ (t E , y +L) (B.1) whereL is now the periodicity of the spatial coordinate y and β, η coincide with the quasi-Rindler parameters. So in Rindler thermodynamics we do not identify retarded time periodically, which is the key difference to quasi-Rindler thermodynamics.

B.1 Rindler horizon and temperature
The Euclidean metric (4.26) has a center at ρ = 0, corresponding to the Rindler horizon in Lorentzian signature. In order for this center to be smooth, the Euclidean time t E ∼ t E + β has to have a periodicity β = 2π/a. Interpreting the inverse of this periodicity as temperature yields the expected Unruh temperature [27] T = a 2π . (B. 2) The same result is obtained from surface gravity. Note that the Unruh temperature (B.2) is independent of the boost parameter η.
There is yet another way to determine the Unruh temperature, namely starting from rotating (non-extremal) BTZ and taking the near horizon limit. The rotating BTZ metric [2] ds 2 = − (r 2 − r 2 + )(r 2 − r 2 − ) 2 r 2 dt 2 + 2 r 2 (r 2 − r 2 + )(r 2 − r 2 − ) leads to a Hawking temperature Now take the near horizon limit by defining ρ = (r 2 − r 2 + )/(2r + ) and dropping higher order terms in ρ, which gives ds 2 = −2aρ dt 2 + dρ 2 2aρ + dx + η dt wherer + = r + / 2 andr − = r − / . Note that in the limit of infinite AdS radius, → ∞, we keep fixed the rescaled parametersr ± , and the coordinate x decompactifies. [ and thus coincides with the Unruh temperature (B.2). Besides verifying this expected result, the calculation above provides expressions for the Rindler parameters a and η in terms of BTZ parameters r ± , which can be useful for other consistency checks as well.
Essentially the same conclusion holds for flat space cosmologies [3], whose metric reads ds 2 = r 2 + 1 − The cosmological temperature T = r 2 + /(2πr 0 ) again coincides with the Unruh temperature (B.2), up to a sign. This sign is explained by inner horizon black hole mechanics [71].
The fact that Hawking/cosmological temperatures coincide with the Rindler temperature is not surprising but follows from kinematics. What is less clear is whether or not extensive quantities like free energy or entropy coincide as well. We calculate these quantities in the next two subsections.

B.2 Rindler free energy
Since we have no Rindler boundary conditions we do not know what the correct on-shell action is, as we have no way of checking the variational principle. However, since the zeromode solutions of quasi-Rindler holography coincide with the zero-mode solutions used in Rindler thermodynamics it is plausible that the action (2.20) can be used again. We base our discussion of free energy and entropy on this assumption.
Evaluating the full action (2.20) on-shell and multiplying it by temperature T = β −1 yields the free energy, The quantityL denotes the range of the coordinate y and physically corresponds to the horizon area. IfL tends to infinity we simply define densities of extensive quantities like free energy or entropy by dividing all such expressions byL. Insertion of the boosted Rindler metric (4.26) into the general expression for free energy (B.10) yields It is worthwhile mentioning that Rindler free energy (B.11) does not coincide with the corresponding BTZ or FSC free energy. Using the identifications (B.6) and (B.9) we find in both cases F BTZ = F FSC = −TL/(8G N ), which differs by a factor 1/2 from the Rindler result (B.11). Nevertheless, as we shall demonstrate below, the corresponding entropies do coincide.

B.3 Rindler entropy
Our result for free energy (B.11) implies that Rindler entropy obeys the Bekenstein-Hawking area law, Note that entropy does not go to zero at arbitrarily small temperature. However, in that regime one should not trust the Rindler approximation since the T → 0 limit is more adequately modelled by extremal horizons rather than non-extremal ones. The result for entropy (B.12) can be obtained within the first order formulation as well. Applying the flat space results of [23] to the present case yields Relating the Chern-Simons level with the inverse Newton constant, k = 1/(4G N ) then reproduces precisely the Bekenstein-Hawking area law (B.12). Interestingly, Rindler entropy (B.12) also follows from near horizon BTZ entropy. The latter is given by (B.14) In the last equality we identified the length of the x-interval using the last relation (B.6) together with ϕ ∼ ϕ + 2π. Thus, the near horizon BTZ entropy coincides with the Rindler entropy, which provides another consistency check on the correctness of our result. The same conclusions hold for the entropy of flat space cosmologies, In the last equality we identified the length of the x-interval using the last relation (B.9) together with ϕ ∼ ϕ + 2π.
The results above confirm that entropy is a near-horizon property, whereas free energy and the conserved charges are a property of the global spacetime.