Holography and AdS 2 gravity with a dynamical aether

Journal of High Energy Physics, Jul 2017

We study two-dimensional Einstein-aether (or equivalently Hořava-Lifshitz) gravity, which has an AdS 2 solution. We examine various properties of this solution in the context of holography. We first show that the asymptotic symmetry group is the full set of time reparametrizations, the one-dimensional conformal group. At the same time there are configurations with finite energy and temperature, which indicate a violation of the Ward identity associated with one-dimensional conformal invariance. These solutions are characterized by a universal causal horizon and we show that the associated entropy of the universal horizon scales with the logarithm of the temperature. We discuss the puzzles associated with this result and argue that the violation of the Ward identity is associated with a type of explicit breaking of time reparametrizations in the hypothetical 0 + 1 dimensional dual system.

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Holography and AdS 2 gravity with a dynamical aether

Received: May AdS2 gravity Christopher Eling 0 1 2 0 We discuss the 1 1 Keble Road , Oxford OX1 3NP , U.K 2 Rudolf Peierls Centre for Theoretical Physics, University of Oxford We study two-dimensional Einstein-aether (or equivalently Horava-Lifshitz) gravity, which has an AdS2 solution. We examine various properties of this solution in the context of holography. We rst show that the asymptotic symmetry group is the full set of time reparametrizations, the one-dimensional conformal group. At the same time there are con gurations with nite energy and temperature, which indicate a violation of the Ward identity associated with one-dimensional conformal invariance. These solutions are characterized by a universal causal horizon and we show that the associated entropy of the universal horizon scales with the logarithm of the temperature. puzzles associated with this result and argue that the violation of the Ward identity is associated with a type of explicit breaking of time reparametrizations in the hypothetical 0 + 1 dimensional dual system. 2D Gravity; Field Theories in Lower Dimensions; Holography and condensed - HJEP07(21)4 1 Introduction 2 3 4 5 6 1 Einstein-aether theory in two dimensions Asymptotic symmetry group Thermodynamics and conserved charges Algebra of charges Discussion Introduction reparametrizations of time t ! f (t), which are the asymptotic symmetries of AdS2 spacetime [1{3]. While physics in lower dimensions is simpler, in this case it appears to be too simple. One way of stating the problem is that the conformal Ward identity, which generically enforces the tracelessness of the stress-tensor, implies in one-dimension that the energy is zero since the stress tensor only has one component T tt. Thus the quantum mechanical theory has no dynamics. On the gravity side this re ected in the fact that pure Einstein gravity is trivial and the Einstein-Hilbert action in two dimensions is a topological term. An early model of a non-trivial two dimensional theory of quantum gravity was studied by Polyakov [4]. The action is given by the non-local Polyakov term which generates the trace anomaly in two-dimensions. In the conformal gauge this reduces to Liouville gravity, which is a special case of a general class of dilaton gravity theories. These theories have solutions with an AdS2 metric plus a non-trivial dilaton in the bulk. Recently, these results have been interpreted in terms of a nearly AdS2/CF T1 (N AdS2/N CF T1) correspondence (see, e.g. [5{9]). On the gravity side, the dilaton explicitly breaks the time reparametrization asymptotic symmetry. The dual description is in terms of the rst nite temperature/energy corrections away from the infrared to a type of quantum mechanical model of interacting fermions, the Sachdev-Ye-Kitaev model. This model has an emergent reparametrization invariance in the T ! 0 limit [10, 11]. { 1 { In this paper we will consider another theory of gravity in two dimensions, Einsteinaether theory [12]. In this theory the metric is coupled to a dynamical unit timelike vector eld, the \aether". One can also recast the theory into a Horava-Lifshitz form,1 which has a preferred foliation of time and therefore is invariant only under foliation preserving di eomorphisms [13]. In higher dimensions this class of theories has been studied as a potential holographic dual to strongly coupled non-relativistic eld theories invariant under Lifshitz scaling symmetries, where Lorentz invariance is broken, see e.g. [16{19]. In section II we discuss the properties of the theory in two-dimensions and show it has solution with an AdS2 metric plus a non-trivial aether pro le in the bulk. As a rst probe of holographic properties, we investigate the asymptotic symmetries of HJEP07(21)4 this solution in section III. We nd that the time reparametrizations of AdS2 are unbroken by the aether. Despite this fact, we show that there are con gurations with nite energy and temperature. In the usual case of a higher dimensional CFT, the presence of a nite temperature introduces a scale which spontaneously breaks the conformal symmetry. One nds an energy associated with the thermal state, but the Ward identity enforcing the traceless stress tensor still holds. However, in 0 + 1 dimensions there is a con ict between spontaneous breaking and the Ward identity. A nite temperature con guration has nite energy, but this is inconsistent with required vanishing T tt as described above. One can think about a nite temperature con guration in 0 + 1 dimensions as either an explicit breaking of the time reparametrization symmetry or as being an anomaly, with the temperature as the background external eld. Indeed, a \central charge" appears in a number of our results. We examine the thermodynamics associated with the AdS2 plus aether solutions in section IV. These are characterized by the presence of a universal horizon, a surface beyond which even signals of arbitrary speed cannot reach in nity. The universal horizon therefore serves as a notion of causal boundary in a non-relativistic theory of gravity. We derive a thermodynamical relation of the form E with the global timelike Killing vector T where E is the Noether charge associated eld and T is the temperature of the universal horizon. Using the thermodynamic relations, we nd there is an entropy associated with the universal horizon S ln( T ), where aspects and possible interpretations of this result. is a new cuto scale. We discuss the puzzling Finally, motivated by the violation of the time reparametrization Ward identity, in section V we study the algebra of charges associated with the asymptotic symmetries, following Brown and Henneaux [21]. Since the boundary geometry is one-dimensional (only time direction), conserved charges are simply evaluated at points and there is no integration over space. This causes problems when de ning the Poisson bracket and a potential central charge independent of where it is evaluated on the boundary. If we instead make the ansatz that the charges are de ned in terms of an integral over time, we nd the potential central charge vanishes. This is at least consistent with the lore that there is no conformal anomaly in one-dimension. We interpret the violation of the Ward identity as being due to a novel 1Speci cally the non-projectable, extended version of the theory without an additional U( 1 ) symmetry [14, 15]. { 2 { type of explicit breaking of the time reparametrizations, caused by the presence of the aether. However, perhaps the \central charge" appearing AdS2 aether system can be seen as an artifact of the null dimensional reduction of an AdS3=CF T2 system. 2 Einstein-aether theory in two dimensions Einstein-aether theory is a theory of gravity where the metric gAB is coupled to a dynamical unit timelike (co-)vector eld uA [22]. The aether eld acts as a preferred frame at every point in spacetime, breaking local Lorentz invariance. To construct the Lagrangian Lae(gAB; uA), one works in e ective theory and writes down all possible terms up to second order in an expansion in derivatives of the metric and the aether. The result in where Lae = R + Lvec, with and Sae = 1 16 Gae Z d x 4 p gLae ; Lvec = KAB CDrAuC rBuD (u2 + 1) ; KAB CD = c1gABgCD + c2 CA DB + c3 DA CB c4uAuBgCD : The coupling constants ci are dimensionless. In [12], Einstein-aether theory in two-dimensions was considered. In this lowerdimensional setting it was shown that the action reduces to the following form Sae = Z d x 2 p g 1 2 FABF AB + (rAuA)2 + (u2 + 1) ; where FAB = rAuB rBuA. In terms of the original ci coupling constants above, c1+c4 and = c1+c2+c3. Also note that in two dimensions the Einstein-Hilbert term leads = to trivial dynamics, since the Ricci scalar is a total derivative. Finally, since the aether is twist-free and hypersurface orthogonal in two-dimensions, it de nes a preferred time slicing. Therefore two-dimensional Einstein-aether theory is equivalent to two-dimensional Horava-Lifshitz gravity [20] . Variation of the Lagrangian with respect to gAB and uA produces the metric equation (2.1) (2.2) (2.3) (2.4) of motion FAC FBC 1 2 gAB 1 2 and the aether eld equation F CDFCD (rC u ) C 2 2 uC rC (rDuD) C + uAuB = 0 (2.5) rBF BA + rA(rC uC ) u A = 0: (2.6) { 3 { 6 = The Lagrange multiplier can be found by multiplying the aether eld equation with uA and using the unit constraint. The solutions to these eld equations were found and analyzed in [12]. In particular, when = there are only at spacetime solutions. When there are non-constant and constant curvature solutions. In the second class an AdS2 solution with an aether eld was found. In Fe erman-Graham like coordinates for the Poincare patch, one nds the solution ds2 = r2dt2 + uAdxA = krdt dr2 r2 r pk2 1 dr; where k = p( ) =( ). We take and to be positive and > . Note that no cosmological constant term is needed for this con guration to be a solution.2 A plot of the ow lines of the aether for this solution on the Penrose diagram of AdS can be found in gure 4 of [12]. The aether eld is regular in the Poincare patch, but becomes singular on the Poincare horizon. In two-dimensions, the boundary of AdS2 is disconnected into two separate boundaries. From the holographic point of view this raises the question of whether the dual description is terms of a single CF T1 or two systems on the boundaries. In this paper we will consider the theory in the Poincare and smaller sub-patches of the spacetime, which appears to restrict us to only one boundary system. 3 Asymptotic symmetry group To investigate the potential holographic dual to this solution, we will analyze the asymptotic symmetries, in the spirit of Brown and Henneaux. To start, we consider the solution in (2.7). We want to nd an asymptotic Killing vector, i.e. a Killing vector A that preserves the following asymptotically AdS boundary conditions gtt = r2 + O( 1 ); ut = kr + O( 1 ); gtr = O(1=r3); ur = (pk2 1)=r + O(1=r2) grr = O(1=r4) The result is t = (t) + r = for arbitrary function (t), associated with an in nitesimal t ! t + (t). This is exactly the asymptotic Killing vector that arises in studies of asymptotic symmetries in pure AdS2, see e.g. [1] . The aether eld does not explicitly break the asymptotic symmetry group, which is the in nite dimensional set of one-dimensional conformal transformations. These can be thought of as \one-half" of the conformal transformations in two-dimensions, which 2Note that we can include a cosmological constant term in the two-dimensional Einstein-aether action. However, this only a ects the solution (2.7) by changing the value of k. { 4 { (2.7) 3 t 1 kr2t + gtt = r2 + stt + ut = kr + pk2 1 t; ur = Under in nitesimal di eomorphisms generated by (3.2) one nds lead to the Virasoro algebra. Here any mapping t ! f (t) takes the metric ds2 = dt2 into We now parametrize the rst order corrections to the metric and aether in the followAsymptotic symmetries are always spontaneously broken. For example, consider the case where stt = t = 0, which corresponds to a choice of vacuum state. This con guration is only invariant under transformations = (1; t), which correspond to in nitesimal time translations and an overall scale transformation. This a ne subgroup A( 1 ) is isomorphic to the Lorentz subgroup of boosts and null rotations (Lorentz transformations preserving null vectors) in three-dimensional Minkowski spacetime. These are the exact symmetries of the metric and aether con guration. Thus there is a spontaneous breaking of time reparametrizations down to A( 1 ). Usually the AdS2 vacuum is invariant also under in nitesimal special conformal transformations generated by (t) = t2, and one has the SL(2) symmetries, but the presence of the aether breaks this down to A( 1 ). In the eld theory we could interpret this as the usual SL(2) invariant vacuum state plus a source associated with the aether eld. 4 Thermodynamics and conserved charges Now suppose we consider the case of a nite di eomorphism of (2.7) preserving the gauge and boundary conditions. One nds where ff; tg is the Schwarzian derivative ds2 = (r2 r02)d 2 + uAdxA = (kr + pk2 1r0)d + ff; tg = dr2 r2 r 2 0 1r0 + pk2 r 2 0 r2 1r + kr0 ! dr; (4.3) and the dot represents a time derivative. Taking, for example, f (t) = er0t one can express the metric to all orders in 1=r as stt = t =  f _ f ; ... f (t) f_(t) 2ff; tg 3 f(t)2 2 f_(t)2 ; kr + pk2 { 5 { (3.3) (3.4) HJEP07(21)4 (4.1) (4.2) which is the AdS2 black hole (or AdS-Rindler coordinates) plus the aether con guration. One can verify that this is indeed a solution to the eld equations. We can also express the above metric in a Horava-Lifshitz gauge associated with the time foliation (slices of constant u) where vanishes ds2 = (r2 r02)du2 + 2Nrdudr + uAdxA = (kr + r0pk2 1)du; 1 r2 Nr22 dr2 r 0 Nr = rpk2 kr + r0pk2 1 + kr0 1 is the shift vector. From this form, we see that there is a universal horizon, de ned as the location where the dot product of the global timelike Killing vector rUH = pk2 k 1 r0 = r r0: TUH = aAsAj j 2 r=rUH ; TUH = r0 2 : The region beyond this horizon is causally disconnected from in nity, even for signals of arbitrary speed and therefore de nes a notion of black hole. In [23, 24] it has been argued there is a Hawking temperature associated with universal horizons, which has the form where aA = uBrBuA and sA is the unit vector orthogonal to uA.3 Evaluating this formula for our solution, we nd (4.4) (4.5) (4.6) (4.7) (4.8) (4.9) HJEP07(21)4 temperature. leads to Note that this is consistent with the exponential relation between the Poincare time t and the Schwarzschild-like time . There is a periodicity in imaginary time with period = 2 =r0. This indicates a potential dual con guration at the boundary is at nite One important question is the nature of the conserved charges corresponding to the asymptotic Killing vectors. One way to extract these charges is to employ the covariant phase space approach of Wald [25]. In general, the variation of the Lagrangian density L 3In [24] it was argued that the Hawking temperature in [23], which was obtained by the tunnelling method, is o by a factor of two. Here we will use the form in [24]. L = Ei i + rA A ; { 6 { where i are the elds in the problem, Ei are the equations of motion, and A( i; the symplectic potential current density. By acting on this equation with two variations, one can show that on-shell where !A = 1 A( 2) 2 A( 1 ). The symplectic form ! is de ned as the integral over a Cauchy slice rA!A = 0; ! = Z d A!A: For di eomorphisms generated by a vector eld A, the eld variations are Lie derivatives. From Hamilton's equations of motion, the variation of the Hamiltonian associated with A is H = Z d A!A( i; L i): This equation can be expressed in rst in terms of the Noether current density J A = (4.10) (4.11) nally in terms of a surface integral and the antisymmetric Noether potential density QAB H = Z QAB [A B] ; where J A = 2rBQAB. The surface element nAB is 2r[AtB], where rA and tA are the unit norms to a surfaces of constant r and t respectively. A Hamiltonian exists for the asymptotic Killing vectors if there is a BA such that R 1 dnABB[A B] = R dnAB [A B]. 1 For Einstein-aether theory, the form of the symplectic current and Noether potentials was found generally in [26, 27]. In the two-dimensional case we nd A( i; L i ) AL, H = Z d A J A 2rB( [A B]) ; H = 2p ( t + _) { 7 { A = p g and (rC uC )(uAgBC 2gABuc) gBC + 2 F AB uB + 2 (rC uC )gAB uB (4.15) QAB = p g F AB(uC C ) + (rC uC )(uA B u B A ) : Computing the Hamiltonian associated with the asymptotic Killing vector (3.2), yields The only contribution to the integral at in nity (here just an evaluation at the boundary) comes from the Noether current density. The last term can be thought of as an integration constant since it does not depend on the variation of the elds. One can re-de ne the H by a shift such that for the background con guration where t = 0 it vanishes, i.e. H We will work with this form from this point forward. Another useful way to compute the charge is via the holographic (Brown-York) stress tensor. Here we consider the on-shell gravitational action, which is a boundary term. For Einstein-aether theory, the e ective action should depend on the boundary metric and boundary aether v . The variation of the e ective action W ( ; v) can be expressed as HJEP07(21)4 where E = p2 W and J = p1 Wv . Demanding di eomorphism invariance of the action W ( ; v), one nds the following Ward identity W = Z ddxp 1 2 E + J v ; Z W = 0 = ddxp (E D + J L v ) ; D (E + J v ) = J D v : T = E + J v ; W = 0 = Z ddx E + J v ; (4.18) (4.19) (4.20) (4.21) (4.22) (4.23) (4.24) where D is the covariant derivative associated with the metric . We can express this equation as In the following we will take the natural de nition of the stress tensor to be Note that this form is equivalent to the (non-symmetric) stress tensor that is obtained via a variation of the vielbein instead of the metric as the fundamental eld (see, e.g. [28]). The Ward identity associated with one-dimensional conformal transformations yields which seems to imply, in one-dimension, a vanishing energy T tt = 0. The associated charge4 is H = Z To compute the stress tensor, we vary the bulk Einstein-aether action and impose the eld equations. The result is W = Sae = Z dt h p h (rC uC )hAB 2r(AuB) rC u C gAB + 2 (rBrBu A gABrC rBuc) + 2 (rBuB)rA uA (4.25) 4This charge has the same value on any surface of constant time since the contribution from the right hand side of (4.21) vanishes at in nity. { 8 { where hAB = gAB T tt. Using gAB = r2 rArB. In two-dimensions the only non-zero part of the stress tensor is tt + , uA = r vt + and htt = r2 tt, we can extract from this expression Ett and J t and nd the value for T tt for the metric (3.3) in the limit as r ! 1. The nal result agrees with (4.18). In the case where 1(t) = 1, the asymptotic Killing vector is a global symmetry and the corresponding charge corresponds to an energy of the system. If we use the Hawking temperature at the universal horizon (4.8), we nd the thermodynamic relation c2dT 2, relating pressure to central charge [29, 30]. One considers a conformal transformation that maps the plane into the cylinder. Using the formula for the transformation of the stress tensor under a conformal mapping, one can show that the vacuum acquires an energy. This can be interpreted as a Casimir energy since the system now has an e ective nite size. Here, if we act with an asymptotic di eomorphism, we nd from (3.4) The rst term has the form of the transformation of a vector current under di eomorphism, while the last is an anomalous term, with a \central charge" of 2 . When we start from the E = 0 vacuum and set f = er0t, this yields the above result. Thus the energy here can be thought of as a Casimir energy arising from the mapping of the system from the line to p a circle. To obtain the entropy of the system we use the thermodynamic relation Inserting the formula for the energy, we nd where is a new \spontaneously generated" scale (integration constant). It acts as a cuto since formally the number of states in the system is in nite. One can also extract the free energy of the system using F = E T S. This yields F = 4 p T 1 ln : In Einstein-aether theory there is no general Wald formula for the entropy but in spherically symmetric black holes in higher dimensions it has been argued the entropy is proportional to the area of the universal horizon [24, 27]. In two-dimensions though the horizon is a point. Therefore our result is a new prediction for horizon entropy in two-dimensional Einstein-aether theory. E = f_E + 2p S = Z dE = T 0 Z T 1 dE dT 0: T 0 dT 0 S = 4 p ln T  f _ f ; T { 9 { (4.27) (4.28) (4.29) (4.30) agree if we identify the factor 4 p A logarithmic dependence of entropy on temperature has been found previously in the Almheiri-Polchinski dilaton model [5], where it is the contribution at one-loop to the thermodynamical entropy from (conformal) scalar matter elds. Spradlin and Strominger also found a logarithmic dependence on temperature in the entanglement entropy for conformal scalar elds outside an AdS2 black hole [31].5 The formulas for two-dimensional entropy as, again, being proportional to a central charge. However, a direct connection with these past results, which are obtained at one-loop, is not clear. It may be that one can consider the aether eld as a type of matter eld on the AdS2 background and the entropy is an entanglement entropy associated with that eld. There are in principle two puzzling features to the logarithmic dependence. For > T the entropy is negative, and as T ! 0 the entropy S ! 1. The zero temperature state is of course the original Poincare vacuum (2.7). One could argue that < T and that as T ! 0, we should also e ectively take the cuto vacuum state vanishes. Essentially T ! 0, such that the entropy in the is where the theory is strongly coupled and the semi-classical picture of Hawking radiation breaks down. However, if we were to take negative entropy seriously, in quantum information theory there is a notion of a conditional entropy H(SjO), which can be negative and has a thermodynamic interpretation [33, 34]. This entropy depends on the amount of information an observer O has about some quantum system S . One could imagine that the entropy associated with the universal horizon is a measure of the ignorance of an observer in the preferred frame about the dual quantum system. Note that in the case of the Poincare vacuum, the universal horizon coincides with the extremal Killing horizon. Here the aether eld becomes singular and in nitely stretched, which could be linked to the divergence of the entropy. 5 Algebra of charges We now investigate whether the violation of the time reparametrization Ward identity could be associated with an anomaly. One way to determine if this is the case is to consider whether the algebra of the conserved charges actually has a central extension. We will rst consider the bracket of two asymptotic Killing vectors, [ 1; 2]A. This is de ned as the potential changes in 2 due to variations L 1 gAB or L 1 uA and visa versa. In this case, these charges are higher order (O(r 4)), so the standard Lie bracket is suitable. One nds One typically expands the function (t) in terms of a basis of polynomials [ 1; 2]t = 1 _2 2 _1: (t) = 1 X m= 1 amtm+1: 5Note that ln(T = ) acts like the dilaton for the spontaneous breaking of conformal symmetry by nite temperature [32]. Perhaps here, where such a spontaneous breaking is explicit, this factor does indeed measure the number of states. (5.1) (5.2) Note that the m = 1 corresponds to time translations, m = 0 to scale transformations, and m = 1 to special conformal transformations. Denoting m = as usual that the Lie bracket leads to the Witt algebra amtm+1, one can show associated with one-dimensional di eomorphisms. In this one-dimensional case we only have one copy of the Witt algebra, instead of the two copies in two-dimensions. The generators ( 1; 0; 1) form a sub-algebra since for these cases the vector elds are nite at zero and in nity. However, in this case, as we noted earlier, one has to be careful because the generator of special conformal transformations is not an exact symmetry of the system. Following original work of Brown and Henneaux, which has been elaborated on in for example, [35{38], one can show that the conserved Noether charges associated with the asymptotic vectors satisfy the following algebra i P h H 1 (g; u); H 2 (g; u) = H[ 1; 2](g; u) + K 1; 2 The bracket on the left hand side represents the Poisson (or Dirac) bracket of the conserved charges. The term K 1; 2 does not depend on the dynamical elds and therefore acts as a central term in the algebra. The Poisson bracket for the charges has been typically de ned as where 2 1 = 0, meaning that the variation acts only on the elds. As a result, On the other hand, h H 1 (g; u); H 2 (g; u) i P = These results do not appear to be consistent with (5.4). In, for example, the AdS3 and BMS cases, one can show that (5.4) holds by evaluating 2 Q 1 and integrating by parts over spatial direction. In those cases the total charges were integrals over space. This is not the case in one-dimension where no spatial integrals are present and one is evaluating at a point on the boundary. One should also have an antisymmetry 2 Q 1 = 1 Q 2 , which is not obviously true above. One way to proceed is to de ne the one-dimensional Poisson bracket so that antisymmetry is made manifest i P h H 1 (g; u); H 2 (g; u) = ( 2 H 1 1 H 2 ) : Then (5.4) does hold and one nds the central-like term However, note that this expression depends on time in that we must evaluate it at some t = t0. Again, comparing to the AdS3 case, the discrepancy is due to the lack of a spatial over R02 d , one of time. integral. If we expand the analogous AdS3 expression into modes eim(t ) and integrate nds that the non-vanishing piece of the central term is independent A possible resolution is to de ne a total time independent charge in terms of an integral over time (and invoking a periodicity in imaginary time) Then if (5.8) holds for H , we nd H = 2p Z 2 0 dt (t) t Z 2 0 If we expand (t) into Fourier modes eimt, for integer m and n, we nd that Km;n vanishes for all (m; n). This indicates this potential charge algebra is without a central term. 6 It is di cult to interpret the non-zero energy via an anomaly in the one-dimensional conformal symmetry. Therefore we instead interpret the violation of the Ward identity as a type of explicit breaking of the time reparametrization symmetry. A nite temperature is a soft breaking, introducing an e ective length scale in T . However, in one dimension scale invariance implies that the density of states must scale like (E ) = A (E ) + B=E [6]. The rst term is a possible zero temperature entropy, while the second is the T 1 term we found from the black hole thermodynamics. This leads to the presence of the logarithm in the entropy and free energy and means there must be another cuto scale generated as well. Thus we have a \spontaneous explicit breaking" supported by the presence of the aether. It would be interesting to understand a potential holographic dual in more detail. Our results may also be useful for the study of various condensed matter systems via AdS2 holography, e.g. [39, 40]. Finally, it is possible that two-dimensional Einstein-aether theory can be realized as a dimensional reduction of a gravity theory in AdS3, along the lines of the Einstein-Maxwelldilaton models discussed in [41, 42]. For example, it is known that non-relativistic theories are the result of a null reduction of gravity on higher dimensional Lorentzian manifolds [43, 44]. Perhaps the central charge and logarithmic scaling found here have their origins in the non-relativistic limit of a two-dimensional CFT. Acknowledgments I would like to thank T. Andrade, Y. Oz, and A. Starinets for valuable discussions. This research was supported by the European Research Council under the European Union's Seventh Framework Programme (ERC Grant agreement 307955). Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. [INSPIRE]. HJEP07(21)4 014 [arXiv:1402.6334] [INSPIRE]. [arXiv:1605.06098] [INSPIRE]. holography, JHEP 07 (2016) 139 [arXiv:1606.03438] [INSPIRE]. [9] G. Mandal, P. Nayak and S.R. 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Christopher Eling. Holography and AdS 2 gravity with a dynamical aether, Journal of High Energy Physics, 2017, 147, DOI: 10.1007/JHEP07(2017)147