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Conformal Holography in Four Dimensions

Daniel Grumiller

Maria Irakleidou

Iva Lovrekovic,

Robert McNees Loyola University Chicago, [email protected]

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Recommended Citation Grumiller, D, M Irakleidou, I Lovrekovic, and R McNees. "Conformal Gravity Holography in Four Dimensions." Physical Review Letters 112, 2014.

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This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 License. © 2014 American Physical Society. week ending PRL 112, 111102 (2014) PHYSICAL REVIEW LETTERS 21 MARCH 2014

Conformal Gravity Holography in Four Dimensions

† ‡ Daniel Grumiller,1,* Maria Irakleidou,1, Iva Lovrekovic,1, and Robert McNees2,§ 1Institute for Theoretical Physics, Vienna University of Technology, Wiedner Hauptstrasse 8–10/136, A-1040 Vienna, Austria 2Department of Physics, Loyola University Chicago, Chicago, Illinois 60660, USA (Received 8 October 2013; published 18 March 2014) We formulate four-dimensional conformal gravity with (anti–)de Sitter boundary conditions that are weaker than Starobinsky boundary conditions, allowing for an asymptotically subleading Rindler term concurrent with a recent model for gravity at large distances. We prove the consistency of the variational principle and derive the holographic response functions. One of them is the conformal gravity version of the Brown–York stress tensor, the other is a “partially massless response”. The on shell action and response functions are finite and do not require holographic . Finally, we discuss phenomenologi- cally interesting examples, including the most general spherically symmetric solutions and rotating black hole solutions with partially massless hair.

DOI: 10.1103/PhysRevLett.112.111102 PACS numbers: 04.20.Ha, 04.50.Kd, 95.35.+d, 98.80.-k

Conformal gravity (CG) in four dimensions is a recurrent (3D) CG [21] suggests that a weaker set of boundary topic in theoretical physics, as it provides a possible conditions should be possible also in four dimensions. resolution to some of the problematic issues with Finding such boundary conditions is interesting for purely Einstein gravity, the established theory of the gravitational theoretical reasons and also phenomenologically. Indeed, interaction, though it usually introduces new ones. the CG analogue of the Schwarzschild solution, the spheri- For instance, like other higher-derivative theories, CG is cally symmetric Mannheim–Kazanas–Riegert (MKR) sol- renormalizable [1,2], but has ghosts [3], whereas Einstein ution [9,22], does not obey the Starobinsky boundary gravity is ghost free, but 2-loop nonrenormalizable [4]. See conditions. A related motivation is to investigate whether [5–8] for important early work on CG. Later, CG was it is true that CG provides an example of a theory that studied phenomenologically by Mannheim in an attempt to allows a nontrivial Rindler term, as suggested in the explain galactic rotation curves without [9–12] discussion of an effective model for gravity at large and emerges theoretically from twistor [13] or distances [23]. The difficulty does not lie in showing that as a counter term in the anti–de Sitter/conformal field the CG equations of motion (EOM) permit a Rindler term theory (AdS/CFT) correspondence [14,15]. More recently, (they do), but in determining a set of boundary conditions ’t Hooft has studied CG in a context [16] consistent with such a term. and Maldacena has shown how Einstein gravity can emerge The main purpose of our Letter is to establish the from CG upon imposing suitable boundary conditions that consistency of a set of (A)dS boundary conditions for CG, weaker than the ones proposed by Starobinsky, that are eliminate ghosts [17]. compatible with the existence of an asymptotic Rindler Physical theories in general require boundary conditions term, the MKR solution, and other solutions with a as part of their definition. In many cases, “natural” condensate of partially massless . boundary conditions—the rapid falloff of all fields near Before starting, we review the most salient features of a boundary or in an asymptotic region—are the appropriate CG. A distinguishing property of CG is that the theory choice, but this is not the case in gravitational theories, since the metric should be nonzero. A prime example is depends only on (Lorentz) angles but not on distances. This gravity in AdS, where the boundary conditions define the means that the theory is invariant under local Weyl behavior of the dual field theory that lives on the conformal rescalings of the metric, boundary of . Gravity in de Sitter (dS) requires 2ω gμν → g~μν ¼ e gμν; (1) similar boundary conditions; they were provided for four- dimensional Einstein gravity by Starobinsky [18]. (See also where the Weyl factor ω is allowed to depend on the [19,20] for a more recent discussion of future boundary coordinates. The bulk action of CG, conditions and conserved charges in dS.) These boundary Z conditions played a crucial role in Maldacena’s reduction pffiffiffiffiffi ¼ α 4 j j βν γλ δτ α μ from CG to solutions of Einstein gravity [17]. ICG CG d x g gαμg g g C βγδC νλτ; (2) It is, however, not clear that the Starobinsky boundary conditions are the most general or phenomenologically is manifestly invariant under Weyl rescalings Eq. (1), since α interesting ones for CG. Experience with three-dimensional the C βγδ is Weyl invariant, and the Weyl factor

0031-9007=14=112(11)=111102(6) 111102-1 © 2014 American Physical Society week ending PRL 112, 111102 (2014) PHYSICAL REVIEW LETTERS 21 MARCH 2014 coming from the square root of the determinant of the As part of the specification of our boundary conditions, metric is precisely canceled by the Weyl factor coming we fix the leading and first-order terms in Eq. (7) on ∂M from the metric terms. The dimensionless coupling con- up to a local Weyl rescaling α stant CG is the only free parameter in the CG action. In most of what follows, we set α ¼ 1 to reduce clutter. The δγð0Þj ¼ 2λγð0Þ δγð1Þj ¼ λγð1Þ CG ij ∂M ij ; ij ∂M ij ; (8) EOM of CG are fourth order and require the vanishing of the Bach tensor, where λ is a regular function on ∂M, while the subleading   terms at second and higher order are allowed to vary freely, δ 1 δ γ δγðnÞj ≠ 0 ≥ 2 ∇ ∇γ þ R γ C αδβ ¼ 0: (3) ∂M for n . An essential difference to the 2 Starobinsky boundary conditions is the presence of a γð1Þ subleading term ij . This term is absent in [18] because There are two especially simple classes of solutions to the the EOM for Einstein gravity force it to vanish. By contrast, γ ð1Þ EOM: conformally flat metrics (C αδβ ¼ 0) and Einstein γ the EOM [Eq. (3)] do not give any conditions on ij , metrics (Rαβ ∝ gαβ) both have vanishing Bach tensor. analogous to 3D CG [21]. Therefore, solutions of Einstein gravity are a subset of To check the consistency of the boundary conditions the broader class of solutions of CG. Eqs. (6)–(8), we consider first the on shell action and then The most general spherically symmetric solution of CG the variational principle. On general grounds, one might is given by the line element [22] expect the bulk action Eq. (2) to be supplemented by two “ – – ” 2 kinds of boundary terms: a Gibbons Hawking York 2 2 dr 2 2 boundary term [24,25] that produces the desired boundary ds ¼ −kðrÞdt þ þ r dΩ 2 ; (4) kðrÞ S value problem (for instance, a Dirichlet boundary value problem), and holographic counterterms [26–31] that Ω2 where d S2 is the line element of the round 2 sphere and guarantee that the action is stationary for all variations that preserve our boundary conditions. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2M kðrÞ¼ 1 − 12aM − − Λr2 þ 2ar: (5) We claim that no such boundary terms are required for r CG, so that the full action is just the bulk action Eq. (2) For a ¼ 0, the solution reduces to Schwarzschild-(A)dS. It Z pffiffiffiffiffi ≪ 1 Γ ¼ ¼ 4 j j λ μσν is noteworthy that for aM , the solution Eqs. (4) and (5) CG ICG d x g C μσνCλ : (9) corresponds to the one presented in [23], derived from an M effective model for gravity at large distances. The first piece of evidence that no counterterms might be Phenomenologically relevant numbers (in Planck units) −123 −61 38 needed comes from the calculation of the on-shell action. It are Λ ≈ 10 , a ≈ 10 , M ≈ 10 M⊙, where M⊙ ¼ 1 −23 is straightforward to show that the on shell action for any for the Sun, so that indeed aM ≈ 10 M⊙ ≪ 1 for all metric behaving like Eqs. (6) and (7),evaluatedona black holes or galaxies in our Universe. compact region ρ ≤ ρ, remains finite as ρ → 0. A related We propose now boundary conditions that admit the c c piece of evidence was provided in [32], where it was shown MKR solution Eqs. (4) and (5). This requires the intro- that the free energy derived from the on shell action Eq. (9) is duction of a length scale l, which in Einstein gravity would consistent with the Arnowitt–Deser–Misner mass and be related to the cosmological constant as Λ ¼ 3σ=l2 (with Wald’s definition of the entropy [33]. The fact that the on σ ¼ −1 for AdS and σ ¼þ1 for dS). Then our asymptotic shell action yields the correct free energy suggests that any (0 < ρ ≪ l) line element reads boundary terms added to the action Eq. (9) should vanish on l2 shell. The simplest possibility is that these terms are ds2 ¼ ð−σdρ2 þ γ dxidxjÞ: identically zero [34]. ρ2 ij (6) A more stringent check of our claim is obtained by proving the consistency of the variational principle and the For simplicity, we partially fix the gauge and use Gaussian finiteness of the holographic response functions. To this coordinates. Close to the conformal boundary at ρ ¼ 0, the end, we first rewrite the Weyl-squared action Eq. (9) as 3D metric has the following asymptotic expansion: Z   pffiffiffiffiffi 2 ρ ρ2 ρ3 Γ ¼ 4 j j 2 μν − 2 þ 32π2χðMÞ ð0Þ ð1Þ ð2Þ ð3Þ CG d x g R Rμν R γ ¼ γ þ γ þ γ þ γ þ (7) M 3 ij ij l ij l2 ij l3 ij Z  pffiffiffiffiffi 4 þ 3 jγj −8σGij þ 3 − 4 ij γð0Þ d x Kij K KK Kij The boundary metric is required to be invertible. All the ∂M 3 γðnÞ  coefficient matrices are allowed to depend on the 8 i “ ” ij k boundary coordinates x , but not on the holographic þ K Kj Kki . (10) coordinate ρ. 3

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The action has been separated into a topological part—the the Lie derivative along the outward- or future-pointing unit Euler characteristic χðMÞ—and a Ricci-squared action, vector nμ normal to ∂M. with the boundary terms in the last two lines canceling In this formulation, the first variation of the action is similar terms that appear in the Euler characteristic for given by with (conformal) boundary; see [35]. Here and Z pffiffiffiffiffi in all subsequent expressions, calligraphic letters indicate δΓ ¼ þ 3 jγjðπijδγ þ Πijδ Þ ij CG EOM d x ij Kij : (11) quantities intrinsic to the 3D surface ∂M. Thus, G is the ∂M 3D Einstein tensor for the metric γij. The extrinsic ij curvature is defined as Kij ¼ −ðσ=2ÞLnγij, where Ln is The momentum π reads

σ σ 1 1 1 πij ¼ ðγij kl −γkl ijÞ þ ρ ðγij − ijÞ− γijD ð kρÞþ Dið ρjÞ− ðγikγjl −γijγklÞL 4 K K fkl 4f ρ K K 2 k nρf 2 nρf 4 nfkl ij ik j ij kl ij 2 ij i kj i j ij kl k þσ½2KR −4K Rk þ2γ KklR −γ KRþ2D K −4D DkK þ2D D K þ2γ ðDkDlK −D DkKÞ 2 1 þ γijKk KlmK −4KikKjlK þ2KijKklK þ γijK3 −2KijK2 −γijKKklK þ4KKi Kjk þi↔j: (12) 3 m kl kl kl 3 kl k

The tensor fμν, which appears in a convenient auxiliary field For metrics that satisfy the boundary conditions Eqs. (6) formulation of the action, is proportional to the four- and (7), their asymptotic expansions are given by ¼ −4ð − 1 Þ dimensional Schouten tensor, fμν Rμν 6 gμνR .The ρ momentum Πij reads ¼ ð2Þ þ ð3Þ þ Eij Eij l Eij ; (17) ij ij ij ij k Π ¼ −8σG − σðf − γ f kÞ þ 4γijð 2 − kl Þ − 8 ij þ 8 i kj (13) l ð1Þ ð2Þ K K Kkl KK K kK : B ¼ B þ B þ; ijk ρ ijk ijk (18) It is noteworthy that we allow the boundary metric and the with extrinsic curvature to vary independently in Eq. (11).   ð1Þ 1 ð1Þ 1 ð0Þ ð1Þ Let us check now the variational principle. Evaluating B ¼ D ψ − γ Dlψ − j↔k; ijk 2l j ik 2 ij kl (19) Eq. (11) on a compact region ρc ≤ ρ, applying the EOM, and making use of the asymptotic expansion Eq. (7) with Eqs. (12) and (13) yields   ð2Þ 1 ð2Þ σ ð0Þ 1 ð0Þ 1 ð1Þ Z ¼ − ψ þ R − γ Rð0Þ þ γð1Þψ qffiffiffiffiffiffiffiffiffiffi Eij 2l2 ij 2 ij 3 ij 8l2 ij ; δΓ j ¼ 3 jγð0Þjðτijδγð0Þ þ ijδγð1ÞÞ CG EOM d x ij P ij : (14) ∂M (20)

ij ij The tensors τ and P are finite as ρc → 0. As we will ð3Þ 3 ð3Þ 1 ð0Þ ð2Þ 1 ð1Þ ð1Þ show below, they satisfy the trace conditions E ¼ − ψ − γ ψ kl ψ − ψ ψ ψ kl ð1Þ ð1Þ 1 ð0Þ ð1Þk ij 4l2 ij 12l2 ij ð1Þ kl 16l2 ij kl ð1Þ (ψ ≔γ − 3 γ γ )  ij ij ij k σ − Rð0Þψ ð1Þ − γð0ÞRð0Þψ kl þ γð0ÞD D ψ kl ij ij kl ð1Þ ij l k ð1Þ γð0Þτij þ 1 ψ ð1Þ ij ¼ 0 γð0Þ ij ¼ 0 12 ij 2 ij P ; ij P ; (15)  3 ð1Þ ð1Þ 1 γ þ D Dkψ − 3D D ψ k þ E þ i↔j; so that the first variation of the action vanishes on shell 2 k ij k i j 24l2 ij when the boundary conditions Eq. (8) are satisfied. (21) Therefore, the action Eq. (9) and our proposed boundary conditions constitute a well-defined variational principle.   ij ij 1 The quantities τ and P appearing in Eq. (14) are the γ ð2Þ ð0Þ ð1Þ kl ð1Þ ð1Þ E ≔ γð1Þ 3ψ þ γ ψ ψ ð1Þ − γð1Þψ þ 5γð2Þψ holographic response functions conjugate to the sources ij ij 2 ij kl ij ij ð0Þ ð1Þ   γ and γ , respectively. We evaluate now the first of these 1 ij ij − σl2 D D γ − γð0ÞDkD γ (22) functions, which is proportional to the usual Brown–York j i ð1Þ 3 ij k ð1Þ : stress tensor. It is useful to introduce the electric Eij and magnetic Bijk parts of the Weyl tensor. In these expressions, the coefficient matrices have been split into trace and trace-free parts as β α α ðnÞ 1 ð0Þ ð Þ ðnÞ Eij ¼ nαn C iβj;Bijk ¼ nαC ijk: (16) γ ¼ γ γ n þ ψ τ ij 3 ij ij . Then the (finite) result for ij (as

111102-3 week ending PRL 112, 111102 (2014) PHYSICAL REVIEW LETTERS 21 MARCH 2014 Z ρ → 0 pffiffiffi c )is 2 i Q½ξ¼ d x huiJ ; (26)    C 2 ð3Þ 1 ð2Þ 4 ð2Þ ð1Þ τ ¼ σ E þ E γð1Þ − E ψ k ij l ij 3 ij l ik j where h is the metric on C and ui is the future-pointing unit C τ γð1Þ 1 ð0Þ ð2Þ 1 ð1Þ ð1Þ vector normal to . The combination of ij, Pij, and ij þ γ E ψ kl þ ψ ψ ψ kl i l ij kl ð1Þ 2l3 ij kl ð1Þ appearing in J is precisely the modified stress tensor of   1 1 Hollands et al. [41]. Thus, the charges Eq. (26) are expected − ψ ð1Þ ψ ð1Þkψ ð1Þl − γð0Þψ ð1Þkψ lm to generate the asymptotic symmetries. The covariant 3 kl i j ij m ð1Þ l 3 divergence of the modified stress tensor satisfies − 4Dk ð1Þ þ ↔ (23) Bijk i j: D ð2τij þ 2 i γð1ÞkjÞ¼ ikDjγð1Þ i P k P ik : (27) We call the function Pij the “partially massless response” (PMR). This name is justified, since it is This ensures that the difference in charges computed on γð1Þ C C V ⊂ ∂M sourced by the term ij in the metric. The latter, when surfaces 1, 2 that bound a region is given by plugged into the linearized CG EOM around an (A)dS Z qffiffiffiffiffiffiffiffiffiffi background, exhibits partial masslessness in the sense Δ ½ξ¼ 3 jγð0ÞjðτijL γð0Þ þ ijL γð1ÞÞ Q d x ξ ij P ξ ij ; (28) of Deser, Nepomechie, and Waldron [36,37].Thisis V expected from the corresponding behavior in 3D [21] and also on general grounds, since the Weyl invariance which vanishes for asymptotic symmetries. Eq. (1) is nothing but the nonlinear completion of We apply now our formulas to three pertinent examples. the gauge enhancement at the linearized level due As a first special case, consider solutions that obey γð1Þ ¼ 0 to partial masslessness; see, for instance, the recent Starobinsky boundary conditions, ij . This includes discussion in [38,39]. (Note that such a nonperturbative the asymptotically (A)dS solutions of Einstein gravity completion of partial masslessness is not generic with a cosmological constant. Then the EOM imply ð2Þ ¼ 0 – to higher derivative theories [40].) Calculating the Eij , so the PMR vanishes and the Brown York stress PMR yields tensor simplifies to

4σ ð2Þ 4σ ð3Þ P ¼ − E ; (24) τ ¼ ij l ij ij l Eij : (29) as ρc → 0.Likeτij, the PMR is finite and does not require This recovers the traceless and conserved stress tensor of holographic renormalization. Einstein gravity [30], in agreement with Maldacena’s Given the expressions Eqs. (23) and (24) for the response analysis [17] and with earlier work by Deser and functions, the trace conditions Eq. (15) follow from trace- Tekin [42]. lessness of the electric and magnetic parts of the Weyl A more interesting example is provided by the MKR γij ð3Þ ¼ ψ ij ð2Þ solution Eqs. (4) and (5). Setting σ ¼ −1 for concreteness, tensor, which give identities ð0ÞEij ð1ÞEij and and defining the traceless matrix pi ¼ diagð1; − 1 ; − 1 Þi γij ð2Þ ¼ γij ð1Þ ¼ 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij 2 2 j ð0ÞEij ð0ÞBijk . Note that for Starobinsky boun- 2 and constants aM ¼ð1 − 1 − 12aMÞ=6 and m ¼ M=l dary conditions the Brown-York stress tensor is traceless, 2 2 yields τi ¼ −8ðm=l Þpi þ 8ðaa =l Þdiagð1; −1; −1Þi but in general, only the PMR is traceless. j j M j i ¼ 8ð l2Þ i To summarize, we have shown the consistency of the and P j aM= p j. For vanishing Rindler accelera- ¼ ¼ 0 boundary conditions Eqs. (6)–(8) by demonstrating that the tion, a aM , the previous Einstein case is recovered. ≠ 0 variational principle is well-defined for the action Eq. (9) For nonvanishing Rindler acceleration, a , the PMR is and by proving finiteness of all 0- and 1-point functions. linear and the trace of the Brown-York stress tensor ≪ 1 This is our main result. quadratic in the Rindler parameter a when aM . Conserved charges may be computed from the currents Thus, the Rindler parameter in the MKR solution can be interpreted as coming from a partially massless ð1Þ condensate. The conserved charge associated with the Ji ¼ð2τi þ 2Pikγ Þξj; (25) j kj Killing vector ∂t may be computed using Eq. (26).If α ¼ð1 64πÞ we normalize the action such that CG = ,we j where ξ is a boundary diffeomorphism associated with an obtain Q½∂t¼m − aaM. The entropy, obtained using asymptotic of the theory. (For now, we consider Wald’s approach or from the on shell action, is σ ¼ −1 ¼ ð4l2Þ ¼ 4π 2 only the AdS case , so that the conformal boundary S Ah= , where Ah rh is the area of the horizon ∂M is a timelike surface.) Given a constant-time surface C kðrhÞ¼0. Remarkably, the entropy obeys an area law in ∂M, the charge is despite the fact that CG is a higher-derivative theory.

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As a third example, we consider rotating black hole §[email protected] solutions in AdS with Rindler hair parametrized by a [1] K. Stelle, Phys. Rev. D 16, 953 (1977). Rindler acceleration μ and rotation parameter a~, but with [2] S. L. Adler, Rev. Mod. Phys. 54, 729 (1982). a vanishing mass parameter; see Eq. (7) of [43]. [3] Generic higher-derivative theories are usually assumed to Interestingly, we find that the absence of a mass parameter suffer from ghosts, but it has been conjectured that CG may admit an alternative quantization that preserves unitarity leads to a vanishing PMR, Pij ¼ 0. This shows that a nonzero Rindler term in the asymptotic expansion Eq. (7), [12,45] (See [46,47] for some criticism). Resolving this γð1Þ ≠ 0 question is beyond the scope of this Letter. Instead, we focus ij , is necessary but not sufficient for a nonzero PMR. Evaluation of the Brown–York stress tensor leads to a on the boundary conditions and variational formulation of conserved energy, E ¼ −a~ 2μ=½l2ð1 − a~ 2=l2Þ2, and con- the classical theory, and on establishing the framework for a served angular momentum, J ¼ El2=a~, both linear in the possible holographic dual. Rindler parameter μ. [4] M. H. Goroff and A. Sagnotti, Nucl. Phys. B266, 709 Finally, it is possible to make a Legendre transformation (1986). of the action Eq. (9) that exchanges the role of the PMR and [5] J. Julve and M. Tonin, Nuovo Cimento Soc. Ital. Fis. 46B, 137 (1978). its source, namely by adding a Weyl invariant boundary [6] E. Fradkin and A. A. Tseytlin, Nucl. Phys. B201, 469 term Z (1982). pffiffiffiffiffi [7] E. T. Tomboulis, Phys. Rev. Lett. 52, 1173 (1984). Γ~ ¼ Γ þ 8 3 jγj ij CG CG d x K Eij: (30) [8] D. G. Boulware, G. T. Horowitz, and A. Strominger, Phys. ∂M Rev. Lett. 50, 1726 (1983). This action is also finite on shell. Its first variation yields [9] P. D. Mannheim and D. Kazanas, Astrophys. J. 342, 635 Z (1989). pffiffiffiffiffi [10] P. D. Mannheim, Prog. Part. Nucl. Phys. 56, 340 (2006). δΓ~ ¼ 3 jγjðτ~ijδγð0Þ þ ~ ijδ ð2ÞÞ ’ CG d x ij P Eij ; (31) [11] P. D. Mannheim and J. G. O Brien, Phys. Rev. Lett. 106, ∂M 121101 (2011). [12] P. D. Mannheim, Found. Phys. 42, 388 (2012). with finite response functions. [13] N. Berkovits and E. Witten, J. High Energy Phys. 08 (2004) 2σ 8σ 009. kl ð1Þ ð0Þ ð2Þ ð1Þ [14] H. Liu and A. A. Tseytlin, Nucl. Phys. B533, 88 (1998). τ~ij ¼ τij þ Eð2Þψ γ þ E γ l kl ij 3l ij [15] V. Balasubramanian, E. G. Gimon, D. Minic, and J. Rahmfeld, 4σ ð2Þ ð1Þk ð2Þ ð1Þk Phys.Rev.D63, 104009 (2001). − ðE ψ þ E ψ Þ; ’ l ik j jk i (32) [16] G. t Hooft, Found. Phys. 41, 1829 (2011). [17] J. Maldacena, arXiv:1105.5632v2. [18] A. A. Starobinsky, JETP Lett. 37, 66 (1983). 4σ [19] D. Anninos, G. S. Ng, and A. Strominger, Classical Quan- ~ ¼ γð1Þ Pij l ij : (33) tum Gravity 28, 175019 (2011). [20] D. Anninos, G. S. Ng, and A. Strominger, J. High Energy The Brown–York stress tensor has zero trace, τ~i ¼ 0. Phys. 02 (2012) 032. i [21] H. Afshar, B. Cvetkovic, S. Ertl, D. Grumiller, and N. To summarize, the results of this Letter provide the basis Johansson, Phys. Rev. D 84, 041502(R) (2011); 85, 064033 for CG holography in four dimensions and show the (2012). viability of the MKR solution and other solutions with [22] R. J. Riegert, Phys. Rev. Lett. 53, 315 (1984). an asymptotic Rindler term. Possible next steps are the [23] D. Grumiller, Phys. Rev. Lett. 105, 211303 (2010). determination of the asymptotic symmetry group, calcu- [24] J. W. York, Jr., Phys. Rev. Lett. 28, 1082 (1972). lation of higher n-point functions, and applications of our [25] G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2752 results to additional solutions of CG. (1977). [26] M. Henningson and K. Skenderis, Fortschr. Phys. 48, 125 We are grateful to H. Afshar, S. Deser, N. Johansson, A. (2000). Naseh, K. Skenderis, A. Waldron, and T. Zojer for [27] V. Balasubramanian and P. Kraus, Commun. Math. Phys. discussions. Many of the calculations presented in this 208, 413 (1999). Letter were performed with the XACT package for [28] R. Emparan, C. V. Johnson, and R. C. Myers, Phys. Rev. D MATHEMATICA [44]. D. G., M. I., and I. L. were supported 60, 104001 (1999). by the START Project No. Y 435-N16 of the Austrian [29] P. Kraus, F. Larsen, and R. Siebelink, Nucl. Phys. B563, 259 Science Fund (FWF) and the FWF Project No. I 952-N16. (1999). [30] S. de Haro, S. N. Solodukhin, and K. Skenderis, Commun. Math. Phys. 217, 595 (2001). [31] I. Papadimitriou and K. Skenderis, J. High Energy Phys. 08 *[email protected] (2005) 004. † [email protected] [32] H. Lu, Y. Pang, C. N. Pope, and J. F. Vazquez-Poritz, Phys. ‡ [email protected] Rev. D 86, 044011 (2012).

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[33] R. M. Wald, Phys. Rev. D 48, R3427 (1993). [41] S. Hollands, A. Ishibashi, and D. Marolf, Phys. Rev. D 72, [34] We stress that vanishing boundary terms are not obviously 104025 (2005). expected for CG. For example, naively extrapolating the [42] S. Deser and B. Tekin, Phys. Rev. Lett. 89, 101101 (2002); result for boundary terms (see appendix A of [48])in Phys. Rev. D 67, 084009 (2003). Lü-Pope [49] would indicate a nonzero [43] H.-S. Liu and H. Lu, J. High Energy Phys. 02 (2013) 139. result for these terms in CG. See also [50]. [44] J. M. Martin-Garcia, XACT: tensor computer algebra http:// [35] R. C. Myers, Phys. Rev. D 36, 392 (1987). www.xact.es/. [36] S. Deser and R. I. Nepomechie, Ann. Phys. (N.Y.) 154, 396 [45] C. M. Bender and P. D. Mannheim, Phys. Rev. Lett. 100, (1984). 110402 (2008). [37] S. Deser and A. Waldron, Phys. Rev. Lett. 87, 031601 [46] A. Smilga, SIGMA 5, 017 (2009). (2001). [47] T.-j. Chen, M. Fasiello, E. A. Lim, and A. J. Tolley, J. [38] S. Deser, E. Joung, and A. Waldron, Phys. Rev. D 86, Cosmol. Astropart. Phys. 02(2013) 042. 104004 (2012). [48] N. Johansson, A. Naseh, and T. Zojer, J. High Energy Phys. [39] S. Deser, M. Sandora, and A. Waldron, Phys. Rev. D 87, 09 (2012) 114. 101501 (2013). [49] H. Lu and C. N. Pope, Phys. Rev. Lett. 106, 181302 [40] S. Deser, S. Ertl, and D. Grumiller, J. Phys. A 46, 214018 (2011). (2013). [50] O. Hohm and E. Tonni, J. High Energy Phys. 04 (2010) 093.

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