We show that the recently found AdS-plane and AdS-spherical wave solutions of
quadratic curvature gravity also solve the most general higher derivative
theory in D-dimensions. More generally, we show that the field equations of
such theories reduce to an equation linear in the Ricci tensor for Kerr-Schild
spacetimes having type-N Weyl and traceless Ricci tensors.Comment: 4 pages, solutions are extended for generic f(Riemann) theories that
  include arbitrary powers of the curvatur

Gurses, Metin

Hervik, Sigbjørn

Sisman, Tahsin Cagri

Tekin, Bayram

English

arXiv

Metin Gürses

Sigbjørn Hervik

Tahsin Çağrı Şişman

Bayram Tekin

Crossref

Physical Review Letters

Anti–de Sitter–Wave Solutions of Higher Derivative Theories

We show that the recently found anti-de Sitter (AdS)-plane and AdS-spherical wave solutions of quadratic curvature gravity also solve the most general higher derivative theory in D dimensions. More generally, we show that the field equations of such theories reduce to an equation linear in the Ricci tensor for Kerr-Schild spacetimes having type-N Weyl and type-N traceless Ricci tensors. © 2013 American Physical Society

Gürses, M.

Hervik, S.

Şişman, T. Ç.

Tekin, B.

Bilkent University Institutional Repository

Anti–de Sitter–Wave Solutions of Higher Derivative TheoriesMetin Gürses,1,* Sigbjørn Hervik,2,† Tahsin Çağr Şişman,3,‡ and Bayram Tekin3,§1Department of Mathematics, Faculty of Sciences Bilkent University, 06800 Ankara, Turkey2Department of Mathematics and Natural Sciences, University of Stavanger, N-4036 Stavanger, Norway3Department of Physics, Middle East Technical University, 06800 Ankara, Turkey(Received 16 May 2013; published 5 September 2013)We show that the recently found anti–de Sitter (AdS)-plane and AdS-spherical wave solutions ofquadratic curvature gravity also solve the most general higher derivative theory in D dimensions. Moregenerally, we show that the field equations of such theories reduce to an equation linear in the Ricci tensorfor Kerr-Schild spacetimes having type-N Weyl and type-N traceless Ricci tensors.DOI: 10.1103/PhysRevLett.111.101101 PACS numbers: 04.50.h, 04.20.Jb, 04.30.wThere is a vast literature on the exact solutions of four-dimensional Einstein’s gravity. But, as more powers ofcurvature are added, or computed in a microscopic theorysuch as string theory, to get a better UV behaved theory, thefield equations become highly nontrivial and so solutionsare not easy to find. In fact, only a few classes of solutionsare known: For example, see [1–5] for solutions in lowenergy string theory.AdS5  S5, which played a major rolein AdS/CFT, is also expected to be an exact solution ofstring theory [6]. In this Letter, we present new asymptoti-cally anti–de Sitter (AdS) solutions, which are AdS-planeand AdS-spherical waves, to D-dimensional, for D  3,generic gravity theories based on the Riemann tensor andits arbitrary number of covariant derivatives which are insome sense natural geometric extensions of Einstein’sgravity. Certain low energy string theory actions constitutea subclass of this theory once all the nongravitational fieldsare turned off [7]. Asymptotically, AdS solutions in higherderivative theories are relevant in the context of genericgravity-gauge theory dualities and holography. Here, weshall provide such solutions.Using a theorem given in [5], we first prove that anyspacetime with type-N Weyl and type-N traceless Riccitensors, where the metric is in the Kerr-Schild form, thefield equations of the most general higher derivative theoryreduce to a linear equation for the traceless Ricci tensor.These spacetimes have constant scalar invariants.Furthermore, using the type-N property of the tracelessRicci tensor in these field equations, in the AdS back-ground, we obtain a linear partial differential equationfor a metric function V. This result implies that the AdS-wave metrics are universal in the sense defined in [5].As a special case, the field equations of the theory whichdepends on the contractions of the Riemann tensor but noton its derivatives, fðRÞ theory, are also highly cumber-some, but using our general result for type-N spacetimesunder certain assumptions, they reduce to those of thequadratic gravity. Then, taking the metric to be in theKundt subclass of type-N spacetimes, we show that AdS-plane wave [8,9] and AdS-spherical wave [10] solutions ofthe quadratic gravity theory are also the solutions of thefðRÞ theory. Logarithmic terms arising in the solutionsof the quadratic gravity exist also in some fðRÞ theoriescorresponding to the generalizations of critical gravity[11,12]. As an application of our result, we show that anytype-N Einstein space (R ¼ ðR=DÞg) solves the fieldequations of the Lanczos-Lovelock theory. In addition, fora special choice of the parameters, any spacetime metrichaving type-N Weyl and type-N traceless Ricci tensorswith a constant Ricci scalar solves a specific Lanczos-Lovelock theory.To find exact asymptotically AdS solutions of a generichigher derivative theory, we shall make use of generalresults of [13–17] which utilize the boost weightformalism.Type-N Weyl and traceless Ricci tensors.—The Weyltensor of type-N spacetimes has been studied in detail in[13–16]. Let ‘, n, and mðiÞ, (i ¼ 2; . . .D 1) be a nulltetrad frame with‘‘ ¼ nn ¼ ‘mðiÞ ¼ nmðiÞ ¼ 0;‘n ¼ 1; mðiÞmðjÞ ¼ ij;(1)where  ¼ 0; 1; 2;   D 1 and i; j ¼ 2; 3;   D 1.Then, the metric takes the formg ¼ ‘n þ ‘n þ ijmðiÞ mðjÞ : (2)The Weyl tensor of type-N spacetimes, expressed in theabove frame, where ‘ is the Weyl aligned null direction,takes the formC ¼ 40ij‘fmðiÞ ‘mðjÞg ; (3)which transforms under scale transformations with boostweight -2 (In the above expressions, we used the notationsof [13]).Type-N property alone is not sufficient to reduce thefield equations of higher derivative gravity theories to asolvable form, we make a further assumption that thespacetime is radiating and the Ricci scalar is constant.PRL 111, 101101 (2013) P HY S I CA L R EV I EW LE T T E R Sweek ending6 SEPTEMBER 20130031-9007=13=111(10)=101101(5) 101101-1  2013 American Physical SocietyFor radiating type-N spacetimes, the Ricci tensor is in theformR ¼ ‘‘ þ RDg; (4)where  is a scalar function and R is the Ricci scalar.Taking R to be a constant and using the Bianchi identity,one obtainsrð‘‘Þ ¼ 0: (5)Notice that the traceless part,S ¼ R  RDg ¼ ‘‘; (6)of the Ricci tensor is of type N. Following [5], and para-phrasing the statement given in that work we have thefollowing result:Theorem.—Consider a Kundt spacetime for which theWeyl and the traceless Ricci tensors be of type N, that ishaving the forms (3) and (6), respectively, and all scalarinvariants be constant. Then, any second rank symmetrictensor constructed from the Riemann tensor and its cova-riant derivatives can be written as a linear combination ofg, S, and higher orders of S (such as, for example,hnS).The proof of the above statement, which we can onlysketch here, depends heavily on the boost-weight formal-ism [5,18,19]. In particular, a Kundt spacetime with type-Ntraceless Ricci and type-N Weyl tensors is degenerate; thatis, all positive boost weight components of the Riemanntensor and its covariant derivatives, rðnÞ (Riemann), arezero [19]. This further implies, that all tensors rðnÞ(Riemann), n  1, are also of maximal boost order 2.Since the components of a two tensor can only have boostweights 2; . . . ;þ2, and using the same arguments as in[5] the traceless part of any second rank symmetric tensorcan only have boost-weight2, and, hence, also be of typeN. Furthermore, using the appendix of [5], the symmetrictensor can only be linear in g or linear in the contractionsof curvature tensors of the form rðnÞ (Riemann). One canput any contraction into the sought form by changing theorder of covariant derivatives with the help of the Ricciidentity;½r;rT... ¼ R	T	... þ    þ R	T...	;and by using the contractions of the Bianchi identity, thatare rR ¼ rR rR and rR ¼ 0. Then,one can now see that the above theorem follows.An immediate consequence of the above theorem is thereduction of the field equation corresponding to the mostgeneral action;I ¼ZdDxffiffiffiffiffiffiffigp fðg; R;rR;. . . ; ðr1r2 . . .rM ÞR; . . .Þ; (7)to the formE ¼ eg þXNn¼0anhnS ¼ 0; (8)where e and an’s are constants depending on the explicitform of the action. Here, N is related to the number ofcovariant derivatives of the Riemann tensor or its contrac-tions in the action: For the low-energy limit of the fullstring theory, N goes to infinity. Hence, we reduced thefield equations of the most general higher derivative theo-ries to a form linear in the Ricci tensor. Furthermore, thetrace part of (8) ise ¼ 0; (9)which gives the relation between the cosmologicalconstant and the parameters of the theory. To have fullsolutions of (8), there must exist at least a real solution to(9). We assume that there is such a solution. The tracelesspart isXNn¼0anhnS ¼ 0: (10)This is still hard to solve: despite its appearance it is anonlinear equation. But, for the Kundt type of Kerr-Schildmetrics g ¼ g þ 2V‘‘ and S ¼ ‘‘ (see[18,20] for more details), one gets hnS ¼ ‘‘Onfor all n ¼ 0; 1; 2; . . . ; N, then contractions of hnS van-ish identically. This nice property of the Kerr-Schildmetrics leads to a further reduction of the field equation.There are two explicit metrics, the AdS-plane [8,9] and theAdS-spherical waves [10], which are of the Kerr-Schildform and belong to the Kundt class of type-N spacetimes.For these metrics, g is the AdS background metric andthe function V satisfies ‘@V ¼ 0. Then,  has the formOV¼hþ2@þ12þ2RðD2ÞDðD1ÞV; (11)where h is the Laplace-Beltrami operator of the AdSbackground, and  arises in r‘ ¼ ‘ðÞ. Becauseof (11), Eq. (10) reduces toXNn¼0anOnþ1V ¼ 0: (12)For N > 1, this equation can be factorized asYNn¼0ðOþ bnÞOV ¼ 0; (13)where some bn’s are real and some are complex constantsin general (complex bn’s come in complex conjugatepairs). Then, the most general solution isV ¼ <XNi¼0Vi; (14)where < represents the real part and Vi’s solve ðOþbiÞVi ¼ 0, i ¼ 0; 1; 2; . . . ; N. Solutions of such linearPRL 111, 101101 (2013) P HY S I CA L R EV I EW LE T T E R Sweek ending6 SEPTEMBER 2013101101-2partial differential equations are given in [8,10] in theAdS background. As a conclusion, we can say that theAdS-wave metrics found recently in [8,10] solve the fieldequations of the most general higher derivative theories. Togive explicit exact solutions, we have to know the constantsbi, i ¼ 0; 1; 2; . . . in terms of the parameters of the theory.For this purpose, we shall consider some special casesbelow. Note that in the above discussion, we have consid-ered only the negative cosmological constant backgroundmetric, AdS. With a positive cosmological constant back-ground metric, dS, it is not possible to obtain analogousexplicit metrics given in [8–10]. For a flat backgroundmetric, the solutions reduce to the plane wave solutionsof generic gravity theories [1,2].Type-N spacetimes in quadratic gravity.—For type-NWeyl and type-N traceless Ricci tensors, and using (5),the field equations of quadratic gravity [21]1R12gRþ0gþ2RR14gRþð2þÞðghrrÞRþhR12gRþ2R14gRRþ2hRR2RRþRR 2RR14gðR2	4R2þR2Þi¼ 0; (15)reduce to the following simplified equations [15,16],ðhþ cÞð‘‘Þ ¼ 0; (16)and a trace equation that gives a relation between theconstant R and the parameters of the theory. Here, c isgiven as [21]c 1þ2Rþ 2ðD2ÞDðD1ÞRþ2ðD3ÞðD4ÞDðD1Þ R: (17)Exact solutions, the AdS-wave metrics, of this have beenreported recently [8–10].fðRÞtheory.—Consider the theory with an action thatdepends on the Riemann tensor and its contractions but noton the derivatives of the Riemann tensor:I ¼ZdDxffiffiffiffiffiffiffigp fðRÞ: (18)For this case, N ¼ 1 and hence (10) reduces toðahþ bÞð‘‘Þ ¼ 0; (19)where a and b are constants. In a generic higher curvaturetheory, the constant a is not zero. However, if the theoryparameters are tuned properly, a can be zero and anexample of this case, the Lanczos-Lovelock theory, isconsidered below. On the other hand, if a is not zero, forAdS-wave metrics in the Kerr-Schild form, the fieldequations reduce to a fourth order linear partial differentialequation with constant coefficients of the formh 2RDðD1ÞM2h 2RDðD1ÞðV‘‘Þ¼0; (20)where h is the Laplace-Beltrami operator of the AdSbackground andM2 ¼ ba 2RDðD 1Þ ; (21)which corresponds to the mass of the spin-2 excitation inthe linearized version of the fðRÞ theory about AdS [22].For the AdS-plane wave,ds2 ¼  RDðD 1Þz22dudvþ XD3m¼1ðdxmÞ2 þ dz2; (22)where u and v are null coordinates, and z is a spatialcoordinate with z > 0. Then,‘ ¼ ð1; 0; 0; . . . ; 0Þ;  ¼ 2z z; (23)and one can find a specific solution [8]Vðu; zÞ ¼ c1ðuÞzD3 þ c2ðuÞ 1z2þ zðD5Þ=2ðc3ðuÞz þ c4ðuÞzÞ; (24)where ci are arbitrary functions and ¼ 12ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðD 1Þ2  4DðD 1ÞM2Rs:Thus, the solution takes the formds2 ¼  RDðD 1Þz22dudvþ XD3m¼1ðdxmÞ2 þ dz2þ Vðu; zÞdu2: (25)In addition, the most general solution of (20) wasalso given in [8]. Note that for the AdS-plane wave, ‘is proportional to the null Killing vector , i.e., ¼ ð1=zÞ‘.On the other hand, for the AdS-spherical wave, oneobtainsds2 ¼  RDðD 1Þz2dt2 þ XD1i¼1ðdxiÞ2þ 2Vð‘dxÞ2; (26)where‘ ¼1;xir;  ¼  1r ‘ þ2rt þ 2z z; (27)with r2 ¼ PD1i¼1 ðxiÞ2 and xD1 ¼ z. The most general V asa solution of (20) was given in [10] whose explicit form isPRL 111, 101101 (2013) P HY S I CA L R EV I EW LE T T E R Sweek ending6 SEPTEMBER 2013101101-3not particularly illuminating to depict here. In this case,there exists no null Killing vector fields. For M2  0, Vdecays sufficiently fast for both AdS-wave solutions andthey are asymptotically AdS. When M2 ¼ 0, there arelogarithmic solutions which spoil the asymptotically AdSstructure in both cases. Let us give two concrete fðRÞtheories as examples.Cubic gravity generated by string theory.—In [7], it wasshown that the bosonic string has the following effectiveLagrangian density at O½ð0Þ2feff ¼Rþ04RR 4R RþR2þð0Þ2242RR	R	þRR	R	; (28)where 0 is the usual inverse string tension. Using theresults of [23], the field equations of (28) for the AdS-plane and AdS-spherical wave metrics reduce to (20) witha and M2 given asa ¼ 702R4DðD 1Þ ; (29)M2 ¼  4DðD 1Þ702R 2ðD 3ÞðD 4Þ70þ 9D 297DðD 1ÞR:(30)Therefore, the solutions quoted above are the solutionsof (28) with this M2.Lanczos-Lovelock theory.—Another special case of thef (Riemann) theory is the Lanczos-Lovelock theory givenwith the Lagrangian densityfL-L ¼X½ðD1Þ=2n¼0an1...2n1...2nYnp¼1R2p12p2p12p ; (31)where an’s are dimensionful constants, [ðD 1Þ=2] corre-sponds to the integer part of ðD 1Þ=2 and 1...2n1...2n is thegeneralized Kronecker delta. In this case, since the con-stant a in (19) vanishes identically, the field equationsreduce tob ¼ 0; (32)where b was calculated in [24] asb ¼ 2ðD 3Þ! X½ðD1Þ=2n¼0annðD 2n 1Þ!2RDðD 1Þn1:(33)Equation (32) gives two subclasses. The first classcorresponds to  ¼ 0 which is of type-N Einstein space.The second class corresponds to b ¼ 0 which gives arelation between the parameters of the theory. In thissubclass, any type-N radiating metric with a constantRicci scalar is an exact solution of the theory. AdS-wavemetrics are exact solutions of the Lancsoz-Lovelock the-ory. We note that the Lanczos-Lovelock theory is free oflogarithmic solutions.Conclusion.—Type-N radiating spacetimes simplify thefield equations of higher derivative theories. This simplifi-cation rests on the result that any second rank symmetrictensor constructed from contractions of type-N Weyl andtype-N traceless Ricci tensors, and their covariant deriva-tives, reduces to a simpler form. To find explicit solutions,we showed that one has to consider a subclass that is theKundt spacetime. Hence, Kerr-Schild metrics reportedrecently as the solutions of the quadratic gravity, theAdS-plane [8] and AdS-spherical wave metrics [10], whichbelong to the Kundt subclass of type-N spacetimes, solvethe field equations of any higher curvature gravity theoryexactly. We gave a cubic theory coming from the stringtheory and the Lanczos-Lovelock theory as explicitexamples. A more detailed version of this work will becommunicated elsewhere.This work is partially supported by the Scientific andTechnological Research Council of Turkey (TUBITAK).*gurses@fen.bilkent.edu.tr†sigbjorn.hervik@uis.no‡tahsin.c.sisman@gmail.com§btekin@metu.edu.tr[1] R. Güven, Phys. Lett. B 191, 275 (1987).[2] G. T. Horowitz and A. R. Steif, Phys. Rev. Lett. 64, 260(1990).[3] G. T. Horowitz and A.A. Tseytlin, Phys. Rev. D 51, 2896(1995).[4] A. A. Coley, Phys. Rev. Lett. 89, 281601 (2002).[5] A. A. Coley, G.W. Gibbons, S. Hervik, and C.N. Pope,Classical Quantum Gravity 25, 145017 (2008).[6] T. Banks and M.B. 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Anti-de sitter-wave solutions of higher derivative theories

We show that the recently found anti-de Sitter (AdS)-plane and AdS-spherical wave solutions of quadratic curvature gravity also solve the most general higher derivative theory in D dimensions. More generally, we show that the field equations of such theories reduce to an equation linear in the Ricci tensor for Kerr-Schild spacetimes having type-N Weyl and type-N traceless Ricci tensors

AdS-Wave Solutions of f(Riemann) Theories

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