Thermodynamics of a BTZ black hole solution with an Horndeski source
aa r X i v : . [ h e p - t h ] M a y Thermodynamics of a BTZ black hole solution with an Horndeski source
Moises Bravo-Gaete ∗ and Mokhtar Hassaine † Instituto de Matem´atica y F´ısica, Universidad de Talca, Casilla 747, Talca, Chile.
In three dimensions, we consider a particular truncation of the Horndeski action that reduces tothe Einstein-Hilbert Lagrangian with a cosmological constant Λ and a scalar field whose dynamicsis governed by its usual kinetic term together with a nonminimal kinetic coupling. Requiring theradial component of the conserved current to vanish, the solution turns out to be the BTZ black holegeometry with a radial scalar field well-defined at the horizon. This means in particular that thestress tensor associated to the matter source behaves on-shell as an effective cosmological constantterm. We construct an Euclidean action whose field equations are consistent with the original onesand such that the constraint on the radial component of the conserved current also appears as a fieldequation. With the help of this Euclidean action, we derive the mass and the entropy of the solution,and found that they are proportional to the thermodynamical quantities of the BTZ solution by anoverall factor that depends on the cosmological constant. The reality condition and the positivityof the mass impose the cosmological constant to be bounded from above as Λ ≤ − l where thelimiting case Λ = − l reduces to the BTZ solution with a vanishing scalar field. Exploiting ascaling symmetry of the reduced action, we also obtain the usual three-dimensional Smarr formula.In the last section, we extend all these results in higher dimensions where the metric turns out tobe the Schwarzschild-AdS spacetime with planar horizon. I. INTRODUCTION
Since the discovery of the BTZ black hole solution[1], three-dimensional Einstein gravity with a negativecosmological constant has become an important field ofinvestigations. A considerable number of papers hasbeen devoted to the physical and mathematical impli-cations of the BTZ solution in particular in the contextof AdS /CFT correspondence. Indeed, it is now wellaccepted that three-dimensional gravity is an excellentlaboratory in order to explore and test some of the ideasbehind the AdS/CFT correspondence [2].It is well-known that the static BTZ geometry whoseline element is given by ds = − (cid:16) r l − M (cid:17) dt + dr r l − M + r dϕ , (1)is a solution of the Einstein equations with a fixed valueof the negative cosmological constant − l − , G µν − l − g µν = 0 . Here, we will exhibit a matter action that sources theBTZ spacetime. In order to achieve this task, the corre-sponding stress tensor T µν of the matter source must be-have on-shell as an effective cosmological constant term,i. e. T on-shell µν = (cid:0) l − + Λ (cid:1) g µν . Indeed, in this case, it is simple to realize that the Ein-stein equations G µν + Λ g µν = T µν , ∗ Electronic address: mbravog-at-inst-mat.utalca.cl † Electronic address: hassaine-at-inst-mat.utalca.cl will automatically be satisfied on the BTZ metric (1).For that purpose, we consider the following three-dimensional action S = Z √− g d x (cid:18) R − −
12 ( αg µν − ηG µν ) ∇ µ φ ∇ ν φ (cid:19) , (2) where R and G µν stand respectively for the Ricci scalarand the Einstein tensor. This model is part of the so-called Horndeski action which is the most general tensor-scalar action yielding at most to second-order field equa-tions in four dimensions [3]. In three dimensions, thecorresponding field equations are also of second order,and are given by G µν + Λ g µν = 12 h αT (1) µν + ηT (2) µν i , (3a) ∇ µ [( αg µν − ηG µν ) ∇ ν φ ] = 0 , (3b)where the stress tensors T ( i ) µν are defined by T (1) µν = (cid:16) ∇ µ φ ∇ ν φ − g µν ∇ λ φ ∇ λ φ (cid:17) . (4) T (2) µν = 12 ∇ µ φ ∇ ν φR − ∇ λ φ ∇ ( µ φR λν ) − ∇ λ φ ∇ ρ φR µλνρ − ( ∇ µ ∇ λ φ )( ∇ ν ∇ λ φ ) + ( ∇ µ ∇ ν φ ) (cid:3) φ + 12 G µν ( ∇ φ ) − g µν (cid:20) −
12 ( ∇ λ ∇ ρ φ )( ∇ λ ∇ ρ φ ) + 12 ( (cid:3) φ ) − ∇ λ φ ∇ ρ φR λρ (cid:21) . The scalar field equation (3b) is a current conservationequation which is a consequence of the shift symmetry ofthe action, φ → φ + const..The first exact black hole solution of these equationswithout cosmological constant was found in [4]. However,in this case, the scalar field becomes imaginary outsidethe horizon. Recently, this problem has been circum-vented by adding a cosmological constant term yieldingto asymptotically locally (A)dS (and even flat for α = 0)black hole solutions with a real scalar field outside thehorizon [5]. The electric charged version of the AdS so-lutions with a Maxwell field have been studied in [6].The field equations (3) admit other interesting solutionswith a nontrivial and regular time-dependent scalar fieldon a static and spherically symmetric spacetime [7]. In-terestingly enough, this solution in the particular case ofΛ = η = 0 reduces to an unexpected stealth configurationon the Schwarzschild metric [7]. There also exist Lifshitzblack hole solutions with a time-independent scalar fieldfor a fixed value of the dynamical exponent z = , [8].Solutions for a more general truncation of the Horndeskiaction that is shift-invariant as well as enjoying the re-flection symmetry φ → − φ have been obtained in [9]. Inall these examples, in order to satisfy the radial part ofthe current conservation (3b) without imposing any con-straints on the radial derivative of the scalar field, thegeometry has been chosen such that αg rr − ηG rr = 0 . (5)We note that this condition simplifies considerably thefield equations in particular in the time-independent casewhere the full conservation equation (3b) is automaticallysatisfied without constraining the radial dependence ofthe scalar field.In the present work, we will show that the BTZ ge-ometry naturally emerges as a solution of this particularHorndeski action (2) in three dimensions subjected to thecondition (5). We will also analyze the thermodynamicalimplications of such solution and extend all our results inarbitrary dimension. The plan of the paper is organizedas follows. In the next section, we present in details thederivation of the solution using the constraint (5). InSec. III, we construct an Euclidean action whose fieldequations turn to be consistent with the original onesand such that the constraint on the radial current (5)naturally appears as a field equation. This constructionwill be useful to obtain the mass and the entropy of thesolution. The usual Smarr formula is also derived by ex-ploiting a scaling symmetry of the reduced action. Therotating version of the solution as well as a particularexample of time-dependent solution will be reported inSec. IV. In Sec. V, we extend all the results to arbitrarydimension where the metric solution is nothing but theSchwarzschild-AdS spacetime with planar horizon. Thelast section is devoted to our conclusions. II. DERIVATION OF THE SOLUTION
Let us derive the most general solution of the fieldequations (3) subjected to the condition (5) with anAnsatz of the form ds = − h ( r ) dt + dr f ( r ) + r dϕ ,φ = φ ( r ) . (6) For clarity, we define ǫ µν := G µν + Λ g µν − h αT (1) µν + ηT (2) µν i . The condition (5) on the geometry becomes f ( r ) = 2 αrh ( r ) ηh ′ ( r ) , (7)and automatically implies that the current conservation(3b) is satisfied. Using this last relation, the radial com-ponent of the Einstein equations ǫ rr = 0 allows to expressthe square of the derivative of the scalar field as( φ ′ ) = − ( α + η Λ) h ′ α rh . The remaining independent Einstein equation, ǫ tt = 0 orequivalently ǫ ϕϕ = 0, yields ǫ tt ∝ ( α − η Λ) [ rh ′′ − h ′ ] = 0 . Hence, it is clear that the point defined by α = η Λ corre-sponds to a degenerate sector [8], while for α = η Λ, thesolution is given by h ( r ) = Cr − M, f ( r ) = αηC ( Cr − M ) , where C and M are two integration constants. In orderto deal with the BTZ metric (1), we choose C = l − andthe coupling constants must be fixed such that αη = l − . (8)Note that the degenerate sector α = η Λ corresponds toa choose of the cosmological constantΛ degenerate = l − . (9)In sum, for αη = l − and Λ = Λ deg , the solution is givenby the BTZ metric (1) together with a radial scalar field ξ ( r ) := ( φ ′ ) = − l + 1) η (cid:0) r l − M (cid:1) (10a) φ ( r ) = ± s − l (Λ l + 1) η ln rl + r r l − M ! , (10b)provided that 2 l (Λ l + 1) η ≤ . (11)Various comments can be made concerning this solution.Firstly, we note that for Λ = − l − , the scalar field van-ishes identically and the solution reduces to the BTZ so-lution. Secondly, the scalar field is well-defined at thehorizon r + = l √ M , and as expected the stress tensor ofthe matter part behaves on-shell as an effective cosmo-logical constant12 h αT (1) µν + ηT (2) µν i on-shell = (cid:0) Λ + l − (cid:1) g µν . As a last comment, we remark that for α > η > ≤ − l − . The limiting case Λ = − l − corre-sponding to the BTZ solution without source.In what follows, we will derive the mass and the en-tropy of the solution (10). III. THERMODYNAMICS OF THE BLACKHOLE SOLUTION
The partition function for a thermodynamical ensem-ble is identified with the Euclidean path integral in thesaddle point approximation around the Euclidean con-tinuation of the classical solution [10]. The Euclideanand Lorentzian action are related by I E = − iI where the periodic Euclidean time is τ = it . The Euclidean contin-uation of the class of metrics considered here is given by[15] ds = N ( r ) F ( r ) dτ + dr F ( r ) + r dϕ . In order to avoid conical singularity at the horizon in theEuclidean metric, the Euclidean time is made periodicwith period β and the Hawking temperature T is givenby T = β − . Since we are only interested in static so-lution with a radial scalar field, it is enough to considera reduced action principle. However, there is an impor-tant subtlety that has to do with the constraint (5) weused in order to derive our solution. Indeed, this con-straint together with the fact of looking for a static scalarfield make the equation associated to the variation of thescalar field (3b) redundant in the sense that the equationis automatically satisfied. Hence, in our reduced action,the constraint (5) should appear as a field equation inorder to deal with an equivalent problem. This can beachieved considering the following Euclidean action I E := I E ( N, F, ξ ) = 2 πβ Z ∞ r + N (cid:20) F ′ + 2Λ r + α rF ξ + 34 ηF F ′ ξ + η F ξ ′ (cid:21) dr + B E , (12)where the dynamical field is chosen to be ξ ( r ) := ( φ ′ ) and not the scalar field itself φ . Note that r + is thelocation of the horizon and B E is a boundary term thatis fixed by requiring that the Euclidean action has anextremum, that is δI E = 0. In this case, the variationwith respect to the dynamical fields N, F and ξ yield E N := F ′ + 2Λ r + α rF ξ + 34 ηF F ′ ξ + η F ξ ′ = 0 ,E F := − N ′ (cid:18) ηF ξ (cid:19) + N (cid:18) α rξ + 14 ηF ξ ′ (cid:19) = 0 ,E ξ := − η N ′ F + N (cid:18) α rF − ηF ′ F (cid:19) = 0 , (13)and the last equation E ξ = 0 is nothing but the constraintused previously (5) to obtain our solution. At the specialpoint α = ηl , the equations (13) turn out to be equivalentto the original ones supplemented by the constraint (5).Indeed, the most general solution of the system (13) canbe derived as follows. For X ( r ) := 4 + 3 ηF ( r ) ξ ( r ) = 0[16], we consider the combination − ηF X E F + ηN FX E N + E ξ = 0 , which permits to obtain ξ ( r ) = − l + 1) ηF ( r ) . Injecting this expression into E N = 0, one obtains that F ( r ) = r /l − M where M is an integration constant,and finally the equation E ξ = 0 implies that N is con-stant which can be chosen to 1 without any loss of gen-erality. Hence, at α = ηl , the most general solution ofthe system (13) is given by N ( r ) = 1 , F ( r ) = r l − M, ξ ( r ) = − l + 1) ηF ( r ) , (14)and corresponds to the solution obtained previously (10).We now determine the boundary term of the Euclideanaction which is given by δB E = − πβ (cid:20) δF (cid:18) ηF ξ (cid:19) + η F δξ (cid:21) ∞ r + . (15)In order to obtain δB E , we need the variations of thefield solutions (14) at infinity δF | ∞ = − δM, ( F δξ ) | ∞ = 2(Λ l + 1) η δF | ∞ , while at the horizon, they are given by δF | r + = − F ′ | r + δr + = − πβ δr + , ( F δξ ) | r + = 2(Λ l + 1) η δF | r + = − l + 1) η πβ δr + . Hence, we have I E = B E ( ∞ ) − B E ( r + ) = 2 π h βM − πr + i (cid:18) − Λ l (cid:19) , and, we can identify the mass M and the entropy S ofthe solution to be given by M = ∂I E ∂β , S = β ∂I E ∂β − I E , yielding M = 2 πM (cid:18) − Λ l (cid:19) , S = 8 π (cid:18) − Λ l (cid:19) r + . (16)Since the Hawking temperature is given by T = π r + ,it is easy to see that the first law d M = T d S holds. Inthe BTZ case Λ = − l − , the scalar field vanishes andthe mass and entropy reduce to the thermodynamicalquantities of the BTZ solution [1].We can now go further by exploiting a scaling symme-try of the reduced action in order to obtain the usualthree-dimensional Smarr formula in the same lines asthose done in Ref. [11]. In fact, it is easy to see thatthe reduced action (12) enjoys the following scaling sym-metry ¯ r = σr, ¯ N (¯ r ) = σ − N ( r ) , ¯ F (¯ r ) = σ F ( r ) , ¯ ξ (¯ r ) = σ − ξ ( r ) , (17)from which one can derive a Noether quantity C ( r ) = N h (cid:18) ηF ξ (cid:19) ( − F + rF ′ ) + η F ( rξ ′ + 2 ξ ) i , which is conserved C ′ ( r ) = 0 by virtue of the field equa-tions (13). Evaluating this quantity at infinity and at thehorizon, one gets C ( r = ∞ ) = M (1 − Λ l ) , C ( r = r + ) = 4 πβ r + (cid:18) − Λ l (cid:19) . Since the Noether charge is conserved, these expressionsmust be equal M (1 − Λ l ) = 4 πβ r + (cid:18) − Λ l (cid:19) , which in turn implies the following Smarr formula M = T S . (18)This latter corresponds to the standard three-dimensional Smarr formula [12]. IV. ROTATING, TIME DEPENDENT ANDSTEALTH SOLUTIONS
Operating a Lorentz boost in the plane ( t, ϕ ), we ob-tain the rotating version of the solution found previously.At the point α = η l − , the metric function turns out tobe the rotating BTZ ds = − F ( r ) dt + dr F ( r ) + r (cid:18) dϕ − J r dt (cid:19) , (19)where the structural function F is given by F ( r ) = r l − M + J r , (20)and, where J corresponds to the angular momentum.The scalar field solution read ξ ( r ) := ( φ ′ ( r )) = − l Λ + 1) ηF ( r ) . (21)We also report two solutions with a linear time scalarfield on the rotating BTZ metric (19). The first one isobtained for α = η l − and given by φ ( t, r ) = q t ± Z s q η − F ( r )(Λ l + 1) ηF ( r ) dr, (22)where q is a constant. The second solution correspondsto a stealth configuration, that is a solution where bothside of the Einstein equations (3a) vanish identically G µν + Λ g µν = 0 = 12 h αT (1) µν + ηT (2) µν i . (23)In fact, for Λ = − /l , a solution of the stealth equations(23) is given by the rotating BTZ metric (19) togetherwith a time-dependent scalar field φ ( t, r ) = q (cid:18) t ± Z drF ( r ) (cid:19) , (24)where q is a constant. This stealth solution is differentfrom the one derived in [13], where in this reference theauthors considered as a source a scalar field nonminimallycoupled. Moreover, in [13], the time-dependent stealthonly exists in the case of the static BTZ metric, that isfor J = 0. V. EXTENSION TO HIGHER DIMENSIONS
We now extend our analysis in arbitrary D dimensionswith the action S = Z √− g d D x (cid:18) R − −
12 ( αg µν − ηG µν ) ∇ µ φ ∇ ν φ (cid:19) , (25) for which the field equations are given by (3). Here, wewill consider an Ansatz metric with a planar horizon anda static radial scalar field ds = − N F dt + dr F + r d~x D − φ = φ ( r ) . The solution of the field equations for this Ansatzsubjected to the constraint (5) is now given by theSchwarzschild-AdS metric with a planar horizon ds = − F ( r ) dt + dr F ( r ) + r d~x D − , (26a) F ( r ) = r l − Mr D − , (26b) ξ ( r ) := ( φ ′ ) = − l Λ + ( D − D − η ( D − D − F ( r ) , (26c)provided that αη = ( D − D − l . (27)Note that as in the three-dimensional case, one can ob-tain an explicit expression of the scalar field φ ( r ) = ± D − s − l (cid:2) l + ( D − D − (cid:3) η ( D − D − × ln " r D − rl + r r l − Mr D − ! . In the case D = 3, the solution reduces to the one previ-ously derived (10) on the BTZ spacetime, and in D = 4,this solution was already reported in Ref. [5]. As before,the stress tensor associated to the variation of the mat-ter source behaves on-shell as an effective cosmologicalconstant term, that is12 h αT (1) µν + ηT (2) µν i on-shell = (cid:18) Λ + ( D − D − l (cid:19) g µν . The reality condition (26c) together with the relation(27) and requiring the standard kinetic term to have theright sign α > ≤ − ( D − D − l . (28)The Euclidean action is now given by I E ( N, F, ξ ) = β Vol(Σ D − ) Z ∞ r + N h ( D − r D − F ′ + 2Λ r D − + α r D − F ξ + 3( D − r D − ηF F ′ ξ + ( D − η r D − F ξ ′ + ( D − D − (cid:16) r D − F + η F ξr D − (cid:17) i dr + B E , (29)where Vol(Σ D − ) stands for the volume of the com-pact ( D − − dimensional planar manifold, and r + =( l M ) / ( d − is the location of the horizon. The vari-ation with respect to the dynamical fields N, F and ξ yield E N := ( D − r D − F ′ + 2Λ r D − + α r D − F ξ + 3( D − r D − ηF F ′ ξ + ( D − η r D − F ξ ′ +( D − D − (cid:16) r D − F + η F ξr D − (cid:17) = 0 , E F := − N ′ (cid:18) ( D − r D − + 34 ( D − r D − ηF ξ (cid:19) + N (cid:18) α r D − ξ + ( D − r D − ηF ξ ′ − ( D − D − r D − F ηξ (cid:19) = 0 ,E ξ := − ( D − η r D − N ′ F + N (cid:18) α r D − F − ( D − r D − η F F ′ − ( D − D − r D − F η (cid:19) = 0 , and the last equation E ξ = 0 is again proportional to theconstraint (5) used previously to obtain the solution. Asbefore, at the special point (27), this system of equationsis equivalent to our original equations supplemented bythe constraint (5) which also appears as a field equation.The most general solution yields to (26) together with N ( r ) = 1.We are now in position to compute the variation δB E = − β Vol(Σ D − )( D − r D − × (cid:20) δF (cid:18) ηF ξ (cid:19) + η F δξ (cid:21) ∞ r + , (30)which permits to obtain that I E = ( D − D − − l Λ2( D − (cid:2) βM − πr D − (cid:3) , We derive the mass M = ∂I E ∂β and the entropy S = β ∂I E ∂β − I E , that read M = (cid:20) ( D − D − − l Λ2( D − (cid:21) M Vol(Σ D − ) (31) S = (cid:20) ( D − D − − l Λ2( D − (cid:21) πr D − Vol(Σ D − ) , and once again, one can easily check that the first lawholds. For the Schwarzschild-AdS case, that is for Λ = − ( D − D − l , these formula reduces to those found in[14]. Finally, the Noether conserved quantity C ( r ) = N r D − ( D − h (cid:18) ηF ξ (cid:19) ( − F + rF ′ )+ η F ( rξ ′ + 2 ξ ) i , (32)which is a consequence of the scaling symmetry of thereduced action (29)¯ r = σr, ¯ N (¯ r ) = σ − D N ( r ) , ¯ F (¯ r ) = σ F ( r ) , ¯ ξ (¯ r ) = σ − ξ ( r ) , (33)permits to derive the following Smarr formula M = 1 D − T S . (34)One may mention that the previous scaling symmetrywill not be possible in the case of spherical or hyperboloidhorizon. VI. CONCLUSIONS
We have been concerned with a particular truncationof the Horndeski theory in three dimensions given by theEinstein-Hilbert piece plus a cosmological constant anda scalar field with its usual kinetic term and a nonmin-imal kinetic coupling. For this model, we have derived the most general solution subjected to the condition (5).In this case, the metric turns to be the BTZ spacetimeand the radial scalar field is shown to be well-defined atthe horizon. We have seen that such solution occurs be-cause the stress tensor associated to the variation of thematter source behaves on-shell as a cosmological constantterm. The constraint on the radial component of the con-served current (5) together with the fact of looking fora static scalar field only impose a restriction on the ge-ometry and not on the scalar field. In other words, thismeans that the field equation associated to the variationof the scalar field is automatically satisfied without im-posing any restriction on the radial dependence of thescalar field. In order to compute the mass and the en-tropy of the solution, we have constructed an Euclideanaction whose field equations turn out to be equivalent tothe original Einstein equations and such that the con-straint on the radial component of the conserved currentappears also as a field equation. This last fact has re-sulted to be primordial to obtain the mass and the en-tropy, and we have verified that the first law was satis-fied. This reduced action has also be useful in order toderive the usual Smarr formula by exploiting a scalingsymmetry. We have extended all these results in arbi-trary dimension where the metric solution is nothing butthe Schwarzschild-AdS spacetime with a planar horizon.In this case also, we have been able to construct an Eu-clidean action whose field equations are equivalent to theoriginal ones supplemented by the constraint conditionon the geometry (5). In all these examples, the horizontopology is planar but this hypothesis is only essentialin order to establish a scaling symmetry of the reducedaction. In fact, in the spherical or hyperboloid cases, onewould be able to construct along the same lines the Eu-clidean action sharing the same features except enjoyingthe scaling symmetry. In higher dimensions, the authorsof Ref. [5] have considered the same model and foundblack hole solutions with spherical and hyperboloid hori-zon topology. In these cases, they have computed thethermodynamical quantities by regularizing the actionwith the use of a regular soliton solution. It will be in-teresting to see wether the Euclidean approach describedhere may yield the same results. It is also appealing thatup to now all the known solutions of the equations (3)are those where the constraint (5) is imposed. It willbe interesting to see wether there exist other solutionsfor which the radial component of the current conserva-tion is not vanishing. Finally, we have also obtained aparticular time-dependent solution where the scalar fielddepends linearly on the time. For such solution, the is-sue concerning the thermodynamical analysis is not clearfor us. Hence, a natural work will consist in providinga consistent Hamiltonian formalism in order to computethe mass, the entropy and also to give a physical inter-pretation on the additional constant q that appears inthe solution. This problem will also be relevant in thecontext of the Lifshitz case where the known solutions[8, 9] are necessarily time-dependent. Acknowledgments
We thank Julio Oliva for useful discussions. MB issupported by BECA DOCTORAL CONICYT 21120271. MH is partially supported by grant 1130423 fromFONDECYT and from CONICYT, Departamento deRelaciones Internacionales “Programa Regional MATH-AMSUD 13 MATH-05”. [1] M. Ba˜nados, C. Teitelboim and J. Zanelli, Phys. Rev.Lett. , 1849 (1992) [hep-th/9204099].[2] J. M. Maldacena, Adv. Theor. Math. Phys. , 231 (1998)[hep-th/9711200].[3] G. W. Horndeski, Int. J. Theor. Phys. , 363 (1974).[4] M. Rinaldi, Phys. Rev. D , 084048 (2012)[arXiv:1208.0103 [gr-qc]].[5] A. Anabalon, A. Cisterna and J. Oliva, arXiv:1312.3597[gr-qc].[6] A. Cisterna and C. Erices, Phys. Rev. D , 084038(2014) [arXiv:1401.4479 [gr-qc]].[7] E. Babichev and C. Charmousis, arXiv:1312.3204 [gr-qc].[8] M. Bravo-Gaete and M. Hassaine, arXiv:1312.7736 [hep-th].[9] T. Kobayashi and N. Tanahashi, arXiv:1403.4364 [gr-qc]. [10] G. W. Gibbons and S. W. Hawking, Phys. Rev. D ,2752 (1977).[11] M. Banados and S. Theisen, Phys. Rev. D , 064019(2005) [hep-th/0506025].[12] L. Smarr, Phys. Rev. Lett. , 71 (1973).[13] E. Ayon-Beato, C. Martinez and J. Zanelli, Gen. Rel.Grav. , 145 (2006) [hep-th/0403228].[14] D. Birmingham, Class. Quant. Grav. , 1197 (1999)[hep-th/9808032].[15] For the Ansatz considered in (6), this will correspond to h ( r ) = N ( r ) F ( r ) and f ( r ) = F ( r ).[16] For X ( r ) = 0, one ends with a particular case of (10) witha fixed value of the cosmological constant Λ = − ll