Cylindrical solutions in metric f(R) gravity
aa r X i v : . [ g r- q c ] D ec Cylindrical solutions in metric f(R) gravity
A. Azadi a ∗ , D. Momeni a † and M. Nouri-Zonoz a,b ‡ a Department of Physics, University of Tehran,North Karegar Ave., Tehran 14395-547, Iran. b School of Astronomy and Astrophysics,Institute for Research in Fundamental Sciences (IPM),P. O. Box 19395-5531 Tehran, Iran.
Abstract
We study static cylindrically symmetric vacuum solutions in Weyl coordinates in the context ofthe metric f(R) theories of gravity. The set of the modified Einstein equations is reduced to a singleequation and it is shown how one can construct exact solutions corresponding to different f ( R )models. In particular the family of solutions with constant Ricci scalar ( R = R ) is found explicitlywhich, as a special case ( R = 0), includes the exterior spacetime of a cosmic string. Another newsolution with constant, non-zero Ricci scalar is obtained and its possible relation to the Linet-Tiansolution in general relativity is discussed. ∗ Electronic address: amir [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] . INTRODUCTION As pointed out nicely by Weinberg in his seminal paper [1] on the cosmological constantproblem, ”
Physics thrives on crisis ”. Perhaps the biggest crisis of the 20-th century physicswhich was carried over to the 21-st century is the so called cosmological constant or darkenergy problem: the 120 orders of magnitude difference between the observational andtheoretical values of the vacuum energy density. To overcome this crisis, different proposalshave been put forward with the hope of obtaining a consistent theoretical background tothe recent observation of an expanding universe which is seemingly not accessible throughthe standard model of cosmology. Obviously one might trace back this lack of a propertheoretical explanation to the basic ingredients of the standard model of cosmology, one ofwhich being the Einstein field equations derived from the Einstein-Hilbert action.
Modified or alternative theories of gravity is the paradigm under which all those theories which differfrom the Einsteinian gravity are studied. One of these non-Einsteinian theories, that arosea lot of enthusiasm recently, is the so called f ( R ) gravity in which a function, f ( R ), replacesthe Einstein-Hilbert (gravitational) Lagrangian R . It seems that f ( R ) actions were firstcontemplated by Eddington [2] and later on rigorously studied by Buchdahl [3] in the contextof nonsingular oscillating cosmologies. These theories could be thought of as a special kindof higher derivative gravitational theories. Having shown that these models are equivalentto scalar-tensor models of gravity it is obvious that one should in the first place check theirconsistency with the solar system tests of Einstein gravity [4]. In most of the models it is notpossible to satisfy these tests and at the same time to account for the accelerated expansionof the Universe without bringing in new degrees of freedom. Recently it was shown that thereare models of f ( R ) gravity in which one could account for both the accelerated expansion ofthe universe and the solar system tests without introducing the cosmological constant [5]-[6].This is why spherically symmetric solutions are the most widely studied exact solutions inthe context of f ( R ) gravity. Apart from this it is interesting, at least from theoretical point ofview, to consider other exact solutions of the modified Einstein equations of f ( R ) theory. Asin the case of the Einstein-Hilbert action one could derive field equations in f ( R ) gravity intwo different approaches, the so called metric and Palatini approaches. But in f ( R ) actions,unlike the Einstein-Hilbert action or its modified version (one with a cosmological constantterm), the field equations obtained by the two approaches are not the same in general. In2hat follows we will be interested only in metric f ( R ) theories of gravity in which connectionis dependent on metric g µν with respect to which, the action is varied. In ordinary GR thereare not that many exact solutions of the field equations for a given symmetry. Being higherderivative theory it is not unexpected to find more exact solutions in f ( R ) gravity and thisturns out to be the case for spherically symmetric solutions [7]. Since cylindrical symmetryis the next symmetry considered normally in the study of exact solutions in GR (not just fortheoretical reasons but also because they might have physical realization in objects such ascosmic strings) it seems natural to extend the studies of exact solutions in f(R) theories in thesame direction. Looking for solutions with a different symmetry, as a first step, we considerstatic cylindrically symmetric vacuum solutions of the f ( R ) modified Einstein equations inthis letter. It is shown how one can reduce the set of equations into a single equation whichcould then be utilized to construct explicit solutions. For constant curvature solutions, usingthe general form of a cylindrically symmetric solution in Weyl coordinates we find, amongpossible solutions, a generalized form of a conical (zero curvature) spacetime as well as twonew (non-zero curvature) solutions with one of their parameters chosen so that it is relatednaively to the cosmological constant (section IV). We discuss possible relation to the socalled Linet-Tian (LT) solution [8]-[9] of the modified Einstein field equations which is thecylindrical analogue of the Schwarzschild-de Sitter spacetime. II. FIELD EQUATIONS IN f (R) GRAVITY In this section we give a brief review of the field equations in f ( R ) gravity. The actionfor f ( R ) gravity is given by S = Z ( f ( R ) + L m ) √− gd x. (1)The field equation resulting from this action in the metric approach, i.e. assuming that theconnection is the Levi-Civita connection and the variation is done with respect to the metric g µν , is given by G µν ≡ R µν − Rg µν = T gµν + T mµν F ( R ) , (2)where the gravitational stress-energy tensor isT gµν = 1 F ( R ) ( 12 g µν ( f ( R ) − RF ( R )) + F ( R ) ; αβ ( g αµ g βν − g µν g αβ )) , (3)3ith F ( R ) ≡ df ( R ) /dR and T mµν the standard matter stress-energy tensor derived fromthe matter Lagrangian L m in the action (1). The vacuum equations of motion, i.e. in theabsence of matter, are given by, F ( R ) R µν − f ( R ) g µν − ∇ µ ∇ ν F ( R ) + g µν (cid:3) F ( R ) = 0 (4)Contraction of the above field equations gives the following relation between f ( R ) and itsderivative F ( R ) F ( R ) R − f ( R ) + 3 (cid:3) F ( R ) = 0 , (5)which will be employed later both to simplify the field equations and to find the generalform of the f ( R ) function. III. CYLINDRICALLY SYMMETRIC VACUUM SOLUTIONS
Interested in the static cylindrically symmetric solutions of the vacuum field equations(4), we start with the general form of such a metric in the cylindrical Weyl coordinates( t, r, ϕ, z ) given by [10]; g µν = diag ( − e k ( r ) − u ( r ) , e k ( r ) − u ( r ) , w ( r ) e − u ( r ) , e u ( r ) ) . (6)The corresponding scalar curvature is R = − wu ′′ + wk ′′ − w ′ u ′ + w ′′ + wu ′ we k , (7)in which ′ ≡ ddr . Using equation (5), the modified Einstein equations become F R µν − ∇ µ ∇ ν F = 14 g µν ( F R − (cid:3) F ( R )) . (8)As in the spherical case, since the metric only depends on the cylindrical radial coordinate r ,one can view equation (8) as a set of differential equations for functions F ( r ), u ( r ), k ( r ) and w ( r ). In this case both sides are diagonal and hence we have four equations. Differentiatingequation (5) with respect to r we have the extra, consistency relation for F ( r ), RF ′ − R ′ F + 3( (cid:3) F ) ′ = 0 . (9)4ny solution of equation (8) must satisfy this relation in order to be also a solution of theoriginal modified Einstein’s equations. From equation (8) it is found that F R µµ − ∇ µ ∇ µ Fg µµ = 14 ( F R − (cid:3) F ( R )) . (10)In other words the combination A µ ≡ F R µµ −∇ µ ∇ µ Fg µµ (with fix indices) is independent of theindex µ and therefore A µ = A ν for all µ, ν . This allows us to write the following independentfield equations; − F ′′ + 2 F ′ ( k ′ − u ′ ) + F ( − k ′ w ′ w + w ′′ w + 2 u ′ ) = 0 (11) F w ( − k ′′ − k ′ w ′ w + w ′′ w ) + F ′ ( k ′ w − ww ′ ) = 0 (12) F w ( − k ′′ + 2 u ′′ − k ′ w ′ w + 2 w ′ u ′ w ) + F ′ ( wk ′ − u ′ w ) = 0 , (13)corresponding to A t = A r , A t = A φ and A t = A z respectively. Therefore, any set of functions u ( r ), k ( r ) and and w ( r ) satisfying the above equations would be a solution of the modifiedEinstein field equations (8) for a given F ( r ) satisfying equation (5). Obviously it is not aneasy task to find a general solution to the above equations, so in the following section wediscuss the simple but important case of solutions with constant curvature. IV. CONSTANT CURVATURE SOLUTIONS
It is known that some of the vacuum constant curvature solutions in f ( R ) gravity areequivalent to vacuum solutions in Einstein theory with the same symmetry. For exampleit is shown in [7] that in the spherically symmetric case the corresponding f ( R ) solutionsinclude the Schwarzschild-de-Sitter space for a specific choice of one of the constants ofintegration. For cylindrical symmetry, in Einstein gravity, static vacuum solutions werefound almost immediately after their spherical counter parts by Levi-Civita [11] but thosewith a cosmological constant have to wait another 60 years to be found independently byLinet [8] and Tian [9]. Their solution reduces to that of a cosmic string in the limit r → f ( R ) gravity, here we consider the simplebut physically important case of static constant curvature spacetimes. To do so, taking R = constant in the field equations (11), (12) and (13) , we arrive at the following set of5quations: 2 u ′ + w ′′ w − k ′ w ′ w = 0 (14) k ′′ + k ′ w ′ w − w ′′ w = 0 (15)2 u ′′ + 2 w ′ u ′ w − k ′ w ′ w − k ′′ = 0 . (16)From equations (15) and (16) one could obtain the following two equations: u ′ = 12 w ′ + c w (17) k ′ = w ′ + c w , (18)in which c and c are integration constants. By substituting equations (17) and (18) in (14)we obtain the following differential equation for w ( r ):12 ( w ′ + c w ) + w ′′ w = 2 w ′ ( c + w ′ ) w . (19)One could solve the above equation by inspection and the solutions discussed below corre-spond to solutions with zero and non-zero values of the Ricci scalar R . Case (1) : solution with R = 0 Looking at equation (19) one could obviously arrange for a solution of w ( r ) linear in r (i.e w ′′ = 0) in which case the metric functions are given as follows; u = c ± r c c ln( w ) (20) k = c + c c ln( wc ) (21) w = c r + c . (22)It could be seen that this is a Ricci flat solution (i.e R = 0 in (7)) in which we should identifywhich one of the constants c to c correspond to physical parameters of the spacetime andwhich ones could be absorbed into the coordinate redefinitions [16]. Substituting the abovefunctions back into the metric form (6) we obtain ds = e − c ± q c c lnρ ) (cid:18) e c + c c lnρ ) ( dρ c − dt ) + ρ dφ (cid:19) + e c ± q c c lnρ ) dz , (23)6n which ˜ c = c − c c lnc and ρ ≡ w = c r is the new radial coordinate by setting c = 0without any loss of generality. One can also show that through the following redefinitionsof the constants and the coordinates m = r c c (24)˜ t = e ˜ c − c A m ( m ∓ t (25)˜ z = e c A ∓ mm ( m ∓ z (26)˜ φ = e − c A − ∓ mm ( m ∓ φ (27)˜ ρ = A m ( m ∓ ρ (28) A = e ˜ c − c c , (29)the above metric reduces to ds = ˜ ρ m ( m ∓ ( d ˜ ρ − d ˜ t ) + ˜ ρ ∓ m d ˜ φ + ˜ ρ ± m d ˜ z , (30)and on applying the following complex transformation [10]˜ t −→ i ˜ z ˜ z −→ i ˜ t, (31)it transforms into the following well known metric ds = ˜ ρ m ( m ∓ ( d ˜ ρ + d ˜ z ) + ˜ ρ ∓ m d ˜ φ − ˜ ρ ± m d ˜ t , (32)which is formally similar to the Levi-Civita’s static cylindrically symmetric solution in GRnormally written without ± sign but with the constant m taking both positive and negativevalues. It should also be noted that the range of the variable ˜ φ is not in general (0 , π ], noteven for the flat cases of m = 0 , m = 0 the spacetime is conical with a deficit angle corresponding to theexterior metric of a cosmic string with the following line element [12] ds = ( d ˜ ρ + d ˜ z ) + a ˜ ρ dφ − d ˜ t , (33)in which a = c e − c is the conical parameter related to the gravitational mass per unitlength of the spacetime, η , as [13] a = c e − c = 1 − η, (34)7uch that 0 < a < < c < ∞ (taking c = 1). This metric, exposing the geometryaround a straight cosmic string, is locally identical to that of flat spacetime however itis not globally Euclidian since the angle ˜ φ = a φ varies in the range 0 ≤ ˜ φ < B where B = 2 πa = 2 π (1 − η ).From equation (5) it is seen that this metric is a solution for any form of f ( R ) for which f ( R = 0) = 0, in other words, f ( R ) must be a linear superposition of R n with n ≥ f ( R ) = R + R Λ and obviously it is not a solutionof the widely studied model for which f ( R ) = R − µ R . Case (2) : solutions with R = constant = 0 We discuss two solutions of this type here: A )-The first solution could be obtained by noting the simplifying fact that equation (16) issatisfied for k = 2 u so that equations (14) and (15) reduce to,2 u ′ + w ′′ w − u ′ w ′ w = 0 (35)2 u ′′ + 2 u ′ w ′ w − w ′′ w = 0 , (36)leading to the following equation, u ′′ + u ′ − u ′ w ′ w = 0 , (37)and consequently to the following relation between functions u and w , e u = D Z wdρ, (38)in which D is a constant to be determined later. On the other hand using (37), from equation(7) we have, R ≡ R = w ′′ w e − u . (39)Substituting from (38) we have; w ′′ w = R ( D Z wdρ ) (40). A solution to this integro-differential equation is given by; w = ρ − / D = ( 1564 R ) / (41)8o that the metric takes the form ds = 4 D ρ − ( − dt + dz + dρ ) + ρ − D dφ , (42)or in a new coordinate system with ¯ ρ = 4 Dρ / ds = 64 D ¯ ρ − ( − dt + dz ) + d ¯ ρ + 64 D ¯ ρ − dφ . (43)Now if one takes R = 4Λ, as in the case of the Einstein field equations in the presence ofΛ, then the constant D introduced above is related to the cosmological constant through D = 15256 Λ − . (44)Of course one should be careful with this kind of identification of the spacetime parameteras will be discussed later. B )-The other solution with constant non-zero Ricci scalar could be found by starting fromthe definition p := p ( w ) = dwdr in terms of which the equation (19) can be integrated toobtain the following first integral:ln( w ) −
13 ln(3 p ( w ) − p ( w ) c − c + 4 p ( w ) c ) − ( c
13 arctan( p ( w ) − c + c p − c + c c − c ) + c
23 arctan( p ( w ) − c + c p − c + c c − c ))( q − c + c c − c ) − + C = 0 . (45)Also in terms of the same function the radial coordinate and the metric functions are givenby; r = Z dwp ( w ) (46) u ( w ) = 1 / Z p ( w ) + c wp ( w ) dw (47) k ( w ) = Z p ( w ) + c wp ( w ) dw. (48)Again it does not seem to be an easy task to find solutions for the complicated equation(45), but in principle for each set of the values for the integration constants c , c and C wehave a solution for p ( w ) and correspondingly solutions for metric functions k ( r ) and u ( r ).One such solution, looking at equation (45), could be obtained by choosing c = 2 c . In thiscase, defining A ≡ e C , we find the following solution for the metric functions k ( w ) = ln ( w ) − √ arctanh (cid:18) √ A w + 4 c c (cid:19) (49)9 ( w ) = 12 ln ( w ) − √ arctanh (cid:18) √ A w + 4 c c (cid:19) , (50)in which w is a solution to the following equation3( dwdr ) − e C w − c = 0 . (51)Apart from the trivial solution w = ( − c ) / e − C (which is indeed equal to zero by equation(19) for a constant w and the fact that c = 2 c ) one could show that there is a solution interms of the Weierstrass ℘ function [17] as follows; w = W eierstrass℘ ( 2 / √ e C r + d, , − c ) e − C / . (52)The constant curvature of the spacetimes given by the functions (49), (50) and (52) could befound through equation (7) in the coordinate system ( t, w, φ, z ), for which we find R = A .Now if again this is going to be compared with the solutions of the Einstein field equationsin the presence of the cosmological constant for which R = 4Λ, then the correspondence R = e C = 4Λ will fix the value of the constant C as C = 13 ln . (53)By looking at equation (5) it is clear that for constant Ricci scalar solutions ( R = R ), nomatter what the symmetry, f ( R ) should satisfy the relation F ( R ) R = 2 f ( R ). Unlikethe solution with R = 0, the above two solutions satisfy this relation for the commonlyconsidered model of f ( R ) = R − µ R with R = 3 µ if we have D = ( 564 ) µ (54) C = 16 ln (3 µ ) (55)in cases A and B respectively. V. DISCUSSION AND SUMMARY
Studies on the exact f ( R ) gravity solutions are mostly restricted to the spherically sym-metric case mainly due to the solvability of the equations and also more importantly the factthat one could compare the results with the solar system observations/experiments basedon the schwarzschild solution as the spacetime metric around a spherical mass such as Sun.10ere we have examined cylindrically symmetric solutions in metric f ( R ) gravity in a generalform given by (6). Restricted by the complexity of the field equations we have only exam-ined solutions with constant Ricci scalar. In the case of Ricci flat solutions we have founda generalized form of a conical spacetime (with zero curvature) which, as a special case,includes the cosmic string spacetime. In the non-zero Ricci scalar case we have obtainedtwo new solutions and in both of them a parameter is identified as the cosmological constantthrough the comparison of their Ricci scalar with that of the modified Einstein field equa-tions (in the presence of the cosmological constant). Obviously neither of these spacetimesare asymptotically flat nor they behave regularly in the limit ρ →
0, and so one can notuse their asymptotic behaviour to compare their parameters with those solutions known inGR having the same symmetry. Therefore one should note that either of the identifications(44) and (53), is a very naive one in the sense that we have not compared the correspondingspacetimes (nor we have studied their specifications) with one already known in the contextof Einstein GR. Even if there are any solutions in GR comparable to these solutions (whichwe are not aware of), since the field equations in f ( R ) gravity are in general of higher ordercompared to their counterparts in GR, the correct identification should be made throughmatchings of different patches of the whole manifold. For example in the cylindrical casethese might correspond to the exterior and interior solutions of a cylindrical shell if one ofthe parameters is going to be interpreted one way or another as the mass of the shell [7].Another point need to be mentioned Incidentally it should be noted that the metric (43)has the same general form as the LT solution introduced in [9] (see also [13]) [18]. Unlessone could find a solution to the equation (40) which exactly corresponds to the LT solutionin Einstein gravity, it seems a reasonable conjecture to say that the solution (43) is the f ( R ) analogue of LT solution in GR. Finally it should be noted that in the vacuum case,for R = constant the metric and the Palatini f ( R ) formulations are dynamically equivalent[15]. Therefore all the above solutions are also solutions in the Palatini gravity with thesame f ( R ) [19]. VI. ACKNOWLEDGEMENT
The authors would like to thank University of Tehran for supporting this project underthe grants provided by the research council. M. N-Z also thanks the University of Tehran’s11enter of excellence in the structure of matter for the partial support. [1] S. Weinberg, Rev. Mod. Phys., Vol. 61, No. 1 (1989)[2] A. S. Eddington, The mathematical theory of relativity, Cambridge University Press, Cam-bridge, (1923)[3] H. A. Buchdahl, Mon. Not. Roy. Astr. Soc., 150, 1 (1970)[4] I. Navarro, K. Van Acoleyen, JCAP 0702 (2007) 022; M. L. Ruggiero, L. Iorio, JCAP 0701(2007) 010; T. Chiba, T. L. Smith, A. L. Erickcek, Phys. Rev. D75 124014 (2007); D. F.Motta, D. J. Shaw, Phys. Rev. D75 063501 (2007); T. Faulkner et al. , Phys. Rev. D76, 063505(2007)[astro-ph/0612569][5] W. Hu, I. Sawicki, arXiv: 0705.1158; S. Nojiri, S. D. Odintsov, arXiv: 0707.1941; S. Nojiri,S. D. Odintsov, arXiv: 0710.1738.[6] A. A. Starobinski, JETP Lett. 86 (2007) 157-163; G. Cognola, et al., arXiv: 0712.4017.[7] T. Multamaki, I. Vilja, Phys. Rev. D74, 064022 (2006) [astro-ph/0606373][8] B. Linet, J. Math. Phys. 27, 1817 (1986)[9] Q. Tian, Phys. Rev. D 33, 3549 (1986)[10] H. Stephani, D. Kramer, M. A. H. MacCallum, C. Hoenselaers and E. Herlt , Exact Solutionsof Einsteins Field Equations, Second edition, CUP (2003)[11] T. Levi-Civita, Rend. Acc. Lincei 27, 183 (1917)[12] A. Vilenkin and E. P. S. Shellard, Cosmic strings and other topological defects, CambridgeUniversity Press (1994)[13] M. Anderson, The mathematical theory of cosmic strings, IOP publishing Ltd., (2003)[14] I. S. Gradshteyn and I. M. Reyzhik, Table of integrals, series and products, Academic press(1994)[15] M. Ferraris, M. Francaviglia, I. Volovich, Class. Quantum Grav. (1994) 1505; G. Magnano,arXiv:gr-qc/9511027[16] Note that now the constants c and c could be written in terms of the constants c and c .[17] Also known as Weierstrass elliptic function, is a doubly periodic function. It is usually writteneither as ( z ; g , g ) or ( z | ω , ω ) in which g and g are called elliptic invariants and are givenin terms of the function’s semi-periods ω and ω [14].
18] It is shown in [9] that for a cosmic string in the presence of a positive cosmological constant,one could write the metric in the following general form ds = cos / ( λρ )( dt − dz ) − dρ − λ − sin ( λρ ) cos / ( λρ ) dφ in which λ = (3Λ) / and other parameters are set equal to 1 [13].[19] We are grateful to M. L. Ruggiero for pointing this out to us.and other parameters are set equal to 1 [13].[19] We are grateful to M. L. Ruggiero for pointing this out to us.