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Exploring Cylindrical Solutions in Modified f(G) Gravity M. J. S. Houndjo(a,b)1, M. E. Rodrigues(c)2, D. Momeni3(d) and R. Myrzakulov(d)4 a Departamento de Engenharia e Ciências Naturais- CEUNES - Universidade Federal do Esp´ırito Santo CEP 29933-415 - Saõ Mateus - ES, Brazil bInstitut de Mathématiques et de Sciences Physiques (IMSP) - 01 BP 613 Porto-Novo, Bénin cUniversidade Federal do Esp´ırito Santo - Centro de Ciências Exatas - Departamento de F´ısica 3 Av. Fernando Ferrari s/n - Campus de Goiabeiras CEP29075-910 - Vitória/ES, Brazil 1 0 dEurasian International Center for Theoretical Physics - Eurasian National University, 2 Astana 010008, Kazakhstan n a J Abstract 0 2 We present cylindrically symmetric solutions for a type of the Gauss-Bonnet gravity, in details. ] We derive the full system of the field equations and show that there exist seven families of exact c q solutions for three forms of viable models. By applying the method based on the effective fluid - r energy momentum tensor components, we evaluatethemass perunit length for thesolutions. From g [ dynamical point of the view, by evaluating the null energy condition for these configurations, we 1 show that in some cases the azimuthal pressure breaks the energy condition. This violation of the v nullenergy condition predictsthe existence of a cylindrical wormhole. 2 4 6 Pacs numbers: 04.50.kd, 04.20.-q 4 . 1 0 1 Introduction 3 1 : Gravity is the most popular fundamental force of the nature. We know many things about the v i X attractive gravitational interaction from 400 years ago. In the weak field, we can describe the long r distance interactions of the solar bodies by classical Newtonian mechanics. But if you want to explain a some non classical facts of gravity , we need a relativistic , general covariance formulation of gravity in terms of the classical gauge fields [1]. According to the equivalence principle, based on the Mach’s idea, the existence of any kind of the matter fields (classical or quantum form) needs a curved space to ignoring the local effects of the gravity in any local Lorentz invariance frame of the coordinates. The simplest one is based on the Ricci scalar R curvature , a second order derivatives of the metric of the spacetime manifold, adopted by the uniqueness theorem. It worksin the best conditions in solarsystem, but it needs some necessarymodifications beyond the galaxy and also inside the disk’s galaxy to explain 1e-mail: [email protected] 2e-mail: [email protected] 3e-mail: [email protected] 4e-mail:[email protected] 1 the problems like dark matter or dark energy. A class of all such alternative gravitational models is called modified gravity (see [2] for an excellent review). Different kinds of this modifications has been proposed. The first extension which remains still now, as a remarkable one is obtained just by replacing the curvature scalar R by an arbitrary function f(R) [3]. The simple case of R2 can be used in the inflationary scenario [4], and also to explain the effects of the observationalconstraints on solar system . Also,awideclassofthecosmologicalaspectshavebeeninvestigatedrecentlybasedonsuchmodifications [5]. But, the modifications of the Einstein gravity is not restricted to the curvature corrections. Gauss- Bonnet topological invariant G gives us another possibility, which has a rich class of the cosmological predictions [6]. Also, the black objects in a theory of gravitywith both R,G in the form of f(R,G) have been obtained in the literatures [7]. The cylindrical symmetry gives us another opportunity to obtain exact solutions. In Einstein gravity it exists a wide class of exact solutions with cylindrical symmetry in vacuum and also with cosmological constant and scalar fields [8]. In modified gravity different kinds of exact solutions with cylindrical symmetryhavebeendiscussedintheliterature,inf(R)[9],Horava-Lifshitzgravity[10],f(T)[11]andetc. Recently [12] we start obtaining exact solutions in cylindrical symmetric spacetimes but in the context of f(R,G) gravity . We derived the field equations for a general form of the modified f(G) models with Lagrangian in the form of f(R,G) R+ f(G). In [12], we proved that there exists a non vacuum ≡ family of exact solutions with R = 0 but with G = 0, corresponding to a fluid with generic forms of 6 the energy density ρ and the pressure components p ,p ,p . The solution resembles the case of the r ϕ z vacuum Levi-Civita (LC) family [13] in the Einstein gravity as the exterior metric of a cosmic string [14, 15, 16]. In that letter , we just expose a very specific solution which correspondsto the LC solution. In this present paper we continue studying the exact solutions for f(R,G) R+f(G), finding more ≡ solutions for some viable forms of f(G), by solving the differential equations of the motion analytically. Moreover,we separatelydiscussedthe solutionsfor three viableforms off(G). Inanycase,we discussed the viability and all possible non trivial solutions of the system which is under review. Also, we examine the solutions from the energy conditions using the modified (effective) gravitational field equations. As a review, we point out here the definitions of the null energy condition (NEC), weak energy condition (WEC), strong energy condition (SEC) and the dominant energy condition (DEC) as follows [17, 18] NEC ρ +p 0. (1) eff eff ⇐⇒ ≥ WEC ρ 0 and ρ +p 0. (2) eff eff eff ⇐⇒ ≥ ≥ SEC ρ +3p 0 and ρ +p 0. (3) eff eff eff eff ⇐⇒ ≥ ≥ DEC ρ 0 and ρ p 0. (4) eff eff eff ⇐⇒ ≥ ± ≥ By starting from any gravitational model and by rewriting it in the forms of a set of effective energy momentum components, we will find that the essence of these energy conditions is independent from 2 the form of the action [19]. It is easy to find that always, NEC implies WEC and WEC implies SEC and DEC. In all the subsequent models we will assume that the regular matter satisfies all the energy conditions separately i.e. ρ 0, ρ p 0, ρ+3p 0. In literature, for example in f(R) gravity it i i ≥ ± ≥ ≥ has been testedfor validity ofthe energyconditions [20]. Also,in f(T)we examinedthese conditions for some viable models [21]. Thus, we must check the validity of these conditions for our cylindrical f(G) solutions. For the importance of the NEC , we will check just this energy condition in any case. Further we obtained the formula for mass per unit length by the method proposed by Israel [22]. The Israel’s formula gives us the mass per length for any isolatedcylindrically symmetric system filled by the energy momentum of the matter as the following formula: Mass per unit length m=2π √ gdr(ρ+Σp ). (5) i ⇐⇒ − Z Recently this formula has been used to calculate the quasi-local mass per unit length for a class of the conformally cylindrical metrics as the interior solutions to the Linet-Tian family [23]. Our plan in this paper is the following: In section II, we present the model and we derive field equations and energy conditions inequalities. In sections III, IV, V we will solve the system of equations for some particular viable models of f(G). In section VI, we analysis the dynamics of the solutions using energy conditions. We conclude and summarize in final section our results. 2 Formalism of f(G) gravity and equations of motion within cylindrical metric We introduce the action for a typical general R+f(G) modification of gravity following [24] R S = d4x√ g +f(G) +S , (6) m − 2κ Z (cid:20) (cid:21) As usual in Einstein gravity, by working with the commutative connections in a Riemannian spacetime, R represents the Ricci scalar, and the modification function f(G) corresponds to a generic globally differentialable function of the Gauss-Bonnet topological invariant G. Also we add the matter action S which induces the energy momentum tensor T . For convention we take κ = 8π , where is m µν N N G G nothing justthe usualclassicalNewtoniangravitationalcoupling. Inmetric formalismandby takingthe metric as the dynamical variable of the model, the equations of motion for the metric g from (6) read µν 1 R Rg +8 R +R g R g R g +R g µν µν µρνσ ρν σµ ρσ νµ µν σρ µσ νρ − 2 − − R h + (g g g g ) ρ σf +(Gf f)g =κT , (7) µν σρ µσ νρ G G µν µν 2 − ∇ ∇ − i Wedenotebyf = df(G) alsotheGauss-Bonnet(GB)termisdefinedasG=R2 R Rµν+R Rµνλσ, G dG − µν µνλσ R and R play the roles of the Ricci tensor and Riemann tensors, respectively. WE adopted the µν µνλσ 3 signature of the Riemannian metric as (+ ) , and V = ∂ V Γλ V and Rσ = ∂ Γσ −−− ∇µ ν µ ν − µν λ µνρ ν µρ − ∂ Γσ +Γω Γσ Γω Γσ for the covariant(curved)derivative andthe Riemann tensor,respectively. We ρ µν µρ ων− µν ωρ have different possibilities for the form of a static cylindrically symmetric metric. The more common and simplest on is the Weyl gauge in which it is easy to interpret the metric functions as the generalized potential functions. This form of the coordinates, called Weyl coordinates (t,r,ϕ,z) is given by [25] g =diag e2u(r), e2k(r) 2u(r), w(r)2e 2u(r), e2k(r) 2u(r) . (8) µν − − − − − − n o Here we must be very careful about any simple interpretation of the coordinates r,t as the usual radial and time. The curvature scalar and the Gauss-Bonnet invariant term G read, respectively w uw R=2 ′′ ′ ′ u′′+u′2+k′′ e2u−2k , (9) w − w − (cid:18) (cid:19) 8e4(u k) G= − k′u′w′′ u′2w′′ 2u′u′′w′+k′u′′w′ u′3w′+3k′u′2w′+k′′u′w′ 2k′2u′w′ w − − − − h +w 3u2u 2k uu +2u4 4k u3 k u2+2k2u2 (10) ′ ′′ ′ ′ ′′ ′ ′ ′ ′′ ′ ′ ′ − − − (cid:0) (cid:1)i Sobyapplyingthestaticcylindricalmetric(8)theequation(7),weobtainthefollowingsystemofcoupled four non linear differential equations 8e4(u k) w− u′w′−k′w′−w u′2−k′u′ fG′′ + u′w′′−k′w′′+u′′w′−2k′u′w′−k′′w′ +2k2w w 2unu(cid:2) k u +u3 (cid:0)3k u2 k(cid:1)u(cid:3) +2kh2u f u2w k uw +2uu w ′ ′− ′ ′′− ′ ′′ ′ − ′ ′ − ′′ ′ ′ ′ G′ − ′ ′′− ′ ′ ′′ ′ ′′ ′ k u w +u3w(cid:0) 3k u2w k uw +2k2uw w 3u2u(cid:1)i 2kuhu +2u4 4k u3 k u2 ′ ′′ ′ ′ ′ ′ ′ ′ ′′ ′ ′ ′ ′ ′ ′ ′′ ′ ′ ′′ ′ ′ ′ ′′ ′ − − − − − − − e2(u k) (cid:16) +2k2u2 f + − 2uw w +w 2u u2 k f =κρ , (11) ′ ′ G ′ ′ ′′ ′′ ′ ′′ w − − − − 8e4(u k) (cid:17)i o h (cid:0) (cid:1)i − 3 u2w +k uw +w u3 k u2 f + u2w k uw +2uu w ku w +u3w w ′ ′ ′ ′ ′ ′ − ′ ′ G′ ′ ′′− ′ ′ ′′ ′ ′′ ′− ′′ ′ ′ ′ 3k′un′2w(cid:2)′ k′′u′w′+2k′2u′w(cid:0)′+w 2k′u(cid:1)′(cid:3)u′′ 3uh′2u′′ 2u′4+4k′u′3+k′′u′2 k′2u′2 fG − − − − − (cid:16) e2(u k) (cid:17)i o + − k′w′ wu′2 +f =κpr , (12) w − h i 8e4(u k) − w k u u2 f +w k u 2uu 3u3+5ku2+k u 2k2u f w ′ ′− ′ G′′ ′ ′′− ′ ′′− ′ ′ ′ ′′ ′− ′ ′ G′ + u′2w′′ k′u′w′′+n2u(cid:0)′u′′w′ k′u(cid:1)′′w′+uh′3w′ 3k′u′2w′ k′′u′w′+2k′2u′w′+w 2k′ui′u′′ − − − − h (cid:16) 3u2u 2u4+4k u3+k u2 2k2u2 f +e2(u k) u2+k +f =κp , (13) ′ ′′ ′ ′ ′ ′′ ′ ′ ′ G − ′ ′′ ϕ − − − 8e4(u k) (cid:17)i o (cid:0) (cid:1) − uw wu2 f + uw +u w +2u2w 3kuw +w 3ku2 2uu 3u3 f w ′ ′− ′ G′′ ′ ′′ ′′ ′ ′ − ′ ′ ′ ′ ′ − ′ ′′− ′ G′ + u2nw(cid:0) k uw +(cid:1)2uu wh k u w +u3w 3k u2w k u(cid:16)w +2k2uw +w 2k(cid:17)uiu ′ ′′ ′ ′ ′′ ′ ′′ ′ ′ ′′ ′ ′ ′ ′ ′ ′ ′′ ′ ′ ′ ′ ′ ′ ′ ′′ − − − − h e2(u k) (cid:16) 3u2u 2u4+4k u3+k u2 2k2u2 f + − w k w +wu2 +f =κp . (14) ′ ′′ ′ ′ ′ ′′ ′ ′ ′ G ′′ ′ ′ ′ z − − − w − (cid:17)i o (cid:0) (cid:1) 4 Using the Eqs.(11-14) we have the following expression for mass per unit length, through the (5), π r0 m = 2κ2 dre−2(k+3u) 4we4(k+u)(2f +w(fG′′u′(k′−u′)+fG′ u′′(k′−2u′))+u′(k′′+5k′u′−2k′2−3u′2)) πZ0 r0 (cid:16) (cid:17) dr 32e8u(f w (k 2u)+f (w (k 2u)+w (k +2k u 2k2 2u 3u2)) −2κ2 G′′ ′ ′− ′ G′ ′′ ′− ′ ′ ′′ ′ ′− ′ − ′′− ′ π Z0r0 (cid:16) + drw(f u(2u k )+f (u( (3k +2)u +k (2k 1)+u2) u (k 4u)))) 2κ2 G′′ ′ ′− ′ G′ ′ − ′ ′ ′ ′− ′ − ′′ ′− ′ Z0 π r0 +2κ2 dre8k(32−fGGw2)+8e2k+6u(wu′′+u′w′) (15) Z0 (cid:17) Here r denotes the radiusofa typicalstring. We canreadthis integralwithoutthe upper boundr asa 0 0 welldefinedexpressionforlocalmass(quasilocal)concentratedinashellwithradiusr ofthe axis. Also, the singularity of the integral as an improper integral at r = 0 understood on the meaning of the thick and thin strings. Also, we used the expressions of the energy-momentum tensor components effective in the integral. This definition is conventional and it can be understood easily if we write the equations in the weak field of a Newtonian distribution of the matter and usage of the classical Poisson’s equation for a non localized distribution of the matter with T in a finite (or compact, spatially) region Ω. So, µν the integral must be evaluated as integral over spacelike hypersurface orthogonal sheet in Ω as dr√h Ω where h denotes the determinant of the positive definite space like slice of the metric manifold ΣR4. Also, the NEC for three different pressure components implies that we must check the following inequalities: NEC for radial pressure p r 1 e2(u k) f G f Gwe4(k u) − (wk k w 2wu 2uw +2wu2+w ) G G − ′′ ′ ′ ′′ ′ ′ ′ ′′ − 8 − w − − − − 8e4(u k) − wf k u +f k w f uw +wf u2+f k w wf k u f k uw w − G′′ ′ ′ G′′ ′ ′− G′′ ′ ′ G′′ ′ G′ ′′ ′− G′ ′ ′′− G′ ′ ′ ′ h +2wf k2u wf k u +f k w 2f k2w f u w f uw 3f u2w G′ ′ ′− G′ ′ ′ G′ ′ ′′− G′ ′ ′− G′ ′′ ′− G′ ′ ′′− G′ ′ ′ −2wfG′ u′3+2wfG′ u′u′′ ≥0 (16) i NEC for azimuthal pressure p ϕ e 4(k+u) − we4(k+u)(w(f u(k u)+f u (k 2u)) 8w G′′ ′ ′− ′ G′ ′′ ′− ′ h +f G+u(k +5k u 2k2 3u2)) G ′ ′′ ′ ′ ′ ′ − − −64e8u(fG′′w′(k′−u′)+fG′ (w′′(k′−u′)+w′(k′′+2k′(u′−k′)−u′′)) +w(f u(u k )+f (2k2u k (u +3u2+u)+u3+2uu ))) G′′ ′ ′− ′ G′ ′ ′− ′ ′′ ′ ′ ′ ′ ′′ +e8k(64 f Gw2)+8e2k+6u(2wu +2uw w ) 0 (17) G ′′ ′ ′ ′′ − − ≥ i 5 NEC for axial pressure p z e 2(k u) 8e4(u k) − − − (wk +k w 2wu 2uw ) wf k u +f k w w ′′ ′ ′− ′′− ′ ′ − w − G′′ ′ ′ G′′ ′ ′ h 2f uw +2wf u2+f k w − G′′ ′ ′ G′′ ′ G′ ′′ ′ −wfG′ k′u′′+5fG′ k′u′w′−6wfG′ k′u′2+2wfG′ k′2u′−wfG′ k′u′+fG′ k′w′′ 2f k2w 2f u w 2f uw +4wf u3 2wf u2+4wf uu 0 (18) − G′ ′ ′− G′ ′′ ′− G′ ′ ′′ G′ ′ − G′ ′ G′ ′ ′′ ≥ i We will check NEC through (16-18) for the exact solutions of the viable models. 3 Solutions for the model f(G) = αG2 This is a special case of the models which have been discussed before [26]. This model is interesting becausetheBig-Ripsingularitymaynotappear. Also,thecosmologyofthismodelpredictstheexistence of a transient phantom epoch, compatible with the observational data. Moreover, this viable model is the dominanttermofamoregeneralcaseoff(G)=c Gβ1+c Gβ2, forβ >β inthe regimeofthe high 1 2 1 2 curvature and also in the primordial formation structure era. In this case the system of the equations of motion(11-14)are integrableinfive differentclassesasfollows: Here we proposesolutions to this model, i.e, finding solutions to the line element and expressions to the energy density and the pressures. We present the solutions to the line element as First case • u(r)=ln[w(r)+u r], k(r)=ln(w(r)), w(r)=w +w r , (19) 1 0 1 where u , w and w are constants. Then, the Gauss-Bonnet invariant and the curvature scalar become 1 0 1 respectively 1 G(r) = 8u2w2(u +w )[4w (w +w r)+u (w +4w r)] (20) −(w +w r)7 1 0 1 1 1 0 1 1 0 1 0 1 n o 1 R(r) = 2u w [3w (w +w r)+u (2w +3w r)] (21) (w +w r)4 1 0 1 0 1 1 0 1 0 1 n o By making use of the field equations (11)-(14), one gets the energy density and pressures as u w ρ(r)= 1 0 (w +w r)11[4w (w +w r)+u (3w +4w r)] −κ2(w +w r)15 0 1 1 0 1 1 0 1 0 1 n 64αu3w3(u +w ) 544w3(w +w r)3+96u w2(w +w r)2(w +17w r) − 1 0 1 1 1 0 1 1 1 0 1 0 1 h +u3 w3+15w2w r+96w w2r2+544w3r3 +3u2w 5w3+69w2w r+608w w2r2+544w3r3 (22) 1 0 0 1 0 1 1 1 1 0 0 1 0 1 1 (cid:0) (cid:1) (cid:0) (cid:1)io 6 u w p (r)= 1 0 (w +w r)10[2w (w +w r)+u (w +2w r)] r κ2(w +w r)14 − 0 1 1 0 1 1 0 1 0 1 n +64αu w (u +w )2 288w4(w +w r)4+36u w3(w +w r)3(9w +32w r) 1 0 1 1 1 0 1 1 1 0 1 0 1 h +4u2w2(w +w r)2 41w2+243w w r+432w2r2 1 1 0 1 0 0 1 1 +u4 w4+10w3w r+164w2(cid:0)w2r2+324w w3r3+288w4r4(cid:1) 1 − 0 0 1 0 1 0 1 1 2u2w 5w4(cid:0)+169w3w r+650w2w2r2+1062w w3r3+576w4r4 (cid:1) (23) 1 1 0 0 1 0 1 0 1 1 (cid:0) (cid:1)io u w p (r)= 1 0 (w +w r)10[2w (w +w r)+u (w +2w r)] ϕ −κ2(w +w r)14 − 0 1 1 0 1 1 0 1 0 1 n 64αu2w2(u +w )2 432w3(w +w r)3+4u w2(w +w r)2( 13w +324w r) − 1 0 1 1 1 0 1 1 1 0 1 − 0 1 h u3 w3+14w2w r+52w w2r2 432w3r3 − 1 0 0 1 0 1 − 1 +2u2w 7w3(cid:0) 59w2w r+596w w2r2+648w3r3 (cid:1) (24) 1 1 − 0− 0 1 0 1 1 (cid:0) (cid:1)io u w p (r)= 1 0 (w +w r)10[2w (w +w r)+u (w +2w r)] z −κ2(w +w r)14 − 0 1 1 0 1 1 0 1 0 1 n 64αu w (u +w )2 96w3[w +(r 1)w ](w +w r)4 − 1 0 1 1 1 0 − 1 0 1 +12u w2(w +w r)3 9w3h+w w (27+41r)+32w2r(r 1) 1 1 0 1 0 0 1 1 − +4u2w (w +w r)2 3w3+4w2w (21r 4)(cid:2)+9w w2r(27+25r)+144w3r2(r 1)(cid:3) 1 1 0 1 0 0 1 − 0 1 1 − +u4 w4+2w3w r(6r (cid:2)7)+8w2w2r2(15r 8)+12w w3r3(27+17r)+96w r4(r 1)(cid:3) 1 − 0 0 1 − 0 1 − 0 1 4 − (cid:2) 2u3w (12r 7)w4+w3w r(186r 71)+2w2w2r2(211+264r(cid:3)) 1 1 − 0 0 1 − 0 1 (cid:16) +42w w3r3(7+13r)+192w4r4(r 1) (25) 0 1 1 − (cid:17)io Now the metric reads: w(r)2 ds2 =[w(r)+u r]2dt2 dr2+dz2+dϕ2 (26) 1 − [w(r)+u r]2 1 (cid:16) (cid:17) This is a conform-stationary metric, non singular, so cannot describe a black string. The mass expression in this case reads from (5), and by integration we obtain ∆ 48u log(r w +u +w ) 1 0 1 1 0 m = +8r −60w (r w +u +w )5 − w 0− 1 0 1 1 0 1 w 2(7u 3+28u 2w +24u w 2+12w 3) 4u 2(87u 4+385u 3w +650u 2w 2+500u w 3+150w 4) 1 1 1 0 1 0 0 1 1 1 0 1 0 1 0 0 + 12(u +w )4 5w (u +w )5 1 0 1 1 0 2u w 48u log(u +w ) 1 1 1 1 0 + + , (27) u +w w 1 0 1 ∆ = 720u 5(r w +u +w )+2400u 4(r w +u +w )2 1 0 1 1 0 1 0 1 1 0 − +45u 3w 3(r w +u +w ) 4800u 3(r w +u +w )3 80u 2w 3(r w +u +w )2 1 1 0 1 1 0 1 0 1 1 0 1 1 0 1 1 0 − − +7200u 2(r w +u +w )4+60u w 3(r w +u +w )3 60w 3(r w +u +w )4 1 0 1 1 0 1 1 0 1 1 0 1 0 1 1 0 − +120u w 2(r w +u +w )4+96u 6 (28) 1 1 0 1 1 0 1 7 Because the mass of the cosmic string is about m 10 7 we can constraint the parameters w ,u ,... as − 1 1 ∼ viable functions of the observationalquantities. Second case • The line element in this case are presented as w(r)=w , k(r)=ln[w(r)], u(r)=ln w(r)+u r2 , (29) 0 0 (cid:2) (cid:3) where u and w is a constant but may not be confused with that of the previous case. With these 0 0 solutions, the Gauss-Bonnet invariant reads 64u3r2 3w +u r2 G(r)= 0 0 0 (30) w4 (cid:0) 0 (cid:1) and the energy density and pressures are the following expressions 4u ρ(r)= 0 11264αu7r8+32768αu6w r6+35840αu5w2r4+18432αu4w3r2+2u w6r2 w7 (31) −κ2w8 0 0 0 0 0 0 0 0 0 − 0 0 (cid:16) (cid:17) 4u2r2 p (r)= 0 23552αu6r6+55296αu5w r4+27648αu4w2r2 w6 (32) r κ2w8 0 0 0 0 0 − 0 0 (cid:16) (cid:17) 4u2r2 p (r)= 0 29696αu6r6+86016αu5w r4+78848αu4w r2+18432αu3w3 w6 (33) ϕ −w8κ2 0 0 0 0 0 0 0 − 0 0 (cid:16) (cid:17) 4u2r2 p (r)= 0 29696αu6r6+8192αu6r7 86016αu5w r4+28672αu5w r5 78848αu4w2r2 z κ2w8 − 0 0 − 0 0 0 0 − 0 0 0 (cid:16) +32768αu4w2r3 18432αu3w3+12288αu3w3r+w6 (34) 0 0 − 0 0 0 0 0 (cid:17) Consequently, the metric reads: ds2 = w +u r2 2dt2 w02 dr2+dz2+dϕ2 (35) 0 0 − [w +u r2]2 0 0 (cid:2) (cid:3) (cid:16) (cid:17) This again represents a conform-stationary metric, non singular, so it can not describe a black string. Like the previous case, mass expression in this case reads from (5), and by integration we obtain 4 r2u2w8 r w6 256αr u4w2 m = 0 0 0 + 0 0A1 + + 0 0 0A3 + (36) w5 (r2u +w )4 640(r2u +w )5 A2 15 A4 0 0 0 0 0 0 0 (cid:16) (cid:17) = 315r8u4+1470r6u3w +2688r4u2w2+2370r2u w3+965w4, (37) A1 0 0 0 0 0 0 0 0 0 0 0 0 = 1/4u w5( 1+w4/(r2u +w )4+(4r w )/(r2u +w ))+2048/15r5( 21+10r )u6α A2 0 0 − 0 0 0 0 0 0 0 0 0 0 − 0 0 +1024/3r3( 16+9r )u5w α (38) 0 − 0 0 0 w 0 = 720+ B (39) A3 − (r2u +w )5 0 0 0 w 5/2(63w 3 32768αu 4(w 32))tan 1(r0√u0) = 0 0 − 0 0− − √w0 (40) 4 A 128√u 0 = 15r8u4( 16+w )+10r6u3(96 5w )w +r4u2(1440 37w )w2 B − 0 0 − 0 0 0 − 0 0 0 0 − 0 0 +15w4(16+w )+5r2u w3(192+5w ) (41) 0 0 0 0 0 0 8 Third case • The line element parameters w(r) and k(r) are fixed as w(r)=w k(r)=ln[w(r)] . (42) 0 Therefore, the Gauss-Bonnet invariant and the curvature become 8u2(r) 2 G(r)= ′ 2u2(r)+u (r) e4u(r), R= u2(r) u (r) e2u(r) . (43) w4 ′ ′′ w2 ′ − ′′ 0 0 (cid:2) (cid:3) (cid:2) (cid:3) Inorderwriteputdownasuitableexpressionforthelineelementparameteru(r),weassumethefollowing relation between G(r) and R(r) G(r)=e2u(u)u′2(r)R(r) . (44) by solving this latter, one gets 12+w2 u(r)=u + 0 ln(r) , (45) 0 w2 8 0 − where u is an integration constant, leading to the final forms of G(r) and R(r) as 0 80 8(w2 48)(w2+12)3e4u0rw02−8 G(r)= 0− 0 , (46) − w4(w2 8)4 0 0 − and 40 4(w2+12)(w2)e2u0rw02−8 R(r)= 0 0 . (47) w2(w2 8)2 0 0 − The energy density and pressures are get respectively as 40 (12+w2)e2u0rw02−8 ρ(r)= 0 w6(w2 8)6(3w2 4) κ2w8(w2 8)8 − 0 0 − 0− 0 0− n 120 +64αe6u0rw02−8(w2+12)4 863232+35072w2 404w4+w6 (48) 0 − 0− 0 0 (cid:0) (cid:1)o 40 (12+w2)2e2u0rw02−8 p (r)= 0 w6(w2 8)6 r κ2w8(w2 8)8 0 0 − 0 0− n 120 +64αe6u0rw02−8(w2+12)4 20736+384w2+w4 (49) 0 − 0 0 (cid:0) (cid:1)o 40 (12+w2)2e2u0rw02−8 p (r)= 0 w6(w2 8)6 ϕ − κ2w8(w2 8)8 − 0 0 − 0 0− n 120 +64αe6u0rw02−8(w2+12)3 1102848 21248w2 84w4+w6 (50) 0 − 0− 0 0 (cid:0) (cid:1)o 40 pz(r) = −(12κ+2ww802()w22e2u08r)w802−8 −w06(w02−8)6+64αe6u0rw1021−28(w02−48)(w02+12)3 0 0 − n 2 w0 8 320(w2 8)rw02−8 +(w4 36w2 22976)rw02−8 (51) × 0− 0 − 0− h io 9 Consequently, the metric reads: ds2 =r˜2βdτ2 r˜ 2βµ2 dr˜2+dz˜2+dϕ˜2 , (52) − − β 12+w02, r˜=µβ−1r,z˜=(cid:16)µβ−1z,ϕ˜=µβ−1(cid:17)ϕ. ≡ w2 8 0− Here µ =wβe βu0 denotes the unique parameter of the solution and it may play the role of the conical 0 − deflection parameter. This again represents a conform-stationary metric, non singular, so it can not describe a black string. Like the previous case, mass expression in this case reads from (5), and by integration we obtain 1 192αβ5(2β 3)(5β(2β 5)+14)e2u0r2β 48e 6u0w 6r8 6β − 0 − m = − − + 6r7 − 2β 7 1 6β (cid:16) − − 256αβ5(2β 3)e6u0 102β2 3β(8r+89)+28(r+6) r6β 9β3e 2u0w 3r5 2β − − + − 0 − +12βr6w(53) − (6β 7)w 5 β+1 0 (cid:0) − 0 (cid:1) (cid:17) Fourth case • In this case, we assume the parameters w(r) and k(r) of the line element as w(r)=√2, k(r)=ln[w(r)]. (54) Asweperformedinthepreviouscase,weneedtofindasuitableexpressionforthelineelementparameter u(r). To do so, we assume the following relation between G(r) and R(r) as G(r)=b (r)R(r)+b (r) , (55) 1 2 such that action algebraic function becomes R+f(G)=[1+2αb b ]R+α b2R2+b2 . (56) 1 2 1 2 (cid:0) (cid:1) Now, by introducing an auxiliary function ψ to be identified to the radial coordinate r (ψ =r), one can rewrite the action algebraic function as R+f(G)=B (ψ)R+B (ψ)R2+V(ψ) , (57) 1 2 such that G(r) and R(r) become G(r)=2e4u(r)u2(r) 2u2+3u (r) , (58) ′ ′ ′′ R(r)=e2u(r)(cid:2)u′2(r) u′′(r)(cid:3) . (59) − (cid:2) (cid:3) Onceagain,relationisrequiredbetweenGandRinordertofindasuitableexpressionforthelineelement parameter u(r). To this end we assume the following relation G(r)=4e2u(r)u2R(r) , (60) ′ 10