Global monopoles and massless dilatons in Einstein-Cartan gravity

A global monopole in dilatonic Einstein-Cartan gravity is presented. A linearized solution representing a global monopole interacting with a massless dilaton is found where Cartan torsion does not interact with the monopole Higgs field.Computation of the geodesic equation shows that the monopole-dilaton system generates a repulsive gravitational field.The solution is shown to break the linear approximation for certain values of torsion. PACS number(s) : 0420,0450 Departamento de F́ısica Teórica Instituto de F́ısica UERJ Rua São Fco. Xavier 524, Rio de Janeiro, RJ Maracanã, CEP:20550-003 , Brasil. E-mail : [email protected] Global monopoles and massless dilatons in Einstein-Cartan gravity. 2 Recently Garcia de Andrade and Barros [1] have found that the presence of global monopoles in linearized Einstein-Cartan gravity [2] lead to modified torsionic effects on physical quantities related to the global monopole such as the deficit angle and the density distribution which are very much in agreement with the predictions of COBE [3].Earlier Dando and Gregory [4] have found the gravitational field of a global monopole within the context of low energy string gravity ,where the global monopole couples with massive and massless dilatons.They found that for massless dilatons the spacetime is generically singular,whereas when the dilaton is massive the monopole induces a long range dilaton cloud.Since dilatons have been recently investigated in the context of Einstein-Cartan gravity [5] it seems interesting to extend their investigated to spacetimes with torsion.Let us consider the simplest model of Barriola-Vilenkin [6] to build global monopoles which is given by the Lagrangean L(ψ) = 1 2 ∇aψ ∇ψ − λ 4 (ψψ − η) (1) where ψ is a triplet of real scalar fields,i = 1, 2, 3 and a, b = 0, 1, 2, 3 are spacetime indices.The model has a global O(3) symmetry spontaneously broken to a global U(1) symmetry by the vacuum choice |ψ| = η.Static spherically symmetric solutiona are search describing a global monopole at rest.The field configuration for this monopole is written according to Dando and Gregory as ψ = ηf(r)e where e is the unit radial vector in the internal space of the monopole.We consider the metric of the monopole as written by ds = B(r)dt −A(r)dr − rdΩ (2) where dΩ = dθ+sinθdφ. Since the linearized global monopole solution of Einstein-Cartan gravity in linear approximation has already been found we will not repeat it here, and go directly to the Global monopole in dilatonic Einstein-Cartan gravity.Its action is given by S = ∫ dx(−g) 1 2 [e(−R − 4(∇φ) − V ′(φ)) + eL] (3) where L is the monopole Lagrangean for the monopole as given in (1),V (φ) is the potential for the dilaton.The action (3) is written in terms of the string metric of the string sigma model.We write the action in terms of the conformal metric gab = e g ab (4) Global monopoles and massless dilatons in Einstein-Cartan gravity. 3 which turns the action into S = ∫ dx(−g) 1 2 [e(−R + 2(∇φ) − V (φ)) + eL(φ, e)] (5) The energy-momentum tensor is decomposed in three parts.The first is the monopole one T ab = e ∇aφ ∇bφ i − gabL (6) the dilaton tensor Sab = 2∇aφ∇bφ+ 1 2 V (φ)− gab(∇φ) 2 (7) and finally the torsion EMT T torsion ab = 3SacdS cd b − 1 2 gabS 2 (8) where S = SabcS abc is the square of the torsion tensor Sabc which S023 is the only non-vanishing torsion component chosen along the radial direction.The Einstein-Cartan equation can then be written in the quasi-Einsteinian form Gab = 1 2 eT ab + Sab + T torsion ab (9) Although the contribution of torsion in the problem considered here seems minor we shall prove that the perturbation method will produce a new torsion dependent global monopole in the Einstein-Cartan dilatonic gravity.Since there is no direct interaction between torsion and the monopole or the dilatonic field the dilaton equation will be the same as the one considered by Dando and Gregory.Once more we take ηλ 1 2 = 1 and the modified energymomentum tensor is T ′t t = α(γ( f ′2 2A + f 2 r2 ) + 1 4 δ) (10) and T ′r r = α(γ(− f ′2 2A + f 2 r2 ) + 1 4 δ) (11) and T ′θ θ = α(γ f ′2 2A + 1 4 δ) (12) Global monopoles and massless dilatons in Einstein-Cartan gravity. 4 where δ = (f 2 − 1),α = e and γ = e.The tt and rr components of Einstein-Cartan-dilatonic-global monopole system now becomes A rA2 + 1 r2 (1− 1 A ) = ǫT ′t t + 1 2 V (φ) + φ′ A + b r2 (13) and −B rB2A2 + 1 r2 (1− 1 A ) = ǫT ′r r + 1 2 V (φ)− φ′ A + b r2 (14) where we have chosen the Barros ansatz for torsion of the global monopole as S = (1−b)η 2 r where b is a constant which when zero reduces the solution to the BarriolaVilenkin solution in the absence of dilatons.To simplify matters we consider in this Letter just massless dilatons leaving the massive dilatons case for a future paper.For massless dilatonic EinsteinCartan gravity V (φ) = 0,where ǫ = η 2 2 and making the expansion A = 1 + ǫA1 + ... (15) and B = 1 + ǫB1 + ... (16) and φ = φ0 + ǫφ1 + ... (17) To the first order we have,A = B = 0 and the equations reduce to φ′0 2 − b r2 = 0 (18) which yields the simple solution φ0 2 = b 1 2 lnr.Substitution of these values into fields equations we obtain A′1 r + A1 r2 = βr( f ′ 2 2 + f 2 r2 ) + 1 4 βr(f 2 − 1) + b r2 (A1 − 1) (19) and −B 1 r + A1 r2 = βr(− f ′ 2 2 + f 2 r2 ) + 1 4 βr(f 2 − 1) + b r2 (A1 + 1) (20) φ1” + 2φ1 ′ r = (a + 1) β r (− f ′ 2 2 ) + a+ 2 4 (f 2 − 1) (21) Global monopoles and massless dilatons in Einstein-Cartan gravity. 5 For large r,f is approximatly unity which greatly simplifies the system to φ1” + 2φ1 ′ r = 0 (22) which can be solved to yield φ1 ′ = er − 2c r3 + d (23) where here we are using β = e0 .The field equations for the metric are A′1 r + A1 r2 = β r + b r2 A1 (24) and −B 1 r + A1 r2 = β r + b r2 A1 (25) Their solution is A1 +B1 = 0 (26) Substitution of the solutions of these equations on the expansion we obtain the following final metric ds = (1− ǫβr 2(b− 1) − ǫ m r )dt − (1 + ǫβr 2(b− 1) + ǫ m r )dr − rdΩ (27) where m is an integration constant.This implies the following geodesics v̇r = β 2(b− 1) + m r2 (28) which very far from the sources shows that the dilaton-global monopoletorsion system is gravitationally repulsive.Note that the torsion factor b is determinant for the repulsive carachter of the global monopole-dilaton system.Notice that due to the presence of βr.Therefore the spacetinme should be bound to values of r < (b − 1) and the need for an exact solution is of prime importance for the better understanding of the problem.A more detailed account of the ideas discussed here may appear elsewhere. Global monopoles and massless dilatons in Einstein-Cartan gravity. 6