Cosmological Lorentzian Wormholes via Noether symmetry approach
aa r X i v : . [ a s t r o - ph . C O ] O c t Cosmological Lorentzian Wormholes via Noether symmetry approach
Abhik Kumar Sanyal † and Ranajit Mandal ‡ ,October 17, 2018 † Dept. of Physics, Jangipur College, Murshidabad, West Bengal, India - 742213. ‡ Dept. of Physics, University of Kalyani, West Bengal, India - 741235.
Abstract
Noether symmetry has been invoked to explore the forms of a couple of coupling parameters and thepotential appearing in a general scalar-tensor theory of gravity in the background of Robertson-Walker space-time. Exact solutions of Einstein’s field equations in the familiar Brans-Dicke, Induced gravity and a Generalnon-minimally coupled scalar-tensor theories of gravity have been found using the conserved current and theenergy equation, after being expressed in a set of new variables. Noticeably, the form of the scale factors remainsunaltered in all the three cases and represents cosmological Lorentzian wormholes, analogous to the Euclideanones. While classical Euclidean wormholes requires an imaginary scalar field, the Lorentzian wormhole do not,and the solutions satisfy the weak energy condition.
Apart from Black holes, Wormholes are yet another extraordinary, exciting and intriguing consequence of Einstein’sGeneral Theory of Relativity (GTR) G µν = R µν − g µν R = κT µν . Since the pioneering works of Lavrelashvili,Rubakov and Tinyakov [1, 2, 3] followed by Giddings and Strominger [4] and thereafter by Morris and Thorne[5, 6], wormholes turn out to be one of the most popular and intensively studied topics in Astronomy. Wormholesare essentially astrophysical objects which connect two asymptotically flat or de-Sitter/ anti-de-Sitter regions bya throat of finite radius. While, microscopic wormholes might provide us with the mechanism that might be ableto solve the cosmological constant problem, macroscopic wormholes might be responsible for the final stage ofevaporation and complete disappearance of black holes. Since the Wheeler-DeWitt equation is independent of thelapse function and as such holds for both the Euclidean and Lorentzian geometry, so following Hawking and Pageformulation [7] Euclidean wormholes may be realized in the early universe, but only for some specified forms ofthe scalar potentials [8]. That is why, most of the efforts have been directed to the study of Lorentzian wormholesin the framework of classical GTR. The striking feature of wormholes is the requirement of the violation of energyconditions. Therefore, realization of Lorentzian wormhole solutions with standard barotropic fluid is not accept-able physically. This implies that the matter supporting the traversable wormholes (wormholes without a horizon)should be exotic [5, 6, 9, 10, 11]. and therefore it should have very strong negative pressure, or even that theenergy density may be negative. Therefore, a lot of efforts have been directed to the study Lorentzian wormholes,in the framework of classical general relativity, sustained by an exotic matter with negative energy density. Ingeneral, these models include both static [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22] and evolving relativistic versions[23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], sustained by a single fluid component. The interest has been mainlydevoted to the study of traversable wormholes, without any horizon, allowing two-way passage through them [34].For static wormholes the fluid requires the violation of the null energy condition (NEC), while in Einstein gravitythere exists non-static Lorentzian wormholes which do not require weak energy condition (WEC) violating matterto sustain them. Such wormholes may exist for arbitrarily small or large intervals of time [23, 24], or even satisfythe dominant energy condition (DEC) in the whole spacetime [35, 36]. Electronic address: ‡ sanyal [email protected]; ‡ [email protected] T µν . In this sense, scalar-tensor theory of gravity in principle should not be treated asa modification of GTR. Nevertheless, in view of the action principle, non-minimally coupled scalar-tensor theoryof gravity may be looked upon as a modification of GTR, since it requires coupling between the Ricci scalar R and some arbitrary function f ( φ ) of the scalar field φ in the form f ( φ ) R , in the action. But in view of thefield equations, it might just again be treated as incorporating a typically different energy-momentum tensoraltogether. Only by modifying Einstein-Hilbert action by introducing different higher-order curvature invariantterms, left hand side of the Einstein’s equation and hence GTR is truly modified. In this sense wormhole solutionsfor scalar-tensor theory of gravity may also be treated as a consequence of GTR. In the context of cosmology, theviolation of energy condition for such matter fields in the early universe, does not in any way affect the late stageof cosmic acceleration.Evolving Lorentzian wormholes in the background of Robertson-Walker metric has already been studied bysome authors [36, 37, 38, 39]. In general, while constructing wormhole geometries, first the form of the redshiftfunction Φ( r ) and the shape function b ( r ), satisfying some general constraints [5, 6] are fixed. This fixes themetric is as well. Thereafter in view of the field equations, components of the energy-momentum tensor requiredto support the spacetime geometry, are explored. For evolving wormholes, one usually generalizes the ansatz forstatic Lorentzian wormhole given by Morris and Thorne [5, 6] in the form, ds = − e r,t ) dt + a ( t ) " dr − b ( r,t ) r + r ( dθ + sin θ dφ ) . (1)However, the evolving Lorentzian wormhole, we are going to study in the present manuscript is different altogether.We follow the same standard method of solving Einstein’s equations, as usually employed in the case of Euclideanwormholes. That is, we do not fix the redshift function Φ( r, t ) or the shape function b ( r, t ), and the reason is,shape of a cosmological wormhole is fixed by the scale factor itself [8, 40, 41, 42, 43]. Practically, our initial aimis just to study the evolution of the early universe in view of the non-minimally coupled scalar-tensor theory ofgravity, corresponding to the action being typically expressed in the following form, A = Z d x √− g (cid:20) f ( φ ) R − πG − ω ( φ ) 12 φ ,µ φ ,µ − V ( φ ) (cid:21) , (2)in the background of isotropic and homogeneous Robertson-Walker metric, ds = − dt + a ( t ) (cid:20) dr − kr + r ( dθ + sin θ dφ ) (cid:21) . (3)Note that since we are interested in the evolution of early universe, so we ignore any form of baryonic matterwhat-so-ever. The action (2) involves the cosmological constant Λ , and so is a generalization of our earlier work[44]. It also involves two coupling parameters f ( φ ) and ω ( φ ) apart from the potential V ( φ ), whose forms arerequired, to explore the evolution. Instead of choosing the forms of these parameters by hand, we apply Noehersymmetry for the purpose. In view of the forms of the parameters, the solutions so obtained have been foundto represent wormholes in the sense that asymptotically ( t → ±∞ ), the scale factor ( a ( t )) tends to de-Sitteruniverse, while for t → f ( φ ), viz. the ‘Brans-Dicke theory’, the ‘Induced theory of gravity’ and the ‘General non-minimally coupledscalar-tensor theory of gravity’, associated with different forms of the coupling parameter ω ( φ ) and the potential V ( φ ). In all the cases the scale factor retains the same form whose nature admits Lorentzian wormholes, whichdon’t violate WEC. It thus appears that Lorentzian wormholes are almost a generic feature of the system (2)under consideration. We conclude in section 5. Noether symmetry approach is a powerful tool in finding unknown parameters e.g. the potential and the couplingparameters appearing in the Lagrangian. Using this method it is possible to obtain a reduction of the fieldequations and sometimes to obtain a full integration of the system, once the cyclic variable of the system is found.The key point related to the Noether symmetry is a Lie algebra presented in the tangent space. Noether theoremstates that, if there exists a vector field X , for which the Lie derivative of given Lagrangian L vanishes, i.e. £ X L = 0 , the Lagrangian admits a Noether symmetry and thus yields a conserved current. If the Lagrangianunder consideration is spanned by the configuration space M = ( a, φ ), then the corresponding tangent space isTM = ( a, φ, ˙ a, ˙ φ ). Hence the generic infinitesimal generator of the Noether symmetry is X = α ( a, φ ) ∂∂a + β ( a, φ ) ∂∂φ + ˙ α ( a, φ ) ∂∂ ˙ a + ˙ β ( a, φ ) ∂∂ ˙ φ where , ˙ α ≡ ∂α∂a ˙ a + ∂α∂φ ˙ φ ; ˙ β ≡ ∂β∂a ˙ a + ∂β∂φ ˙ φ. (4)In the above, α, β are both generic functions of a and φ and the Lagrangian is invariant under the transformation X i.e. £ X L = XL = α ∂L∂a + β ∂L∂φ + ˙ α ∂L∂ ˙ a + ˙ β ∂L∂ ˙ φ = 0 , (5)where £ X L is the Lie derivative of the point Lagrangian with respect to X. The above equation may be solved tofind the unknown parameters of the theory e.g. the potential. However, if there are several unknown parameters,relations connecting those are found, which may be used to select the form of the parameters from physicalargument. Now, in view of the Cartan’s one form θ L = ∂L∂ ˙ a da + ∂L∂ ˙ φ dφ, (6)the constant of motion Q = i X θ L , which is essentially the conserved current, is expressed as Q = α ( a, φ ) ∂L∂ ˙ a + β ( a, φ ) ∂L∂ ˙ φ . (7)It is well known that the cyclic variable helps a lot in exploring the exact description of the dynamical system.So, once X is found in view of the solutions of the Noether equation (5), it is possible to change the variables to u ( a, φ ) and v ( a, φ ), such that i X du = 1; i X dv = 0 i.e. i X du = α ∂u∂a + β ∂u∂φ = 1; i X dv = α ∂v∂a + β ∂v∂φ = 0 . (8)The Lagrangian when expressed in term of the new variables, u becomes the cyclic variable, and the constant ofmotion Q is its canonically conjugate momentum, i.e. Q = P u . Thus the conserved current assumes a very sim-ple form so that exact integration is often found leading to exact solution of the filed equations under consideration.3evertheless, as mentioned in the introduction, there is an insidious problem associated with Noether sym-metric solutions for gravitational system in particular. Noether symmetry is not on-shell for constrained systemlike gravity. This means, in general it neither satisfy the field equations nor its solutions by default. Due todiffeomorphic invariance gravity constrains the Hamiltonian (it also constrains the momenta in general, whenevertime-space components exists in the metric) of the system to vanish, i.e. the total energy ( E ) of a gravitatingsystem is always zero. This is essentially the ( ) component of the Einstein’s field equations, when expressedin terms of configuration space-variable. Noether equation does not recognize the said constraints (energy andmomenta). Therefore, except in some particular case, often it does not satisfy the ( ) component of Einstein’sfield equations [47, 48, 49]. Also, sometimes it leads to degeneracy in the Lagrangian [50, 51]. Further, often it failsto explore the known symmetries of the system [52, 53]. Finally, different canonical forms (the point Lagrangianobtained through Lagrange multiplier method, and under reduction to Jordan’s and Einstein’s frames) of F ( R )theory of gravity yield different conserved currents [54]. A possible resolution to the problems [55] is not to fixthe gauge, viz. the lapse function N, a-priori, but to keep it arbitrary, so that the ( ) component of Einstein’sequation is recognized by Noether equation. In the process, the symmetry generator X should be modified to X = α ∂∂a + β ∂∂φ + γ ∂∂N + ˙ α ∂∂ ˙ a + ˙ β ∂∂ ˙ φ + ˙ γ ∂∂ ˙ N . (9)Likewise, when the metric would contain time-space components, one should keep the shift vector ( N i ) arbitrary.However, with the introduction of lapse function (and shift vector as well) the Hessian determinant vanishes andthe point Lagrangian becomes singular [56]. It is therefore required to follow Dirac’s constraint analysis, whichcomplicates the problem. Under such circumstances, another proposal has been placed very recently and that isto modify the existence condition for Noether symmetry as [57], £ X L − ηE − X i δ i P i = 0 , (10)where, E is the total energy of the gravitating system which is constrained to vanish, and essentially is the( ) component of Einstein’s equations. P i are the momenta which are also constrained to vanish, and are the( i ) components of Einstein’s field equations. In the above, i runs from 1 through to 3 and η = η ( a, φ ) and δ i = δ i ( a, φ ) are generic functions of a and φ . It has been possible to remove the problems associated with F ( R )theory of gravity [57] and Einstein’s field equations are automatically satisfied in all circumstances.In the present manuscript however, we take a different route to explore Noether symmetry of the action (2),which has been used earlier [44] and found to be a very powerful technique to find explicit solutions of the fieldequations. First, we use the standard symmetry generator (4) and consequently the standard Noether equation (5)to find Noether solutions. Since, there are five unknown parameters of the theory (2), viz. a, φ, f ( φ ) , ω ( φ ) , V ( φ ),consequently there are five unknown parameters α ( a, φ ) , β ( a, φ ) , f ( φ ) , ω ( φ ) , V ( φ ) involved in four Noether equa-tions. So, Noether equations when solved would lead to relations amongst the parameters. To find the formsexplicitly, we therefore would require to make yet another choice. We shall therefore assume known standardforms of f ( φ ), and solve Noether equations to explore the forms of ω ( φ ) and V ( φ ) and consequently α ( a, φ ) and β ( a, φ ). We then express the scale factor a , the scalar field φ , the Lagrangian, the conserved current Q and theenergy equation, viz. the ( ) equation of Einstein in terms of the new variables u and v . Next, we solve for u and v in view of the last two equations, viz. the conserved current and the energy equations, and transform backto find the explicit solutions for a ( t ) and φ ( t ). Since, ( ) equation is used for the purpose, so all the Einstein’sequations are automatically satisfied. We study three different cases corresponding to three different physicalchoices of the parameter f ( φ ), viz. the ‘Brans-Dicke form ( f ( φ ) = φ )’, the ‘Induced gravity theory ( f ( φ ) = ǫφ )’and the ‘General non-minimal coupled theory ( f ( φ ) = 1 − ǫφ )’ and surprisingly observe that all the three caseslead to the same forms of the scale factor ( a ( t )), which are Lorentzian wormhole solutions. It therefore appearsthat Lorentzian wormhole is a natural outcome of non-minimally coupled theory (2) under consideration.4 Action and Noether symmetric approach
In the Friedmann-Robertson-Walker minisuperspace (3) under consideration the Ricci scalar reads as R = 6 (cid:0) ¨ aa + ˙ a a + ka (cid:1) and therefore the action (2) takes the following form A = Z (cid:18) m p ( − f a ˙ a − a ˙ a ˙ φf ′ + 3 kaf ) + 12 ω ˙ φ a − a V − m p a Λ (cid:19) dt (11)in the unit ~ = c = 1 ) while, m p ( m p = πG ) is the Planck mass. The above action is canonical, provided theHessian determinant: W = P ∂ L∂ ˙ a∂ ˙ φ = − a (cid:0) f ′ + 2 ωf (cid:1) = 0 . The point Lagrangian is expressed (in the unit8 πG = 1 ) as, L = − f a ˙ a − a ˙ a ˙ φf ′ + 3 kaf + 12 ω ˙ φ a − a V − a Λ . (12)The field equations are, (cid:18) aa + ˙ a a + ka (cid:19) + (cid:18) ¨ φ + 2 ˙ aa ˙ φ (cid:19) f ′ f + 1 f (cid:18)
12 ˙ φ ω − V (cid:19) + f ′′ f ˙ φ − Λ f = 0 (13) (cid:18) ¨ φ + 3 ˙ aa ˙ φ (cid:19) ω f ′ − (cid:18) ¨ aa + ˙ a a + ka (cid:19) + (cid:18)
12 ˙ φ ω ′ + V ′ (cid:19) f ′ = 0 (14) (cid:18) ˙ a a + ka (cid:19) + ˙ aa ˙ φ f ′ f − f (cid:18)
12 ˙ φ ω + V (cid:19) − Λ3 f = 0 (15)where, dot denotes derivative with respect to time while prime represents derivative with respect to φ . Theexpressions for the effective energy density ( ρ e ) and the effective pressure ( p e ) are, ρ e = 1 f (cid:20) ω φ + V − aa ˙ φf ′ + Λ (cid:21) ; p e = 1 f (cid:20) ω φ − V + (cid:18) ¨ φ + 2 ˙ aa ˙ φ (cid:19) f ′ + f ′′ ˙ φ − Λ (cid:21) . (16)Consequently, one can also compute the sum as, ρ e + p e = 1 f (cid:20) ω ˙ φ + (cid:18) ¨ φ − ˙ aa ˙ φ (cid:19) f ′ + f ′′ ˙ φ (cid:21) . (17)As mentioned, to explore the form of the unknown parameters involved in the point Lagrangian, let us now demandNoether symmetry by imposing the condition from (5) to find the following Noether equation α (cid:16) − a ˙ a ˙ φf ′ − a f + 3 kf + 3 ω φ a − a V − a (cid:17) + β (cid:16) − a ˙ a ˙ φf ′′ − a ˙ a f ′ + 3 kaf ′ + ω ′ φ a − a V ′ (cid:17) + (cid:16) ∂α∂a ˙ a + ∂α∂φ ˙ φ (cid:17) ( − a ˙ φf ′ − a ˙ af ) + (cid:16) ∂β∂a ˙ a + ∂β∂φ ˙ φ (cid:17) ( − a ˙ af ′ + ω ˙ φa ) . (18)Naturally, equation (18) is satisfied provided the co-efficient of ˙ a , ˙ φ , ˙ a ˙ φ and the term free from time derivativevanish separately, i.e. α + 2 a ∂α∂a + a ∂β∂a f ′ f + aβ f ′ f = 0 , (19)3 α − f ′ ω ∂α∂φ + 2 a ∂β∂φ + aβ ω ′ ω = 0 , (20)5 α + a ∂α∂a + a ∂β∂φ (cid:1) + a f ′′ f ′ β + 2 ff ′ ∂α∂φ − ω f ′ a ∂β∂a = 0 , (21)3 kf (cid:0) α + aβ f ′ f (cid:1) = a (cid:0) V α + βV ′ a + 3Λ α (cid:1) . (22)We now look for the conditions on the integrability of this set of above equations (19) - (22). Since, here thenumber of equations are four while the number of unknown parameters are five ( α, β, ω, f, V ), so the set of aboveequations can not be solved exactly unless extra condition is imposed. Rather, we obtain restrictions on theforms of α , β , f , ω and V . This will leave large freedom of choice, so that all the interesting cases may beaccommodated. However, as mentioned, we shall in the present manuscript restrict ourselves to study only threecases of particular importance. The set of particular differential equations are solved under the assumption that α and β are separable (and non null), i.e. α ( a, φ ) = A ( a ) B ( φ ); β ( a, φ ) = A ( a ) B ( φ ) . (23)With these assumptions, the integrability conditions are (See Appendix) A = − cla ; B = c ff ′ B ; A = la ; V = V o f − Λ; B ′ = 23 ωf ′ B ; 3 f ′ + 2 ωf = n ωf (24)where, c, l, n, V are all arbitrary constants. Note that since f ( φ ) = 0 and ω ( φ ) = 0 , so the nondegeneracycondition remains satisfied provided n = 0 . Clearly, we need to solve the above six equation (24), for sevenunknowns, viz. ( A , A , B , B , f, V, ω ). It is important to mention that while general conserved current alwaysexists for V ∝ f [47, 48, 49, 58, 59], Noether symmetry exists for V ∝ f , in the absence of Lambda. Thisclearly depicts that Noether symmetry procedure is unable to explore all the available symmetries of a theory.For ω = 1 , a general nonminimally coupled case we get in view. The last relation of equation (24) then gives anelliptic integral, which can be solved for f in closed form only under the assumption n = 0 . But this makes theHessian determinant W = 0 , and so the Lagrangian turns out to be degenerate. Also, for n = 0 , the generalsolution of (24) is, f = − ( φ − φ ) , which makes the Newtonian gravitational constant G negative [42]. Thus weomit the case ω = 1 . f ( φ ) Let us make things clear yet again, for a consistency check. To get a picture of evolution of the early universe inview of the action (2), we need to solve the set of Einstein’s field equations (13), (14) and (15) exactly. Out ofwhich only two are independent and they involve 5 unknowns ( a ( t ) , φ ( t ) , f ( φ ) , ω ( φ ) , V ( φ )) altogether. Clearly, onerequires 3 physically reasonable assumptions for the purpose, and the standard followup is to choose some typicalforms of f ( φ ) , ω ( φ ) and V ( φ ). Instead we impose Noether symmetry i.e. £ X L = 0 , as our first assumption,since nothing is more physical in the world than symmetry. As a result we find four equations (19), (20), (21) and(22), with five unknown parameters viz. α, β, f, ω, V . Thus at this stage we require just one more assumptionto exactly solve the above set of Noether equations. To handle the above set of partial differential equations (19)through to (22) we consider separation of variables and ended up with yet another set of equations (24). One canclearly notice that already A and A are found exactly as functions of the scale factor a ( t ), while we are left withfour relations in (24) with five parameters B , B , f, ω, V . Hence still one needs one more assumption to explicitlyfind the forms of these five parameters. This proves everything is consistent so far. One can generate indefinitelylarge number of symmetries and hence exact cosmological solutions, by making different choices of one of theparameters. In this section however, we shall study only three different cases making reasonable assumptions onthree different forms of f ( φ ), since as already known, different forms of f ( φ ) leads to different physical theory.The three cases represent ‘Brans-Dicke theory of gravity’, ‘Induced theory of gravity’ and ‘General non-minimaltheory of gravity’. As a result we find α ( a, φ ), β ( a, φ ), ω ( φ ) and V ( φ ), and hence the conserved current. Weshall then express the Lagrangian in terms of the new variables u and v , u being cyclic, and use the conservedcurrent and the energy equation expressed in terms of the new variables as Q = ∂L∂ ˙ u ; E L = ∂L∂ ˙ u ˙ u + ∂L∂ ˙ v ˙ v − L = 0 , (25)to solve the Einstein’s field equations exactly in all the three different cases.6 ase 1. Brans-Dicke theory. First we consider the well-known Brans-Dicke theory by choosing f ( φ ) = φ . Inview of Equation (24), we therefore obtain the following solutions V = V φ − Λ , ω = 12 nφ − φ , B = B p nφ − φ , B = cB p nφ − , (26)where, B is yet another constant. In view of equations (24) and (26) α and β are obtained as, α = − C p nφ − aφ , β = C p nφ − a , (27)where the constant C = c B l . So the conserved current (7) is found as, Q = 3 Ca p nφ − (cid:18) ˙ aa + nφ − nφ − φφ (cid:19) . (28)Using the forms of f ( φ ) = φ and the forms of ω ( φ ) , V ( φ ) presented in equation (26), the point Lagrangian (12)may now be expressed as, L = − a ˙ a φ − a ˙ a ˙ φ + 3 kaφ + 6 a ˙ φ nφ − φ − V a φ . (29)At this stage let us perform the change of variables to obtain the corresponding cyclic coordinate associated withthe conserved current (28). Equation (8) is solved exactly under the following choice, u = a φ p nφ − v = aφ, (30)which may be inverted to yield a = nv − u v ; φ = 8 v nv − u . (31)Being always a > L = 6 ˙ u v − nv ˙ v + 3 kv − V v , (32)u being cyclic, the conserved current (25) reads as, Q = 12 ˙ uv , (33)and the energy equation (25) leads to the following first order differential equation for v , (cid:18) Q − k (cid:19) + V v = 3 n v , (34)which may be integrated to obtain the following solution for v as, v = e pt + 4 F V e − pt V , (35)7here, F = 3 k − Q . We may also obtain in view of equation (33), the form of the cyclic coordinate u as, u = (cid:18) √ nQ √ V (cid:19)(cid:18) e pt − F V e − pt (cid:19) + u (36)where, p = q V n . Setting the integration constant u = 0 , for the origin of time, the exact solution for a ( t ) and φ ( t ) are found as a ( t ) = (cid:18) n ( a e pt + a e − pt ) − a e pt − a e − pt ) ( a e pt + a e − pt ) (cid:19) , (37) φ ( t ) = ( a e pt + a e − pt ) (cid:16) n ( a e pt + a e − pt ) − a e pt − a e − pt ) (cid:17) , (38)Where, a = V , a = 3 k − Q , a = √ nQ √ V and a = √ n √ V Q (cid:16) k − Q (cid:17) are constants. As t → ∞ , the scalefactor a → p n a e pt , while as t → −∞ , the scale factor a → p n a e pt , and finally as t → a → q n ( a + a ) − (cid:0) a − a a + a (cid:1) . Therefore asymptotically ( t → ±∞ ) the universe is de-Sitter, while as t → t → ±∞ ), the scalar field turns out to be a constant φ → q n . As aresult in the present case in view of equations (16) and (17), asymptotically one finds ρ e → V φ > p e → − V φ and ρ e + p e → Case 2. Induced theory of gravity
Let us now consider induced theory of gravity by the choice f ( φ ) = ǫφ ,where ǫ is the coupling constant. Under this choice, we obtain the following solutions in view of Equation (24), V = V ǫ φ − Λ , ω = 48 ǫnǫ φ − , B = D s nǫ φ − nǫ φ , B = cD s nǫ φ − nǫ φ , (39)where c, D are constant. As a result we also find α = − N p nǫ φ − aφ , β = N p nǫ φ − a φ , (40)where the constant N = cD lǫ √ n . The conserved current in the present case reads as, Q = 3 N ǫa p nǫ φ − (cid:18) ˙ aa + 2 nǫ φ − nǫ φ − φφ (cid:19) . (41)The Lagrangian (12) takes the form, L = − ǫa ˙ a φ − ǫa ˙ aφ ˙ φ + 3 ǫkaφ + 24 ǫa ˙ φ nǫ φ − − V ǫ a φ . (42)As before, let us now perform the change of variables to obtain the corresponding cyclic coordinate u . Theequation (8) is satisfied under the choice u = a φ p nǫ φ − v = aφ , (43)8hich may be inverted to obtain a = nǫ v − u v , φ = √ v √ nǫ v − u , (44)while the Lagrangian (42) in view of the new variables now takes the following form, L = 6 ǫ ˙ u v − nǫ v ˙ v + 3 ǫkv − V ǫ v . (45)Now, u being cyclic, the conserved current (25) reads as, Q = 12 ǫ ˙ uv . (46)In view of the energy equation (25) we find the following first order differential equation in v , (cid:16) Q ǫ − kǫ (cid:17) + V v = 3 n v . (47)The above first order differential equation in v may be integrated to find the following form of v , v = e pt + 4 F V e − pt V , (48)where, F = kǫ − Q ǫ . The cyclic variable u may be found as well in view of (46) as, u = (cid:18) Q √ n V ǫ √ V (cid:19)(cid:18) e pt − F V e − pt (cid:19) + u , (49)where, p = q V n . Setting the integration constant u = 0 as before, we finally obtain exact solutions of a ( t ) and φ ( t ) as, a ( t ) = (cid:18) nǫ ( b e pt + b e − pt ) − b e pt − b e − pt ) ( b e pt + b e − pt ) (cid:19) , (50) φ ( t ) = ( b e pt + b e − pt ) (cid:18) nǫ ( b e pt + b e − pt ) − b e pt − b e − pt ) (cid:19) , (51)where, b = V , b = kǫ − Q ǫ , b = Q √ n V ǫ √ V , b = (cid:0) kǫ − Q ǫ (cid:1)(cid:0) Q √ n ǫ √ V (cid:1) are constants as specified. Clearly,the form of the scale factor remains unaltered from the previous case, and as such represents Lorentzian wormholesolution yet again. Weak energy condition ρ e > ρ e + p e ≥ Case 3. Non-minimally coupled theory of gravity.
Finally, let us consider f ( φ ) = (1 − εφ ), ε being acoupling constant. Under this choice, Equation (24) yields the following set of solutions, V = V (1 − εφ ) − Λ , ω = 48 ε φ (1 − εφ )[ n (1 − εφ ) − , B = B p n (1 − εφ ) − √ n (1 − εφ ) , B = − cB p n (1 − εφ ) − √ nεφ , (52)9here c, B are constants. As a result we find α = − N p n (1 − εφ ) − a (1 − εφ ) , β = − N p n (1 − εφ ) − εa φ , (53)where N = cB l √ n is a constant. The conserved current turns out to be, Q = 3 N a p n (1 − εφ ) − (cid:18) ˙ aa + 2 ε φ ˙ φ − εφ n (1 − εφ ) − n (1 − εφ ) − (cid:19) , (54)while the Lagrangian (12) takes the following form, L = − a ˙ a (1 − εφ ) + 6 εa ˙ aφ ˙ φ + 3 ka (1 − εφ ) + 24 ε a φ ˙ φ (1 − εφ ) (cid:0) n (1 − εφ ) − (cid:1) − V a (1 − εφ ) . (55)As before, we now perform the change of variables to obtain the corresponding cyclic coordinate u . Equation (8)may be solved to find, u = a (1 − εφ )8 p n (1 − εφ ) − v = a (1 − εφ ) , (56)which may be inverted to obtain a = nv − u v , φ = 1 ε (cid:18) − v nv − u (cid:19) . (57)The Lagrangian (55) in terms of the new variables ( u and v ) takes the following simplified form L = 6 ˙ u v − nv ˙ v + 3 kv − V v . (58)Since u is cyclic, the conserved current may be found in view of (25) as, Q = 12 ˙ uv , (59)while the energy equation (25) reads as, (cid:16) Q − k (cid:17) + V v = 3 n v , (60)which may be integrated to yield v = e pt + 4 F V e − pt V , (61)where, F = 3 k − Q . We can also find the cyclic coordinate u in view of (59) as, u = (cid:18) √ nQ √ V (cid:19)(cid:18) e pt − F V e − pt (cid:19) + u , (62)10here p = q V n . As before we set the integration constant u to zero for the origin of time, and solve the scalefactor a ( t ) and the scalar field φ ( t ) exactly as a ( t ) = (cid:18) n ( a e pt + a e − pt ) − a e pt − a e − pt ) ( a e pt + a e − pt ) (cid:19) , (63) φ ( t ) = 1 √ ε (cid:18) − ( a e pt + a e − pt ) n ( a e pt + a e − pt ) − a e pt − a e − pt ) (cid:19) , (64)where, a = V , a = 3 k − Q , a = √ nQ √ V , and a = √ n √ V Q (cid:18) k − Q (cid:19) are constants. Here again we observethat the form of the scale factor remains unaltered from the earlier ones and therefore represents Lorentzianwormhole solution. The weak energy condition is not violated here again. Excitement raised after Ruggiero et al [45, 46] for the first time applied Noether symmetry in the scalar-tensortheory of gravity, to find a form of potential which naturally led to ‘Inflation’. Thereafter, many people worked inthe field and proved it to be a very powerful tool to explore the parameters and the potential involved in a theory.It also make things much easier to solve the Einstein’s field equations, particularly in view of the cyclic coordinate.The technique has been applied here again for a general non-minimally coupled scalar-tensor theory of gravity, inthe presence of cosmological constant. Three cases of particular interest have been studied, viz. the ‘Brans-Dicketheory’, the ‘Induced theory of gravity’ and the ‘General non-minimal scalar-tensor theory of gravity’. Whileonly an imaginary scalar field admits classical Euclidean wormhole solution [8], here, its Lorentzian counterpartadmits wormhole solutions for real scalar field. Noticeably, all the cases having different coupling parametersand potentials yield the same form of the scale factor, which represents cosmological Lorentzian wormholes. Suchwormholes admit weak energy condition. While Euclidian wormholes do not exist in general for arbitrary potential[8], evolving cosmological Lorentzian wormholes on the contrary, appear to be a generic feature of non-minimallycoupled Scalar-tensor theory of gravity. Such solutions depicts that, the universe itself evolved as a Lorentzianwormhole, which initiates inflation thereafter.In the process, it removes cosmological singularity arising from GTReven at the classical level. It is now required to check if the inflationary behaviour is at par with the currentlyreleased data [60], which we pose in future.
A Appendix
In the appendix we explicitly solve the set of Noether equations (19) - (22), under separation of variables (23), todemonstrate that the solutions lead to the set of equation (24). Primarily, equation (19) takes the form A + 2 aA ′ a ( A + aA ′ ) = − B B f ′ f = − C , (65)under the condition a ( A + aA ′ ) = 0 . Next, equation (20) takes the form3 B − f ′ ω B ′ B ′ + ω ′ ω B = − a A A = − C , (66)and of-course we need to fix (2 B ′ + ω ′ ω B ) = 0 . Now, using the relations (65) and (66), we get the followingrelation from equation (22)3 kf (1 + C C ) = a (cid:18) V + V ′ C C ff ′ + 3Λ (cid:19) , (67)11hich, for k = 0 implies that: C C = − , V = V f − Λ . (68)In order to obtain A and A , let us set C = c = C , and use it in equations (65) and (66). As a result, weobtain, A = la , A = − cla , B = c ff ′ B . (69)Finally, using the equations (21) and (66), we obtain the last two relations appearing in (24), viz. B ′ = 2 ω f ′ B , f ′ + 2 ωf = n ωf . (70)It is interesting to note that the equations (69) and (70) naturally lead to the following general relation between α ( a, φ ) and β ( a, φ ), viz. α = − aβ f ′ f . (71) References [1] G. Lavrelashvili, A. Rubakov and G. Tinyakov, JETP Lett. , 167 (1987).[2] G. Lavrelashvili, A. Rubakov and G. Tinyakov, Nucl. Phys. B 299 , 757 (1988).[3] G. Lavrelashvili, A. Rubakov and G. Tinyakov, Mod. Phys. Lett.
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