Euclidean wormholes with Phantom field and Phantom field accompanied by perfect fluid
aa r X i v : . [ g r- q c ] M a r Euclidean wormholes with Phantom field andPhantom field accompanied by perfect fluid
F. Darabi ∗ Department of Physics, Azarbaijan University of Tarbiat Moallem, 53714-161 Tabriz, Iran
November 6, 2018
Abstract
We study the classical Euclidean wormhole solutions for the gravitational systems withminimally coupled pure Phantom field and minimally coupled Phantom field accompaniedby perfect fluid. It is shown that such solutions do exist and then the general forms of thePhantom field potential are obtained for which there are classical Euclidean wormholesolutions.
Keywords:
Euclidean Wormholes; Phantom field
PACS: ∗ [email protected] Introduction
Classical wormholes are usually considered as Euclidean metrics that consist of two asymp-totically flat regions connected by a narrow throat (handle). Wormholes have been studiedmainly as instantons, namely solutions of the classical Euclidean field equations. In general,such wormholes can represent quantum tunneling between different topologies. They are pos-sibly useful in understanding black hole evaporation [1]; in allowing nonlocal connections thatcould determine fundamental constants; and in vanishing the cosmological constant Λ [2]-[4].They are even considered as an alternative to the Higgs mechanism [5]. Consequently, suchsolutions are worth finding.The reason why classical wormholes may exist is related to the implication of a theoremof Cheeger and Glommol [6] which states that a necessary (but not sufficient) condition for aclassical wormhole to exist is that the eigenvalues of the Ricci tensor be negative somewhere onthe manifold. Unfortunately, there exists certain special kinds of matter for which the abovenecessary condition is satisfied. For example, the energy-momentum tensors of an axion fieldand of a conformal scalar field are such that, when coupled to gravity, the Ricci tensor hasnegative eigenvalues. However, for pure gravity or a (real) minimally coupled scalar field it iseasy to show that the Ricci tensor can never be negative. It is shown that the (pure imaginary)minimally coupled scalar field, conformally coupled scalar field, a third rank antisymmetrictensor field and some special kinds of matter sources have wormhole solutions [7]-[12].In the present paper we shall investigate the possibility for the existence of classical Eu-clidean wormholes in the gravitational models minimally coupled with the pure Phantom field,and Phantom field accompanied by perfect fluid. Historically, Phantom fields were first intro-duced in Hoyles version of the steady state theory. In adherence to the perfect cosmologicalprinciple, a creation field (Cfield) was introduced by Hoyle to reconcile the model with thehomogeneous density of the universe by the creation of new matter in the voids caused bythe expansion of the universe [13]. It was further refined and reformulated in the Hoyle andNarlikar theory of gravitation [14]. From the cosmological point of view, the Phantom field is agood candidate for exotic matter which is characterized by the equation of state ω = p/ρ < − No wormholes with minimally coupled real scalar field
It is well known that classical Euclidean wormholes can occur if the Ricci tensor has negativeeigenvalues somewhere on the manifold. Actually, this is necessary but not sufficient conditionfor their existence and is related to the implication of a theorem of Cheeger and Glommol [6].For example, the energy-momentum tensors of an axion field and of a conformal scalar fieldare such that, when coupled to gravity, the Ricci tensor has negative eigenvalues. However,the story of minimally coupled real scalar field was somehow controversial. Costakis et al [18],had already claimed that Euclidean wormholes could be obtained with minimally coupled realscalar field, in contradiction with widely belief that wormholes require rather unusual mattersources [19]. However, Coule commented on this claim and showed that an imaginary scalarfield is necessary for Euclidean wormholes to occur [20].Let us consider the action of an ordinary scalar field minimally coupled to gravity [9] S = Z d x √− g (cid:20) R − ∇ µ φ ∇ µ φ − V ( φ ) (cid:21) , (1)where V ( φ ) is the potential of the scalar field. The energy momentum tensor of the field isderived by varying the action in terms of g µν T µν = − √− g δ S δg µν (2)= ∇ µ φ ∇ ν φ − g µν (cid:20) ∇ λ φ ∇ λ φ + V ( φ ) (cid:21) . The trace is also obtained as T = −∇ µ φ ∇ µ φ − V ( φ ) . (3)The Einstein equations R µν = T µν − g µν T, (4)give R µν = ∇ µ φ ∇ ν φ + g µν V ( φ ) , (5)which shows R µν can never be negative in Euclidean space g µν = (+ + ++) for V ( φ ) ≥ φ → iφ [20]. We consider theFriedmann-Robertson-Walker (FRW) metric in its Euclidean form ds = dτ + a ( τ ) dr (1 − kr ) + r ( dθ + sin θdφ ) ! , (6)where k = +1 , , − a a = 1 a + (cid:20)
12 ˙ φ − V ( φ ) (cid:21) , (7) We use the units in which πG = 1. φ + 3 ˙ aa ˙ φ = dV ( φ ) dφ , (8)where a dot denotes derivatives with respect to Euclidean time τ . By taking the ansatz [18]˙ φ = Aa p , (9)where A is a constant and p is a positive integer, the potential takes on the following form V ( φ ( a )) = A ( p − pa p . (10)Substituting this form into Eq.(7) gives˙ a a = 1 a + 3 A pa p . (11)In spatially closed FRW models, wormholes are typically described by a constraint equationof the form ˙ a a = 1 a − Consta n . (12)In order an asymptotically Euclidean wormhole to exist it is necessary that ˙ a > a which requires n >
2. Then, this wormhole has two asymptotically flat regions connected bya throat where ˙ a = 0. Comparing Eq.(11) with Eq.(12) reveals that the real scalar field cannot represent Euclidean wormholes and in order to have Euclidean wormholes we should let A → iA (or φ → iφ ) so that ˙ a a = 1 a − A pa p , (13)which now agrees with the typical known wormholes such as p = 2 or n = 4 (conformal scalarfield [8]) and p = 3 or n = 6 (axion field [7]). In the previous section we realized that the negative eigenvalues of Ricci tensor for an imaginaryscalar field is caused, in principle, by the the derivative part alone being negative definite,although the potential is also affected by φ → iφ . However, one may consider a model of realscalar field whose kinetic term is intrinsically negative definite. This is the so called Phantom field, and we will show that the wormhole solutions like those obtained by Costakis et al holdfor this model.The action for the Phantom field minimally coupled to gravity is written [9] S = Z d x √− g (cid:20) R + 12 ∇ µ φ ∇ µ φ − V ( φ ) (cid:21) . (14)4he energy momentum tensor and its trace are given by T µν = −∇ µ φ ∇ ν φ + g µν (cid:20) ∇ λ φ ∇ λ φ − V ( φ ) (cid:21) , (15) T = ∇ µ φ ∇ µ φ − V ( φ ) . (16)The Einstein equations read R µν = −∇ µ φ ∇ ν φ + g µν V ( φ ) . (17)It turns out that R µν may be negative in Euclidean space g µν = (+ + ++) if and only if thefollowing condition is satisfied −∇ µ φ ∇ ν φ + g µν V ( φ ) < , (18)where the potential may take one of the following cases ( V ( φ ) ≥ ,V ( φ ) < . (19)The field equations for the closed universe are obtained˙ a a = 1 a − (cid:20)
12 ˙ φ + V ( φ ) (cid:21) , (20)¨ φ + 3 ˙ aa ˙ φ = − dV ( φ ) dφ . (21)By choosing the ansatz (9) the potential casts in the following form V ( φ ( a )) = A (3 − p )2 pa p , (22)which after substituting into Eq.(20) gives˙ a a = 1 a − A pa p , (23)which is in agreement with the typical wormholes equation (12) provided that p > φ → iφ . However, the point is that this map affects the potential as well and theequations become different again. In fact, as explained in [20], the major cause for the missingminus sign of the kinetic term in [18] arises from the use of a different convention for theenergy-momentum energy tensor and a crucial mistake about a + sign in front of the potentialin Eq.(12). In the present paper, similar to [20] we have used the conventions of Wald [21] for5he energy-momentum tensor of the scalar field which provides us with the correct forms ofEqs.(20) and (21) which result in Eqs.(22) and (23).Now, we may realize the important differences between the results obtained here and thoseof obtained in [18]. The equation (22) is the same as the one obtained in [18], but the equation(23) is different from the corresponding one in [18] in that the functional dependence of ˙ a /a on p has now been changed. Moreover, the one condition (9) in [18] is replaced here by thetwo conditions considered in (19). This is because the missing minus sign in Eq.(8) of Ref.[18]requires mistakenly that the potential must always be positive, whereas in the correct form ofthis equation for the case of Phantom field, namely (17), the potential can take the negativevalues as well. Therefore, the results obtained in [18] are just the V ( φ ) ≥ V ( φ ) ≥ It is clear that in order the potential (22) satisfies the condition V ( φ ) ≥ p ≤
3. We then consider the three cases for p as follows. p = 3 : This case corresponds to a vanishing potential but the scalar field is still dynamical, namely˙ φ = 0. The Friedmann equation (23) is written˙ a a = 1 a − A a , (24)which is in the form of Euclidean wormhole (12). This equation solves to [18]1 √ F ( δ, r ) − √ E ( δ, r ) + 1 u √ u − / √− A ( τ − τ ) , (25)where F ( δ, r ) and E ( δ, r ) are the elliptic integrals of the first and second kind, respectively, with δ = arccos(1 /u ), r = 1 / √
2, and u = (2 / / √− A ) a . By inverting Eq.(25) the correspondingform of a ( τ ) is obtained. p = 2 : In this case, using the same ansatz (9) the system of equations (20), (21) leads to a ( τ ) = 3 A τ − c ) , (26) φ ( τ ) = 2 √ A √ τ − c ) ! + c , (27) V ( φ ) = 49 A cos √ φ − c )2 ! . (28)It turns out that (26) represents a wormhole solution with the minimum radius (throat) of thesize a min ∼ | A | which is connected to asymptotically flat space as τ → ±∞ .6 .1.3 Case p = 1 : The system of equations (20), (21) for the ansatz (9) leads to [18] a ( τ ) = a + a τ, (29) φ ( τ ) = Aa ln( a + a τ ) + a , (30) V ( φ ) ∼ exp( φ − a ) , (31)where a , a , a , and a are constants of integrations. The solution (29) does not show awormhole. This is because the boundary condition ˙ a = 0 kills the coefficient a and we loosethe time dependence of the solution. Moreover, the radius a becomes negative for τ → −∞ .Actually, the reason why this case does not lead to a wormhole is simple: the cases p ≤ p < : It is obvious that this case is not supported by Eq.(23), but if we take an open universe k = − ˙ a a = − a − A pa p . (32)Hence, we can allow p to take the negative values and Eq.(32) is written as˙ a a = 3 A | p | a | p | − a . (33)This equation is describing wormhole solutions because it gives a throat at ˙ a = 0 and ˙ a remains positive at large a . Therefore, the wormhole solutions are obtained as τ − τ = Z da q A | p | a | p | +2 − , (34) φ ( τ ) = Z Aa ( τ ) | p | dτ, (35)with a throat size at ˙ a = 0 a = | p | A ! | p | +2 . (36) The case p = 0 is not physically viable, so we have not considered this case throughout the paper. .2 Case: V ( φ ) < In this case, using the ansatz (9) in Eq.(22) we find p > τ − τ = Z da q − A pa p − , (37) φ ( τ ) = Z A [ a ( τ )] p dτ, (38)with a throat size at ˙ a = 0 a = A p ! p − . (39) ˙ φ = g ( a ) , k = ± We now consider the general case by assuming the ansatz [18] and k = ± φ = g ( a ) . (40)Equation (21) becomes g ′ ˙ a + 3 ˙ aa g = − dVdφ , (41)where a prime denotes d/da . Using (40) in the form ˙ a = g dadφ , we obtain V ( a ) = − Z " gg ′ + 3 g a da (42)= c − g − g ln a + 6 Z gg ′ ln ada, where an integration by part has been used and c is the constant of integration. Putting thisform of potential into equation (20) gives˙ a a = ka + 3 Z g a da, (43)or τ − τ = Z da h k + 3 a hR g a da ii / , (44)where τ as the integration constant denotes the time at which a becomes a ≡ a min . Noticethat equations (43), (44) are different from the corresponding ones obtained in [18] due to theabsence of a g term (because of the previously mentioned missing minus sign). If we demandthe wormhole solutions of the form a = a + sinh τ − τ m , (45)8here m is a constant, then comparing (45) with (44) results in g ( a ) = " aa − a + m − km a ! . (46)Substituting (46) into (42) gives the potential in terms of the radius aV ( a ) = c + c a + c a . (47)The scalar field as a function of a can also be obtained by using (40) and (44) φ ( a ) = Z g ( a ) da h k + 3 a hR g a da ii / , (48)where g ( a ) is given by (46). The action for the Phantom field minimally coupled to gravity and a perfect fluid source isgiven by [22] S = Z d x √− g (cid:20) R + 12 ∇ µ φ ∇ µ φ − V ( φ ) + L matter (cid:21) . (49)The energy momentum tensor and the corresponding Einstein equations are given by T µν = ( ρ b − p b ) u µ u ν + p b g µν − ∇ µ φ ∇ ν φ + g µν (cid:20) ∇ λ φ ∇ λ φ − V ( φ ) (cid:21) , (50) R µν = ( ρ b − p b ) u µ u ν − ∇ µ φ ∇ ν φ − g µν ( ρ b + p b − V ( φ )) . (51)The Ricci tensor may be negative in Euclidean space if and only if the following condition issatisfied ( ρ b − p b ) u µ u ν − ∇ µ φ ∇ ν φ − g µν ( ρ b + p b − V ( φ )) < , (52)where the potential can take the following cases as before ( V ( φ ) ≥ ,V ( φ ) < . (53)One may define the pressure and energy density of the Phantom field φ in Euclidean signature ρ φ = ˙ φ V ( φ ) , p φ = ˙ φ − V ( φ ) . (54)9herefore, the conditions (53) are rewritten as ( ρ φ ≥ p φ ,ρ φ < p φ . (55)To justify that the condition (52) may be generally satisfied, one may for instance take amonotonic function φ ( τ ) for which we obtain R = 12 ( ρ b + ρ φ − p b − p φ ) − ˙ φ . (56)It is easily seen that R may become negative provided that some suitable equations of statefor matter and Phantom field are taken so that either the terms including the energy densitiesand pressures become negative, or otherwise, the (Euclidean) kinetic term precedes these terms.Unlike the pure Phantom field case where the wormholes are possible just for the closed andopen universe, in the present model which includes the Phantom field accompanied by mattersource we shall examine the existence of wormholes for the three cases of closed, flat and openuniverses. k = 1 : The field equations in this case are obtained˙ a a = 1 a − [ ρ b + ρ φ ] , (57)¨ φ + 3 ˙ aa ˙ φ = − dV ( φ ) dφ , (58)where the background energy density due to the matter and radiation is given by ρ b = ρ m a + ρ r a . (59)If we limit ourselves to the ansatz (9), and use (22) and (54) in (57) we obtain˙ a a = 1 a − " ρ m a + ρ r a + 3 A pa p , (60)It is seen that ˙ a remains positive at large a provided that p > p > p >
0, respectively. Therefore, the wormhole solutions areobtained τ − τ = Z da r − (cid:16) ρ m a + ρ r a + A pa p − (cid:17) , (61)10 ( τ ) = Z A [ a ( τ )] p dτ, (62)where the throat size at ˙ a = 0 is given by the solution of the following equation2 p ( a p − − ρ m a p − − ρ r a p − ) − A = 0 , (63)which depends on the properties of the matter sources, namely ρ m and ρ r . k = 0 : The field equations are obtained ˙ a a = − " ρ m a + ρ r a + 3 A pa p , (64)¨ φ + 3 ˙ aa ˙ φ = − dV ( φ ) dφ , (65)It is easily seen that for positive values of p we can not cast the equation (64) in the form ofwormhole equation (12) . In other words, ˙ a can not be positive for any positive value of p .Therefore, positive values of p are excluded to be candidates for wormhole solutions. However,for the negative values of p one can rewrite (64) as˙ a a = 3 A a | p | | p | − (cid:20) ρ m a + ρ r a (cid:21) . (66)This equation is describing wormhole solutions since ˙ a remains positive at large a and resultsin asymptotically Euclidean wormholes τ − τ = Z da r A | p | a | p | +2 − h ρ m a + ρ r a i , (67) φ ( τ ) = Z Aa ( τ ) | p | dτ, (68)whose typical throat size at ˙ a = 0 is given by3 A | p | a | p | +4 − ρ m a − ρ r = 0 . (69) k = − : The field equations are given by˙ a a = − a − " ρ m a + ρ r a + 3 A pa p , (70)¨ φ + 3 ˙ aa ˙ φ = − dV ( φ ) dφ . (71)11he equation (70), for positive values of p , does not describe wormhole solutions since ˙ a cannever be positive. Therefore, the positive values of p are again excluded from consideration.However, for negative values of p one can rewrite (70) as˙ a a = 3 A a | p | | p | − (cid:20) a + ρ m a + ρ r a (cid:21) . (72)Similar to (66), the equation (72) may describe wormhole solutions τ − τ = Z da r A | p | a | p | +2 − h ρ m a + ρ r a i , (73) φ ( τ ) = Z Aa ( τ ) | p | dτ, (74)whose throat size is given by the solution of the following equation3 A | p | a | p | +4 − a − ρ m a − ρ r = 0 . (75) ˙ φ = g ( a ) , k = ± , We again consider the general case by assuming the ansatz (40). Moreover, we shall considerthe three cases of closed, flat and open universes indicated by k = ± ,
0. In the same way asdiscussed in the subsection 3.3, we obtain the general field equation˙ a a = ka − ρ m a − ρ r a + 3 Z g a da, (76)or τ − τ = Z da h k − ρ m a − ρ r a + 3 a hR g a da ii / . (77)Demanding the wormhole solutions of the form (45) leads to g ( a ) = " aa − a + m − km a ! − (cid:18) ρ m a + 4 ρ r a (cid:19) , (78) V ( a ) = c + c a + c a + c a + c a . (79)and φ ( a ) = Z g ( a ) da h k − ρ m a − ρ r a + 3 a hR g a da ii / . (80)12 Conclusion
The possible forms of matter sources which may result in classical Euclidean wormholes arevery limited. The axion field, conformal scalar field, (pure imaginary) minimally coupledscalar field, conformally coupled scalar field, a third rank antisymmetric tensor field and somespecial kinds of matter sources result in such wormhole solutions. However, pure gravity ora (real) minimally coupled scalar field do not represent Euclidean wormholes. Any effortto obtain new forms of matter representing Euclidean wormhole solutions are of particularimportance in quantum gravity. In this paper, we have considered the Phantom field minimallycoupled to gravity and studied the possibility of Euclidean wormholes to occur in the spatiallyclosed and open Friedmann-Robertson-Walker universes. It is shown that these solutions mayappear due to the negative kinetic energy of the Phantom field. Then, we have obtained somewormhole solutions in this model and found the general form of the corresponding Phantomfield potential. Then, we have studied the possibility of Euclidean wormholes to occur inthe system of a Phantom field accompanied by perfect fluid, minimally coupled to gravity.The existence of these solutions is explicitly shown and wormhole solutions together withthe corresponding Phantom field potentials are obtained for spatially closed, flat and openFriedmann-Robertson-Walker universes.It is appealing to study other models in which the kinetic term may effectively get wrongsign similar to the Phantom field. Kinetically driven inflation or k-Inflation models are ex-amples of this kind where a large class of higher-order (i.e. non-quadratic) scalar kineticterms can, without the help of potential terms, drive an inflationary evolution starting fromrather generic initial conditions [23]. We aim to study the possibility of obtaining Euclideanwormholes in such models as well.
Acknowledgment
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