The simplest wormhole in Rastall and k-essence theories
Kirill A. Bronnikov, Vinícius A.G. Barcellos, Laura P. de Carvalh, Júlio C. Fabris
aa r X i v : . [ g r- q c ] F e b The simplest wormhole in Rastall and k-essence theories
Kirill A. Bronnikov Center fo Gravitation and Fundamental Metrology, VNIIMS, Ozyornaya ul. 46, Moscow 119361, Russia;Institute of Gravitation and Cosmology, Peoples’ Friendship University of Russia (RUDN University), ul.Miklukho-Maklaya 6, Moscow 117198, Russia;National Research Nuclear University “MEPhI”, Kashirskoe sh. 31, Moscow 115409, Russia
Vin´ıcius A.G. Barcellos and Laura P. de Carvalho
N´ucleo Cosmo-ufes & Departamento de F´ısica, CCE, Universidade Federal do Esp´ırito Santo,Vit´oria, ES, CEP29075-910, Brazil
J´ulio C. Fabris N´ucleo Cosmo-ufes & Departamento de F´ısica, CCE, Universidade Federal do Esp´ırito Santo,Vit´oria, ES, CEP29075-910, BrazilNational Research Nuclear University “MEPhI”, Kashirskoe sh. 31, Moscow 115409, Russia
The geometry of the Ellis-Bronnikov wormhole is implemented in the Rastall and k-essence theoriesof gravity with a self-interacting scalar field. The form of the scalar field potential is determinedin both cases. A stability analysis with respect to spherically symmetric time-dependent pertur-bations is carried out, and it shows that in k-essence theory the wormhole is unstable, like theoriginal version of this geometry supported by a massless phantom scalar field in general relativity.In Rastall’s theory, it turns out that a perturbative approach reveals the same inconsistency thatwas found previously for black hole solutions: time-dependent perturbations of the static configu-ration prove to be excluded by the equations of motion, and the wormhole is, in this sense, stableunder spherical perturbations.
Black holes and wormholes are remarkable predictions of the General Relativity theory (GR). Thedetection of gravitational waves emitted by merging of compact objects [1] and the recent image ofa supermassive object at the center of the galaxy M87 [2] have brought black holes to the status ofastrophysical objects whose existence in nature leaves little doubt. On the other hand, wormholesremain a hypothetical prediction of GR. In its simplest configuration, a wormhole is composedof two asymptotically flat Minkowskian space-times connected by a kind of tunnel. The two flatasymptotic regions are usually considered as different universes that are connected by a throat. Oneof the problematic aspects of wormhole configurations is the necessity of having negative energy, atleast in the vicinity of the throat, in order that they could exist. Negative energy, which impliesviolation of the standard energy conditions, brings two main problems: the configuration can beunstable; or generally, the throat may not be traversable, in the sense that tidal forces may be huge,and possibly only pointlike objects may cross it from one universe to the other, except for somespecial cases. For a pedagogical description of wormhole properties, see Ref. [3]. e-mail: [email protected] e-mail: [email protected] The Ellis-Bronnikov (EB) wormhole [4, 5] is one of the simplest solutions of GR leading to astructure of two flat asymptotics connected by a throat. As a matter content, the EB wormholesolution uses a free massless scalar field with negative energy. Such field is normally denoted as aphantom scalar field. The configuration is, as could be expected, unstable due to the repulsive natureof the scalar field, see, e.g., [6, 7] and references therein. Studies of static, spherically symmetricconfigurations in the presence of scalar fields have a long history, see [8,9] for the first seminal workson these lines. In parallel, there has been much effort to obtain wormhole solutions which, besidesbeing traversable, would be stable and do not require exotic matter. However, it is hard to fulfillthese requirements in the context of GR and even in its extensions for a simple reason: in orderto cross the throat by coming from one region and arriving in the other, the geodesics must firstconverge and later diverge, and this property requires repulsive properties of matter which shouldthus violate at least some of the standard energy conditions. Still in the framework of GR there are,on the one hand, an example [10] of a stable wormhole supported by some kind of phantom matter,and, on the other hand, examples of phantom-free rotating cylindrically symmetric wormholes whosestability properties are yet unknown [11, 12].It is well known that a given metric may be a solution of the field equations of different theoriesof gravity or even in a single theory with different matter sources. An example is [10] where theEB wormhole in GR is supported by a particular kind of phantom perfect fluid instead of a scalarfield as in [4, 5]. In the case of different theories, the matter content should naturally depend onthe theory under consideration. In this paper we explore the EB wormhole metric in two differenttheories. The first one is Rastall’s theory of gravity [13] that abandons one of the cornerstones ofGR, the usual conservation law for matter fields. The second one is the k-essence theory [14] whichmodifies the matter sector by introducing non-canonical forms for the kinetic term of a scalar field.The k-essence proposal may be connected with some fundamental theories inspired by quantumgravity. In both cases our goal is to verify if it is possible to avoid the usual difficulties in wormholeconstruction and to obtain stable solutions.Previously, both Rastall and k-essence theories with a self-interacting scalar field have beenstudied in attempts to obtain static, spherically symmetric black hole solutions [15, 16]. The solu-tions turned out to be quite exotic, mainly due to the asymptotic properties at infinity. A stabilityanalysis has shown that those k-essence solutions were unstable [17]. However, surprisingly, theperturbation analysis of the Rastall solutions was shown to be inconsistent, and the stability issueremained unclear [18]. It has been speculated that this property of the Rastall solutions is connectedwith the absence of a Lagrangian formulation of this theory. A curious aspect of these k-essenceand Rastall solutions is that they share some duality properties, in spite of quite different structuresof the theories themselves [19].Here we show that the EB wormhole metric can be a solution of both Rastall and k-essencetheories under the condition that the potential describing the self-interaction of the scalar field isnonzero. We determine the form of this potential in each case. In the k-essence theory we usea power-law expression of the kinetic term, as in Ref. [16]. We perform a perturbation analysisof these solutions using a gauge-invariant approach, and we find that the k-essence solution isunstable. Unlike that, in Rastall gravity the inconsistency found previously for black hole solutionsre-appears here, and no time-dependent spherically symmetric perturbations can exist. Thus theEB metric in this framework may be said to be stable under such perturbations, but the existence ofnonperturbative time-dependent solutions cannot be excluded, to say nothing of possible instabilitiesunder less symmetric perturbations.The paper is organized as follows. In Section 2 some general expression to be used in thecalculations are settled out. In Section 3, the EB wormhole solution in GR is reproduced forcomparison. In Section 4, the corresponding wormhole solution and the stability issue is presentedfor Rastall gravity. A similar analysis is carried out in k-essence theory in Section 5. In Section 6we present our conclusions.
The goal of the present section is to give some general relations that will be used in the rest of thepaper. We assume spherical symmetry but not necessarily static. This allows us to easily considera static configuration which we will call the background and linear perturbations around it.Spherical symmetry can be described by a metric of the form ds = e γ ( t,x ) dt − e α ( t,x ) dx − e β ( t,x ) d Ω , (1)where d Ω is the metric on a unit 2-sphere. If the configuration besides being spherically symmetricis also static, the metric coefficients α, β and γ depend only on the radial coordinate x . There isfreedom to reparametrize the radial coordinate, and its particular choice can be made by postulatinga condition connecting the coefficients α , β and γ .For the metric (1) the components of the Ricci tensor and expression for the d’Alambertianoperator acting on a scalar field are given by R = e − γ (cid:2) ¨ α + 2 ¨ β + ˙ α + 2 ˙ β − ˙ γ ( ˙ α + 2 ˙ β )] − e − α (cid:2) γ ′′ + γ ′ ( γ ′ − α ′ + 2 β ′ ) (cid:3) ,R = e − γ (cid:2) ¨ α + ˙ α ( ˙ α − ˙ γ + 2 ˙ β ) (cid:3) − e − α (cid:2) γ ′′ + 2 β ′′ − α ′ ( γ ′ + 2 β ′ ) + γ ′ + 2 β ′ (cid:3) ,R = e − γ (cid:2) ¨ β + ˙ β ( ˙ α − ˙ γ + 2 ˙ β ) (cid:3) − e − α (cid:2) β ′′ + β ′ ( γ ′ − α ′ + 2 β ′ ) (cid:3) + e − β ,R = 2 ˙ β ′ + 2( β ′ − γ ′ ) ˙ β − β ′ ˙ α, (2) ✷ φ = e − γ (cid:2) ¨ φ + ( ˙ α − ˙ γ + 2 ˙ β ) ˙ φ (cid:3) − e − α (cid:2) φ ′′ + ( γ ′ − α ′ + 2 β ′ ) φ ′ (cid:3) . (3)where dots denote /. t. and primes /. x. .In the case of a static space-time, all time derivatives disappear. However, in the study of smalltime-dependent perturbations around a given static solution at linear order, the linear terms withtime derivatives become relevant.In what follows we will discuss wormhole configurations in GR, Rastall’s theory of gravity in thepresence of a scalar field, and k-essence theories. In all these cases, the gravitational field equationscan be written as the Einstein equations with appropriate stress-energy tensors T νµ , R νµ − δ νµ R = − T νµ , (4)or alternatively, R νµ = − T νµ + 12 δ νµ T ρρ , (5)where we are using units in which (in usual notations) c = 8 πG = 1. These expressions are alsovalid in Rastall gravity, under a suitable redefinition of the stress-energy tensor. Let us begin with recalling a derivation of the Ellis-Bronnikov wormhole solution in the context ofGR. The equations in the presence of a free massless scalar field φ are given by R νµ − δ νµ R = − ǫ (cid:18) φ ; µ φ ; ν − δ νµ φ ; ρ φ ; ρ (cid:19) , (6) ✷ φ = 0 , (7)where the parameter ǫ indicates if the scalar field is of ordinary (canonical) ( ǫ = 1) or phantom( ǫ = −
1) type. The Einstein equations rewritten in the form (5) read R νµ = − ǫφ ; µ φ ; ν . (8)Let us consider the static metric (1) and a scalar field φ = φ ( x ). The set of equations (6) and(7) is then most conveniently solved using the harmonic coordinate condition α = 2 β + γ [5] (underwhich we will denote the radial coordinate by u ). Indeed, under this condition, the scalar fieldequation (7) and two independent equations among (8) (specifically, R = 0 and R + R = 0) takethe form φ ′′ = 0 , γ ′′ = 0 , β ′′ + γ ′′ = e β +2 γ , (9)(the prime stands here for d/du ). All of them are immediately integrated giving φ = Cu, γ = − mu, m, C = const , (10)(where two other integration constants are suppressed by choosing the scale of t and the zero pointof φ ), and( β ′ + γ ′ ) = e β +2 γ + k sign k, k = const , (11)where one more integration constant is suppressed by choosing the zero point of u . The solution of(11) depends on the sign of k : e − β − γ = k − sinh( ku ) , k > ,e − β − γ = u, k = 0 ,e − β − γ = k − sin( ku ) , k < , (12)which can be jointly written as e − β − γ = s ( k, u ) ≡ k − sinh( ku ) , k > ,u, k = 0 ,k − sin( ku ) , k < . (13)Lastly, substituting (10) and (11) into the (cid:0) (cid:1) component of Eqs. (6) (which is an integral of othercomponents), we obtain a relation between the integration constants: k sign k = m + 12 ǫC . (14)The metric takes the form ds = e − mu dt − e mu s ( k, u ) (cid:18) du s ( k, u ) + d Ω (cid:19) , (15)The constants m and C have the meaning of the Schwarzschild mass and the scalar charge, respec-tively. The coordinate u is defined (without loss of generality) at u >
0, and u = 0 corresponds tospatial infinity (since there r ( u ) ≡ e β → ∞ ), at which the metric is asymptotically flat.Equations (10), (14) and (15) give a joint representation of all static, spherically symmetricsolutions to Eqs. (6), (7): Fisher’s solution [8] of 1948 (repeatedly rediscovered afterwards) corre-sponding to ǫ = 1 (hence k >
0) and all three branches of the solution for ǫ = − k (sometimes called anti-Fisher solutions). Detailed descriptions of the correspondinggeometries can be found, e.g., in [5, 6, 20, 21]. Note that the instability of Fisher’s solution undersmall radial perturbations was shown in [22], and that of anti-Fisher solutions in [6, 7].Our interest here is with wormhole solutions, which form the branch ǫ = − , k <
0: in this case,we have two flat spatial infinities at u = 0 and u = π/ | k | . The solution looks more transparentafter the radial coordinate transformation x = b cot( bu ) , b := | k | , (16)which brings the solution to the form ds = e − m [ π/ − arctan( x/b )] dt − e m [ π/ − arctan( x/b )] dx − ( x + b ) d Ω , (17) φ = C (cid:2) π/ − arctan( x/b ) (cid:3) , (18)where x is the so-called quasiglobal coordinate corresponding to the “gauge” α + γ = 0 in termsof the metric (1). The simplest configuration is obtained in the case of zero mass, m = 0: ds = dt − dx − ( x + b ) d Ω , (19) φ = ± b √ (cid:2) π/ − arctan( x/b ) (cid:3) . (20)It is this solution that is called the Ellis wormhole [4], or the Ellis-Bronnikov (EB) wormhole, sincethis and more general scalar-vacuum and scalar-electrovacuum configurations were obtained anddiscussed in [5]. In terms of the metric (1), we have in (19) α ≡ γ ≡ , β ≡ log r ( x ) = 12 log( x + b ) . (21) Consider now linear time-dependent spherically symmetric perturbations of the EB wormhole, de-scribed by additions δα , δβ , δγ and δφ of the corresponding static (background) quantities, char-acterized by some smallness parameter ε . Following [6, 21, 22], we choose the perturbation gauge δβ = 0. Then the perturbation equations following from (7) and (6) in the order O ( ε ) can bewritten as e α − γ ) δ ¨ φ − δφ ′′ − [2 β ′ + γ ′ − α ′ ] δφ ′ − [ δγ ′ − δα ′ ] φ ′ = 0 . (22) e α − γ ) δ ¨ α − δγ ′′ − δγ ′ (2 γ ′ − α ′ + 2 β ′ ) + γ ′ δα ′ = 0 , (23) e α − γ ) δ ¨ α − δγ ′′ + δα ′ ( γ ′ + 2 β ′ ) + ( α ′ − γ ′ ) δγ ′ = 2 ǫφ ′ δφ ′ , (24) β ′ ( δγ ′ − δα ′ ) − e α − β ) δα = 0 , (25) − β ′ δ ˙ α = − ǫ φ ′ δ ˙ φ. (26)These equations are written with an arbitrary radial coordinate x in the background static metricbut with a particular choice ( δβ = 0) of the perturbation gauge fixing the reference frame inperturbed space-time. We see that Eq. (26) can be integrated in t giving δα = − η δφ + ξ ( x ) , η = φ ′ β ′ , (27)with an arbitrary function ξ ( x ); we will put ξ ( x ) ≡ γ = α = 0 and ǫ = −
1, the remainingequations read δ ¨ φ − δφ ′′ − β ′ δφ ′ − φ ′ ( δγ ′ − δα ′ ) = 0 . (28) δ ¨ α − δγ ′′ − β ′ δγ ′ = 0 , (29) δ ¨ α − δγ ′′ + 2 β ′ δα ′ = − φ ′ δφ ′ , (30) β ′ ( δγ ′ − δα ′ ) − e − β δα = 0 , (31)Subtracting equations (29) and (30) and using (27), we obtain δγ ′ = 12 ( η ′ δφ − ηδφ ′ ) . (32)Knowing δα and δγ ′ , or equivalently using directly (31), we can eliminate the metric perturbationsfrom the scalar field equation, which results in the following master equation: δ ¨ φ − δφ ′′ − β ′ δφ ′ − η ′ φ ′ δφ = 0 . (33)Assuming the time dependence of δφ as a single spectral mode, δφ ∝ e iωt , δφ ′′ + 2 β ′ δφ ′ + ( ω + η ′ φ ′ ) δφ = 0 . (34)Eliminating δφ ′ by the substitution δφ = e − β y ( x ), we arrive at the Schr¨odinger-like equation y ′′ + (cid:26) ω + η ′ φ ′ − β ′′ − β ′ (cid:27) y = 0 , (35)which coincides with the master equation found in [6] in the special case where α = γ = 0 and noscalar field potential is present. Using our expressions for φ and β in the Ellis wormhole solution,we find y ′′ + (cid:26) ω − (cid:20) b (3 x + 2 b ) x ( x + b ) (cid:21)(cid:27) y = 0 . (36)The stability analysis requires imposing boundary condition. In our case, for x → ±∞ it isreasonable to require δφ →
0, or y = o ( | x | ). We can note that, asymptotically, Eq. (36)) hassolutions in terms of Bessel functions, y ( x ) = A ± p | x | J ± ν ( ω | x | ) , ν = p b + 1 / , A ± = const . (37)If ω = i ¯ ω (an imaginary frequency describing an instability), the solutions become y ( x ) = p | x | h A K ν (¯ ω | x | ) + A I ν (¯ ω | x | ) i , A , = const . (38)where K ν and I ν are modified Bessel functions. The function K ν tends to zero at large | x | ,therefore, correct boundary conditions with imaginary ω are compatible with an instability. Onthe other hand, the positive nature of the effective potential V eff ( x ) in Eq. (36) (the expression inbrackets) seems to exclude “energy levels” ω <
0. However, this argument cannot be directlyapplied because of a pole of this effective potential near the wormhole throat x = 0, V eff ≈ /x .This potential can be regularized by the appropriate Darboux transformation as described in [6, 7].The regularized potential turns out to contain a sufficiently deep well leading to the existence of anunstable perturbation mode, related to an evolving throat radius. The same result was previouslyobtained by a numerical study [23] which proved that an Ellis wormhole can either collapse to ablack hole or inflate, depending on the sign of the initial perturbation.The gauge condition we are using, δβ = 0, seems to prevent considering perturbations connectedwith a changing throat radius. But a more thorough investigation shows [6, 7] that the unknown δφ in the master equation Eq. (33) is actually a gauge-invariant quantity. Indeed, a perturbationgauge may be described as a small coordinate shift x µ → x µ + ξ µ with ξ µ = O ( ε ), or, in the ( x, t )subspace, t = ¯ t + ξ ( x, t ) , x = ¯ x + ξ ( x, t ) . Then it can be directly verified that quantities like β ′ δφ − φ ′ δβ do not change under such coordinateshifts and are thus gauge-invariant, as well as their products with any background quantities, forexample, 1 /β ′ . It follows that δφ in our consideration is the specific form of the gauge-invariantquantity ψ = δφ − φ ′ δβ ′ /β ′ in the gauge δβ = 0. Other functions involved in (33) are combinationsof the background quantities, therefore we can safely replace there δφ with ψ and conclude that thewhole master equation is gauge-invariant. It can thus be used for considering any perturbations,including those with an evolving throat radius.The gauge invariance issue is presented in more detail in [6, 7, 21], and its analogue for pertur-bations in cosmology is discussed in [24]. In our further consideration we obtain gauge-invariantmaster equations for spherical perturbations in a similar way. In Rastall’s theory, if the source of gravity is a scalar field φ with a self-interaction potential V ( φ ),the field equations can be written as [15, 18] R νµ − δ νµ R = − ǫ (cid:26) φ ; µ φ ; ν + 2 − a δ νµ φ ; ρ φ ; ρ (cid:27) − (3 − a ) δ νµ V ( φ ) , (39) ✷ φ + ( a − φ ; ρ φ ; σ φ ; ρ ; σ φ ; α φ ; α = − ǫ (3 − a ) V φ , (40)where a is a constant parameter of the theory, and at its special value a = 1 we return to GR.Thus the effective stress-energy tensor of the scalar field reads T νµ = ǫ (cid:26) φ ; µ φ ; ν − − a δ νµ φ ; ρ φ ; ρ (cid:27) + δ νµ W ( φ ) , (41)where W ( φ ) = (3 − a ) V ( φ ). The modified Einstein equations can be rewritten as R µν = − ǫ (cid:26) φ ; µ φ ; ν + 1 − a g µν φ ; ρ φ ; ρ (cid:27) + g µν W ( φ ) . (42)For the static metric (1) and φ = φ ( x ), the Rastall equations reduce to aφ ′′ + [ γ ′ − aα ′ + 2 β ′ ] φ ′ = ǫe α W φ , (43) γ ′′ + γ ′ ( γ ′ − α ′ + 2 β ′ ) = − ǫ − a ) φ ′ − e α W, (44) γ ′′ + 2 β ′′ − α ′ ( γ ′ + 2 β ′ ) + γ ′ + 2 β ′ = − ǫ − a ) φ ′ − e α W, (45) β ′′ + β ′ ( γ ′ − α ′ + 2 β ′ ) − e α − β ) = − ǫ − a ) φ ′ − e α W, (46)where W φ = dW/dφ .These equations become identical to the GR equations with a massless scalar field if we put ǫ − a ) φ ′ = − e α W. (47)where we should take into account that W φ = W ′ /φ ′ . Then all solutions for α , β , γ and φ ′ arethe same as in GR. But, a new element in Rastall gravity is that one needs a nonzero potentialin order to create these solutions. For any given special solution, the potential can be determinedfrom Eq. (47) or from any of the equations (44)–(46).In particular, for the Ellis wormhole solution (19), (20) the potential is found to be W ( φ ) ≡ (3 − a ) V ( φ ) = b (1 − a )( x + b ) = 1 − ab cos ( φ/ √ . (48)Thus we have the simplest Ellis wormhole solution in Rastall gravity for any value of the Rastallparameter a . To obtain the linear perturbation equations, we are consider Eqs. (40) and (42) using the expressions(2) for the Ricci tensor components, the gauge δβ = 0 and the potential (48) as a function of φ .The equations read e α − γ ) δ ¨ φ − aδφ ′′ − ( γ ′ − aα ′ + 2 β ′ ) δφ ′ − φ ′ ( δγ ′ − aδα ′ ) = − ǫe α (2 W φ δα + W φφ δφ ) , (49) e α − γ ) δ ¨ α − δγ ′′ − (2 γ ′ − α ′ + 2 β ′ ) δγ ′ + γ ′ δα ′ = ǫ (1 − a ) φ ′ δφ ′ + e α (2 W δα + W φ δφ ) , (50) e α − γ ) δ ¨ α − δγ ′′ + ( γ ′ + 2 β ′ ) δα ′ + ( α ′ − γ ′ ) δγ ′ = ǫ (3 − a ) φ ′ δφ ′ + e α (2 W δα + W φ δφ ) , (51) β ′ ( δγ ′ − δα ′ ) − e α − β ) δα = − ǫ (1 − a ) φ ′ δφ ′ − e α (2 W δα + W φ δφ ) , (52) − β ′ δ ˙ α = − ǫ φ ′ δ ˙ φ. (53)For our simplest case γ = α = 0, ǫ = −
1, the equations read δ ¨ φ − aδφ ′′ − β ′ δφ ′ − φ ′ ( δγ ′ − aδα ′ ) = − ǫ (2 W φ δα + W φφ δφ ) , (54) δ ¨ α − δγ ′′ + 2 β ′ δγ ′ = ǫ (1 − a ) φ ′ δφ ′ + (2 W δα + W φ δφ ) , (55) δ ¨ α − δγ ′′ − β ′ δα ′ = ǫ (3 − a ) φ ′ δφ ′ + (2 W δα + W φ δφ ) , (56) β ′ ( δγ ′ − δα ′ ) − e − β δα = − ǫ (1 − a ) φ ′ δφ ′ − (2 W δα + W φ δφ ) , (57) β ′ δ ˙ α = − φ ′ δ ˙ φ. (58)In [18] it has been shown that the stability problem for the Rastall theory in static, sphericallysymmetric configurations is inconsistent unless all perturbations are zero. It turns out that here wecome across the same problem, as could be expected in view of those results. Indeed, from Eq. (58)we obtain, as previously in GR, δα = − ηδφ, η = φ ′ β ′ . (59)From the difference of (55) and (56) we obtain δγ ′ = ǫ ηδφ ′ − η ′ δφ ) . (60)On the other hand, from Eqs. (57) and (59) it follows δγ ′ = − (cid:16) ηδφ ′ − η ′ δφ (cid:17) + (1 − a ) η (cid:20) δφ ′ + (cid:18) η φ ′ + φ ′′ β ′ (cid:19) δφ (cid:21) . (61)In this expression we have separated the terms contained in (59) from the others.The expressions (59) and (61) coincide only if a = 1, that is, when the Rastall theory reducesto GR, or if the quantity in brackets in (61) vanishes. In the second case, we can find explicitly thebehavior of δφ : δφ = φ ( t ) exp (cid:18) − x + 2 b arctan xb (cid:19) , φ ( t ) = arbitrary function . (62)It is easy to see that, according to Eqs. (54) and (59), in the solution (62) the only possibility is φ ( t ) = 0. Hence, there is no perturbation at linear level, the same result as in [18]. Quite similarlyto [18], it implies the absence of perturbations in all orders of smallness. Let us consider the theory defined by the Lagrangian density L = √− g h R + f ( X ) − V ( φ ) i , (63)with the definitions X = ηφ ; ρ φ ; ρ , η = ± . (64)The scalar field equation has the form ηf X ✷ φ + 2 f XX φ ,ρ φ ,σ φ ; ρ ; σ = V φ , (65)where the subscripts X and φ denote derivatives with respect to the corresponding variables. TheEinstein equations have the form (4) with the stress-energy tensor T νµ = f X ηφ ,µ φ ,ν − δ νµ f + δ νµ V. (66)In the form (5) they can be written as R νµ = ηf X φ ,µ φ ,ν − δ νµ ( − f + Xf X + 2 V ) . (67)0Let us now consider static, spherically symmetric space-times with the metric (1) and φ = φ ( x )and choose f ( X ) = ǫf X n , n > , f > , ǫ = ± . (68)To avoid the possibility of complex values of f ( X ), we must then fix η = −
1, so that X = e − α φ ′ . (69)The resulting equations of motion are nf e − nα φ ′ n − n (2 n − φ ′′ + [2 β ′ + γ ′ − (2 n − α ′ ] φ ′ o = − ǫV φ , (70) γ ′′ + γ ′ (2 β ′ + γ ′ − α ′ ) = − ǫf n − e − n ) α φ ′ n − e α V, (71) γ ′′ + 2 β ′′ − α ′ ( γ ′ + 2 β ′ ) + γ ′ + 2 β ′ = f n + 1) e − n ) α φ ′ n − e α V, (72) β ′′ + β ′ (2 β ′ + γ ′ − α ′ ) − e α − β ) = − ǫf n − e − n ) α φ ′ n − e α V. (73)If we assume that the Ellis wormhole is a solution to Eqs. (70)-(73), we substitute there theexpressions (21) and find that the sum and difference of (71) and (72) leads to the relations V = ǫf n − e nγ φ ′ n , nf ǫφ n = 2 r ′′ r = 2 b ( x + b ) . (74)It follows that ǫ = −
1, which is natural for a wormhole solution that must violate the Null EnergyCondition, so that T − T <
0. As a result, we obtain the following explicit expressions for φ ′ andthe potential V : φ ′ = C ( x + b ) − /n , C = (cid:18) b nf (cid:19) / (2 n ) , (75) V = f − n ) φ ′ n = f − n ) C n ( x + b ) . (76)Substituting the expression for φ ′ to (70) to find V φ , one can verify that the latter coincides with V φ = V ′ /φ ′ obtained directly from (76), thus confirming the correctness of the solution.One can integrate φ ′ given by (75) to obtain φ = Cxb − /n F (cid:16) , n ,
32 ; − x b (cid:17) + φ , φ = const . (77)It is not simple to obtain a closed expression for V ( φ ) (to do that, we must invert the hypergeometricfunction). However, V ( φ ) is well defined since φ ′ > x . Also, at some special values of n the hypergeometric function can reduce to simpler expressions.Thus the Ellis wormhole solution is consistent with k-essence theory with a potential.1 The perturbation equations in the gauge δβ = 0, under the condition α = γ = 0 (but theirperturbations are nonzero) can be written as δ ¨ φ − (2 n − δφ ′′ − β ′ δφ ′ − (cid:26) δγ ′ − (2 n − δα ′ (cid:27) φ ′ = 1 nf φ ′ (1 − n ) (cid:26) V φφ φ ′ δφ + V φ (cid:20) nφ ′ δα + 2(1 − n ) δφ ′ (cid:21)(cid:27) , (78) δ ¨ α − δγ ′′ − β ′ δγ ′ = f ( n − φ ′ n − h (1 − n ) φ ′ δα + nδφ ′ i + (cid:16) V δα + V φ δφ (cid:17) , (79) δ ¨ α − δγ ′′ − β ′ δα ′ = − f ( n + 1) φ ′ n − h (1 − n ) φ ′ δα + nδφ ′ i + (cid:16) V δα + V φ δφ (cid:17) , (80) β ′ ( δγ ′ − δα ′ ) − e − β δα = f (1 − n ) φ ′ n − h (1 − n ) φ ′ δα + nδφ ′ i − (cid:16) V δα + V φ δφ (cid:17) , (81) − β ′ δ ˙ α = n f φ ′ n − δ ˙ φ. (82)From Eq. (82) one obtains δα = − n ηδφ, ¯ η = f φ ′ n − β ′ . (83)Using this result, and combining Eqs. (79) and (80), we obtain δγ ′ = n − n )¯ ηδφ ′ + n (cid:20) ¯ η ′ + (1 − n ) n ¯ η φ ′ (cid:21) δφ. (84)This expression is consistent with (61) if n = a = 1 and ǫ = − δα ′ by combining (81) with the difference of (79)and (80) as δα ′ = 12 β ′ h − nf φ ′ n − − e − β δα − − n ) f β ′ φ ′ n − δφ i . (85)This expression coincides with the one obtained by directly differentiating (83), which verifies thecorrectness of the model and the calculations.Now, to obtain the master equation for δφ , we use the previous results and insert into them(78), along with the relations due to the background equations, V φ = − n ¯ ηβ ′ (cid:20) (2 n − φ ′′ φ ′ + 2 β ′ (cid:21) = 4 b Cn ( n − x ( b + x ) − /n , (86) V φφ = − n (cid:26) ηβ ′′ β ′ φ ′ + ¯ η ′ β ′ φ ′ (cid:20) (2 n − φ ′′ φ ′ + 2 β ′ (cid:21) +(2 n − η (cid:20) β ′′ φ ′′ φ ′ + β ′ (cid:18) φ ′′′ φ ′ − φ ′′ φ ′ (cid:19)(cid:21)(cid:27) = 4 b ( n − C n ( b + x ) − /n ( nb + (2 − n ) x ) . (87)2The final form of the master equation is − δ ¨ φ + (2 n − δφ ′′ + (cid:26) β ′ + 2 (1 − n ) n V φ ¯ ηβ ′ (cid:27) δφ ′ + (cid:26) n ¯ η ′ φ ′ + n (1 − n )2 ¯ η φ ′ − n φ ′ β ′ V φ + φ ′ V φφ n ¯ ηβ ′ (cid:27) δφ = 0 , (88)or explicitly, δ ¨ φ − (2 n − δφ ′′ − β ′ (cid:20) − n − n (cid:21) δφ ′ + U ( x ) δφ = 0 ,U ( x ) = 2 b (2 n − x ( x + b ) + 2 n ( x + b ) h nb ( n + n − − x (5 n − n + 2) i . (89)In general, the analysis of Eq. (89) is quite complicated. Let us begin with a simple examplewhich shows a particular case where it is possible to explicitly prove the instability. Particularly,let us fix n = 1 /
2. This case has been investigated in [16, 18] in search for black hole solutions anda study of their stability. In fact, there are some exotic types of black hole, but they are unstable.Now we are considering the same problem for the Ellis wormhole.With n = 1 /
2, Eq. (89) greatly simplifies and reads δ ¨ φ − x + b ( x + b ) δφ = 0 , (90)which is easily integrated giving δφ = K ( x ) e H ( x ) t + K ( x ) e − H ( x ) t , H ( x ) = √ x + b x + b , (91)where K ( x ) and K ( x ) are arbitrary functions. This evidently demonstrates the instability of thebackground configuration since the expression (91) exponentially grows with time if K = 0.If n = 1, we return to the situation in GR.If n < /
2, Eq. (89) loses its hyperbolic nature, and the system is hydrodynamically unstablefor the same reason as described in [17] and other papers.Of more interest is the situation where n > /
2, in which Eq. (89) has a wave nature. It is thenreasonable to get rid of the term containing δφ ′ by putting δφ = e − pβ y ( x, t ) , p = 2 n − n + 2 n (1 − n ) , (92)after which the equation acquires the form¨ y − (2 n − y ′′ + [ U ( x ) + (2 n − β ′′ + pβ ′ )] y = 0 , (93)or, assuming a single spectral mode, y ∝ e iωt , so that ¨ y = − ω y , y ′′ + (cid:20) ω n − − V eff ( x ) (cid:21) y = 0 , V eff ( x ) = U ( x )2 n − pβ ′′ + p β ′ . (94)It is the Schr¨odinger-like equation usually appearing in stability studies, for which the corre-sponding boundary-value problem should be solved in order to obtain stability conclusions. For3perturbations of wormholes with phantom scalar fields, the effective potentials V eff ( x ) contain asingularity on the throat which can be regularized with proper Darboux transformations [6, 7, 21]under the condition that V eff ( x ) = 2 /x + O (1) near the throat (where x is the “tortoise” coordinatein the wormhole space-time, and x = 0 is the throat). This condition is generally satisfied for worm-holes supported by phantom scalar fields with arbitrary potentials [21]. Surprisingly, this conditionalso holds for the effective potential V eff ( x ) in our equation (94) for wormholes in k-essence theory,so that the stability problem can be solved along the lines of [6, 7, 21]. This requires a separatestudy, to be performed in the near future. The Ellis-Bronnikov solution represents the simplest analytical wormhole solution that can be ob-tained in GR. It consists of two asymptotically flat regions connected by a throat. This wormholerequires a massless, minimally coupled phantom scalar field: this means that all space, not onlythe throat, is filled with a field having negative energy density. In spite of being a very elegant andsimple solution, the EB wormhole has a major drawback: it is unstable under linear perturbations.To look for a simple wormhole solution like the EB one that may not require phantom fields and/orthat are stable is a challenge, even if some extensions of GR are employed.It is well known that the same metric can be a solution of different gravitational theories. Wehave exploited this fact in order to investigate the conditions at which the EB wormhole metriccan be a solution in the context of two extended gravity theories, Rastall gravity and k-essence.Rastall gravity is a more radical departure from GR since it modifies the usual expression for theconservation law. In some sense, Rastall gravity may be recast in the structure similar to GRwhere the expression for the energy-momentum tensor must be modified. Unlike that, k-essence isessentially a modification of the matter sector, keeping a Lagrangian formulation, by generalizingthe usual kinetic expression. Both theories have applications, for example, in cosmology [14, 25, 26]and black hole physics [15, 16].We have shown that the EB metric can be a static, spherically symmetric solution in bothRastall and k-essence theories. To achieve that, a potential must be added in both cases, implyingthat, as opposed to GR, a self-interacting scalar field is required. The next step was to investigatethe stability of these solutions in Rastall and k-essence cases. In Rastall gravity we face the samefeature that was already found for black hole solutions: the usual perturbative approach leads toinconsistencies forcing to set all fluctuations near the background solution equal to zero. Perhapsthis curious property is connected with the absence of a Lagrangian formulation. In k-essencetheory, we have shown that the wormhole is unstable with respect to linear perturbation at leastfor the parameter n in the range n ≤ / Acknowledgements:
V.B., L.C. and J.F thank FAPES (Brazil) and CNPq (Brazil) for partialfinancial support. K.B. was supported by the RUDN University Strategic Academic Leadership4Program. The research of K.B. was also funded by the Ministry of Science and Higher Education ofthe Russian Federation, Project “Fundamental properties of elementary particles and cosmology”N 0723-2020-0041, and by RFBR Project 19-02-00346.
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