Novel Einstein-Scalar-Gauss-Bonnet Wormholes without Exotic Matter

Novel wormholes are obtained in Einstein-scalar-Gauss-Bonnet theory for several coupling functions. The wormholes may feature a single-throat or a double-throat geometry and do not demand any exotic matter. The scalar field may asymptotically vanish or be finite, and it may possess radial excitations. The domain of existence is fully mapped out for several forms of the coupling function.

Introduction.-Einstein-scalar-Gauß-Bonnet (EsGB) theories represent interesting alternative theories of gravity, where the Einstein-Hilbert action is supplemented by a scalar field, non-minimally coupled to quadratic curvature terms in the form of the Gauß-Bonnet (GB) term. The resulting field equations are then of second order, avoiding Ostrogradski instability and ghosts.
Motivated by string theory with the dilaton as the scalar field, the Einstein-dilaton-Gauß-Bonnet (EdGB) theory features an exponential coupling between the scalar field and the Gauß-Bonnet term [1][2][3]. Black-hole solutions arising in the context of the EdGB theory differ from the Schwarzschild or Kerr black holes since they possess a non-trivial dilaton field and thus carry dilaton hair [4,5]. The extended family of EsGB theories, where different coupling functions of the scalar field to the GB term may be employed, has attracted recently considerable attention [6][7][8][9]. In these theories, spontaneous scalarization of black holes arises, and new scalarized solutions appear beside the Schwarzschild solution. The stability of scalarized black holes of EsGB theories has been addressed in detail by analyzing their radial perturbations and revealing a distinct dependence on the coupling function [10]. Besides spontaneous scalarization, also induced scalarization is possible in these EsGB models where the scalar field does not vanish at infinity. In that case, the black holes of General Relativity (GR) are not recovered, since they would require an everywhere vanishing scalar field.
A particularly interesting property emerging in the EdGB solutions is the presence of regions with negative effective energy density -this is due to the presence of the higher-curvature GB term and is therefore of purely gravitational nature [4,5]. Consequently, the EdGB theory allows for Lorentzian, traversable wormhole solutions without the need for exotic matter [11,12]. It is tempting to conjecture that the more general EsGB theories should also allow for wormhole solutions. Thus, in the context of this work, we consider a general class of EsGB theories with an arbitrary coupling function for the scalar field. We first readdress the case of the exponential coupling function, and show that the EdGB theory is even richer than previously thought, since it features also wormhole solutions with a double throat and an equator in between. Then, we consider alternative forms of the scalar coupling function, and demonstrate that the EsGB theories always allow for wormhole solutions, featuring both single and double throats. The scalar field may vanish or be finite at infinity, and it may have nodes. As expected, no scalarization of GR solutions takes place, since these would need the presence of exotic matter. We also map the domain of existence of these wormholes in several exemplifications, evaluate their global charges and throat areas, and illustrate the transition from single-throat to double-throat geometry.
EsGB Theory.-We consider the following effective action describing a general EsGB theory where φ is the scalar field with a coupling function F (φ) to the GB term R 2 GB = R µνρσ R µνρσ − 4R µν R µν + R 2 . The Einstein and scalar field equations are obtained by variation of the action with respect to the metric, respectively the scalar field, where and the dot denotes derivative with respect to the scalar field.
In this work, we consider only static, spherically-symmetric solutions of the field equations. In our previous analyses [11,12], we employed the line-element for which the circumferential radius is R c (l) = l 2 + r 2 0 . Clearly, R c possesses a minimum at l = 0, corresponding to the wormhole throat, but it does not allow for a local maximum, i.e. an equator. Consequently, wormholes with an equator were missed in [11,12]. Here, we employ a different set of coordinates which allow for the following type of wormhole solutions, In these coordinates, the circumferential radius is expressed as R c (η) = e f1/2 η 2 + η 2 0 . The condition for an extremum at η = 0 yields f ′ 1 (0) = 0. The substitution of the metric (5) in the Einstein and field equations, given in Eqs. (2), leads to three second-order, ordinary differential equations (ODEs) and one constraint. In order to find asymptotically-flat, regular wormhole solutions, we have to impose appropriate boundary conditions at asymptotic infinity and at the throat/equator. These have the form: where the last condition results from the constraint. In addition, we can choose a fixed value at η = 0 for any of the functions f 0 , f 1 or φ. Thus, for a given coupling function, the wormhole solutions form a two-parameter family. The mass M and the scalar charge D of the wormhole can be read off the asymptotic behaviour of the metric and scalar functions, namely If the throat/equator is located at η = 0, then its area is given by A t,e = 4πR 2 c (0) = 4πη 2 0 e f1(0) , while for a double-throat wormhole with the throat located at η t , A t = 4πR 2 c (η t ) = 4π(η 2 t + η 2 0 )e f1(ηt) . Several forms of the coupling function have been considered, including exponential F = αe −γφ , F = αe −γφ 2 , power-law F = αφ n , with n = 0, and logarithmic F = α ln(φ) functions. In this paper, due to the limited space, we will mainly focus on the coupling functions F = αe −φ and F = αφ 2 . On the side of our quest for wormhole solutions, spontaneously scalarized black holes also emerge.
Smarr relation.-For the exponential coupling function F = αe −γφ , the Smarr relation reads where κ th is the surface gravity at the throat, κ th = − 1 2 (∇ µ ξ ν )(∇ µ ξ ν ), with timelike Killing vector ξ µ . The quantities S th and D th are defined on the throat as where φ th ,f 0th denote the function values at the throat. Also, h th is the induced spatial metric on the throat, R th the scalar curvature of h th , and the vector n µ th is normal on the throat. The general form of the Smarr relation holds both for single-throat and double-throat wormholes; in the latter case, we only need to replace the throat by the equator, where κ eq , S eq , and D eq are defined analogously to κ th , S th , and D th . The difference of Eqs. (9) and (12) yields a relation between quantities defined at the throat and the equator: For a general coupling function, it is not possible to find a Smarr relation which involves only boundary terms.
Null energy condition.-The null energy condition (NEC) has the form T µν n µ n ν ≥ 0 for any null vector n µ . For a spherically-symmetric spacetime, the NEC takes the form −T t t + T η η ≥ 0 ∧ −T t t + T θ θ ≥ 0. Using the expansion of the wormhole solution at the throat/equator, we find −T t t + T η η ηt,e = −2 e −f1 R ′′ c /R c ηt,e . Consequently, the NEC is violated at the throat, since R c possesses a minimum at the throat, implying R ′′ c (η t ) > 0. Moreover, since any wormhole with an equator also possesses throats, the NEC is violated for double throat wormholes, too. In terms of the coupling function Junction conditions.-When single-throat and double-throat wormholes are extended symmetrically at the throat, respectively at the equator, jumps are induced in the Einstein and scalar equations, which can be attributed to a distribution of matter located there, where s µ ν denotes the stress-energy tensor of the matter at the throat, resp. equator, and s scal a source term for the scalar field. We assume a perfect fluid with pressure p and energy density ρ, and a scalar charge ρ scal at the throat, resp. equator, together with the action where λ 1 , λ 0 are constants,h ab is the three-dimensional induced metric at the throat, resp. equator, andR is the corresponding Ricci scalar. Substitution of the metric then yields the junction conditions where all quantities are taken at η = 0. Numerical solutions.-For the numerical integration of the ODEs, we use the compactified coordinate x = η/(η + η 0 ) to cover the range 0 ≤ η ≤ ∞. The software package COLSYS is then used to solve the Einstein and scalar-field equations for given boundary conditions.  As an example, in Fig. 1(a) we depict the metric component −g tt = e f0 , the scalar field φ and the scaled circumferential radius R c /R t for several coupling functions. All solutions are characterised by the same boundary values f 0 (0) = −5 and φ(0) = 0.5 but we have allowed for two different asymptotic values for φ, namely φ ∞ = 0 and φ ∞ = 1. We observe that the behaviour of the metric component −g tt and the circumferential radius R c /R t depends rather mildly on the form of the coupling function or the asymptotic value φ ∞ . On the other hand, both of these factors considerably affect the profile of the scalar field as may be clearly seen from the plot.
In Fig. 1(b), we compare single and double-throat wormholes for the same values of the scaled scalar charge D/M and scaled throat area A t /16πM 2 , and for the case of a quadratic coupling function, F (φ) = αφ 2 . We note that, for the single-throat wormhole, the derivatives of the −g tt and φ do not vanish at the throat, a feature which leads to the introduction of physically-acceptable matter around the throat and thus to the junction conditions Eqs. (18)-(20). For the double-throat wormholes, the metric and scalar field are smooth at the throat, and junction conditions arise instead at the equator. As an indicative example of the geometry of the double-throat wormholes, in Fig. 2(a) we depict the isometric embedding of such a solution that clearly features an equator and two throats. The isometric embedding follows from the equation e f1 [dη 2 + (η 2 + η 2 0 ) dϕ 2 ] = dz 2 + dρ 2 + ρ 2 dϕ 2 , where the equatorial plane, defined as the line-element (5) for t = const. and θ = π/2, is equated with a hypersurface in the three-dimensional, Euclidean space. Considering (z, ρ) as functions of η, the above equation readily yields ρ(η) = e f1/2 η 2 + η 2 0 while z(η) follows via an integration. In Fig. 2(b), we also show the geometry transition between single and double-throat wormholes, by plotting ρ vs. z, for a sequence of solutions for fixed α/r 2 0 = 0.25. With increasing scaled throat area, the double-throat wormholes develop a degenerate throat and turn into single-throat ones. If the scaled throat area is increased further, a second transition takes place where the single-throat wormholes turn in double-throat ones.
Next, we discuss the domain of existence of the wormhole solutions, in terms of the scaled scalar charge and the scaled throat area, and restrict our discussion to the exponential and quadratic coupling functions. In Fig. 3(a), we show the domain for the exponential case, F = αe −φ . The different curves correspond to families of wormholes for a fixed value of α with single throat (dashed) and double throat (solid). The boundary of the domain of existence is formed by the black hole solution (solid black), the wormhole solutions with a degenerate throat (dotted black), configurations with cusp singularities outside the throat (dashed black) and configurations with singularities at the equator (dashed-dotted black). We note that the part of the domain of existence above the dashed-dotted curve comprises both single-throat and double-throat wormholes. The sequence of solutions of Fig. 2(b), showing the transition between double-throat and single-throat, are indicated by the red dots in Fig. 3(a). In addition, the single-throat wormholes can be obtained from the double-throat ones by cutting the latter at the throat and continuing symmetrically to the left. The region of the domain of existence below the dashed-dotted curve contains only single-throat wormholes which are not related to double-throat solutions.
We now turn to the case of the quadratic coupling function, F = αφ 2 . For φ(∞) = 1, the coupling functioṅ F assumes a finite asymptotic value, as in the exponential case. The domain of existence in this case is then quite similar to the one depicted in Fig. 3(a), with wormhole solutions arising for arbitrarily small (but finite) values of α up to some maximal value. In contrast, if φ(∞) = 0, thenḞ vanishes asymptotically and the range of α, for which wormholes arise, is also limited from below. The domain of existence in this case is shown in Fig. 3(b). The Schwarzschild black holes are now part of the boundary of the domain of existence, as indicated by the dot in Fig. 3(b), since the constant configuration φ ≡ φ ∞ solves the scalar field equation trivially. Moreover, wormhole solutions exists for which the scalar field possesses N nodes. The boundary of the domain of existence for N = 1 is shown in the inlet in Fig. 3(b). Note that the range of α in this case is