Novel traversable wormhole in General Relativity and Einstein-Scalar-Gauss-Bonnet theory supported by nonlinear electrodynamics

The wormholes are fascinating predictions arising from the geometrical description of gravity. They involve a topological spacetime configuration as a shortcut between distant points or regions in spacetime. The first wormhole interpretation originally came from the work of Einstein and Rosen in 1935, with their solution known as the Einstein-Rosen bridge einstein35 , which is, in essence, the maximally extended Schwarzschild black hole solution kruskal60 . However, the “throat” of this wormhole is dynamic and hence non-traversable, meaning that its radius expands to a maximum and quickly contracts to zero so fast that even a photon cannot pass through kruskal60 . Further, in 1988, a solution to the wormhole traversability problem was established by Morris and Thorne morris88 . They obtained one type of wormhole metric and the necessary conditions (absence of horizons and the flare-out condition) that can guarantee the traversability of a wormhole spacetime. Moreover, they showed that in the context of general relativity (GR), the throat of these types of wormholes only could be kept open by some form of “exotic” matter morris88-2 having negative energy density and whose energy-momentum tensor violates the null-energy condition (NEC). This kind of exotic fields have been intensely discussed, principally, in the context of electromagnetic and scalar fields Gibbons1996 . Although an explanation about the fundamental origin of phantom fields is still in discussion, they are frequently used, for instance; in the construction of novel cosmological models known as Phantom Cosmologies Pha_Cos ; Ph_SF1 ; Ph_SF . Also, they have been used to explain the current period of accelerated expansion of the Universe. Indeed, an accelerated expansion of the Universe can be accounted by assuming the existence of an effective field generating repulsive gravity between its elements, e.g., a repulsive field would correspond in Einstein gravity to a fluid with negative pressure, like dark energy, which is an essential ingredient of the Standard Cosmological Model, one of the most popular models in modern physics. A source of repulsive gravity, in the context of exotic fields, would be represented by a matter distribution with negative energy density, i.e. a phantom field. In fact, comparison with observational data suggests it as a strong candidate for dark energy explanation Hannenstadt2006 ; Dunkle2009 ; Caldwell2003 ; Dutta2009 ; Alestas2020 ; Cedeno2021 . Phantom fields are also considered in the astrophysics context. For instance, Ref. Galactic show the possibility that galactic dark matter exists in a scenario where a phantom field is responsible for the dark energy distribution.
Therefore, is very important to understand the physical properties of phantom fields in the framework of gravity theories. In particular, a phantom scalar field could be used to generate the negative kinetic energy density that allows traversable wormholes. For instance, one of the first and simplest examples of traversable wormholes is the static, spherically symmetric and asymptotically flat (SSS-AF) Ellis wormhole Ellis whose energy-momentum tensor can be represented by a massless phantom scalar field. This solution has been extensively studied, and its properties like gravitational lensing EllisLensing , quasi-normal modes QNMEllis , shadows EllisShadows and stability EllisStability have been thoroughly investigated. Recently, bronnikov13 shows that the source of the Ellis wormhole as a perfect fluid with negative energy density and a source-free radial electric or magnetic field is also possible.

The construction of traversable wormholes has been studied using non-linear electrodynamics (NLED) as source. NLED theories are derived from Lagrangians $\mathcal{L}=\mathcal{L}(\mathcal{F},\mathcal{G})$ that depend arbitrarily on the two electromagnetic invariants, $\mathcal{F}=2(\bm{\mathcal{E}}^{2}-\bm{\mathcal{B}}^{2})$ and $\mathcal{G}=\bm{\mathcal{E}}\cdot\bm{\mathcal{B}}$, where $\bm{\mathcal{E}}$ and $\bm{\mathcal{B}}$ are the electric and magnetic fields, respectively. Albeit this form is arbitrary, there exist two outstanding Lagrangians: the Born-Infeld theory (BI)

where $b$ is a constant which has the physical interpretation of a critical field strength BI ; and the Euler-Heisenberg theory (EH),

which corresponds to the weak field approximation of Heuler_35 ; Heisenberg_36 , and the coupling constant $\mu$ is written as $\mu=2\alpha^{2}/(45m^{4}_{e})$, where $m_{e}$ is the mass of the electron and $\alpha$ is the fine structure constant. Considering NLED Lagrangian of the form $\mathcal{L}(\mathcal{F})$ coupled to gravity, interesting solutions arise, like regular black holes or traversable wormholes, among others Ayon_Garcia ; Arellano ; Bronnikov2017 ; EH_BH ; Novello .

By using NLED as a source in the Einstein field equations, two kind of T-WH have been investigated: dynamic Arellano ; Bronnikov2017 and static Bronnikov_Walia . While, dynamic T-WHs are possible in the GR context by using NLED as the only source, the SSS T-WHs static wormholes are not possible in the NLED context Arellano2006 . To date, the SSS T-WHs studied requires an additional scalar field Bronnikov_Walia . Howerver, the common NLED models used for the construction of dynamical and static traversable wormholes do not satisfy plausible physical conditions, like how to reduce them to Maxwell electrodynamics in the weak field limit Bronnikov2017 ; Bronnikov_Walia ; Canate_Breton ; Canate_Breton_Ortiz ; Canate_Magos_Breton .

In this paper we will construct a SSS-AF T-WH, to do this we consider a Euler-Heisenberg-like electrodynamics model as the NLED source, with the Lagrangian density defined by:

where $\mu_{{}_{0}}$ and $\mu_{{}_{1}}$ are real parameters of the model. Moreover, we use a self-interacting scalar field, described by the potential,

being $\mathscr{U}_{{}_{0}}=\mathscr{U}(0)$, whereas $\beta_{{}_{0}}$, $\beta_{{}_{1}}$ and $\beta_{{}_{2}}$ are real parameters. This power-law potential has interesting applications in cosmology Ramirez ; Senoguz ; Amit ; Takyi .

This paper is structured as follows: In Section II we derive the field equations for the GR-NLED-SF, in Section III we present the canonical metric of the traversable wormhole spacetime and discuss the Ellis wormhole solution, in Section IV we construct our novel traversable wormhole solution within the framework of GR-SF-NLED. Whereas, in Section V, we show that our T-WH spacetime is also a exact solution of EsGB-NLED, but now with a real scalar field having a positive kinetic term. In Section VI the null trajectories and the capture cross-section of massless (photon) by this T-WH spacetime, are analyzed. In the last section final conclusions are presented. Through this paper we will use the system of units where $G=k_{{}_{B}}=c=\hbar=1$, and the metric signature $(-+++)$.

The GR-NLED-SF theory is defined by the following action,

where $R$ is the scalar curvature, $\phi$ is a scalar field coupled to gravity, and $\mathscr{U}=\mathscr{U}(\phi)$ is the scalar potential; whereas $\mathcal{L}=\mathcal{L}(\mathcal{F})$ is a function of the electromagnetic invariants $\mathcal{F}\equiv\frac{1}{4}F_{\alpha\beta}F^{\alpha\beta}$, being $F_{\alpha\beta}=2\partial_{[\alpha}A_{\beta]}$ the components of the electromagnetic field tensor $\bm{F}=\frac{1}{2}F_{\alpha\beta}\bm{dx^{\alpha}}\wedge\bm{dx^{\beta}}$ and $A_{\alpha}$ are the components of the electromagnetic potential.

Using the notation $\mathcal{L}_{{}_{\mathcal{F}}}\equiv\frac{d\mathcal{L}}{d\mathcal{F}}$ and $\dot{\mathscr{U}}=\frac{d\mathscr{U}}{d\phi}$, the GR-NLED-SF field equations arising from (5) takes the form,

where, $G_{\alpha}{}^{\beta}=R_{\alpha}{}^{\beta}-\frac{R}{2}\delta_{\alpha}{}^{\beta}$ are the components of the Einstein tensor, $(E_{\alpha}{}^{\beta})\!_{{}_{{}_{S\!F}}}$ are the components of the energy-momentum tensor of self-interacting scalar field,

and $(E_{\alpha}{}^{\beta})\!_{{}_{{}_{N\!L\!E\!D}}}$ are the components of the NLED energy-momentum tensor,

Our aim is to find a static, spherically symmetric, and asymptotically flat, charged wormhole solution for the set Eqs. (6) with a non-trivial scalar field. To do this, we assume the scalar field is static and spherically symmetric, $\phi=\phi(r)$, and also the metric has the static and spherically symmetric form,

with $A=A(r)$ and $B=B(r)$ unknown functions to be determined.

In terms of $A$ and $B$, the non-vanishing components of the Einstein tensor are,

where ^′ denotes the derivative respect to the radial coordinate $r$, i.e. $A^{\prime}=\frac{dA}{dr}$. Whereas, for the non-trivial components of the energy-momentum tensor of self-interacting scalar field we have,

Regarding the electromagnetic field tensor, since the spacetime is SSS, we can restrict ourselves to purely magnetic field, i.e., $\mathcal{E}=0$ and $\mathcal{B}\neq 0$.

With this restriction, the electromagnetic field tensor has the form, $F_{\alpha\beta}=\mathcal{B}\left(\delta^{\theta}_{\alpha}\delta^{\varphi}_{\beta}-\delta^{\varphi}_{\alpha}\delta^{\theta}_{\beta}\right)$. In this way, for a static and spherically symmetric spacetime with line element (9), the general solution of the equations $\nabla_{\mu}(\mathcal{L}_{{}_{\mathcal{F}}}F^{\mu\nu})=0$ is,

This means $\bm{F}=r^{4}\mathcal{Q}(r)\sin\theta\hskip 2.84544pt\bm{d\theta}\wedge\bm{d\varphi}$, and therefore $d\bm{F}=0=(r^{4}\mathcal{Q}(r))^{\prime}\sin\theta\hskip 2.84544pt\bm{dr}\wedge\bm{d\theta}\wedge\bm{d\varphi}$. This implies $\mathcal{Q}(r)~{}=~{}\sqrt{2}\hskip 1.70709ptq/r^{4}$, where $\sqrt{2}\hskip 1.70709ptq$ is an integration constant, which plays the role of the magnetic charge. Hence, the components of the electromagnetic field tensor $F_{\alpha\beta}$ and the invariant $\mathcal{F}$ are given by;

Finally, the energy-momentum tensor components for NLED, assuming the SSS spacetime with metric (9), the purely magnetic field (13), and a generic Lagrangian density $\mathcal{L}(F)$, are written as

Inserting the above components in the field equations $C_{\alpha}{}^{\beta}=G_{\alpha}{}^{\beta}-8\pi[(E_{\alpha}{}^{\beta})\!_{{}_{{}_{S\!F}}}+(E_{\alpha}{}^{\beta})\!_{{}_{{}_{N\!L\!E\!D}}}]=0$, we obtain the GR-NLED-SF field equations for the metric ansatz (9) and the magnetic field (13):

whereas the scalar field must satisfies,

This ends the general treatment of the static, spherically symmetric and pure magnetic solutions. In what follows we will discuss the general properties of the wormhole spacetime solution we have obtained.

The canonical metric of a (3+1)-dimensional SSS-WH solution morris88 ; morris88-2 is given by,

where $\Phi(r)$ and $b(r)$ are smooth functions, known as redshift and shape functions respectively. The domain for radial coordinate $r$ has a minimum at $r=r_{{}_{0}}$, where the WH throat is defined by $b(r_{{}_{0}})=r_{{}_{0}}$, and is unbounded for $r>r_{{}_{0}}$. This coordinate has a special geometric interpretation, as $4\pi r^{2}$ is the area of a sphere centered on the WH throat. On the other hand, for the WH to be traversable, one must demand:

with ^′ denoting derivative with respect to $r$, are satisfied (see morris88 ; morris88-2 for details).

Traversability and violation of Null Energy Condition.
Let us consider the null vector $\bm{n}~{}=~{}(e^{-\Phi(r)},\sqrt{1-b(r)/r},0,0)$ , identify $e^{A(r)}~{}=~{}e^{2\Phi(r)}$ and $e^{B(r)}=\frac{1}{1-\frac{b(r)}{r}}$ in the spacetime (19). Using (10) and assuming the flaring out condition is fulfilled, after contracting the Einstein tensor with $\bm{n}$ and evaluating at $r=r_{{}_{0}}$, yields:

Thus, in GR, $G_{\alpha\beta}=\kappa T_{\alpha\beta}$, from the balance between the matter and the curvature quantities, the fulfillment of the flaring out condition implies that the NEC (which states that $T_{\alpha\beta}n^{\alpha}n^{\beta}\geq 0$ for any null vector $n^{\alpha}$) is violated, therefore, the presence of exotic matter is unavoidable for having a T-WH in GR.

In this section we will study the NLED-SF theory defined by NLED model and a scalar potential, given respectively by,

admits the following metric,

for $\mu_{{}_{0}}=-q^{2}/8$, $\mu_{{}_{1}}=2|q|/3$, $\mathscr{U}_{{}_{0}}=-1/(12q^{2})$, $\beta_{{}_{0}}=\beta_{{}_{1}}=\beta_{{}_{2}}=1/q^{2}$, together with the scalar field,

as a pure magnetic exact solution of the GR-NLED-SF field equations (15)-(18). The metric (40) admits a T-WH interpretation since satisfies the properties (20)-(22), because this metric has the form (32) but with non vanishing redshift function given by $\Phi(r)=-\frac{q^{2}}{r^{2}}$, can be interpreted as a non-trivial redshift modification of the Ellis WH metric, with WH throat at $r=r_{{}_{0}}=|q|$. In other words, the line element (40) is not of type (24), therefore it describes a non-ultra-static spherically symmetric asymptotically flat traversable wormhole.

The absence of curvature singularities in the WH domain, $r\geq|q|$, can be deduced from the analytical expressions of its curvature invariants,

which are all regular in the whole WH domain $r\geq|q|$ .

Here, it is important to emphasise that the Lagrangian density (38) reduces to Maxwell theory in the limit of weak field, i.e. $\mathcal{L}\rightarrow\kappa\mathcal{F}$ and $\mathcal{L}_{\mathcal{F}}\rightarrow\kappa$ (being $\kappa$ a constant) as $\mathcal{F}\rightarrow 0$.

Besides the Maxwell limit condition, another important physical requirement is desired for the electromagnetic Lagrangian (38) to fulfill, it is the WEC. We can guarantee the validity of WEC in a limited region of the spacetime determined by,

which includes the wormhole throat $r=r_{{}_{0}}=|q|$. See Appendix B for details.

Phantom scalar field: Defining a new scalar field by $\psi=i\phi$ (phantom field), the theory for the which the metric (40) is a pure magnetic exact solution, arises from the action,

with nonlinear electromagnetic field described by (38) and and scalar field potential,

with $\psi$ given by,

Einstein-scalar-Gauss-Bonnet (EsGB) theories belong to a class of alternative theories of gravity, in which the Einstein-Hilbert action with scalar field, $\phi$, that is (5), is modified by including a quadratic curvature correction given by the product of a function of the scalar field, $\bm{f}(\phi)$, and the Gauss-Bonnet term $R_{{}_{GB}}^{2}=R_{\alpha\beta\mu\nu}R^{\alpha\beta\mu\nu}-4R_{\alpha\beta}R^{\alpha\beta}+R^{2}$. Thus, the dynamical equations of EsGB theory minimally coupled to matter fields are derived from the following action,

where $\bm{f}(\phi)R_{{}_{GB}}^{2}$ stands for the scalar field non-minimally coupled to the Gauss-Bonnet invariant, or Scalar field-Gauss-Bonnet (SFGB) term, $\mathcal{L}_{\rm matter}$ is the matter fields Lagrangian and $\psi_{a}$ describes any matter field included in the EsGB theory. Although the Gauss-Bonnet term includes quadratic curvature components, the resulting field equations arising from it are of second order, and therefore avoid the Ostrogradski instability and ghosts. Various aspects of the EsGB theory have been studied in detail lately. To name just a few applications in cosmology, it was shown in cosmic_acceleration that the theory can describe the present stage of cosmic acceleration, and can provide an exit from a scaling matter-dominated epoch to a late-time accelerated expansion Tsujikawa2006 . The possible reconstruction of the coupling and potential functions for a given scale factor was considered in recon , and the consequences of the EsGB in an inflationary setting have been considered in Odintsov2020 . In our case the coupling between the scalar field and the Gauss-Bonnet term allows the violation of the null-energy condition required for the traversable Ellis wormhole, and so in a sense the effective negative energy density comes from the geometry itself instead of the matter source as in GR. Recently, in antoniou19 various novel wormhole solutions in EsGB theory have been obtained, numerically, for several coupling functions. Other numerical traversable wormhole solutions have been obtained earlier kanti11 in Einstein-dilaton-Gauss-Bonnet theory kanti96 , which involve an exponential coupling between the scalar field representing the dilaton and the Gauss-Bonnet term. Exact traversable wormhole solutions have been discussed within EsGB, however, at present only ultra-static are available canate19 ; canate19b . From now on, we will consider the EsGB theory in the presence of NELD described by $\mathcal{L}(\mathcal{F})$. The EsGB-NLED field equations arising from the action (48) with $\mathcal{L}_{\rm matter}=\mathcal{L}(\mathcal{F})$, are given by,

where $(E_{a}{}^{b})\!_{{}_{{}_{S\!F\!G\!B}}}$, denotes the components of tensor,

with $\tilde{R}^{\rho\gamma}{}_{\mu\nu}=\epsilon^{\rho\gamma\sigma\tau}R_{\sigma\tau\mu\nu}/\sqrt{-g}$, and $\dot{\bm{f}}=\frac{d\bm{f}}{d\phi}$. We shall refer to $(E_{\alpha}{}^{\beta})\!_{{}_{{}_{S\!F\!G\!B}}}$ as the Scalar field-Gauss-Bonnet (SFGB) tensor, since it represents the contribution to the spacetime curvature due to the effects of the SFGB term. In fact, $(E_{\alpha}{}^{\beta})\!_{{}_{{}_{S\!F\!G\!B}}}$ can be written as, $(E_{\alpha}{}^{\beta})\!_{{}_{{}_{S\!F\!G\!B}}}=(E_{\alpha}{}^{\beta})\!_{{}_{{}_{S\!F}}}+(E_{\alpha}{}^{\beta})\!_{{}_{{}_{G\!B}}}$ where $(E_{\alpha}{}^{\beta})\!_{{}_{{}_{S\!F}}}$ is given by (7) whereas $(E_{\alpha}{}^{\beta})\!_{{}_{{}_{G\!B}}}=-\frac{1}{2}(g_{\alpha\rho}\delta_{\lambda}{}^{\beta}+g_{\alpha\lambda}\delta_{\rho}{}^{\beta})\eta^{\mu\lambda\nu\sigma}\tilde{R}^{\rho\xi}{}_{\nu\sigma}\nabla_{\xi}\partial_{\mu}\bm{f}(\phi).$ The structure of the field equations (49) motives the definition of the effective energy-momentum tensor, $\mathscr{E}_{a}{}^{b}$, as $\mathscr{E}_{a}{}^{b}=(E_{a}{}^{b})\!_{{}_{{}_{S\!F\!G\!B}}}+(E_{a}{}^{b})\!_{{}_{{}_{N\!L\!E\!D}}}$, thus, the EsGB-NLED theory can be written in a GR-like form, $G_{a}{}^{b}=8\pi\mathscr{E}_{a}{}^{b}$. Therefore, by taking the divergence of Eq.(49), and taking into account that $\nabla_{\beta}(E_{\alpha}{}^{\beta})\!_{{}_{{}_{N\!L\!E\!D}}}=0$, and $\nabla_{\beta}G_{\alpha}{}^{\beta}=0$, from Bianchi identities, it follows that $\nabla_{\beta}(E_{\alpha}{}^{\beta})\!_{{}_{{}_{S\!F\!G\!B}}}=0$. Our aim here, is to introduce the SSS-AF metric (40) as a pure-magnetic exact solution in EsGB-NLED gravity with a real scalar field in the whole T-WH spacetime, i.e. $\phi(r)\in\mathbb{R}$ for all $r\geq|q|$.