Sources of matter for wormholes in a k-essence theory

In this work, we analyze some matter sources associated with wormhole models within a k-essence theory coupled to the gravitational sector through a phantom scalar field. We adopt a spherically symmetric background in (3+1) dimensions and consider two types of systems: electrically and magnetically charged. In the first case, we consider the generalized Ellis-Bronnikov model, in which we fix the k-essence field exponent to $n=1/2$ and take the parameter $m\ge 2$, where $m$ is the parameter that modifies the area of this wormhole model. From this we obtained the expression for the scalar field, the potential, and the associated electromagnetic functions for any values of the parameter $m\geq{2}$. In the second and third models, we consider the scenario of two wormholes that are structured according to the adjustment of the parameters that define their area function $Σ^2$, and in both cases we adopt $n=1/2$. Finally, we show that the violation of the null energy conditions is conditioned by the parameters of the area function.

💡 Research Summary

This paper presents a detailed investigation into the matter sources that can support traversable wormhole geometries within the framework of a k-essence theory coupled to gravity via a phantom scalar field. The primary goal is to construct analytical models where exotic matter, necessary to violate the Null Energy Condition (NEC) at the wormhole throat, is explicitly provided by a combination of a k-essence scalar field and Nonlinear Electrodynamics (NED).

The theoretical foundation is laid out in Section II, starting from an action principle that minimally couples gravity to a k-essence field (with a function F(X,φ)=F0*X^n - 2V(φ)) and an electromagnetic Lagrangian L(f). The phantom nature of the scalar field is enforced by setting η=-1 to ensure a positive kinetic term X. The authors derive the field equations for a general static, spherically symmetric metric in the “quasi-global” gauge (Eq. 14), distinguishing carefully between the magnetically charged case (where the only non-zero component of the field strength is H_23) and the electrically charged case (with H_01). They highlight a methodological tool for the electric case: introducing an auxiliary field P_μν = L_f H_μν and its invariant P, which often yields simpler analytical expressions than working directly with the invariant f.

The core of the work applies this formalism to three specific wormhole models. The first and most extensively analyzed model (Section III) is the Generalized Ellis-Bronnikov (GEB) wormhole, defined by the metric functions A(x)=1 and Σ(x) = (x^m + a^m)^(1/m), where ‘a’ is the throat radius and m≥2 is a shape parameter (m=2 recovers the classic Ellis-Bronnikov wormhole). For the magnetic case and with a specific choice of the k-essence exponent n=1/2, the authors successfully derive exact analytical expressions for the scalar field φ(x) (Eq. 26), its potential V(x) (Eq. 27), and the NED functions L(f) and L_f (Eq. 28, 29). A key observation here is that the electromagnetic functions depend solely on the geometric parameters ’m’ and ‘a’, not on the k-essence parameter ’n’. The Lagrangian L(f) is also rewritten in terms of the electromagnetic invariant f (Eq. 30, 31). The process for the electric case in the same geometry is also outlined.

Sections IV and V are dedicated to two additional wormhole models, where the area function Σ(x) is defined by different polynomial structures. The same procedure—fixing n=1/2 and solving the field equations—is employed to derive the corresponding matter sources (scalar field, potential, electromagnetic functions) for these geometries, demonstrating the generality of the approach.

The final analytical section (Section VI) addresses the crucial issue of energy conditions. It explicitly shows that the violation of the NEC, a prerequisite for a traversable wormhole throat, is inherently “conditioned” by the parameters defining the wormhole’s area function Σ(x). In the GEB model, for instance, the specific value of ’m’ directly influences the sign of the NEC expression. This establishes a concrete link between the wormhole’s geometric shape and the specific properties of the exotic matter required to sustain it.

In conclusion, this work provides a set of exact, analytical matter source configurations for several wormhole spacetimes within a phantom k-essence theory coupled to NED. By fixing n=1/2, it achieves tractable solutions and clearly demonstrates how the necessary exotic energy is distributed and how its properties are governed by the underlying geometry of the wormhole itself. The research strengthens the theoretical underpinnings of wormhole physics by offering concrete field-theoretic models for the exotic matter source.