Traversable wormholes in light of class I approach
Introduction
In recent years the so called Karmarkar condition or class I embedding approach [1], [2] has been used as a building block to find well behaved and physically admissible solutions to Einstein field equations describing compact structures, such as neutron stars or quark stars (see [3], [4], [5], [6] and references contained therein). Roughly speaking, the approach states that any -dimensional (pseudo)-Riemannian space can be embedded into a -dimensional pseudo-Euclidean pace. Then, the embedded space is of class . In the case of a -dimensional pseudo-Riemannian manifold describing a spherically symmetric and static space–time, the embedding procedure into a -dimensional pseudo-Euclidean space leads to an equation linking the temporal and radial components of the metric tensor. In this respect, if (or ) is known the geometry is full determined. It is worth mentioning that the Karmakar approach has been employed to build wormhole solutions (see Ref. [7], for example) but in this work, the author concentrated on the study of general aspects of the shape function. More recently the same scheme has been extended into the arena of gravity [8], [9]. In considering wormhole solutions, Morris–Thorne pioneering works [10], [11] established the first foundations in the understanding of how the so-called time machines work. In few words, a wormhole solution to Einstein field equations consists in a tunnel (usually called throat) connecting two asymptotically flat or infinite regions [12]. Now, in order to be traversable the solution must satisfy some geometric requirements [10], [11], [12] and additionally, the matter distribution sustaining the throat of the wormhole must violate the null energy condition (NEC) [13]. Regardingly, the wormhole is sustained by exotic fields [10], [11]. This last statement is crucial to ensure the existence of well posed wormhole solutions in the framework of Einstein gravity theory. Indeed, in more general and complexes situations in which the wormhole space–time is non symmetric and is time dependent (dynamic solution) NEC is also violated [14]. In this respect, wormhole solutions of Einstein field equations satisfying NEC are not possible. Despite being unrealistic material fields compared to those matter distributions that fulfill the so-called energy conditions, recently, this kind of matter content has been widely used in studies that attempt to explain the accelerated expansion of the Universe. In this regard, the exotic phantom energy field has been used to build the energy–momentum threading the throat of the wormhole [15], [16], [17]. In the manufacture of wormhole solutions an interesting technology is the well known thin-shell methodology [18]. In this direction thin-shell wormholes with cylindrically symmetric have been investigated [19], [20] and spherically symmetric thin-shell wormhole in the arena of linear electrodynamics [21] as well as non-linear electrodynamics [22], [23], [24]. Besides, wormhole solutions with other symmetries have been studied in [25], [26], [27] and also traversable Schwarzschild-like wormholes were considered in [28]. It is worth mentioning that the study of wormhole solutions has been carried out in the arena of modified or alternative gravity theories such as Brans–Dicke scalar field [29], [30] and [31], [32], [33], [34], [35]. An interesting point to be noted is that in such studies the solutions satisfied NEC (in general these solutions satisfied all energy conditions) which can be attributed to the corrections introduced by the additional fields into the theory.
The motivation and spirit of the present article is to build wormhole geometries employing the class I condition in the background of general relativity. In this concern we have obtained two models representing wormhole solutions accompanied with an exotic energy–momentum tensor. Particularly, the first model satisfies all the criteria [10], [11], [12], [13], [14] to describe a well posed wormhole solution. The second solution, although is an extension of the Morris–Thorne wormhole, the resulting line element does not reproduce the Minkowski space–time at large distances. To complement the study we have analyzed the hydrostatic equilibrium by using the modified relativistic balance equation [16].
The article is organized as follows. In Section 2 the main properties of a traversable wormhole solution are listed. In Section 3 Einstein fields equations for anisotropic matter distributions are presented and also class I condition is revisited. Sections 4 Model #1, 5 Model #2 present the resulting models. In Section 6 the main physical and geometric features are discussed and finally Section 7 concludes the work.
Section snippets
Wormhole morphology
In canonical coordinates the metric of a static and spherically symmetric wormhole can be parametrized as where and represent the red-shift and the so-called shape function, respectively. Now, in order to ensure that such a geometry corresponds to a traversable wormhole connecting two asymptotically flat or infinite spaces, the metric (1) must satisfy the following requirements [10], [11], [12],
- •
The size of the tunnel connecting the regions is
Einstein’s field equations
In Schwarzschild like-coordinates the most general line element describing a static and spherically symmetric space–time is given by where and depend on the radial coordinate only. Now, assuming that (2) is a solution of the Einstein field equations1 where with , and the density, radial and the transverse pressure respectively, we obtain
Model #1
In this section we shall use the Karmakar condition previously described in order to construct a wormhole geometry. Regardingly, we need to rewrite the pair in Eq. (1) in terms of in Eq. (2), namely Now, the flare-out condition implies at the throat, which means that must be zero at . Moreover, to ensure one needs to impose, Furthermore, to guarantee an asymptotically flat space–time and to avoid event
Model #2
In this section we construct another wormhole solution but this time we specify the shape function in order to obtain the ref-shift function through Eq. (19). To this end, we set to be the well known Morris–Thorne model [10], [11] which describes a wormhole solution with a null red-shift and driven by an exotic matter distribution. Now replacing (28) into Eq. (19) we obtain Note that, at the throat Eq. (29) provides so
Model #1
In order to analyze the behavior of the matter distribution supporting the wormhole geometry, it is convenient to express the field equations (11)–(13) in terms of and , namely Note that from Eqs. (31), (32) the null energy condition (NEC) reads, Now, since and we have which means that at and its neighborhood the NEC is violated and as it is well
Remarks
In this work we have implemented the embedding class I approach to find traversable wormhole solutions sustained by an imperfect energy–momentum tensor in the context of general relativity. Firstly, to obtain the wormhole geometry, we have proposed a suitable red-shift function . With this information at hand the shape function is immediately computed from Eq. (18) together with Eqs. (1), (2). The resulting shape function satisfies all the geometrical requirements in order to
CRediT authorship contribution statement
Francisco Tello-Ortiz: Conceptualization, Formal analysis, Investigation, Methodology, Software, Validation, Visualization, Writing - original draft. E. Contreras: Formal analysis, Methodology, Visualization, Writing - review & editing, Validation, Software.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
F. Tello-Ortiz thanks the financial support by the CONICYT PFCHA/DOCTORADO-NACIONAL/2019-21190856 , grant Fondecyt No. 1161192, Chile and projects ANT-1856 and SEM 18–02 at the Universidad de Antofagasta, Chile.
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