Elsevier

Annals of Physics

Volume 419, August 2020, 168217
Annals of Physics

Traversable wormholes in light of class I approach

https://doi.org/10.1016/j.aop.2020.168217Get rights and content

Highlights

  • Analytical wormhole solutions by embedding class I in general relativity.

  • Morris–Thorne wormhole space–time with a non null red-shift function.

  • The solutions fulfill the geometrical requirements to represent traversable wormholes.

Abstract

In this work, we employ the class I approach to obtain wormhole solutions in the framework of general relativity in two different ways. Firstly, we propose a suitable red-shift function in order to find its associated shape function. Afterwards, we solve the inverse problem, namely, we impose the well known Morris–Thorne shape function to obtain the corresponding red-shift. It is found that, on one hand, the first model satisfies all the general requirements of a traversable wormhole. On the other hand, although the second solution violates the null energy condition at the throat as expected, the solution is not asymptotically flat. The study is complemented by analyzing the hydrostatic balance of the system by means of the modified relativistic hydrostatic equilibrium equation.

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 n-dimensional (pseudo)-Riemannian space can be embedded into a n+p-dimensional pseudo-Euclidean pace. Then, the embedded space is of class p. In the case of a 4-dimensional pseudo-Riemannian manifold describing a spherically symmetric and static space–time, the embedding procedure into a 5-dimensional pseudo-Euclidean space leads to an equation linking the temporal gtt and radial grr components of the metric tensor. In this respect, if gtt (or grr) 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 f(R,T) 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 f(R,T) [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 ds2=eΦdt2dr21brr2dθ2+sin2θdϕ2,where Φ=Φ(r) and b=b(r) 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 ds2=eνdt2eλdr2r2dθ2+sin2θdϕ2,where ν=ν(r) and λ=λ(r) depend on the radial coordinate only. Now, assuming that (2) is a solution of the Einstein field equations1 Rμν12gμνR=8πTμν,where Tνμ=diag(ρ,pr,pt,pt) with ρ, pr and pt the density, radial and the transverse pressure respectively, we obtain 8πρ=1

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 {Φ,b} in Eq. (1) in terms of {ν,λ} in Eq. (2), namely ν(r)=Φ(r)andb(r)=r1eλ(r).Now, the flare-out condition implies b(r0)=r0 at the throat, which means that eλ must be zero at r=r0. Moreover, to ensure limr=b(r)r=0,one needs to impose, limreλ(r)=1.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 b(r) in order to obtain the ref-shift function Φ(r) through Eq. (19). To this end, we set b(r) to be the well known Morris–Thorne model [10], [11] b(r)=r02r,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 Φ(r)=Ln{C+Dr0Lnr+r2r022}.Note that, at the throat Eq. (29) provides eΦ(r)|r=r0=C+Dr0Ln2(r0),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 b(r) and Φ(r), namely ρ=b8πr2,pr=18πΦr1brbr3,pt=rb8πrΦ2+2Φ4+Φ2rrbr2rrbΦ2+1r. Note that from Eqs. (31), (32) the null energy condition (NEC) reads, ρ+pr=18πΦr1brbr3+br2.Now, since b(r0)=r0 and b(r0)<1 we have ρ+pr|r=r0=18πbr21r2<0,which means that at r0 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 Φ(r). With this information at hand the shape function b(r) is immediately computed from Eq. (18) together with Eqs. (1), (2). The resulting shape function b(r) 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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