Stability of thin shell wormhole in theory of gravity
Introduction
Morris and Thorne [1] investigated the traversable Lorentzian wormholes with a throat as a geometrical object, like tunnels (or bridge) that connect two space–times whether similar or different. A well-studied class of wormholes is that of thin shell constructed by cutting and pasting two manifolds, [2], formed a complete geodesic , a new one with a shell placed on the joining a hypersurface (throat) performed, showing the formation of wormholes required an exotic matter to violate energy conditions, [3].
Moreover, an exotic matter has received more attention in cosmology and this attention has led to the explanation of the observed accelerated expansion of the universe [4], [5]. Yet, cosmological observation confirmed that our Universe is undergoing a phase of an accelerated expansion, [6] and [7]. Further, this phenomenon can be interpreted as the dark energy models [8], [9], [10], [11], [12] or modified gravity theories [13], [14], [15], [16], [17].
Dark energy, as a new source of exotic matter, has several studies to explore the corresponding wormhole in various kinds of cosmological theories, [5]. The stability analysis of TSWs with a particular equation of state (EoS) (for instance, Phantom energy, Chaplygin gas, generalized Chaplygin gas, Viscous fluid, etc.) at the throat under linear radial perturbation has been investigated by several authors Lobo [18], [19], Wang and Meng [20], Eid [21], Poisson and Visser [22], Eiroa [23], and Dias and Lemos [24].
There are several theories of modified gravity, which extend in some way general relativity, for example, Brans–Dick theory [25], string theory [26], scalar tensor theory [27], Gauss–Bonnet theory [28], higher dimensional Lovelock theories [29] and theories of gravity. Among those theories, the most reliable modification is regarding, a class of theories as candidate to replace general relativity using the Einstein’s–Hilbert gravitational action. This may be found by taking the Ricci scalar as an arbitrary function of the scalar invariant, [30], [31] and [27]. This theory gives a larger set of solutions and provides the natural gravitational alternative for dark energy, [11] and [32].
In a sequence of papers the stability of wormholes associated with gravity have been studied, such as, [33] proposed the unstable wormhole solutions in gravity. While [34] discussed the geometry of wormhole in -gravity and [35] studied the black hole stability in gravity. Myung [36] investigated the instability of Kerr black holes in gravity. Bambi et al. [37] studied wormholes in the Palatini formulation of gravity.
Moreover, [38] analyzed the necessary conditions of wormhole in theory. Khaybullina and Tuleganova [39] studied the stability of Schwarzschild TSW in theory. Also, [40] studied the stability of cylindrical TSW with MGCG in gravity. While, [41] investigated the construction of thin shell wormholes from metric theory, taking into consideration that [42] analyzed the cosmological wormhole in gravity.
Moreover, [43] introduced the wormholes supported by hybrid metric Palatini gravity. Also, [44] studied the wormhole solutions that obey the null energy condition everywhere in the generalized hybrid metric Palatini gravity without exotic matter. Eventually, [45] discussed the stability of Kerr black holes in generalized hybrid metric Palatini gravity.
From this perspective, it is well known that, the stability of Reissner–Nordstrom (RN)–de Sitter TSWs with a MGCG EoS have been performed in two different configurations of gravity models, such as quadratic type model and a cubic type model [46], [47], [48] and [49].
The paper is organized as follows: Section 2 we display dynamics of TSWs in gravity supported by MGCG. While in Section 3 we explain the necessity of stability analysis through radial perturbation using two examples of a quadratic and a cubic type model. Finally, in Section 4 we present a remarking conclusion.
Section snippets
Dynamics of thin shell wormhole in gravity
The theory of gravity is considered as a modification of general relativity using a function instead of the Ricci scalar, in which the well known Einstein–Hilbert action of general relativity [30], to be expressed in terms of to become, where is the Lagrangian for the matter distribution, and is the determinant of the metric . The variation of the action (1) with respect to the metric tensor , leads to the following fourth order partial
Stability analysis and results
For the stability analysis, one uses the Taylor expansion of at , up to the second order:
Accordingly, the first and the second derivatives of become
Moreover, Eqs. (26), (27) may be written by plugging Eq. (20) and the derivative of (15), with taking into consideration that , to get and
Conclusion
The dynamics and stability analysis of RN-DS TSW through Visser cut and paste approach in the framework of -gravity supported by MGCG equation of state are investigated, using a linear radial perturbation around the throat of a wormhole.
Moreover, the RN-DS TSWs is stable at , while it unstable at . The numerical analysis is used to explain the relation between square speed of sound and the radius of the wormhole throat with different values of the parameters (
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.
Acknowledgment
I would like to thank Dr. M.E. Kahil, MSA University for his helpful comments.
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