Advances in High Energy Physics

Volume 2016, Article ID 2868750, 13 pages

http://dx.doi.org/10.1155/2016/2868750

## Stability of the Regular Hayward Thin-Shell Wormholes

Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore 54590, Pakistan

Received 23 May 2016; Revised 1 July 2016; Accepted 10 July 2016

Academic Editor: Luis A. Anchordoqui

Copyright © 2016 M. Sharif and Saadia Mumtaz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

The aim of this paper is to construct regular Hayward thin-shell wormholes and analyze their stability. We adopt Israel formalism to calculate surface stresses of the shell and check the null and weak energy conditions for the constructed wormholes. It is found that the stress-energy tensor components violate the null and weak energy conditions leading to the presence of exotic matter at the throat. We analyze the attractive and repulsive characteristics of wormholes corresponding to and , respectively. We also explore stability conditions for the existence of traversable thin-shell wormholes with arbitrarily small amount of fluid describing cosmic expansion. We find that the space-time has nonphysical regions which give rise to event horizon for and the wormhole becomes nontraversable producing a black hole. The nonphysical region in the wormhole configuration decreases gradually and vanishes for the Hayward parameter . It is concluded that the Hayward and Van der Waals quintessence parameters increase the stability of thin-shell wormholes.

#### 1. Introduction

One of the most interesting attributes of general relativity is the possible existence of hypothetical geometries having nontrivial topological structure. Misner and Wheeler [1] described these topological features of space-time as solutions of the Einstein field equations known as wormholes. A “wormhole” having a tunnel with two ends allows a short way associating distant regions of the universe. Besides the lack of some observational lines of evidence, wormholes are regarded as a part of black holes (BH) family [2]. The simplest example is the Schwarzschild wormhole that connects one part of the universe to another through a bridge. This wormhole is not traversable as it does not allow a two-way communication between two regions of the space-time leading to the contraction of wormhole throat.

Physicists have been motivated by the proposal of Lorentzian traversable wormholes given by Morris and Thorne [3]. In case of traversable wormholes, the wormhole throat is threaded by exotic matter which causes repulsion against the collapse of the wormhole throat. The most distinguishing property of these wormholes is the absence of event horizon which enables observers to traverse freely across the universe. It was shown that a BH solution with horizons could be converted into wormhole solution by adding some exotic matter which makes the wormhole stable [4]. Traversable wormhole solutions must satisfy the flare-out condition preserving their geometry due to which the wormhole throat remains open. The existence of exotic matter yields the violation of null energy condition (NEC) and weak energy condition (WEC) which is the basic property for traversable wormholes. Null energy condition is the weakest one whose violation gives rise to the violation of WEC and strong energy conditions (SEC). The exotic matter is characterized by stress-energy tensor components determined through Israel thin-shell formalism [5].

Thin-shell wormholes belong to one of the wormhole classes in which exotic matter is restricted at the hypersurface. To make sure that the observer does not encounter nonphysical zone of BH, a thin shell strengthens the wormhole provided that it has an exotic matter for its maintenance against gravitational collapse. The physical viability of thin-shell wormholes is a debatable issue due to inevitable amount of exotic matter which is an essential ingredient for the existence as well as stability of wormholes. The amount of exotic matter can be quantified by the volume integral theorem which is consistent with the concept that a small quantity of exotic matter is needed to support wormhole [6]. Visser [7, 8] developed an elegant cut-and-paste technique to minimize the amount of exotic matter by restricting it at the edges of throat in order to obtain a more viable thin-shell wormhole solution.

It is well known that thin-shell wormholes are of significant importance if they are stable. The stable/unstable wormhole models can be investigated either by applying perturbations or by assuming equation of state (EoS) supporting exotic matter at the wormhole throat. In this context, many authors constructed thin-shell wormholes following Visser’s cut-and-paste procedure and discussed their stability. Kim and Lee [9] investigated stability of charged thin-shell wormholes and found that charge affects stability without affecting the space-time itself. Ishak and Lake [10] analyzed stability of spherically symmetric thin-shell wormholes. Lobo and Crawford [11] studied spherically symmetric thin-shell wormholes with cosmological constant () and found that stable solutions exist for positive values of . Eiroa and Romero [12] studied linearized stability of charged spherical thin-shell wormholes and found that the presence of charge significantly increases the possibility of stable wormhole solutions. Sharif and Azam [13] explored both stable and unstable configurations for spherical thin-shell wormholes. Sharif and Mumtaz analyzed stable wormhole solutions from regular ABG [14] and ABGB [15] space-time in the context of different cosmological models for exotic matter.

It is found that one may construct a traversable wormhole theoretically with arbitrarily small amount of fluid describing cosmic expansion. In order to find any realistic source for exotic matter, different candidates of dark energy have been proposed like tachyon matter [16], family of Chaplygin gas [17, 18], phantom energy [19], and quintessence [20]. Eiroa [21] assumed generalized Chaplygin gas to study the dynamics of spherical thin-shell wormholes. Kuhfittig [22] analyzed stability of spherical thin-shell wormholes in the presence of and charge by assuming phantom like EoS at the wormhole throat. Sharif and collaborators discussed stability analysis of Reissner-Nordström [23] and Schwarzschild de Sitter as well as anti-de Sitter [24] thin-shell wormholes in the vicinity of generalized cosmic Chaplygin gas (GCCG) and modified cosmic Chaplygin gas (MCCG). Some physical properties of spherical traversable wormholes [25] as well as stability of cylindrical thin-shell wormholes [26, 27] have been studied in the context of GCCG, MCCG, and Van der Waals (VDW) quintessence EoS. Recently, Halilsoy et al. [28] discussed stability of thin-shell wormholes from regular Hayward BH by taking linear, logarithmic, and Chaplygin gas models and found stable solutions for increasing values of Hayward parameter.

This paper is devoted to the construction of thin-shell wormholes from regular Hayward BH by considering three different models of exotic matter at the throat. The paper is organized as follows. In Section 2, we construct regular Hayward thin-shell wormholes and analyze various physical aspects of these constructed thin-shell wormholes. Section 3 deals with stability formalism of the regular Hayward thin-shell wormholes in the vicinity of VDW quintessence EoS and Chaplygin gas models. We find different throat radii numerically and show their expansion or collapse with different values of parameters. Finally, we provide summary of the obtained results in the last section.

#### 2. Regular Hayward Black Hole and Wormhole Construction

The static spherically symmetric regular Hayward BH [29] is given bywhere and and and are positive constants. This regular BH is chosen for thin-shell wormhole because a regular system can be constructed from finite energy and its evolution is more acceptable. This reduces to de Sitter BH for , while the metric function for the Schwarzschild BH is obtained as . Its event horizon is the largest root of the equationThis analysis of the roots shows a critical ratio and radius such that, for and , the given space-time has no event horizon yielding a regular particle solution. The regular Hayward BH admits a single horizon if and , which represents a regular extremal BH. At and , the given space-time becomes a regular nonextremal BH with two event horizons.

We implement the standard cut-and-paste procedure to construct a timelike thin-shell wormhole. In this context, the interior region of the regular Hayward BH is cut with . The two 4D copies are obtained which are glued at the hypersurface . In fact, this technique treats the hypersurface as the minimal surface area called wormhole throat. The exotic matter is concentrated at the hypersurface making the wormhole solution a thin shell. We can take coordinates at the shell. The induced 3D metric at with throat radius is defined aswhere is the proper time on the shell.

This construction requires the fulfillment of flare-out condition by the throat radius ; that is, the embedding function in (1) should satisfy the relation . The thin layer of matter on causes the extrinsic curvature discontinuity. In this way, Israel formalism is applied for the dynamical evolution of thin shell which allows matching of two regions of space-time partitioned by . We find nontrivial components of the extrinsic curvature aswhere dot and prime stand for and , respectively. To determine surface stresses at the shell, we use Lanczos equations, which are the Einstein equations given bywhere and . The surface energy-momentum tensor yields the surface energy density and surface pressures . Solving (4) and (5), the surface stresses are obtained asIn order to prevent contraction of wormhole throat, matter distribution of surface energy-momentum tensor must be negative which indicates the existence of exotic matter making the wormhole traversable [7, 8]. The amount of this matter should be minimized for the sake of viable wormhole solutions. We note from (6) and (7) that and showing the violation of NEC and WEC for different values of , , and . In Figure 1, we plot a graph for pressure showing that pressure is a decreasing function of the throat radius (a), while (b) shows violation of energy conditions associated with regular Hayward thin-shell wormholes.