# Thin-shell wormholes from the regular Hayward black hole

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## Abstract

We revisit the regular black hole found by Hayward in \(4\)-dimensional static, spherically symmetric spacetime. To find a possible source for such a spacetime we resort to the nonlinear electrodynamics in general relativity. It is found that a magnetic field within this context gives rise to the regular Hayward black hole. By employing such a regular black hole we construct a thin-shell wormhole for the case of various equations of state on the shell. We abbreviate a general equation of state by \(p=\psi (\sigma )\) where \(p\) is the surface pressure which is a function of the mass density \((\sigma )\). In particular, linear, logarithmic, Chaplygin, etc. forms of equations of state are considered. In each case we study the stability of the thin shell against linear perturbations. We plot the stability regions by tuning the parameters of the theory. It is observed that the role of the Hayward parameter is to make the TSW more stable. Perturbations of the throat with small velocity condition are also studied. The matter of our TSWs, however, remains exotic.

## Keywords

Black Hole Black Hole Solution Extremal Black Hole Exotic Matter Regular Black Hole## 1 Introduction

Thin-shell wormholes (TSWs) constitute one of the wormhole classes in which the exotic matter is confined on a hypersurface and therefore can be minimized [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16] (the \(d\)-dimensional thin-shell wormhole is considered in [17] and the case with a cosmological constant is studied in [18]). Finding a physical (i.e. non-exotic) source to wormholes of any kind remains as ever a challenging problem in Einstein’s general relativity. In this regard we must add that modified theories of gravity present more alternatives with their extra degrees of freedom. We recall, however, that each modified theory partly cures things, while it partly adds its own complications. Staying within Einstein’s general relativity and finding remedies seems to be the prominent approach, provided the proper spacetimes are employed. An interesting class of spacetimes that may serve the purpose is the spacetimes of regular black holes.

*Our motivation for choosing a regular black hole in the wormhole construction can be justified by the fact that a regular system can be established from a finite energy.* *In high energy collision experiments for instance, the formation of such regular objects is more tenable.* Such a black hole was discovered first by Bardeen and came to be known as Bardeen black hole [19, 20, 21, 22]. Ayon-Beato and Garcia in [22] introduced a nonlinear electric field source for the Bardeen black hole. Bronnikov, later on, showed that the regular ‘electric’ black hole, e.g., the one considered by Ayon-Beato and Garcia, is not a quite correct solution to the field equations, because in these solutions the electromagnetic Lagrangian is inevitably different in different parts of space. On the contrary, quite correct solutions of this kind (and even with the same metric) can be readily obtained with a magnetic field (since in nonlinear electrodynamics (NED) there is no such duality as in the linear Maxwell theory). All this is described in detail in [23, 24, 25]. A similar type of black hole solution was given by Hayward [26], which provides the main motivation and fuel to the present study. This particular black hole solution has well-defined asymptotic limits, namely it is Schwarzschild for \(r\rightarrow \infty \) and de Sitter for \(r\rightarrow 0.\) In order to make a better account of the Hayward black hole we attempt first to explore its physical source. For this reason we search for the NED and find that a magnetic field within this theory accounts for such a source. Note that every NED does not admit a linear Maxwell limit and indeed this is precisely the case that we face in the present problem. In other words, if our NED model did have a Maxwell limit, then the Hayward spacetime should coincide with the Reissner–Nordström (RN) limit. Such a limit does not exist in the present problem. Once we fix our bulk spacetime the next step is to locate the thin shell which must lie outside the event horizon of the black hole. The surface energy-momentum tensor on the shell must satisfy the Israel junction conditions [27, 28, 29, 30, 31]. As the Equation of State (EoS) for the energy-momentum on the shell we choose different models, which are abbreviated by \(p=\psi (\sigma )\). Here \(p\) stands for the surface pressure, \(\sigma \) is the mass (energy) density and \(\psi (\sigma )\) is a function of \(\sigma .\) We consider the following cases: (1) linear gas (LG) [32, 33], where \(\psi (\sigma )\) is a linear function of \(\sigma \); (2) Chaplygin gas (CG) [34, 35], where \(\psi (\sigma ) \sim \frac{1}{\sigma };\) (3) generalized Chaplygin gas (GCG) [36, 37, 38, 39, 40], where \(\psi (\sigma ) \sim \frac{1}{\sigma ^{\nu }}\) (\(\nu =\) constant); (4) modified generalized Chaplygin gas (MGCG) [41, 42, 43, 44], where \(\psi (\sigma ) \sim \) LG+GCG; and (5) logarithmic gas (LogG), where \(\psi (\sigma ) \sim \ln \vert \sigma \vert \).

For each of the cases we plot the second derivative of the derived potential function \(V^{\prime \prime }(a_{0})\), where \(a_{0}\) stands for the equilibrium point. The region where the second derivative is positive (i.e. \(V^{\prime \prime }(a_{0}) >0\)) yields the regions of stability which are all depicted in figures. This summarizes the strategy that we adopt in the present paper for the stability of the thin-shell wormholes constructed from the Hayward black hole.

The organization of the paper is as follows. Section 2 reviews the Hayward black hole and determines a Lagrangian for it. A derivation of the stability condition is carried out in Sect. 3. Particular examples of the equation of state follow in Sect. 4. Small velocity perturbations are the subject of Sect. 5. The paper ends with the our conclusion in Sect. 6.

## 2 Regular Hayward black hole

### 2.1 Magnetic monopole field as a source for the Hayward black hole

## 3 Stable thin-shell wormhole condition

## 4 Some models of exotic matter supporting the TSW

Recently two of us analyzed the effect of the Gauss–Bonnet parameter on the stability of TSW in higher-dimensional EGB gravity [46]. In that paper some specific models of matter have been considered such as LG, CG, GCG, MGCG, and LogG. In this work we get closely to the same EoSs and we analyze the effect of Hayward’s parameter in the stability of the TSW constructed above.

### 4.1 Linear gas (LG)

### 4.2 Chaplygin gas (CG)

### 4.3 Generalized Chaplygin gas (GCG)

### 4.4 Modified generalized Chaplygin gas (MGCG)

### 4.5 Logarithmic gas (LogG)

## 5 Stability analysis for small velocity perturbations around the static solution

### 5.1 The Schwarzschild example

### 5.2 The Hayward example

## 6 Conclusion

Thin-shell wormholes are constructed from the regular black hole (or non-black hole for a certain range of parameters) discovered by Hayward. We show first that this solution is powered by a magnetic monopole field within the context of NED. The nonlinear Lagrangian in the present case can be expressed in a non-polynomial form of the Maxwell invariant. Such a Lagrangian does not admit a linear Maxwell limit. By employing the spacetime of Hayward and different equations of state of generic form, \(p=\psi (\sigma )\), on the thin shell we plot possible stable regions. Amongst these, linear, logarithmic, and different Chaplygin gas forms are used, and stable regions are displayed. The method of identifying these regions relies on the reduction of the perturbation equations to a harmonic equation of the form \(\ddot{x}+\frac{1}{2}V^{\prime \prime } (a_{0}) x=0\) for \(x=a-a_{0}.\) Stability simply amounts to the condition \(V^{\prime \prime } (a_{0}) >0\), which is plotted numerically. In all different equations of state we obtained stable regions and observed that the Hayward parameter \(\ell \) plays a crucial role in establishing the stability. That is, for higher \(\ell \) value we have enlargement in the stable region. The trivial case, \(\ell =0\), corresponds to the Schwarzschild case and is well known. We have considered also perturbations with small velocity. It turns out that our TSW is no more stable against such a kind of perturbations. We would like to add here that a stable spherically symmetric wormhole in general relativity has been introduced in [47]. Finally, we admit that in each case our energy density happens to be negative so that we are confronted with exotic matter. In a separate study we have shown that not to have exotic matter to thread the wormhole we have to abandon spherical symmetry and consider prolate/oblate spheroidal sources [48].

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