Thinshell wormholes with charge in F(R) gravity
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Abstract
In this article, we construct a class of constant curvature and spherically symmetric thinshell Lorentzian wormholes in F(R) theories of gravity and we analyze their stability under perturbations preserving the symmetry. We find that the junction conditions determine the equation of state of the matter at the throat. As a particular case, we consider configurations with mass and charge. We obtain that stable static solutions are possible for suitable values of the parameters of the model.
Keywords
Fundamental Form Cosmological Horizon Junction Condition Exotic Matter Surface Energy Density1 Introduction
Traversable Lorentzian wormholes are solutions of gravitational theories which have a throat that connects two regions of the same universe or two different universes [1, 2]. In General Relativity, they are threaded by matter that violates the null energy condition [1, 2, 3, 4, 5]; the amount of this exotic matter can be made arbitrary small [6], but at the expense of large pressures at the throat [7]. Traversable wormholes can be constructed [2] by cutting and pasting two manifolds to form a new one, with a shell at the joining surface corresponding to the throat, where the flareout condition is fulfilled. These thinshell wormholes have been extensively studied in the literature because of their simplicity, which makes the analysis of stability easier, and the exotic matter can be confined to the shell. Wormholes with a continuous energystress tensor at the throat usually also need a cut and paste procedure to confine the exotic matter or to obtain a suitable asymptotic behavior. Stability studies of spherically symmetric thinshell wormholes, with a linearized equation of state at the throat, have been performed under radial perturbations by several authors ([8, 9, 10, 11, 12, 13, 14, 15, 16] and references therein). Plane and cylindrical thinshell wormholes were also analyzed [16, 17, 18, 19, 20, 21, 22, 23, 24]. The Chaplygin gas and its generalizations were used to model the exotic matter supporting wormholes [25, 26, 27, 28, 29, 30]. The linearized stability of Schwarzschild thinshell wormholes with variable equations of state has been recently considered [31].
Within the context of General Relativity, the observed accelerated expansion of the universe during the matter dominated epoch requires of the presence of dark energy. Instead of this nonstandard fluid, modifications of General Relativity were proposed in order to solve both the problems of dark energy and dark matter, required by the concordance (\(\Lambda \)CDM) model. One of the simplest possible modifications corresponds to the socalled F(R) gravity [32, 33, 34], in which the Einstein–Hilbert lagrangian is replaced by a function F(R) of the Ricci scalar R. The F(R) theories can provide an alternative for an unified picture of both inflation and the accelerated expansion at later times. Besides the cosmological aspects, it is of interest to study compact objects in these alternative theories. Static and spherically symmetric black hole solutions in F(R) have been found [35, 36, 37] in the last decade. Traversable wormholes in F(R) were also studied in recent years [38, 39, 40, 41].
Thin shells in General Relativity are modeled by using the wellknown Darmois–Israel [42, 43, 44] formalism. The junction conditions allow to match two solutions onto a hypersurface under different conditions, for example the interior and exterior solutions corresponding to stars, galaxies, etc. They are also useful for the study of thin layers of matter and in braneworld cosmology. In the last decade, the junction conditions have been generalized to F(R) theories of gravity [45, 46]. The junction conditions are more stringent in F(R) gravity than in General Relativity. For nonlinear F(R), they always require continuity of the trace of the second fundamental form at the matching hypersurface and, with the exception of quadratic F(R), the continuity of the curvature scalar R. Quadratic F(R) has some specific features: the curvature scalar R can be discontinuous at the matching hypersurface and, as a consequence, the shell will have, besides the standard energymomentum tensor, an external energy flux vector, an external scalar pressure (or tension) and another energymomentum contribution resembling classical dipole distributions [47, 48]. The last one can be interpreted as a gravitational double layer. All these contributions should be present in order to make the whole energymomentum tensor divergencefree [47, 48]. Recently, these results were extended to the most general gravitational theory with a Lagrangian quadratic in the curvature [49].
In the present work, we construct thinshell wormholes with spherical symmetry in F(R) theory with constant curvature and we study their stability under radial perturbations. The paper is organized as follows: in Sect. 2, the wormhole construction is done; in Sect. 3, the stability of static configurations is analyzed; in Sect. 4 the formalism is applied to charged wormholes; finally, in Sect. 5 the conclusions are presented. We adopt units in which \(G=c=1\), where G and c denote the gravitational constant and the speed of light, respectively.
2 Wormhole construction
3 Stability of static configurations
4 Wormholes with charge

For \(R_0\le 0\), no static solutions are present if \(Q \le Q_c\). As the charge grows, i.e. \(Q>Q_c\), two static solutions appear: one stable and the other unstable. Then the static solutions fuse into one; finally for larger values of charge they disappear.

For \(R_0>0\), when \(Q>Q_c\) there exist two static solutions (one stable and the other unstable) with a similar behavior as in the \(R_0<0\) case, but also a third static solution with a larger value of \(a_0/M\) is always present. This third solution is unstable for any value of the charge.
5 Conclusions
We have constructed a class of spherically symmetric wormholes by using the thinshell formalism in F(R) theories; the surface that joins the two equal copies of a solution with constant curvature \(R_0\) corresponds to the throat. We have shown that the matter at the throat should satisfy the equation of state \(p=\sigma /2\). The condition \(F'(R_0)>0\), required to have a positive effective gravitational constant and a nonghost graviton, results in exotic matter at the throat. We also have obtained the condition for the stability of static configurations under perturbations preserving the symmetry. In particular, we have applied the formalism to wormholes with mass M and charge Q. As we have assumed that \(F'(R_0)>0\), the term associated with the charge has the same sign as in General Relativity, and the matter at the throat is exotic. We have found that stable solutions are possible for appropriate values of the parameters, for both positive and nonpositive \(R_0\). In the first case, one static solution is always also present, being always unstable for any value of the charge; for large values of \(Q/ (M\sqrt{F'(R_0)})\) two static solutions are also present within a short range of charge, one of them is stable, while the other is unstable. In the second case, a large value of \(Q/ (M\sqrt{F'(R_0)})\) is required to have two static solutions for a short range of charge, one of them is stable and the other is unstable. The qualitative aspects of the results do not depend on the particular form of the function F(R), each theory only manifests itself through the constant \(F'(R_0)\), in the form of an effective charge \(Q/F'(R_0)\).
Notes
Acknowledgments
This work has been supported by CONICET and Universidad de Buenos Aires.
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