Abstract
The geometry of the Ellis–Bronnikov wormhole is implemented in the Rastall and k-essence theories of gravity with a self-interacting scalar field. The form of the scalar field potential is determined in both cases. A stability analysis with respect to spherically symmetric time-dependent perturbations is carried out, and it shows that in k-essence theory the wormhole is unstable, like the original version of this geometry supported by a massless phantom scalar field in general relativity. In Rastall’s theory, it turns out that a perturbative approach reveals the same inconsistency that was found previously for black hole solutions: time-dependent perturbations of the static configuration prove to be excluded by the equations of motion, and the wormhole is, in this sense, stable under spherical perturbations.
Highlights
The standard energy conditions, brings two main problems: the configuration can be unstable; or generally, the throat may not be traversable, in the sense that tidal forces may be huge, and possibly only pointlike objects may cross it from one universe to the other, except for some special cases
The Ellis–Bronnikov solution represents the simplest analytical wormhole solution that can be obtained in General Relativity theory (GR)
It consists of two asymptotically flat regions connected by a throat. This wormhole requires a massless, minimally coupled phantom scalar field: this means that all space, the throat, is filled with a field having negative energy density
Summary
The Ellis–Bronnikov (EB) wormhole [4,5] is one of the simplest solutions of GR leading to a structure of two flat asymptotics connected by a throat. The k-essence proposal may be connected with some fundamental theories inspired by quantum gravity In both cases our goal is to verify if it is possible to avoid the usual difficulties in wormhole construction and to obtain stable solutions. Both Rastall and k-essence theories with a self-interacting scalar field have been studied in attempts to obtain static, spherically symmetric black hole solutions [15,16]. We show that the EB wormhole metric can be a solution of both Rastall and k-essence theories under the condition that the potential describing the self-interaction of the scalar field is nonzero.
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