Abstract
In this paper we show that wormholes in (2+1) dimensions (3-D) cannot be sourced solely by both Casimir energy density and tension, differently from what happens in a 4-D scenario, in which case it has been shown recently, by the direct computation of the exact shape and redshift functions of a wormhole solution, that this is possible. We show that in a 3-D spacetime the same is not true since the arising of at least an event horizon is inevitable. We do the analysis for massive and massless fermions, as well as for scalar fields, considering quasi-periodic boundary conditions and find that a possibility to circumvent such a restriction is to introduce, besides the 3-D Casimir energy density and tension, a cosmological constant, embedding the surface in a 4-D manifold and applying a perpendicular weak magnetic field. This causes an additional tension on it, which contributes to the formation of the wormhole. Finally, we discuss the possibility of producing the condensed matter analogous of this wormhole in a graphene sheet and analyze the electronic transport through it.
Highlights
Negative energies of free quantum fields subject to certain boundary conditions, has been increasingly examined in the context of wormholes [12]
In this paper we have studied 3-D traversable wormholes and explicitly shown that they cannot be sourced by only the Casimir energy density, radial and lateral tensions
It has been demonstrated [6] that in 4-D case this is possible by the direct computation of the redshift and shape functions based on a Morris–Thorne wormhole solution, in extended theories of gravitation [21,22]
Summary
Recent works considering the Casimir effect in spacetimes around of wormholes have been published [17,18,19,20], as well as others which analyze how traversable wormholes can be produced and sustained by means of both the Casimir energy and tension, in the context of General Relativity and extended theories of gravitation, in semiclassical approaches [6,21,22]. We will investigate 3-D traversable wormholes and show that this construction is not possible, since at least an event horizon appears when one considers only the Casimir quantities as gravity source. We will do this analysis by considering massive and massless fermions, as well as scalar fields, adopting quasi-periodic boundary conditions. We analyze if it is possible, as in the 4-D case, to sustain a traversable wormhole in a 3-D spacetime from the Casimir quantities, namely, energy density and tension. We will consider the possible sources for λ0, λ1
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