Abstract
We find a family of exact solutions to the Einstein–Maxwell equations for rotating cylindrically symmetric distributions of a perfect fluid with the equation of state p=wρ (|w|<1), carrying a circular electric current in the angular direction. This current creates a magnetic field along the z axis. Some of the solutions describe geometries resembling that of Melvin’s static magnetic universe and contain a regular symmetry axis, while some others (in the case w>0) describe traversable wormhole geometries which do not contain a symmetry axis. Unlike Melvin’s solution, those with rotation and a magnetic field cannot be vacuum and require a current. The wormhole solutions admit matching with flat-space regions on both sides of the throat, thus forming a cylindrical wormhole configuration potentially visible for distant observers residing in flat or weakly curved parts of space. The thin shells, located at junctions between the inner (wormhole) and outer (flat) regions, consist of matter satisfying the Weak Energy Condition under a proper choice of the free parameters of the model, which thus forms new examples of phantom-free wormhole models in general relativity. In the limit w→1, the magnetic field tends to zero, and the wormhole model tends to the one obtained previously, where the source of gravity is stiff matter with the equation of state p=ρ.
Highlights
Cylindrical symmetry is the second simplest space-time symmetry making it possible to obtain numerous exact solutions in general relativity and its extensions, characterizing local strong gravitational field configurations
We have shown that a free longitudinal magnetic field cannot support a vortex gravitational field with the metric (1)
Some of them (Solutions 1–3) contain a symmetry axis which can be made regular by properly choosing the solution parameters
Summary
Cylindrical symmetry is the second (after the spherical one) simplest space-time symmetry making it possible to obtain numerous exact solutions in general relativity and its extensions, characterizing local strong gravitational field configurations. A large number of static cylindrically symmetric solutions have been obtained and studied since the advent of general relativity, including vacuum, electrovacuum, perfect fluid and others, see reviews in [1,2,3] and references therein. For instance, Birch [4] has reported the discovery of polarization anisotropy in radio signals from extragalactic sources which could be a signature of a slow rotation of the Universe This gave rise to the emergence of numerous cosmological models with rotation, most of which possess cylindrical symmetry; see [5,6] and references therein. Melvin’s famous solution to the Einstein–Maxwell equations, an “electric or magnetic geon” [13], is a completely regular static, cylindrically symmetric solution with a longitudinal electric or magnetic field as the only source of gravity.
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