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In the present paper we discuss properties of a model of a ring wormhole, recently proposed by Gibbons and Volkov Phys. Rev. D 96, 024053 (2017) ; J. Cosmol. Astropart. Phys. 05 (2017) 039; Phys. Lett. B 760, 324 (2016). Such a wormhole connects two flat spacetimes which are glued through disks of the radius a bounded by the string with negative angle deficit -2. The presence of the string's matter violating the null-energy condition makes the wormhole static and traversable. We study gravitational field of static sources in such a spacetime in the weak-field approximation. In particular, we discuss how a field of an oblate thin massive shell surrounding one of the wormhole's mouths is modified by its presence. We also obtain a solution of a similar problem when both mouths of the wormhole are located in the same space. This approximate solution is found for the case when the distance L between these mouths is much larger than the radius a of the ring. We demonstrate that the corresponding locally static gravitational field in such a multiply connected space is nonpotential. As a result of this, the proper time gap for the clock's synchronization linearly grows with time and closed timelike curves are formed. This process inevitably transforms such a traversable ring wormhole into a time machine. We estimate the timescale of this process.
Frolov et al. (Fri,) studied this question.