On the cylindrically symmetric wormholes WhCR^e: The motion of test particles

Abstract: In this article we partially implement the program outlined in the previous paper of the authors [A. V. Aminova and P. I. Chumarov, Phys. Rev. D 88, 044005 (2013)]. The program owes its origins to the following comment in paper [M. Cveti\v{c} and D. Youm, Nucl. Phys. B, 438, 182 (1995), Addendum-ibid. 449,146 (1995)], where a class of static spherically symmetric solutions in $(4+n)$-dimensional Kaluza--Klein theory was studied: "...We suspect that the same thing [as for spherical symmetry] will happen for axially symmetric stationary configurations, but it remains to be proven". We study the radial and non-radial motion of test particles in the cylindrically symmetric wormholes found in the authors'paper of type $\rm WhCR^e$ in 6-dimensional reduced Kaluza--Klein theory with Abelian gauge field and two dilaton fields, with particular attention to the extent to which the wormhole is traversable. In the case of non-radial motion along a hypersurface z=const we show that, as in the Kerr and Schwarzschild geometries, we should distinguish between orbits with impact parameters greater resp. less than a certain critical value $D_c$, which corresponds to the unstable circular orbit of radius $u_c$ $(r_c)$. For $D^2>D_c^2$ there are two kinds of orbits: orbits of the first kind arrive from infinity and have pericenter distances greater than $u_c$, whereas orbits of the second kind have apocenter distances less than $u_c$ and terminate at the singularity at $u=-\infty$ $(r=0)$. For $D=D_c$ orbits of the first and second kinds merge and both orbits spiral an infinite number of times toward the unstable circular orbit $u=u_c$. For $D^2<D_c^2$ we have only orbits of one kind: starting at infinity, they cross the wormhole throat and terminate at the singularity.