Particle motion around traversable wormholes: Possibility of closed timelike geodesics

The present work investigates the general wormhole solution in Einstein gravity with an exponential shape function around an ultrastatic and a finite redshift geometry. The geodesic motion around the wormholes is studied in which the deflection angle of the orbiting photon sphere is found to be negative after a certain region, indicating the presence of repulsive effect of gravity in both the ultrastatic and finite redshift wormholes. Various unbounded and bounded timelike trajectories are presented on the wormhole embedding diagrams, in which some of the bound orbits involve intersection points that may lead to causality violating geodesics. Another class of closed timelike geodesics are obtained in the unstable circular trajectory that appeared at the wormhole throat. Finally, the trajectories are classified in terms of the family of CTG orbits.