Probing naked singularities in the charged and uncharged \(\gamma - metrics\) with quantum wave packets

The non-trivial naked singularities that possess directional behavior in the charged and uncharged Zipoy-Voorhees (ZV) spacetimes, known as \( \gamma - metrics \) are investigated within the context of quantum mechanics. Classically singular spacetime is understood as a geodesic incompleteness with respect to a particle probe, while quantum singularity is understood as a non-unique evolution of test quantum wave packets. In this study, quantum wave packets obeying Klein–Gordon equation are used to probe timelike naked singularities. It is shown by rigorous mathematical calculations that the outermost singularity developed in the charged and uncharged ZV spacetime on the equatorial plane is quantum mechanically singular for all values of the deformation parameter \(\gamma \). However, directional singularities that develop on the symmetry axis is shown to be healed partially for specific range of the parameter \(\gamma \), if the analysis is restricted purposely to only specific mode (s-wave mode). Allowing arbitrary modes, classical directional singularities remains quantum singular.