Motion of test particles in the field of a naked singularity

TL;DR

This paper explores how a quadrupole moment affects the motion of particles and photons around a naked singularity, revealing significant differences from Schwarzschild black holes, including disk structure and geodesic behavior.

Contribution

It introduces the quadrupole-metric to analyze test particle motion near naked singularities, highlighting the impact of quadrupole parameters on geodesics and accretion disk properties.

Findings

Quadrupole parameter influences perihelion distance.

Accretion disk structure varies with quadrupole value.

Presence of repulsive gravity affects disk formation.

Abstract

We investigate the motion of test particles in the gravitational field of a static naked singularity generated by a mass distribution with quadrupole moment. We use the quadrupole-metric ($q-$metric) which is the simplest generalization of the Schwarzschild metric with a quadrupole parameter. We study the influence of the quadrupole on the motion of massive test particles and photons and show that the behavior of the geodesics can drastically depend on the values of the quadrupole parameter. In particular, we prove explicitly that the perihelion distance depends on the value of the quadrupole. Moreover, we show that an accretion disk on the equatorial plane of the quadrupole source can be either continuous or discrete, depending on the value of the quadrupole. The inner radius of the disk can be used in certain cases to determine the value of the quadrupole parameter. The case of a discrete accretion is interpreted as due to the presence of repulsive gravity generated by the naked singularity. Radial geodesics are also investigated and compared with the Schwarzschild counterparts.