Geodesic deviation in the $q$-metric

TL;DR

This paper investigates how the quadrupole deformation parameter affects tidal forces and geodesic deviation of test particles near a static, axially symmetric compact object, revealing deformation-dependent stretching or compression effects.

Contribution

It provides a detailed analysis of geodesic deviation in the $q$-metric, highlighting the influence of quadrupole deformation on tidal forces compared to Schwarzschild spacetime.

Findings

Deformation parameter $q$ alters geodesic deviation behavior.

Deviation vectors can stretch or compress depending on $q$ and $ heta$.

Behavior near the singularity at $r=2m$ is significantly affected.

Abstract

We consider the tidal forces between test particles falling along geodesics in the exterior spacetime generated by a static and axially symmetric compact matter source with non-vanishing mass quadrupole. Specifically, we analyze the radial and angular geodesic deviation, compare it with that of the Schwarzschild spacetime, and investigate the impact of the deformation parameter $q$, at different polar angles $\theta$ with respect to the vertical symmetry axis. Furthermore, we examine the geodesic deviation for the case of non-constant $\theta$ during the radial fall. It is shown that the presence of the deformation parameter affects the behavior of the geodesic deviation vectors, depending on its value. In particular, we observe that for arbitrary values of $q$ and $\theta$ the behavior of the deviation vector differs as it approaches the singularity at $r = 2m$. Above all, we can witness either stretching or compressing of the deviation vector for various combinations of $q$ and $\theta$. These findings provide insight into the effects of quadrupole deformation on the motion of test particles in the vicinity of the central object.