Geodesic flows in rotating black hole backgrounds

TL;DR

This paper analyzes the behavior of geodesic flows in rotating black hole spacetimes, deriving and solving evolution equations for expansion, shear, and rotation, revealing how black hole parameters influence focusing and defocusing of geodesics.

Contribution

It provides analytical solutions for BTZ black holes and numerical analysis for Kerr black holes, highlighting the effects of rotation and initial conditions on geodesic evolution.

Findings

Analytical solutions for BTZ geodesic equations.

Existence of a critical initial expansion in Kerr black holes.

Black hole spin parameter influences geodesic focusing times.

Abstract

We study the kinematics of timelike geodesic congruences, in the spacetime geometry of rotating black holes in three (the BTZ) and four (the Kerr) dimensions. The evolution (Raychaudhuri) equations for the expansion, shear and rotation along geodesic flows in such spacetimes are obtained. For the BTZ case, the equations are solved analytically. The effect of the negative cosmological constant on the evolution of the expansion ($\theta$), for congruences with and without an initial rotation ($\omega_0$) is noted. Subsequently, the evolution equations, in the case of a Kerr black hole in four dimensions are written and solved numerically, for some specific geodesics flows. It turns out that, for the Kerr black hole, there exists a critical value of the initial expansion below (above) which we have focusing (defocusing). We delineate the dependencies of the expansion, on the black hole angular momentum parameter, $a$, as well as on $\omega_0$. Further, the role of $a$ and $\omega_0$ on the time (affine parameter) of approach to a singularity (defocusing/focusing) is studied. While the role of $\omega_0$ on this time of approach is as expected, the effect of $a$ leads to an interesting new result.