Geodesic flows in rotating black hole backgrounds

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 ($θ$), for congruences with and without an initial rotation ($ω_0$) 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 $ω_0$. Further, the role of $a$ and $ω_0$ on the time (affine parameter) of approach to a singularity (defocusing/focusing) is studied. While the role of $ω_0$ on this time of approach is as expected, the effect of $a$ leads to an interesting new result.

Sixteen pages, five figures