Anirvan DasGupta, Hemwati Nandan, Sayan Kar
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.
View original:
http://arxiv.org/abs/1202.5370
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