Geodesic flows in rotating black hole backgrounds

We study the kinematics of timelike geodesic congruences, in the spacetime geometry of rotating black holes in three [the Bañados-Teitelboim-Zanelli (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 Bañados-Teitelboim-Zanelli 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 singularity (defocusing/focusing) is studied. While the role of ${\omega }_0$ on the time to singularity is as expected, the effect of $a$ leads to an interesting new result.

Figure 1

Effect of $\Lambda$ on the evolution of the expansion scalar $\theta (\lambda )$ for the cases (a) without initial rotation and (b) with initial rotation (${\omega }_0=0.2$).

Figure 2

Orbits around the BTZ black hole without initial rotation for (a) with horizon ($\Lambda =-0.4$) and (b) without horizon ($\Lambda =-0.45$). The dotted and dashed lines indicate, respectively, the outer $(r_{+})$ and inner $(r_{-})$ horizons in the left panel above.

Figure 3

Orbits around the BTZ black hole with initial rotation (${\omega }_0=0.2$) for (a) with horizon ($\Lambda =-0.4$) and (b) without horizon ($\Lambda =-0.45$).

Figure 4

Defocusing (for ${\theta }_0>{\theta }_c$) for the cases (a) without initial rotation and (b) with initial rotation (${\omega }_0=0.1$). The effect of (c) angular momentum and (d) initial rotation on time of approach to singularity (${\lambda }_s$). The initial shear in all cases is zero.

Figure 5

Focusing (for ${\theta }_0<{\theta }_c$) for the cases (a) without initial rotation and (b) with initial rotation $({\omega }_0=0.2)$. The effect of (c) angular momentum and (d) initial rotation on time of approach to singularity (${\lambda }_s$). The initial shear is ${\sigma }_0=0.28$.