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.Comment: Sixteen pages, five figure

We investigate the kinematics of deformations in two and three dimensional media by explicitly solving (analytically) the evolution equations (Raychaudhuri equations) for the expansion, shear and rotation associated with the deformations. The analytical solutions allow us to study the dependence of the kinematical quantities on initial conditions. In particular, we are able to identify regions of the space of initial conditions that lead to a singularity in finite time. Some generic features of the deformations are also discussed in detail. We conclude by indicating the feasibility and utility of a similar exercise for fluid and geodesic flows in flat and curved spacetimes.

We study the kinematics of timelike geodesic congruences in two and four dimensions in spacetime geometries representing stringy black holes. The Raychaudhuri equations for the kinematical quantities (namely, expansion, shear and rotation) characterising such geodesic flows are written down and subsequently solved analytically (in two dimensions) and numerically (in four dimensions) for specific geodesics flows. We compare between geodesic flows in dual (electric and magnetic) stringy black hole backgrounds in four dimensions, by showing the differences that arise in the corresponding evolutions of the kinematic variables. The crucial role of initial conditions and the spacetime curvature on the evolution of the kinematical variables is illustrated. Some novel general conclusions on geodesic focusing are obtained from the analytical and numerical findings. We also propose new quantifiers in terms of (a) the time (affine parameter) of approach to a singularity and (b) the location of extrema in the functional evolution of the kinematic variables, which may be used to distinguish between flows in different geometries. In summary, our quantitative findings bring out hitherto unknown features of the kinematics of geodesic flows, which, otherwise, would have remained overlooked, if we confined ourselves to only a qualitative analysis.Comment: Revised with several changes, 24 pages, 7 figure

We investigate test particle trajectories in warped spacetimes with a thick brane warp factor, a cosmological on--brane line element and a time dependent extra dimension. The geodesic equations are reduced to a first order autonomous dynamical system. Using analytical methods, we arrive at some useful general conclusions regarding possible trajectories. Oscillatory motion, suggesting confinement about the location of the thick brane, arises for a growing warp factor. On the other hand, we find runaway trajectories (exponential-like) for a decaying warp factor. Variations of the extra dimensional scale factor yield certain quantitative differences. Results obtained from explicit numerical evaluations match well with the qualitative conclusions obtained from the dynamical systems analysis.Comment: 20 pages, 9 figures, title changed, contents added, presentation improved, results are largely unchange

Abstract. We investigate the geodesic motion in the background of Kerr-Sen Black Hole arising in the heterotic string theory. The nature of effective potential is discussed in radial as well as latitudinal direction. A special class of spherical photon orbits is obtained along with the expression for the turning point for radial photons. Dependence of photon motion within this class of solution is discussed explicitly in view of the different Black Hole parameters. We have discussed the allowed regions for geodesic motion of massless test particles around KerrSen Black Hole in more generalised way by including non-equatorial motion of the photons into the account. The conditions for different types of possible orbits are discussed with specific parameter values along with the corresponding orbit structure. No terminating orbits are possible for photons due to non-zero Black Hole charge. Observables on the angular plane (viz. bending of light and perihelion precession for massive test particles) are analysed as special cases. We have also calculated the rotation and mass parameters for Kerr-Sen Black Hole in terms of the red/blue shifts of the photons in circular and equatorial orbits emitted by the massive test particles which represent stars or other probable sources of photons.