Kinematics of geodesic flows in stringy black hole backgrounds

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) characterizing such geodesic flows are written down and subsequently solved analytically (in two dimensions) and numerically (in four dimensions) for specific geodesic 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 caustic formation and geodesic focusing are obtained from the analytical and numerical findings. We also propose a new quantifier in terms of the time (affine parameter) of approach to a singularity, 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.