Hemwati Nandan

Confinement of test particles in warped spacetimes

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.

KINEMATICS OF FLOWS ON CURVED, DEFORMABLE MEDIA

Kinematics of geodesic flows on specific, two-dimensional, curved surfaces (the sphere, hyperbolic space and the torus) are investigated by explicitly solving the evolution (Raychaudhuri) equations for the expansion, shear and rotation, for a variety of initial conditions. For flows on the sphere and on hyperbolic space, we show the existence of singular (within a finite value of the time parameter) as well as non-singular solutions. We illustrate our results through a phase diagram which demonstrates under which initial conditions (or combinations thereof) we end up with a singularity in the congruence and when, if at all, we can obtain non-singular solutions for the kinematic variables. Our analysis portrays the differences which arise due to positive or negative curvature and also explores the role of rotation in controlling singular behavior. Subsequently, we move on to geodesic flows on two-dimensional spaces with varying curvature. As an example, we discuss flows on a torus. Characteristic oscillatory features, dependent on the ratio of the two radii of the torus, emerge in the solutions for the expansion, shear and rotation. Singular (within a finite time) and non-singular behavior of the solutions are also discussed. Finally, we conclude with a generalization to three-dimensional spaces of constant curvature, a summary of some of the generic features obtained and a comparison of our results with those for flows in flat space.

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.

Kinematics of deformable media

Abstract 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 a flat and curved spacetimes.

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 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 (

Null geodesics and observables around the Kerr–Sen black hole

theta

Geodesic motion in R-charged black hole spacetimes

), for congruences with and without an initial rotation (

Bending angle of light in equatorial plane of Kerr–Sen Black Hole

omega_0

Null geodesics and red–blue shifts of photons emitted from geodesic particles around a noncommutative black hole space–time

) 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,

Confinement and focusing of geodesics in warped spacetimes

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