Geometry of homothetic Killing trajectories and stationary limit surfaces

Publisher Website
Google Scholar
Abstract
We investigate the intrinsic geometry of timelike and null homothetic Killing trajectories. We do this through the use of two different Frenet–Serret formalisms developed by Synge and Bonnor. The curvature of such a nongeodesic timelike curve passing through a spacetime point is a direct measure of the expansion of the congruence at that point. Moreover, the rotation of this congruence is directly related to the torsions of its individual curves. For both classes of curves, timelike and null, the intrinsic scalars can be expressed as exponential functions of suitable parameters. We prove the following theorem twice, first by using the timelike formalism, and then through the null one. A homothetic stationary limit surface (where the homothetic Killing vector becomes null) is a null geodesic hypersurface if and only if the rotation of the homothetic Killing congruence vanishes on that hypersurface. The second, ’’null’’ approach allows us to derive this theorem directly on the hypersurface. Finally, we note that by setting a parameter equal to zero we recover all of the corresponding results for ordinary Killing vectors.

This publication has 4 references indexed in Scilit: