Geodesics in some exact rotating solutions of Einstein's equations

Abstract

In examining some exact solutions of Einstein's field equations, the main approach used here is to study the geodesic motion of light, and sometimes test particles. Difficulties in solving the geodesic equations are avoided by using computer algebra to solve the equations numerically and to plot them in two- or three-dimensional diagrams. Interesting features revealed by these diagrams may then be investigated analytically. Application of this technique to the van Stockum solution for a rotating dust cylinder and to Bonnor's rotating dust cloud seems to reveal different constraints on the spatial distribution of geodesics with different parameters. Analysis then continns that, in the highest mass van Stockum case, null geodesics in the vacuum exterior are radially confined according to their initial conditions. Null geodesics plotted in Bonnor's dust cloud seem to be repelled before they can reach the centre. Although there is no event horizon, analysis reveals a central region which cannot be penetrated by light from spatial infinity and from which light cannot escape to spatial infinity. The gravitomagnetic clock effect is studied in van Stockum spacetime. The effect is found to be frame dependent and can be reduced to zero by a suitable coordinate transformation.