Radiating Levi-Civita metric

Abstract

A generalization of Levi-Civita's static solution for the exterior gravitational field of a cylinder of infinite length and finite cross section, corresponding to the field equations Tab=qkakb, kaka=0, is discussed. The expression for q is of the form f'(u)/ rho , which shows the typical cylindrical fall-off over the null hypersurface u=constant. It is pointed out that the passage of an outgoing wave affects permanently the static Levi-Civita space-time and the energy for this class of solutions is a particular case of the C-energy defined by Thorne. The geodesic equations for a test particle moving in the radial direction display a gravitational 'induction field' which is associated with a changing mass in the Newtonian field and is directed towards the axis of the cylinder. In contrast with the spherically symmetric case the induction field always acts to decrease the energy of a test particle on which it acts. It is shown that the mass per unit length of the static cylinder strongly affects the shear and divergence of the null congruence. The asymptotic behaviour of the Weyl tensor is analysed and a peeling theorem proved for this case.