On general plane fronted waves.: Geodesics

A general class of Lorentzian metrics, M-0 x R-2, [., .](x) + 2 du dv + H(x, u) du(2), with (M-0, [., .](x)) any Riemannian manifold, is introduced in order to generalize classical exact plane fronted waves. Here, we start a systematic study of their main geodesic properties: geodesic completeness, geodesic connectedness and multiplicity causal character of connecting geodesics. These results are independent of the possibility of a full integration of geodesic equations. Variational and geometrical techniques are applied systematically. In particular, we prove that the asymptotic behavior of H(x, u) with x at infinity determines many properties of geodesics. Essentially, a subquadratic growth of H ensures geodesic completeness and connectedness, while the critical situation appears when H(x, u) behaves in some direction as \x\(2), as in the classical model of exact gravitational waves.