On General Plane Fronted Waves. Geodesics

A general class of Lorentzian metrics, M0 ×R, 〈·, ·〉z = 〈·, ·〉x + 2 du dv + H (x, u) du2, with (M0, 〈·, ·〉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.