On General Plane Fronted Waves. Geodesics

A general class of Lorentzian metrics, $\mo \times \R^2$, $\langle\cdot,\cdot\rangle_z = \langle\cdot,\cdot\rangle_x + 2\ du\ dv + H(x,u)\ du^2$, with $(\mo, \langle\cdot,\cdot\rangle_x)$ 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.