On General Plane Fronted Waves. Geodesics

Abstract

A general class of Lorentzian metrics, $\mathcal{M}_0 \times \mathbb{R}^2$, $\langle \cdot ,\; \cdot \rangle _z = \langle \cdot ,\; \cdot \rangle _x + 2\;\;du\;\;dv + H(x,u)\;du^2$, with $(\mathcal{M}_0 ,\;\langle \cdot ,\; \cdot \rangle _x )$ 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 $|{\kern 1pt} x{\kern 1pt} |^2$, as in the classical model of exact gravitational waves.