On General Plane Fronted Waves. Geodesics

TL;DR

This paper introduces a broad class of Lorentzian metrics generalizing classical plane fronted waves and systematically studies their geodesic properties using variational and geometric methods, highlighting the influence of the asymptotic behavior of H.

Contribution

It extends the analysis of geodesic properties to a general class of Lorentzian metrics, independent of explicit solutions, emphasizing the role of asymptotic behavior of H.

Findings

Subquadratic growth of H ensures geodesic completeness and connectedness.

Critical behavior occurs when H grows quadratically, as in classical gravitational waves.

Asymptotic behavior of H at infinity determines key geodesic properties.

Abstract

A general class of Lorentzian metrics, $M_0 x R^2$, $ds^2 = <.,.> + 2 du dv + H(x,u) du^2$, with $(M_0, <.,.>$ 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