On General Plane Fronted Waves. Geodesics

Abstract

A general class of Lorentzian metrics, $M_{0}x{R}^2$, $d{s}^2=<.,.>+2dudv+H(x,u)d{u}^2$, 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

Final version with minor errata corrected. 19 pages, Latex. To appear in Gen. Relat. Gravit. (2003)

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• The gravitational wave rocket
• Geodesic connectedness of semi-Riemannian manifolds
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