On General Plane Fronted Waves. Geodesics

Details On General Plane Fronted Waves. Geodesics On General Plane Fronted Waves. Geodesics

2002-11-05

arxiv.org/abs/gr-qc/0211017v2

Gen.Rel.Grav. 35 (2003) 631-649

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

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

Scientific paper

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

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