ar X iv : g r - qc / 0 21 10 17 v 2 5 M ar 2 00 3 On general Plane Fronted Waves . Geodesics
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چکیده
A general class of Lorentzian metrics, M0 × R 2 , ·, ·z = ·, ·x + 2 du dv + H(x, u) du 2 , with (M0, ·, ·x) any Riemannian mani-fold, 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 or 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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تاریخ انتشار 1983