On General Plane Fronted Waves. Geodesics

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

Einstein Spaces with a Conformal Group

“…A particular class is the class of polarized exact plane waves with a potential H(u, x, y) = h(u)(x 2 − y 2 ) [7]. Isometric, homothetic and conformal vector fields of pp-waves were classified in a kind of a recursive normal form in [39], starting from the possible Killing fields.…”

On the Geometry of pp-Wave Type Spacetimes

“…In this ambient, a name as generalized ppwave or "M -fronted wave with parallel rays" (M p-wave) seems natural for our spacetimes (1.1). Nevertheless, we will maintain the name PFW (plane fronted wave) in agreement with previous references [17,26] or the nomenclature in [4], but with no further pretension.…”

“…The authors, in collaboration with A.M. Candela [17], considered the following class of spacetimes, say PFW, which essentially include classical pp-waves (and, thus, plane waves):…”

“…In this framework, our goals in [17,26] were, essentially: (i) to introduce the class of reasonably generic waves (1.1), (ii) to justify that, for a physically reasonable asymptotic behaviour of the wave, |H(·, u)| must be "subquadratic" (plane waves lie in the limit quadratic case), and (iii) to show that, in this case, the geometry of the wave presents good global properties: global hyperbolicity [26] and geodesic connectedness [17]. Even more, the unstability of these geometric properties in the quadratic case implied interesting questions in Riemannian Geometry, studied in [15].…”

“…In the present article, we explain the role of the mathematical tools introduced in [17,15,26] in relation to both, classical problems on waves as [39,21], and posterior developments [34,31,33,32,35,45]. The proofs are referred to the original articles, or sketched in the case of further results.…”