On the Geometry of PP-Wave Type Spacetimes

Global geometric properties of product manifolds M = M × R2, endowed with a metric type 〈·, ·〉 = 〈·, ·〉R + 2dudv + H(x, u)du 2 (where 〈·, ·〉R is a Riemannian metric on M and H : M × R → R a function), which generalize classical plane waves, are revisited. Our study covers causality (causal ladder, inexistence of horizons), geodesic completeness, geodesic connectedness and existence of conjugate points. Appropiate mathematical tools for each problem are emphasized and the necessity to improve several Riemannian (positive definite) results is claimed. The behaviour of H(x, u) for large spatial component x becomes essential, being a spatial quadratic behaviour critical for many geometrical properties. In particular, when M is complete, if −H(x, u) is spatially subquadratic, the spacetime becomes globally hyperbolic and geodesically connected. But if a quadratic behaviour is allowed (as happen in plane waves) then both global hyperbolicity and geodesic connectedness maybe lost. From the viewpoint of classical General Relativity, the properties which remain true under generic hypotheses on M (as subquadraticity for H) become meaningful. Natural assumptions on the wave -finiteness or asymptotic flatness of the frontimply the spatial subquadratic behaviour of |H(x, u)| and, thus, strong results for the geometry of the wave. These results not always hold for plane waves, which appear as an idealized non-generic limit case.