On the Geometry of pp-Wave Type Spacetimes

Abstract

Abstract.Global geometric properties of product manifolds M = M × R 2 , 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 front-imply 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.

Geodesics in nonexpanding impulsive gravitational waves with Λ. II

“…In our present analysis we aim at establishing geodesic completeness of the impulsive wave spacetime (M, g) with generalized metric g. So we have to prove that all solutions of the geodesic equations (14) are global. In this section we will prove global existence and uniqueness of solutions given any initial data that forces them to run into the impulsive wave.…”

“…with data constructed from the seed family with data (17), see (24). This net hence constitutes a solution candidate for the system (14) of geodesic equations in the generalized spacetime (M, g) with data constructed from the seed data (17) and defined on the interval J ε = [α ε , α ε + η]. First, for the purpose of constructing a local solution candidate we assume the data (17) to be constant in ε and to be given by (20).…”

The Ehlers-Kundt conjecture about gravitational waves and dynamical systems

“…Remark 3.5. In the case that the polynomial V is homogeneous, the inequalities (17)- (20) with δ = 0 and ǫ = 0 are enough in order to prove the (sharper) result stated in Remark 3.2. In fact, in contrast with the non-homogeneous case explained in the previous remark, the case ǫ = 0 can be proved easily in the homogeneous case, as lower order terms of the polynomial do not exist.…”

“…Finally, chosen 0 < θ − < θ + < π/(2n), one can find A > 0, ρ 0 > 1 (bigger than any prescribed constant) 8 such that all the previous inequalities hold for ρ ≥ ρ 0 , by replacing ǫ and δ by A/ρ and 1/ρ, resp., in (17)- (20).…”

“…To get (20) repeat the process above by replacing all the bounds involving θ 1 by analogous ones with −θ 1 . Then, obtain new values of ǫ 1 (< ǫ), ρ 1 (ǫ 1 ), ρ 2 (ǫ 1 ), and choose some ρ 0 bigger (resp.…”