On the geometry of pp-wave type spacetimes

Abstract

Global geometric properties of product manifolds $\mathcal{M}=M\times {\mathbb{R}}^2$, endowed with a metric type $<\cdot ,\cdot >=<\cdot ,\cdot {>}_R+2dudv+H(x,u)d{u}^2$ (where $<\cdot ,\cdot {>}_R$ is a Riemannian metric on $M$ and $H:M\times \mathbb{R}\to \mathbb{R}$ 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.

19 pages, Proceedings of the March-2004 Heraeus Seminar "Mathematical Relativity: New ideas and developments"

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• Completeness of first and second order ODE flows and of Euler-Lagrange equations
• The globally hyperbolic metric splitting for non-smooth wave-type space-times
• Twistor initial data characterisation of pp-waves
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