COLLISIONS OF PARTICLES IN LOCALLY ADS SPACETIMES

Abstract. We investigate 3-dimensional globally hyperbolic AdS manifolds containing “particles”, i.e., cone singularities along a graph Γ. We impose physically relevant conditions on the cone singularities, e.g. positivity of mass (angle less than 2pi on time-like singular segments). We construct examples of such manifolds, describe the cone singularities that can arise and the way they can interact (the local geometry near the vertices of Γ). The local geometry near an “interaction point ” (a vertex of the singular locus) has a simple geometric description in terms of polyhedra in the extension of hyperbolic 3-space by the de Sitter space. We then concentrate on spaces containing only (interacting) massive particles. To each such space we associate a graph and a finite family of pairs of hyperbolic surfaces with cone singularities. We show that this data is sufficient to recover the space locally (i.e., in the neighborhood of a fixed metric). This is a partia