Intersecting hypersurfaces in AdS and Lovelock gravity

TL;DR

This paper explores the behavior of intersecting hypersurfaces in Lovelock gravity within AdS and dS backgrounds, revealing unique intersection configurations, matter requirements at collisions, and conditions for singularity-free limits.

Contribution

It extends the understanding of hypersurface intersections in Lovelock gravity, highlighting new configurations and the impact of energy conditions on collision solutions.

Findings

Higher co-dimension membranes can intersect at co-dimension 1 hypersurfaces.

Collision solutions require matter at the intersection, often conflicting with energy conditions.

Infinite intersections at the boundary lead to singularity-free spacetimes.

Abstract

Colliding and intersecting hypersurfaces filled with matter (membranes) are studied in the Lovelock higher order curvature theory of gravity. Lovelock terms couple hypersurfaces of different dimensionalities, extending the range of possible intersection configurations. We restrict the study to constant curvature membranes in constant curvature AdS and dS background and consider their general intersections. This illustrates some key features which make the theory different to the Einstein gravity. Higher co-dimension membranes may lie at the intersection of co-dimension 1 hypersurfaces in Lovelock gravity; the hypersurfaces are located at the discontinuities of the first derivative of the metric, and they need not carry matter. The example of colliding membranes shows that general solutions can only be supported by (spacelike) matter at the collision surface, thus naturally conflicting with the dominant energy condition (DEC). The imposition of the DEC gives selection rules on the types of collision allowed. When the hypersurfaces don't carry matter, one gets a soliton-like configuration. Then, at the intersection one has a co-dimension 2 or higher membrane standing alone in AdS-vacuum spacetime \emph{without conical singularities.} Another result is that if the number of intersecting hypersurfaces goes to infinity the limiting spacetime is free of curvature singularities if the intersection is put at the boundary of each AdS bulk.