Intersecting hypersurfaces, topological densities and Lovelock Gravity

TL;DR

This paper develops a geometric framework for Lovelock gravity involving intersecting hypersurfaces, using a multi-parameter connection approach to handle discontinuities and intersection terms in the action.

Contribution

It introduces a novel geometric method with multi-parameter connections to systematically include intersection terms in Lovelock gravity actions.

Findings

Efficiently handles space-time division into honeycomb networks.

Generalizes Lovelock Lagrangian to simplices for intersection contributions.

Shows matter can be naturally supported at intersections without singularities.

Abstract

Intersecting hypersurfaces in classical Lovelock gravity are studied exploiting the description of the Lovelock Lagrangian as a sum of dimensionally continued Euler densities. We wish to present an interesting geometrical approach to the problem. The analysis allows us to deal most efficiently with the division of space-time into a honeycomb network of cells produced by an arbitrary arrangement of membranes of matter. We write the gravitational action as bulk terms plus integrals over each lower dimensional intersection. The spin connection is discontinuous at the shared boundaries of the cells, which are spaces of various dimensionalities. That means that at each intersection there are more than one spin connections. We introduce a multi-parameter family of connections which interpolate between the different connections at each intersection. The parameters live naturally on a simplex. We can then write the action including all the intersection terms in a simple way. The Lagrangian of Lovelock gravity is generalized so as to live on the simplices as well. Each intersection term of the action is then obtained as an integral over an appropriate simplex. Lovelock gravity and the associated topological (Euler) density are used as an example of a more general formulation. In this example one finds that singular sources up to a certain co-dimensionality naturally carry matter without introducing conical or other singularities in spacetime geometry.