Intersecting hypersurfaces, topological densities and Lovelock Gravity
Journal
Journal of Geometry and Physics
Date Issued
August 2007
Author(s)
DOI
10.1016/j.geomphys.2007.03.005
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.
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.