Singular sources in gravity and homotopy in the space of connections
Journal
Journal of Mathematical Physics
Date Issued
December 2009
Author(s)
DOI
10.1063/1.3250196
Abstract
Suppose a Lagrangian is constructed from its fields and their derivatives.
When the field configuration is a distribution, it is unambiguously defined as
the limit of a sequence of smooth fields. The Lagrangian may or may not be a
distribution, depending on whether there is some undefined product of
distributions. Supposing that the Lagrangian is a distribution, it is
unambiguously defined as the limit of a sequence of Lagrangians. But there
still remains the question: Is the distributional Lagrangian uniquely defined
by the limiting process for the fields themselves? In this paper a general
geometrical construction is advanced to address this question. We describe
certain types of singularities, not by distribution valued tensors, but by
showing that the action functional for the singular fields is (formally)
equivalent to another action built out of \emph{smooth} fields. Thus we manage
to make the problem of the lack of a derivative disappear from a system which
gives differential equations. Certain ideas from homotopy and homology theory
turn out to be of central importance in analyzing the problem and clarifying
finer aspects of it.
The method is applied to general relativity in first order formalism, which
gives some interesting insights into distributional geometries in that theory.
Then more general gravitational Lagrangians in first order formalism are
considered such as Lovelock terms (for which the action principle admits
space-times more singular than other higher curvature theories).
When the field configuration is a distribution, it is unambiguously defined as
the limit of a sequence of smooth fields. The Lagrangian may or may not be a
distribution, depending on whether there is some undefined product of
distributions. Supposing that the Lagrangian is a distribution, it is
unambiguously defined as the limit of a sequence of Lagrangians. But there
still remains the question: Is the distributional Lagrangian uniquely defined
by the limiting process for the fields themselves? In this paper a general
geometrical construction is advanced to address this question. We describe
certain types of singularities, not by distribution valued tensors, but by
showing that the action functional for the singular fields is (formally)
equivalent to another action built out of \emph{smooth} fields. Thus we manage
to make the problem of the lack of a derivative disappear from a system which
gives differential equations. Certain ideas from homotopy and homology theory
turn out to be of central importance in analyzing the problem and clarifying
finer aspects of it.
The method is applied to general relativity in first order formalism, which
gives some interesting insights into distributional geometries in that theory.
Then more general gravitational Lagrangians in first order formalism are
considered such as Lovelock terms (for which the action principle admits
space-times more singular than other higher curvature theories).