Diffeomorphism Invariant Actions for Partial Systems

TL;DR

This paper introduces a new class of diffeomorphism invariant actions for partial systems, allowing the study of localized regions in spacetime without boundary-preserving restrictions, and without solving original equations.

Contribution

It proposes actions invariant under all diffeomorphisms for regions, enabling localized analysis independent of boundary conditions and prior solutions.

Findings

Actions are invariant under all diffeomorphisms.

Dynamics outside the region remain undetermined.

Construction does not require solving original equations.

Abstract

Local action principles on a manifold $\M$ are invariant (if at all) only under diffeomorphisms that preserve the boundary of $\M$. Suppose, however, that we wish to study only part of a system described by such a principle; namely, the part that lies in a bounded region $R$ of spacetime where $R$ is specified in some diffeomorphism invariant manner. In this case, a description of the physics within $R$ should be invariant under {\it all} diffeomorphisms regardless of whether they preserve the boundary of this region. The following letter shows that physics in such a region can be described by an action principle that $i$) is invariant under both diffeomorphisms which preserve the boundary of $R$ and those that do not, $ii$) leaves the dynamics of the part of the system {\it outside} the region $R$ completely undetermined, and $iii$) can be constructed without first solving the original equations of motion.