Diffeomorphism Invariant Actions for Partial Systems

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 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 outside the region R completely undetermined, and iii) can be constructed without first solving the original equations of motion. Typeset using REVTEX 1