Minimally-coupled Quasilinear Symmetric Hyperbolic Systems, Naturally

Abstract

We consider symmetric hyperbolicity and well-posedness in the context of natural equations, which are closed subobjects of jet bundles over natural bundles. We show how non-natural equations can be "naturalized", and we discuss how a precise version of "minimal coupling" offers a (provably unique, up to isomorphism) method for constructing natural equations from certain equations defined on a single manifold. We discuss how to define symmetrizers, hyperbolizations, and constraints for minimally coupled natural equations, and we show how these can be used to recover standard existence and uniqueness results, relative to any choice of background structure (a metric, for minimally coupled systems). Additionally, we show that minimal coupling and naturality together imply that the causal cones associated with a (first order, quasilinear, minimally coupled) naturally symmetric hyperbolic natural equation must always coincide with the metric lightcones, and that a certain rank-2 tensor, associates with a natural hyperbolization, must always be divergence free relative to the associated Levi-Civita derivative operator.