Hyperbolicity of Physical Theories with Application to General Relativity

We consider gauge theories from the free evolution point of view, in which initial data satisfying constraints of a theory are given. Because the constraints are compatible with the field equations they remain so. We study a model constrained Hamiltonian theory and identify a particular structure in the equations of motion which we call the standard gauge freedom. The pure gauge subsystem of this model theory is identified and the manner in which the gauge variables couple to the field equations is presented. We demonstrate that the set of gauge choices that can be coupled to the field equations to obtain a, properly defined, wave-like formulation is exactly the set of wave-like pure gauges. Consequently we analyze a parametrized family of formulations of general relativity. The generalization of the harmonic gauge formulation to a five parameter family of gauge conditions is obtained.