Einstein and Yang-Mills Theories in Hyperbolic Form without Gauge Fixing

The evolution of physical and gauge degrees of freedom in the Einstein and Yang-Mills theories are separated in a gauge-invariant manner. We show that the equations of motion of these theories can be written in flux-conservative first-order symmetric hyperbolic form where the only nonzero characteristic speed is that of light. This dynamical form is ideal for global analysis, analytic approximation methods such as gauge-invariant perturbation theory, and numerical solution.