  The evolution equations of Einstein's theory and of Maxwell's theory---the
latter used as a simple model to illustrate the former--- are written in gauge
covariant first order symmetric hyperbolic form with only physically natural
characteristic directions and speeds for the dynamical variables. Quantities
representing gauge degrees of freedom [the spatial shift vector
$\beta^{i}(t,x^{j})$ and the spatial scalar potential $\phi(t,x^{j})$,
respectively] are not among the dynamical variables: the gauge and the physical
quantities in the evolution equations are effectively decoupled. For example,
the gauge quantities could be obtained as functions of $(t,x^{j})$ from
subsidiary equations that are not part of the evolution equations. Propagation
of certain (``radiative'') dynamical variables along the physical light cone is
gauge invariant while the remaining dynamical variables are dragged along the
axes orthogonal to the spacelike time slices by the propagating variables. We
obtain these results by $(1)$ taking a further time derivative of the equation
of motion of the canonical momentum, and $(2)$ adding a covariant spatial
derivative of the momentum constraints of general relativity (Lagrange
multiplier $\beta^{i}$) or of the Gauss's law constraint of electromagnetism
(Lagrange multiplier $\phi$). General relativity also requires a harmonic time
slicing condition or a specific generalization of it that brings in the
Hamiltonian constraint when we pass to first order symmetric form. The
dynamically propagating gravity fields straightforwardly determine the
``electric'' or ``tidal'' parts of the Riemann tensor.
