  We discuss several explicitly causal hyperbolic formulations of Einstein's
dynamical 3+1 equations in a coherent way, emphasizing throughout the
fundamental role of the ``slicing function,'' $\alpha$---the quantity that
relates the lapse $N$ to the determinant of the spatial metric $\bar{g}$
through $N = \bar{g}^{1/2} \alpha$. The slicing function allows us to
demonstrate explicitly that every foliation of spacetime by spatial time-slices
can be used in conjunction with the causal hyperbolic forms of the dynamical
Einstein equations. Specifically, the slicing function plays an essential role
(1) in a clearer form of the canonical action principle and Hamiltonian
dynamics for gravity and leads to a recasting (2) of the Bianchi identities
$\nabla_\beta G^\beta\mathstrut_\alpha \equiv 0$ as a well-posed system for the
evolution of the gravitational constraints in vacuum, and also (3) of
$\nabla_\beta T^\beta\mathstrut_\alpha \equiv 0$ as a well-posed system for
evolution of the energy and momentum components of the stress tensor in the
presence of matter, (4) in an explicit rendering of four hyperbolic
formulations of Einstein's equations with only physical characteristics, and
(5) in providing guidance to a new ``conformal thin sandwich'' form of the
initial value constraints.
