  We find a choice of variables for the 3+1 formulation of general relativity
which casts the evolution equations into (flux-conservative)
symmetric-hyperbolic first order form for arbitrary lapse and shift, for the
first time. We redefine the lapse function in terms of the determinant of the
3-metric and a free function U which embodies the lapse freedom. By rescaling
the variables with appropriate factors of 1/c, the system is shown to have a
smooth Newtonian limit when the redefined lapse U and the shift are fixed by
means of elliptic equations to be satisfied on each time slice. We give a
prescription for the choice of appropriate initial data with controlled
extra-radiation content, based on the theory of problems with different
time-scales. Our results are local, in the sense that we are not concerned with
the treatment of asymptotic regions. On the other hand, this local theory is
all what is needed for most problems of practical numerical computation.
