  We propose a new formulation for 3+1 numerical relativity, based on a
constrained scheme and a generalization of Dirac gauge to spherical
coordinates. This is made possible thanks to the introduction of a flat
3-metric on the spatial hypersurfaces t=const, which corresponds to the
asymptotic structure of the physical 3-metric induced by the spacetime metric.
Thanks to the joint use of Dirac gauge, maximal slicing and spherical
components of tensor fields, the ten Einstein equations are reduced to a system
of five quasi-linear elliptic equations (including the Hamiltonian and momentum
constraints) coupled to two quasi-linear scalar wave equations. The remaining
three degrees of freedom are fixed by the Dirac gauge. Indeed this gauge allows
a direct computation of the spherical components of the conformal metric from
the two scalar potentials which obey the wave equations. We present some
numerical evolution of 3-D gravitational wave spacetimes which demonstrates the
stability of the proposed scheme.
