First-order symmetric hyperbolic Einstein equations with arbitrary fixed gauge

We find a one-parameter family of variables which recast the 3 + 1 Einstein equations into first-order symmetric hyperbolic form for any fixed choice of gauge. Hyperbolicity considerations lead us to a redefinition of the lapse in terms of an arbitrary factor times a power of the determinant of the 3-metric; under certain assumptions, the exponent can be chosen arbitrarily, but positive, with no implication of gauge fixing.