Foliation of the schwarzschild and reissner-nordström space-times by flat space-like hypersurfaces

It is known that spherically symmetric space-times admit flat space-like foliations. In this paper a simple procedure for the complete foliation of the Schwarzschild space-time by flat space-like hypersurfaces is developed, using the fact that unforced geodesics are orthogonal to such hypersurfaces. The method is then extended to obtain a complete foliation of the Reissner-Nordström space-time by such hypersurfaces. In this case, as there is a barrier beyond which the geodesies do not go, a complete foliation is obtained by analytically continuing the hypersurfaces beyond the barrier. A duality between imaginary time-like geodesies and space-like hypersurfaces is noted, which also provides complete foliation - but the hypersurfaces are not flat. The features of our foliation are discussed.