Truncation of Einstein equations through Gravitational Foliation

Abstract

In previous works, we suggested considering a (3+1)D quantum gravitational field as an evolution of a (2+1)D renormalized quantum gravitational field along the direction of the gravitational force. The starting point of the suggestion is a derivation of a unique hypersurface which looks effectively like (2+1)D from the point of view of Einstein equations in (3+1)D. In this paper, we derive such unique hypersurfaces for different kinds of stationary spherical metrics. We find that these hypersurfaces exist whenever all the components of the gravitational force field vanish on the hypersurface. We discuss the implication of this result and the necessary further work.