Can gravity generate the time arrow?

Gravitational force is known, in some conditions, to be more intense than any other force leading to formation of black holes. The characteristic feature of the latter is the so-called T-regions where the radial coordinate of a distant observer becomes time-like. This implies that the inversion of this coordinate, instead of trivially meaning the spherical symmetry of the reference frame, is, in fact, inversion of time, and, hence, solutions of the Einstein equations does not have to be invariant under such transformation.