For this purpose, in our four previous papers [8-10, 16] we studied a quasilocal family of "Evolving Null Horizons" (ENHs) of (1 + n)-splitting space-time, explained the reason for such a study, constructed a variety of examples of ENHs, and in some cases established their relation with black hole isolated horizons introduced by Ashtekar et al.
In this paper, for the first time in the literature we have studied the existence of a family of time-dependent ENHs in GRW space-time which, in general, is not (1 + n)-splitting space-time.
Also, observe that in this paper we have only related ENHs to black hole event horizons represented by a Killing horizon, whereas in our previous works [8-10, 16] we related ENHs to one of the three types of isolated horizons which, unlike event horizons, may not be represented by a Killing horizon.
For this reason, one may try different approach to show the existence of a family of ENHs of a prescribed class of GRW space-time which relate to isolated horizons.
A null hypersurface (H, h, l) of the family F of space-time ([bar.M], [bar.g]) is called an Evolving Null Horizon, briefly denoted by ENH, if
Moreover, it is easy to see that condition (b) of Theorem 1 will hold for an ENH. This clearly shows that there exists a "Physical Model" of a class F = (([H.sub.u]), ([h.sub.u]), ([l.sub.u])) of a family of totally umbilical null hypersurfaces of the space-time ([bar.M], [bar.g]), satisfying the hypothesis and three conclusions of Theorem 1, such that each of its members is an Evolving Null Horizon (ENH) which may evolve into a fixed null hypersurface (H, [gamma]) with zero expansion, [[theta].sub.(l)] = 0.