Totally geodesic null hypersurfaces have been widely used in the study of isolated black holes. In this paper, we introduce a new quasilocal notion of a family of totally umbilical null hypersurfaces called evolving null horizons (ENH) of a dynamical spacetime, satisfied under an appropriate energy condition. We focus on a variety of examples of ENHs and in some cases establish their relation with event and isolated horizons. We also present two specific physical models of an ENH in a black hole spacetime. Beside the examples, for further study we propose two open problems on possible general existence of an ENH in a black hole spacetime and its canonical or unique existence. The results of this paper have ample scope of working on totally umbilical null hypersurfaces of Lorentzian and, in general, semiRiemannian manifolds. 1. Introduction It is well-known that null hypersurfaces play an important roll in the study of black hole horizons. A black hole is a region of spacetime which contains a huge amount of mass compacted into an extremely small volume. Shortly after Einstein's first version of the theory of gravitation that was published in , in , Karl Schwarzschild computed the gravitational fields of stars using Einstein's field equations. He assumed that the star is spherical, gravitationally collapsed, and nonrotating. His solution is called a Schwarzschild solution which is an exact solution of static vacuum fields of the point-mass. Since then, considerable work has been done on black hole physics of asymptotically flat and time-independent spacetimes. Such isolated black holes deal with the following concepts of event and isolated horizons. 1.1. Event Horizons A boundary of a spacetime is called an event horizon, briefly denoted by EH, beyond which events cannot affect the observer. Note that an event horizon is intrinsically a global concept since its definition requires the knowledge of the whole spacetime to determine whether null geodesics can reach null infinity. EHs have played a key role and this includes Hawking's area increase theorem, black hole thermodynamics, black hole perturbation theory, and the topological censorship results. The most important family is the Kerr-Newman black holes. Moreover, an EH always exists in black hole asymptotically flat spacetime under a weak cosmic censorship condition. We refer Hawking’s paper on “event horizon” [1], three papers of Hájí？ek’s work [2–4] on “perfect horizons” (later called “nonexpanding horizons” by Ashtekar et al. [5]), and a paper by Galloway [6] in which he has shown that the null
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