 Ashtekar et al. (1999, 2000) offer a different, related generalization of the idea of a black hole, based on what they call isolated horizons. This definition is somewhat more restrictive than Hayward's in so far as, as the name suggests, it requires that no stress-energy cross such a horizon. Subsequent work by Ashtekar and Krishnan (2003), Ashtekar (2007) and Hayward (2006, in OIR,  2009) clarified the relationship between the two, showing that the isolated horizon can be considered, in a sense, a special case of the trapping horizon. (See Hayward 2013 for a recent comprehensive review, and Faraoni 2013 for one with special attention to its relevance to cosmology.) For lack of a better term, we shall call black holes defined by trapping or isolated horizons “quasi-local black holes”. (‘Local’ because they are not global objects in the sense that black holes as tradionally defined are, and ‘quasi’ because they still can extend arbitrarily far throughout spacetime.) 