6 Applications in Mathematical Physics

For simplicity, in Section 4 we restricted our review of the laws of black hole mechanics to the Einstein–Maxwell theory. However, there is a large body of literature on black holes in more general theories with dilatonic, Yang–Mills, Higgs, Proca, and Skyrme fields. These fields are not expected to be physically significant in the macroscopic, astrophysical world. However, they are of considerable interest from a mathematical physics perspective because their inclusion brings about qualitative, structural changes in the theory. The most dramatic of these is that the uniqueness theorems that play a central role in the Einstein–Maxwell theory are no longer valid. Consequently, the structure of these ‘colored’ or ‘hairy’ black-holes is much more complicated than those in the Einstein–Maxwell theory, and most of the work in this area has been carried out through a combination of analytic and numerical methods. By now a very large body of facts about stationary black holes with hair has accumulated. A major challenge is to unify this knowledge through a few, general principles.

The isolated horizon framework has provided surprising insights into the properties of hairy black holes in equilibrium [18, 76, 74, 25, 20, 75]. While the zeroth and first laws go through in a straightforward manner, the notion of the horizon mass now becomes much more subtle and its properties have interesting consequences. The framework also suggests a new phenomenological model of colored black holes as bound states of ordinary, uncolored black holes and solitons. This model successfully explains the qualitative behavior of these black holes, including their stability and instability, and provides unexpected quantitative relations between colored black holes and their solitonic analogs.

In these theories, matter fields are minimally coupled to gravity. If one allows non-minimal couplings, the first law itself is modified in a striking fashion: Entropy is no longer given by the horizon area but depends also on the matter fields. For globally stationary space-times admitting bifurcate Killing horizons, this result was first established by Jacobson, Kang, and Myers [124], and by Iyer and Wald [183, 123] for a general class of theories. For scalar fields non-minimally coupled to gravity, it has been generalized in the setting of Type II WIHs [21]. While the procedure does involve certain technical subtleties, the overall strategy is identical to that summarized in Section 4.1. Therefore we will not review this issue in detail.

This section is divided into two parts. In the first, we discuss the mechanics of weakly isolated horizons in presence of dilatons and Yang–Mills fields. In the second, we discuss three applications. This entire discussion is in the framework of isolated horizons because the effects of these fields on black hole dynamics remain largely unexplored.