Let us begin with Einstein–Yang–Mills theory considered in the last Section 6.1. As we saw, the ADM mass fails to be a good measure of the horizon mass for colored black holes. The failure of black hole uniqueness theorems also prevents the isolated horizon framework from providing a canonical notion of horizon mass on the full phase space. However, one can repeat the strategy used for dilatonic black holes to define horizon mass unambiguously for the static solutions [76, 25].
Consider a connected component of the known static solutions, labelled by . Using for the surface gravity of the properly normalized static Killing vector, in this sector one can construct a live vector field and obtain a first law. The energy is well-defined on the full phase space and can be naturally interpreted as the horizon mass for colored black holes with ‘quantum number’ . The explicit expression is given by, we have used [129, 182] under consideration tends to the solitonic solution with the same ‘quantum numbers’ . Hence, by taking this limit, we conclude . Therefore, on any -static sector, we have the following interesting relation between the black hole and solitonic solutions:
The relation (80) was first proposed for spherical horizons in , verified in , and extended to distorted horizons in . It provided impetus for new work by mathematical physicists working on colored black holes. The relation has been confirmed in three more general and non-trivial cases:
Isolated horizon considerations suggested the following simple heuristic model of colored black holes : A colored black hole with quantum numbers should be thought of as ‘bound states’ of a ordinary (colorless) black hole and a soliton with color quantum numbers , where can be more general than considered so far. Thus the idea is that an uncolored black hole is ‘bare’ and becomes ‘colored’ when ‘dressed’ by the soliton.
The mass formula (80) now suggests that the total ADM mass has three components: the mass of the bare horizon, the mass of the colored soliton, and a binding energy given by):
The predictions for fixed have recently been verified beyond spherical symmetry: for the distorted, axially symmetric Einstein–Yang–Mills solutions in  and for the distorted ‘dipole’ solutions in Einstein–Yang–Mills–Higgs solutions in . Taken together, the predictions of this model can account for all the qualitative features of the plots of the horizon mass and surface gravity as functions of the horizon radius and quantum numbers. More importantly, they have interesting implications on the stability properties of colored black holes.
One begins with an observation about solitons and deduces properties of black holes. Einstein–Yang–Mills solitons are known to be unstable ; under small perturbations, the energy stored in the ‘bound state’ represented by the soliton is radiated away to future null infinity . The phenomenological model suggests that colored black holes should also be unstable and they should decay into ordinary black holes, the excess energy being radiated away to infinity. In general, however, even if one component of a bound system is unstable, the total system may still be stable if the binding energy is sufficiently large. An example is provided by the deuteron. However, an examination of energetics reveals that this is not the case for colored black holes, so instability should prevail. Furthermore, one can make a few predictions about the nature of instability. We summarize these for the simplest case of spherically symmetric, static black holes for which there is a single quantum number :
Expectation 1 of the model is known to be correct . Prediction 2 has been shown to be correct in the , colored black holes in the sense that the frequency of all unstable modes is a decreasing function of the area, whence the characteristic decay time grows with area [182, 49]. To our knowledge a detailed analysis of instability, needed to test Predictions 3 and 4 are yet to be made.
Finally, the notion of horizon mass and the associated stability analysis has also provided an ‘explanation’ of the following fact which, at first sight, seems puzzling. Consider the ‘embedded Abelian black holes’ which are solutions to Einstein–Yang–Mills equations with a specific magnetic charge . They are isometric to a family of magnetically charged Reissner–Nordström solutions and the isometry maps the Maxwell field strength to the Yang–Mills field strength. The only difference is in the form of the connection; while the Yang–Mills potential is supported on a trivial bundle, the Maxwell potential requires a non-trivial bundle. Therefore, it comes as an initial surprise that the solution is stable in the Einstein–Maxwell theory but unstable in the Einstein–Yang–Mills theory [50, 59]. It turns out that this difference is naturally explained by the WIH framework. Since the solutions are isometric, their ADM mass is the same. However, since the horizon mass arises from Hamiltonian considerations, it is theory dependent: It is lower in the Einstein–Yang–Mills theory than in the Einstein–Maxwell theory! Thus, from the Einstein–Yang–Mills perspective, part of the ADM mass is carried by the soliton and there is positive which can be radiated away to infinity. In the Einstein–Maxwell theory, is zero. The stability analysis sketched above therefore implies that the solution should be unstable in the Einstein–Yang–Mills theory but stable in the Einstein–Maxwell theory. This is another striking example of the usefulness of the notion of the horizon mass.
We will now briefly summarize the most interesting result obtained from this framework in more general theories. When one allows Higgs or Proca fields in addition to Yang–Mills, or considers Einstein–Skyrme theories, one acquires additional dimensionful constants which trigger new phenomena [48, 179, 182]. One of the most interesting is the ‘crossing phenomena’ of Figure 11 where curves in the ‘phase diagram’ (i.e., a plot of the ADM mass versus horizon radius) corresponding to the two distinct static families cross. This typically occurs in theories in which there is a length scale even in absence of gravity, i.e., even when Newton’s constant is set equal to zero [154, 20].
In this case, the notion of the horizon mass acquires further subtleties. If, as in the Einstein–Yang–Mills theory considered earlier, families of static solutions carrying distinct quantum numbers do not cross, there is a well-defined notion of horizon mass for each static solution, although, as the example of ‘embedded Abelian solutions’ shows, in general its value is theory dependent. When families cross, one can repeat the previous strategy and use Equation (77) to define a mass along each branch. However, at the intersection point of the th and th branches, the mass is discontinuous. This discontinuity has an interesting implication. Consider the closed curve in the phase diagram, starting at the intersection point and moving along the th branch in the direction of decreasing area until the area becomes zero, then moving along to the th branch and moving up to the intersection point along the th branch (see Figure 11). Discontinuity in the horizon mass implies that the integral of along this closed curve is non-zero. Furthermore, the relation between the horizon and the soliton mass along each branch implies that the value of this integral has a direct physical interpretation:solitons to the knowledge of horizon properties of the corresponding black holes! Because of certain continuity properties which hold as one approaches the static Einstein–Yang–Mills sector in the space to static solutions to Einstein–Yang–Mills–Higgs equations , one can also obtain a new formula for the ADM masses of Einstein–Yang–Mills solitons. If for the black hole solutions of this theory is integrable over the entire positive half line, one has :  have been verified numerically in the spherically symmetric case , but the axi-symmetric case is still open.
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