Structure of colored, static black holes

6.2 Structure of colored, static black holes

We will briefly summarize research in three areas in which the isolated horizon framework has been used to illuminate the structure of static, colored black holes and associated solitons⁷ .

6.2.1 Horizon mass

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 $⃗n0 ≡ (n01,n02)$ . Using for $κ0$ the surface gravity of the properly normalized static Killing vector, in this sector one can construct a live vector field $ta 0$ and obtain a first law. The energy $Et0 Δ$ is well-defined on the full phase space and can be naturally interpreted as the horizon mass $(⃗n0) M Δ$ for colored black holes with ‘quantum number’ $⃗n0$ . The explicit expression is given by

where, following a convention in the literature on hairy black holes [76

], we have used

rather than the surface gravity $κ⃗n0(R )$ of static solutions. Now, as in the Einstein–Maxwell case, the Hamiltonian generating evolution along $a t0$ contains only surface terms: $t0 t0 Ht0 = EADM − E Δ$ and is constant on each connected, static sector if $ta0$ coincides with the static Killing field on that sector. By construction, our $ta0$ has this property for the $⃗n0$ -sector under consideration. Now, in the Einstein–Maxwell case, since there is no constant with the dimension of energy, it follows that the restriction of $t0 H ADM$ to the static sector must vanish. The situation is quite different in Einstein–Yang–Mills theory where the Yang–Mills coupling constant $g$ provides a scale. In $c = 1$ units, $√ -- (g G )−1 ∼ mass$ . Therefore, we can only conclude that

for some $⃗n0$ -dependent constant $C(⃗n0)$ . As the horizon radius shrinks to zero, the static solution [129 , 182

] under consideration tends to the solitonic solution with the same ‘quantum numbers’ $⃗n0$ . Hence, by taking this limit, we conclude $√ -- −1 (⃗n ) sol,(⃗n0) (g G ) C 0 = M ADM$ . Therefore, on any $⃗n$ -static sector, we have the following interesting relation between the black hole and solitonic solutions:

Thus, although the main motivation behind the isolated horizon framework was to go beyond globally time-independent situations, it has led to an interesting new relation between the ADM masses of black holes and their solitonic analogs already in the static sector.

The relation (80 ) was first proposed for spherical horizons in [76 ], verified in [74 ], and extended to distorted horizons in [25 ]. 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:

Non-spherical, non-rotating black holes parameterized by two quantum numbers [133 ].
Non-spherical solutions to the more general Einstein–Yang–Mills–Higgs theory [109 ], where distortions are caused by ‘magnetic dipole hair’ [131 ].
Static solutions in the Born–Infeld theory [60 ].

6.2.2 Phenomenological model of colored black holes

Isolated horizon considerations suggested the following simple heuristic model of colored black holes [20 ]: A colored black hole with quantum numbers $⃗n$ should be thought of as ‘bound states’ of a ordinary (colorless) black hole and a soliton with color quantum numbers $⃗n$ , where $⃗n$ 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 $(⃗n) M ADM$ has three components: the mass $M Δ(0)$ of the bare horizon, the mass $M AsoDl,M⃗n$ of the colored soliton, and a binding energy given by

If this picture is correct, being gravitational binding energy, $E binding$ would have to be negative. This expectation is borne out in explicit examples. The model has several predictions on the qualitative behavior of the horizon mass (77

), the surface gravity, and the relation between properties of black holes and solitons. We will illustrate these with just three examples (for a complete list with technical caveats, see [20

]):

For any fixed value of $RΔ$ and of all quantum numbers except $n$ , the horizon mass and surface gravity decrease monotonically with $n$ .
For fixed values of all quantum numbers $⃗n$ , the horizon mass $M (Δ⃗n)(R Δ)$ is non-negative, vanishing if and only if $R Δ$ vanishes, and increases monotonically with $R Δ$ . $β (⃗n)$ is positive and bounded above by $1$ .
For fixed $⃗n$ , the binding energy decreases as the horizon area increases.

The predictions for fixed $⃗n$ have recently been verified beyond spherical symmetry: for the distorted, axially symmetric Einstein–Yang–Mills solutions in [133 ] and for the distorted ‘dipole’ solutions in Einstein–Yang–Mills–Higgs solutions in [109 ]. 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.

Figure 10:

An initially static colored black hole with horizon $Δin$ is slightly perturbed and decays to a Schwarzschild-like isolated horizon $Δ fin$ , with radiation going out to future null infinity $+ ℐ$ .

One begins with an observation about solitons and deduces properties of black holes. Einstein–Yang–Mills solitons are known to be unstable [174 ]; under small perturbations, the energy stored in the ‘bound state’ represented by the soliton is radiated away to future null infinity $ℐ+$ . 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 $n$ :

All the colored black holes on the $n$ th branch have the same number (namely, $2n$ ) of unstable modes as the $n$ th soliton. (The detailed features of these unstable modes can differ especially because they are subject to different boundary conditions in the two cases.)
For a given $n$ , colored black holes with larger horizon area are less unstable. For a given horizon area, colored black holes with higher value of $n$ are more unstable.
The ‘available energy’ for the process is given by

Part of it is absorbed by the black hole so that its horizon area increases and the rest is radiated away to infinity. Note that $(n) E available$ can be computed knowing just the initial configuration.
In the process the horizon area necessarily increases. Therefore, the energy radiated to infinity is strictly less than $(n) E available$ .

Expectation 1 of the model is known to be correct [173 ]. Prediction 2 has been shown to be correct in the $n = 1$ , 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 $0 P Δ$ . 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 $SU (2)$ bundle, the Maxwell potential requires a non-trivial $U (1)$ 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 $YM E available$ which can be radiated away to infinity. In the Einstein–Maxwell theory, $EEaMvailable$ 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.

6.2.3 More general theories

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 ].

Figure 11:

The ADM mass as a function of the horizon radius $R Δ$ in theories with a built-in non-gravitational length scale. The schematic plot shows crossing of families labelled by $n = 1$ and $n = 2$ at $inter RΔ = RΔ$ .

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 $(n) M Δ$ along each branch. However, at the intersection point $inter RΔ$ of the $n$ th and $(n + 1)$ th branches, the mass is discontinuous. This discontinuity has an interesting implication. Consider the closed curve $C$ in the phase diagram, starting at the intersection point and moving along the $(n + 1)$ th branch in the direction of decreasing area until the area becomes zero, then moving along $aΔ = 0$ to the $n$ th branch and moving up to the intersection point along the $n$ th branch (see Figure 11 ). Discontinuity in the horizon mass implies that the integral of $β (RΔ )$ 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:

This is a striking prediction because it relates differences in masses of 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 [75

], one can also obtain a new formula for the ADM masses of Einstein–Yang–Mills solitons. If $(1 − β(n)(r))$ for the black hole solutions of this theory is integrable over the entire positive half line, one has [20

Both these predictions of the phenomenological model [20 ] have been verified numerically in the spherically symmetric case [75 ], but the axi-symmetric case is still open.