In dilaton gravity, the Einstein–Maxwell theory is supplemented with a scalar field – called the dilaton – and (in the Einstein frame) the Maxwell part of the action is replaced by
At spatial infinity, one now has three charges: the ADM mass , the usual electric charge , and another charge :is conserved along whence is well-defined.
It is straightforward to extend the Hamiltonian framework of Section 4.1 to include the dilaton. To define energy, one can again seek live time-translation vector fields , evolution along which is Hamiltonian. The necessary and sufficient condition now becomes the following: There should exist a phase space function , constructed from horizon fields, such that
The major difference arises in the next step, when one attempts to construct a preferred . With the dilatonic coupling, the theory has a unique three parameter family of static solutions which can be labelled by [101, 94, 95, 148]. As in the Reissner Nordström family, these solutions are spherically symmetric. In terms of these parameters, the surface gravity of the static Killing field , which is unit at infinity, is given byand agree with the time-translation Killing field in all static solutions. It was the availability of such live vector fields that provided a canonical notion of the horizon mass in the Einstein–Maxwell theory in Section 4.1.3.
One can weaken the requirements by working on sectors of phase space with fixed values of . On each sector, trivially depends only on and . So one can set , select a canonical , and obtain a mass function . However, now the first law (72) is satisfied only if the variation is restricted such that . For general variation, one has the modified law . From the restricted perspective of static sector, this is just a fact. The isolated horizon framework provides a deeper underlying reason: In these theories, there is no evolution vector field defined for all points of the phase space, which coincides with the properly normalized Killing field on all static solutions, and evolution along which is Hamiltonian on the full phase space.
In the Einstein–Maxwell theory, with and without the dilaton, one can not construct a quantity with the dimensions of mass from the fundamental constants in the theory. The situation is different for Einstein–Yang–Mills theory because the coupling constant has dimensions . The existence of such a dimensionful quantity has interesting consequences.
For simplicity, we will restrict ourselves to Yang–Mills fields, but results based on the isolated horizon framework go through for general compact groups . Let us begin with a summary of the known static solutions. First, the Reissner–Nordström family constitutes a continuous 2-parameter set of static solutions of the Einstein–Yang–Mills theory, labelled by . In addition, there is a 1-parameter family of ‘embedded Abelian solutions’ with (a fixed) magnetic charge , labelled by . Finally, there are families of ‘genuinely non-Abelian solutions’. For these, the analog of the Israel theorem for Einstein–Maxwell theory fails to hold [129, 128, 130]; the theory admits static solutions which need not be spherically symmetric. In particular, an infinite family of solutions labelled by two integers is known to exist. All static, spherically symmetric solutions are known and they correspond to the infinite sub-family , labelled by a single integer. However, the two parameter family is obtained using a specific ansatz, and other static solutions also exist. Although the available information on the static sector is quite rich, in contrast to the Einstein–Maxwell-dilaton system, one is still rather far from having complete control.
However, the existing results are already sufficient to show that, in contrast to the situation in the Einstein–Maxwell theory, the ADM mass is not a good measure of the black hole (or horizon) mass even in the static case. Let us consider the simplest case, the spherically symmetric static solutions labelled by a single integer (see Figure 9). Let us decrease the horizon area along any branch . In the zero area limit, the solution is known to converge point-wise to a regular, static, spherical solution, representing an Einstein–Yang–Mills soliton [37, 172, 58]. This solution has, of course, a non-zero ADM mass , which equals the limiting value of . However, in this limit, there is no black hole at all! Hence, this limiting value of the ADM mass can not be meaningfully identified with any horizon mass. By continuity, then, can not be taken as an accurate measure of the horizon mass for any black hole along an branch. Using the isolated horizon framework, it is possible to introduce a meaningful definition of the horizon mass on any given static branch.
To establish laws of black hole mechanics, one begins with appropriate boundary conditions. In the Maxwell case, the gauge freedom in the vector potential is restricted on the horizon by requiring on . The analogous condition ensuring that the Yang–Mills potential is in an ‘adapted gauge’ on is more subtle . However, it does exist and again ensures that (i) the action principle is well defined, and (ii) the Yang–Mills electric potential is constant on the horizon, where the absolute sign stands for the norm in the internal space. The rest of the boundary conditions are the same as in Section 2.1.1. The proof of the zeroth law and the construction of the phase space is now straightforward. There is a well-defined notion of conserved horizon chargesnot enter the statement of the first law. In the Abelian case, a non-zero magnetic charge requires non-trivial bundles, and Chern numbers characterizing these bundles are discrete. Hence the magnetic charge is quantized and, if the phase space is constructed from connections, vanishes identically for any variation . In the non-Abelian case, one can work with a trivial bundle and have non-zero . Therefore, does not automatically vanish and absence of this term in the first law is somewhat surprising.
A more significant difference from the Abelian case is that, because the uniqueness theorem fails, one can not use the static solutions to introduce a canonical function on the entire phase space, whence as in the dilatonic case, there is no longer a canonical horizon mass function on the entire phase space. In the next Section 6.2 we will see that it is nonetheless possible to introduce an extremely useful notion of the horizon mass for each static sequence.
© Max Planck Society and the author(s)