Beyond Einstein–Maxwell theory

6.1 Beyond Einstein–Maxwell theory

Our primary purpose in this section is to illustrate the differences from the Einstein–Maxwell theory. These stem from ‘internal charges’ and other ‘quantum numbers’ that are unrelated to angular momentum. Therefore, for simplicity, we will restrict ourselves to non-rotating weakly isolated horizons. Extension to include angular momentum is rather straightforward.

6.1.1 Dilatonic couplings

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

where $α ≥ 0$ is a free parameter which governs the strength of the coupling of $φ$ to the Maxwell field $Fab$ . If $α = 0$ , one recovers the standard Einstein–Maxwell–Klein–Gordon system, while $α = 1$ occurs in a low energy limit of string theory. For our illustrative purposes, it will suffice to consider the $α = 1$ case.

At spatial infinity, one now has three charges: the ADM mass $MADM$ , the usual electric charge $Q ∞$ , and another charge $&tidle;Q ∞$ :

$&tidle; Q ∞$ is conserved in space-time (i.e., its value does not change if the 2-sphere of integration is deformed) while $Q ∞$ is not. From the perspective of weakly isolated horizons, it is more useful to use $aΔ$ , $Q Δ$ , and $&tidle;QΔ$ as the basic charges:

where $S$ is any cross-section of $Δ$ . Although the standard electric charge is not conserved in space-time, it is conserved along $Δ$ whence $Q Δ$ 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 $ta$ , evolution along which is Hamiltonian. The necessary and sufficient condition now becomes the following: There should exist a phase space function $Et Δ$ , constructed from horizon fields, such that

Thus, the only difference from the Einstein–Maxwell case is that $Q Δ$ is now replaced by $Q&tidle;Δ$ . Again, there exists an infinite number of such live vector fields and one can construct them systematically starting with any (suitably regular) function $κ0(aΔ, &tidle;Q Δ)$ of the horizon area and charge and requiring $κ (t)$ should equal $κ 0$ .

The major difference arises in the next step, when one attempts to construct a preferred $a t0$ . With the dilatonic coupling, the theory has a unique three parameter family of static solutions which can be labelled by $(aΔ, Q Δ,Q&tidle;Δ)$ [101 , 94 , 95 , 148 ]. As in the Reissner Nordström family, these solutions are spherically symmetric. In terms of these parameters, the surface gravity $κ (ξ)$ of the static Killing field $ξa$ , which is unit at infinity, is given by

The problem in the construction of the preferred $a t0$ is that we need a function $κ0$ which depends only on $a Δ$ and $Q&tidle;Δ$ and, since $κ(ξ)$ depends on all three horizon parameters, one can no longer set $κ0 = κ (ξ)$ on the entire phase space. Thus, there is no live vector field $ta0$ which can generate a Hamiltonian evolution and 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 $Q Δ$ . On each sector, $κ (ξ)$ trivially depends only on $aΔ$ and $&tidle;QΔ$ . So one can set $κ0 = κ (ξ)$ , select a canonical $t0$ , and obtain a mass function $M Δ = Et0 Δ$ . However, now the first law (72 ) is satisfied only if the variation is restricted such that $δQ Δ = 0$ . For general variation, one has the modified law [25 ]

where $κ = κ(t) 0$ , $ˆΦ2 = (Q ΔQ&tidle;Δ∕R2 ) Δ$ and $ˆQ2 = Q ΔQ&tidle;Δ Δ$ . Thus, although there is still a first law in terms of $a t0$ and $M Δ$ , it does not have the canonical form (72

) because $a t0$ does not generate Hamiltonian evolution on the entire phase space. More generally, in theories with multiple scalar fields, if one focuses only on static sectors, one obtains similar ‘non-standard’ forms of the first law with new ‘work terms’ involving scalar fields [100 ]. 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 $a t$ 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.

6.1.2 Yang–Mills fields

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 $g$ has dimensions $−1∕2 (LM )$ . The existence of such a dimensionful quantity has interesting consequences.

Figure 9:

The ADM mass as a function of the horizon radius $RΔ$ of static spherically symmetric solutions to the Einstein–Yang–Mills system (in units provided by the Yang–Mills coupling constant). Numerical plots for the colorless ( $n = 0$ ) and families of colored black holes ( $n = 1,2$ ) are shown. (Note that the $y$ -axis begins at $M = 0.7$ rather than at $M = 0$ .)

For simplicity, we will restrict ourselves to $SU (2)$ Yang–Mills fields, but results based on the isolated horizon framework go through for general compact groups [25 ]. 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 $YM (aΔ, Q Δ )$ . In addition, there is a 1-parameter family of ‘embedded Abelian solutions’ with (a fixed) magnetic charge $0 PΔ$ , labelled by $0 (aΔ,P Δ)$ . 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 $(n1,n2 )$ is known to exist. All static, spherically symmetric solutions are known and they correspond to the infinite sub-family $(n1, n2 = 0)$ , 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 $n$ (see Figure 9 ). Let us decrease the horizon area along any branch $n ⁄= 0$ . 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 $M sol ADM$ , which equals the limiting value of $M BH ADM$ . 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, $BH M ADM$ can not be taken as an accurate measure of the horizon mass for any black hole along an $n ⁄= 0$ 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 $ℒ ℓAa = 0$ on $Δ$ . The analogous condition ensuring that the Yang–Mills potential $A$ is in an ‘adapted gauge’ on $Δ$ is more subtle [25 ]. However, it does exist and again ensures that (i) the action principle is well defined, and (ii) the Yang–Mills electric potential $Φ (ℓ) := − |ℓ ⋅ A |$ 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 charges

where $|F | := [(εabFi )(εabFj )Kij ]12 ab ab$ , with $Kij$ being the Cartan–Killing metric on $SU(2)$ , $εab$ the alternating tensor on the cross-section $S$ of $Δ$ , and where $⋆ |F |$ is defined by replacing $F$ with $⋆ F$ . Finally, one can again introduce live vector fields $a t$ , time evolution along which generates a Hamiltonian flow on the phase space, and establish a first law for each of these $ta$ :

Note that, even though the magnetic charge $P YM Δ$ is in general non-zero, it does not enter the statement of the first law. In the Abelian case, a non-zero magnetic charge requires non-trivial $U (1 )$ bundles, and Chern numbers characterizing these bundles are discrete. Hence the magnetic charge is quantized and, if the phase space is constructed from connections, $δP EM$ vanishes identically for any variation $δ$ . In the non-Abelian case, one can work with a trivial bundle and have non-zero $P YΔM$ . Therefore, $δP YM Δ$ 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 $κ0$ on the entire phase space, whence as in the dilatonic case, there is no longer a canonical horizon mass $M Δ$ 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.