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
is a free parameter which governs the
strength of the coupling of
to the Maxwell field
. If
, one recovers the standard
Einstein-Maxwell-Klein-Gordon system, while
occurs in a low energy limit of string theory. For
our illustrative purposes, it will suffice to consider the
case.
At spatial infinity, one now has three charges:
the ADM mass
, the usual electric charge
, and another charge
:
is conserved in space-time (i.e., its value does not
change if the 2-sphere of integration is deformed) while
is not. From the perspective of weakly isolated
horizons, it is more useful to use
,
, and
as the basic charges:
where
is any cross-section of
. Although the
standard electric charge is not conserved in space-time, it 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
Thus, the only difference from the Einstein-Maxwell case is that
is now replaced by
. Again, there
exists an infinite number of such live vector fields and one can
construct them systematically starting with any (suitably regular)
function
of the horizon area and charge and
requiring
should equal
.
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, 95, 96, 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 by
The problem in the construction of the preferred
is that we need a function
which depends only on
and
and, since
depends on all three horizon
parameters, one can no longer set
on the
entire phase space. Thus, there is no live vector field
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
. 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 [26
]
where
,
and
. Thus, although there is still a first
law in terms of
and
, it does not have
the canonical form (72) because
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
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
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 [26
]. 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 [127
, 128, 129]; 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 [38, 171, 60]. 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 [26
]. 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 charges
where
, with
being the Cartan-Killing
metric on
,
the alternating tensor on
the cross-section
of
, and where
is defined by replacing
with
. Finally, one can again introduce live vector fields
, time evolution along which generates a Hamiltonian
flow on the phase space, and establish a first law for each of
these
:
Note that, even though the magnetic charge
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
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.