The coset structure of the stationary field equations is shared by various self-gravitating matter
models with massless scalars (moduli) and Abelian vector fields. For scalar mappings into a
symmetric target space , say, Breitenlohner et al. [31
] have classified the models admitting a
symmetry group, which is sufficiently large to comprise all scalar fields arising on the effective
level10
within one coset space,
. A prominent example of this kind is the EM-dilaton-axion system, which is
relevant to
supergravity and to the bosonic sector of four-dimensional heterotic string theory: The
pure dilaton-axion system has an
symmetry, which persists in dilaton-axion gravity with an
Abelian gauge field [114]. Like the EM system, the model also possesses an
symmetry, arising
from the dimensional reduction with respect to the Abelian isometry group generated by the Killing field.
However, Gal’tsov and Kechkin [116, 117] have shown that the full symmetry group is larger than
: The target space for dilaton-axion gravity with a
vector field is the coset
[113]. Using the fact that
is isomorphic to
,
Gal’tsov and Kechkin [118] were also able to give a parametrization of the target space in
terms of
(rather than
) matrices. The relevant coset space was shown to be
; for the generalization to the dilaton-axion system with multiple vector fields we refer
to [119, 121].
Common to the black-hole solutions of the above models is the fact that their Komar mass can be
expressed in terms of the total charges and the area and surface gravity of the horizon [153]. The reason for
this is the following: Like the EM equations (6.26
), the stationary field equations consist of the
three-dimensional Einstein equations and the
-model equations,
The complete set of Smarr type formulae can be used to get rid of the horizon-values of the
scalar potentials. In this way one obtains a relation, which involves only the Komar mass, the
charges and the horizon quantities. For the EM-dilaton-axion system one finds, for instance [153],
A very simple illustration of the idea outlined above is the static, purely electric EM system. In this
case, the electrovacuum coset reduces to
. The matrix
is parameterized in terms of the electric potential
and the gravitational potential
.
The
-model equations comprise
differential conservation laws, of which
is
redundant:
In the “extreme” case, the Bogomol’nyi–Prasad–Sommerfield (BPS) bound [128] for the static
EM-dilaton-axion system, , was previously obtained by constructing null
geodesics of the target space [86]. For spherically-symmetric configurations with non-degenerate horizons
(
), Eq. (7.5
) was derived by Breitenlohner et al. [31]. In fact, many of the spherically-symmetric
black-hole solutions with scalar and vector fields [126, 131, 122] are known to fulfill Eq. (7.5
), where the
left-hand side is expressed in terms of the horizon radius (see [120] and references therein).
Using the generalized first law of black-hole thermodynamics, Gibbons et al. [130] obtained
Eq. (7.5
) for spherically-symmetric solutions with an arbitrary number of vector and moduli
fields.
The above derivation of the mass formula (7.5) is neither restricted to spherically-symmetric
configurations, nor are the solutions required to be static. The crucial observation is that the coset structure
gives rise to a set of Smarr formulae, which is sufficiently large to derive the desired relation.
Although the result (7.5
) was established by using the explicit representations of the EM and
EM-dilaton-axion coset spaces [153], similar relations are expected to exist in the general case. More
precisely, it should be possible to show that the Hawking temperature of all asymptotically-flat (or
asymptotically NUT) non-rotating black holes with massless scalars and Abelian vector fields is given by
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Living Rev. Relativity 15, (2012), 7
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