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.
[14
] have classified the models admitting a symmetry group which is
sufficiently large to comprise
all
scalar fields arising on the effective level
within
one
coset space,
G
/
H
. 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 [61]. Like the EM system, the model also possesses an
SO
(1,2) symmetry, arising from the dimensional reduction with
respect to the Abelian isometry group generated by the Killing
field. Gal'tsov and Kechkin [63], [64] have shown that the full symmetry group is, however, larger
than
: The target space for dilaton-axion gravity with an
U
(1) vector field is the coset
[62]. Using the fact that
SO
(2,3) is isomorphic to
, Gal'tsov and Kechkin [65] were also able to give a parametrization of the target space in
terms of
(rather than
) matrices. The relevant coset was shown to be
.
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 [89]. The reason for this is the following: Like the EM equations (32
), the stationary field equations consist of the
three-dimensional Einstein equations and the
-model equations,
The current one-form
is given in terms of the hermitian matrix
, which comprises all scalar fields arising on the effective
level. The
-model equations,
, include
differential current conservation laws, of which
are redundant. Integrating all equations over a space-like
hyper-surface extending from the horizon to infinity, Stokes'
theorem yields a set of relations between the charges and the
horizon-values of the scalar potentials.
The crucial observation is that Stokes' theorem provides
independent
Smarr relations, rather than only
ones. (This is due to the fact that all
-model currents are
algebraically
independent, although there are
differential identities which can be derived from the
field equations.)
The
complete
set of Smarr type formulas 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 [89],
where
and
are the surface gravity and the area of the horizon, and the RHS
comprises the asymptotic flux integrals, that is, the total mass,
the NUT charge, the dilaton and axion charges, and the electric
and magnetic charges, respectively.
A very simple illustration of the idea outlined above is the
static, purely electric EM system. In this case, the electrovac
coset
reduces to
. The matrix
is parametrized in terms of the electric potential
and the gravitational potential
. The
-model equations comprise
differential conservation laws, of which
is redundant:
[It is immediately verified that Eq. (39) is indeed a consequence of the Maxwell and Einstein Eqs. (38
).] Integrating Eqs. (38
) over a space-like hyper-surface and using Stokes' theorem
yields
which is the well-known Smarr formula. In a similar way, Eq. (39) provides an
additional
relation of the Smarr type,
which can be used to compute the horizon-value of the electric
potential,
. Using this in the Smarr formula (40
) gives the desired expression for the total mass,
.
In the ``extreme'' case, the BPS bound [75] for the static EM-dilaton-axion system,
, was previously obtained by constructing the null geodesics of
the target space [45]. For spherically symmetric configurations with non-degenerate
horizons (
), Eq. (37
) was derived by Breitenlohner
et al.
[14]. In fact, many of the spherically symmetric black hole
solutions with scalar and vector fields [73], [76], [69] are known to fulfill Eq. (37
), where the LHS is expressed in terms of the horizon radius (see
[67] and references therein). Using the generalized first law of
black hole thermodynamics, Gibbons
et al.
[72] recently obtained Eq. (37
) for spherically symmetric solutions with an arbitrary number of
vector and moduli fields.
The above derivation of the mass formula (37) 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 formulas which is sufficiently large to derive the desired
relation. Although the result (37
) was established by using the explicit representations of the EM
and EM-dilaton-axion coset spaces [89], 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
provided that the stationary field equations assume the form (36), where
is a map into a
symmetric space,
G
/
H
. Here
and
denote the charges of the scalars (including the gravitational
ones) and the vector fields, respectively.
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Stationary Black Holes: Uniqueness and Beyond
Markus Heusler http://www.livingreviews.org/lrr-1998-6 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |