With respect to the adapted time coordinate
t, defined by
, the metric of a stationary spacetime is parametrized in terms
of a three-dimensional (Riemannian) metric
, a one-form
, and a scalar field
, where stationarity implies that
,
and
are functions on
:
Using Cartan's structure equations (see, e.g. [165]), it is a straightforward task to compute the Ricci scalar for
the above decomposition of the spacetime metric
. The result shows that the Einstein-Hilbert action of a
stationary spacetime reduces to the action for a scalar field
and an Abelian vector field
a, which are coupled to three-dimensional gravity. The fact that
this coupling is
minimal
is a consequence of the particular choice of the conformal
factor in front of the three-metric
in the decomposition (8
). The vacuum field equations are, therefore, equivalent to the
three-dimensional Einstein-matter equations obtained from
variations of the effective action
with respect to
,
and
a
. (Here and in the following
and
denote the Ricci scalar and the inner product
with respect to
.)
It is worth noting that the quantities
and
a
are related to the norm and the twist of the Killing field as
follows:
where
and
denote the Hodge dual with respect to
and
, respectively
. Since
a
is the connection of a fiber bundle with base space
and fiber
G, it behaves like an Abelian gauge potential under coordinate
transformations of the form
. Hence, it enters the effective action in a gauge-invariant
way, that is, only via the ``Abelian field strength'',
.
![]() |
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 |