With respect to an adapted coordinate , so that
, the metric of a stationary spacetime can
be parameterized in terms of a three-dimensional (Riemannian) metric
, a one-form
, and a scalar field
, where stationarity implies that
,
and
are functions on
:
The notation suggests that
is a time coordinate,
, but this restriction does not
play any role in the local form of the equations that we are about to derive. Similarly the local calculations
that follow remain valid regardless of the causal character of
, provided that
is not null everywhere,
and then one only considers the region where
does not change sign. On any connected
component of this region
is either spacelike or timelike, as determined by the sign of
, and then the
metric
is Lorentzian, respectively Riemannian, there. In any case, both the parameterization of the
metric and the equations become singular at places where
has zeros, so special care is required
wherever this occurs.
Using Cartan’s structure equations (see, e.g., [300]), it is a straightforward task to compute the Ricci
scalar for the above decomposition of the spacetime metric; see, e.g., [155] for the details of the derivation.
The result is that the Einstein–Hilbert action of a stationary spacetime reduces to the action for a scalar
field and a vector field
, 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 (6.1
). The vacuum field equations are thus seen to be equivalent to the
three-dimensional Einstein-matter equations obtained from variations of the effective action
It is worth noting that the quantities and
are related to the norm and the twist of the Killing
field as follows:
http://www.livingreviews.org/lrr-2012-7 |
Living Rev. Relativity 15, (2012), 7
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