4.2 Stationary toroidal Kaluza–Klein black holes
The four-dimensional vacuum Einstein equations simplify considerably in the stationary and
axisymmetric setting by reducing to a harmonic map into the hyperbolic plane (see Sections 8
and 3.2.3). A similar such reduction in
-dimensions works when the isometry group
includes
, i.e., besides the stationary vector there exist
commuting axial Killing
vectors.
Since the center
of
has dimension
in the asymptotically flat case the existence of such a group of isometries is only possible for
or
. However, one can move beyond the usual asymptotic-flatness and consider
instead
-asymptotically-flat spacetimes, in the sense of Section 2.2, with asymptotic ends
, satisfying
and
, with the isometry group containing
,
. Here one takes
, with the reference metric of the
form
, where
is the flat
-torus metric. Finally, the action of
on
by isometries is assumed in the exterior region
to take the form
Such metrics will be referred to as stationary toroidal Kaluza–Klein metrics.