4.4 Lensing in axisymmetric
stationary spacetimes
Axisymmetric stationary spacetimes are of interest in view of
lensing as general-relativistic models for rotating deflectors. The
best known and most important example is the Kerr metric which
describes a rotating black hole (see Section 5.8). For non-collapsed rotating
objects, exact solutions of Einstein’s field equation are known
only for the idealized cases of infinitely long cylinders
(including string models; see Section 5.10) and disks (see Section 5.9). Here we collect, as a
preparation for these examples, some formulas for an unspecified
axisymmetric stationary metric. The latter can be written in
coordinates
, with capital indices
taking the values 1 and 2, as
where all metric coefficients depend on
only.
We assume that the integral curves of
are closed, with
the usual
-periodicity, and that the 2-dimensional
orbits spanned by
and
are timelike. Then
the Lorentzian signature of
implies that
is positive definite. In general, the vector field
need not be timelike and the hypersurfaces
need not be spacelike. Our assumptions allow for
transformations
with a constant
. If, by such a
transformation, we can achieve that
everywhere,
we can use the purely spatial formalism for light rays in terms of
the Fermat geometry (recall Section 4.2). Comparison of Equation (96) with
Equation (61) shows that the
redshift potential
, the Fermat metric
, and the Fermat one-form
are
respectively. If it is not possible to make
negative on the entire spacetime domain under
consideration, the Fermat geometry is defined only locally and,
therefore, of limited usefulness. This is the case, e.g., for the
Kerr metric where, in Boyer-Lindquist coordinates,
is positive in the ergosphere (see Section 5.8).
Variational techniques related to Fermat’s
principal in stationary spacetimes are detailed in a book by
Masiello [218]. Note
that, in contrast to standard terminology, Masiello’s definition of
stationarity includes the assumption that the surfaces
are spacelike.
For a rotating body with an equatorial plane
(i.e., with reflectional symmetry), the Fermat metric of the
equatorial plane can be represented by an embedding diagram, in
analogy to the spherically symmetric static case (recall
Figure 11). However, one should
keep in mind that in the non-static case the lightlike geodesics do
not correspond to the geodesics of
but are affected, in addition, by a sort of Coriolis
force produced by
. For a review on embedding diagrams,
including several examples (see [160
]).