### 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 [219]. 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 [159]).