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 (y1, y2,φ,t), with capital indices A, B, ... taking the values 1 and 2, as
g = gtt(y)dt2 + 2gtφ (y)dtdφ + gφφ(y)dφ2 + gAB (y)dyAdyB, (96 )
where all metric coefficients depend on 1 2 y = (y ,y ) only. We assume that the integral curves of ∂φ are closed, with the usual (2π )-periodicity, and that the 2-dimensional orbits spanned by ∂φ and ∂t are timelike. Then the Lorentzian signature of g implies that gAB (y ) is positive definite. In general, the vector field ∂t need not be timelike and the hypersurfaces t = constant need not be spacelike. Our assumptions allow for transformations (φ, t) ↦→ (φ + Ωt, t) with a constant Ω. If, by such a transformation, we can achieve that gtt < 0 everywhere, we can use the purely spatial formalism for light rays in terms of the Fermat geometry (recall Section 4.2). Comparison of Equation (96View Equation) with Equation (61View Equation) shows that the redshift potential f, the Fermat metric ˆg, and the Fermat one-form ˆϕ are
e2f = − g , (97 ) tt 2 gAB- A B gtφ-−-gttgφφ- 2 ˆg = − gtt dx dx + g2 dφ , (98 ) gtφ tt ˆϕ = − ---d φ, (99 ) gtt
respectively. If it is not possible to make gtt 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, gtt 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 t = constant 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 11View Image). However, one should keep in mind that in the non-static case the lightlike geodesics do not correspond to the geodesics of ˆg but are affected, in addition, by a sort of Coriolis force produced by ˆϕ. For a review on embedding diagrams, including several examples (see [159Jump To The Next Citation Point]).

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