Historical notes.

The Kerr metric was found by Kerr [180]. The coordinate representation (118) is due to Boyer and
Lindquist [36]. The literature on lightlike (and timelike) geodesics of the Kerr metric is abundant (for an
overview of the pre-1979 literature, see Sharp [305]). Detailed accounts on Kerr geodesics can be found in
the books by Chandrasekhar [54] and O’Neill [248].

Fermat geometry.

The Killing vector field is not timelike on that part of the domain of outer communication where
. This region is known as the ergosphere. Thus, the general results of Section 4.2 on
conformally stationary spacetimes apply only to the region outside the ergosphere. On this region, the Kerr
metric is of the form (61), with redshift potential

First integrals for lightlike geodesics.

Carter [53] discovered that the geodesic equation in the Kerr metric admits another independent constant
of motion , in addition to the constants of motion and associated with the Killing vector fields
and . This reduces the lightlike geodesic equation to the following first-order form:

Light cone.

With the help of Equations (124, 125, 126, 127), the past light cone of any observation event can be
explicitly written in terms of elliptic integrals. In this representation the light rays are labeled by the
constants of motion and . In accordance with the general idea of observational
coordinates (4), it is desirable to relabel them by spherical coordinates on the observer’s celestial
sphere. This requires choosing an orthonormal tetrad at . It is convenient to choose
, , and, thus, perpendicular to the hypersurface
(“zero-angular-momentum observer”). For an observation event in the equatorial plane, , at
radius , one finds

Lensing by a Kerr black hole.

For an observation event and a light source with worldline , both in the domain of outer
communication of a Kerr black hole, several qualitative features of lensing are unchanged in comparison
to the Schwarzschild case. If is past-inextendible, bounded away from the horizon and
from (past lightlike) infinity, and does not meet the caustic of the past light cone of , the
observer sees an infinite sequence of images; for this sequence, the travel time (e.g., in terms
of the time coordinate ) goes to infinity. These statements can be proven with the help of
Morse theory (see Section 3.3). On the observer’s sky the sequence of images approaches the
apparent boundary of the black hole which is shown in Figure 22. This follows from the fact that

- the infinite sequence of images must have an accumulation point on the observer’s sky, by compactness, and
- the lightlike geodesic with this initial direction cannot go to infinity or to the horizon, by assumption on .

If meets the caustic of ’s past light cone, the image is not an Einstein ring, unless is on the axis of rotation. It has only an “infinitesimal” angular extension on the observer’s sky. As always when a point source meets the caustic, the ray-optical calculation gives an infinitely bright image. Numerical studies show that in the Kerr spacetime, where the caustic is a tube with astroid cross-section, the image is very bright whenever the light source is inside the tube [285]. In principle, with the lightlike geodesics given in terms of elliptic integrals, image positions on the observer’s sky can be calculated explicitly. This has been worked out for several special wordlines . The case that is a circular timelike geodesic in the equatorial plane of the extreme Kerr metric, , was treated by Cunningham and Bardeen [67, 17]. This example is of relevance in view of accretion disks. Viergutz [329] developed a formalism for the case that has constant and coordinates, i.e., for a light source that stays on a ring around the axis. One aim of this approach, which could easily be rewritten in terms of the exact lens map (recall Section 2.1), was to provide a basis for numerical studies. The case that is an integral curve of and that and are at large radii is treated by Bozza [38] under the additional assumption that source and observer are close to the equatorial plane and by Vazquez and Esteban [328] without this restriction. All these articles also calculate the brightness of images. This requires determining the cross-section of infinitesimally thin bundles with vertex, e.g., in terms of the shape parameters and (recall Figure 3). For a bundle around an arbitrary light ray in the Kerr metric, all relevant equations were worked out analytically by Pineault and Roeder [277]. However, the equations are much more involved than for the Schwarzschild case and will not be given here. Lensing by a Kerr black hole has been visualized (i) by showing the apparent distortion of a background pattern [278, 306] and (ii) by showing the visual appearence of an accretion disk [278, 282, 306]. The main difference, in comparison to the Schwarzschild case, is in the loss of the left-right symmetry. In view of observations, Kerr black holes are considered as candidates for active galactic nuclei (AGN) since many years. In particular, the X-ray variability of AGN is interpreted as coming from a “hot spot” in an accretion disk that circles around a Kerr black hole. Starting with the pioneering work in [67, 17], many articles have been written on calculating the light curves and the spectrum of such “hot spots”, as seen by a distant observer (see, e.g., [75, 12, 174, 168, 108]). The spectrum can be calculated in terms of a transfer function that was tabulated, for some values of , in [65] (cf. [329, 330]). A Kerr black hole is also considered as the most likely candidate for the supermassive object at the center of our own galaxy. (For background material see [106].) In this case, the predicted angular diameter of the black hole on our sky, in the sense of Figure 22, is about 30 microarcseconds; this is not too far from the reach of current VLBI technologies [107]. Also, the fact that the radio emission from our galactic center is linearly polarized gives a good motivation for calculating polarimetric images as produced by a Kerr black hole [44]. The calculation is based on the geometric-optics approximation according to which the polarization vector is parallel along the light ray. In the Kerr spacetime, this parallel-transport law can be explicitly written with the help of constants of motion [61, 277, 316] (cf. [54], p. 358). As to the large number of numerical codes that have been written for calculating imaging properties of a Kerr black hole the reader may consult [175, 329, 285, 108].

The Kerr metric with describes a naked singularity and is considered as unphysical by most authors. Its lightlike geodesics have been studied in [47, 49] (cf. [54], p. 375). The Kerr–Newman spacetime (charged Kerr spacetime) is usually thought to be of little astrophysical relevance because the net charge of celestial bodies is small. For the lightlike geodesics in this spacetime the reader may consult [48, 50].

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