Redshift and Fermat geometry.

Comparison of Equation (69) with the general form (61) of a conformally stationary spacetime shows that
here the redshift potential is a function of only, the Fermat one-form vanishes, and the Fermat
metric is of the special form

Index of refraction and embedding diagrams.

Transformation to an isotropic radius coordinate via

Light cone.

In a spherically symmetric static spacetime, the (past) light cone of an event can be written in terms
of integrals over the metric coefficients. We first restrict to the equatorial plane . The
-geodesics are then determined by the Lagrangian

Exact lens map.

Recall from Section 2.1 that the exact lens map [122] refers to a chosen observation event and a
chosen “source surface” . In general, for we may choose any 3-dimensional submanifold that is
ruled by timelike curves. The latter are to be interpreted as wordlines of light sources. In a spherically
symmetric and static spacetime, we may take advantage of the symmetry by choosing for a
sphere with its ruling by the -lines. This restricts the consideration to lensing for
static light sources. Note that all static light sources at radius undergo the same redshift,
. Without loss of generality, we place the observation event on the 3-axis
at radius . This gives us the past light cone in the representation (81). To each ray from the
observer, with initial direction characterized by , we can assign the total angle the ray
sweeps out on its way from to (see Figure 6). is given by Equation (86),

For calculating image distortion (see Section 2.5) and the brightness of images (see Section 2.6) we have to consider infinitesimally thin bundles with vertex at the observer. In a spherically symmetric and static spacetime, we can apply the orthonormal derivative operators and to the representation (81) of the past light cone. Along each ray, this gives us two Jacobi fields and which span an infinitesimally thin bundle with vertex at the observer. points in the radial direction and points in the tangential direction (see Figure 7). The radial and the tangential direction are orthogonal to each other and, by symmetry, parallel-transported along each ray. Thus, in contrast to the general situation of Figure 3, and are related to a Sachs basis simply by and . The coefficients and are the extremal angular diameter distances of Section 2.4 with respect to a static observer (because the -grid refers to a static observer). In the case at hand, they are called the radial and tangential angular diameter distances. They can be calculated by normalizing and , These formulas have been derived first for the special case of the Schwarzschild metric by Dwivedi and Kantowski [84] and then for arbitrary spherically symmetric static spacetimes by Dyer [85]. (In [85], Equation (92) is erroneously given only for the case that, in our notation, .) From these formulas we immediately get the area distance for a static observer and, with the help of the redshift , the luminosity distance (recall Section 2.4). In this way, Equation (91) and Equation (92) allow to calculate the brightness of images according to the formulas of Section 2.6. Similarly, Equation (91) and Equation (92) allow to calculate image distortion in terms of the ellipticity (recall Section 2.5). In general, is a complex quantity, defined by Equation (49). In the case at hand, it reduces to the real quantity . The expansion and the shear of the bundles under consideration can be calculated from Kantowski’s formula [172, 84], to which Equation (27) reduces in the case at hand. The dot (= derivative with respect to the affine parameter ) is related to the derivative with respect to by Equation (83). Evaluating Equations (91, 92) in connection with the exact lens map leads to quite convenient formulas, for static light sources at . Setting and and comparing with Equation (87) yields (cf. [271]) These formulas immediately give image distortion and the brightness of images if the map is known.

Caustics of light cones.

Quite generally, the past light cone has a caustic point exactly where at least one of the extremal angular
diameter distances , vanishes (see Sections 2.2, 2.3, and 2.4). In the case at hand, zeros of
are called radial caustic points and zeros of are called tangential caustic points (see Figure 8).
By Equation (92), tangential caustic points occur if is a multiple of , i.e., whenever a light ray
crosses the axis of symmetry through the observer (see Figure 8). Symmetry implies that a point source is
seen as a ring (“Einstein ring”) if its worldline crosses a tangential caustic point. By contrast, a
point source whose wordline crosses a radial caustic point is seen infinitesimally extended in
the radial direction. The set of all tangential caustic points of the past light cone is called the
tangential caustic for short. In general, it has several connected components (“first, second,
etc. tangential caustic”). Each connected component is a spacelike curve in spacetime which projects to
(part of) the axis of symmetry through the observer. The radial caustic is a lightlike surface
in spacetime unless at points where it meets the axis; its projection to space is rotationally
symmetric around the axis. The best known example for a tangential caustic, with infinitely
many connected components, occurs in the Schwarzschild spacetime (see Figure 12). It is also
instructive to visualize radial and tangential caustics in terms of instantaneous wave fronts,
i.e., intersections of the light cone with hypersurfaces . Examples are shown in
Figures 13, 18, and 19. By symmetry, a tangential caustic point of an instantaneous wave front can
be neither a cusp nor a swallow-tail. Hence, the general result of Section 2.2 implies that the
tangential caustic is always unstable. The radial caustic in Figure 19 consists of cusps and is, thus,
stable.

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