Figure 1:
Illustration of the exact lens map. is the chosen observation event, is the chosen source surface. is a hypersurface ruled by timelike curves (worldlines of light sources) which are labeled by the points of a 2dimensional manifold . The lens map is defined on the observer’s celestial sphere , given by Equation (1), and takes values in . For each , one follows the lightlike geodesic with this initial direction until it meets and then projects to . For illustrating the exact lens map, it is an instructive exercise to intersect the light cones of Figures 12, 24, 25, and 29 with various source surfaces . 

Figure 2:
Wave fronts that are locally stable in the sense of Arnold. Each picture shows the projection into 3space of a wavefront, locally near a caustic point. The projection is made along the integral curves of a timelike vector field. The qualitative features are independent of which timelike vector field is chosen. In addition to regular, i.e., noncaustic, points , there are five kinds of stable points, known as fold , cusp , swallowtail , pyramid , and purse . The and notation refers to a relation to exceptional groups (see [11]). The picture is taken from [149]. 

Figure 3:
Crosssection of an infinitesimally thin bundle. The Jacobi matrix (19) relates the Jacobi fields and that span the bundle to the Sachs basis vectors and . The shape parameters , , and determine the outline of the crosssection; the angle that appears in Equation (19) does not show in the outline. The picture shows the projection into the 2space (“screen”) spanned by and ; note that, in general, and have components perpendicular to the screen. 

Figure 4:
Pastoriented lightlike geodesic from an observation event to an emission event . is the worldline of the observer, is the worldline of the light source. is the 4velocity of the observer at and is the 4velocity of the light source at . 

Figure 5:
Distortion pattern. The picture shows, in a Mercator projection with as the horizontal and as the vertical coordinate, the celestial sphere of an observer at a spacetime point where the Weyl tensor is of Petrov type . The pattern indicates the elliptical images of spherical objects to within lowest nontrivial order with respect to distance. The length of each line segment is a measure for the eccentricity of the elliptical image, the direction of the line segment indicates its major axis. The distortion effect vanishes at the north pole which corresponds to the fourfold principal null direction. Contrary to the other Petrov types, for type N the pattern is universal up to an overall scaling factor. The picture is taken from [56] where the distortion patterns for the other Petrov types are given as well. 

Figure 6:
Illustration of the exact lens map in spherically symmetric static spacetimes. The picture shows a spatial plane. The observation event (dot) is at , static light sources are distributed at . is the colatitude coordinate on the observer’s sky. It takes values between 0 and . is the angle swept out by the ray with initial direction on its way from to . It takes values between 0 and . In general, neither existence nor uniqueness of is guaranteed for given . A similar picture is in [271]. 

Figure 7:
Thin bundle around a ray in a spherically symmetric static spacetime. The picture is purely spatial, i.e., the time coordinate is not shown. The ray is contained in a plane, so there are two distinguished spatial directions orthogonal to the ray: the “radial” direction (in the plane) and the “tangential” direction (orthogonal to the plane). For a bundle with vertex at the observer, the radial diameter of the crosssection gives the radial angular diameter distance , and the tangential diameter of the crosssection gives the tangential angular diameter distance . In contrast to the general situation of Figure 3, here the angle is zero (if the Sachs basis is chosen appropriately). Recall that and are positive up to the first caustic point. 

Figure 8:
Tangential and radial caustic points. Tangential caustic points, , occur on the axis of symmetry through the observer. A (point) source at a tangential caustic point is seen as a (1dimensional) Einstein ring on the observer’s sky. A point source at a radial caustic point, , appears “infinitesimally extended” in the radial direction. 

Figure 9:
The function for the Schwarzschild metric. Light rays that start at with initial direction are confined to the region where . The equation defines for each a critical value . A light ray from with asymptotically approaches . 

Figure 10:
Index of refraction , given by Equation (105), for the Schwarzschild metric as a function of the isotropic coordinate . 

Figure 11:
Fermat geometry of the equatorial plane of the Schwarzschild spacetime, embedded as a surface of revolution into Euclidean 3space. The neck is at (i.e., ), the boundary of the embeddable part at (i.e., ). The geodesics of this surface of revolution give the light rays in the Schwarzschild spacetime. A similar figure can be found in [4] (also cf. [159]). 

Figure 12:
Past light cone in the Schwarzschild spacetime. One sees that the light cone wraps around the horizon, then forms a tangential caustic. In the picture the caustic looks like a transverse selfintersection because one spatial dimension is suppressed. (Only the hyperplane is shown.) There is no radial caustic. If one follows the light rays further back in time, the light cone wraps around the horizon again and again, thereby forming infinitely many tangential caustics which alternately cover the radius line through the observer and the radius line opposite to the observer. In spacetime, each caustic is a spacelike curve along which ranges from to , whereas ranges from to some maximal value and then back to . Equaltime sections of this light cone are shown in Figure 13. 

Figure 13:
Instantaneous wave fronts of the light cone in the Schwarzschild spacetime. This picture shows intersections of the light cone in Figure 12 with hypersurfaces for four values, with . The instantaneous wave fronts wrap around the horizon and, after reaching the first caustic, have two caustic points each. If one goes further back in time than shown in the picture, the wave fronts another time wrap around the horizon, reach the second caustic, and now have four caustic points each, and so on. In comparison to Figure 12, the representation in terms of instantaneous wave fronts has the advantage that all three spatial dimensions are shown. 

Figure 14:
Escape cones in the Schwarzschild metric, for five values of . For an observer at radius , light sources distributed at a radius with and illuminate a disk whose angular radius is given by Equation (107). The boundary of this disk corresponds to light rays that spiral towards the light sphere at . The disk becomes smaller and smaller for . Figure 9 illustrates that the notion of escape cones is meaningful for any spherically symmetric and static spacetime where has one minimum and no other extrema [253]. For the Schwarzschild spacetime, the escape cones were first mentioned in [249, 224], and explicitly calculated in [319]. A picture similar to this one can be found, e.g., in [54]. 

Figure 15:
Lens map for the Schwarzschild metric. The observer is at , the light sources are at . is the colatitude on the observer’s sky and is the angle swept out by the ray (see Figure 6). was calculated with the help of Equation (87). is restricted by the opening angle of the observer’s escape cone (see Figure 14). Rays with asymptotically spiral towards the light sphere at . The first diagram (cf. [118], Figure 5) shows that ranges from 0 to if ranges from 0 to . So there are infinitely many Einstein rings (dashed lines) whose angular radius approaches . One can analytically prove [212, 246, 39] that the divergence of for is logarithmic. This is true whenever light rays approach an unstable light sphere [37]. The second diagram shows over a logarithmic axis. The graph of approaches a straight line which was called the “strongfield limit” by Bozza et al. [39, 37]. The picture illustrates that it is a good approximation for all light rays that make at least one full turn. The third diagram shows over a logarithmic axis. For every source position one can read the position of the images (dotted line). There are infinitely many, numbered by their order (89) that counts how often the light ray has crossed the axis. Images of odd order are on one side of the black hole, images of even order on the other. For the sources at and one can read the positions of the Einstein rings. 

Figure 16:
Radial angular diameter distance , tangential angular diameter distance and travel time in the Schwarschild spacetime. The data are the same as in Figure 15. For the definition of and see Figure 7. can be calculated from with the help of Equation (94) and Equation (95). For the Schwarzschild case, the resulting formulas are due to [84] (cf. [85, 118]). Zeros of indicate Einstein rings. If and have different signs, the observer sees a sideinverted image. The travel time (= Fermat arclength) can be calculated from Equation (85). One sees that, over the logarithmic axis used here, the graph of approaches a straight line. This illustrates that diverges logarithmically if approaches its limiting value . This can be verified analytically and is characteristic of all cases where light rays approach an unstable light sphere [40]. 

Figure 17:
Luminosity distance and ellipticity (image distortion) in the Schwarzschild spacetime. The data are the same as in Figures 15 and 16. If point sources of equal bolometric luminosity are distributed at , the plotted function gives their magnitude on the observer’s sky, modulo an additive constant . For the calculation of one needs and (see Figure 16), and the general relations (41) and (48). This procedure follows [84] (cf. [85, 118]). For source and observer at large radius, related calculations can also be found in [212, 246, 200, 336]. Einstein rings have magnitude in the rayoptical treatment. For a light source not on the axis, the image of order is fainter than the image of order by magnitudes, see [212, 246]. (This is strictly true in the “strongfield limit”, or “strongbending limit”, which is explained in the caption of Figure 15.) The above picture is similar to Figure 6 in [246]. Note that it refers to point sources and not to a radiating spherical surface of constant surface brightness; by Equation (54), the latter would show a constant intensity. The lower part of the diagram illustrates image distortion in terms of . Clearly, is infinite at each Einstein ring. The doublelogarithmic representation shows that beyond the second Einstein ring all images are extremely elongated in the tangential direction, . Image distortion in the Schwarzschild spacetime is also treated in [85, 120, 119], an approximation formula is derived in [241]. 

Figure 18:
Instantaneous wave fronts in the spacetime of a nontransparent Barriola–Vilenkin monopole with . The picture shows in 3dimensional space the intersections of the past light cone of some event with four hypersurfaces , at values . Only one half of each instantaneous wave front and of the monopole is shown. When the wave front passes the monopole, a hole is pierced into it, then a tangential caustic develops. The caustic of each instantaneous wave front is a point, the caustic of the entire light cone is a spacelike curve in spacetime which projects to part of the axis in 3space. 

Figure 19:
Instantaneous wave fronts in the spacetime of a transparent Barriola–Vilenkin monopole with . In addition to the tangential caustic of Figure 18, a radial caustic is formed. For each instantaneous wave front, the radial caustic is a cusp ridge. The radial caustic of the entire light cone is a lightlike 2surface in spacetime which projects to a rotationally symmetric 2surface in 3space. 

Figure 20:
Metric coefficient for the Janis–Newman–Winicour metric. For , is similar to the Schwarzschild case (see Figure 9). For , has no longer a minimum, i.e., there is no longer a light sphere which can be asymptotically approached by light rays. 

Figure 21:
The region , defined by Equation (128), in the Kerr spacetime. The picture is purely spatial and shows a meridional section , with the axis of symmetry at the lefthand boundary. Through each point of there is a spherical geodesic. Along each of these spherical geodesics, the coordinate oscillates between extremal values, corresponding to boundary points of . The region meets the axis at radius , given by . Its boundary intersects the equatorial plane in circles of radius (corotating circular light ray) and (counterrotating circular light ray). are determined by and . In the Schwarzschild limit the region shrinks to the light sphere . In the extreme Kerr limit the region extends to the horizon because in this limit both and ; for a caveat, as to geometric misinterpretations of this limit (see Figure 3 in [16]). 

Figure 22:
Apparent shape of a Kerr black hole for an observer at radius in the equatorial plane. (For the Schwarzschild analogue, see Figure 14.) The pictures show the celestial sphere of an observer whose 4velocity is perpendicular to a hypersurface . (If the observer is moving one has to correct for aberration.) The dashed circle is the celestial equator, , and the crossing axes indicate the direction towards the center, . Pastoriented light rays go to the horizon if their initial direction is in the black disk and to infinity otherwise. Thus, the black disk shows the part of the sky that is not illuminated by light sources at a large radius. The boundary of this disk corresponds to light rays that asymptotically approach a spherical light ray in the region of Figure 21. For an observer in the equatorial plane at infinity, the apparent shape of a Kerr black hole was correctly calculated and depicted by Bardeen [16] (cf. [54], p. 358). Earlier work by Godfrey [141] contains a mathematical error. 

Figure 23:
On a cone with deficit angle , the point can be connected to every point in the doubleimaging region (shaded) by two geodesics and to a point in the singleimaging region (nonshaded) by one geodesic. 

Figure 24:
Past light cone of an event in the spacetime of a nontransparent string of finite radius with and . The metric (133) is considered on the region , and the light rays are cut if they meet the boundary of this region. The coordinate is not shown, the vertical coordinate is time . The “chimney” indicates the region which is occupied by the string. The light cone has no caustic but a transverse selfintersection (cut locus). The cut locus, in the (2 + 1)dimensional picture represented as a curve, is actually a 2dimensional spacelike submanifold. When passing through the cut locus, the lightlike geodesics leave the boundary of the chronological past . Note that the light cone is not a closed subset of the spacetime. 

Figure 25:
Past light cone of an event in the spacetime of a transparent string of finite radius with and . The metric (133) is matched at to an interior metric, and light rays are allowed to pass through the interior region. The perspective is analogous to Figure 24. The light rays which were blocked by the string in the nontransparent case now form a caustic. In the (2 + 1)dimensional picture the caustic consists of two lightlike curves that meet in a swallowtail point (see Figure 26 for a closeup). Taking the dimension into account, the caustic actually consists of two lightlike 2manifolds (fold surfaces) that meet in a spacelike curve (cusp ridge). The third picture in Figure 2 shows the situation projected to 3space. Each of the pastoriented lightlike geodesics that form the caustic first passes through the cut locus (transverse selfintersection), then smoothly slips over one of the fold surfaces. The fold surfaces are inside the chronological past , the cusp ridge is on its boundary. 

Figure 26:
Closeup of the caustic of Figure 25. The string is not shown. Taking the dimension into account, the swallowtail point is actually a spacelike curve (cusp ridge). 

Figure 27:
Instantaneous wave fronts in the spacetime of a nontransparent string of finite radius with and . The picture shows in 3dimensional space the intersections of the light cone of Figure 24 with three hypersurfaces , at values . The vertical coordinate is the coordinate which was suppressed in Figure 24. Only one half of each instantaneous wave front is shown so that one can look into its interior. There is a transverse selfintersection (cut locus) but no caustic. 

Figure 28:
Instantaneous wave fronts in the spacetime of a transparent string of finite radius with and . The picture is related to Figure 25 as Figure 27 is related to Figure 24. Instantaneous wave fronts that have passed through the string have a caustic, consisting of two cusp ridges that meet in a swallowtail point. This caustic is stable (see Section 2.2). The caustic of the light cone in Figure 25 is the union of the caustics of its instantaneous wave fronts. It consists of two fold surfaces that meet in a cusp ridge, like in the third picture of Figure 2. 

Figure 29:
Past light cone of an event in the spacetime (156) of a plane gravitational wave. The picture was produced with profile functions and . Then there is focusing in the direction and defocusing in the direction. In the (2 + 1)dimensional picture, with the coordinate not shown, the past light cone is completely refocused into a single point , with the exception of one generator . It depends on the profile functions whether there is a second, third, and so on, caustic. In any case, the generators leave the boundary of the chronological past when they pass through the first caustic. Taking the coordinate into account, the first caustic is not a point but a parabola (“astigmatic focusing”) (see Figure 30). An electromagnetic plane wave (vanishing Weyl tensor rather than vanishing Ricci tensor) can refocus a light cone, with the exception of one generator, even into a point in 3 + 1 dimensions (“anastigmatic focusing”) (cf. Penrose [259] where a handdrawing similar to the picture above can be found). 

Figure 30:
“Small wave fronts” of the light cone in the spacetime (156) of a plane gravitational wave. The picture shows the intersection of the light cone of Figure 29 with the lightlike hyperplane for three different values of the constant: (a) exactly at the caustic (parabola), (b) at a larger value of (hyperbolic paraboloid), and (c) at a smaller value of (elliptic paraboloid). In each case, the hyperplane does not intersect the one generator tangent to ; all other generators are intersected transversely and exactly once. 
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