List of Figures

View Image Figure 1:
Illustration of the exact lens map. p O is the chosen observation event, đ’¯ is the chosen source surface. đ’¯ is a hypersurface ruled by timelike curves (worldlines of light sources) which are labeled by the points of a 2-dimensional manifold đ’Ŧ. The lens map is defined on the observer’s celestial sphere 𝒮O, given by Equation (1View Equation), and takes values in đ’Ŧ. For each w ∈ 𝒮O, one follows the lightlike geodesic with this initial direction until it meets đ’¯ and then projects to đ’Ŧ. 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 đ’¯.
View Image Figure 2:
Wave fronts that are locally stable in the sense of Arnold. Each picture shows the projection into 3-space of a wave-front, 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., non-caustic, points (A ) 1, there are five kinds of stable points, known as fold (A2 ), cusp (A3 ), swallow-tail (A4 ), pyramid − (D 4 ), and purse + (D 4 ). The Ak and Dk notation refers to a relation to exceptional groups (see [11]). The picture is taken from [149].
View Image Figure 3:
Cross-section of an infinitesimally thin bundle. The Jacobi matrix (19View Equation) relates the Jacobi fields Y1 and Y2 that span the bundle to the Sachs basis vectors E1 and E2. The shape parameters D+, D −, and χ determine the outline of the cross-section; the angle ψ that appears in Equation (19View Equation) does not show in the outline. The picture shows the projection into the 2-space (“screen”) spanned by E 1 and E 2; note that, in general, Y 1 and Y 2 have components perpendicular to the screen.
View Image Figure 4:
Past-oriented lightlike geodesic λ from an observation event pO to an emission event pS. γO is the worldline of the observer, γS is the worldline of the light source. UO is the 4-velocity of the observer at p O and U S is the 4-velocity of the light source at p S.
View Image Figure 5:
Distortion pattern. The picture shows, in a Mercator projection with Φ as the horizontal and Θ as the vertical coordinate, the celestial sphere of an observer at a spacetime point where the Weyl tensor is of Petrov type N. The pattern indicates the elliptical images of spherical objects to within lowest non-trivial 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 Θ = 0 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.
View Image 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 r = rO, static light sources are distributed at r = r S. Θ is the colatitude coordinate on the observer’s sky. It takes values between 0 and π. Φ (Θ ) is the angle swept out by the ray with initial direction Θ on its way from rO to rS. It takes values between 0 and ∞. In general, neither existence nor uniqueness of Φ (Θ) is guaranteed for given Θ. A similar picture is in [271].
View Image Figure 7:
Thin bundle around a ray in a spherically symmetric static spacetime. The picture is purely spatial, i.e., the time coordinate t 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 cross-section gives the radial angular diameter distance D+, and the tangential diameter of the cross-section gives the tangential angular diameter distance D−. In contrast to the general situation of Figure 3, here the angle χ is zero (if the Sachs basis (E ,E ) 1 2 is chosen appropriately). Recall that D+ and D − are positive up to the first caustic point.
View Image Figure 8:
Tangential and radial caustic points. Tangential caustic points, D − = 0, occur on the axis of symmetry through the observer. A (point) source at a tangential caustic point is seen as a (1-dimensional) Einstein ring on the observer’s sky. A point source at a radial caustic point, D+ = 0, appears “infinitesimally extended” in the radial direction.
View Image Figure 9:
The function R (r) for the Schwarzschild metric. Light rays that start at rO with initial direction Θ are confined to the region where R (r) ≥ R (rO) sin Θ. The equation R (3m ) = R (rO )sinδ defines for each rO a critical value δ. A light ray from rO with Θ = δ asymptotically approaches r = 3m.
View Image Figure 10:
Index of refraction n (&tidle;r), given by Equation (105View Equation), for the Schwarzschild metric as a function of the isotropic coordinate &tidle;r.
View Image Figure 11:
Fermat geometry of the equatorial plane of the Schwarzschild spacetime, embedded as a surface of revolution into Euclidean 3-space. The neck is at r = 3m (i.e., &tidle;r ≈ 1.87m), the boundary of the embeddable part at r = 2.25m (i.e., &tidle;r = m). 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]).
View Image 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 self-intersection because one spatial dimension is suppressed. (Only the hyperplane 𝜗 = π ∕2 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 r ranges from 2m to ∞, whereas t ranges from − ∞ to some maximal value and then back to − ∞. Equal-time sections of this light cone are shown in Figure 13.
View Image 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 t = constant for four t-values, with t1 > t2 > t3 > t4. 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.
View Image Figure 14:
Escape cones in the Schwarzschild metric, for five values of rO. For an observer at radius rO, light sources distributed at a radius rS with rS > rO and rS > 3m illuminate a disk whose angular radius δ is given by Equation (107View Equation). The boundary of this disk corresponds to light rays that spiral towards the light sphere at r = 3m. The disk becomes smaller and smaller for rO → 2m. Figure 9 illustrates that the notion of escape cones is meaningful for any spherically symmetric and static spacetime where R has one minimum and no other extrema [253]. For the Schwarzschild spacetime, the escape cones were first mentioned in [249224], and explicitly calculated in [319]. A picture similar to this one can be found, e.g., in [54].
View Image Figure 15:
Lens map for the Schwarzschild metric. The observer is at rO = 5m, the light sources are at rS = 10m. Θ is the colatitude on the observer’s sky and Φ(Θ ) is the angle swept out by the ray (see Figure 6). Φ (Θ ) was calculated with the help of Equation (87View Equation). Θ is restricted by the opening angle δ of the observer’s escape cone (see Figure 14). Rays with Θ = δ asymptotically spiral towards the light sphere at r = 3m. The first diagram (cf. [118], Figure 5) shows that Φ (Θ ) ranges from 0 to ∞ if Θ ranges from 0 to δ. So there are infinitely many Einstein rings (dashed lines) whose angular radius approaches δ. One can analytically prove [21224639] that the divergence of Φ (Θ ) for Θ → δ is logarithmic. This is true whenever light rays approach an unstable light sphere [37]. The second diagram shows Φ (Θ) over a logarithmic Θ-axis. The graph of Φ approaches a straight line which was called the “strong-field limit” by Bozza et al. [3937]. The picture illustrates that it is a good approximation for all light rays that make at least one full turn. The third diagram shows cosΦ (Θ ) over a logarithmic Θ-axis. For every source position 0 < 𝜗 < π one can read the position of the images (dotted line). There are infinitely many, numbered by their order (89View Equation) 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 𝜗 = π and 𝜗 = 0 one can read the positions of the Einstein rings.
View Image Figure 16:
Radial angular diameter distance D+ (Θ ), tangential angular diameter distance D − (Θ ) and travel time T (Θ) in the Schwarschild spacetime. The data are the same as in Figure 15. For the definition of D+ and D − see Figure 7. D ±(Θ ) can be calculated from Φ (Θ) with the help of Equation (94View Equation) and Equation (95View Equation). For the Schwarzschild case, the resulting formulas are due to [84] (cf. [85118]). Zeros of D − indicate Einstein rings. If D + and D − have different signs, the observer sees a side-inverted image. The travel time T (Θ) (= Fermat arclength) can be calculated from Equation (85View Equation). One sees that, over the logarithmic Θ-axis used here, the graph of T approaches a straight line. This illustrates that T (Θ ) diverges logarithmically if Θ approaches its limiting value δ. This can be verified analytically and is characteristic of all cases where light rays approach an unstable light sphere [40].
View Image Figure 17:
Luminosity distance D (Θ ) lum and ellipticity 𝜀(Θ ) (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 r = rS, the plotted function 2.5log10(Dlum (Θ)2) gives their magnitude on the observer’s sky, modulo an additive constant m0. For the calculation of Dlum one needs D+ and D − (see Figure 16), and the general relations (41View Equation) and (48View Equation). This procedure follows [84] (cf. [85118]). For source and observer at large radius, related calculations can also be found in [212246200336]. Einstein rings have magnitude − ∞ in the ray-optical treatment. For a light source not on the axis, the image of order i + 2 is fainter than the image of order i by 2.5 log10(e2π) ≈ 6.8 magnitudes, see [212246]. (This is strictly true in the “strong-field limit”, or “strong-bending 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 r = rS of constant surface brightness; by Equation (54View Equation), the latter would show a constant intensity. The lower part of the diagram illustrates image distortion in terms of 𝜀 = D−-− D+- D+ D−. Clearly, |𝜀| is infinite at each Einstein ring. The double-logarithmic representation shows that beyond the second Einstein ring all images are extremely elongated in the tangential direction, |𝜀| > 100. Image distortion in the Schwarzschild spacetime is also treated in [85120119], an approximation formula is derived in [241].
View Image Figure 18:
Instantaneous wave fronts in the spacetime of a non-transparent Barriola–Vilenkin monopole with k = 0.8. The picture shows in 3-dimensional space the intersections of the past light cone of some event with four hypersurfaces t = constant, at values t1 > t2 > t3 > t4. 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 3-space.
View Image Figure 19:
Instantaneous wave fronts in the spacetime of a transparent Barriola–Vilenkin monopole with k = 0.8. 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 2-surface in spacetime which projects to a rotationally symmetric 2-surface in 3-space.
View Image Figure 20:
Metric coefficient R (r) for the Janis–Newman–Winicour metric. For 1 < γ < 1 2, R (r) is similar to the Schwarzschild case γ = 1 (see Figure 9). For 1 γ ≤ 2, R(r) has no longer a minimum, i.e., there is no longer a light sphere which can be asymptotically approached by light rays.
View Image Figure 21:
The region đ’Ļ, defined by Equation (128View Equation), in the Kerr spacetime. The picture is purely spatial and shows a meridional section φ = constant, with the axis of symmetry at the left-hand boundary. Through each point of đ’Ļ there is a spherical geodesic. Along each of these spherical geodesics, the coordinate 𝜗 oscillates between extremal values, corresponding to boundary points of đ’Ļ. The region đ’Ļ meets the axis at radius rc, given by r3c − 3mr2c + a2rc + ma2 = 0. Its boundary intersects the equatorial plane in circles of radius rph + (corotating circular light ray) and ph r− (counter-rotating circular light ray). ph r± are determined by ph ph 2 2 r± (r± − 3m ) = 4ma and r+ < rp+h< 3m < rp−h< 4m. In the Schwarzschild limit a → 0 the region đ’Ļ shrinks to the light sphere r = 3m. In the extreme Kerr limit a → m the region đ’Ļ extends to the horizon because in this limit both rph→ m + and r → m +; for a caveat, as to geometric misinterpretations of this limit (see Figure 3 in [16]).
View Image Figure 22:
Apparent shape of a Kerr black hole for an observer at radius rO in the equatorial plane. (For the Schwarzschild analogue, see Figure 14.) The pictures show the celestial sphere of an observer whose 4-velocity is perpendicular to a hypersurface t = constant. (If the observer is moving one has to correct for aberration.) The dashed circle is the celestial equator, Θ = π∕2, and the crossing axes indicate the direction towards the center, Θ = π. Past-oriented 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 đ’Ļ 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.
View Image Figure 23:
On a cone with deficit angle 0 < δ < π, the point p can be connected to every point q in the double-imaging region (shaded) by two geodesics and to a point in the single-imaging region (non-shaded) by one geodesic.
View Image Figure 24:
Past light cone of an event pO in the spacetime of a non-transparent string of finite radius ρ∗ with k = 0.8 and a = 0. The metric (133View Equation) is considered on the region ρ > ρ∗, and the light rays are cut if they meet the boundary of this region. The z coordinate is not shown, the vertical coordinate is time t. The “chimney” indicates the region ρ < ρ ∗ which is occupied by the string. The light cone has no caustic but a transverse self-intersection (cut locus). The cut locus, in the (2 + 1)-dimensional picture represented as a curve, is actually a 2-dimensional spacelike submanifold. When passing through the cut locus, the lightlike geodesics leave the boundary of the chronological past I− (pO). Note that the light cone is not a closed subset of the spacetime.
View Image Figure 25:
Past light cone of an event p O in the spacetime of a transparent string of finite radius ρ∗ with k = 0.8 and a = 0. The metric (133View Equation) is matched at ρ = ρ ∗ 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 non-transparent case now form a caustic. In the (2 + 1)-dimensional picture the caustic consists of two lightlike curves that meet in a swallow-tail point (see Figure 26 for a close-up). Taking the z-dimension into account, the caustic actually consists of two lightlike 2-manifolds (fold surfaces) that meet in a spacelike curve (cusp ridge). The third picture in Figure 2 shows the situation projected to 3-space. Each of the past-oriented lightlike geodesics that form the caustic first passes through the cut locus (transverse self-intersection), then smoothly slips over one of the fold surfaces. The fold surfaces are inside the chronological past − I (pO), the cusp ridge is on its boundary.
View Image Figure 26:
Close-up of the caustic of Figure 25. The string is not shown. Taking the z-dimension into account, the swallow-tail point is actually a spacelike curve (cusp ridge).
View Image Figure 27:
Instantaneous wave fronts in the spacetime of a non-transparent string of finite radius ρ∗ with k = 0.8 and a = 0. The picture shows in 3-dimensional space the intersections of the light cone of Figure 24 with three hypersurfaces t = constant, at values t1 > t2 > t3. The vertical coordinate is the z-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 self-intersection (cut locus) but no caustic.
View Image Figure 28:
Instantaneous wave fronts in the spacetime of a transparent string of finite radius ρ∗ with k = 0.8 and a = 0. 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 swallow-tail 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.
View Image Figure 29:
Past light cone of an event pO in the spacetime (156View Equation) of a plane gravitational wave. The picture was produced with profile functions f (u ) > 0 and g(u) = 0. Then there is focusing in the x-direction and defocusing in the y-direction. In the (2 + 1)-dimensional picture, with the y-coordinate not shown, the past light cone is completely refocused into a single point q, with the exception of one generator λ. 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 − I (pO) when they pass through the first caustic. Taking the y-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 hand-drawing similar to the picture above can be found).
View Image Figure 30:
“Small wave fronts” of the light cone in the spacetime (156View Equation) of a plane gravitational wave. The picture shows the intersection of the light cone of Figure 29 with the lightlike hyperplane u = constant for three different values of the constant: (a) exactly at the caustic (parabola), (b) at a larger value of u (hyperbolic paraboloid), and (c) at a smaller value of u (elliptic paraboloid). In each case, the hyperplane u = constant does not intersect the one generator λ tangent to ∂v; all other generators are intersected transversely and exactly once.