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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
2-dimensional 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 . |
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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 , there are five kinds of stable points,
known as fold , cusp , swallow-tail , pyramid , and
purse . The and notation refers to a relation
to exceptional groups (see [11]). The picture
is taken from [150]. |
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Figure 3:
Cross-section 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
cross-section; the angle that appears in Equation (19) does not show in the outline. The picture shows
the projection into the 2-space
(“screen”) spanned by and ; note that, in general, and have components perpendicular to the screen. |
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Figure 4:
Past-oriented 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 4-velocity of the observer at and is the 4-velocity of the
light source at . |
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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
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 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. |
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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]. |
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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 cross-section gives the radial
angular diameter distance , and the
tangential diameter of the
cross-section 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. |
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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
(1-dimensional) Einstein ring on the observer’s sky. A point source
at a radial caustic point, , appears
“infinitesimally extended” in the radial
direction. |
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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 . |
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Figure 10:
Index of
refraction , given by
Equation (105), for the Schwarzschild metric as a function of the isotropic coordinate . |
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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 (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. [160]). |
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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 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 . Equal-time sections of this light cone are shown in Figure 13. |
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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. |
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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, 223], and
explicitly calculated in [320]. A picture
similar to this one can be found, e.g., in [54]. |
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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. [119], 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 [211, 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 “strong-field 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. |
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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, 119]). Zeros
of
indicate Einstein rings. If and have different signs, the
observer sees a side-inverted 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]. |
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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, 119]). For source
and observer at large radius, related calculations can also be found in [211, 246, 201, 337]. Einstein
rings have magnitude in the ray-optical
treatment. For a light source not on the
axis, the image of order is fainter than
the image of order by magnitudes, see [211, 246]. (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 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
double-logarithmic 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, 121, 120], an
approximation formula is derived in [240]. |
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Figure 18:
Instantaneous wave fronts
in the spacetime of a non-transparent Barriola-Vilenkin
monopole with . The picture shows in 3-dimensional 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 3-space. |
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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 2-surface in spacetime which projects to a
rotationally symmetric 2-surface in 3-space. |
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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. |
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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 left-hand 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 (counter-rotating 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]). |
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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 4-velocity 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, . 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
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 [142] contains a
mathematical error. |
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Figure 23:
On a cone with deficit
angle , the point can be connected to every point in the double-imaging region (shaded) by two
geodesics and to a point in the single-imaging region (non-shaded) by one geodesic. |
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Figure 24:
Past light cone of an
event
in the spacetime of a non-transparent 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
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 . Note that the light cone is not a closed subset
of the spacetime. |
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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 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 -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 , the cusp ridge is on
its boundary. |
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Figure 26:
Close-up of the caustic
of Figure 25. The string is not shown. Taking the -dimension into
account, the swallow-tail point is actually a spacelike curve (cusp
ridge). |
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Figure 27:
Instantaneous wave fronts
in the spacetime of a non-transparent string of finite
radius
with and . The picture shows in 3-dimensional 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 self-intersection (cut locus) but no
caustic. |
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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 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. |
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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
hand-drawing similar to the picture above can be
found). |
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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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