4.3 Lensing in spherically
symmetric and static spacetimes
The class of spherically symmetric and static spacetimes is of
particular relevance in view of lensing, because it includes models
for non-rotating stars and black holes (see Sections 5.1, 5.2, 5.3), but also for more exotic
objects such as wormholes (see Section 5.4), monopoles (see Section 5.5), naked singularities (see
Section 5.6), and Boson or Fermion stars (see
Section 5.7). Here we collect the relevant
formulas for an unspecified spherically symmetric and static
metric. We find it convenient to write the metric in the form
As Equation (69) is a special case of
Equation (61), all results of
Section 4.2 for conformally stationary
metrics apply. However, much stronger results are possible because
for metrics of the form (69) the geodesic equation
is completely integrable. Hence, all relevant quantities can be
determined explicitly in terms of integrals over the metric
coefficients.
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
By Fermat’s principle, the geodesics of
coincide with the
projection to 3-space of light rays. The travel time (in terms of
the time coordinate
) of a lightlike curve coincides with
the
-arclength of its projection. By symmetry, every
-geodesic stays in a plane through the origin. From
Equation (70) we read that the
sphere of radius
has area
with respect
to the Fermat metric. Also, Equation (70) implies that the
second fundamental form of this sphere is a multiple of its first
fundamental form, with a factor
. If
the sphere
is totally geodesic, i.e., a
-geodesic that starts tangent to this sphere remains
in it. The best known example for such a light sphere or photon
sphere is the sphere
in the
Schwarzschild spacetime (see Section 5.1). Light spheres also occur in the
spacetimes of wormholes (see Section 5.4). If
, the
circular light rays in a light sphere are stable with respect to
radial perturbations, and if
, they
are unstable like in the Schwarschild case. The condition under
which a spherically symmetric static spacetime admits a light
sphere was first given by Atkinson [13
].
Abramowicz [1] has
shown that for an observer traveling along a circular light orbit
(with subluminal velocity) there is no centrifugal force and no
gyroscopic precession. Claudel, Virbhadra, and Ellis [59
] investigated, with
the help of Einstein’s field equation and energy conditions, the
amount of matter surrounded by a light sphere. Among other things,
they found an energy condition under which a spherically symmetric
static black hole must be surrounded by a light sphere. A purely
kinematical argument shows that any spherically symmetric and
static spacetime that has a horizon at
and is
asymptotically flat for
must
contain a light sphere at some radius between
and
(see Hasse and Perlick [153
]). In the same
article, it is shown that in any spherically symmetric static
spacetime with a light sphere there is gravitational lensing with
infinitely many images. Bozza [37
] investigated a
strong-field limit of lensing in
spherically symmetric static spacetimes, as opposed to the
well-known weak-field limit, which applies to light rays that come
close to an unstable light sphere. (Actually, the term
“strong-bending limit” would be more appropriate because the
gravitational field, measured in terms of tidal forces, need not be
particularly strong near an unstable light sphere.) This limit
applies, in particular, to light rays that approach the sphere
in the Schwarzschild spacetime (see [39
] and, for
illustrations, Figures 15, 16, and 17).
Index of refraction and
embedding diagrams.
Transformation to an isotropic radius
coordinate
via
takes the Fermat metric (70) to the form
where
On the right-hand side
has to be expressed by
with the help of Equation (72). The results of
Section 4.2 imply that the lightlike
geodesics in a spherically symmetric static spacetime are
equivalent to the light rays in a medium with index of
refraction (74) on Euclidean 3-space.
For arbitrary metrics of the form (69), this result is due
to Atkinson [13
]. It reduces the
lightlike geodesic problem in a spherically symmetric static
spacetime to a standard problem in ordinary optics, as treated,
e.g., in [212], §27, and [199],
Section 4. One can combine this result with our earlier
observation that the integral in Equation (67) has the same form as
the functional in Maupertuis’ principle in classical mechanics.
This demonstrates that light rays in spherically symmetric and
static spacetimes behave like particles in a spherically symmetric
potential on Euclidean 3-space (cf., e.g., [105
]). If the embeddability condition
is satisfied, we define a function
by
Then the Fermat metric (70) reads
If restricted to the equatorial plane
, the
metric (77) describes a surface
of revolution, embedded into Euclidean 3-space as
Such embeddings of the Fermat geometry have been visualized for
several spacetimes of interest (see Figure 11 for the Schwarzschild
case and [160
] for other
examples). This is quite instructive because from a picture of a
surface of revolution one can read the qualitative features of its
geodesics without calculating them. Note that Equation (72) defines the isotropic
radius coordinate uniquely up to a multiplicative constant. Hence,
the straight lines in this coordinate representation give us an
unambiguously defined reference grid for every spherically
symmetric and static spacetime. These straight lines have been
called triangulation lines
in [62, 63], where their
use for calculating bending angles, exactly or approximately, is
outlined.
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
For fixed radius value
, initial conditions
determine a unique solution
,
of the Euler-Lagrange equations.
measures the initial direction with respect to the
symmetry axis (see Figure 6). We get all light
rays issuing from the event
,
,
,
into the past
by letting
range from 0 to
and applying rotations around the symmetry axis. This
gives us the past light cone of this event in the form
and
are spherical coordinates on the
observer’s sky. If we let
float over
, we get the observational coordinates (4) for an observer on a
-line, up to two modifications. First,
is not the same as proper time
; however, they are related just by a constant,
Second,
is not the same as the affine parameter
; along a ray with initial direction
, they are related by
The constants of motion
give us the functions
,
in terms of
integrals,
Here the notation with the dots is a short-hand; it means that the
integral is to be decomposed into sections where
is a monotonous function of
, and that the absolute value of the integrals over
all sections have to be added up. Turning points occur at radius
values where
and
(see Figure 9). If the metric
coefficients
and
have been specified, these
integrals can be calculated and give us the light cone (see
Figure 12 for an example).
Having parametrized the rays with
-arclength (= travel
time), we immediately get the intersections of the light cone with
hypersurfaces
(“instantaneous wave
fronts”); see Figures 13, 18, and 19.
Exact lens
map.
Recall from Section 2.1 that the exact lens
map [123
] 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),
where the same short-hand notation is used as in Equation (86).
is not necessarily defined for all
because not all light rays that start at
may reach
. Also,
may be multi-valued because a light ray may
intersect the sphere
several times.
Equation (81) gives us the
(possibly multi-valued) lens map
It assigns to each point on the observer’s sky the position of the
light source which is seen at that point.
may take all values between 0 and infinity. For each
image we can define the order
which counts how often the ray has met the axis. The standard
example where images of arbitrarily high order occur is the
Schwarzschild spacetime (see Section 5.1). For many, though not all,
applications one may restrict to the case that the spacetime is
asymptotically flat and that
and
are so large that the spacetime is almost flat at
these radius values. For a light ray with turning point at
, Equation (87) can then be
approximated by
If the entire ray remains in the region where the spacetime is
almost flat, Equation (90) gives the usual
weak-field approximation of light bending with
close to
. However, Equation (90) does not require that
the ray stays in the region that is almost flat. The integral in
Equation (90) becomes arbitrarily
large if
comes close to an unstable light
sphere,
and
. This
situation is well known to occur in the Schwarzschild spacetime
with
(see Section 5.1, in particular Figures 9, 14, and 15). The divergence of
is always logarithmic [37
]. Virbhadra and
Ellis [337
] (cf. [339
] for an earlier
version) approximately evaluate Equation (90) for the case that
source and observer are almost aligned, i.e., that
is close to an odd multiple of
. This corresponds to replacing the sphere at
with its tangent plane. The resulting “almost exact
lens map” takes an intermediary position between the exact
treatment and the quasi-Newtonian approximation. It was originally
introduced for the Schwarzschild metric [337
] where it
approximates the exact treatment remarkably well within a wide
range of validity [119
]. On the other hand,
neither analytical nor numerical evaluation of the “almost exact
lens map” is significantly easier than that of the exact lens map.
For situations where the assumption of almost perfect alignment
cannot be maintained the Virbhadra-Ellis lens equation must be
modified (see [70
]; related material
can also be found in [38
]).
4.3.0.1 Distance measures, image distortion and brightness
of images.
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 [173, 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.