5.1 Schwarzschild
spacetime
The (exterior) Schwarzschild metric
has the form (69) with
It is the unique spherically symmetric vacuum solution of
Einstein’s field equation. At the same time, it is the most
important and best understood spacetime in which lensing can be
explicitly studied without approximations. Schwarzschild lensing
beyond the weak-field approximation has astrophysical relevance in
view of black holes and neutron stars. The increasing evidence that
there is a supermassive black hole at the center of our Galaxy
(see [107
] for background
material) is a major motivation for a detailed study of
Schwarzschild lensing (and of Kerr lensing; see Section 5.8). In the following we consider
the Schwarzschild metric with a constant
and we ignore the region
for which the
singularity at
is naked. The Schwarzschild metric is
static on the region
.
(The region
for
is equivalent
to the region
for
. It is
usually considered as unphysical but has found some recent interest
in connection with lensing by wormholes; see Section 5.4.)
Historical
notes.
Shortly after the discovery of the Schwarzschild metric by
Schwarzschild [302]
and independently by Droste [80], basic
features of its lightlike geodesics were found by Flamm [114],
Hilbert [158
], and
Weyl [347
]. Detailed studies
of its timelike and lightlike geodesics were made by
Hagihara [146] and
Darwin [72
, 73
]. For a fairly
complete list of the pre-1979 literature on Schwarzschild geodesics
see Sharp [306
]. All modern
text-books on general relativity include a section on Schwarzschild
geodesics, but not all of them go beyond the weak-field
approximation. For a particularly detailed exposition see
Chandrasekhar [54
].
Redshift and Fermat
geometry.
The redshift potential
for the Schwarzschild metric
is given in Equation (101). With the help of
we can directly calculate the redshift via
Equation (68) if observer and light
source are static (i.e.,
-lines). If the light source
or the observer does not follow a
-line, a Doppler
factor has to be added. Independent of the velocity of observer and
light source, the redshift becomes arbitrarily large if the light
source is sufficiently close to the horizon. For light source and
observer freely falling, the redshift formula was discussed by
Bażański and Jaranowski [24]. If projected to
3-space, the light rays in the Schwarzschild spacetime are the
geodesics of the Fermat metric which can be read from
Equation (70)
(cf. Frankel [116]),
The metric coefficient
, as given by
Equation (101), has a strict minimum
at
and no other extrema (see Figure 9). Hence, there is an
unstable light sphere at this radius (recall Equation (71)). The existence of
circular light rays at
was noted already by
Hilbert [158]. The
relevance of these circular light rays in view of lensing was
clearly seen by Darwin [72
, 73] and
Atkinson [13
]. They realized, in
particular, that a Schwarzschild black hole produces infinitely
many images of each light source, corresponding to an infinite
sequence of light rays that asymptotically spiral towards a
circular light ray. The circular light rays at
are also associated with other physical effects such
as centrifugal force reversal and “locking” of gyroscopes. These
effects have been discussed with the help of the Fermat geometry (=
optical reference geometry) in various articles by Abramowicz and
collaborators (see, e.g., [5, 4
, 6, 2]).
Index of refraction and embedding
diagrams.
We know from Section 4.3 that light rays in any spherically
symmetric and static spacetime can be characterized by an index of
refraction. This requires introducing an isotropic radius
coordinate
via Equation (72). In the Schwarzschild
case,
is related to the Schwarzschild radius coordinate
by
ranges from
to infinity if
ranges from
to infinity. In terms of the isotropic
coordinate, the Fermat metric (102) takes the form
with
Hence, light propagation in the Schwarzschild metric can be
mimicked by the index of refraction (105); see Figure 10. The index of
refraction (105) is known since
Weyl [349]. It was employed
for calculating lightlike Schwarzschild geodesics, exactly or
approximately, e.g., in [13
, 231, 106, 204]. This index of
refraction can be modeled by a fluid flow [288]. The
embeddability condition (75) is satisfied for
(which coincides with the so-called Buchdahl limit). On this domain the Fermat
geometry, if restricted to the equatorial plane
, can be represented as a surface of revolution in
Euclidean 3-space (see Figure 11). The entire region
can be embedded into a space of constant negative
curvature [3
].
Lensing by a Schwarzschild black hole.
To get a Schwarzschild black hole, one joins at
the static Schwarzschild region
to the non-static Schwarzschild region
in such a way that ingoing light rays can cross this surface
but outgoing cannot. If the
observation event
is at
, only the
region
is of relevance for lensing, because
the past light cone of such an event does not intersect the
black-hole horizon at
. (For a Schwarzschild white
hole see below.) Such a light cone is depicted in Figure 12 (cf. [183
]). The picture was
produced with the help of the representation (81) which requires
integrating Equation (85) and Equation (86). For the
Schwarzschild case, these are elliptical integrals. Their numerical
evaluation is an exercise for students (see [45] for a MATHEMATICA
program). Note that the evaluation of Equation (85) and Equation (86) requires knowledge of
the turning points. In the Schwarzschild case, there is at most one
turning point
along each ray (see Figure 9), and it is given by
the cubic equation
The representation (81) in terms of Fermat
arclength
(= travel time) gives us the
intersections of the light cone with hypersurfaces
. These “instantaneous wave fronts” are depicted in
Figure 13 (cf. [147
]). With the light
cone explicitly known, one can analytically verify that every
inextendible timelike curve in the region
intersects the light cone infinitely many times,
provided it is bounded away from the horizon and from (past
lightlike) infinity. This shows that the observer sees infinitely
many images of a light source with this worldline. The same result
can be proven with the help of Morse theory (see Section 3.3), where one has to exclude the
case that the worldline meets the caustic of the light cone. In the
latter case the light source is seen as an Einstein ring. For
static light sources (i.e.,
-lines), either all images are
Einstein rings or none. For such light sources we can study lensing
in the exact-lens-map formulation of Section 4.3 (see in particular Figure 6). Also,
Section 4.3 provides us with formulas for
distance measures, brightness, and image distortion which we just
have to specialize to the Schwarzschild case. For another treatment
of Schwarzschild lensing with the help of the exact lens map,
see [119
]. We place our
static light sources at radius
. If
and
, only light rays with
,
can reach the radius value
(see Figure 9). Rays with
asymptotically spiral towards the light sphere at
.
lies between 0 and
for
and between
and
for
. The escape cone defined by Equation (107) is depicted, for
different values of
, in Figure 14. It gives the domain
of definition for the lens map. The lens map is graphically
discussed in Figure 15. The pictures are
valid for
and
.
Qualitatively, however, they look the same for all cases with
and
. From the diagram one can
read the position of the infinitely many images for each light
source which, for the two light sources on the axis, degenerate
into infinitely many Einstein rings. For each fixed source, the
images are ordered by the number
(
) which counts how often the ray has met the axis.
This coincides with ordering according to travel time. With
increasing order
, the images come closer and closer to
the rim at
(see Figure 15). They are
alternately upright and side-inverted (see Figure 16), and their
brightness rapidly decreases (see Figure 17). These basic
features of Schwarzschild lensing are known since pioneering papers
by Darwin [72] and
Atkinson [13]
(cf. [211
, 246
, 201
]). A detailed study
of Schwarzschild lensing was carried through by Virbhadra and
Ellis [337
] with the help of an
“almost exact lens map” (see Section 4.3). The latter assumes that observer
and light source are in the asymptotic region and almost aligned,
but the light rays are allowed to make arbitrarily many turns
around the black hole. Various methods of how multiple imaging by a
black hole could be discovered, directly or indirectly, have been
discussed [211
, 201
, 15
, 14
, 274, 76].
Related work has also been done for Kerr black holes (see
Section 5.8). An interesting suggestion was
made in [162]. A Schwarzschild black
hole, somewhere in the universe, would send photons originating
from our Sun back to the vicinity of our Sun (“boomerang
photons” [316]). If the
black hole is sufficiently close to our Solar system, this would
produce images of our own Sun on the sky that could be detectable.
The lensing effect of a Schwarzschild black hole has been
visualized in two ways:
- by showing the visual appearance of some
background pattern as distorted by the black hole [66, 296, 233] (only
primary images,
, are considered), and
- by showing the visual appearence of an
accretion disk around the black hole [211
, 130, 15, 14] (higher-order
images are taken into account).
Numerous ray tracing programs have been
developed for the Schwarzschild metric and, more generally, for the
Kerr metric (see Section 5.8).
Lensing by a non-transparent Schwarzschild
star.
To model a non-transparent star of radius
one has to restrict the exterior Schwarzschild
metric to the region
. Lightlike geodesics
terminate when they arrive at
. The star’s
radius cannot be smaller than
unless it is allowed to be
time-dependent. The qualitative features of lensing depend on
whether
is bigger than
. Stars with
are called ultracompact [166]. Their
existence is speculative. The lensing properties of an ultracompact
star are the same as that of a Schwarzschild black hole of the same
mass, for observer and light source in the region
. In particular, the apparent angular radius
on the observer’s sky of an ultracompact star is
given by the escape cone of Figure 14. Also, an
ultracompact star produces the same infinite sequence of images of
each light source as a black hole. For
, only
finitely many of the images survive because the other lightlike
geodesics are blocked. A non-transparent star has a finite focal
length
in the sense that parallel light from
infinity is focused along a line that extends from radius value
to infinity.
depends on
and on
. For the values of our Sun
one finds
au (1 au = 1 astronomical unit =
average distance from the Earth to the Sun). An observer at
can observe strong lensing effects of the Sun on
distant light sources. The idea of sending a spacecraft to
was occasionally discussed in the
literature [103, 234
, 326]. The lensing
properties of a non-transparent Schwarzschild star have been
illustrated by showing the appearance of the star’s surface to a
distant observer. For
bigger than but of the same
order of magnitude as
, this has relevance for
neutron stars (see [352, 256, 129, 287, 221, 239]).
may be chosen time-dependent, e.g., to model a
non-transparent collapsing star. A star starting with
cannot reach
in finite
Schwarzschild coordinate time
(though in finite proper time
of an observer at the star’s surface), i.e., for a collapsing star
one has
for
. To a distant observer, the total luminosity of a
freely (geodesically) collapsing star is attenuated exponentially,
. This formula was first derived by
Podurets [279] with an
incorrect factor 2 under the exponent and corrected by Ames and
Thorne [8].
Both papers are based on kinetic photon theory (Liouville’s
equation). An alternative derivation of the luminosity formula,
based on the optical scalars, was given by Dwivedi and
Kantowski [84]. Ames and Thorne also
calculated the spectral distribution of the radiation as a function
of time and position on the apparent disk of the star. All these
analyses considered radiation emitted at an angle
against the normal of the star as measured by a
static (Killing) observer. Actually, one has to refer not to a
static observer but to an observer comoving with the star’s
surface. This modification was worked out by Lake and
Roeder [198].
Lensing by a transparent
Schwarzschild star.
To model a transparent star of radius
one has to join the
exterior Schwarzschild metric at
to an interior
(e.g., perfect fluid) metric. Lightlike geodesics of the exterior
Schwarzschild metric are to be joined to lightlike geodesics of the
interior metric when they arrive at
. The radius
of the star can be time-independent only if
. For
(ultracompact star), the lensing
properties for observer and light source in the region
differ from the black hole case only by the possible
occurrence of additional images, corresponding to light rays that
pass through the star. Inside such a transparent ultracompact star,
there is at least one stable photon sphere, in addition to the
unstable one at
outside the star (cf. [153
]). In principle,
there may be arbitrarily many photon spheres [177]. For
, the lensing properties depend on
whether there are light rays trapped inside the star. For a perfect
fluid with constant density, this is not the case; the resulting
spacetime is then asymptotically simple, i.e., all inextendible
light rays come from infinity and go to infinity. General results
(see Section 3.4) imply that then the number of
images must be finite and odd. The light cone in this
exterior-plus-interior Schwarzschild spacetime is discussed in
detail by Kling and Newman [183].
(In this paper the authors constantly refer to their interior
metric as to a “dust” where obviously a perfect fluid with constant
density is meant.) Effects on light rays issuing from the star’s
interior have been discussed already earlier by Lawrence [203]. The
“escape cones”, which are shown in Figure 14 for the exterior
Schwarzschild metric have been calculated by Jaffe [167] for points
inside the star. The focal length of a transparent star with
constant density is smaller than that of a non-transparent star of
the same mass and radius. For the mass and the radius of our Sun,
one finds 30 au for the transparent case, in contrast to the
above-mentioned 550 au for the non-transparent case [234]. Radiation from a
spherically symmetric homogeneous dust star that collapses to a
black hole is calculated in [305], using
kinetic theory. An inhomogeneous spherically symmetric dust
configuration may form a naked singularity. The redshift of light
rays that travel from such a naked singularity to a distant
observer is discussed in [83].
Lensing by a
Schwarzschild white hole.
To get a Schwarzschild white hole one joins at
the static Schwarzschild region
to the non-static Schwarzschild region
at
in such a way
that outgoing light rays can cross
this surface but ingoing cannot. The
appearance of light sources in the region
to an observer in the region
is discussed in [112, 232, 81, 196
, 197].