Historical notes.

Shortly after the discovery of the Schwarzschild metric by Schwarzschild [301] and independently by
Droste [80], basic features of its lightlike geodesics were found by Flamm [113], Hilbert [157], and
Weyl [347]. Detailed studies of its timelike and lightlike geodesics were made by Hagihara [145] and
Darwin [72, 73]. For a fairly complete list of the pre-1979 literature on Schwarzschild geodesics see
Sharp [305]. 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 BaΕΌaΕski 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 [115]),

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

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. [182]). 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

- by showing the visual appearance of some background pattern as distorted by the black hole [66, 295, 234] (only primary images, , are considered), and
- by showing the visual appearence of an accretion disk around the black hole [212, 129, 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 [165]. 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 [340, 235, 325]. 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, 128, 287, 222, 240]). 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 [280] 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 [197].

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. [152]). In principle, there may be arbitrarily many photon spheres [176]. 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 [182]. (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 [166] 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 [235]. Radiation from a spherically symmetric
homogeneous dust star that collapses to a black hole is calculated in [304], 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 [111, 233, 81, 195, 196].

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