2.1 Light cone and exact lens
map
In an arbitrary spacetime
, what an observer at an
event
can see is determined by the lightlike geodesics
that issue from
into the past. Their union gives the
past light cone of
. This is the central geometric object for lensing
from the spacetime perspective. For a point source with worldline
, each past-oriented lightlike geodesic
from
to
gives rise to an image of
on the observer’s sky. One should view any such
as the central ray of a thin bundle that is focused by the
observer’s eye lens onto the observer’s retina (or by a telescope
onto a photographic plate). Hence, the intersection of the past
light cone with the world-line of a point source (or with the
world-tube of an extended source) determines the visual appearance
of the latter on the observer’s sky.
In mathematical terms, the observer’s sky or celestial
sphere
can be viewed as the set of all
lightlike directions at
. Every such direction
defines a unique (up to parametrization) lightlike geodesic through
, so
may also be viewed as a subset of the
space of all lightlike geodesics in
(cf. [209
]). One may choose at
a future-pointing vector
with
, to be interpreted as the 4-velocity of the
observer. This allows identifying the observer’s sky
with a subset of the tangent space
,
If
is changed, this representation changes according to
the standard aberration formula of special relativity. By
definition of the exponential map
, every affinely parametrized geodesic
satisfies
.
Thus, the past light cone of
is the image of the map
which is defined on a subset of
. If we
restrict to values of
sufficiently close to 0, the
map (2) is an embedding,
i.e., this truncated light cone is an embedded submanifold; this
follows from the well-known fact that
maps a
neighborhood of the origin, in each tangent space,
diffeomorphically into the manifold. However, if we extend the
map (2) to larger values of
, it is in general neither injective nor an immersion;
it may form folds, cusps, and other forms of caustics, or transverse self-intersections.
This observation is of crucial importance in view of lensing. There
are some lensing phenomena, such as multiple imaging and image
distortion of (point) sources into (1-dimensional) rings, which can
occur only if the light cone fails to be an embedded submanifold
(see Section 2.8). Such lensing phenomena are
summarized under the name strong
lensing effects. As long as the light cone is an embedded
submanifold, the effects exerted by the gravitational field on the
apparent shape and on the apparent brightness of light sources are
called weak lensing effects. For examples of light cones
with caustics and/or transverse self-intersections, see
Figures 12, 24, and 25. These pictures show
light cones in spacetimes with symmetries, so their structure is
rather regular. A realistic model of our own light cone, in the
real world, would have to take into account numerous irregularly
distributed inhomogeneities (“clumps”) that bend light rays in
their neighborhood. Ellis, Bassett, and Dunsby [99
] estimate that such
a light cone would have at least
caustics which
are hierarchically structured in a way that reminds of fractals.
For calculations it is recommendable to introduce
coordinates on the observer’s past light cone. This can be done by
choosing an orthonormal tetrad
with
at the observation event
. This parametrizes
the points of the observer’s celestial sphere by spherical
coordinates
,
In this representation, map (2) maps each
to a spacetime point. Letting the observation event
float along the observer’s worldline, parametrized by proper time
, gives a map that assigns to each
a spacetime point. In terms of coordinates
on the spacetime manifold, this map is
of the form
It can be viewed as a map from the world as it appears to the
observer (via optical observations) to the world as it is. The
observational coordinates
were introduced by Ellis [98] (see [100] for a
detailed discussion). They are particularly useful in cosmology but
can be introduced for any observer in
any spacetime. It is useful to
consider observables, such as distance measures (see
Section 2.4) or the ellipticity that
describes image distortion (see Section 2.5) as functions of the
observational coordinates. Some observables, e.g., the redshift and
the luminosity distance, are not determined by the spacetime
geometry and the observer alone, but also depend on the
4-velocities of the light sources. If a vector field
with
has been fixed, one may restrict to an
observer and to light sources which are integral curves of
. The above-mentioned observables, like redshift and
luminosity distance, are then uniquely determined as functions of
the observational coordinates. In applications to cosmology one
chooses
as tracing the mean flow of luminous matter (“Hubble
flow”) or as the rest system of the cosmic background radiation;
present observations are compatible with the assumption that these
two distinguished observer fields coincide [32].
Writing map (4) explicitly requires
solving the lightlike geodesic equation. This is usually done,
using standard index notation, in the Lagrangian formalism, with
the Lagrangian
, or in the Hamiltonian
formalism, with the Hamiltonian
.
A non-trivial example where the solutions can be explicitly written
in terms of elementary functions is the string spacetime of
Section 5.10. Somewhat more general, although
still very special, is the situation that the lightlike geodesic
equation admits three independent constants of motion in addition
to the obvious one
. If, for any pair of the
four constants of motion, the Poisson bracket vanishes (“complete
integrability”), the lightlike geodesic equation can be reduced to
first-order form, i.e., the light cone can be written in terms of
integrals over the metric coefficients. This is true, e.g., in
spherically symmetric and static spacetimes (see Section 4.3).
Having parametrized the past light cone of the
observation event
in terms of
, or more
specifically in terms of
, one may set up an exact lens map. This exact lens map is
analogous to the lens map of the quasi-Newtonian approximation
formalism, as far as possible, but it is valid in an arbitrary
spacetime without approximation. In the quasi-Newtonian formalism
for thin lenses at rest, the lens map assigns to each point in the
lens plane a point in the source plane (see, e.g., [299
, 275
, 343
]). When working in
an arbitrary spacetime without approximations, the observer’s sky
is an obvious substitute for the lens plane. As a
substitute for the source plane we choose a 3-dimensional
submanifold
with a prescribed ruling by timelike
curves. We assume that
is globally of the form
, where the points of the 2-manifold
label the timelike curves by which
is ruled. These
timelike curves are to be interpreted as the worldlines of light
sources. We call any such
a source surface.
In a nutshell, choosing a source surface means choosing a
two-parameter family of light sources.
The exact lens map is a map from
to
. It is defined by following, for each
, the past-pointing geodesic with initial vector
until it meets
and then projecting to
(see Figure 1). In other words, the
exact lens map says, for each point on the observer’s celestial
sphere, which of the chosen light sources is seen at this point.
Clearly, non-invertibility of the lens map indicates multiple
imaging. What one chooses for
depends on the situation. In
applications to cosmology, one may choose galaxies at a fixed
redshift
around the observer. In a
spherically-symmetric and static spacetime one may choose static
light sources at a fixed radius value
. Also, the
surface of an extended light source is a possible choice for
.
The exact lens map was introduced by Frittelli and
Newman [123
] and further
discussed in [91, 90]. The
following global aspects of the exact lens map were investigated
in [270
]. First, in general
the lens map is not defined on all of
because not all
past-oriented lightlike geodesics that start at
necessarily meet
. Second, in general the lens
map is multi-valued because a lightlike geodesic might meet
several times. Third, the lens map need not be differentiable and
not even continuous because a lightlike geodesic might meet
tangentially. In [270
], the notion of a
simple lensing neighborhood is
introduced which translates the statement that a deflector is
transparent into precise mathematical language. It is shown that
the lens map is globally well-defined and differentiable if the
source surface is the boundary of such a simple lensing
neighborhood, and that for each light source that does not meet the
caustic of the observer’s past light cone the number of images is
finite and odd. This result applies, as a special case, to
asymptotically simple and empty spacetimes (see Section 3.4).
For expressing the exact lens map in coordinate
language, it is recommendable to choose coordinates
such that the source surface
is given by the equation
, with a constant
, and that the worldlines of the light sources are
-lines. In this situation the remaining coordinates
and
label the light sources and the exact
lens map takes the form
It is given by eliminating the two variables
and
from the four equations (4) with
and fixed
. This is the way in
which the lens map was written in the original paper by Frittelli
and Newman; see Equation (6) in [123
]. (They used complex
coordinates
for the observer’s celestial sphere
that are related to our spherical coordinates
by stereographic projection.) In this explicit
coordinate version, the exact lens map can be succesfully applied,
in particular, to spherically symmetric and static spacetimes, with
,
,
, and
(see Section 4.3 and the Schwarzschild example in
Section 5.1). The exact lens map can also be
used for testing the reliability of approximation techniques.
In [184] the authors find
that the standard quasi-Newtonian approximation formalism may lead
to significant errors for lensing configurations with two lenses.