If a spacetime is globally conformal to an open
subset of Minkowski spacetime, the past light cone of every event
is an embedded submanifold. Hence, multiple imaging cannot occur
(recall Section 2.8). For instance, multiple imaging
occurs in spatially closed but not in spatially open
Robertson-Walker spacetimes. In any conformally flat spacetime,
there is no image distortion, i.e., a sufficiently small sphere
always shows a circular outline on the observer’s sky (recall
Section 2.5). Correspondingly, every
infinitesimally thin bundle of light rays with a vertex is
circular, i.e., the extremal angular diameter distances and
coincide (recall Section 2.4). In addition,
also coincides with the area distance
, at least up to sign.
changes sign
at every caustic point. As
has a zero if and only if
has a zero, all caustic points of an infinitesimally
thin bundle with vertex are of multiplicity two (anastigmatic focusing), so all images have
even parity.
The geometry of light bundles can be studied
directly in terms of the Jacobi equation (= equation of geodesic
deviation) along lightlike geodesics. For a detailed investigation
of the latter in conformally flat spacetimes, see [273]. The more
special case of Friedmann-Lemaître-Robertson-Walker spacetimes
(with dust, radiation, and cosmological constant) is treated
in [101].
For bundles with vertex, one is left with one scalar equation for
, that is the focusing equation (44
) with
. This equation can be explicitly integrated for
Friedmann-Robertson-Walker spacetimes (dust without cosmological
constant). In this way one gets, for the standard observer field in
such a spacetime, relations between redshift and (area or
luminosity) distance in closed form [219]. There are
generalizations for a Robertson-Walker universe with dust plus
cosmological constant [178] and dust
plus radiation plus cosmological constant [71]. Similar formulas can
be written for the relation between age and redshift [322].