Lensing in conformally flat spacetimes

4.1 Lensing in conformally flat spacetimes

By definition, a spacetime is conformally flat if the conformal curvature tensor (= Weyl tensor) vanishes. An equivalent condition is that every point admits a neighborhood that is conformal to an open subset of Minkowski spacetime. As a consequence, conformally flat spacetimes have the same local conformal symmetry as Minkowski spacetime, that is they admit 15 independent conformal Killing vector fields. The global topology, however, may be different from the topology of Minkowski spacetime. The class of conformally flat spacetimes includes all (kinematic) Robertson–Walker spacetimes. Other physically interesting examples are some (generalized) interior Schwarzschild solutions and some pure radiation spacetimes. All conformally flat solutions to Einstein’s field equation with a perfect fluid or an electromagnetic field are known (see [310 ], Section 37.5.3).

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 $D+$ and $D −$ coincide (recall Section 2.4). In addition, $D+ = D −$ also coincides with the area distance $D area$ , at least up to sign. $D = D + −$ changes sign at every caustic point. As $D+$ has a zero if and only if $D −$ 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 $D+ = D − = ±Darea$ , that is the focusing equation (44 ) with $σ = 0$ . 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 [220 ]. There are generalizations for a Robertson–Walker universe with dust plus cosmological constant [177 ] and dust plus radiation plus cosmological constant [71 ]. Similar formulas can be written for the relation between age and redshift [321 ].