Lensing in Arbitrary Spacetimes

2 Lensing in Arbitrary Spacetimes

By a spacetime we mean a 4-dimensional manifold $ℳ$ with a ( $∞ C$ , if not otherwise stated) metric tensor field $g$ of signature $(+, +,+, − )$ that is time-oriented. The latter means that the non-spacelike vectors make up two connected components in the entire tangent bundle, one of which is called “future-pointing” and the other one “past-pointing”. Throughout this review we restrict to the case that the light rays are freely propagating in vacuum, i.e., are not influenced by mirrors, refractive media, or any other impediments. The light rays are then the lightlike geodesics of the spacetime metric. We first summarize results on the lightlike geodesics that hold in arbitrary spacetimes. In Section 3 these results will be specified for spacetimes with conditions on the causal structure and in Section 4 for spacetimes with symmetries.

2.1 Light cone and exact lens map
2.2 Wave fronts
2.3 Optical scalars and Sachs equations
2.4 Distance measures
2.5 Image distortion
2.6 Brightness of images
2.7 Conjugate points and cut points
2.8 Criteria for multiple imaging
2.9 Fermat’s principle for light rays