The Jacobi map for gravitational lensing: the role of the exponential map

TL;DR

This paper derives the fundamental equations of gravitational lensing using Jacobi fields and the exponential map, clarifying singular behaviors and reaffirming previous conclusions in the field.

Contribution

It provides a formal derivation of lensing equations in arbitrary space-times based on Jacobi fields and the exponential map, addressing potential singularity issues.

Findings

Clarifies the role of the exponential map in lensing equations

Analyzes singular behavior of the optical deformation matrix

Confirms the validity of previous lensing arguments despite potential singularities

Abstract

We present a formal derivation of the key equations governing gravitational lensing in arbitrary space-times, starting from the basic properties of Jacobi fields and their expressions in terms of the exponential map. A careful analysis of Jacobi fields and Jacobi classes near the origin of a light beam determines the nature of the singular behavior of the optical deformation matrix. We also show that potential problems that could arise from this singularity do not invalidate the conclusions of the original argument presented by Seitz, Schneider & Ehlers (1994).