A Note on the Overall Magnification of a Gravitational Point-Source-Point-Lens System

TL;DR

This paper derives a formula for the total magnification in gravitational point-source-point-lens systems, clarifying how magnification varies with observer position and resolving apparent contradictions with photon conservation.

Contribution

It introduces a kinematic approach to accurately compute the primary image magnification for any ray trajectory in gravitational lensing.

Findings

Magnification dips below unity far from the optical axis.

Magnification returns to unity directly behind the source.

Provides a first-order accurate formula for magnification.

Abstract

The total magnification due to a point lens has been of particular interest as the theorem that gravitational lensing results in light amplification for all observers appears to contradict the conservation of photon number. This has been discussed several times, and various resolutions have been offered. In this note, we use a kinematic approach to provide a formula for the magnification factor for the primary image accurate to first order and valid for rays leaving the source at any trajectory. We thus determine the magnification over a sphere surrounding the system. A new result found is that while the magnification dips below unity far from the optical axis as noted by others, it returns to unity directly behind the source.