The geometry of gravitational lensing magnification

TL;DR

This paper introduces a new, general definition of gravitational lensing magnification based on the van Vleck determinant and exponential map, applicable to arbitrary convex neighborhoods in Lorentzian manifolds, without relying on traditional lens-source plane models.

Contribution

It provides a novel, mathematically rigorous definition of magnification in gravitational lensing that extends beyond classical models, using geometric tools like the van Vleck determinant.

Findings

Magnification is monotonically increasing along geodesics in convex neighborhoods.

The new definition applies to arbitrary spacetimes, including Schwarzschild.

It avoids the traditional lens and source plane framework.

Abstract

We present a definition of unsigned magnification in gravitational lensing valid on arbitrary convex normal neighborhoods of time oriented Lorentzian manifolds. This definition is a function defined at any two points along a null geodesic that lie in a convex normal neighborhood, and foregoes the usual notions of lens and source planes in gravitational lensing. Rather, it makes essential use of the van Vleck determinant, which we present via the exponential map, and Etherington's definition of luminosity distance for arbitrary spacetimes. We then specialize our definition to spacetimes, like Schwarzschild's, in which the lens is compact and isolated, and show that our magnification function is monotonically increasing along any geodesic contained within a convex normal neighborhood.