The Optical Origin of the Mass-Sheet Transformation

TL;DR

This paper reveals that the mass-sheet transformation in gravitational lensing originates from a fundamental scaling symmetry in the ray-trace relation when expressed in proper-distance coordinates, providing a clear physical interpretation.

Contribution

It demonstrates that the MST arises from a scaling symmetry in the ray-trace relation, clarifying its physical origin in gravitational lensing.

Findings

MST follows from a scaling symmetry in the ray-trace relation.

Proper-distance coordinates isolate a geometric focusing term.

The Image-Selection Relation (ISR) determines image properties.

Abstract

In gravitational lensing, the Mass-Sheet Transformation (MST)-or mass-sheet degeneracy-leaves image positions unchanged while scaling magnifications and time delays. The transformation scales the lens mass distribution and superposes a uniform mass sheet, but this formulation offers no clear physical interpretation. Here I show that the MST follows directly from a scaling symmetry that becomes apparent when the ray-trace relation is written in proper-distance coordinates. In this form, the ray-trace relation isolates a geometric focusing term. Subtracting this term from the deflection law defines the Image-Selection Relation (ISR), which determines image positions, magnifications, and differential time delays. The ISR exhibits a scaling symmetry that leaves image positions unchanged while scaling magnifications and time delays. Restoring the geometric focusing term then gives a ray-trace relation related to the original one by the Mass-Sheet Transformation.