The Optical Origin of the Mass-Sheet Transformation

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.

References in corpus (4)

• Dissecting the Gravitational Lens B1608+656. II. Precision Measurements of the Hubble Constant, Spatial Curvature, and the Dark Energy Equation of State
• TDCOSMO IV: Hierarchical time-delay cosmography -- joint inference of the Hubble constant and galaxy density profiles
• Mass-sheet degeneracy, power-law models and external convergence: Impact on the determination of the Hubble constant from gravitational lensing
• TDCOSMO VI: Distance Measurements in Time-delay Cosmography under the Mass-sheet transformation