Equivalence between definitions of the gravitational deflection angle of light for a stationary spacetime

TL;DR

This paper proves the equivalence of two different definitions of the gravitational deflection angle of light in stationary spacetimes, clarifying their relationship and implications for gravitational lensing analysis.

Contribution

It demonstrates the equivalence between the Huang-Cao line integral and the Ono-Ishihara-Asada definitions of the deflection angle, regardless of asymptotic regions.

Findings

The two definitions are mathematically equivalent.

The equivalence holds for any asymptotic regions.

Remarks on the light ray direction in practical applications.

Abstract

The Gibbons-Werner-Ono-Ishihara-Asada method for gravitational lensing in a stationary spacetime has been recently reexamined [Huang and Cao, arXiv:2306.04145], in which the gravitational deflection angle of light based on the Gauss-Bonnet theorem can be rewritten as a line integral of two functions $H$ and $T$. The present paper proves that the Huang-Cao line integral definition and the Ono-Ishihara-Asada one [Phys. Rev. D 96, 104037 (2017)] are equivalent to each other, whatever asymptotic regions are. A remark is also made concerning the direction of a light ray in a practical use of these definitions.