Correspondence between two gravitational lens equations in a static and spherically symmetric spacetime

TL;DR

This paper clarifies the relationship between two gravitational lens equations in static, spherically symmetric spacetimes, showing that the VE equation can be improved and made equivalent to the OB equation through a specific transformation.

Contribution

It demonstrates the existence of an unphysical branch in the VE equation and provides a transformation to align it with the OB equation, clarifying their connection.

Findings

The VE equation contains an unphysical branch.

Removing the unphysical branch improves the VE equation.

The improved VE equation is equivalent to the OB equation after a transformation.

Abstract

Virbhadra and Ellis have proposed a very accurate equation (referred to as VE equation) for the gravitational lens in a static and spherically symmetric spacetime [Phys. Rev. D 62,084003 (2000)], whereas an improved equation (referred to as OB equation) has been derived by Bozza [Phys. Rev. D 78, 103005 (2008)] based on a relation found by Ohanian [Am. J. Phys. 55, 428 (1987)]. The OB equation was rediscovered later by Takizawa, Ono and Asada [Phys. Rev. D, 102, 064060 (2020)]. VE and OB equations seem to be very different from each other. The present paper shows that there exists an unphysical branch in the VE equation. Consequently, the VE equation can be improved by removing the unphysical branch. The improved version of the VE equation is found to be the same as the OB equation when a suitable transformation is made between the deflection angles defined differently in the two formulations. An explicit expression of the transformation is found. We also argue possible numerical differences when the transformation between the deflection angles is ignored.