Comparison of approximate gravitational lens equations and a proposal for an improved new one

Keeping the exact general relativistic treatment of light bending as a reference, we compare the accuracy of commonly used approximate lens equations. We conclude that the best approximate lens equation is the Ohanian lens equation, for which we present a new expression in terms of distances between observer, lens, and source planes. We also examine a realistic gravitational lensing case, showing that the precision of the Ohanian lens equation might be required for a reliable treatment of gravitational lensing and a correct extraction of the full information about gravitational physics.

Figure 1

Generic gravitational lensing configuration. Note that only the angles $\gamma$, $\theta$, and $\overline{\theta }$ are well-defined in the curved spacetime, whereas all other geometrical quantities need to be defined referring to the Minkowski space associated with the original curved space.

Figure 2

The exact position of the image $\theta$ as a function of the source position $\beta$ in a linear-log plot. The dashed line is the shadow of the black hole ${\theta }_{\min }$. For $\beta >0$ the plot represents the primary image, for $\beta <0$ it represents the secondary image.

Figure 3

Relative error in the position of the images for the weak deflection lens equation (WD), the small angles lens equation (SA), the Virbhadra and Ellis lens equation (VE), and the improved Ohanian lens equation (OB) in a log scale. The dashed vertical lines bound the region with $|\beta |<{\theta }_E$, in which gravitational lensing is mostly interesting.

Figure 4

Comparison among the relative error in the position of the images for the Virbhadra and Ellis lens equation (VE) and what we expect from the asymptotic approximation (asym) and the approximation that the intersection point $C$ lies on the lens plane (cross).

Figure 5

Fractional displacement of the first pair of higher order images from the shadow border as a function of $\beta$ as defined in Eq. (46).

Figure 6

Errors in the estimate of the fractional displacement of the first pair of higher order images from the shadow borders. SA refers to the small angles lens equation, VE to the Virbhadra and Ellis lens equation, OB to the improved Ohanian lens equation. The dashed lines bound the range in which $|\beta |<{\theta }_{E,1}$, where ${\theta }_{E,1}$ is the angular radius of the first order Einstein ring.