Spacetime perspective of Schwarzschild lensing

TL;DR

This paper develops an exact lens equation in Schwarzschild spacetime, compares it with traditional weak-field and strong-field approximations, and demonstrates the accuracy of the strong-field thin-lens equation for large bending angles.

Contribution

It introduces a background-independent exact lens equation and evaluates the accuracy of various approximations in Schwarzschild spacetime.

Findings

Exact expressions for angular-diameter distance, magnification, and arrival time.

Weak-field thin-lens approximation is inadequate at small impact parameters.

Strong-field thin-lens equation remains accurate even with multiple lens turns.

Abstract

We propose a definition of an exact lens equation without reference to a background spacetime, and construct the exact lens equation explicitly in the case of Schwarzschild spacetime. For the Schwarzschild case, we give exact expressions for the angular-diameter distance to the sources as well as for the magnification factor and time of arrival of the images. We compare the exact lens equation with the standard lens equation, derived under the thin-lens-weak-field assumption (where the light rays are geodesics of the background with sharp bending in the lens plane, and the gravitational field is weak), and verify the fact that the standard weak-field thin-lens equation is inadequate at small impact parameter. We show that the second-order correction to the weak-field thin-lens equation is inaccurate as well. Finally, we compare the exact lens equation with the recently proposed strong-field thin-lens equation, obtained under the assumption of straight paths but without the small angle approximation, i.e., with allowed large bending angles. We show that the strong-field thin-lens equation is remarkably accurate, even for lightrays that take several turns around the lens before reaching the observer.