Spacetime perspective of Schwarzschild lensing - art. no. 064021

We propose a definition of an exact lens equation without reference to a ba ckground 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 fo r the magnification factor and time of arrival of the images. We compare th e exact lens equation with the standard lens equation, derived under the th in-lens-weak-field assumption (where the light rays are geodesics of the ba ckground with sharp bending in the lens plane, and the gravitational field is weak), and verify the fact that the standard weak-field thin-lens equati on is inadequate at small impact parameter. We show that the second-order c orrection to the weak-field thin-lens equation is inaccurate as well. Final ly, we compare the exact lens equation with the recently proposed strong-fi eld thin-lens equation, obtained under the assumption of straight paths but without the small angle approximation, i.e., with allowed large bending an gles. We show that the strong-held thin-lens equation is remarkably accurat e, even for light rays that take several turns around the lens before reach ing the observer.