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.Comment: 22 pages, 6 figures, to appear in Phys. Rev.

We introduce the idea of shape parameters to describe the shape of the pencil of rays connecting an observer with a source lying on his past light cone. On the basis of these shape parameters, we discuss a setting of image distortion in a generic ͑exact͒ spacetime, in the form of three distortion parameters. The fundamental tool in our discussion is the use of geodesic deviation fields along a null geodesic to study how source shapes are propagated and distorted on the path to an observer. We illustrate this nonperturbative treatment of image distortion in the case of lensing by a Schwarzschild black hole. We conclude by showing that there is a nonperturbative generalization of the use of Fermat's principle in lensing in the thin-lens approximation.

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The images of many distant galaxies are displaced, distorted and often multiplied by the presence of foreground massive galaxies near the line of sight; the foreground galaxies act as gravitational lenses. Commonly, the lens equation, which relates the placement and distortion of the images to the real source position in the thin-lens scenario, is obtained by extremizing the time of arrival among all the null paths from the source to the observer (Fermat's principle). We show that the construction of envelopes of certain families of null surfaces consitutes an alternative variational principle or version of Fermat's principle that leads naturally to a lens equation in a generic spacetime with any given metric. We illustrate the construction by deriving the lens equation for static asymptotically flat thin lens spacetimes. As an application of the approach, we find the bending angle for moving thin lenses in terms of the bending angle for the same deflector at rest. Finally we apply this construction to cosmological spacetimes (FRW) by using the fact they are all conformally related to Minkowski space.

We present an analysis of weak lensing observations for RXJ1347-1145 over a 43 ′ × 43 ′ field taken in B and R filters on the Blanco 4m telescope at CTIO. RXJ1347-1145 is a massive cluster at redshift z = 0.45. Using a population of galaxies with 20 < R < 26, we detect a weak lensing signal at the p < 0.0005 level, finding best-fit parameters of σ v = 1400 +130 −140 km s −1 for a singular isothermal sphere model and r 200 = 3.5 +0.8 −0.2 Mpc with c = 15 +64 −10 for a NFW model in an Ω m = 0.3, Ω Λ = 0.7 cosmology. In addition, a mass to light ratio M/L R = 90±20 M ⊙ /L R⊙ was determined. These values are consistent with the previous weak lensing study of RXJ1347-1145by Fischer & Tyson (1997, giving strong evidence that systemic bias was not introduced by the relatively small field of view in that study. Our best-fit parameter values are also consistent with recent X-ray studies by Allen et al. (2002) and Ettori et al. (2001) but are not consistent with recent optical velocity dispersion measurements by Cohen & Kneib (2002).