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).

In this work we investigate aspects of light cones in a Schwarzschild geometry, making connections to gravitational lensing theory and to a new approach to general relativity, the null surface formulation. By integrating the null geodesics of our model, we obtain the light cone from every space-time point. We study three applications of the light cones. First, by taking the intersection of the light cone from each point in the space-time with null infinity, we obtain the light cone cut function, a four parameter family of cuts of null infinity, which is central to the null surface formulation. We examine the singularity structure of the cut function. Second, we give the exact gravitational lens equations, and their specialization to the Einstein ring. Third, as an application of the cut function, we show that the recently introduced coordinate system, the ''pseudo Minkowski'' coordinates, are a valid coordinate system for the space-time.

Background: Peer-cooperative learning has been shown in the literature to improve student success in gateway science and mathematics courses. Such studies typically demonstrate the impact of students' attending peer-led learning sessions on their learning or grades in an individual course. In this article, we examine the effects of introducing a required, comprehensive peer-cooperative learning system across five departments simultaneously at a master's public university, looking not only at students' success in supported classes, but also their retention within STEM fields two years hence. Combining institutional demographic data with students' course grades and retention rates, we compare outcomes between 456 students who took their major's introductory course in the two years prior to implementation of the program, and 552 students who did so after implementation. Results: While these two student groups did not significantly differ in either their demographic profile or their SAT scores, the post-implementation group earned significantly higher grades in their introductory courses in each major, due largely to an erasure of the mediating effect of SAT scores on course grades. Further, this increase in introductory course grades was also associated with an increase in the two-year retention rate of students in STEM majors. Conclusions: This finding is significant as it suggests that implementing comprehensive educational reform using required peer-led cooperative learning may have the proximate effect of mitigating differences in academic preparation (as measured by SAT scores) for students in introductory STEM courses. Furthermore, this increase in success leads to increased retention rates in STEM, expanding the pipeline of students retained in such fields.

We examine the accuracy of strong gravitational lensing determinations of the mass of galaxy clusters by comparing the conventional approach with the numerical integration of the fully relativistic null geodesic equations in the case of weak gravitational perturbations on Robertson-Walker metrics. In particular, we study spherically symmetric, three-dimensional singular isothermal sphere models and the three-dimensional matter distribution of Navarro and coworkers which are both commonly used in gravitational lensing studies. In both cases we study two different methods for mass-density truncation along the line of sight: hard truncation and conventional (no truncation). We find that the relative error introduced in the total mass by the thin-lens approximation alone is less than 0.3% in the singular isothermal sphere model and less than 2% in the model of Navarro and coworkers. The removal of hard truncation introduces an additional error of the same order of magnitude in the best case and up to an order of magnitude larger in the worst case studied. Our results ensure that the future generation of precision cosmology experiments based on lensing studies will not require the removal of the thin-lens assumption, but they may require a careful handling of truncation.