We study strong gravitational lensing due to a Schwarzschild black hole. Apart from the primary and the secondary images we find a sequence of images on both sides of the optic axis; we call them relativistic images. These images are formed due to large bending of light near $r=3M$ (the closest distance of approach ${r}_{o}$ is greater than $3M).$ The sources of the entire universe are mapped in the vicinity of the black hole by these images. For the case of the Galactic supermassive ``black hole'' they are formed at about $17$ microarcseconds from the optic axis. The relativistic images are not resolved among themselves, but they are resolved from the primary and secondary images. However the relativistic images are very much demagnified unless the observer, lens and source are very highly aligned. Because of this and some other difficulties the observation of these images does not seem to be feasible in the near future. However, it would be a great success of the general theory of relativity in a strong gravitational field if they ever were observed and it would also give an upper bound, ${r}_{o}=3.21M,$ to the compactness of the lens, which would support the black hole interpretation of the lensing object.

Schwarzschild black hole lensing

The theory of gravitational lensing is reviewed from a spacetime perspective, without quasi-Newtonian approximations More precisely, the review covers all aspects of gravitational lensing where light propagation is described in terms of lightlike geodesics of a metric of Lorentzian signature It includes the basic equations and the relevant techniques for calculating the position, the shape, and the brightness of images in an arbitrary general-relativistic spacetime It also includes general theorems on the classification of caustics, on criteria for multiple imaging, and on the possible number of images The general results are illustrated with examples of spacetimes where the lensing features can be explicitly calculated, including the Schwarzschild spacetime, the Kerr spacetime, the spacetime of a straight string, plane gravitational waves, and others

https://link.springer.com/content/pdf/10.12942%2Flrr-2004-9.pdf

Gravitational Lensing from a Spacetime Perspective

Relativistic Cosmology: The standard model and extensions

We model massive dark objects at centers of many galaxies as Schwarzschild black hole lenses and study gravitational lensing by them in detail We show that the ratio of mass of a Schwarzschild lens to the differential time delay between outermost two relativistic images (both of them either on the primary or on the secondary image side) is extremely insensitive to changes in the angular source position as well as the lens-source and lens-observer distances Therefore, this ratio can be used to obtain very accurate values for masses of black holes at centers of galaxies Similarly, angular separations between any two relativistic images are also extremely insensitive to changes in the angular source position and the lens-source distance Therefore, with the known value of mass of a black hole, angular separation between two relativistic images would give a very accurate result for the distance of the black hole Accuracies in determination of masses and distances of black holes would however depend on accuracies in measurements of differential time delays and angular separations between images Deflection angles of primary and secondary images as well as effective deflection angles of relativistic images on the secondary image side are always positive However, the effective deflection angles of relativistic images on the primary image side may be positive, zero, or negative depending on the value of angular source position and the ratio of mass of the lens to its distance We show that effective deflection angles of relativistic images play significant role in analyzing and understanding strong gravitational field lensing

Relativistic images of Schwarzschild black hole lensing

We give the formulation of the gravitational lensing theory in the strong field limit for a Schwarzschild black hole as a counterpart to the weak field approach. It is possible to expand the full black hole lens equation to work a simple analytical theory that describes the physics in the strong field limit at a high accuracy degree. In this way, we derive compact and reliable mathematical formulae for the position of additional critical curves, relativistic images and their magnification, arising in this limit.

Strong Field Limit of Black Hole Gravitational Lensing

(November 29, 1999)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 theSchwarzschild case, we give exact expressions for the angular-diameter distance to the sources aswell as for the magniﬁcation factor and time of arrival of the images We compare the exactlens 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, andthe gravitational ﬁeld is weak), and verify the fact that the standard weak-ﬁeld thin-lens equationis 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 therecently proposed strong-ﬁeld thin-lens equation, obtained under the assumption of straight pathsbut without the small angle approximation, ie, with allowed large bending angles We show thatthe strong-ﬁeld thin-lens equation is remarkably accurate, even for lightrays that take several turnsaround the lens before reaching the observerI INTRODUCTION

Spacetime perspective of Schwarzschild lensing

We present an analysis of weak-lensing observations for RX J1347-1145 over a 43' × 43' field taken in B and R filters on the Blanco 4 m telescope at CTIO. RX J1347-1145 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 σv = 1400 km s-1 for a singular isothermal sphere model and r200 = 3.5 Mpc with c = 15 for an NFW model in an Ωm = 0.3, ΩΛ = 0.7 cosmology. In addition, a mass-to-light ratio M/LR = 90 ± 20 M☉/LR☉ was determined. These values are consistent with the previous weak-lensing study of RX J1347-1145 by Fischer & Tyson, 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. and Ettori et al. but are not consistent with recent optical velocity dispersion measurements by Cohen & Kneib.

http://iopscience.iop.org/article/10.1086/428891/pdf

Wide-field weak lensing by rx j1347 - 1145

We present an analysis of weak lensing observations for RXJ1347-1145 over a 43' X 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 \sigma_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 \Omega_m = 0.3, \Omega_\Lambda = 0.7 cosmology. In addition, a mass to light ratio M/L_R =90 \pm 20 M_\odot / L_{R\odot} was determined. These values are consistent with the previous weak lensing study of RXJ1347--1145 by Fischer and 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 and Kneib, 2002.

http://export.arxiv.org/pdf/astro-ph/0502281

Wide-field weak lensing by RXJ1347-1145

In a previous article concerning image distortion in non-perturbative gravitational lensing theory we described how to introduce shape and distortion parameters for small sources. We also showed how they could be expressed in terms of the scalar products of the geodesic deviation vectors of the source's pencil of rays in the past lightcone of an observer. In the present work we give an alternative approach to the description of the shape and distortion parameters and their evolution along the null geodesic from the source to the observer, but now in terms of the optical scalars (the convergence and shear of null vector field of the observer's lightcone) and the associated optical equations, which relate the optical scalars to the curvature of the spacetime.

Image distortion from optical scalars in nonperturbative gravitational lensing

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. 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. 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.

/pdf/required-peer-cooperative-learning-improves-retention-of-2bqwfvp8sj.pdf

Thomas P. Kling

Papers

Spacetime perspective of Schwarzschild lensing

Wide-field weak lensing by rx j1347 - 1145

Wide-field weak lensing by RXJ1347-1145

Image distortion from optical scalars in nonperturbative gravitational lensing

Required Peer-cooperative Learning Improves Retention of STEM Majors