Light's Bending Angle due to Black Holes: From the Photon Sphere to Infinity

The bending angle of light is a central quantity in the theory of gravitational lensing. We develop an analytical perturbation framework for calculating the bending angle of light rays lensed by a Schwarzschild black hole. Using a perturbation parameter given in terms of the gravitational radius of the black hole and the light ray's impact parameter, we determine an invariant series for the strong-deflection bending angle that extends beyond the standard logarithmic deflection term used in the literature. In the process, we discovered an improvement to the standard logarithmic deflection term. Our perturbation framework is also used to derive as a consistency check, the recently found weak deflection bending angle series. We also reformulate the latter series in terms of a more natural invariant perturbation parameter, one that smoothly transitions between the weak and strong deflection series. We then compare our invariant strong deflection bending-angle series with the numerically integrated exact formal bending angle expression, and find less than 1% discrepancy for light rays as far out as twice the critical impact parameter. The paper concludes by showing that the strong and weak deflection bending angle series together provide an approximation that is within 1% of the exact bending angle value for light rays traversing anywhere between the photon sphere and infinity.Comment: 22 pages, 5 figure

Strong Gravitational Lensing by Sgr A*

In recent years, there has been increasing recognition of the potential of the galactic center as a probe of general relativity in the strong field. There is almost certainly a black hole at Sgr A* in the galactic center, and this would allow us the opportunity to probe dynamics near the exterior of the black hole. In the last decade, there has been research into extreme gravitational lensing in the galactic center. Unlike in most applications of gravitational lensing, where the bending angle is of the order of several arc seconds, very large bending angles are possible for light that closely approaches a black hole. Photons may even loop multiple times around a black hole before reaching the observer. There have been many proposals to use light's close approach to the black hole as a probe of the black hole metric. Of particular interest is the property of light lensed by the S stars orbiting in the galactic center. This paper will review some of the attempts made to study extreme lensing as well as extend the analysis of lensing by S stars. In particular, we are interested in the effect of a Reissner-Nordstrom like 1/r^2 term in the metric and how this would affect the properties of relativistic images.Comment: 13 pages, 9 figures. Submitted as invited review article for the GR19 issue of CQ

A Characteristic Planetary Feature in Double-Peaked, High-Magnification Microlensing Events

A significant fraction of microlensing planets have been discovered in high-magnification events, and a significant fraction of these events exhibit a double-peak structure at their peak. However, very wide or very close binaries can also produce double-peaked high-magnification events, with the same gross properties as those produced by planets. Traditionally, distinguishing between these two interpretations has relied upon detailed modeling, which is both time-consuming and generally does not provide insight into the observable properties that allow discrimination between these two classes of models. We study the morphologies of these two classes of double-peaked high-magnification events, and identify a simple diagnostic that can be used to immediately distinguish between perturbations caused by planetary and binary companions, without detailed modeling. This diagnostic is based on the difference in the shape of the intra-peak region of the light curves. The shape is smooth and concave for binary lensing, while it tends to be either boxy or convex for planetary lensing. In planetary lensing this intra-peak morphology is due to the small, weak cusp of the planetary central caustic located between the two stronger cusps. We apply this diagnostic to five observed double-peaked high-magnification events to infer their underlying nature. A corollary of our study is that good coverage of the intra-peak region of double-peaked high-magnification events is likely to be important for their unique interpretation.Comment: 6 pages, 3 figure

Estimating the parameters of the Sgr A* black hole

The measurement of relativistic effects around the galactic center may allow in the near future to strongly constrain the parameters of the supermassive black hole likely present at the galactic center (Sgr A*). As a by-product of these measurements it would be possible to severely constrain, in addition, also the parameters of the mass-density distributions of both the innermost star cluster and the dark matter clump around the galactic center.Comment: Accepted for publication on General Relativity and Gravitation, 2010. 11 Pages, 1 Figur

Particle motion and gravitational lensing in the metric of a dilaton black hole in a de Sitter universe

We consider the metric exterior to a charged dilaton black hole in a de Sitter universe. We study the motion of a test particle in this metric. Conserved quantities are identified and the Hamilton-Jacobi method is employed for the solutions of the equations of motion. At large distances from the black hole the Hubble expansion of the universe modifies the effective potential such that bound orbits could exist up to an upper limit of the angular momentum per mass for the orbiting test particle. We then study the phenomenon of strong field gravitational lensing by these black holes by extending the standard formalism of strong lensing to the non-asymptotically flat dilaton-de Sitter metric. Expressions for the various lensing quantities are obtained in terms of the metric coefficients.Comment: 8 pages, RevTex, 1 eps figures; discussion improved; typos corrected; references adde

Gravitational lensing by a charged black hole of string theory

We study gravitational lensing by the Gibbons-Maeda-Garfinkle-Horowitz-Strominger (GMGHS) charged black hole of heterotic string theory and obtain the angular position and magnification of the relativistic images. Modeling the supermassive central object of the galaxy as a GMGHS black hole, we estimate the numerical values of different strong-lensing parameters. We find that there is no significant string effect present in the lensing observables in the strong-gravity scenario.Comment: 6 page

Interacting Cosmic Fluids in Brans-Dicke Cosmology

We provide a detailed description for power-law scaling FRW cosmological models in Brans-Dicke theory dominated by two interacting fluid components during the expansion of the universe.Comment: 13 pages, 8 figure

Gravitational Lensing by Black Holes

We review the theoretical aspects of gravitational lensing by black holes, and discuss the perspectives for realistic observations. We will first treat lensing by spherically symmetric black holes, in which the formation of infinite sequences of higher order images emerges in the clearest way. We will then consider the effects of the spin of the black hole, with the formation of giant higher order caustics and multiple images. Finally, we will consider the perspectives for observations of black hole lensing, from the detection of secondary images of stellar sources and spots on the accretion disk to the interpretation of iron K-lines and direct imaging of the shadow of the black hole.Comment: Invited article for the GRG special issue on lensing (P. Jetzer, Y. Mellier and V. Perlick Eds.). 31 pages, 12 figure

Strong deflection limit of black hole gravitational lensing with arbitrary source distances

The gravitational field of supermassive black holes is able to strongly bend light rays emitted by nearby sources. When the deflection angle exceeds $\pi$, gravitational lensing can be analytically approximated by the so-called strong deflection limit. In this paper we remove the conventional assumption of sources very far from the black hole, considering the distance of the source as an additional parameter in the lensing problem to be treated exactly. We find expressions for critical curves, caustics and all lensing observables valid for any position of the source up to the horizon. After analyzing the spherically symmetric case we focus on the Kerr black hole, for which we present an analytical 3-dimensional description of the higher order caustic tubes.Comment: 20 pages, 8 figures, appendix added. In press on Physical Review