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

Quasi-Equatorial Gravitational Lensing by Spinning Black Holes in the Strong Field Limit

Spherically symmetric black holes produce, by strong field lensing, two infinite series of relativistic images, formed by light rays winding around the black hole at distances comparable to the gravitational radius. In this paper, we address the relevance of the black hole spin for the strong field lensing phenomenology, focusing on trajectories close to the equatorial plane for simplicity. In this approximation, we derive a two-dimensional lens equation and formulae for the position and the magnification of the relativistic images in the strong field limit. The most outstanding effect is the generation of a non trivial caustic structure. Caustics drift away from the optical axis and acquire finite extension. For a high enough black hole spin, depending on the source extension, we can practically observe only one image rather than two infinite series of relativistic images. In this regime, additional non equatorial images may play an important role in the phenomenology.Comment: 13 pages, 9 figures. Improved version with detailed physical discussio

The weakly perturbed Schwarzschild lens in the strong deflection limit

We investigate the strong deflection limit of gravitational lensing by a Schwarzschild black hole embedded in an external gravitational field. The study of this model, analogous to the Chang & Refsdal lens in the weak deflection limit, is important to evaluate the gravitational perturbations on the relativistic images that appear in proximity of supermassive black holes hosted in galactic centers. By a simple dimensional argument, we prove that the tidal effect on the light ray propagation mainly occurs in the weak field region far away from the black hole and that the external perturbation can be treated as a weak field quadrupole term. We provide a description of relativistic critical curves and caustics and discuss the inversion of the lens mapping. Relativistic caustics are shifted and acquire a finite diamond shape. Sources inside the caustics produce four sequences of relativistic images. On the other hand, retro-lensing caustics are only shifted while remaining point-like to the lowest order.Comment: 12 pages, 1 figure

Microlensing Detections of Moons of Exoplanets

We investigate the characteristic of microlensing signals of Earth-like moons orbiting ice-giant planets. From this, we find that non-negligible satellite signals occur when the planet-moon separation is similar to or greater than the Einstein radius of the planet. We find that the satellite signal does not diminish with the increase of the planet-moon separation beyond the Einstein radius of the planet unlike the planetary signal which vanishes when the planet is located well beyond the Einstein radius of the star. We also find that the satellite signal tends to have the same sign as that of the planetary signal. These tendencies are caused by the lensing effect of the star on the moon in addition to the effect of the planet. We determine the range of satellite separations where the microlensing technique is optimized for the detections of moons. By setting an upper limit as the angle-average of the projected Hill radius and a lower limit as the half of the Einstein radius of the planet, we find that the microlensing method would be sensitive to moons with projected separations from the planet of $0.05 {\rm AU} \lesssim d_{\rm p} \lesssim 0.24 {\rm AU}$ for a Jupiter-mass planet, $0.03 {\rm AU}\lesssim d_{\rm p} \lesssim 0.17 {\rm AU}$ for a Saturn-mass planet, and $0.01 {\rm AU} \lesssim d_{\rm p} \lesssim 0.08 {\rm AU}$ for a Uranus-mass planet. We compare the characteristics of the moons to be detected by the microlensing and transit techniquesComment: 6pages, 6 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

Red alert: labile heme is an alarmin

This publication hasn't any creative commons license associated.This deposit is composed by the main article, and it hasn't any supplementary materials associated. There is no public supplementary material available.The deposited article is a pre-print version.Alarmins are a heterogeneous group of endogenous molecules that signal cellular damage when sensed extracellularly. Heme is an endogenous molecule that acts as a prosthetic group of hemoproteins, such as hemoglobin and myoglobin. When released from damaged red blood cells or muscle cells, oxidized hemoglobin and myoglobin release their prosthetic heme groups, respectively. This generates labile heme, which is sensed by pattern recognition receptors (PRR) expressed by innate immune cells and possibly regulatory T cells (TREG). The ensuing adaptive response, which alerts for the occurrence of red blood cell or muscle cell damage, regulates the pathologic outcome of hemolysis or rhabdomyolysis, respectively. In conclusion, we propose that labile heme is an alarmin.Fundação para a Ciência e Tecnologia grants: (PTDC/SAU TOX/116627/2010, HMSP-ICT/0022/2010, RECI/IMI-IMU/0038/2012); Fundação de Amparo a Pesquisa do Estado do Rio de Janeiro; Conselho Nacional de Pesquisa, INCTDengue, Coordenacao de Aperfeicoamento de Pessoal de Nivel Superior.info:eu-repo/semantics/publishedVersio

A comparison of approximate gravitational lens equations and a proposal for an improved new one

Keeping the exact general relativistic treatment of light bending as a reference, we compare the accuracy of commonly used approximate lens equations. We conclude that the best approximate lens equation is the Ohanian lens equation, for which we present a new expression in terms of distances between observer, lens and source planes. We also examine a realistic gravitational lensing case, showing that the precision of the Ohanian lens equation might be required for a reliable treatment of gravitational lensing and a correct extraction of the full information about gravitational physics.Comment: 11 pages, 6 figures, to appear on Physical Review

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

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

Kerr black hole lensing for generic observers in the strong deflection limit

We generalize our previous work on gravitational lensing by a Kerr black hole in the strong deflection limit, removing the restriction to observers on the equatorial plane. Starting from the Schwarzschild solution and adding corrections up to the second order in the black hole spin, we perform a complete analytical study of the lens equation for relativistic images created by photons passing very close to a Kerr black hole. We find out that, to the lowest order, all observables (including shape and shift of the black hole shadow, caustic drift and size, images position and magnification) depend on the projection of the spin on a plane orthogonal to the line of sight. In order to break the degeneracy between the black hole spin and its inclination relative to the observer, it is necessary to push the expansion to higher orders. In terms of future VLBI observations, this implies that very accurate measures are needed to determine these two parameters separately.Comment: 17 pages, 4 figures, one section added, to appear on Physical Review