A review on analytical studies in gravitational lensing

Abstract

In this study, we review some current studies on gravitational lensing for black holes, mainly in the context of general relativity. We mainly focus on the analytical studies related to lensing with references to observational results. We start with reviewing lensing in spherically symmetric Schwarzschild spacetime, showing how to calculate deflection angles before moving to the rotating counterpart, the Kerr metric. Furthermore, we extend our studies for a particular class of newly proposed solutions called black-bounce spacetimes and discuss throughout the review how to explore lensing in these spacetimes and how the various parameters can be constrained using available astrophysical and cosmological data.

Horizon-scale tests of gravity theories and fundamental physics from the Event Horizon Telescope image of Sagittarius A ∗

“…This result confirms that it is genuinely the thin shell of matter which is responsible for the space-time casting a shadow, even in the absence of a photon sphere. Of course, it is clear from equation (45) that the shadow size depends on χ e and R b2 separately, so that different choices of χ e and R b2 such that χ e R b2 = 2 would lead to results which differ from those we have shown here. Nonetheless, the qualitative features we have shown and discussed would remain unchanged: this is sufficient for the purposes of the present discussion with the aim of showing that a thin shell of matter can cast a shadow.…”

“…The Sen BH is a non-regular space-time appearing in the low-energy limit of heterotic string theory [742], whose field content includes the electromagnetic field A µ with field-strength tensor F µν , the second-rank antisymmetric tensor gauge field B µν (also known as Kalb-Ramond field [743]), and the dilaton field Φ, 45 as described by the following action in the string frame [744] (see also [745][746][747][748] for other examples of BH and black p-brane solutions in the low-energy limit of string theory):…”

“…On the plane of a distant observer, the boundary of the BH shadow marks the apparent image of the photon region (the boundary of the region of space-time which supports closed spherical photon orbits) 22 , and separates capture orbits from scattering orbits: for detailed reviews on BH shadows, see e.g. [41][42][43][44][45]. Under certain conditions and after appropriate calibration, the radius of the bright ring can serve as a proxy for the BH shadow radius, with very little dependence on the details of the surrounding accretion flow [11,12,28,[46][47][48][49][50][51][52].…”

Gravitational lensing by a stable rotating regular black hole

“…Moreover, he extended the method to the situation of rotation [75] and studied the time delay [76] between different relativistic images formed by the gravitational lensing of black holes in the strong deflection limit. Since then, this method has been widely used in the studies of strong deflection of light [77][78][79][80][81][82][83][84][85][86][87][88] and time delay [89][90][91][92][93]. When the distance between the photons and the black hole is large, the gravitational lensing effect of the black hole should be depicted by the weak deflection limit.…”