Abstract
A perturbative method to compute the deflection angle of both timelike and null rays in arbitrary static and spherically symmetric spacetimes in the strong field limit is proposed. The result takes a quasi-series form of (1-b_c/b) where b is the impact parameter and b_c is its critical value, with coefficients of the series explicitly given. This result also naturally takes into account the finite distance effect of both the source and detector, and allows to solve the apparent angles of the relativistic images in a more precise way. From this, the BH angular shadow size is expressed as a simple formula containing metric functions and particle/photon sphere radius. The magnification of the relativistic images were shown to diverge at different values of the source-detector angular coordinate difference, depending on the relation between the source and detector distance from the lens. To verify all these results, we then applied them to the Hayward BH spacetime, concentrating on the effects of its charge parameter l and the asymptotic velocity v of the signal. The BH shadow size were found to decrease slightly as l increases to its critical value, and increase as v decreases from light speed. For the deflection angle and the magnification of the images however, both the increase of l and decrease of v will increase their values.
Highlights
It is necessary to understand the general geodesic behavior of null/timelike rays near the gravitational center
The deflection of null rays in the strong field limit (SFL) has been systematically studied by Bozza et al for static and spherically symmetric (SSS) spacetimes and equatorial motion in stationary and axisymmetric (SAS) spacetimes [8,9,10] and has since been used to study the deflection and GL in the SFL for many interesting spacetimes [11,12,13,14,15,16,17,18,19,20,21]
Note that most of the deflection angles found in these works assumed that the source and detectors are located at infinite distance
Summary
The null geodesics in this limit has been well studied and understood in many spacetimes. Note that most of the deflection angles found in these works assumed that the source and detectors are located at infinite distance. Through these studies, it is found that when the impact parameter b approaches the critical value bc, the deflection angle diverges as α(b → bc+) ≈ alog 1− b bc. With this deflection angle, the single most remarkable consequence is the existence of a series of relativistic image with infinite magnification for rays near the critical photon sphere. The single most remarkable consequence is the existence of a series of relativistic image with infinite magnification for rays near the critical photon sphere The messenger for these GL images and for GL in the weak field limit were traditionally light rays.
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