Perturbative deflection angle, gravitational lensing in the strong field limit and the black hole shadow

TL;DR

This paper introduces a perturbative method for calculating gravitational lensing and black hole shadow sizes in strong field regimes, accounting for finite distances and applied to Hayward black holes, revealing how charge and velocity affect observable features.

Contribution

It proposes a novel perturbative approach for strong field gravitational lensing that explicitly includes finite distance effects and provides simple formulas for black hole shadow sizes.

Findings

Black hole shadow size decreases with increasing charge parameter l.

Relativistic image magnifications diverge at different source-detector configurations.

Shadow size increases as signal velocity decreases from light speed.

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 $l$ increase 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.