More on the relativistic images produced in gravitational lensing

TL;DR

This work develops a strong-field gravitational lensing framework and applies it to charged and rotating black holes, providing numerical predictions for relativistic images around Sgr A*. By solving the strong deflection limit and computing observables such as deflection angles, image positions, magnifications, and time delays up to $n=2$, the authors quantify how charge $q$ and spin $a$ reshape the photon sphere and the resulting lensing signals. The key findings show that increasing $q$ or prograde spin weakens the overall lensing scale (smaller $x_p$ and $J_p$) but can enhance magnifications and generate distinctive asymmetries and timing signatures, notably a spin-induced shift between prograde and retrograde images and a sign change in $DeltaT^o_{1,1}$. These results offer prospects for constraining black hole properties with future very-long-baseline interferometry, linking observable relativistic images to the spacetime geometry around compact objects.

Abstract

We investigate the gravitational lensing properties and the formation of relativistic images associated with black holes modeled by the Reissner-Nordström and Kerr space-time geometries. In particular, we perform numerical computations of the deflection angles, image angular positions, magnification factors (including demagnification), and time delays between distinct images, and we systematically quantify the dependence of these observables on the electric charge and spin parameters of the black hole.

Figures (13)

• Figure 1: Illustration showing the formation of the two relativistic image sequences due to the looping around the photon sphere of the black hole. This also conveys the location of the primary and secondary images obtained.
• Figure 2: Lens diagram with lens $L$, observer $O$, source $S$ and image $I$. Distances $D_{ds}$, $D_d$ and $D_s$ are the respective distances from the lens to source, observer to lens and observer to source. $OL$ is the optical axis in which the angular positions of the source $\beta$ and the image $\theta$ are measured; $\hat{\alpha}$ is the deflection angle of the light ray (in blue). $SQ$ and $OI$ are tangents to the light ray (in blue) at the $S$ and $O$ respectively. $LN$ and $LT$ are perpendiculars to $OI$ and $SQ$ respectively which are the impact parameter $J$. This figure is an adaptation to the figure presented in virbhadra_schwarzschild_2000.
• Figure 3: The relativistic images.
• Figure 4: Illustration showing the looping of light rays and the formation of the higher order relativistic images with their positions around the photon sphere of the black hole.
• Figure 5: Illustration showing the different relativistic image locations. On each side of the lens $L$ is an infinite series of relativistic images formed. $I_{r1}$ and $I_{r2}$ stand for the first and second order relativistic images. $I_p$ and $I_s$ are the primary and secondary images respectively.