Relativistic images of Schwarzschild black hole lensing

TL;DR

This paper models gravitational lensing by Schwarzschild black holes, revealing that certain ratios and angular separations are highly insensitive to source position, enabling precise measurements of black hole masses and distances.

Contribution

It introduces a method to accurately determine black hole masses and distances using relativistic image properties, highlighting the role of effective deflection angles in strong lensing analysis.

Findings

Mass-to-time delay ratio is insensitive to source position.

Angular separations between relativistic images are highly stable.

Effective deflection angles vary with source position and influence lensing analysis.

Abstract

We model massive dark objects at centers of many galaxies as Schwarzschild black hole lenses and study gravitational lensing by them in detail. We show that the ratio of mass of a Schwarzschild lens to the differential time delay between outermost two relativistic images (both of them either on the primary or on the secondary image side) is extremely insensitive to changes in the angular source position as well as the lens-source and lens-observer distances. Therefore, this ratio can be used to obtain very accurate values for masses of black holes at centers of galaxies. Similarly, angular separations between any two relativistic images are also extremely insensitive to changes in the angular source position and the lens-source distance. Therefore, with the known value of mass of a black hole, angular separation between two relativistic images would give a very accurate result for the distance of the black hole. Accuracies in determination of masses and distances of black holes would however depend on accuracies in measurements of differential time delays and angular separations between images. Deflection angles of primary and secondary images as well as effective deflection angles of relativistic images on the secondary image side are always positive. However, the effective deflection angles of relativistic images on the primary image side may be positive, zero, or negative depending on the value of angular source position and the ratio of mass of the lens to its distance. We show that effective deflection angles of relativistic images play significant role in analyzing and understanding strong gravitational field lensing.