Relativistic images of Schwarzschild black hole lensing

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.

Gravitational lensing by a structure in a cosmological background

“…Hence, the time delays in these two coordinates are equal, as expected due to the fact that both metrics tends to FRW metric at infinity. Now, neglecting the second term and applying the Fermat principle, we obtain the lens equation [23]:…”

Finite-distance corrections to the gravitational bending angle of light in the strong deflection limit

“…For instance, strong-field gravitational lensing in a Schwarzschild black hole was investigated by Frittelli, Kling and Newman [8], by Virbhadra and Ellis [9] and more comprehensively by Virbhadra [10]; distinctive lensing features of naked singularities were studied by many authors [11][12][13][14][15][16][17][18][19][20]. Kitamura, Nakajima and Asada proposed a lens model in the inverse powers of the distance as 1/r n [21], such that the Schwarzschild lens, the Ellis wormhole lens and a gravitational lens associated with exotic matter (or energy) that might follow a non-standard equation of state can be discussed in a unified manner of describing these models as a one parameter family [22][23][24][25].…”

“…In order to understand a strong gravitational field, a more rigorous formulation of the bending angle has been investigated [8][9][10][11][12][13][14][15]. For instance, strong-field gravitational lensing in a Schwarzschild black hole was investigated by Frittelli, Kling and Newman [8], by Virbhadra and Ellis [9] and more comprehensively by Virbhadra [10]; distinctive lensing features of naked singularities were studied by many authors [11][12][13][14][15][16][17][18][19][20].…”

Erratum to: Strong Gravitational Lensing in a Brane-World Black Hole

“…We also used a term relativistic images and ignored that this was coined by Virbhadra and Ellis [1] and relativistic images were comprehensively studied by Virbhadra [6]. In view of this, we clarify that it is truly important for us to state an erratum, because we ignored stating the provenance of those equations in the published paper.…”