Extreme amplification regimes of the Schwarzschild gravitational lens

¹ Research Centre for Computational Physics and Data Processing, Silesian University in Opava, Bezručovo nám. 13, 746 01 Opava, Czech Republic
² M. R. Štefánik Observatory and Planetarium, Sládkovičova 41, 920 01 Hlohovec, Slovak Republic
³ Institute of Physics of the Czech Academy of Sciences, Na Slovance, 1999/2 Prague, Czech Republic
⁴ Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical Engineering, Břehová 7, Prague, Czech Republic
⁵ Dipartimento di Fisica, Università La Sapienza, Piazzale Aldo Moro 5, 00185 Roma, Italy
⁶ INAF – Osservatorio Astronomico di Roma, Via Frascati 33, 00078 Monteporzio Catone, Roma, Italy
e-mail: luigi.stella@inaf.it
⁷ Dipartimento di Fisica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma, Italy
⁸ International Space Science Institute (ISSI), Hallerstrasse 6, 3012 Bern, Switzerland
⁹ Physikalisches Institut, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland

Received: 10 November 2022
Accepted: 8 February 2023

Abstract

We investigated a complete set of relativistic images of a small source located at an arbitrary distance from a Schwarzschild black hole gravitational lens. This paper offers a description of a simple and efficient fully relativistic method for calculating the bolometric intensity amplification. We focused our analysis primarily on sources located at small radii and close angular distance from the caustic line, both behind and in front of the compact lens. We term the corresponding large deflection regime ‘extreme lensing’. We approximated the regime of fully-relativistic, extreme amplification of point sources by simple analytical formulae valid over a wide range of source distances. Using such approximations, we also derived formulae for the maximum amplification of extended sources close to or intercepted by the caustic line. Simple analytical approximations of the time delay between the brightest consecutive images in extreme amplification regimes are also presented.