Strong field gravitational lensing of particles by a black-bounce-Schwarzschild black hole
Phys. Rev. D 113, 124082 – Published 25 June, 2026
DOI: https://doi.org/10.1103/4xdv-kszx
Abstract
The gravitational lensing of relativistic and nonrelativistic neutral massive particles in the black-bounce-Schwarzschild black hole spacetime is investigated in the strong deflection limit. Beginning with the explicit equations of motion of a massive particle in the regular spacetime, we achieve the equation of the particle sphere and thus the radius of the unstable timelike circular orbit. It is interesting to find that the particle sphere equation can reduce to the well-known photon sphere equation, when the particle’s initial velocity is equal to the speed of light. We adopt the strong field limit approach to calculate the black-bounce-Schwarzschild deflection angle of the particle subsequently, and obtain the strong-deflection lensing observables of the relativistic images of a pointlike particle source. The observables mainly include the apparent angular particle sphere radius, the angular separation between the outermost relativistic image and the other ones that are packed together, and the ratio between the particle-flux magnification of the outermost image and that of the packed ones (or equivalently, their resulting magnitudelike difference). The velocity effects induced by the deviation of the initial velocity of the particle from light speed on the corresponding strong-field lensing observables of the images of a pointlike light source in the regular geometry, along with these on the strong deflection limit coefficients and the critical impact parameter of the lightlike case, are then formulated. The influence of the spacetime bounce on the Schwarzschild lensing properties of the images of a massive-particle-emission source in the strong field limit is also considered. Serving as an application of the results, we finally concentrate on evaluating the astronomical detectability of the velocity- and bounce-induced effects on the lensing observables and analyzing their dependence on the parameters, by modeling the supermassive black hole at the galactic center (i.e., Sgr ) and the one at the center of galactic M87 (i.e., ) as the lens, respectively.
