Gravitational lensing of massive particles in Schwarzschild gravity

TL;DR

This paper studies the gravitational lensing effects on massive particles in Schwarzschild spacetime, deriving a unified formula for their bending angles and exploring implications for astrophysical observations.

Contribution

It introduces a master formula for the bending angle of massive particles, unifying massless and massive cases, and analyzes their gravitational lensing effects in detail.

Findings

Derived a simple particle sphere radius as a function of velocity and black hole mass.

Established a master elliptic function formula for the bending angle of particles.

Analyzed weak and strong gravitational lensing corrections for massive particles.

Abstract

Both massless light ray and objects with nonzero mass experience trajectory bending in a gravitational field. In this work the bending of trajectories of massive objects in a Schwarzschild spacetime and the corresponding gravitational lensing (GL) effects are studied. A particle sphere for Schwarzschild black hole (BH) is found with its radius a simple function of the particle velocity and proportional to the BH mass. A single master formula for both the massless and massive particle bending angle is found, in the form of an elliptic function depending only on the velocity and impact parameter. This bending angle is expanded in both large and small velocity limits and large and small impact parameter limits. The corresponding deflection angle for weak and strong GL of massive particles are analyzed, and their corrections to the light ray deflection angles are obtained. The dependence of the deflection angles on the source angle and the particle speed is investigated. Finally we discuss the potential applications of the results in hypervelocity star observations and in determining mass/mass hierarchy of slow particles/objects.