Strong deflection of massive particles in spherically symmetric spacetimes

TL;DR

This paper derives a general analytical approximation for the strong deflection angles of massive particles near compact objects in any static, spherically symmetric, asymptotically flat spacetime, extending previous specific metric results.

Contribution

It provides a universal strong deflection limit formula for massive particles applicable to any such spacetime, including Schwarzschild, Reissner-Nordström, and Janis-Newman-Winicour metrics.

Findings

Derived a general strong deflection limit formula for massive particles.

Applied formulas to Schwarzschild, Reissner-Nordström, and Janis-Newman-Winicour metrics.

Validated the approximation against known exact solutions.

Abstract

Near a gravitating compact object, massive particles traveling along timelike geodesics are gravitationally deflected similarly to light. In this paper, we study the deflection angles of these particles in the strong deflection limit. This analytical approximation applies when particles in unbound orbits approach the compact object very closely, circle around it at a radius close to that of the unstable circular orbit, and eventually escape. While previous studies have provided results for particular metrics, we offer a general solution applicable to any static, spherically symmetric and asymptotically flat spacetime. After briefly reviewing the exact expression for the deflection angle of massive particles, we present a strong deflection limit analysis for this general case. The developed formulas are then applied to three particular metrics: Schwarzschild, Reissner-Nordstr\"om and Janis-Newman-Winicour.