Strong deflection of massive particles in spherically symmetric spacetimes

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-Nordström and Janis-Newman-Winicour.

💡 Research Summary

This paper extends the well‑known strong deflection limit (SDL), originally developed for light rays, to massive particles moving on timelike geodesics in any static, spherically symmetric, asymptotically flat spacetime. The authors begin by writing the generic line element
( ds^{2}= -A(r)dt^{2}+B(r)dr^{2}+C(r)d\Omega^{2} )
with the usual asymptotic conditions (A\to1), (B\to1), (C\to r^{2}) as (r\to\infty). Using the Lagrangian formalism they identify two conserved quantities: the energy per unit rest mass (E) (measured at infinity) and the angular momentum per unit rest mass (L). The normalization condition for timelike geodesics yields a radial equation that can be cast in the form
( \dot r^{2}= \frac{1}{B(r)}\bigl