Strong deflection of massive particles via the geodesic deviation equation

TL;DR

This paper formulates the strong deflection limit for massive particles in static, spherically symmetric spacetimes, linking divergence behavior to local geometric and matter properties via the geodesic deviation equation.

Contribution

It introduces a covariant method using the geodesic deviation equation to analyze strong deflection of massive particles, relating divergence coefficients to local curvature and matter content.

Findings

Deflection angle diverges logarithmically near unstable circular orbits.

The divergence coefficient is determined by the radial instability exponent.

Matter influences the divergence through a specific local scalar involving energy density and pressures.

Abstract

We develop a formulation of the strong deflection limit for the scattering of particles following timelike geodesics in asymptotically flat, static, and spherically symmetric spacetimes. For fixed specific energy, as the angular momentum approaches its critical value from above, the particle passes arbitrarily close to the associated unstable circular orbit, undergoes many windings around it, and the deflection angle diverges logarithmically. Using the geodesic deviation equation, we show covariantly that the coefficient of this logarithmic divergence is determined by the radial instability exponent of the critical trajectory, defined per unit azimuthal angle. We express this instability exponent in terms of local curvature data on the unstable circular orbit, thereby providing both kinematic and geometric interpretations of the strong deflection limit. In general relativity, its matter dependence enters only through a single local scalar combination constructed from the static-frame energy density and the principal radial and tangential pressures.