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Strong deflection of massive particles via the geodesic deviation equation

Takahisa Igata1,* and Yohsuke Takamori2,†

  • *Contact author: takahisa.igata@gakushuin.ac.jp
  • †Contact author: takamori@wakayama-nct.ac.jp
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Phys. Rev. D 113, 124083 – Published 25 June, 2026

DOI: https://doi.org/10.1103/fbfh-tqpj

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.

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Phys. Rev. D 113, 124083– Published 25 June, 2026
  • Received 12 March 2026
  • Accepted 8 June 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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