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Strong-deflection expansion of the deflection angle near a degenerate photon sphere

Takahisa Igata1,*, Tadashi Sasaki2,†, and Naoki Tsukamoto3,‡

  • *Contact author: takahisa.igata@gakushuin.ac.jp
  • †Contact author: ta-sasaki@kumagaku.ac.jp
  • ‡Contact author: tsukamoto@rikkyo.ac.jp
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Phys. Rev. D 113, 104022 – Published 12 May, 2026

DOI: https://doi.org/10.1103/b47c-8rdg

Abstract

We present a strong-deflection expansion for the deflection angle of light rays scattered near a degenerate photon sphere in asymptotically flat, static, and spherically symmetric spacetimes. Our prescription isolates the divergent contribution to the deflection-angle integral arising from the ray’s passage near the marginal orbit in a way that remains nonsingular at marginality, thereby yielding a unique leading power-law term. When expressed in terms of the radius of closest approach, the leading coefficient in the strong deflection limit factorizes into a universal branch constant and a local factor determined by the third derivative of the effective potential at the degenerate photon sphere. When the expansion is rewritten in terms of the impact parameter, the coefficient is simply multiplied by an additional local conversion factor. We show that the local factor in the closest-approach expansion admits an invariant representation through the areal-radius derivative of a dimensionless tidal measure constructed from the electric part of the Weyl tensor. In general relativity, we further relate this quantity to the areal-radius derivative of a weighted null-energy density profile. Analytic examples validate this factorization and yield closed-form expressions for the leading divergent coefficients in representative marginal configurations.

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See Also

Deflection angle in the strong deflection limit: A perspective from local geometrical invariants and matter distributions

Takahisa Igata
Phys. Rev. D 113, 044042 (2026)

Article Text

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Phys. Rev. D 113, 104022– Published 12 May, 2026
  • Received 12 March 2026
  • Accepted 17 April 2026
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