Universality Classes of Higher-Order Degenerate Circular Null Orbits: Critical Dynamics and Strong-Deflection Scaling

Abstract Circular null orbits are fundamental structures in relativistic gravitational lensing, photon trapping, black-hole shadows, and the propagation of high-frequency radiation. For an ordinary unstable photon sphere, the associated radial potential possesses a nondegenerate maximum, producing exponential approach to the critical orbit and the familiar logarithmic divergence of the strong-deflection angle. A marginally unstable photon sphere occurs when the quadratic curvature of the effective potential vanishes, producing instead algebraic critical dynamics and power-law divergence of the deflection angle. In this work, we develop a systematic higher-order classification of degenerate circular null orbits based on the order of the first nonvanishing derivative of the photon effective potential. If the first nonzero derivative occurs at order n > 2, we show locally that the critical trajectory obeys the universal algebraic law r − r꜀ ∼ |φ|⁻²ⁿ⁻² (that is, r − r꜀ ∼ |φ| raised to the power −2/(n−2)), while, under the appropriate scattering assumptions, the strong-deflection angle scales with impact parameter as α_div ∼ |b − b꜀| raised to the power −(n−2)/(2n). These exponents satisfy the universal relation χₙηₙ = 1/n, where χₙ = 2/(n − 2) characterizes critical-orbit relaxation and ηₙ = (n − 2)/(2n) characterizes strong-deflection divergence. The familiar logarithmic case is recovered as the nondegenerate n = 2 limit in a singular rather than algebraic sense, while the established cubic marginal case corresponds to n = 3. We then construct an explicit asymptotically flat, static, spherically symmetric spacetime realizing the quartic class (n = 4), A₄(r) = 1 − 2/r + 3/(2r²) − 2/(5r³), for which the photon potential possesses an exact quartic maximum at r꜀ = 1. The corresponding critical trajectory obeys r − r꜀ ∼ |φ|⁻¹, and the strong-deflection angle exhibits the new power law α_div ∼ |b − b꜀|⁻¹ᐟ⁴. The model is asymptotically flat, possesses a regular exterior region r ≥ 1, and its effective stress-energy tensor satisfies the null, weak, strong, and dominant energy conditions throughout that exterior. The results establish a hierarchy of universality classes for degenerate circular null trapping and identify the quartic class as the first even-order extension beyond the currently established cubic marginal case.