Universal small-spin formulas for the shadow area, the capture cross-section and the spin-down of Kerr-like rotating black holes

Many rotating black-hole metrics studied in the literature are generated from a static, spherically symmetric seed ds² = −f dt² + dr²/f + r² dΩ² by the Newman–Janis algorithm in the form of Azreg-Aïnou, which yields the Kerr metric with 2Mr replaced by r²(1 − f(r)). For this whole class we give closed formulas, in terms of f and its derivatives at a single radius, for the leading effects of a small spin a on three observables: (i) the area of the black-hole shadow seen by a distant observer at any inclination, to order a² and, with a term-by-term finite-part method, to orders a⁴, a⁶ and a⁸; (ii) the capture cross-section for neutral massive particles of any speed arriving from any direction, to order a² (and to order a⁴ along the spin axis); and (iii) the mean angular momentum swallowed per unit captured energy, which controls the spin-down of a black hole accreting collisionless matter. The order-a² shadow formula depends on f only through f(R) and the curvature combination κ = 2f − R²f″ at the photon sphere R; the massive-particle formula contains the combination 3ff′ + rff″ − 2rf′², whose zero defines the innermost stable circular orbit. Specialising to Kerr–Newman gives explicit expressions valid at arbitrary charge, including the shadow area to order a⁴; for Kerr the method reproduces the known low-order terms and yields the order-a⁶ and a⁸ coefficients of the shadow area at any inclination. All results were checked against direct high-precision computations of the exact shadow contour or of the exact capture boundary for Kerr, Kerr–Newman, rotating Bardeen and rotating Hayward black holes, with agreement to 11–22 significant digits. AI disclosure: This work was carried out with substantial assistance from an AI system, Claude (Anthropic), used through the Claude Code software, under the author's direction. The AI system proposed research directions, wrote and ran the computer-algebra and numerical code, performed the derivations and checks, searched the literature, and drafted the text. The author directed the work and is responsible for its publication. Novelty assessments are based on literature searches and may be incomplete.