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    Scalaron-modified null focusing and radial monotonicity in static f(R) gravity

    Maickol Muñoz-Palma1,*, Francisco S. N. Lobo2,3,†, Jean Báez Cuevas4,‡, and Francisco Tello-Ortiz1,§

    • *Contact author: m.munoz58@ufromail.cl
    • †Contact author: fslobo@ciencias.ulisboa.pt
    • ‡Contact author: jean.baez@pucv.cl
    • §Contact author: francisco.tello@ufrontera.cl

    Phys. Rev. D 114, 064088 – Published 28 September, 2026

    DOI: https://doi.org/10.1103/hnw1-gl3z

    Abstract

    We derive an exact radial monotonicity law for static, spherically symmetric spacetimes in metric f(R) gravity. For ds2=A(r)dt2−dr2/B(r)−r2dΩ2, the matter contribution and the scalaron Hessian combine into an effective radial-convergence numerator that fixes the derivative of Q=B/A. On every connected static interval with A>0, B>0, and fR≡df/dR>0, its sign therefore determines the monotonicity of Q. The integrated identity retains the finite, generally nonzero value of B/A at a regular nondegenerate Killing horizon, consequently, a fixed-sign convergence condition orders the horizon end point ratios rather than excluding two horizons. A zero-integral obstruction arises for equal end point values, including boundaries where B→0 while A remains finite and nonzero. Saturation is equivalent to B/A=const, and in vacuum requires a scalaron profile linear in the areal radius. We illustrate the strict nonsaturated branch, within the general relativity sector, using the exact constant-density stellar interior, and apply the equality and consistency diagnostics to the constant-X and power-law solutions of Multamäki and Vilja [Spherically symmetric solutions of modified field equations in f(R) theories of gravity, Phys. Rev. D 74, 064022 (2006)]. In particular, in the Schwarzschild–de Sitter two-horizon parameter range, one constant-X solution crosses fR=0 inside the complete static patch, while a direct substitution into the original radial equation exposes an unresolved exponent mismatch in the displayed power-law family. The results provide a model-independent static-sector diagnostic and a precise starting point for a future horizon-regular treatment of black hole interiors.

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