Scalaron-modified null focusing and radial monotonicity in static gravity
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 gravity. For , the matter contribution and the scalaron Hessian combine into an effective radial-convergence numerator that fixes the derivative of . On every connected static interval with , , and , its sign therefore determines the monotonicity of . The integrated identity retains the finite, generally nonzero value of 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 while remains finite and nonzero. Saturation is equivalent to , 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- and power-law solutions of Multamäki and Vilja [Spherically symmetric solutions of modified field equations in theories of gravity, Phys. Rev. D 74, 064022 (2006)]. In particular, in the Schwarzschild–de Sitter two-horizon parameter range, one constant- solution crosses 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.