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

Abstract

We derive an exact radial monotonicity law for static, spherically symmetric spacetimes in metric \(f(R)\) gravity. For \(ds^2=A(r)dt^2-dr^2/B(r)-r^2dΩ^2\), 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 \(f_R\equiv df/dR>0\), 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 endpoint ratios rather than excluding two horizons. A zero-integral obstruction arises for equal endpoint values, including boundaries where \(B\to0\) while \(A\) remains finite and nonzero. Saturation is equivalent to \(B/A=\mathrm{const}\), and in vacuum requires a scalaron profile linear in the areal radius. We illustrate the strict non-saturated 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. In particular, in the Schwarzschild--de Sitter two-horizon parameter range, one constant-\(X\) solution crosses \(f_R=0\) 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.