Horizon-regular cross-focusing and inner-horizon obstructions in spherical f(R) gravity

Abstract

We formulate a horizon-regular double-null criterion for regular inner marginal horizons in spherically symmetric metric $f(R)$ gravity. Using normalized outgoing and ingoing radial null vectors $\ell^\mu$ and $n^\mu$, with $n^\mu$ affinely parametrized, we derive an exact evolution law for the area-weighted outgoing expansion $r^2\theta_{(\ell)}$. Its source is controlled by the scalaron $F\equiv f_R>0$ and by a mixed quantity $\mathcal{P}_{\ell n}$ containing matter, scalaron derivatives, and the curvature potential. If $\mathcal{P}_{\ell n}\leq F/r^2$ along a regular ingoing null segment issuing from a nondegenerate future outer marginal sphere, then the outgoing expansion cannot return to zero, and no second regular marginal sphere of the same family can occur on that generator. Conversely, a nondegenerate future inner marginal sphere requires the reverse inequality, so an outer--inner pair necessarily entails a source reversal and an exact integral balance. No trapped-region assumption is required. In the static limit, the criterion reduces to a horizon-regular relation involving the radial derivative of the metric function and remains valid in the degenerate case under the stated regularity conditions. It reproduces the Reissner--Nordstr\"om classification and is verified in an exact charged, nonconstant-curvature $f(R)$ black hole with a nonconstant scalaron. The resulting Cauchy-horizon statement is conditional and applies only when the candidate boundary is also a regular nondegenerate future inner marginal horizon.