球对称f(R)引力中的视界正则交叉聚焦与内视界障碍
Horizon-regular cross-focusing and inner-horizon obstructions in spherical f(R) gravity
AI总结：
本研究在球对称f(R)引力中构建视界正则双零判据，推导面积加权外向膨胀演化律，验证其在带电非恒定曲率黑洞中的有效性，明确柯西-视界陈述的适用条件。
AI中文摘要：
我们针对球对称度量f(R)引力中的正则内边缘视界，构建了一种视界正则的双零判据。利用归一化的外向和内向零矢ℓ^μ与n^μ（其中n^μ为仿射参数化），我们推导了面积加权外向膨胀r²θ_(ℓ)的精确演化律。该演化律的源由标量子F≡f_R>0以及包含物质、标量子导数和曲率势的混合量𝒫_(ℓn)控制。若从非退化未来外边缘球面发出的正则内向零矢段上满足𝒫_(ℓn)≤F/r²，则外向膨胀无法回到零，且在该生成元上不会出现同一家族的第二个正则边缘球面。反之，非退化未来内边缘球面需要反向不等式，因此外-内对必然伴随源的反转和精确积分平衡。本研究无需 trapped-region假设。在静态极限下，该判据简化为涉及度量函数径向导数的视界正则关系，且在给定正则条件下于退化情形中仍有效。它再现了Reissner-Nordström分类，并在具有非恒定标量子的精确带电、非恒定曲率f(R)黑洞中得到验证。所得柯西-视界陈述是有条件的，仅当候选边界同时为正则非退化未来内边缘视界时适用。
英文摘要：
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 $\ell^μ$ and $n^μ$, with $n^μ$ affinely parametrized, we derive an exact evolution law for the area-weighted outgoing expansion $r^2θ_{(\ell)}$. 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--Nordström 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.
