AI 中文总结
该研究推导静态球对称f(R)引力中精确径向单调律，结合物质与标量龙海森项确定Q=B/A的导数符号，给出视界端点比值约束，诊断已知解的一致性，为黑洞内部视界正则处理提供基础。
AI 中文摘要
我们推导了度规f(R)引力中静态球对称时空的精确径向单调性定律。对于线元ds²=A(r)dt²−dr²/B(r)−r²dΩ²，物质贡献与标量龙海森项结合为有效径向收敛分子，该分子决定了Q=B/A的导数。在每个满足A>0、B>0且f_R≡df/dR>0的连通静态区间上，其符号因此决定了Q的单调性。积分恒等式保留了正则非退化基灵视界处B/A的有限且通常非零的值；因此，固定符号的收敛条件会对视界端点的比值进行排序，而非排除两个视界的存在。当端点值相等时会出现零积分阻碍，包括B→0而A保持有限且非零的边界情况。饱和等价于B/A为常数，且在真空情形下要求标量龙的轮廓随面半径线性变化。我们利用精确的常密度恒星内部解，在广义相对论分支内阐释了严格非饱和分支，并将等式与一致性诊断应用于Multamäki和Vilja提出的常X解与幂律解。具体而言，在史瓦西-德西特双视界参数范围内，一个常X解在完整静态区域内穿过f_R=0的位置，而将幂律族直接代入原始径向方程时，发现其显示的幂律族存在未解决的指数不匹配问题。这些结果提供了一种不依赖模型的静态区域诊断方法，并为未来黑洞内部的视界正则处理提供了精确的起点。
英文摘要
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.
Journal refPhys. Rev. D 114, 064088 (2026)
