标题： 所有F(R)引力理论的通用黑洞解 显示英文标题

标题： Universal black hole solutions for all F(R) gravitational theories

摘要： 扩展的引力模型在过去几十年中受到了广泛关注。 在本工作中，我们研究了$F(R)$理论中真空、静态和球对称时空的解空间，引入了将真空方程简化为一个二阶方程的新方法。 首次，我们推导了度规函数关于任意函数 $F(R)$的解析表达式，提供了关于引力作用如何影响时空结构的详细见解。 我们分析了解在何时是渐近平坦的、在核心处是正则的，并且包含事件视界，得到了熵、温度和比热关于 $F(R)$的显式表达式。 通过使用一个度规自由度，我们确定了最一般的解并研究了其(不)物理性质，表明在真空情况下，无法在此限制框架内解决奇点问题。 对于涉及两个度规函数的一般情况，我们使用多种近似方案探讨了对施瓦茨希尔德-(反)德西特时空的修正，发现$F(R)$对广义相对论的扩展会引发不可忽略的不稳定性。 最后，通过轴向扰动的分析，我们推导了黑洞准正则模态势能的一般表达式，作为任意拉格朗日量的函数。 显示英文摘要

摘要： Extended gravitational models have gained large attention in the last couple of decades. In this work, we examine the solution space of vacuum, static, and spherically symmetric spacetimes within $F(R)$ theories, introducing novel methods that reduce the vacuum equations to a single second-order equation. For the first time, we derive analytic expressions for the metric functions in terms of the arbitrary functional $F(R)$, providing detailed insight into how the gravitational action impacts the structure of spacetime. We analyze conditions under which solutions are asymptotically flat, regular at the core, and contain an event horizon, obtaining explicit expressions for entropy, temperature, and specific heat in terms of $F(R)$. By using a single metric degree of freedom, we identify the most general solution and examine its (un)physical properties, showing that resolving singularities is not possible within this restricted framework in vacuum. For the general case involving two metric functions, we use several approximation schemes to explore corrections to Schwarzschild-(anti)de Sitter spacetimes, finding that $F(R)$ extensions to General Relativity induce instabilities that are not negligible. Finally, through an analysis of axial perturbations, we derived a general expression for the potential of quasinormal modes of a black hole as a function of the arbitrary Lagrangian.