$f(R)$引力中的无限简并慢旋转解
Infinitely degenerate slowly rotating solutions in $f(R)$ gravity
AI总结：
针对一类幂律$f(R)$模型，本文直接用度量形式构造旋转参数至二阶的多个慢旋转真空黑洞解，揭示其对耦合常数的依赖及无毛定理的扩展必要，并比较轨道与阴影性质。
摘要：
本工作检验$f(R)$引力模型中的无毛猜想。无毛猜想断言，广义相对论中与任何物质耦合的所有黑洞都必须是Kerr-Newman型。然而，在存在非线性物质源的某些情况下，该猜想并不成立。在此，我们通过针对一类$f(R)$模型（$f(R)=(\alpha_{0}+\alpha_{1}R)^{p}$，$p>1$）显式构造旋转参数至二阶的多个慢旋转黑洞解来解决这一问题。这样的$f(R)$包含$R$的所有高次幂。我们通过解析方式表明，多个真空解在旋转参数至二阶时满足场方程。换言之，我们表明多个真空解依赖于任意常数，而这些常数又依赖于模型的耦合参数。因此，我们的结果表明，修正引力理论的无毛定理值得扩展以纳入耦合常数。我们结果的独特性源于这些解直接由度量形式体系获得，而未使用共形变换。我们讨论了这些黑洞解的运动学性质，并将其与慢旋转Kerr黑洞进行比较。具体而言，我们表明$f(R)$中黑洞的圆轨道比Kerr黑洞的更小。这意味着$f(R)$中黑洞的最内稳定圆轨道小于Kerr黑洞；因此，阴影半径也可能更小。最后，我们讨论了研究结果对未来观测的启示。
英文摘要：
This work tests the no-hair conjecture in $f(R)$ gravity models. No-hair conjecture asserts that all black holes in General Relativity coupled to any matter must be Kerr-Newman type. However, the conjecture fails in some cases with non-linear matter sources. Here, we address this by explicitly constructing multiple slow-rotating black hole solutions, up to second order in rotational parameter, for a class of $f(R)$ models ( $f(R) =(α_{0} + α_{1}\,R)^{p}, p > 1$). Such an $f(R)$ includes all higher-powers of $R$. We analytically show that multiple vacuum solutions satisfy the field equations up to the second order in the rotational parameter. In other words, we show that the multiple vacuum solutions depend on arbitrary constants, which depend on the coupling parameters of the model. Hence, our results indicate that the no-hair theorem for modified gravity theories merits extending to include the coupling constants. The uniqueness of our result stems from the fact that these are obtained directly from metric formalism without conformal transformation. We discuss the kinematical properties of these black hole solutions and compare them with slow-rotating Kerr. Specifically, we show that the circular orbits for the black holes in $f(R)$ are smaller than that of Kerr. This implies that the inner-most stable circular orbit for black holes in $f(R)$ is smaller than Kerr's; hence, the shadow radius might also be smaller. Finally, we discuss the implications of our results for future observations.
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