Combining with the small-large black hole phase transition, the thermodynamic geometry has been well applied to study the microstructure for the charged anti–de Sitter (AdS) black hole. In this paper, we extend the geometric approach to the rotating Kerr-AdS black hole and aim to develop a general approach for the Kerr-AdS black hole. Treating the entropy and pressure as the fluctuation coordinates, we construct the Ruppeiner geometry for the Kerr-AdS black hole by making use of the Christodoulou-Ruffini-like squared-mass formula, which is quite different from the charged case. Employing the empirical observation of the corresponding scalar curvature, we find that, for the near-extremal Kerr-AdS black hole, the repulsive interaction dominates among its microstructure, while for far-from-extremal Kerr-AdS black hole, the attractive interaction dominates. The critical phenomenon is also observed for the scalar curvature. These results uncover the characteristic microstructure of the Kerr-AdS black hole. Such general thermodynamic geometry approach is worth generalizing to other rotating AdS black holes, and more interesting microstructure is expected to be discovered.

中文翻译：

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旋转克尔反德西特黑洞的一般热力学几何方法

结合小到大的黑洞相变，热力学几何已被很好地应用于研究带电反德西特（AdS）黑洞的微观结构。在本文中，我们将几何方法扩展到旋转的 Kerr-AdS 黑洞，并旨在为 Kerr-AdS 黑洞开发一种通用方法。以熵和压强为涨落坐标，我们利用类Christodoulou-Ruffini的质量平方公式构造Kerr-AdS黑洞的Ruppeiner几何，这与带电情况有很大不同。利用相应标量曲率的经验观察，我们发现，对于接近极值的 Kerr-AdS 黑洞，排斥相互作用在其微观结构中占主导地位，而对于远离极值的 Kerr-AdS 黑洞，有吸引力的互动占主导地位。对于标量曲率也观察到临界现象。这些结果揭示了 Kerr-AdS 黑洞的特征微观结构。这种通用的热力学几何方法值得推广到其他旋转的 AdS 黑洞，并有望发现更有趣的微观结构。