In this work we explore the connections between Newman–Penrose scalars, including the Penrose–Rindler $\mathcal{K}$-curvature, with the equation of state of asymptotically Anti-de Sitter Reissner–Nordström black holes. After briefly reviewing the equation of state for these black holes from the point of view of both the Extended Phase Space and the Horizon Thermodynamics approaches, a geometric splitting is given for such an equation in terms of the non vanishing Newman–Penrose scalars which define the $\mathcal{K}$-curvature at the horizon. From this splitting, a possible thermodynamical interpretation is developed for such scalars in the context of the black hole thermodynamics approaches initially discussed. Afterwards, the square root of the Bel–Robinson tensor is employed to propose conditions at the horizons in terms of pressures or energy densities, which can be understood as alternative thermodynamical definitions of these surfaces.

在这项工作中，我们探索了纽曼–彭罗斯标量，包括彭罗斯–林德勒之间的联系 $\mathcal{ķ}$曲率，具有渐近的反德西斯特·赖斯纳·诺德斯特伦黑洞的状态方程。从扩展相空间和地平线热力学方法的角度简要回顾了这些黑洞的状态方程后，根据不消失的Newman-Penrose标量对这种方程进行了几何拆分。$\mathcal{ķ}$弯曲的地平线。通过这种分裂，在最初讨论的黑洞热力学方法的背景下，为此类标量开发了一种可能的热力学解释。然后，使用Bel–Robinson张量的平方根根据压力或能量密度提出地平线上的条件，这可以理解为这些表面的替代热力学定义。