Thiago Campos

While the quasi-local thermodynamics of spherically symmetric black holes is well described by pressure and volume, extending this framework to rotating spacetimes poses a significant challenge. Rotation induces an oblate deformation of the horizon, breaking the direct functional dependence between geometric volume and area. In this work, we resolve this difficulty by establishing a quasi-local thermodynamic framework for Kerr-Newman black holes. We demonstrate that accommodating this kinematic deformation requires extending the thermodynamic phase space to include a geometric eccentricity parameter $Y$ and its conjugate, a thermodynamic shear tension $X$. Consequently, the rotational contribution is incorporated into the first law with a shear work term $X dY$. We derive the generalized first laws and Smarr formulas (Euler relations) for both the internal energy and enthalpy representations, showing that these thermodynamic potentials can be obtained through Legendre transformations that isolate the quasi-local energy from the rotational energy. Thus, this framework provides a novel perspective on the thermodynamics of rotating black holes, integrating the geometric deformation of the horizon into a quasi-local description.