From Specific Heat Bounds to Penrose-like Inequalities: A Unified Thermodynamic Perspective

Under explicit and physically motivated assumptions, we identify a local-to-global thermodynamic mechanism that organizes several previously disconnected inequalities on black holes. We first focus on asymptotically flat, static, and spherically-symmetric black holes, and provide a unified thermodynamic perspective on various Penrose-like inequalities. We also discuss the generalizations in asymptotically Schwarzschild anti-de Sitter black holes/branes. In holographic settings, these bounds imply monotonic behavior of normalized thermodynamic quantities in the dual field theory. Notably, one central integrated inequality can be established directly under the NEC in general static planar symmetric asymptotically Schwarzschild anti-de Sitter black branes. Our purpose is not to claim a fully general proof of the proposed local bounds, but to expose a unified structural perspective connecting local thermodynamic constraints to global geometric inequalities. Establishing the local bounds beyond the classes considered here remains an open problem.