The First Law of Black Hole Thermodynamics in Arbitrary Gravity and Matter Theories Using Differential Forms

Abstract

Black hole thermodynamics plays a central role in probing the microscopic structure of spacetime. Since the 1990s, a general method for deriving the first law of black hole thermodynamics in arbitrary diffeomorphism-invariant theories has been available within the metric formalism. Differential forms, however, provide a natural and powerful language for formulating gravitational theories, particularly those extending beyond General Relativity. In this work, we develop a unified prescription for deriving the first law using differential forms defined on spacetime. Our framework applies to a broad class of theories, including those with torsion and generic matter fields, and yields concrete expressions for the first law.