A Constraint-Free Formulation of Black Hole Thermodynamics from the Field Equations

TL;DR

This paper introduces a new, constraint-free approach to derive the first law of black hole thermodynamics directly from Einstein's equations, applicable to multi-horizon black holes and higher-derivative theories.

Contribution

It generalizes previous methods by allowing unconstrained variations of all black hole parameters, avoiding restrictions to the outer horizon and extending validity to complex black hole solutions.

Findings

Method remains valid in higher-derivative gravity theories.

Unconstrained variations can be applied directly in thermodynamic state space.

Traditional horizon-based variation schemes fail for black holes with three independent parameters.

Abstract

We develop a constraint-free formulation that generalizes Padmanabhan's method for deriving the first law of black hole thermodynamics directly from the Einstein field equations. In previous studies, even for multi-horizon black holes, variations were restricted to the outer horizon by imposing an additional constraint, and the PdV term was introduced by multiplying the field equations evaluated at the outer horizon by the corresponding volume variation dV. However, since general variations of the black hole parameters shift both horizons, variations at both horizons must be taken into account. To this end, we propose multiplying the horizon field equations by the entropy variation dS under such unconstrained variations. We show that this method remains valid even in higher-derivative theories of gravity. In addition, we find that $r_{\pm}$-based variation schemes generically break down for black holes characterized by three independent parameters (M,J,Q). By working directly in the thermodynamic state space (M,J,Q), we show that the Einstein field equations evaluated at the outer horizon can be also interpreted as the first law of black hole thermodynamics for general variations without imposing any additional constraints.