A New Derivation of Classical Gravitational Second Law of Thermodynamics

Abstract

It is established that black holes have entropy and behave as thermodynamical systems. Associating entropy to gravitational fields has not remained limited to black holes, necessitating the notion of the second law of thermodynamics in gravitating systems. There have been many ideas and attempts to prove the second law in gravitational systems starting from first principles. Within the covariant phase space formalism, we define gravitational entropy as the charge associated with the local boosts, detaching the gravitational entropy from horizons or trapped surfaces. Our definition encompasses and generalizes the existing notions of entropy. Using this definition for the Einstein gravity case, we compute variations of the entropy along the path of any causal observer and establish that the entropy variations are always non-negative if the matter content satisfies the strong energy condition integrated along any segment of the causal path.

Figures (2)

• Figure 1: The future-oriented causal curve $\gamma$, the normal and tangent vectors to it $s^\mu, v^\mu$ and the two future-oriented null vectors fields $l^\mu, n^\nu$ on the $2d$ plane ${\cal D}$. At each point on $\gamma$ we have a codimension-2 spacelike hypersurface $\Sigma$, denoted by a point on ${\cal D}$.
• Figure 2: The codimension-1 causal surface $\Gamma$ which is topologically $\Sigma \times \gamma$. At a given $\lambda$, $\Gamma$ is limited to $\Sigma(\lambda)$.