Statistical mechanics of gravity and the thermodynamical origin of time

The lack of a statistical and thermodynamical theory of the gravitational field is pointed out. The possibility of developing such a theory is discussed, as well as its theoretical and cosmological relevance. The connection between this problem and the problem of physical time in (compact space) general relativity is emphasized. The idea that a preferred physical time variable is singled out by the statistical properties of the state is proposed. A scheme for a generally covariant statistical thermodynamics is put forward, by extending the Gibbs formalism to the presymplectic, or constrained, dynamical systems. This scheme is based on three equations. The first characterizes any statistical state; the second is the definition of a vector flow in terms of the statistical state; the third is an intrinsic definition of equilibrium. The vector flow of an equilibrium state is denoted thermodynamical time, and the suggestion is made that thermodynamical time is the internal time that carries all the 'common sense' characterizations of the notion of time. Finally, it is suggested that the issue of the second law of the thermodynamics could be reconsidered within this generally covariant framework.