Microcanonical Phase Space and Entropy in Curved Spacetime

Abstract

We discuss the structure of microcanonical ensembles in inertial and non-inertial frames attached to a confined system of positive energy particles in curved spacetime. Under certain physically reasonable assumptions that ensure the existence of such ensembles, we obtain, for microcanonical ensembles, exact analytical results in certain stationary spacetimes such as Rindler, Schwarzschild, and de Sitter along with leading curvature corrections in arbitrary curved spacetimes. For de Sitter, the exact results have interesting limits when the size of the system is comparable to $Λ^{-1/2}$. We further highlight two generic characteristics of the leading curvature corrections for a point particle system confined to a spherical or cubical box: (1) they are characterized by Ricci and Einstein tensors, and (2) their contribution is proportional to the bounding area. We argue that the area scaling in (2) does not hold for arbitrary box geometries. We also present a general argument to highlight two distinct sources of divergences in the phase space volume, coming from redshift and spatial geometry, and illustrate this by comparing and contrasting the results for (i) geodesic box in de Sitter, (ii) geodesic box in Schwarzschild, and (iii) uniformly accelerated box in Minkowski. Finally, we extend these results to $N$ particle systems in the restricted case of massless (ultra-relativistic) limit for static spacetimes, for which the results follow very simply from single particle results. Furthermore, we show that the ultra-relativistic expression for equipartition of energy in flat spacetimes continues to hold in static spacetimes.