Caloric curves of classical self-gravitating systems in general relativity

We determine the caloric curves of classical self-gravitating systems at statistical equilibrium in general relativity. In the classical limit, the caloric curves of a self-gravitating gas depend on a unique parameter nu = GNm/Rc(2), called the compactness parameter, where N is the particle number and R the system's size. Typically, the caloric curves have the form of a double spiral. The "cold spiral," corresponding to weakly relativistic configurations, is a generalization of the caloric curve of nonrelativistic classical self-gravitating systems. The "hot spiral," corresponding to strongly relativistic configurations, is similar (but not identical) to the caloric curve of the ultrarelativistic self-gravitating black-body radiation. We introduce two types of normalization of energy and temperature to obtain asymptotic caloric curves describing, respectively, the cold and the hot spirals in the limit nu -> 0. As the number of particles increases, the cold and the hot spirals approach each other, merge at nu'(S) = 0.128, form a loop above nu(S) = 0.1415, reduce to a point at nu(max) = 0.1764, and finally disappear. Therefore, the double spiral shrinks when the compactness parameter nu increases, implying that general relativistic effects render the system more unstable. We discuss the nature of the gravitational collapse at low and high energies with respect to a dynamical (fast) or a thermodynamical (slow) instability. We also provide an historical account of the developments of the statistical mechanics of classical self-gravitating systems in Newtonian gravity and general relativity.