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 ν=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 ν→0. As the number of particles increases, the cold and the hot spirals approach each other, merge at ν_{S}^{'}=0.128, form a loop above ν_{S}=0.1415, reduce to a point at ν_{max}=0.1764, and finally disappear. Therefore, the double spiral shrinks when the compactness parameter ν 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.