General Symmetries: From Homogeneous Thermodynamics to Black Holes

We consider the topic of symmetries as classifying tools for thermodynamic systems. We adopt the contact geometry approach, in a general framework including standard homogeneous thermodynamics but not limited to it, and we focus our attention on the problem of the existence of a general symmetry, to be defined as a symmetry which is the same for a class of thermodynamic systems. Homogeneity symmetry of standard equilibrium thermodynamics is the paradigmatic example of general symmetry, and we point out its being associated with a multi-class thermodynamics, whose mathematical characterization is taken into account. Furthermore, quasi-homogeneity symmetry, which describes some non-extensive systems, is shown to give rise to a general symmetry, in the above sense, in the case of non-relativistic self-gravitating fermions. In the latter case, it is also conjectured to give rise to a multi-class structure. An analysis of the behavior of transversal symmetries under the partial Legendre involutions enhances a special role of quasi-homogeneity symmetry, as well as the role of special thermodynamic limit is pointed out as a tool for investigating the topic of general symmetries.