Abstract
Conditions for entropy extremum of self-bound relativistic systems of particles, in stationary axisymmetric motion, are obtained under the following constraints: (i) The system is kept isolated in a geometrical sense, implying that the total mass-energy and total angular momentum, defined by the asymptotic behavior of the metric, are kept constant, (ii) the total number of particles is kept constant, and (iii) Einstein's constraint equations are imposed on a spacelike hypersurface. It is shown that if the system is in mechanical equilibrium, the total entropy is an extremum for all trial nonequilibrium configurations that satisfy the constraints and respect the symmetry, if and only if (i) Einstein's dynamical equations are satisfied, (ii) the temperature and the chemical free energy, as seen from infinity, are constant, and (iii) the system is rigidly rotating. The proof does not depend on a particular functional expression for the mass or for the angular momentum; consequently, no related Lagrange multipliers have been used.
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