Previous applications of relativistic thermodynamics have been either to systems in static thermodynamic equilibrium or to non-static systems in which reversible processes are taking place. The present article deals with non-static systems in which irreversible processes take place. It is shown that the failure of the principle of energy conservation to hold in relativistic mechanics in its older simple form removes a classical limitation on the irreversible increases in entropy which can take place in an isolated system. The removal of this limitation provides possibilities in relativistic thermodynamics for irreversible processes to take place in an isolated system without ever reaching an unsurpassable state of maximum entropy and minimum free energy where further change would be impossible. Such possibilities are found to be illustrated by non-static models of the universe which could undergo a continued series of irreversible expansions and contractions without ever arriving at a state of rest. Special attention is given to a non-static model of the universe in which the irreversible annihilation of matter would take place in the later stages of expansion and the irreversible formation of matter out of radiation in the later stages of contraction. Finally, remarks are made concerning the bearing of the findings on the interpretation of phenomena in the actual universe.

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