This article investigates, on the basis of relativistic mechanics and relativistic thermodynamics, the theoretical requirements for nonstatic models of the universe to exhibit a behaviour which is periodic in time. The discussion is limited to models which can be regarded from a large-scale point of view as filled with a uniform distribution of fluid, and the time coordinate is purposely chosen so as to agree with the proper time that would be used by local observers at rest in this fluid. It is first shown that the analytical requirements which would correspond to the continuous expansion and contraction of such models between definite maximum and minimum limits, together with the thermodynamic requirement that the expansion and contraction must be reversible, could not both be simultaneously satisfied with any fluid for filling the model which has reasonable properties. Attention is then turned to models of the universe having line elements which would be quasi-periodic in the time. A wide range of possibilities is found for models which would expand from zero proper volume to a maximum and then return, the treatment failing however to provide the analytical conditions for a minimum at the lower limit. It is pointed out, nevertheless, that from a physical point of view contraction to zero proper volume could only be followed by renewed expansion, so that we might expect for such models a continued series of expansions and contractions. Furthermore, it is found possible to satisfy the analytical requirements for such quasi-periodic behaviour by models which would expand and contract at a finite rate reversibly without increase in entropy, so that we might then expect a continued series of identical expansions and contractions. It is pointed out that Einstein's recent model of a nonstatic universe filled with incoherent matter, exerting no pressure and unaccompanied by radiation, would satisfy the thermodynamic as well as the analytical requirements for such a series of identical repetitions, and models filled solely with black-body radiation and with an equilibrium mixture of radiation and perfect gas are discussed which also satisfy these requirements. In conclusion some remarks are made concerning the bearing of such findings on the behaviour of the actual universe.

• Received 18 September 1931

DOI:https://doi.org/10.1103/PhysRev.38.1758