On the problem of the entropy of the universe as a whole

The well-known problem of the entropy of the universe as a whole arises from the difficulties encountered by classical thermodynamics—first in failing to account for the presumed fact that the entropy of the universe has always been increasing at an enormous rate and nevertheless has not yet reached its maximum value—and second in failing to allow an emotionally satisfactory feeling towards our universe whose ultimate fate would be the stagnation of "heat-death." The purpose of the present article is to examine this problem from the point of view of the extension of thermodynamics to general relativity which has previously been made by the author. A number of earlier contributions to the solution of the problem, which have been made from the standpoint of classical thermodynamics or statistical mechanics, are first briefly described in order to emphasize the very different character of the contribution to the problem made in the present article. It is then pointed out that the problem of the entropy of the universe arises in the classical thermodynamics because of the presumption that thermodynamic processes cannot take place both reversibly and at a finite rate, and that the general nature of the contribution to the problem offered by relativistic thermodynamics consists in showing the possibility of thermodynamic changes which could take place at a finite rate and at the same time reversibly without increase in entropy. The principles of relativistic thermodynamics are then reviewed, and this difference between the classical and relativistic thermodynamics is shown by considering the possibilities of carrying out reversible changes at a finite rate in the properties of a thermodynamic fluid. In the classical thermodynamics it is found that no change in the thermodynamic properties could be allowed to take place at a finite rate, the entropy density of the fluid necessarily remaining constant in accordance with the equation dφ0/dt=0. On the other hand, in relativistic thermodynamics it is found possible to allow changes to take place at a finite rate in the proper volume of the fluid, due to changes in the gravitational potentials gμν, and still maintain reversibility provided the changes satisfy the relation ∂/∂x4(φ0sqrt[-g]dx4/ds)=0. To exhibit the nature of the reversible changes at a finite rate thus permitted in relativistic thermodynamics, consideration is given to the highly idealized model of a non-static universe filled with black-body radiation as a thermodynamic fluid, and it is shown that the radius, total proper volume, and entropy density of such a universe could be changing at a finite rate and yet reversibly without increase in entropy. Furthermore, in the case of an expanding model of the kind considered, it is shown that an ordinary observer, who marks out with rigid meter sticks a small region of this universe in his immediate vicinity for study, would find the energy density, energy content, and the temperature of this region decreasing with the time, and would find the number of quanta leaving the region per second greater than the number entering, and the average frequency of the quanta that leave greater than that of those that return. These phenomena would be interpreted by the observer, from a classical point of view, as due to radiation from his neighborhood into the colder surroundings of space and hence as leading to an increase in entropy, in spite of the fact that all the processes taking place in such a model would actually be reversible from the point of view of the relativistic thermodynamics which should be applied to such a problem. In conclusion remarks are made concerning the disparity between the above model, which was chosen for purposes of illustration because of its mathematical simplicity, and models which would be more suitable to serve as representations of the actual universe. And some indication is given of the further developments that should be undertaken.