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VOL. 17, 1931 PHYSICS: R. C. TOLMAN 153 2 A. W. Rucker, "On the Suppressed Dimensions of Physical Quantities," Phil. Mag., 1889, pp. 104-114. 3 Carl Hering, Conversion Tables, New York, 1904. 4E. Bennett, "A Digest of the Relations between the Electrical Units and the Laws Underlying the Units," Univ. of Wisconsin Bull., 1917. 6 Smithsonian Physical Tables, Sixth Edition, Washington, 1916. B Bureau of Standards, "Electric Units and Standards," Circular 60, 1920. 7 A. E. Kennelly, "Magnetic Circuit Units," Trans. Am. Inst. El. Engrs., Jan., 1930, 49, No. 2, Apr., 1930, pp. 486-510. Discussion by Gokhale, p. 503, Tables II and III. ON THERMOD YNAMIC EQUILIBRI UM IN A STA TIC EINSTEIN UNIVERSE By RICHARD C. TOLMAN NORMAN BRIDGE LABORATORY, CALIFORNIA INSTITUTE OF TECHNOLOGY Communicated February 10, 1931 § 1. Introduction.-It has now become evident that the transformation of matter into radiation taking place throughout the universe and the red-shift observed in the light from the extra-galactic nebulae appear to imply a non-static quality in the universe which can be treated with some success with the help of a non-static cosmological line element.' If the quantity which gives the dependence of this non-static line element on the time is set equal to a constant, it is found that the line element then becomes the same as Einstein's original line element for a static universe. Hence the Einstein static universe may be regarded as a special case of the more general non-static universe, and we must continue to be interested in the properties of the Einstein universe not only because it is a limiting case of the more general model for the universe, but also because it represents a situation which might arise in the course of the evolution of the actual universe. The present article will deal with the thermodynamics of the Einstein universe, and in particular will treat the conditions for thermodynamic equilibrium between matter and radiation in such a universe assuming the possibility of their transformation into each other. Treatments of the general problem of the equilibrium between matter and radiation have already been given for the case of a perfect monatomic gas interacting with black body radiation both in the absence and presence of gravitational fields. In the absence of any appreciable gravitational field, it was shown by the work of Stern2 and myself3 that the number of monatomic molecules of mass m present in unit volume at equilibrium at temperature T would be given by a formula of the form mcS N = bT31/2 e kT (1) PHYSICS: R. C. TOLMAN 154 PROC. N. A. S. where b is a constant whose value cannot be determined solely from the first and second laws of thermodynamics, c is the velocity of light and k is Boltzmann's constant. On account of the large effect of the exponent -mc2/kT the equilibrium concentration of matter given by this formula would be exceedingly low, even for masses as small as that of the electron and for temperatures as high as the 40,000,0000 assumed in the interior of the stars, unless indeed the constant b could be shown to have an enormous value. Also in the presence of the gravitational field of a spherical distribution of fluid, I have recently been able to show4 that the equilibrium concentration of monatomic gas would again be given by a formula of the same form (1) as for flat space-time. But in the presence of the gravitational field of the Einstein static universe I originally found5 a slightly different formula, which however still had a similar large exponential dependence on -mc2/kT. Since that time, however, the development of the non-static line element for the universe has made it easier to understand the process by which the Einstein universe could be regarded as changed from one stati