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VOL,. 14, 1928 PHYSICS: R. C. TOLMAN 701 FURTHER REMARKS ON THE SECOND LAW OF THERMODYNAMICS IN GENERAL RELATIVITY By RICHARD C. TOLMAN NORMAN BRIDGE LABORATORY OF PHYSICS, CALIFORNIA INSTITUTS OF TECHNOLOGY Communicated July 16, 1928 1. Introduction.-In a previous article' I have attempted an extension of thermodynamics to general relativity, and in following articles2 have discussed certain applicatibns of this extension. The postulate that was taken in this work as the analogue of the ordinary first law of thermodynamics was the same as the modified principle of the conservation of energy given by Einstein3 as applying in the case of general relativity considerations. And in spite of the early criticisms of Einstein's principle, based on the fact that it could not be expressed in the form of a tensor equation, I feel that Einstein's treatment is now safely regarded as satisfactory, since the principle is expressible by an equation which is true for all sets of coordinates. The postulate, which was taken in the work as the analogue of the second law of thermodynamics, was however a new one and the main purpose of the present article is to present some further reasons which led me to the principle chosen. In the previous article the new postulate was stated in the form which it takes when applied to an isolated finite system, and was justified by showing that it reduced to the ordinary second law in the limiting case of flat space-time, and by showing that it agreed with the principle of covariance because of its expression in tensor form. In the present article, however, the new postulate will be presented in the form which it takes when applied to a non-isolated infinitesimal system, and will be justified by showing that it can be regarded as a very natural result of generalizing the older thermodynamics. To do this we shall first obtain from the older thermodynamics an expression which embodies the results of the second law as applied to an infinitesimal four-dimensional region in flat space-time. Proceeding on the basis of the equivalence hypothesis we shall then regard this expression as true for an infinitesimal region even in curved space-time. And, finally, we shall generalize so as to put this expression in covariant form, and thus obtain the desired modification of the second law. It will also be shown as a further justification that the new form of the second law leads to an expression for the entropy of a system in a stationary state, which agrees with what is to be expected on the basis of the usual relation between entropy and probability. And the article will in addition present an opportunity to make a number of incidental remarks which may be of a clarifying nature. PHYSICS: R. C. TOLMAN 702 PROC. N. A. S. 2. Application of Ordinary Thermodynamics to an Infinitesimal Region. Let us first examine the application of the classical thermodynamics to an infinitesimal region in flat space-time, employing Galilean co6rdinates, x, y, z and t, corresponding to the line element ds2 = -dX2 - dy2 - dz2 + dt2 (1) If we consider the material contained in the infinitesimal spatial volume ex6y8z, it would seem to be the essence of the second law that the increase in the entropy of this region which occurs in a time interval At should be equal to or greater than the entropy which is brought into the region during that interval by the actual convection of material or by the flow of heat. Hence, we may write as an expression of the requirements of the second law, the inequality, /t - (Qw) bx8y6z8t + aQ (2) x5y&z6t ) - - (4u) + - (4v) + bz T bxy where 4 is the density of entropy, u, v and w are the velocities of macroscopic flow at the point in question, and 6Q/T is the quotient of the he