Carnot Cycles for General Relativistic Systems

Abstract

A generalization of ordinary Carnot cycles is given for thermodynamic systems with stationary gravitational fields. The two heat reservoirs are assumed to be located at different points in space. In addition to the standard change of thermodynamic quantities the Carnot engine is allowed to change its position during the cycle. A ‘generalized Carnot cycle’ is then defined by the following process: (1) Connection of the Carnot engine with the first heat reservoir (ex-changing heat), (2) Change of position of the Carnot engine from the first to the second heat reservoir, (3) Connection of the Carnot engine with the second heat reservoir (exchanging heat), (4) Change of position of the Carnot engine from the second to the first heat reservoir, after which the cycle repeats. In all changes of position the presence of the gravitational field has to be considered. The special case of an ordinary Carnot cycle is obtained when there is no gravitational field or when the heat reservoirs are located at the same point. Under the assumption that gravitation can be described by general relativity the efficiency of these generalized Carnot cycles is calculated for stationary fields. Thermodynamic equilibrium exists when the efficiency of a generalized Carnot cycle operating between any two parts of the system is zero. For this case we find that T • ||ξ|| is a constant independent of position. As used here T is the ordinary thermodynamic temperature and ||ξ|| denotes the norm field of the Killing vector field ξ, representing the stationarity of the gravitational field. The proof is independent of the field equations of general relativity. Consequently equilibrium consists of a temperature field which depends on the gravitational field. For static fields with spherical symmetry Tolman has proved this relation by using the field equation of general relativity. Our results show that this relation holds quite generally for arbitrary stationary fields. © 1970, Walter de Gruyter. All rights reserved.