Relativistic Thermodynamics, a Lagrangian Field Theory for general flows including rotation

Any theory that is based on an action principle has a much greater predictive power than one that does not have such a formulation. The formulation of a dynamical theory of General Relativity, including matter, is here viewed as a problem of coupling Einstein’s theory of pure gravity to an independently chosen and well defined field theory of matter. It is well known that this is accomplished in a most natural way when both theories are formulated as relativistic, Lagrangian field theories, as is the case with Einstein-Maxwell theory. Special matter models of this type have ben available; here a more general thermodynamical model that allows for vortex flows is used. In a wider context, the problem of subjecting hydrodynamics and thermodynamics to an action principle is one that has been pursued for at least 150 years. A solution to this problem has been known for some time, but only under the strong restriction to potential flows. A variational principle for general flows has become available. It represents a development of the Navier-Stokes-Fourier approach to fluid dynamics. The principal innovation is the recognition that two kinds of flow velocity fields are needed, one the gradient of a scalar field and the other the time derivative of a vector field that is closely associated with vorticity. In the relativistic theory that is presented here the latter is the Hodge dual of an exact 3-form, well known as the notoph field of Ogievetskij and Palubarinov, the B-field of Kalb and Ramond and the vorticity field of Lund and Regge. The total number of degrees of freedom of a unary system, including the density and the two velocity fields is 4, as expected. The present paper deals with the relativistic context, Special Relativity and General Relativity. The energy momentum tensor has a structure that is more general than that of Tolman, and different from proposed generalizations; it appears to be well suited to represent rotational flows in General Relativity. The current of mass flow is conserved: the theory incorporates the hydrodynamical equation of continuity.