Relativistic Field Theory of Fluids

Abstract

Classical relativistic field theory is applied to perfect and magneto-hydrodynamic flows. The fields for Hamilton's principle are shown to be the Lagrangian coordinates of the fluid elements, which are potentials for the matter current 4-vector and the electromagnetic field 2-form. The energy momentum tensor and equations of motion are derived from the fields. In this way the theory of continua is shown to have the same form as other field theories, such as electromagnetism and general relativity. Waves are treated as an example of the power of field theoretic methods. The average or background flow and the waves are considered as two interacting components of the system. The wave-background interaction involves the transfer of energy and momentum between the waves and the average flow, but the total energy and momentum are conserved. The average Lagrangian for the total wave-background system is found by expanding the Lagrangian about the background flow and averaging over the phase. The total energy-momentum tensor is constructed, and the conservation of energy and momentum are discussed. Varying the wave amplitude gives the dispersion and polarization relations for the waves, and varying the phase gives the rays and conservation of wave quanta (or wave action). The wave quanta move with the group velocity along the bi-characteristic rays.