Invariants of flows in 4-d space time

Abstract

Differential forms and Lie derivatives are widely used in the study of invariance of physical quantities. The vanishing of the Lie derivative of a quantity signifies its invariance as one moves along the curves that are integrals of the flow. In this paper, we consider incompressible flows and study their integral invariants. Techniques employed include the invariance of forms (both relative and absolute) in the Euclidean space-time framework. The study yields five categories of local invariants corresponding to five possible exterior forms in 4-D space-time. In the case of viscous flow, we develop methods to calculate the rate of change of circulation, helicity, parity, and vorticity flux. We arrive at a conclusion that the absolute integral invariance of the parity four form yields Ertel's potential vorticity theorem. An attempt is made to study the above invariants using the technique of geometric algebra also.