Symmetry evolution for the imperfect fluid under perturbations

Abstract

Ever since a new symmetry was found for the imperfect fluid with vorticity, the question of the effect of perturbations on the symmetry itself has been raised. This new symmetry arose when realizing that local four-velocity gauge-like transformations would render the left-hand side of the Einstein equations invariant because the metric tensor would be invariant under this new kind of local transformations. Then the point was raised about the invariance of such a kind of transformations of the stress–energy tensor on the right-hand side of the Einstein equations in curved four-dimensional Lorentz spacetimes. It was verified that these invariances do not work with plain perfect fluid but they do work for imperfect fluids. The imperfect fluid stress–energy tensor will be invariant under local four-velocity gauge-like transformations when additional transformations are introduced for several variables included in the stress–energy tensor itself. This local invariance was also the criteria introduced in order to present a new stress–energy tensor for vorticity as well. New tetrads are at the core of the realization of the existence of this new symmetry because it is through these new tetrads that this new symmetry is realized. It is through the local transformation of these new tetrad vectors that we can prove that the metric tensor is invariant. This new kind of symmetry has its origins in a similar tetrad formulation as to the Einstein–Maxwell spacetimes formalism presented in previous manuscripts. In this paper, we will introduce local perturbations by external agents to the relevant objects in the imperfect fluid geometry. We will demonstrate a theorem that proves that the symmetries under four-velocity gauge-like transformations will be instantaneously broken but at the same time transformed into new symmetries. Because the local orthogonal planes determined by these new tetrads, which happen to be the local planes of symmetry will tilt under local perturbations. There will be a symmetry evolution under perturbations.