Matter shear and vorticity in conformally flat spacetimes

In this paper we consider conformally flat perturbations on the Friedmann-Lemaitre-Robertson-Walker spacetime containing a general matter field. Working with the linearized field equations, we unearth some important geometrical properties of matter shear and vorticity and how they interact with the thermodynamical quantities in the absence of any free gravity powered by the Weyl curvature. As there are hardly any physically realistic rotating exact conformally flat solutions in general relativity, these covariant and gauge invariant results bring out transparently the role of vorticity in the linearized regime. Most interestingly, we demonstrate that the matter shear obeys a transverse traceless tensor wave equation, and the vorticity obeys a vector wave equation in this regime. These shear and vorticity waves replace the gravitational waves in the sense that they causally carry the information about local change in the curvature of these spacetimes. We also look at the heat transport equation in this case and how this varies from the Newtonian case.