Role of Weyl tensor and spacetime shear in relativistic fluids.

Abstract

The main gravitational theory in which we develop this work is general relativity. We study the role of the Weyl tensor in general relativistic fluid motion including the e↵ects of spacetime shear. Firstly we consider conformally flat perturbations on the Friedmann Lemaitre RobertsonWalker (FLRW) spacetime containing a general matter field. Working with the linearised field equations, we find some important geometrical properties of matter shear and vorticity, and show how they interact with the thermodynamic quantities in the absence of any free gravity powered by the Weyl curvature. We demonstrate that the matter shear obeys a transverse traceless tensor wave equation and the vorticity obeys a vector wave equation in this linearised regime. These shear and vorticity waves replace the gravitational waves in the sense that they causally carry information about local change in the curvature of these spacetimes. We also study the heat transport equation in this case, and show how this varies from the Newtonian case. Secondly we show that a general but shear-free perturbation of homogeneous and isotropic universes are necessarily silent, without any gravitational waves. We prove this in two steps. First, we establish that a shear-free perturbation of these universes are acceleration-free and the fluid flow geodesics of the background universe map onto themselves in the perturbed universe. This e↵ect then decouples the evolution equations of the electric and magnetic part of the Weyl tensor in the perturbed spacetimes and the magnetic part no longer contains any tensor modes. Although the electric part, that drives the tidal forces, does have tensor modes sourced by the anisotropic stress, these modes have homogeneous oscillations at every point on a time slice without any wave propagation. This analysis shows the critical role of the shear tensor in generating cosmological gravitational waves.