On Shear-Free perturbations of FLRW Universes
  A surprising exact result for the Einstein Field Equations is that if
pressure-free matter is moving in a shear-free way, then it must be either
expansion-free or rotation-free. It has been suggested this result is also true
for any barotropic perfect fluid, but a proof has remained elusive. We consider
the case of barotropic perfect fluid solutions linearized about a
Robertson-Walker geometry, and prove that the result remains true except for
the case of a specific highly non-linear equation of state. We argue that this
equation of state is non-physical, and hence the result is true in the
linearized case for all physically realistic barotropic perfect fluids. This
result, which is not true in Newtonian cosmology, demonstrates that the
linearized solutions, believed to result in standard local Newtonian theory, do
not always give the usual behaviour of Newtonian solutions.
