Conserved quantities in expanding Gödel cosmology

At linear order we study perturbations to a Gödel background spacetime which includes expansion in addition to rotation. We investigate the transformation behavior of these perturbations under gauge transformations and construct gauge-invariant quantities. Using the perturbed energy conservation equation we find that there are conserved quantities in expanding Gödel cosmology, in particular a spatial metric trace perturbation, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msub> <a:mi>ζ</a:mi> <a:mrow> <a:mi>SMTP</a:mi> </a:mrow> </a:msub> </a:math> , which is conserved on large scales for pressureless dust. We finally extend our discussion to a perfect fluid matter content to also obtain conserved quantities in this context.