Appendix L Stress tensor in comoving coordinates

With the help of transformation (6.8) and (6.9) we can transform the stress tensor for a newtonian fluid in cartesian coordinates

\displaystyle\sigma^{\prime}_{ij}=2\eta{\left[\frac{1}{2}{\left(\frac{\partial% }{\partial r_{j}}v_{i}+\frac{\partial}{\partial r_{i}}v_{j}\right)}-\frac{1}{n% }\delta_{ij}\frac{\partial}{\partial r_{k}}v_{k}\right]}+\zeta\delta_{ij}\frac% {\partial}{\partial r_{k}}v_{k}.

(L.1)

into the stress tensor for a newtonian fluid in comoving coordinates

$\displaystyle\sigma^{\prime}_{ij}=$	$\displaystyle 2\eta{\left[\frac{1}{2}{\left(\frac{1}{a}\frac{\partial}{% \partial x_{j}}u_{i}+\frac{\dot{a}}{a}\delta_{ij}+\frac{1}{a}\frac{\partial}{% \partial x_{i}}u_{j}+\frac{\dot{a}}{a}\delta_{ji}\right)}-\frac{1}{n}\delta_{% ij}{\left(\frac{1}{a}\frac{\partial}{\partial x_{k}}u_{k}+n\frac{\dot{a}}{a}% \right)}\right]}$	(L.2)
	$\displaystyle+\zeta\delta_{ij}{\left(\frac{1}{a}\frac{\partial}{\partial x_{k}% }u_{k}+n\frac{\dot{a}}{a}\right)}$	(L.3)
$\displaystyle=$	$\displaystyle 2\eta{\left[\frac{1}{2a}{\left(\frac{\partial}{\partial x_{j}}u_% {i}+\frac{\partial}{\partial x_{i}}u_{j}\right)}+\frac{\dot{a}}{a}\delta_{ij}-% \frac{n}{n}\frac{\dot{a}}{a}\delta_{ij}-\frac{1}{na}\delta_{ij}\frac{\partial}% {\partial x_{k}}u_{k}\right]}$	(L.4)
	$\displaystyle+\zeta\delta_{ij}{\left(\frac{1}{a}\frac{\partial}{\partial x_{k}% }u_{k}+n\frac{\dot{a}}{a}\right)}$	(L.5)
$\displaystyle=$	$\displaystyle\frac{1}{a}{\left\{2\eta{\left[\frac{1}{2}{\left(\frac{\partial}{% \partial x_{j}}u_{i}+\frac{\partial}{\partial x_{i}}u_{j}\right)}-\frac{1}{n}% \delta_{ij}\frac{\partial}{\partial x_{k}}u_{k}\right]}+\zeta\delta_{ij}{\left% (\frac{\partial}{\partial x_{k}}u_{k}+n\dot{a}\right)}\right\}}.$	(L.6)