The flat FRW metric using standard cosmological time $t$ and Cartesian spatial co-ordinates $(x,y,z)$ (with $c=1$) is given by: $$ds^2=dt^2-a^2(t)(dx^2+dy^2+dz^2).$$ I understood that the FRW metric describes a physically expanding Universe whatever co-ordinate system one uses. But do some co-ordinate systems expand with the Universe whereas others do not?
A co-moving observer using the $(t,x,y,z)$ co-ordinate system has a lightclock of fixed proper length ($d\tau=dt$) so he can detect that the Universe is expanding. Thus his 4-d co-ordinate system $(t,x,y,z)$ can be said to be not expanding as a whole.
However the same flat FRW metric can be expressed using conformal time $\eta$ and Cartesian spatial co-ordinates: $$ds^2=a^2(\eta)(d\eta^2-dx^2-dy^2-dz^2).$$ A co-moving observer using the $(\eta,x,y,z)$ co-ordinate system has a lightclock of expanding proper length ($d\tau=a(\eta)\ d\eta$) with a fixed co-moving/conformal length $d\eta$. Thus he cannot detect that the Universe is expanding. Therefore his 4-d co-ordinate system $(\eta,x,y,z)$ is expanding along with the Universe.
Is this right?