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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?

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The co-moving coordinate system factors out the scale factor and, thus, ignores the expansion of space. As you rightfully pointed out, any observer using only this coordinate system would not notice that the universe is expanding because their definition of distance is, indeed, expanding with the universe. I suppose that makes perfect sense; you don't notice something when you ignore it. However, it should be pointed out that it's unrealistic to have an observer that operates solely in the co-moving coordinate system. Too many measurements rely on effects that take place using proper scales. An observer using only the co-moving system would measure a reduction in the speed of light; the further it traveled, the slower it would go. Yet, at the same time, the light from standard candles would still appear to redshift even though it was "slowing" down. Two observations that are easily explained outside co-moving coordinates but would be contradictory from purely within them. You can point out the conformal time term and say that this resolves the issues I laid out here, but that then causes issues locally. Where expansion does not create a significant or any difference, the arbitrary lengthening of a standard proper temporal unit would be perplexing to anyone that did not know its exact basis in expansion.

Co-moving coordinates are more of a mathematical notion that can be translated to physical reality. They are not coordinates that one would stumble upon before being aware of the universe's expansion like one would with Cartesian coordinates. The co-moving system represents active ignorance of expansion, not passive ignorance. You could correctly say that co-moving coordinates represents a system that expands with the universe and, thus, ignores expansion entirely. But I wouldn't go so far as to say that anyone using purely this coordinate system is unaware of expansion. There would certainly be indirect signs of its presence that would point to expansion. Whether or not that counts as detecting the universe expand is a moot point and too open to opinion for me to get into here.

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Yet is it not true that one of the deepest principles of relativity is that no coordinate system can ever lead you to any contradictions with experiment? I think it is must be true that any language stimulated by any coordinate system has to provide a self-consistent language for talking about what is happening, agreeing with all observations. So although I agree that the conformal coordinates seem odd relative to our normal picture of the situation, but I think it should be usable without difficulty for all observations.

It seems to me the natural physical interpretation one gives to the expanding universe, in the language of conformal coordinates, is that rulers (and with them all bound systems) are shrinking and time is speeding up, maintaining the constancy of the local determinations of the speed of light as required by experiment. Why they are shrinking is not answered any better than why is space expanding. So in conformal coordinates, one does not say that space is expanding, but distances are still increasing and redshifts are naturally explained without anything happening to space. It seems to me the lesson is that language comes from coordinates, and is in some sense arbitrary, within the constraints of the invariant measurables.

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