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I have a question on co-moving and synchronized reference systems. I read the corresponding section (97) in Landau/Lifshitz's second book "Field Theory" on it. In particular it is said that space filling matter cannot be in rest in such a system except in case of dust-like matter with pressure $p=0$. This is the case I am interested in. Because further on in section 103 of Landau/Lifshitz a solution of Einstein's field equations (EFE) is discussed with just this assumption. Would an observer in such a system observe all surrounding collapsing particles in rest with respect to him? What is with the shells/layers of dust above and below the observer? Don't they experience a different acceleration towards the point where everything is collapsing to?

I also seem to have an example of matter moving a comoving synchronized system. A Robertson-Walker metric seems to describe a comoving synchronized system of the cosmic matter in the universe. If cosmic matter would not move WRT such a system, how would it be possible to notice the expansion of the universe ? So if matter apparently moves in such a system, what distinguishes it from a simple rest frame of single chosen particle (participating in an expansion or a collapse) while the other particles of the matter distribution move with respect to it ?

EDIT:

Another feature I strongly wonder about is the fact that points observed in such a system keep staying on geodesic lines have the same time-coordinate (the eigen-time) according the definition of the synchronised respectively co-moving reference system:

$$ds^2 =dt^2 -\gamma_{ab}(t) dx^a dx^b \quad \text{with} \quad a,b=1,2,3$$

Nevertheless -- using again the example of the expanding universe -- 2 galaxies of large distance among them would have a different time due to their relative motion due to the cosmic expansion. Another example are particles participating in a gravitational collapse which is also described by a synchronised/co-moving reference system:

$$ds^2 = dt^2 - e^\lambda(t,R)dR^2 -r^2(t,R)d\Omega^2 $$

To recap:

1) What means "co-moving" ? Is it meant to be only locally or even globally ?

2) Why is it possible to use the same time coordinate in such a system, although different points in such a system exert relative motion to each other which leads to different times at these points ?

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You need to be clearer about the distinction between coordinates and measured quantities. Coordinates are just a map, and like a map of the world, they contain scaling distortions described by the metric.

Here is an illustration of a comoving coordinate system for an FLRW universe with positive curvature. FLRW universe with positive curvature

The galaxies on geodesics maintain constant position in these coordinates, but the scaling means that they appear to shrink over time so that distances between them increase (this is exactly equivalent to saying that the universe expands relative to local distances). Time is a function of cosmic time - time as measured by each galaxy from the initial singularity. In this diagram the time axis has been scaled so that radial light speed is a constant.

According to my notes $r = \sin(\chi), \chi, \sinh(\chi)$ for positive, zero and negative curvature where $r$ is the RW radial coordinate (I never actually use the RW form of the metric). The metric shown has $$ds^2=a^2(\tau)[d\tau^2 - d\chi^2 - f^2(\chi) d\Omega^2 ] $$ with $$d\Omega^2 = d\theta^2 + \sin^2\theta d\phi^2.$$ The expansion is contained in the factor $a$. The substitution $$ad\tau = dt$$ gives $$ds^2 = dt^2 - a^2(t)[ d\chi^2 + f^2(\chi) d\Omega^2 ]. $$

The meaning of $\chi$ can be seen for positive curvature by wrapping the map at constant time around a sphere, which also shows that the centre of the map is arbitrary and makes clear the geodesics determined by symmetry. $a$ used to be known as "the radius of the universe"

mapping the universe onto a sphere

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  • $\begingroup$ The graph is amazing. Let me ask some details. How is the coordinate $\chi$ related to the coordinates in the FLRW $ds^2 = dt^2 -\mathcal{R}(t)(\frac{dr^2}{1-r^2} + r^2\Omega^2)$ ? So conclusion is: observed in the right (co-moving?) coordinates there is no motion of galaxies WRT each other ? How can I see formally that the coordinates shrink over time? Thank you. $\endgroup$ – Frederic Thomas Mar 31 at 12:12
  • $\begingroup$ I added a little. I hope this helps. $\endgroup$ – Charles Francis Mar 31 at 15:22
  • $\begingroup$ Thank you very much. I will go through all this and may be ask another question. Independent from this I will award the bounty to you. Great! $\endgroup$ – Frederic Thomas Mar 31 at 17:38
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If cosmic matter would not move WRT such a system, how would it be possible to notice the expansion of the universe ?

Observationally we see the light is redshifted so this is caused by the expansion of the universe. Other theories such as static universe or tired light theory has been ruled out. And you can show that in a static matter-filled universe (which you described) the static universe is at unstable equilibirium.

So if matter apparently moves in such a system, what distinguishes it from a simple rest frame of single chosen particle (participating in an expansion or a collapse) while the other particles of the matter distribution move with respect to it ?

The main point is that the expansion of the universe should be homogeneous and isotropic. This expansion is naturally described by the comoving coordinates. I am not sure about the usage of the other coordinates systems but the main point is to keep this homogeneous and isotropic expansion.

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  • $\begingroup$ My question is not about a world model. It's about the synchronised reference system where all points stay on geodesic worldlines and have the same time coordinate as the metric is $ds^2=dt^2 -\gamma_{ab}(t)dx^a dx^b$. $(a,b)=1,2,3$.So for instance in the milky way (MW) and a far-distant galaxy (moving WRT to the MW due to cosmic expansion) the same time coordinate is used although they would not have the same time due to the relative motion. Remains also the question what "co-moving " really means, only locally co-moving or globally co-moving (the latter seems impossible)? $\endgroup$ – Frederic Thomas Mar 28 at 12:50

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