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In FLRW metric, I am reading that all galaxies are given a comoving coordinate $\lbrace x^i, i=0,1, 2,3\rbrace$( Schutz) . Since GR can be done in any coordinate system, using comoving coordinates is fine.

What confuses me is that, when Local Inertial Frames exist only Locally, why are comoving coordinates given to everypoint in the manifold.

Comoving coordinates should have been given just locally around then origin of the comoving coordinate system centred at any point in the manifold. And the book talks about Locally Comoving Frames.

What is the link between these Globally given comoving coordinates and Locally Comoving Frames.

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I do not have the book, but probably the global comoving frame simply demands, that the spatial grid is comoving at each point, while local comoving frame is probably frame where only the origin is comoving. For local comoving frames it makes sense to demand other properties, like locally reducing the metric to its standard Minkowski form, i.e. that locally comoving frame will be local inertial frame as it is usually understood.

For global comoving frame, we cannot demand the "minkowskization" property, but we can demand other useful properties, like time coordinate corresponding to proper time of any comoving observer.

If you imagine that at some moment (with properly chosen definition of "moment" that respects the symmetry) you lay down some spatial coordinates in the universe and that the universe is filled by some dust particles at each point (Dust particles are of course used only for imagination, they should not be thought of as having physical reality). As time evolves, the dust moves through spacetime and comoving frame simply demands that every dust particle will have the same spatial coordinate for all times and that the time coordinate is simply given by proper time of the dust particle. Thus to get coordinates of event "Sun was born" is to find which dust particles crosses the event and read of its spatial coordinate and its proper time. In this frame, the comoving distance will not be the same as proper distance. In fact, in an expanding universe, the proper distance between particles will increase even thought spatial coordinates (and thus also comoving distance) will keep the same.

On the other hand, in local inertial frame, the coordinate distances are same as proper distances, but then the dust particles that are not at the origin will not keep the same coordinates along the evolution of universe.

Answering the comment

Thanks for the answer. It makes a bit sense now but I am still a bit confused as to how coordinates ( both space and time) are give in FLRW metric. How is the time coordinate in particular given. What is meant by slicing the space time at a particular time. We can only construct these space like hypersurfaces in SR. So in GR we should be able to do for every Local Lorentz Frame and the time coordinate should be local. Moreover each observer will dran his own space like hypersurface ( because of simultaneity issues). But the time coordinate in FLRW metric is given globally. Same for space - like coordinates. Could you please add in the answer as to how precisely the coordinates both space and time are given in FLRW metric. I think I have been misunderstanding how coordinates are given in the first place. So if you could just add a brief overview from the beginning as to how the coordinates are given in the first place ( and things like why coordinate time is the proper time here), it will be really very useful. Thanks in advance

Well yes, in general you cannot create spacelike hypersurfaces in GR that would have the same nice interpretations as in SR. But you can construct some "random" spacelike hypersurface in GR, that has no nice physical significance. It is just some submanifold of spacetime where each event is spacelike separated with every other. Beyond that, the hypersurface is fully general.

So in FLRW, create set of all spatial hypersurfaces that exist in FLRW. Now, FLRW is not just some arbitrary spacetime, it has very strong symmetries - the space is homogeneous and isotropic. This is not property of every spacelike hypersurface, but it is property of some. Pick one and stick with it.

Ok, so now you have spacelike hypersurface, lets call it H that is homogeneous and isotropic. Great. Lay down some coordinates there. Any coordinates will do, just pick 3 numbers for any point of the hypersurface (granting some mathematical "consistency" properties, in particular the coordinate function $H\rightarrow \mathbb{R}^3$ needs to be a homeomorphism). Also pick a time value for the whole hypersurface, lets say it is $t_0$. So every event on our hypersurface was given 4 numbers $(t_0,x,y,z)$. Again, they are just some numbers, names attached to every event on our hypersurface. Do not seek any physical meaning in those numbers.

Then let us drop our dust particles everywhere in the hypersurface so that they have no spatial velocity (they respect the symmetry). Each dust particle is uniquely identified by its 3 space coordinates $(x,y,z)$. Now let them evolve along the geodesics and stop the evolution once the dust particle reaches time $t$ according to its own clocks. Every dust particle reaches somewhere in spacetime and all the events of type "dust particle from $(t_0,x,y,z)$ reached time $t$ of its own clocks" form some kind of subset of spacetime. We can label these new events with 4 numbers $(t,x,y,z)$. We can do this for any valid value of $t$ and we say, that these numbers define a coordinates called global comoving coordinates. Saying it won't make it so, but we can check that because of niceness of FLRW spacetime, these numbers will indeed be valid choice for a coordinate system (i.e. the function will be a homeomorphism) and it will in fact cover whole spacetime uniquely.

So we have a coordinate system that gives a name to every event in spacetime. Because FLRW is so nice that it allowed us to do that, we know by construction that the time $t$ is proper time that dust particle from hypersurface $t_0$ reached hypersurface $t$. We were in luck, this times, we found nice interpretation for the coordinate $t$. Of course, this is only proper time for our dust particles. Other observer, who would not follow trajectory of the dust particles would measure different time. In particular, observer who has some speed wrt dust particles and is thus not comoving with the spacetime (not respecting the symmetry), would measure different time upon reaching hypersurface $t$.

Also, our construction did not make use of Einstein synchronization of clocks. We did not demand, that dust particles consider the events on the hypersurface $t$ as simultaneous in the sense of SR. In fact, the universe is expanding/contracting, the space can be curved and therefore the whole method of sending a signal and waiting for its return to compare time at different places needs to be modified, if it can be applied at all. Our spatial hypersurfaces and our time coordinate $t$ simply have different meanings than traditional spatial hypersurfaces and time coordinates of SR.

I hope this helps.

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  • $\begingroup$ Thanks for the answer. It makes a bit sense now but I am still a bit confused as to how coordinates ( both space and time) are give in FLRW metric. How is the time coordinate in particular given. What is meant by slicing the space time at a particular time. We can only construct these space like hypersurfaces in SR. So in GR we should be able to do for every Local Lorentz Frame and the time coordinate should be local. Moreover each observer will dran his own space like hypersurface ( because of simultaneity issues). But the time coordinate in FLRW metric is given globally. Same for space $\endgroup$ – Shashaank Dec 28 '20 at 11:11
  • $\begingroup$ cont- like coordinates. Could you please add in the answer as to how precisely the coordinates both space and time are given in FLRW metric. I think I have been misunderstanding how coordinates are given in the first place. So if you could just add a brief overview from the beginning as to how the coordinates are given in the first place ( and things like why coordinate time is the proper time here), it will be really very useful. Thanks in advance $\endgroup$ – Shashaank Dec 28 '20 at 11:16
  • $\begingroup$ @Shashaank I edited the answer $\endgroup$ – Umaxo Dec 28 '20 at 13:14
  • $\begingroup$ That's a wonderful explanation (thus the upvote). Many Thanks for it. It cleared almost everything. Just 2 confirmations and 2 small issue to finish off. Confirmations-1) In the end t turned out be physical in that it is proper time but the spacelike coordinate still need not be anything physical.2) We can construct many such FLRW spacelike hypersurfaces for different values of $t_0$ where t_0 need not be necessarily physical... Are these 2 points right...Please Kindly confirm. $\endgroup$ – Shashaank Dec 28 '20 at 15:26
  • $\begingroup$ And the small issue - 1) Why will t_0 be the same everywhere on the spacelike hypersurface ( where these spacelike hypersurfaces are different from the SR ones). Shouldn't t_0 be a local variable then. Why will it be the same all over the hypersurface (is it because it need not be physical time, just a time like coordinate which will be constant for the space like hypersurface - but if it so we do in the ens find that t is proper time). 2) How will the Local Comoving frames fit into the edited picture you have described. Kindly let me know these 2 point as well, please. Thanks in advance. $\endgroup$ – Shashaank Dec 28 '20 at 15:31

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