Undergrad Cosmology Comoving coordinates and observers

[AHSAN MUJTABA]

TL;DR

I am unable to understand the concept of comoving coordinates though I know what the proper an coordinate quantities mean in cosmology but still I am missing the visualization of these coordinates with the expansion of universe. Can I get an example or explanation which would be easier for me to at least start observing some sense? does it mean, if we are in a rest frame patch of coordinate system, then this patch is moving with the expansion(that's what I am picturizing)

I just want to visualize the math, any help would be appreciated.
TIY

 

[Halc]

It is pretty simple actually. The distance between any two points is 'currently' the proper distance between the two points. I put 'currently' in quotes because both points (events) are simultaneous, such that the apparent age of the universe to a comoving observer at the other point is the same age as we see here. Most observers are comoving, meaning that they have negligible velocity (called peculiar velocity) relative to the coordinate system.

The proper distance between objects at times other than now is the current proper distance multiplied by the scale factor, which ranges from zero at the big bang, to 1 now, to arbitrarily high values in the future. It is not a linear function, so scale factor is not 0.1 when the universe was 1.38 billion years old.

If we are at rest (have zero peculiar velocity), then we're not moving at all with the expansion. Motion is relative to something, and 'the expansion' cannot be that thing. Yes, other comoving objects increase their proper separation from us as a function of that expansion, but that isn't really motion in the usual sense of relativistic velocity or anything. That's why there's no problem with the rate being greater than speed of light. It isn't a velocity the way velocity is usually used.

 

[PeterDonis]

Most observers are comoving, meaning that they have negligible velocity (called peculiar velocity) relative to the coordinate system.

This probably needs to be clarified a bit. We, here on Earth, can detect that we are not comoving: we see a strong dipole anisotropy in the CMB, which indicates that we are moving, relative to a comoving observer at the same spatial location (and therefore relative to the comoving coordinate system), at about 600 kilometers per second (IIRC). While this velocity is small compared to the speed of light, and does not really affect any everyday observations, it can't really be considered "negligible" since we can observe it. (In fact our observations of the CMB now are several orders of magnitude more precise than would be required to detect this motion.)

Most observers, whatever observers may be in the rest of the universe, will probably have a similar state of motion to us on Earth, meaning that, assuming they develop technology at least as good as ours, they will be able to detect that they are not exactly comoving. Over large enough distance scales, these motions average out, so that the average motion of all the matter in the universe is comoving. But individual pieces of matter on small scales will in general not be.

 

[jambaugh]

Begin with understanding that in unified space-time the metric defines the proper distance or proper time between two events and not two physical objects. Two physical objects are represented by world lines, paths of event points in the space-time manifold.

That being clarified, how do you describe distances between two objects which may be moving relative to each other as well as being effectively accelerated by the curvature of that space-time manifold... answer is you can't uniquely for general cases. But in considering a single object you can choose from among many coordinate systems where the object's world-line is a coordinate curve. Such would be a co-moving coordinate system for that object. In terms of cosmology one can do the same for the COM of an matter. And within that class of coordinates you can minimize other quantities such as relative motion in some sense.

I don't agree that co-moving coordinates are appropriate for certain cosmological models since they aren't necessarily geodesic coordinates. E.g. deSitter models show the universe first contracting and then expanding in one set of coordinates that are (potentially) co-moving. But, in more naturally geodesic coordinates oriented toward one free-falling observer the spatial universe has a fixed circumferential radius. In such coordinates time evolution is a symmetry transformation.

 

[Halc]

This probably needs to be clarified a bit. We, here on Earth, can detect that we are not comoving: we see a strong dipole anisotropy in the CMB

Agree. I'm not claiming that we're comoving to the point of undetectability.

... which indicates that we are moving, relative to a comoving observer at the same spatial location (and therefore relative to the comoving coordinate system), at about 600 kilometers per second (IIRC).

More like 400 km/sec. 600 is the peculiar velocity of our galaxy and thus our average speed over a very long time, but we're currently on the side where our orbital motion vector subtracts almost directly off that. In 100M years we'll double that to 800ish.

While this velocity is small compared to the speed of light, and does not really affect any everyday observations, it can't really be considered "negligible" since we can observe it.

Absolutely, it is detectable. It being negligible in my opinion is due to the fact that it doesn't alter the apparent age of the universe, which we know to fewer significant digits than the time dilation you get from moving at 0.0014c. It seems an anomaly to find a sizeable object with a peculiar velocity of over 3% of light speed, and even that isn't enough to alter the apparent age much.
I saw something about a cannon that fired Jupiter size masses at nearly light speed. Impressive power! If an observer were to ride that, the universe would look much younger, and of course the night sky would be massively red/blue shifted from one side to the other.