[Halc]

Crumbs, this is a long post.  Sorry.  Some stuff has been trimmed out

For long posts, I try to make my replies shorter than the post to which I’m replying, but I failed on my last one. Guess that means the conversation is interesting.

Comoving only has meaning relative to an expanding metric, so no, it cannot be done relative to a static metric.

That's a disposable comment.  A rectangle doesn't stop being a rectangle when it's a square.

OK, I’ll grant that. ‘Comoving’ is defined as a measurement and thus is a relationship with the material all around you and isn’t really a coordinate system (CS) dependent thing as my comment implies. My house is always comoving with the ground under it, and thus is always stationary relative to that ground regardless of any alternate choice of CS is made, and I can be doing physics in the alternate CS but still measure if I’m comoving with the nearby ground.

We discussed geodesically incomplete spacetime such as when a black hole stops an observer from seeing the CMBR beyond it

But it doesn’t. You see the entire CMBR (and more!) from an accelerated position near a black hole. There’s no obstruction despite the fact that all the light comes from the direction in which you’re accelerating. Light from the other side bends around and finds you. It can only be blocked by physical obstruction like the Earth under you.

You (Halc) also mentioned gravitational potential wells. This will influence redshift but we're going to keep your pebbles or my emitter and receiver identical in mass and radius,    i.e. we're just going to idealise the situation and ignore gravitational redshift of this kind.

I used pebbles in an effort to minimize mass, because they’re supposedly in a zero-energy solution scenario, so they must be arbitrarily small.
Our two obervation points on opposite sides of a galaxy do need to be at identical potentials. That scenario was the real universe, not the Minkowskian one.
You don’t need to say “You (Halc)”. You’re responding to my quotes. Everybody knows who “You” is in this context. Second-person sentence structure is appropriate, but you switch to 3rd person sometimes, I suppose to address the perceived audience.

My view of red-shift was stated as:
I maintain that red-shift is caused by the properties of the space through which a photon has travelled. In particular, if space is expanding then photons lose energy.
...
Yes [that] is a co-ordinate effect, if you want to look upon it that way.

It is critical to my point, so yes, I want to look at it both ways.

However, the co-ordinates are not completely arbitrary.  They can be identified or singled out by observations in the real universe (upto translations and spatial rotations).  My observers can try to use a different frame but they will know if it is co-moving because they can observe the CMBR and check for isotropy.

No argument. I’m just saying light doesn’t slowly redshift relative to an inertial frame like the one in which both pebbles are stationary. The redshift only occurs relative to the expanding CS, that being a property of a hyperbolic CS like that.

This is important, so I'm going to say it another way:  Co-moving co-ordinates are not arbitrary or completely abstract.

Agree, but the rotating frame in which my house is stationary is also not completely abstract. I don’t mean to trivialize it with that comment. Comoving coordiantes are universe whereas the frame of my house is not. I can think of no other CS that foliates most of the (real) universe like that, but any arbitrary inertial frame does foliate the entire zero-energy universe in which we put the pebbles, even events not foliated by the comoving CS.

We can construct a local frame that has many of the properties of the co-ordinate system that is used in the FLRW metric and large-scale models of FRW universes.

I thought the coordinate system used by the FLRW metric was not local at all. I was unaware that the FLRW metric referenced a local CS at all except in the usual GR way that says spacetime is locally Minkowskian except at a physical singularity.

At any point in open space (away from gravitational sources) we can identify a local inertial frame such that an observer remaining at the origin of our frame will observe the CMBR isotropically. This is the local CMB frame.

Some choose the term “CMB frame” to mean the universal comoving coordinate system, which is why I balk at the term. Of course “local CMB frame” implies a local inertial frame, but the adjective is usually left off, leaving the reader to wonder if we’re talking about the comoving frame or a local inertial one. You cannot speak of a local frame when discussing expansion across a galaxy since it is exactly the divergence from locality that we’re trying to measure.
On the other hand, in the zero energy scenario, all inertial frames are indefinite sized, and there is no ‘local’ about it. Your discussion that follows seems to reference the zero-energy scenario since you’re specifying “away from gravitational sources” which is impossible except in the absence of gravitational sources.

For an observer remaining at the origin of this CMB frame all of the following hold:
…
(ii) Local co-ordinate time, t  and universe-wide co-moving time, T  show no dilation (they pass at the same rate) 

Hence the importance of the absence of gravitational sources since even a perfectly uniform distribution of matter will sink your local clock deep into a gravity well. The only way universe-wide co-moving time is then meaningful is if it is defined as the time on a clock at average gravitational depth, but that average depth keeps changing as the density drops with expansion. Fun stuff.
Anyway, I’ll grant this point, lest our goal become out of reach.

(iv) We can improve the correspondance between the local co-ordinate system and the co-moving system.  The scale factor is quite arbitrary in most models, we're usually only concerned with the ratio of two scale factors at different times.  We can set  a(time=now) to be 1,  so that we have x ≈ X etc. for times close to "now".  Then, locally (in both space and time) our CMB frame is a good representation of the co-moving co-ordinate system with a(now) = 1.  (We can do a similar trick by insisting distances in both the co-moving co-ordinates and local co-ordinates are measured in metres - defined as distance along a null path over a fraction of a second.  This is slightly more complicated to explain and everyone has stopped reading already and more Maths isn't going to bring the audience back).

Can only do that locally. Distances, even without time progressing, are very different in the two coordinate systems, but they do match locally. This is why the radius of the visible universe can be ~48 BLY today whereas in a universe where an inertial frame applies, it could only have a radius of 13.8 BLY if that’s what T is. Of course, in the zero-energy solution, the size of the visible universe isn’t limited to 48 BLY. There are no event horizons.

Exactly as you stated, measuring a photon's energy in the local rest frames, the emitter and receiver would report the same frequency.

Good. Now I’ll add that it is not necessarily the case in the real universe, only our special (pebble) one with no gravity involved.

They have no way of knowing if space was Minkowski or if it was expanding.

Just a coordinate difference, so there’s no physical difference between the two. And they can always look at the CMB which gets you a peculiar-velocity meter for the one CS and gets you a location of the reference event for the Minkowskian CS. But space isn’t one or the other in the special case since the difference is purely abstract. In the real case, there is gravity and such so of course Minkowski spacetime does not describe the universe.

I'm not sure that [these comments are consistent]:

This is different than maintaining constant separation in an expanding metric which requires constant proper acceleration of at least one of the pebbles.

Given perfect linear expansion (a linear scalefactor), two objects (say a pair of pebbles a considerable distance apart) that are stationary relative to each other will remain at a constant proper separation forever in the absence of external forces

Good catch. 2nd comment is a crude approximation and thus wrong. The latter comment is true only in an inertial CS. Those same points are increasing their proper separation in the expanding CS, so in order to keep that distance constant in that CS, proper acceleration is needed outward to slow the inbound inertial velocity required. That is what the first comment references.
So to word it another way, using the expanding CS: You have one pebble stationary and a 2nd one with a peculiar velocity towards the first, maintaining a fixed distance from it at time T. In the absence of acceleration outward or any other forces acting on it, the 2nd pebble will reach the first eventually. Hence the necessity of outward proper acceleration to keep it at that fixed distance.

- - - - -
I suggested that expansion doesn't have to be happening in dense regions of space like galaxies:
…
I just can't see how you (Halc) can interpret that quote (from the paper of Francis et.al) in this way.

I’d rather consider what can be measured, and not what coordinate system is best chosen to describe a local collection of mass.
Put a comoving observer on each side of the galaxy, and a tape measure between them. Are they moving apart, together, or staying essentially at constant distance?  If the latter, then I’d accept that expansion is absent in a galaxy, but I think I can demonstrate the first case.

They say:  There is no expansion for the galaxy to over-come, since the metric of the local universe has already been altered by the presence of the mass of the galaxy.
You (Halc) say:  It doesn’t say that space isn’t expanding there... only stuff about the metric

Keep in mind that I consider ‘expanding’ to be a coordinate effect, so whether space is expanding is nothing but an abstract choice. The paper says there’s no reason to choose the expanding CS.

I can only say:  The metric is all about the expansion.

The metric is an abstract choice. Or so I suggest at least. Hence I think an empirical test would server better than a choice of one’s favorite CS. My whole point with the pebbles is to demonstrate that expansion or not is just a coordinate difference, not a physical one, so long as scalefactor is linear. Hence the CS choice is arbitrary.

Where I have said "the CMB frame" I am referring to a whole class of different inertial frames,  one such frame existing at each point in space.

That would make it more or less the way I use the phrase. Such a frame is the cosmological frame, or comoving frame, and it isn’t inertial at all, so doesn’t have the usual inertial properties.

Thus, the emitter pebble measures in their CMB frame

I had the emitter measure it in the emitter frame. If I have a laser that puts out frequency F, then it will be measured at F in the emitter’s inertial frame regardless of its peculiar motion. Anything else is a calculation, not a direct measurement. Let’s at least be explicit with the frame references so we’re at least clear.

Then there's this section:

With Euclidean geometry, when one object is at rest in the CMB frame and another object is at a distance but has no velocity relative to the first then we expect the second object to also be at rest in the CMB frame.

Nope. We expect the 2nd object to have a nonzero peculiar velocity since its unaccelerated worldline does not intersect the selected reference event. This is essential to our disagreement I think. I would hesitate to say ‘Euclidean geometry’. We’re talking Minkowskian geometry in which space is Euclidean, but spacetime is not. The frame rotations are different.

Which is not right.  However, it comes down to the use of the phrase "Euclidean geometry" which you have picked up on.  Some people do say "Euclidean" to mean "Minkowskian" or that the spacetime has a standard "Lorentzian" metric and I am one of those people.

Fine. I’m not. Euclidean spacetime does not have invariant intervals for instance. I know people use the word differently, so I’d rather we just kept away from depending on the term and one’s local interpretation of it.
Perhaps my disagreement stems from the ambiguous reference to “the CMB frame” used in your comment. Is it the CMB frame of the 1st or 2nd object? See why I don’t like the term?

Summary
I maintain these views:
  1.    Space doesn't have to be expanding inside a galaxy.  There are at least two good reasons to think that it isn't:
       (i)  The FLRW metric is not a good approximation to the metric of space in a dense region.
(ii)  Using the FRW universe models and the Friedmann equations we can see that increasing matter density causes decelleration of expansion.  We can naively assume that what happens on a universe-wide scale should also happen locally.

Increasing matter density of the whole universe causes deceleration of the whole universe. The equations don’t imply that local collections of matter, emptying out nearby regions to do so, has any effect on the local expansion. So I have one better reason to think that it is expanding in a galaxy:
(iii) Two comoving observers on opposite sides of the galaxy will find themselves receding from each other. If they do, then it demonstrates local-density-independent expansion. Each views the CMB frequency as isotropic, so there is no arbitrary selection of equations going on. I haven’t demonstrated this, but I can. If a measuring rod is already in place, they can quickly view their motion relative to it.

2.    There are ways that we can "know" if space is expanding inside a galaxy but they'll take a few years to actually do.

Just a few years?  I was thinking 6+ digits of years if any signals need to be passed. Want to do it quicker?  Get a smaller lab and more precise instruments. Hence my planet that was a 1 meter dense ball surrounded by a radius 5000 km glass shell.

I adjust my position on this issue:
3.    Having worked through the issues and based on comments from others.  I acknowledge that using red-shift from emitters and receivers or two pebbles is not the easiest way to determine if space in a galaxy is expanding.   

There was no galaxy in the pebble example. The scenario was meant to illustrate something else.

[Eternal Student]

Thanks again @Halc  I've enjoyed the discussion.

   There's nothing wrong with short replies.  There are several other threads that need more urgent attention.

Bye for now and best wishes to you.

LATE EDITING:   I've had even more time to read through what you've said Halc.    There are a few points I could pick up on but I'm thinking that the thread has drifted off topic from the Original Post quite a lot already.  So I was going to leave the thread alone now.
      The argument about whether expansion is happening inside a galaxy or mainly in the voids of space between galaxies (along with exactly what difference would appear in a co-ordinate independent measurement) still seems to be a frequent topic of discussion.  I've seen a couple of related threads appearing in some other forums (I won't name those other forums here) in the last week.  There are pundits (experts and/or self-proclaimed experts) making several different statements.

[Europa]

Am I correct in thinking that there is space, and into this space has emerged our universe? And through this space the mass of our universe is expanding? Warping and rippling this space as it does so?

There is insufficient data to define what the universe is

[yor_on]

Not as I get it Europe.  Space and time becomes interlocked together with mass, relative motion and accelerations.

[yor_on]

It also depends on definitions I should add. Locally defined, as from yourself, time is of one invariant magnitude, and space is what change with speeds and accelerations (mass is a equivalence to a acceleration in relativity). And your cm is always a cm.
=

If you want it even more complicated you can add a inflation and a accelerating expansion to it.

[TommyJ]

The earliest mention of ‘space’ in relation to ‘outer space’ is attributed to the 17th century. It was used in a description of the Universe and its bodies.
Speaking of the Universe size and expansion/infinity/beyond concepts.
The observation starting point is defined as unknown (although all researches are built on the Big Bang). That’s why the main focus is understanding what is going on and what is crucial to humanity’s nearest proximity.
Now observed expansion is increasing speed, but there are 3 options: it will stop at a certain balanced point, slow down but never stop (infinitely), start to collapse back.
Existing observations have several approaches.
Including it’s expansion as a sphere or a flat matter expansion. In theory gravity should slow down this expansion (this is where dark matter and its mass matters).
Spherical theory tells us geography, geometry and the evident conclusion, that no matter how long you travel on the sphere surface, you will never find an ending obstacle.