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# Cosmological Redshift and metric expansion

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If the axes in the images above (see https://www.scienceforums.net/topic/131720-cosmological-redshift-and-metric-expansion/?do=findComment&comment=1248849) were labeled with metric units, e.g., 1 km, 2 km, etc., then they could represent a metric. In that case, it would be immediately obvious that they are just two different pictures of the same thing, i.e., a 2 km by 2 km area.

But without units, they can mean anything, for example degrees of latitude and longitude. In this case, they show two shapes, both taking up 2 degrees of latitude and 2 degrees of longitude. Which is larger? This is impossible to tell as it depends where on Earth are they. Moreover, if they are on different planets, it also depends on the planets' radii.

Edited by Genady

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On 8/25/2023 at 6:46 PM, AbstractDreamer said:

Here's a neat trick.  Which green polygon has greater volume?

What about the scale factor?

If the green polygon represent the universe,then they are two different universes..with different metrics...meaning you are making wrong volume comparison.

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On 8/27/2023 at 5:39 PM, Genady said:

Yes, I do. So, the rest of the conclusions are wrong. Coordinates don't say anything about the length, area, and volume. Coordinates are arbitrary.

Expansion of space is a solution of the field equation in these conditions.

You're very wrong.  Coordinates are sufficient for comparison.   2<3.   Two is less than Three.  You don't need any units of meters, seconds, degrees or apples, square roots or coloured pixels.

On 8/27/2023 at 8:01 PM, Genady said:

Wanted to add a couple of examples to my previous comment and to demonstrate why this ^^^ is incorrect.

One example is metric that measures number of colored pixels. In this metric, the second shape is larger than the first.

Another metric is square of difference between the pixels in vertical and in horizontal directions. In this metric, the first shape is larger than the second.

Wrong again.  The number of coloured pixels are the same in both, even if you use a metric of number of coloured pixels.  Count them.  There are 4 pixels in both.

The difference between the pixels in both vertical and horiztonal direction are also the same.  There are two pixels in the vertical direction for both, and 2 pixels in the horizontal directions for both.

Your mistake is assuming the grid lines have significance.  They dont.  Only the numbers on the axes have meaning.

4 hours ago, Genady said:

If the axes in the images above (see https://www.scienceforums.net/topic/131720-cosmological-redshift-and-metric-expansion/?do=findComment&comment=1248849) were labeled with metric units, e.g., 1 km, 2 km, etc., then they could represent a metric. In that case, it would be immediately obvious that they are just two different pictures of the same thing, i.e., a 2 km by 2 km area.

But without units, they can mean anything, for example degrees of latitude and longitude. In this case, they show two shapes, both taking up 2 degrees of latitude and 2 degrees of longitude. Which is larger? This is impossible to tell as it depends where on Earth are they. Moreover, if they are on different planets, it also depends on the planets' radii.

Labelling with units is irrelevant.  Which number is bigger?  The number 2 or the number 3?

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29 minutes ago, AbstractDreamer said:

You're very wrong.  Coordinates are sufficient for comparison.   2<3.   Two is less than Three.  You don't need any units of meters, seconds, degrees or apples, square roots or coloured pixels.

Wrong again.  The number of coloured pixels are the same in both, even if you use a metric of number of coloured pixels.  Count them.  There are 4 pixels in both.

The difference between the pixels in both vertical and horiztonal direction are also the same.  There are two pixels in the vertical direction for both, and 2 pixels in the horizontal directions for both.

Your mistake is assuming the grid lines have significance.  They dont.  Only the numbers on the axes have meaning.

Labelling with units is irrelevant.  Which number is bigger?  The number 2 or the number 3?

You can say whatever you want about your pictures. But it is not how GR works. In GR, distances are determined via metric.

Take two events in Minkowski spacetime, for example, events A and B:

Which one is closer to 0?

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34 minutes ago, Genady said:

You can say whatever you want about your pictures. But it is not how GR works. In GR, distances are determined via metric.

Take two events in Minkowski spacetime, for example, events A and B:

Which one is closer to 0?

Neither.  As neither axes have any values other than zero.   In this manifold, zero is the only value that can be taken.  Everything has the value zero.

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3 minutes ago, AbstractDreamer said:

Neither.  As neither axes have any values other than zero.   In this manifold, zero is the only value that can be taken.  Everything has the value zero.

And now?

Define "closer"

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1 minute ago, AbstractDreamer said:

Define "closer"

IOW, define metric. Exactly.

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Exactly you're wrong and just trolling.

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4 hours ago, AbstractDreamer said:

You're very wrong.  Coordinates are sufficient for comparison.   2<3.   Two is less than Three.  You don't need any units of meters, seconds, degrees or apples, square roots or coloured pixels.

Wrong again.  The number of coloured pixels are the same in both, even if you use a metric of number of coloured pixels.  Count them.  There are 4 pixels in both.

The difference between the pixels in both vertical and horiztonal direction are also the same.  There are two pixels in the vertical direction for both, and 2 pixels in the horizontal directions for both.

Your mistake is assuming the grid lines have significance.  They dont.  Only the numbers on the axes have meaning.

Labelling with units is irrelevant.  Which number is bigger?  The number 2 or the number 3?

You have to establish common ground for arguments...i.e which number base are you using when comparing 2 and 3.

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On 8/27/2023 at 5:30 PM, AbstractDreamer said:

so space expansion doesn't change volume

Can you make precise what exactly you mean by “volume” here? Is it a 3-volume of space, as in a geodesic ball for example? Or a 4-volume of spacetime?

Both of these depend on the metric, as do all measurements of lengths, angles, areas, and n-volumes in general on a differentiable manifold.

For 3-volumes this is immediately obvious without any further ado, since FLRW spacetime is not a vacuum solution, and thus the Ricci tensor does not vanish. Therefore 3-volumes are not, in general, conserved as they age into the future, irrespective of your choice of coordinates. The only way to get them to be remain constant would be to choose a cosmological constant of just the right value so that you end up with a constant scale factor. But that is not compatible with observations.

For 4-volumes of spacetimes, the determinant of the metric enters as part of the volume element, so when you perform the integration to find the total volume of a given spacetime region, the result will explicitly depend on the times you use as integration limits, since the scale factor will be in there. So again, expansion has an effect here.

All of these are general mathematical results in differential geometry, and not specific to just GR as a theory.

I’m happy to show you the mathematical expressions if you need to see them (but I’m sure you’ve seen enough of me around here to know that I’m not just making this up). Otherwise Misner/Thorne/Wheeler has a good overview on how to construct general volumes on differentiable manifolds, or you can take a quick look on Wiki as well.

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• 2 weeks later...
On 8/29/2023 at 4:38 PM, Markus Hanke said:

Can you make precise what exactly you mean by “volume” here? Is it a 3-volume of space, as in a geodesic ball for example? Or a 4-volume of spacetime?

I'm not entirely certain what I mean by volume.

I have two thoughts.

Firstly, it is the "space expansion", whatever "space" or "volume" it is that is expanding.   So I'm guessing the kinematic interpretation of redshift suggests only a 3volumes expansion, and the (at least partly) gravitational interpretation suggests a 4-volume expansion.   But the accepted interpretation is kinematic only - a 3volume expansion?

Secondly it is about temperature.  Mordred mentioned "increasing volume":

On 5/24/2023 at 7:15 PM, Mordred said:

It may help to consider that the other major evidence of expansion isn't simply redshift. The most important evidence is the temperature decrease due to an increasing volume.

I don't know what he means by volume here.  I guess its 3-volume space.

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5 hours ago, AbstractDreamer said:

kinematic interpretation of redshift

Another way to look at this is that in curved spacetimes (of which FLRW is a specific example), energy-momentum is not - in general - a globally conserved quantity, even though it remains conserved everywhere locally. Thus it is not surprising that light does not retain its original frequency when traversing large regions of non-flat spacetime.

I personally think this is a better way to view this, since, after all, these galaxies remain in free fall and do not undergo proper acceleration at any time, despite the velocity-distance correlation.

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• 2 months later...
On 9/8/2023 at 7:22 AM, Markus Hanke said:

Another way to look at this is that in curved spacetimes (of which FLRW is a specific example), energy-momentum is not - in general - a globally conserved quantity, even though it remains conserved everywhere locally. Thus it is not surprising that light does not retain its original frequency when traversing large regions of non-flat spacetime.

I personally think this is a better way to view this, since, after all, these galaxies remain in free fall and do not undergo proper acceleration at any time, despite the velocity-distance correlation.

My point is not about refuting kinematic interpretation of redshift.   A phenomenon of space expanding would certainly cause the cosmological redshift observations that we measure.   This does not mean cosmological redshift observations are entirely and completely explained by space expansion.

If we measure a redshift of 3, why must the entirety of that redshift be caused by space expansion and nothing else?   What evidence do we have that nothing else causes cosmological redshift?

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3 hours ago, AbstractDreamer said:

My point is not about refuting kinematic interpretation of redshift.   A phenomenon of space expanding would certainly cause the cosmological redshift observations that we measure.   This does not mean cosmological redshift observations are entirely and completely explained by space expansion.

If we measure a redshift of 3, why must the entirety of that redshift be caused by space expansion and nothing else?   What evidence do we have that nothing else causes cosmological redshift?

Biggest for me, is that you would expect to also see some blueshifting, if it had some other cause.

It's also tied to distance. The further something is, the greater the recession velocity. Not what you might expect from a random sample.

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4 hours ago, AbstractDreamer said:

If we measure a redshift of 3, why must the entirety of that redshift be caused by space expansion and nothing else?

Because it is directly correlated to the distance of the source object in question, and that relationship is the exact same no matter in which direction we look, and no matter what else is/is not between here and there. Also, objects don’t just recede from us, but also from each other

One must also remember that redshift is only one of several consequences of metric expansion; it’s by no means the only data point.

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9 hours ago, Endy0816 said:

Biggest for me, is that you would expect to also see some blueshifting, if it had some other cause.

It's also tied to distance. The further something is, the greater the recession velocity. Not what you might expect from a random sample.

Why would we expect to see blueshifting?  We theorised space expansion from the discovery of cosmological redshift.  Not the other way around.  If cosmological redshift had some other cause, it wouldn't change what we measure at all.  It would change our theory.

9 hours ago, Markus Hanke said:

Because it is directly correlated to the distance of the source object in question, and that relationship is the exact same no matter in which direction we look, and no matter what else is/is not between here and there. Also, objects don’t just recede from us, but also from each other

One must also remember that redshift is only one of several consequences of metric expansion; it’s by no means the only data point.

According the Hubble-Lemaitre law, yes it is correlated to distance.  But why does the Hubble-Lemaitre law attribute 100% of observable cosmological redshift to space expansion, and 0% attributed to cosmological time dilation?  Where is the evidence that cosmological time dilation does not exist?

Before cosmological redshift, there was no such thing as either space expansion or cosmological time dilation.
After we discovered cosmological redshift, space expansion was accepted/invented/theorised, but why not "cosmological time dilation"?

As far as I have read, redshift IS the only source data point.  Space expansion was theorised from discovering the redshift.   All other data points derive from the theory.  We can't use a derivation to prove a premise.

Hubbles law: v = H0D

Why can't it be

v = H0D-PDT

Where T is the proper time difference and PD is a constant of proportionality of T which can change over relative proper distance.

In the Hubble=Lemaitre law, T simply has the value of zero.  So yes the further away something is, the faster it is receding, necessarily reaching superluminal velocities above certain distances.

If T has  a non zero then it could be that the further away something is, the faster or slower time is ticking where they are relative to the observer, instead of receding faster, potentially never breaking the limit of c, but not necessarily so.

Edited by AbstractDreamer
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1 hour ago, AbstractDreamer said:

But why does the Hubble-Lemaitre law attribute 100% of observable cosmological redshift to space expansion, and 0% attributed to cosmological time dilation?

Because it is based on the FLRW solution to the EFE, and therein only the spatial part of the metric is non-trivial and carries an expansion factor, there is no time dilation in this cosmological spacetime.

1 hour ago, AbstractDreamer said:

Before cosmological redshift, there was no such thing as either space expansion or cosmological time dilation.

Both GR in general and its particular solution for this case, the FLRW metric, predate Hubble’s observational findings. Metric expansion is a direct consequence of the laws of gravity, for any homogenous and isotropic distribution of energy-momentum that meets certain criteria; it’s not an independent, stand-alone idea.

1 hour ago, AbstractDreamer said:

After we discovered cosmological redshift, space expansion was accepted/invented/theorised

Accepted as the basis for a model in the context of cosmology, yes, but not invented or theorised - see above.

1 hour ago, AbstractDreamer said:

As far as I have read, redshift IS the only source data point.

Redshift was historically the first observational evidence that became available to us, but nowadays the Lambda-CDM model covers many other observations too, which were made after Hubble, eg the CMBR with its polarisation, the Ly-wavelength g-wave background, large-scale structure, acoustic baryonic oscillations, ratios of light/heavy elements etc. Among all proposed cosmological models, it is the one that fits the body of all available data the best - though it almost certainly won’t be the last word, I dare predict, because it also does have its problems.

1 hour ago, AbstractDreamer said:

If T has  a non zero then it could be that the further away something is, the faster or slower time is ticking where they are relative to the observer

You are welcome to try and find a solution (that isn’t just a trivial diffeomorphism) to the EFE that leads to such a law. Remember that it should also be compatible with all other observations, in order to be a useful cosmological model.

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18 hours ago, AbstractDreamer said:

Why would we expect to see blueshifting?  We theorised space expansion from the discovery of cosmological redshift.  Not the other way around.  If cosmological redshift had some other cause, it wouldn't change what we measure at all.  It would change our theory.

Would just expect to see a more typical distribution of directions in the first place. Some apparently headed towards us(blue), some away(red), and others approximately keeping pace.

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On 11/16/2023 at 3:41 PM, Markus Hanke said:

Because it is based on the FLRW solution to the EFE, and therein only the spatial part of the metric is non-trivial and carries an expansion factor, there is no time dilation in this cosmological spacetime.

Both GR in general and its particular solution for this case, the FLRW metric, predate Hubble’s observational findings. Metric expansion is a direct consequence of the laws of gravity, for any homogenous and isotropic distribution of energy-momentum that meets certain criteria; it’s not an independent, stand-alone idea.

Accepted as the basis for a model in the context of cosmology, yes, but not invented or theorised - see above.

Redshift was historically the first observational evidence that became available to us, but nowadays the Lambda-CDM model covers many other observations too, which were made after Hubble, eg the CMBR with its polarisation, the Ly-wavelength g-wave background, large-scale structure, acoustic baryonic oscillations, ratios of light/heavy elements etc. Among all proposed cosmological models, it is the one that fits the body of all available data the best - though it almost certainly won’t be the last word, I dare predict, because it also does have its problems.

You are welcome to try and find a solution (that isn’t just a trivial diffeomorphism) to the EFE that leads to such a law. Remember that it should also be compatible with all other observations, in order to be a useful cosmological model.

But why does the FLRW solution assume that "only the spatial part of the metric is non-trivial and carries an expansion factor, there is no time dilation in this cosmological spacetime."

On what basis and evidence independent of this assumption do we have that this disposition is true.  Why is it only the spatial metric and not the temporal metric that expands in the case of cosmological scale?

The FLRW solution contains within it the Hubble constant.  The constant describes adiabatic space expansion.  We cant then use the FLRW solution to justify that space expansion exists, because it is a solution that requires it to exist!   Newton's law of gravitation solution does not prove that gravity is force.  It is a solution that requires a force he called gravity.

"The Friedmann–Lemaître–Robertson–Walker metric (FLRW; /ˈfrdmən ləˈmɛtrə .../) is a metric based on the exact solution of the Einstein field equations of general relativity. The metric describes a homogeneous, isotropic, expanding (or otherwise, contracting) universe that is path-connected, but not necessarily simply connected" from Wiki

It is describing an expanding universe.  In other words, as it's premises it pre-assumes that the universe is homogenous, isotropic and expanding.  I'm not denying that any observations that subsequently fit the model certainly support the assumptions.  And I'm in no way suggesting there is no space expansion at all. But what about observations that don't fit the model such as data from the JWST?

The lamba-CDM model also uses the Hubble constant presumably derived from the FLRW metric.  Again, it premises that space expands.  A derivation cannot prove a premise.  A geocentric theory does not prove the sun circles around the earth, because it already assumes it does.   We can also add arbitrary complex formulas to make the geocentrism fit new conflicting data such as retrograde precession, just as we can add new mechanisms such as dark energy to make an expanding universe fit redshift and other observations.

CMBR polarisation suggests some space expansion occurred.  And we can arbitrarily parametise a formula to fit exactly what we observe and fit how we understand things work.  But making everything fit comes with the danger of confirmation bias, especially when the fit is arbitrary.   If there are any other mechanisms that we don't understand or haven't yet identified, then making things fit will with certainty blind us to those mechanisms.

Gravitation waves being stretched also fits an assumption of space expansion in the same way as redshifted EM spectra, but the same argument stands.  Why can some unknown from of time dilation not also contribute to gravitation wave stretching -  why must all stretching of gravitational waves be solely caused by space expansion, other than it nicely fits the FLRW metric which is already orientated to only the spatial metric expanding.

Large-scale structure patterns tell us of space expansion vs gravity vs time.  Most theories support the idea of a "force" (dark energy) that counteracts gravity to give us the patterns we see today.  Again we can arbitrarily parametise a formula to fit what we observe with how we understand things, which in this case is:  something (dark energy) is working against gravity, and it does so at different rates depending on when (time).   But why is all dark energy due to space expansion?  Just because we have a solution that takes the position of "only the spatial part of the metric carries an expansion factor"?  So if we are using large-scale structure as evidence for only space expansion, that is a fallacy of circular logic: "Given a solution where only the space metric expands (FLRW), then...  ...only space expands"  Well of course!  It's already given!

I have no response regarding acoustic baryonic oscillations and BB nucleosynthesis right now as I have no understanding at all on those topics.  But evidence for gravity as a force does not refute gravity as spacetime curvature.  In the same way, observations that fit space expansion does not specifically refute other mechanisms.  On the other hand there is quite a lot of refuting evidence against Lamda CDM especially since JWST.

Again, we go back to Hubble's law and cosmological redshift, as the only empirical data source that is not derived from the FLRW premise that only the spatial metric expands, or derived from a parametisation of that metric

Why is there a 1:1 causal link between redshift and space expansion and 1:0 (zero) causal link with redshift and time dilation, when we know it is a single spacetime manifold?  What empirical evidence did Hubble and Lemaitre have in 1920 to believe only space expansion and not time dilation causes redshift.  Why does the FLWR metric choose that only the spatial metric and not the temporal metric expands?

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14 hours ago, AbstractDreamer said:

But why does the FLRW solution assume that "only the spatial part of the metric is non-trivial and carries an expansion factor

It doesn’t assume this, it’s how the maths work out. The FLRW metric is a solution to the Einstein equations - so you start by putting in your initial and boundary conditions, and solve the equations; the result is FLRW.

14 hours ago, AbstractDreamer said:

In other words, as it's premises it pre-assumes that the universe is homogenous, isotropic and expanding.

The initial assumptions are homogeneity and isotropy, which taken together already constrain the final metric enough to give it its general form. To obtain the exact form, you begin with the standard energy-momentum tensor for a perfect fluid, feed it into the field equations, and solve. The result is that all coordinate dependencies drop out, except the time dependence of the spatial part.

FLRW is a Petrov-type O spacetime; metric expansion (meaning: measurements of distance depend on when they are performed) is a general geometric property of many such spacetimes, and by no means exclusive to FLRW.

14 hours ago, AbstractDreamer said:

But what about observations that don't fit the model such as data from the JWST?

Yes, Lambda-CDM doesn’t provide a perfect fit to all data, but it provides the best fit amongst all currently known cosmological models. But I agree that it probably won’t be the last word on the subject.

15 hours ago, AbstractDreamer said:

If there are any other mechanisms that we don't understand or haven't yet identified

There is of course always the possibility that new physics exist which we are not yet aware of. If so, our cosmological models will need to change.

FLRW provides the best fit to the data, based on the physics (ie GR and fluid dynamics) which we currently know.

15 hours ago, AbstractDreamer said:

Just because we have a solution that takes the position of "only the spatial part of the metric carries an expansion factor"?

Again, you can try and find a solution to the field equations that has a different form, yet still fits the available observational data. I would be interested in seeing it.

15 hours ago, AbstractDreamer said:

Why does the FLWR metric choose that only the spatial metric and not the temporal metric expands?

Again, the FLRW metric is a solution to the field equations of GR, it has not just been invented “by hand” - so the best answer to your question is that the laws of gravity determine it to be so. So far as I can see (having worked through this entire solution process myself), every time you start with homogeneity and isotropy as basic symmetries, you’ll end up with a metric of the general form of FLRW, and the precise spatial coordinate functions will depend on the energy-momentum tensor you use in the field equations. For this to come out different, you’d have to amend the laws of gravity itself.

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22 hours ago, AbstractDreamer said:

But why does the FLRW solution assume that "only the spatial part of the metric is non-trivial and carries an expansion factor, there is no time dilation in this cosmological spacetime."

Actually, one can perform a coordinate transformation on the FLRW metric to produce a metric in which time and space expand equally. The resultant metric is a scalar function multiple of the Minkowskian metric, and therefore FLRW spacetime is conformally flat, or as Markus Hanke said above, "a Petrov-type O spacetime". One advantage of such a coordinate system is that it simplifies light-like trajectories as well as the cosmological redshift.

Edited by KJW
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On 11/22/2023 at 1:59 PM, KJW said:

Actually, one can perform a coordinate transformation on the FLRW metric to produce a metric in which time and space expand equally. The resultant metric is a scalar function multiple of the Minkowskian metric, and therefore FLRW spacetime is conformally flat, or as Markus Hanke said above, "a Petrov-type O spacetime". One advantage of such a coordinate system is that it simplifies light-like trajectories as well as the cosmological redshift.

Yes, such a transformation would just be a diffeomorphism, which is of course always allowed. The disadvantage though is that it changes the physical meaning of the time coordinate - it then no longer corresponds to a clock co-moving with the cosmological fluid. I also suspect (not sure though) that this would introduce off-diagonal terms into the metric?

I think everything considered, the usual Gaussian normal coordinates probably give the simplest and easiest to understand form of the metric.

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3 hours ago, Markus Hanke said:

Yes, such a transformation would just be a diffeomorphism, which is of course always allowed.

I actually prefer the term "coordinate transformation" because that's what they are. I think the term "diffeomorphism" is a bit too esoteric for me. There are also "point transformations" which are conceptually distinct from coordinate transformations though mathematically identical, and it's not clear to me to which of these the term "diffeomorphism" actually refers.

3 hours ago, Markus Hanke said:

The disadvantage though is that it changes the physical meaning of the time coordinate - it then no longer corresponds to a clock co-moving with the cosmological fluid.

The OP enquired as to why it is only space that expands and not time. The answer is that the difference between time expanding with space and time not expanding with space is just a coordinate transformation, which means that there is no physical difference. However, as you correctly point out, the time coordinate of the time-expanding metric doesn't correspond to anything, in particular, not a co-moving clock, whereas the spatial coordinates actually do correspond to the co-moving cosmological fluid.

3 hours ago, Markus Hanke said:

I also suspect (not sure though) that this would introduce off-diagonal terms into the metric?

No, it doesn't. Only the time coordinates are involved in the coordinate transformation. If the original metric describes a flat three-dimensional space, then the transformed metric will be a scalar function multiple of the Minkowskian metric:

guv = ƒ(t) ηuv

3 hours ago, Markus Hanke said:

I think everything considered, the usual Gaussian normal coordinates probably give the simplest and easiest to understand form of the metric.

It depends on how the metric is used. For example, if one is describing light-like trajectories in spacetime, then they have a simple straight-line form.

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10 hours ago, KJW said:

the time coordinate of the time-expanding metric doesn't correspond to anything

Would it not correspond to a clock that is in motion relative to the cosmological fluid?

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