Cosmological principle paradox?

[AWA]

What I'm saying is that the universe at low redshifts (say, z=0.01, for instance) looks very different from the universe at z=1 when you look at the details, such as the separation between galaxies and the sorts of galaxies you see. The differences increase as you go further back.

If you want to talk about the detalis , that's OK but it's got nothing to do with the paradox I was referring to, the details are totally inhomogenous of course, it only takes a look at my garage to confirm that.
We are concerned here with large scale only, not with the evolution of a star, a galaxy or fish out of the fridge.

 

[Chalnoth]

If you want to talk about the detalis , that's OK but it's got nothing to do with the paradox I was referring to, the details are totally inhomogenous of course, it only takes a look at my garage to confirm that.
We are concerned here with large scale only, not with the evolution of a star, a galaxy or fish out of the fridge.

What I mean is that the z=0.01 universe is statistically distinguishable from the z=1 universe. That is, many of the average properties of the universe at z=0.01 are very different from the universe at z=1. So yes, at large scales, low redshifts are very different from high redshifts.

 

[AWA]

What I mean is that the z=0.01 universe is statistically distinguishable from the z=1 universe. That is, many of the average properties of the universe at z=0.01 are very different from the universe at z=1. So yes, at large scales, low redshifts are very different from high redshifts.

Ahaaa, but that is because we are not talking about homogenous scales yet, I can assure you the z=0.000001 universe is very diferent from the z=1 universe. That is trivial and solves nothing.

 

[George Jones]

Let's suppose we had super-advance telescopes (let's forget technical and time limitations for the sake of the argumen) with ultra high deep field that allows us to make a map similar to the SDSS map but up to a redshift z from a little after decoupling, according to standard cosmology, at this scale the map surely would show homogeneity (if we don't find it at this scale I wonder at what scale we might expect to).
But this map is also a look-back time map of the time dimension of the last 13 bly, so it would also be showing us homogeneity in the time dimension.

As a matter of fact we don't need that supertelescope, we are watching an isotropic and homogenous to more than a part in 10^5 map from further time back already, the CMB, so we seem to have homogeneity in time at least up to 13.64 bly which for a universe 13.7 bly old is a good proportion of the total.

But this is just observation of spatial homogeneity at different epochs. I think the problem is that your definition of time homogeneity is different than everyone else's definition of time homogeneity.

Well for me this seems to conflict with this statement from wikipedia:"The Perfect Cosmological Principle is an extension of the Cosmological Principle, which accepts that the universe changes its gross feature with time, but not in space." I mean if it doesn't change in space, it shouldn't change in time to keep congruence with the lookback time we see when we look at the space surrounding us at great distances.

The only solution I find is that our universe follows the "perfect cosmological principle" except at the initial singularity, which could mean that ultimately it doesn't.

 

[Chalnoth]

Ahaaa, but that is because we are not talking about homogenous scales yet, I can assure you the z=0.000001 universe is very diferent from the z=1 universe. That is trivial and solves nothing.

Huh? These scales are most certainly homogeneous scales. The z=0.01 universe represents a spherical shell approximately 22,000Mpc^2 in surface area, while the z=1 universe represents a spherical shell approximately 138Gpc^2 in surface area. If we set the length scale of homogeneity to be ~80Mpc or so, then the area scale would be ~6,400Mpc^2 or so. In either case, both of these spherical shells are far beyond that, and thus we'd only need to observe a decent but not too huge fraction of the sky at z=0.01 to get homogeneity, and only a tiny fraction of the sky at z=1.

 

[AWA]

Huh? These scales are most certainly homogeneous scales. The z=0.01 universe represents a spherical shell approximately 22,000Mpc^2 in surface area, while the z=1 universe represents a spherical shell approximately 138Gpc^2 in surface area. If we set the length scale of homogeneity to be ~80Mpc or so, then the area scale would be ~6,400Mpc^2 or so. In either case, both of these spherical shells are far beyond that, and thus we'd only need to observe a decent but not too huge fraction of the sky at z=0.01 to get homogeneity, and only a tiny fraction of the sky at z=1.

Let's set this straight for accuracy sake. Maybe my source is misleading or I am misinterpreting it.

From the image in this page: http://www.sdss.org/includes/sideimages/sdss_pie2.html
I interpret that the SDSS galaxy map has data up to about a z=0.14, much higher than the z=0.01 that you mention. But most likely the data is scarce at the outer zone of the map.

Authors have differing opinions about whether we can already say we are observing homogeneity at this distances, depending on the statistical analysis they perform on the data, but for many we haven't reach yet what we can call properly statiscal homogeneity.

I would suspect that at redshift z=1 there is certainly homogeneity as you say. And obviously a sphere at z=2 also, and at 3,4...20,200, etc. And logically the spatial spherical surfaces between z=1 and 2: 1.0000...1 up to 1.999...9. All this arbitrarily(well, not arbitrary, it's given by the resolution of our instruments but in the OP I ignored technical difficulties) large number of homogenous spherical spatial surfaces represent arbitrarily small time interval snapshots towards the past. And yet you claim they don't tell us anything about the homogenity of the time dimension. But this set of snapshots give you a timeline, and every point on this timeline with an arbitrarily small separation of the points is homogenous. My impression is that this makes the timeline homogenous at least from z=1 onwards. Otherwise I'd like to have someone explain me why not.
Source: http://cas.sdss.org/public/en/sdss/default.asp#time

 

[Chalnoth]

I would suspect that at redshift z=1 there is certainly homogeneity as you say. And obviously a sphere at z=2 also, and at 3,4...20,200, etc. And logically the spatial spherical surfaces between z=1 and 2: 1.0000...1 up to 1.999...9. All this arbitrarily(well, not arbitrary, it's given by the resolution of our instruments but in the OP I ignored technical difficulties) large number of homogenous spherical spatial surfaces represent arbitrarily small time interval snapshots towards the past. And yet you claim they don't tell us anything about the homogenity of the time dimension.

I do claim they tell us something about the homogeneity of the time dimension. I claim that they tell us that there isn't homogeneity in the time dimension, that we see a variety of trends in the statistical behavior of the universe across redshift.

 

[AWA]

I do claim they tell us something about the homogeneity of the time dimension. I claim that they tell us that there isn't homogeneity in the time dimension, that we see a variety of trends in the statistical behavior of the universe across redshift.

Where do you see those trends across redshift? If you claim to observe inhomogeneity at large scale across redshift, your claiming spatial inhomogeneity, 'cause radial distances from here are defined by redshift.

 

[Calimero]

No, distances are not defined by redshift.

 

[Chalnoth]

Where do you see those trends across redshift? If you claim to observe inhomogeneity at large scale across redshift, your claiming spatial inhomogeneity, 'cause radial distances from here are defined by redshift.

Not at all, because the spatial homogeneity is only claimed for equal-time slices, and the radial direction is looking across a succession of equal-time slices. So we expect statistical isotropy as a consequence of homogeneity, but we do not expect the radial direction to appear homogeneous. And it doesn't.

As for the particular trends, well, you've got structure formation, you've got evolution of galaxy populations, you've got density evolution. Going deep into the past you've got reionization, the "dark ages", and, of course, the phase transition that emitted the CMB.

 

[Chalnoth]

No, distances are not defined by redshift.

Sort of. Redshift is typically used as a proxy for distance in galaxy surveys. And it's a reasonable enough proxy once you're out high enough in redshift that the local motion is small compared to the cosmological redshift (which isn't actually that far...typical peculiar redshifts max out at around $\Delta z = 0.003$, with most being much much smaller).

 

[AWA]

Not at all, because the spatial homogeneity is only claimed for equal-time slices, and the radial direction is looking across a succession of equal-time slices. So we expect statistical isotropy as a consequence of homogeneity, but we do not expect the radial direction to appear homogeneous. And it doesn't.

We don't expect it. But it shows up, a succesion of homogenous equal-time slices is what I call the time dimension.

As for the particular trends, well, you've got structure formation, you've got evolution of galaxy populations, you've got density evolution. Going deep into the past you've got reionization, the "dark ages", and, of course, the phase transition that emitted the CMB.

Formation and evolution of galaxies are rather local events compared to the scales we are dealing with;the others are my solution to paradox.

 

[Calimero]

Sort of. Redshift is typically used as a proxy for distance in galaxy surveys. And it's a reasonable enough proxy once you're out high enough in redshift that the local motion is small compared to the cosmological redshift (which isn't actually that far...typical peculiar redshifts max out at around $\Delta z = 0.003$, with most being much much smaller).

Yes, but what I mean is that object at z=4 is not twice as far as object at z=2.

 

[AWA]

Yes, but what I mean is that object at z=4 is not twice as far as object at z=2.

Sure, the distance-redshift relation is not linear, so?Borrowing a little from cosmology textbook stuff, this is from Hobson's General Relativity:

"In general relativity the concept of a ‘moment of time’ is ambiguous and is replaced by the notion of a three-dimensional spacelike hypersurface. To define a ‘time’ parameter that is valid globally, we ‘slice up’ spacetime by introducing a series of non-intersecting spacelike hypersurfaces that are labelled by some parameter t. This parameter then defines a universal time in that ‘a particular time’ means a given spacelike hypersurface. We may construct the hypersurfaces t = constant in any number of ways. In a general spacetime there is no preferred ‘slicing’ and hence no preferred ‘time’coordinate t.
According to Weyl’s postulate, there is a unique worldline passing through each (non-singular) spacetime point. The set of worldlines is sometimes described as providing threading for the spacetime."

In our case one of the three spatial dimensions of the spacelike hypersurfaces represents visually (thanks to light's nature) the timelike worldline passing thru each spacetime point defined by a specific redshift.

 

[Chalnoth]

Yes, but what I mean is that object at z=4 is not twice as far as object at z=2.

Yes, that is very true.

 

[Chalnoth]

Formation and evolution of galaxies are rather local events compared to the scales we are dealing with;the others are my solution to paradox.

As I keep saying, the population of galaxies at z=1 is, for instance, very different from the population of galaxies at z=0.01. The reason for this is that there are more older galaxies in the nearby universe, and more younger galaxies in the early universe. Clusters are more numerous and larger in the nearby universe. Active galactic nuclei are more common a bit further away (that is, AGN's are typically characteristic of younger galaxies).

 

[AWA]

As I keep saying, the population of galaxies at z=1 is, for instance, very different from the population of galaxies at z=0.01. The reason for this is that there are more older galaxies in the nearby universe, and more younger galaxies in the early universe. Clusters are more numerous and larger in the nearby universe. Active galactic nuclei are more common a bit further away (that is, AGN's are typically characteristic of younger galaxies).

I thought we agreed before that the universe at z=0.01 is not yet homogenous so it is obviously different than the universe at z=1.
Besides the examples you are giving about galactic age shouldn't affect the universe density at large scale. The clusters part I would have to check it. I'm not sure that has been statistically shown to happen.

 

[Chalnoth]

I thought we agreed before that the universe at z=0.01 is not yet homogenous so it is obviously different than the universe at z=1.

You said it. I didn't agree to it.

What you have to bear in mind is that there is a fundamental difference between the typical length scale of homogeneity and actually statistically demonstrating it (given an appropriate threshold). To statistically demonstrate it, you need a region much larger than the scale of homogeneity (because you have to show that all such regions of said size are statistically identical, to within some pre-defined threshold).

It's not such a surprise to me that there's argument about this, because it's mathematically a difficult thing to demonstrate, and the threshold of homogeneity is arbitrary anyway.

Besides the examples you are giving about galactic age shouldn't affect the universe density at large scale. The clusters part I would have to check it. I'm not sure that has been statistically shown to happen.

Well, density is one of the most significant things to evolve with redshift. The average density of the matter in our universe at z=1 is eight times the density today. Baryon Acoustic Oscillation observations, which measure the typical separation between galaxies at different redshifts, are a good measurement of how this density changes with redshift.

 

[Calimero]

Sure, the distance-redshift relation is not linear, so?.

So it depends on input parameters.

Borrowing a little from cosmology textbook stuff, this is from Hobson's General Relativity:

"In general relativity the concept of a ‘moment of time’ is ambiguous and is replaced by the notion of a three-dimensional spacelike hypersurface. To define a ‘time’ parameter that is valid globally, we ‘slice up’ spacetime by introducing a series of non-intersecting spacelike hypersurfaces that are labelled by some parameter t. This parameter then defines a universal time in that ‘a particular time’ means a given spacelike hypersurface. We may construct the hypersurfaces t = constant in any number of ways. In a general spacetime there is no preferred ‘slicing’ and hence no preferred ‘time’coordinate t.
According to Weyl’s postulate, there is a unique worldline passing through each (non-singular) spacetime point. The set of worldlines is sometimes described as providing threading for the spacetime."

In our case one of the three spatial dimensions of the spacelike hypersurfaces represents visually (thanks to light's nature) the timelike worldline passing thru each spacetime point defined by a specific redshift.

Thanks. Very nice quotation.

I thought we agreed before that the universe at z=0.01 is not yet homogenous so it is obviously different than the universe at z=1..

There is the source of your confusion. Please look http://www.math.lsa.umich.edu/mmss/coursesONLINE/Astro/Ex2.2/" . You are comparing homogeneity of two different time slices, which, obviously, when compared are not the same, and talking about 'not yet' homogeneous universe. Universe is just as much homogeneous now as it ever was, just on different scale.

 

[AWA]

There is the source of your confusion. Please look http://www.math.lsa.umich.edu/mmss/coursesONLINE/Astro/Ex2.2/" . You are comparing homogeneity of two different time slices, which, obviously, when compared are not the same, and talking about 'not yet' homogeneous universe. Universe is just as much homogeneous now as it ever was, just on different scale.

Thanks, cool site.
A couple of remarks: first, that is a simulation school exercise and they stress it that as a simulation it is not expected to match the real universe.
Second, according to GR (and as pointed out in the quote from my last post) you can slice up spacetime arbitrarily (general covariance, remember?) : "We may construct the hypersurfaces t = constant in any number of ways. In a general spacetime there is no preferred ‘slicing’ and hence no preferred ‘time’coordinate t."
Given a statistically significant number of slices you should find homogeneity across the sufficiently long time-like worldline formed by the statistically large stack of spacelike slices if each of the different time spacelike slices is itself homogenous. If you don't agree with this, I should remind you that GR is to this day the best theory to understand the universe that we have.

 

[AWA]

The average density of the matter in our universe at z=1 is eight times the density today.

Please, back up that figure with some reliable reference. Specifying how do you exactly measure the universe density time spacelike slice at precisely z=1.

 

[Chalnoth]

Please, back up that figure with some reliable reference. Specifying how do you exactly measure the universe density time spacelike slice at precisely z=1.

Matter density scales as $a^{-3}$. But $1+z = 1/a$, so matter density scales as $(1+z)^3$.

Basically, proposing that the matter density wasn't eight times its current value at z=1 requires proposing a universe that has radial-dependent density for an equal-time slicing. This is, in principle, a rather difficult thing to accurately determine, but suffice it to say our theories that use a homogeneous matter distribution work, while alternative theories proposed to explain certain unpleasant aspects of the homogeneous theories don't.

 

[AWA]

Matter density scales as $a^{-3}$. But $1+z = 1/a$, so matter density scales as $(1+z)^3$.

Right, that is a model-dependent calculation, not an observation. That is what is apparently contradicted (and thus where I see the paradox) by the hypothetical future SDSS 3D galaxy map up to a high redshift, that we expect to be homogenous. But perhaps, I'm misunderstanding something and cosmologists don't expect to find that map statistically homogenous. If so, please explain.

Basically, proposing that the matter density wasn't eight times its current value at z=1 requires proposing a universe that has radial-dependent density for an equal-time slicing.

I see what you mean, yes, that seems to follow logically.
The universe is a strange place, and is full of apparent contradictions, wish we knew it better, but let's be humble (and honest), we are barely starting to grasp it.

This is, in principle, a rather difficult thing to accurately determine, but suffice it to say our theories that use a homogeneous matter distribution work, while alternative theories proposed to explain certain unpleasant aspects of the homogeneous theories don't.

Ultimately, it seems to come down to a practical matter.
BTW, I don't know what are those theories proposed to explain unpleasant aspects of homogenous theories, and what these unpleasnt aspects are. Would you elaborate a little on this?

 

[Chalnoth]

Right, that is a model-dependent calculation, not an observation. That is what is apparently contradicted (and thus where I see the paradox) by the hypothetical future SDSS 3D galaxy map up to a high redshift, that we expect to be homogenous. But perhaps, I'm misunderstanding something and cosmologists don't expect to find that map statistically homogenous. If so, please explain.

You have to look into it in a bit more detail. For example, if you combine nearby measurements of the Hubble constant and supernovae with WMAP data, you end up with a nearly-flat universe.

If you then proceed with the assumption of flatness and make use of BAO data, you get a connection between length scales at different redshifts. If this link between length scales at different redshifts doesn't line up, then that would be evidence that the assumption of homogeneity was wrong. Basically, this length scale being at the expected place with the assumption of flatness is a reasonably direct test of the relationship $z+1 = 1/a$.

There are all sorts of different ways you can do this sort of experiment, but the basic idea here is that if you make a series of assumptions, and multiple independent experiments measure the same set of parameters based upon those assumptions, you gain confidence that those assumptions are, in fact, true, at least in an approximate sense. One of those foundational assumptions is homogeneity.

Ultimately, it seems to come down to a practical matter,it seems.
BTW, I don't know what are those theories proposed to explain unpleasant aspects of homogenous theories, and what these unpleasnt aspects are. Would you elaborate a little on this?

In this case, some have attempted to explain away the accelerated expansion by proposing a universe that has radial-dependent density. It turns out that such proposals are ruled out by observation.

 

[AWA]

You have to look into it in a bit more detail. For example, if you combine nearby measurements of the Hubble constant and supernovae with WMAP data, you end up with a nearly-flat universe.

If you then proceed with the assumption of flatness and make use of BAO data, you get a connection between length scales at different redshifts. If this link between length scales at different redshifts doesn't line up, then that would be evidence that the assumption of homogeneity was wrong. Basically, this length scale being at the expected place with the assumption of flatness is a reasonably direct test of the relationship $z+1 = 1/a$.

There are all sorts of different ways you can do this sort of experiment, but the basic idea here is that if you make a series of assumptions, and multiple independent experiments measure the same set of parameters based upon those assumptions, you gain confidence that those assumptions are, in fact, true, at least in an approximate sense. One of those foundational assumptions is homogeneity.

This is all understood and fine, I'm just taking that assumption to its last logical consequences if we take relativity seriously, and if we agree that if you observe long distances spaces you are also observing the past, one cannot be homogenous if the other isn't too, and viceversa. As they say, you can't have one without the other.

This leads to some contradiction with standard cosmology, so when in doubt, of course we choose standard cosmology, right?

 

[AWA]

You have to look into it in a bit more detail.

If you then proceed with the assumption of flatness and make use of BAO data, you get a connection between length scales at different redshifts. If this link between length scales at different redshifts doesn't line up, then that would be evidence that the assumption of homogeneity was wrong. Basically, this length scale being at the expected place with the assumption of flatness is a reasonably direct test of the relationship $z+1 = 1/a$.

Searching with the word BAO in arxiv, the first random paper I read casts shadows over BAO signal measures.: http://arxiv.org/PS_cache/arxiv/pdf/1009/1009.1232v1.pdf

 

[Chalnoth]

This is all understood and fine, I'm just taking that assumption to its last logical consequences if we take relativity seriously, and if we agree that if you observe long distances spaces you are also observing the past, one cannot be homogenous if the other isn't too, and viceversa. As they say, you can't have one without the other.

This leads to some contradiction with standard cosmology, so when in doubt, of course we choose standard cosmology, right?

This is getting tiring. We do not observe homogeneity in the radial direction. We don't expect to, because the radial direction is also looking backwards in time.

What we observe instead is a universe that looks like the nearby universe is a later version of the far away universe. In other words, it's as if looking outward in space is looking through a succession of homogeneous equal-time slices. This is the standard cosmology. This is what we observe. None of our observations contradict this, and it is fully self-consistent.

As for the BAO paper, if you look at their data, the apparent $3\sigma$ deviation is represented in figure 6, where you can clearly see that the discrepancy comes down to the signal being rather noisier than their simulations estimate, which would be indicative of not properly taking something into account in the simulations.

A perhaps better paper for this particular issue is this one:
http://arxiv.org/abs/0705.3323

...because this shows the combination of WMAP, supernova, and BAO data. The relevant plot is fig. 13, where you see that the contours all converge on the same point in parameter space when these data are taken together.

 

[AWA]

This is getting tiring. We do not observe homogeneity in the radial direction.

Oh, but you don't have to answer if you get tired. There are more people in this forum.

We do not observe homogeneity in the radial direction.We don't expect to

Are you sure? we expect to find spatial homogeneity. why on Earth would you want to leave out one spatial dimension just beats me. I mean that's pretty bizarre, how do you keep one spatial direction inhomogenous and the others homogenous, and still keep isotropy?

What we observe instead is a universe that looks like the nearby universe is a later version of the far away universe. In other words, it's as if looking outward in space is looking through a succession of homogeneous equal-time slices.

See posts 49 and 55.

 

[Chalnoth]

Are you sure? we expect to find spatial homogeneity. why on Earth would you want to leave out one spatial dimension just beats me. I mean that's pretty bizarre, how do you keep one spatial direction inhomogenous and the others homogenous, and still keep isotropy?

1. Many cosmologists, early on, expected homogeneity in both time and space. This was disproven when Hubble measured the expansion of the universe.
2. Since when we look far away, we are looking back in time, we do not expect to see homogeneity in that direction, because an expanding universe changes with time.

See posts 49 and 55.

So, you're still confused about the simultaneity thing? The expansion of the universe itself creates a notion of "universal time". If you use coordinates that move with the expansion, then observers that are stationary with respect to the coordinate system each see the universe as being isotropic from their point of view.

This is, ultimately, what we mean by spatial homogeneity: if I go anywhere else in the visible universe, and adjust my velocity to move along with the local matter there, the universe will look isotropic to me. A homogeneous universe is defined as one in which you can do this: you can move anywhere within the universe, set your velocity to some value, and see an isotropic universe. You can then define the time coordinate globally in such a way that at the same time, separated observers see the same properties of the universe (such as the CMB temperature). In these coordinates, the properties of the universe are the same everywhere in space, but change with time.

You can change to a different set of coordinates, of course, and things won't necessarily look constant in space any longer. You'll still get the right answers for any observable you calculate, but you won't see the homogeneity.

 

[AWA]

Since when we look far away, we are looking back in time, we do not expect to see homogeneity in that direction, because an expanding universe changes with time.

This indeed is getting repetitive,once again this makes no sense in GR. You have some source where this is explicitly stated? that homogeneity is forbidden in one spatial direction?

This is, ultimately, what we mean by spatial homogeneity: if I go anywhere else in the visible universe, and adjust my velocity to move along with the local matter there, the universe will look isotropic to me. A homogeneous universe is defined as one in which you can do this: you can move anywhere within the universe, set your velocity to some value, and see an isotropic universe. You can then define the time coordinate globally in such a way that at the same time, separated observers see the same properties of the universe (such as the CMB temperature). In these coordinates, the properties of the universe are the same everywhere in space, but change with time.

You can change to a different set of coordinates, of course, and things won't necessarily look constant in space any longer. You'll still get the right answers for any observable you calculate, but you won't see the homogeneity.

You won't? spatial homogeneity is not a physical observable? it is just a convenient perspective only watchable with some privileged coordinates?

 

[Chalnoth]

You won't? spatial homogeneity is not a physical observable? it is just a convenient perspective only watchable with some privileged coordinates?

Yes, spatial homogeneity is only something that is watchable in some privileged coordinates. The only sort of homogeneity that would be visible in any coordinates is space-time homogeneity. We don't get to do that for homogeneity that is only in space, unfortunately.

This doesn't mean that spatial homogeneity is meaningless, however. Yes, it only appears in some special choice of coordinates. However, it isn't something that you can do in any sort of universe you might conceive. Remember the definition I laid down previously: if, at any point in space, one can construct a hypothetical observer that will see an isotropic universe, then we can call that universe homogeneous in space.

I could easily construct a universe that doesn't have this property. For instance, if we imagine a universe that is very dense in the direction of both poles of the Earth, but has very little matter in the directions outward from the Earth's equator, that would be a very anisotropic universe. The north/south direction would be picked out as a special direction. But what's more, there is no choice of observer located on Earth that could see that distribution as being isotropic.

In the end, this model of a homogeneous universe isn't a direct observable (because we can't move far enough away to check isotropy from different spatial locations), but it does have observable consequences. Namely, it states that the expansion of the universe should follow the Friedmann equations. When we measure the expansion of our universe using many different sorts of observations, and continually come up with the same answer every time, we gain confidence that the Friedmann equations are valid, at least approximately, which means we gain confidence that our universe is genuinely homogeneous in space (for a specific choice of coordinates).

 

[AWA]

Yes, spatial homogeneity is only something that is watchable in some privileged coordinates. The only sort of homogeneity that would be visible in any coordinates is space-time homogeneity. We don't get to do that for homogeneity that is only in space, unfortunately.

This doesn't mean that spatial homogeneity is meaningless, however. Yes, it only appears in some special choice of coordinates. However, it isn't something that you can do in any sort of universe you might conceive. Remember the definition I laid down previously: if, at any point in space, one can construct a hypothetical observer that will see an isotropic universe, then we can call that universe homogeneous in space.

I could easily construct a universe that doesn't have this property. For instance, if we imagine a universe that is very dense in the direction of both poles of the Earth, but has very little matter in the directions outward from the Earth's equator, that would be a very anisotropic universe. The north/south direction would be picked out as a special direction. But what's more, there is no choice of observer located on Earth that could see that distribution as being isotropic.

In the end, this model of a homogeneous universe isn't a direct observable (because we can't move far enough away to check isotropy from different spatial locations), but it does have observable consequences. Namely, it states that the expansion of the universe should follow the Friedmann equations. When we measure the expansion of our universe using many different sorts of observations, and continually come up with the same answer every time, we gain confidence that the Friedmann equations are valid, at least approximately, which means we gain confidence that our universe is genuinely homogeneous in space (for a specific choice of coordinates).

Ok, I see now clearly the source of our disagreement and of my "false paradox". Actually there is no paradox at all.

I guess the moral of the story is that one must not take GR to seriously because that is considered naive at best and against standard cosmology at worse.
But all books on GR stress general invariance (covariance), all of them say we can choose coordinates arbitrarily, which also means of course we can privilege some coordinates for the sake of convenience, but that convenience in no way means the results be get with that coordinates are physically real unless they can be reproduced with other choices of coordinates and metrics. I guess I also took too seriously the interchangeability of spacetime dimensions that relativity teaches us.

Now you tell me that spatial homogeneity, even though it is a property as physical as it can be, only appears with a determinate choice of coordinates that produce a certain privileged slicing of spacelike hypersurfaces, and that this homogeneity disapears if we try to make it coordinate invariant when we change the coordinates, appearing instead a sort of statistical homogeneity wrt both space and time (spacetime) and inhomogeneity or radial density dependence in the purely spatial hypersurface, and both of this things are forbidden by standard cosmology and astronomical observations and I have to take your word on this, no matter what GR says because you know more than me and standard cosmology says so and I'm a responsible citizen.

I declare the paradox solved unless someone else finds this a bit odd too or has some new input. Thanks a lot.

 

[Chalnoth]

General Relativity itself respects general covariance. But the specific distribution in our universe does not. In fact, it's pretty easy to prove that normal matter/radiation cannot respect general covariance, because the only covariant stress-energy tensor is one that behaves like vacuum energy.

Therefore, the very existence of matter ensures that the universe will look different in different coordinate systems. The general covariance of General Relativity ensures that you get the same results for the behavior of said matter no matter what coordinate system you use. And the math will be made easier if we use coordinates that follow any symmetries that exist in the matter distribution.

For instance, if you are doing physics on the surface of the Earth and not moving very far, it is convenient to approximate the Earth as a perfectly-flat surface. This set of coordinates will start to be wrong if we move too far along the Earth's surface or too far above it, but it is a convenient choice as long as we don't do these things.

If you are instead, for example, attempting to put a satellite into low-Earth orbit, it becomes convenient to use spherical coordinates centered at the center of the Earth, and ignore the effect of bodies further away. This set of coordinates will start to be wrong if you get too close to the Moon, or far enough from the Earth that the Sun's gravity becomes more important.

If you are instead interested in describing the motions of the planets, it becomes convenient to use spherical coordinates centered on the Sun.

And so on and so forth. You can use General Relativity in each case. In each case, exploiting the symmetries of the physical matter distribution makes the math easier.

This is what we are doing when we use FRW coordinates: we are exploiting a particular symmetry of the average matter distribution of our universe, namely spatial homogeneity. Allowing our coordinate choice to follow this symmetry makes the math easier.

 

[AWA]

General Relativity itself respects general covariance. But the specific distribution in our universe does not. In fact, it's pretty easy to prove that normal matter/radiation cannot respect general covariance, because the only covariant stress-energy tensor is one that behaves like vacuum energy.

I wish some expert relativist would confirm this, maybe some guy from the relativity forum, as I consider it not exact but that might be due to my poor knowledge of GR. I'll try to think about it some more.

Therefore, the very existence of matter ensures that the universe will look different in different coordinate systems.

We must have some different understanding of isotropy as different coordinates systems can mean rotating the observer point of view and this should be invariant if there is isotropy.

This is what we are doing when we use FRW coordinates: we are exploiting a particular symmetry of the average matter distribution of our universe, namely spatial homogeneity. Allowing our coordinate choice to follow this symmetry makes the math easier

Only remember this coordinate-dependent spatial homogeneity hasn't been completely confirmed by empirical observations. Close but not yet.

 

[Chalnoth]

I wish some expert relativist would confirm this, maybe some guy from the relativity forum, as I consider it not exact but that might be due to my poor knowledge of GR. I'll try to think about it some more.

Well, it's really trivial to see that this is true in special relativity. The Lorentz transformations in special relativity are the set of transformations that leave the following matrix unchanged:
$$\begin{array}{rrrr} 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1 \end{array}$$

(This can also be identified as the metric of Minkowski space-time.)

Since the stress-energy tensor transforms between coordinate systems in the same way as the metric, to get a stress-energy tensor that also doesn't change when you perform a Lorentz transform, you need that stress-energy tensor to be proportional to the metric. That is:

$$\begin{array}{rrrr} \rho & 0 & 0 & 0\\ 0 & -\rho & 0 & 0\\ 0 & 0 & -\rho & 0\\ 0 & 0 & 0 & -\rho \end{array}$$

In other words, you need pressure that is equal to the negative of the energy density, a condition which no known matter field satisfies, but which vacuum energy does (some scalar fields get close, but the relationship isn't exact).

We must have some different understanding of isotropy as different coordinates systems can mean rotating the observer point of view and this should be invariant if there is isotropy.

Perhaps I wasn't entirely clear. The point is that the existence of matter ensures that at least some coordinate transformations lead to changes a different-looking universe. Obviously there can still be other symmetries in the universe such that certain particular types of coordinate change may leave everything looking the same. As you mention, isotropy means that rotating your coordinate change has no effect. And homogeneity means that performing a spatial translation on your coordinate system has no effect (for a particular equal-time slicing of the universe).

Only remember this coordinate-dependent spatial homogeneity hasn't been completely confirmed by empirical observations. Close but not yet.

Isotropy has, however, and that is also coordinate-dependent. One need only have a different velocity and the isotropy no longer appears.