Exploring the Impact of FRW Coordinates on SR Effects in Cosmology

[nutgeb]

Homogeneity just means that observers at different points in the field see the same thing when they look around them, it doesn't imply that each observer will see the visual density as even when they look at different distances from themselves as you seem to be suggesting here.

I disagree. Normal perspective makes faraway objects look smaller than nearby objects. Because of the Pythagorean Theorem, each object's apparent angular size changes in exact proportion to its radial distance.

However, if you look at a field of homogeneously distributed statues through a telephoto lens (such as a telescope), the perspective becomes distorted. The relative angular size of a distant statue is exaggerated compared to a nearby statue, so the radial separation looks compressed as distance increases. This creates the appearance of increasing density with increasing distance.

But we know that this distortion caused by a telescope is merely visual, not physical, so we correct for it in our measurements. We conclude that after correcting for this known visual effect, the distribution of statues is homogeneous. We can test this correction by measuring the radial separations of objects locally with a ruler. This shows us that the distortion we saw was not physical. We conclude that the proper distance separation of objects is physically homogeneous if, and only if, we factor out the telescope's visual distortion of perspective.

If we add spatial curvature (e.g. by performing observations near a black hole), the distortion is not merely visual, it is physical. If we measure positively curved space with rulers, we will measure that the radial separation of distant objects at rest looks compressed relative to nearby objects because it actually is compressed. (Or conversely that that the radial separation between nearby objects has increased a lot while the radial separation between distant objects has increased only a little.) There is no visual distortion occurring here (as long as we don't look through a telescope), there is actual physical distortion.

Let's start with a homogeneous field of statues in flat space at infinite distance from a black hole. Then when we move the statue field nearby the BH, our rulers tell us that a field of statues that was homogeneously distributed when infinitely distant from the BH is no longer homogeneously distributed in the direction radial to the BH. The radial separation, in terms of proper distance, has decreased as a function of distance from (or increased as a function of proximity to) the BH.

In order to restore homogeneity near the BH, we would need to decrease the radial separation between statutes as a function of their proximity to the BH. But then if we later drag our redistributed field of statues far away from the BH, they will no longer be homogeneously distributed.

Changing spatial curvature has the same physical effect on angular size as it does on radial homogeneity. Exactly like changing the focal length of a telescope has the same visual effect on apparent angular size as it does on apparent radial homogeneity.

 

[JesseM]

I disagree.

Are you disagreeing with the fact that this is how "homogeneity" is defined in cosmology? Or are you just arguing you don't think there's a way to reconstruct the "actual" distribution from visual appearances in order to check if it's homogeneous?

If we add spatial curvature (e.g. by performing observations near a black hole), the distortion is not merely visual, it is physical.

Why are you talking about black holes rather than FRW spacetimes? Pretty sure the curvature of space isn't uniform in the case of a black hole spacetime (at least not for a hypersurface of constant t in Schwarzschild coordinates, and I don't think it would be for a surface of constant time in Eddington-Finkelstein or Kruskal-Szekeres coordinates either, at least not judging from the embedding diagrams), it approaches flatness as radial distance approaches infinity (since the black hole spacetime is treated as 'asymptotically flat') but gets more curved as you approach the black hole. In contrast, with a FRW spacetime the idea is that in a "hypersurface of homogeneity" (constant t in comoving coordinates) the spatial curvature is the same everywhere. In the case of positive curvature the 2D analogue is the surface of a sphere, which you can see has the same curvature everywhere, whereas the 2D analogue for a black hole would be something like the embedding diagram shown near the bottom of this page:

 

[nutgeb]

Are you disagreeing with the fact that this is how "homogeneity" is defined in cosmology?

Yes. Homogeneity in cosmology means equal spatial separation when measured locally with rulers, i.e. equal proper distance. (In the simplified example where all objects have the same mass.) Otherwise the mass density would vary as a function of location.

Why are you talking about black holes rather than FRW spacetimes? Pretty sure the curvature of space isn't uniform in the case of a black hole spacetime.

I'm specifically trying to use the non-uniformity of the BH curvature to show a plausible example of changing from less to more spatial curvature and back again by moving the statue field toward and away from the BH. The statue field is presumed to be small enough that the difference in curvature between the inner and outer boundaries is negligible. In a previous post I already used an FRW example to make the same point, but obviously one could object to the notion of 'turning on' and 'turning off' the curvature in an FRW model. As you know, the sign of the curvature can never actually change in an FRW model.

 

[JesseM]

Yes. Homogeneity in cosmology means equal spatial separation when measured locally with rulers, i.e. equal proper distance. (In the simplified example where all objects have the same mass.)

Separation between what and what? The idealized FRW model assumes the universe is filled with a perfectly uniform fluid, not galaxies at discrete locations. Of course this is just intended as an approximation for a real universe where the density of clumps of matter like stars and galaxies becomes increasingly homogeneous as you pick larger and larger regions of space to compare. Anyway, if we assume the distance from one clump to its neighbor is always the same in the comoving frame, then doesn't this imply that the criterion I mentioned, which says that the universe around you will look the same no matter where you are located, would also hold true? Can you come up with any possible ways this criterion could be true but the universe not be homogeneous according to your definition when you look at the "true" distribution in the comoving frame?

Also, I don't understand how your definition above relates to your earlier discussion of a failure of "homogeneity" in a closed unvierse, where you seemed to be saying that a mere visual lack of consistent apparent separation with distance represented a lack of "homogeneity"; such visual distortions don't mean that spatial separation between neighboring stars isn't uniform in the comoving frame. Or did I misunderstand, and you were not actually arguing that these visual distortions contradict homogeneity?

By the way, for some evidence that it is not unusual to define homogeneity in terms of visual appearance at different points in spacetime, here's a discussion from pages 134-135 of Hawking and Ellis' The Large Scale Structure of Space-Time:

In the earliest cosmologies, man placed himself in a commanding position at the centre of the universe. Since the time of Copernicus we have been steadily demoted to a medium sized planet going round a medium sized star on the outer edge of a fairly average galaxy, which is itself simply one of a local group of galaxies. Indeed we are so democratic that we would not claim that our position in space is specially distinguished in any way. We shall, following Bondi (1960), call this assumption the Copernican principle.

A reasonable interpretation of this somewhat vague principle is to understand it as implying that, when viewed on a suitable scale, the universe is approximately spatially homogeneous.

By spatially homogeneous, we mean there is a group of isometries which acts freely on M, and whose surfaces of transitivity are space-like three-surfaces; in other words, any point on one of these surfaces is equivalent to any other point on the same surface. Of course, the universe is not exactly spatially homogeneous; there are local irregularities, such as stars and galaxies. Nevertheless it might seem reasonable to suppose that the universe is spatially homogeneous on a large enough scale.

While one can build mathematical models fulfilling this requirement of homogeneity (see next section), it is difficult to test homogeneity directly by observation, as there is no simply way of measuring the separation between us and distant objects. This difficulty is eased by the fact that we can, in principle, fairly easily observer isotropies in extragalactic observations (i.e. we can see if these observations are the same in different directions, or not), and isotropies are closely connected with homogeneity. These observational investigations of isotropy which have been carried out so far support the conclusion that the universe is approximately spherically symmetric about us.

...

It is possible to write down and examine the metrics of all space-times which are spherically symmetric; particular examples are the Schwarzschild and Reissner-Nordstrom solutions (see 5.5); however these are asymptotically flat spaces. In general, there can exist at most two points in a spherically symmetric space from which the space looks spherically symmetric ... The exceptional cases are those in which the universe is isotropica about every point in space time; so we shall interpret the Copernican principle as stating that the universe is approximately spherically symmetric about every point (since it is approximately spherically symmetric about us).

As has been shown by Walker (1944), exact spherical symmetry about every point would imply that the universe is spatially homogeneous and admits a six-parameter group of isometries whose surfaces of transitivity are spacelike three-surfaces of constant curvature. Such a space is called a Robertson-Walker (or Friedmann) space (Minkowski space, de Sitter space and anti-de Sitter space are all special cases of the general Robertson-Walker spaces).

So they do say that homogeneity can be defined in terms of observations at different points in space, although the definition is a little different from what I remembered; they don't define it in terms of the universe looking the same from every point, but in terms of it looking isotropic everywhere (I was thinking that you could have a universe isotropic everywhere but not necessarily homogenous, like if everyone saw the universe's appearance not varying with angle but they did see different relationships between visual density and distances from themselves, but apparently this is impossible).

I'm specifically trying to use the non-uniformity of the BH curvature to show a plausible example of changing from less to more spatial curvature and back again by moving the statue field toward and away from the BH. In a previous post I already used an FRW example to make the same point, but obviously one could object to the notion of 'turning on' and 'turning off' the curvature in an FRW model. As you know, the sign of the curvature can never actually change in an FRW model.

Since it can't change in an FRW model, then what does this question about black holes have to do with the question of whether observational evidence can favor one type of cosmological curvature over another, or whether it can be consistent or inconsistent with homogeneity? I don't really follow what you're trying to show overall, maybe someone more well-versed in GR can comment, but it might also help if you'd provide a concise outline of whatever it is you're arguing and the logical connections between different elements you've brought up like black hole spacetimes/FRW spacetimes/visual appearances and angular density as a function of distance/supposed difficulties in determining spatial curvature visually.

 

[nutgeb]

Anyway, if we assume the distance from one clump to its neighbor is always the same in the comoving frame, then doesn't this imply that the criterion I mentioned, which says that the universe around you will look the same no matter where you are located, would also hold true?

Your question is too simplistic. I am looking at the two aspects of apparent distortion relevant to the WMAP project, radial homogeneity and angular size, and asking whether one can look distorted while the other does not. If they are flip sides of the same characteristic, then that can't occur. If they are effects that can be manipulated independent of each other, then it can occur.

Also, I don't understand how your definition above relates to your earlier discussion of a failure of "homogeneity" in a closed unvierse, where you seemed to be saying that a mere visual lack of consistent apparent separation with distance represented a lack of "homogeneity"; such visual distortions don't mean that spatial separation between neighboring stars isn't uniform in the comoving frame.

OK, you noticed that I refined the explanation in the later post. At first I described it as a visual effect (because a website suggested that interpretation). But later I realized that it is solely a physical effect; there is no visual distortion (such as the purely visual distortion caused by gravitational lensing). Sorry for the confusion.

Since it can't change in an FRW model, then what does this question about black holes have to do with the question of whether observational evidence can favor one type of cosmological curvature over another, or whether it can be consistent or inconsistent with homogeneity?

Once an FRW model has been 'launched', the sign of its curvature can't change. But we can readily imagine different FRW scenarios that are 'launched' with different curvature signs. We can then predict how that would change observations. The BH example was another mechanism to get to a change in sign. If it isn't helpful, ignore it.

I don't really follow what you're trying to show overall...

The WMAP objective is to try to determine what curvature sign our universe was 'launched' with, and therefore has today. The WMAP technique assumes that the apparent distribution of galaxies can be homogeneous while at the same time the apparent angular sizes are distorted. I'm just trying to parse the FRW metric to understand whether a universe with apparently distorted angular sizes can ever appear to be homogeneous at the same time. I don't see how it can.

 

[JesseM]

Your question is too simplistic. I am looking at the two aspects of apparent distortion relevant to the WMAP project, radial homogeneity and angular size, and asking whether one can look distorted while the other does not. If they are flip sides of the same characteristic, then that can't occur. If they are effects that can be manipulated independent of each other, then it can occur.

OK, but what is supposed to be the significance of distortions in angular size? Again, an outline of the connections between the various topics you've brought up would be helpful.

Once an FRW model has been 'launched', the sign of its curvature can't change. But we can readily imagine different FRW scenarios that are 'launched' with different curvature signs. We can then predict how that would change observations. The BH example was another mechanism to get to a change in sign. If it isn't helpful, ignore it.

It might be helpful if I understood better what specifically you are arguing about WMAP and different FRW universes (or if I understood GR better, of course)--are you saying you think observations from a given point in the universe wouldn't change if we first calculated what an observer would see in a universe with one curvature value, and then calculated what an observer would see in a universe with a different curvature value? If that's not what you're arguing, then why do you think there is a problem using our actual observations to try to determine the curvature?

The WMAP objective is to try to determine what curvature sign our universe was 'launched' with, and therefore has today. The WMAP technique assumes that the apparent distribution of galaxies can be homogeneous while at the same time the apparent angular sizes are distorted. I'm just trying to parse the FRW metric to understand whether a universe with apparently distorted angular sizes can ever appear to be homogeneous at the same time. I don't see how it can.

I'm not clear on how WMAP measurements are used to determine curvature myself, but I thought that what was being measured was supposed to be actual inhomogeneties in the early universe (thought to be due to quantum fluctuations magnified by inflation), not visual distortions of some kind. When they talk about the "angular spectrum" on p. 4 http://www.pma.caltech.edu/Courses/ph12/papers/WMAP.pdf, I think they're looking at spots of different temperatures in the CMBR whose differences in angular size represent actual differences in the size of regions with higher or lower temperatures.

 

[nutgeb]

OK, but what is supposed to be the significance of distortions in angular size?

As I said in earlier posts, the WMAP project is looking for a variance from the angular diameters predicted for the baryon acoustical peaks (which are believed to be governed by the speed of sound through the early universe before recombination), after applying corrections for various known or theoretically predicted visual distortions. If an unaccounted-for variance remains after all the corrections are applied, it will be interpreted as affirmative evidence of physical spatial curvature.

The corrections being applied are very complex, and I can't get into the intricacies.

Instead I am asking one simple question: is it ever possible to observe an apparent distortion of angular size without observing (or correctly predicting, in the absence of definitive observations) an exactly proportional apparent distortion in the radial homogeneity of the matter distribution?

If not (which is the answer I expect), then a finding of unaccounted-for variance in the angular size should be interpreted as either (1) an unexpected kind or degree of visual distortion which also proportionally affects the apparent radial homogeneity, or (2) a variance from homogeneity, which in turn should be interpreted as a violation of the cosmological principle. Conversely, if (as is more likely) apparent radial homogeneity is verified by observations (or is correctly predicted, in the absence of definitive observations), then no unaccounted-for variance in angular size should ever be observed, regardless of the actual spatial curvature.

So if the answer to my question is what I expect, then it would never be valid to interpret an unaccounted-for finding of variance, or a finding that such variance is absent, as affirmative evidence of what the actual spatial curvature is.

That's the essence of what my question is and why it's relevant. This is a repeat of what I've already said.

 

[JesseM]

As I said in earlier posts, the WMAP project is looking for a variance from the angular diameters predicted for the baryon accoustical peaks (which are believed to be governed by the speed of sound through the early universe before recombination), after applying corrections for various known or theoretically predicted visual distortions. If an unaccounted-for variance remains after all the corrections are applied, it will be interpreted as affirmative evidence of physical spatial curvature.

OK, so are you talking about something similar to what's discussed on p. 80 of this book? It's an older book so they're discussing evidence about fluctuations in the CMBR from the BOOMERANG balloon rather than the WMAP satellite, but I'd imagine the principle is basically the same:

The much higher resolution of BOOMERANG compared to COBE has enabled a fundamental test to be made of the nature of the fluctuations. The primordial fluctuations are enhanced by the astrophysics of the early universe on small angular scales, of around a degree. This corresponds to how far a radiation pressure-driven fluctuation propagates in the early universe. This distance is limited by the age of the universe at last scattering, about 300,000 years. This so-called last scattering surface, or the horizon of the universe at last scattering of matter and radiation, has a physical scale of about 30 megaparsecs. The distance of the last scattering surface to us is about 6000 megaparsecs. From this, we infer that the characteristic angular scale is 45 arc-minutes in a flat universe. This enhancement, by about a factor of three, predicted by theory because of the effects of gravity, was measured by BOOMERANG. It constitutes a confirmation of the primordial origin of the fluctuations.

The fundamental result came, however, with the precise determination of the angular scale of the peak. The physical scale associated with the horizon of the universe at last scattering translates on the sky to an angular scale that depends on the curvature of the universe. If the universe is negatively curved, as in a lower density universe, the predicted peak shifts to small angular scales. In effect, the gravity field of the universe acts like a lens.

In fact the peak measured by BOOMERANG corresponds precisely to the expectation for a flat universe. The location of the peak indicates that the density is within a few per cent of the critical value.

This page from NASA's WMAP site has a similar explanation:

Inflation (and subsequent theories that complement it) predicts that the cosmic microwave background will have a series of "bumps" or very specific peaks in temperature fluctuations at very specific angular scales. The biggest bump will be at 1 degree on the sky.

Why 1 degree? First, remember that a degree is a physical distance, about twice the space covered by a full moon, and that inflation makes the Universe flat -- no funky curves and hidden distances. Next, imagine the moment of inflation a mere instant after the Big Bang when the Universe was nearly infinitely dense. Now think of the fog-bound Universe over the next 400,000 years. Inflation sets the conditions for vibrations to wobble through the fog. Protons, attracted by gravity, would roll towards each other like marbles in a ditch. Photons trying to shine create radiation pressure pushing the protons out. This pushing and shoving in the fog creates what are essentially sound waves.

One degree corresponds to the distance a sound wave could travel in 400,000 years in a flat Universe. The temperature (and thus density) differences caused by inflation would be spaced across the sky in one-degree patches.

Picture a triangle. We are measuring one angle. From this we can determine the length "across" this angle on the sky because we know the speed of our sound wave and approximately how long it's been traveling (400,000 years). What we are after is the distance that the microwave radiation has traveled, the distance "out and away" from us, if you will.

A flat Universe determines one distance; a curved universe determine another. If instruments measuring the cosmic microwave background don't see a peak temperature fluctuation at one degree, then inflation is wrong. If the peaks are at two degrees, for example, then space must be curved like a sphere (much like the Earth). Then matter and sound can travel a greater distance in 400,000 years.

So then is your claim that this last paragraph is wrong somehow, and that even for a positively-curved universe we should expect to still see the peak temperature fluctuation at one degree?

Instead I am asking one simple question: is it ever possible to observe an apparent distortion of angular size without observing (or correctly predicting, in the absence of definitive observations) an exactly proportional apparent distortion in the radial homogeneity of the matter distribution?

Can you elaborate on what you mean by "apparent distortion of angular size" and "apparent distortion in the radial homogeneity of the matter distribution"? "Distortion" of the angular size of what--changes in the angular scale of temperature fluctuations in the CMBR? Do you mean the distortions caused by departures from flatness, what the first book I quoted was referring to when it said "If the universe is negatively curved, as in a lower density universe, the predicted peak shifts to small angular scales. In effect, the gravity field of the universe acts like a lens"? And when you say "distortion in radial homogeneity of the matter distribution", are you still talking about the CMBR, or are you talking about matter from more recent times (like galaxies) which lies in the same angular direction in the sky as a given temperature fluctuation in the CMBR?

If not (which is the answer I expect), then a finding of unaccounted-for variance in the angular size should be interpreted as either (1) and unexpected visual distortion

"Unexpected" how? Like a visual distortion that contradicts what general relativity would predict?

or (2) a variance from homogeneity, which in turn should be interpreted as a violation of the cosmological principle.

But the cosmological principle only says that the universe approaches homogeneity on larger and larger scales--it's understood that temperature fluctuations in the CMBR are supposed to be due to real differences in density (compression waves) of the matter/energy filling the universe at the time of last scattering, no? And these differences are supposed to be "seeds" for later structure formation. For example, this page says:

One theory for the origin of these irregularities is that spontaneous fluctuations in the pre-inflationary epoch were greatly magnified by inflation. In the post-inflationary cosmos, these fluctuations produced regions just slightly denser than their surroundings. The differences in density are in turn amplified by gravity, which pulls matter into the denser regions. This process of amplification, cosmologists believe, sowed the "seeds" on which our present-day structures--including the enormous sheets of galaxies--could have formed.

Are you somehow suggesting that the universe at the time of last scattering might actually have been perfectly homogenous, and that the CMBR fluctuations might be a purely visual effect?

 

[nutgeb]

This page from NASA's WMAP site[/url] has a similar explanation:
So then is your claim that this last paragraph is wrong somehow, and that even for a positively-curved universe we should expect to still see the peak temperature fluctuation at one degree?

I'm not claiming anything. I'm suggesting the possibility it is wrong and asking why not.

Can you elaborate on what you mean by "apparent distortion of angular size" and "apparent distortion in the radial homogeneity of the matter distribution"? "Distortion" of the angular size of what--changes in the angular scale of temperature fluctuations in the CMBR?

Yes.

Do you mean the distortions caused by departures from flatness, what the first book I quoted was referring to when it said "If the universe is negatively curved, as in a lower density universe, the predicted peak shifts to small angular scales. In effect, the gravity field of the universe acts like a lens"?

I'm not saying that gravitating matter in the intervening space between the emitter and observer doesn't act like a lens, it does. (I emphasize that the visual lensing is caused directly by transverse gravitational acceleration, not by the spatial curvature.) I'm suggesting that if the apparent angular scale varies from the predicted size, after applying all known and predicted visual corrections, then the apparent radial distribution of matter should vary in the same proportion.

And when you say "distortion in radial homogeneity of the matter distribution", are you still talking about the CMBR, or are you talking about matter from more recent times (like galaxies) which lies in the same angular direction in the sky as a given temperature fluctuation in the CMBR?

The latter.

"Unexpected" how?

Unexpected in the sense that after all known and predicted visual distortion effects (other than spatial curvature of course) are taken into account, a variance remains.

But the cosmological principle only says that the universe approaches homogeneity on larger and larger scales--it's understood that temperature fluctuations in the CMBR are supposed to be due to real differences in density (compression waves) of the matter/energy filling the universe at the time of last scattering, no?

I'm talking about radial homogeneity, not transverse homogeneity.

Are you somehow suggesting that the universe at the time of last scattering might actually have been perfectly homogeneous, and that the CMBR fluctuations might be a purely visual effect?

No.

 

[JesseM]

I'm not saying that gravitating matter in the intervening space between the emitter and observer doesn't act like a lens, it does. (I emphasize that the visual lensing is caused directly by transverse gravitational acceleration, not by the spatial curvature.)

Is it the standard understanding among cosmologists "that the visual lensing is caused directly by transverse gravitational acceleration, not by spatial curvature"? Does what you mean by "transverse gravitational acceleration" depend on inhomogeneities, or would you expect it even in an ideal FRW universe filled with a fluid that has a perfectly uniform density at each moment in comoving coordinates?

I'm suggesting that if the apparent angular scale varies from the predicted size, after applying all known and predicted visual corrections, then the apparent radial distribution of matter should vary in the same proportion.

Can you give a concise summary of why you think this should be true, or quote the post where you explained this?

Unexpected in the sense that after all known and predicted visual distortion effects (other than spatial curvature of course) are taken into account, a variance remains.

Are there any other predicted visual distortion effects besides curvature that would be expected to shift the angular scale of temperature fluctuations in the CMBR from that predicted by flatness? If so, what are they?

I'm talking about radial homogeneity, not transverse homogeneity.

So when you talked about the possibility of "a variance from homogeneity", you were talking about inhomogeneities along the radial direction? But do you mean inhomogeneities in what we see along a radial direction, which would naturally be expected in an expanding universe since density and structure change over time and looking further out means looking further back in time (not to mention we are also looking at different regions of space, so if variations in density at the time of last scattering cause inhomogeneties in the transverse direction they should also cause inhomogeneities in the radial direction), or do you mean inhomogeneities along a radial direction from us in a given surface of constant time in comoving coordinates, or both/neither?

 

[nutgeb]

You ask a lot of questions !

Is it the standard understanding among cosmologists "that the visual lensing is caused directly by transverse gravitational acceleration, not by spatial curvature"?

Yes, as far as I know.

Does what you mean by "transverse gravitational acceleration" depend on inhomogeneities, or would you expect it even in an ideal FRW universe filled with a fluid that has a perfectly uniform density at each moment in comoving coordinates?

Both of the above.

Can you give a concise summary of why you think this should be true, or quote the post where you explained this?

I covered it in the first three posts on this page. I really don't want to repeat it again.

Are there any other predicted visual distortion effects besides curvature that would be expected to shift the angular scale of temperature fluctuations in the CMBR from that predicted by flatness? If so, what are they?

An important visual lensing effect is the one caused by gravitational acceleration by the intervening matter field.

But do you mean inhomogeneities in what we see along a radial direction, which would naturally be expected in an expanding universe since density and structure change over time and looking further out means looking further back in time (not to mention we are also looking at different regions of space, so if variations in density at the time of last scattering cause inhomogeneties in the transverse direction they should also cause inhomogeneities in the radial direction), or do you mean inhomogeneities along a radial direction from us in a given surface of constant time in comoving coordinates, or both/neither?

The former, with the effects you mentioned (and others) having already been corrected for. With the only thing not having been corrected for being the change in angular size and radial separations caused by spatial curvature. As I said, I want to focus on one single question, not on all the other complexities, please!

 

[JesseM]

I covered it in the first three posts on this page. I really don't want to repeat it again.

Those posts were the ones that seemed to conflate visual homogeneity with physical homogeneity in comoving coordinates, though (all that stuff about visual inhomogeneities seen through a telephoto lens and such). For example, in the first post on this page you said:

The apparent angular size of distant features is an example of the seeming inability to directly observe the visual effects of spatial curvature of an apparently homogeneous FRW model. In the 'closed' FRW metric, the 'intrinsic' distribution of comoving features would be radially denser as distance from the origin increases, thereby decreasing their apparent angular size to an observer at any large comoving coordinate.

How could the "intrinsic" distribution vary by distance from the origin? The whole point of the FRW model is that that the distribution is intrinsically perfectly uniform on any surface of constant comoving time.

Maybe I'm just misunderstanding you because I don't have a lot of knowledge of GR or cosmology, and others more well-versed in these subjects would follow your meaning. But no one else is responding to this thread, which might suggest that even those on this forum who are well-versed in this stuff can't really follow your point, in which case it might be of benefit to restate the essentials of your argument in concise form (perhaps starting a new thread to do so).

One other thing:

Is it the standard understanding among cosmologists "that the visual lensing is caused directly by transverse gravitational acceleration, not by spatial curvature"?

Yes, as far as I know.

But then at the end of the post you say:

The former, with the effects you mentioned (and others) having already been corrected for. With the only thing not having been corrected for being the change in angular size and radial separations caused by spatial curvature

You said "the visual lensing" has nothing to do with spatial curvature, so if you're saying that "the change in angular size and radial separations caused by spatial curvature" are things that need to be corrected for, you mean that these visual effects are distinct from what you call "lensing"? The book I quoted earlier seemed to equate changes in angular size with lensing, and say they depended on the curvature of space:

If the universe is negatively curved, as in a lower density universe, the predicted peak shifts to small angular scales. In effect, the gravity field of the universe acts like a lens.

 

[nutgeb]

You are overly fixated on whether apparent distortion of angular size arises from physical or visual effects. I've found that sources often use the term 'gravitational lens' so loosely in this context that you can't figure out what specifically they mean by it. Whether or not curved space itself (e.g., the empty FRW model) acts as a lens, bending light rays that pass through it (which I strongly doubt), it is clearly also a physical effect, because spatial curvature causes physical changes in radial separations and relative angular sizes. Also, visual distortions are excellent analogies for the behavior of physical distortions. That's why the telephoto-wide angle lens analogy is particularly helpful on this subject.

I'll repeat the logic of my argument again, in the most straightforward way:

1. Logic suggests that any observation of apparent variation in angular size (from the predicted size) will be accompanied by an exactly proportional apparent radial inhomogeneity of the matter distribution. This is equally true regardless of whether the cause of the apparent angular distortion is non-homogeneous physical distribution, temporal evolution in the object's location or size, or visual lensing.

2. Any radial inhomogeneity in the large-scale matter distribution is unacceptable, because it would violate the cosmological principle and preclude use of the FRW metric. Therefore, by application of principle #1 above, we will reject as (almost certainly) invalid any methodology that interprets an observed variation in angular size as a physical effect. Instead, the observed variation must be resolved as (a) a measurement error, (b) a flaw in our temporal evolution model, or (c) a visual lensing effect that has not been correctly compensated for. (This principle #2 is stated a bit strongly for my taste, but I want it to be clear.)

3. The variance in angular size caused by spatial curvature is a physical effect.

4. By application of principles #1, #2, and #3, interpretation of observed angular size variation in the CMB as a physical effect must be rejected as invalid. The methodology must be re-evaluated and resolved per #2(a), (b) or (c).

The good news is that the apparent variance in angular size of CMB peaks is very small after all corrections are applied. The bad news is that (according to the above logic), any remaining discrepency cannot be validly interpreted as evidence of spatial curvature.

 

[JesseM]

Logic suggests that any observation of apparent variation in angular size (from the predicted size) will be accompanied by an exactly proportional apparent radial inhomogeneity of the matter distribution.

Do you mean a radial inhomogeneity at each single instant of comoving time, or just a radial inhomogeneity in the matter at all the points in spacetime which the light reaching us now has passed through in the billions of years it took to get from the surface of last scattering to us? If the latter, I still don't understand why you think this conflicts with the FRW model, which would naturally predict that the density of matter is continually decreasing as the universe expands, so the light reaching us now has been passing through successively less dense regions on its long path to reach us. And if it's the former--if you think that somehow a variation in angular size predicted by flatness requires that the universe be radially inhomogeneous at each moment in comoving time (to a degree that would significantly affect the angular size of temperature spots, as opposed to a minor degree)--then can you elaborate on why you think "logic suggests" this?

 

[nutgeb]

Do you mean a radial inhomogeneity at each single instant of comoving time, or just a radial inhomogeneity in the matter at all the points in spacetime which the light reaching us now has passed through in the billions of years it took to get from the surface of last scattering to us? ... And if it's the former--if you think that somehow a variation in angular size predicted by flatness requires that the universe be radially inhomogeneous at each moment in comoving time (to a degree that would significantly affect the angular size of temperature spots, as opposed to a minor degree)--then can you elaborate on why you think "logic suggests" this?

The former. I already explained why! Distortion of angular size and homogeneous spacing are flip sides of the same effect, they are not separate effects. Please read the Wikipedia page on lensing effects, and re-read my post about the field of statues being dragged into and out of a BH's gravitational field. Despite your objections, as I explained the latter is a good analogy for what happens in the FRW metric if the curvature sign is changed.

 

[JesseM]

The former. I already explained why!

In which post? The telephoto lens one? (edit: I see you added some sentences to your post saying you are talking about lensing type effects) It still seems to me like you're conflating visual homogeneity with intrinsic homogeneity there. In that post you say:

Positive spatial curvature distortion (or telephoto distortion) makes an intrinsically homogeneous distribution of objects appear to be increasingly radially compressed, or overdense, as radial distance increases. At some point the radial density will approach infinity, like the north-south visual compression of continents near the equator on a globe viewed from a polar perspective. At the same time, the relative angular size of distant objects compared to nearby objects increases as their radial separation increases. In other words, there is no east-west visual compression of the equatorial continents as viewed from the polar perspective. (There is of course the normal, undistorted perspective effect that makes features near the equator look slightly smaller in both the east-west and north-south directions than features at the pole, because the equator is farther from the observer than the pole is.)

If space is positively curved, evidently the only way to achieve a homogeneous distribution is to distribute the objects in an intrinsically non-homogeneous pattern, with increasing radial separation between objects as a function of distance.

How can a "homogeneous distribution" be the same as an "intrinsically non-homogeneous pattern", unless by "homogeneous distribution" you're talking about visual effects like the telephoto effect? (and if this is what you mean I think this is probably a misuse of terminology, which may be why people are having trouble understanding you) But if that's what you mean, then where are you getting the idea that this type of "visual homogeneity" is in fact seen observationally in cosmology? Isn't the whole point of measuring curvature via the angular size of temperature spots based on looking to see if observationally we get the sort of "telephoto effect" you describe, where we do see the angular size of things being distorted rather than being "visually homogeneous"? The book I quoted earlier seemed to be saying something like this here:

The primordial fluctuations are enhanced by the astrophysics of the early universe on small angular scales, of around a degree. This corresponds to how far a radiation pressure-driven fluctuation propagates in the early universe. This distance is limited by the age of the universe at last scattering, about 300,000 years. This so-called last scattering surface, or the horizon of the universe at last scattering of matter and radiation, has a physical scale of about 30 megaparsecs. The distance of the last scattering surface to us is about 6000 megaparsecs. From this, we infer that the characteristic angular scale is 45 arc-minutes in a flat universe. This enhancement, by about a factor of three, predicted by theory because of the effects of gravity, was measured by BOOMERANG. It constitutes a confirmation of the primordial origin of the fluctuations.

Not sure what the "enhancement" is being measured relative to, perhaps relative to the angular scale we'd expect to see for objects 30 megaparsecs in size that are 6000 megaparsecs away in a flat Minkowski spacetime.

 

[nutgeb]

It still seems to me like you're conflating visual homogeneity with intrinsic homogeneity there. ... How can a "homogeneous distribution" be the same as an "intrinsically non-homogeneous pattern", unless by "homogeneous distribution" you're talking about visual effects like the telephoto effect?

Again, you missed the point. If you read my earlier posts, I used the terminology "intrinsically non-homogeneous pattern" to mean what the pattern would have needed to look like if space had been flat instead of curved. It's not a complicated concept, but it is an abstract one: Imagine an FRW model with flat curvature, and a spatial curvature that is radially non-homogeneous (in a Lorentz-equivalent way). Then "add" spatial curvature (e.g., relaunch the FRW model with the same physical distribution but this time apply spatial curvature). Presto, the formerly non-homogeneous distribution becomes physically homogeneous, as a consequence of curvature alone. Not a telephoto effect.

It's just like the homogeneity of the statue field changes when you drag it toward and away from the BH.

In other words, physical radial homogeneity is not "conserved" over changes in the sign of curvature. That basic concept (without those words) is discussed in my even earlier posts.

I use the telephoto effect as an analogy only because its such a superb way to visualize curvature effects, regardless of whether the curvature is visual or physical in nature.

 

[JesseM]

Again, you missed the point. If you read my earlier posts, I used the terminology "intrinsically non-homogeneous pattern" to mean what the pattern would have needed to look like if space had been flat instead of curved.

Do you mean the visual pattern, or the intrinsic pattern of how objects are distributed? If the latter, "needed to" in order for what to be true?

It's not a complicated concept, but it is an abstract one: Imagine an FRW model with flat curvature, and a spatial curvature that is radially non-homogeneous (in a Lorentz-equivalent way).

How can "flat curvature" be compatible with "a spatial curvature that is radially non-homogeneous"? Perhaps in the first case you are talking about spacetime curvature rather than spatial curvature, if so you need to be more precise in your terminology. And how can an "FRW model" be radially non-homogeneous, if by this you mean the spatial curvature is non-homogeneous on a single spacelike surface of constant comoving time (which was how you define radial inhomogeneity in post 50), since by definition all FRW universes are perfectly homogeneous in any such surface of constant time?

Then "add" spatial curvature (e.g., relaunch the FRW model with the same physical distribution but this time apply spatial curvature).

Didn't you just say the spatial curvature was radially non-homogeneous in the first version, meaning you were already applying spatial curvature? Perhaps you meant to say the matter distribution is radially non-homogeneous in the first version? You really aren't communicating in a way that makes your meaning remotely clear.

Presto, the formerly non-homogeneous distribution becomes physically homogeneous, as a consequence of curvature alone.

Does it? Why?

It's just like the homogeneity of the statue field changes when you drag it toward and away from the BH.

Here you seem to be talking about post 36, but your argument there was very vague as well:

Let's start with a homogeneous field of statues in flat space at infinite distance from a black hole. Then when we move the statue field nearby the BH, our rulers tell us that a field of statues that was homogeneously distributed when infinitely distant from the BH is no longer homogeneously distributed in the direction radial to the BH. The radial separation, in terms of proper distance, has decreased as a function of distance from (or increased as a function of proximity to) the BH.

In order to restore homogeneity near the BH, we would need to decrease the radial separation between statutes as a function of their proximity to the BH. But then if we later drag our redistributed field of statues far away from the BH, they will no longer be homogeneously distributed.

The phrase "then we move the statue field nearby the BH" tells us nothing about how we move them, or why you think that the process of moving them would cause the radial separation as measured by rulers to change (obviously we could intentionally move them in a way that would preserve this measured separation if we chose too). Do you mean that they are moved in such a way that they are all equally far apart in Schwarzschild coordinates or something? Along the same lines, the statement that we "would need to decrease the radial separation between statues as a function of their proximity to the BH" in order to "restore homogeneity" makes little sense on the surface--if you're talking about their separation as measured by rulers, then if the separation changes as a function of distance doesn't that mean the distribution is not homogeneous, by definition? Again, are you talking about changing the separation in a particular coordinate system? If so you really need to refer to it by name. In any case I have no idea what connection this example is supposed to have to the cosmological example and the mysterious statements about "applying spatial curvature".

In other words, physical radial homogeneity is not "conserved" over changes in the sign of curvature.

No idea what this sentence means, and I doubt anyone else reading the thread does either.

I use the telephoto effect as an analogy

I understand you aren't talking literally about telephoto lenses, I was also speaking in a metaphorical way when I wrote Isn't the whole point of measuring curvature via the angular size of temperature spots based on looking to see if observationally we get the sort of "telephoto effect" you describe in the previous post. Basically I just meant the angular size of distant objects (like temperature spots) is different from what you'd expect based on their physical size and distance.

 

[nutgeb]

"Needed to" in order for what to be true?

Needed to avoid a violation of the cosmological principle.

How can "flat curvature" be compatible with "a spatial curvature that is radially non-homogeneous"? Perhaps in the first case you are talking about spacetime curvature rather than spatial curvature...

I don't mean that.

And how can an "FRW model" be radially non-homogeneous, if by this you mean the spatial curvature is non-homogeneous on a single spacelike surface of constant comoving time (which was how you define radial inhomogeneity in post 50), since by definition all FRW universes are perfectly homogeneous in any such surface of constant time?

We've been over that ground before. That's why I introduced the example with the BH and statue field. I explained the difficulty with changing sign and introducing inhomogeneity in the FRW model before you ever mentioned it, so please don't lecture me about it!

Didn't you just say the spatial curvature was radially non-homogeneous in the first version, meaning you were already applying spatial curvature? Perhaps you meant to say the matter distribution is radially non-homogeneous in the first version?

Yes that's what I meant. Excuse me for a typo, but thank goodness it was obvious what I really meant.

Does it? Why?

A matter distribution that is Lorentz contracted will look homogeneous in negative space, etc., etc., as I've said repeatedly in earlier posts. If you need to learn more about this subject, read my earlier posts.

The phrase "then we move the statue field nearby the BH" tells us nothing about how we move them, or why you think that the process of moving them would cause the radial separation as measured by rulers to change (obviously we could intentionally move them in a way that would preserve this measured separation if we chose too).

There's nothing complicated about my statement. The statues are all attached to an inflexible ground plane, as you would expect statues to be. You attach tethers to the ground plane and pull the whole structure close to the BH, and then bring it to a halt. I didn't do anything to the distribution of individual statues -- their physical separation changed simply because the spatial curvature increased. It would make no sense to construct this hypothetical any other way.

Along the same lines, the statement that we "would need to decrease the radial separation between statues as a function of their proximity to the BH" in order to "restore homogeneity" makes little sense on the surface--if you're talking about their separation as measured by rulers, then if the separation changes as a function of distance doesn't that mean the distribution is not homogeneous, by definition?

Of course, that's my point. The distribution was homogeneous, then we towed the statue field close to the BH, causing the distribution of individual statues to become radially inhomogeneous. As measured by rulers!

Again, are you talking about changing the separation in a particular coordinate system? If so you really need to refer to it by name.

I already said in an earlier post that I was talking in terms of proper distance coordinates.

No idea what this sentence means, and I doubt anyone else reading the thread does either.

The sentence means exactly what it says. In the BH statue field example, when the sign of spatial curvature changes from 0 (flat) to positive (because we drag the statute field close to the BH), the formerly homogeneous distribution of the statues becomes inhomogeneous. Thus homogeneity is not conserved when the sign of curvature changes.

 

[JesseM]

Needed to avoid a violation of the cosmological principle.

Doesn't the cosmological principle say that the intrinsic distribution of matter at a given moment of comoving time should be uniform? But putting the pieces together, you're saying an "intrinsically non-homogeneous pattern" means what the intrinsic distribution of objects would need to look like if space were flat rather than curved, in order not to violate the cosmological principle. Still doesn't make any sense, if space were flat rather than curved then in order not to violate the cosmological principle it would have to be distributed in a homogenous way on a given surface of constant comoving time, not an "intrinsically non-homogenous" way (edit: also see the very last sentence of this post about there being no single correct way to map points in one spacetime to points in another).

How can "flat curvature" be compatible with "a spatial curvature that is radially non-homogeneous"? Perhaps in the first case you are talking about spacetime curvature rather than spatial curvature...

I don't mean that.

OK, so according to your clarification below you meant to say "a matter distribution that is radially non-homogeneous", so presumably "flat curvature" just referred to spatial flatness. Well, how could flat space be compatible with a non-homogeneous distribution of matter? The only ways I can think of to make sense of this are either to assume we are talking about SR rather than GR where matter has no effect on the curvature of spacetime, or to assume we are talking about a field of statues whose masses are negligible, in the context of a spacetime that is curved as in the flat FRW model by a uniform fluid (separate from the statues) filling spacetime. If you can imagine some third alternative that does not require us to invent new laws of physics different than either GR or SR, please explain.

And how can an "FRW model" be radially non-homogeneous, if by this you mean the spatial curvature is non-homogeneous on a single spacelike surface of constant comoving time (which was how you define radial inhomogeneity in post 50), since by definition all FRW universes are perfectly homogeneous in any such surface of constant time?

We've been over that ground before. That's why I introduced the example with the BH and statue field.

But the example of the BH, which presumably is meant to work within the laws of GR, sheds no light on how I'm supposed to make sense of a scenario that seems to be blatantly incompatible with GR.

I explained the difficulty with changing sign and introducing inhomogeneity in the FRW model before you ever mentioned it, so please don't lecture me about it!

Merely pointing out that you understand that your scenario doesn't make sense in GR doesn't help me to make sense of it. It's a little like those questions people sometimes ask about what you would see if you accelerated to the speed of the light, where the only answer one can really give is "your premise is impossible in SR, so it wouldn't be possible to answer this question without inventing a new theory to supplant it".

A matter distribution that is Lorentz contracted will look homogeneous in negative space, etc., etc., as I've said repeatedly in earlier posts. If you need to learn more about this subject, read my earlier posts.

When referring to earlier posts it would help if you would actually tell me which post to look at, since this is a long thread. I very much doubt that your earlier explanations would make sense reading them again when they didn't make sense the first time, just as your statues-near-a-black-hole example still doesn't make any sense and therefore doesn't shed light on your cosmological scenario.

There's nothing complicated about my statement. The statues are all attached to an inflexible ground plane, as you would expect statues to be. You attach tethers to the ground plane and pull the whole structure close to the BH, and then bring it to a halt.

This is totally meaningless, you can't have rigid extended bodies in GR. You need to actually specify how the plane behaves as it is moved from one area of curved spacetime to another. Of course if you choose a spacelike path between nearby statues that lies entirely in one surface of simultaneity in Schwarzschild coordinates (or whatever coordinate system you prefer), and entirely in the radial direction, then as with all paths through spacetime the metric will give you an objective definition of this path's length. So, I suppose you could move the statues so that this distance remains constant (I believe this is also the distance you'd get if you took a bunch of rulers so tiny that spacetime curvature in each ruler's neighborhood was negligible, lined them all up end-to-end at constant Schwarzschild radii, and measured the distance between statues this way). Alternately, as I said earlier, you could move them so that their coordinate distance remains constant in some coordinate system like Schwarzschild coordinates. But if you don't want either of these then you need to give a definition of what you do mean that is actually compatible with GR.

I didn't do anything to the distribution of individual statues -- their physical separation changed simply because the spatial curvature increased.

What does "physical separation" mean in this case? Does it refer to either of the two ways of defining the distance between statues that I mentioned above? If not then this, too, needs to be defined.

Of course, that's my point. The distribution was homogeneous, then we towed the statue field close to the BH, causing the distribution of individual statues to become radially inhomogeneous. As measured by rulers!

Again, your notion of the statues being "towed" by a rigid platform is just not possible in GR.

Again, are you talking about changing the separation in a particular coordinate system? If so you really need to refer to it by name.

I already said in an earlier post that I was talking in terms of proper distance coordinates.

"Proper distance" is a term I've usually seen in SR, used to refer to the distance between two events with a spacelike separation in the inertial frame where the events are simultaneous. For obvious reasons this doesn't make sense in the context of curved spacetimes. "Proper distance" is also sometimes used for comoving distance in cosmology (or maybe comoving distance with a scale factor applied), but this doesn't make sense in a black hole spacetime either. I would guess you probably mean something like the coordinate-invariant distance along a spacelike path in this spacetime, but in this case the details of what path you want to use are important--as I suggested earlier, you might use a path that lies entirely in a surface of constant t in Schwarzschild coordinates, and also entirely along the radial direction.

The sentence means exactly what it says. In the BH statue field example, when the sign of spatial curvature changes from 0 (flat) to positive (because we drag the statute field close to the BH), the formerly homogeneous distribution of the statues becomes inhomogeneous.

You haven't properly specified what physical process is implied by that word "becomes" though--your notion of dragging them towards the BH is ill-defined. In any case, here you are clearly talking about some physical process of moving objects within a GR context, so it's not at all clear what this has to do with the cosmological scenario where you seem to be imagining suddenly changing the laws of physics in some way so that mass goes from not curving spacetime to curving it. It's also a totally ill-defined question if you're asking where a statue at a given point in flat space would be if you abruptly "turned on" the curvature of space, since here we are talking about two different GR spacetimes and there's no natural way to map points in one to points in the other.

 

[nutgeb]

Jesse, some of your early comments helped me clarify my thoughts, and I want to express my appreciation for that. Now let's agree to disagree.

 

[JesseM]

Jesse, some of your early comments helped me clarify my thoughts, and I want to express my appreciation for that. Now let's agree to disagree.

OK, but if you want to actually learn about cosmology in the context of general relativity, I'd think you'd want to know when some of your assumptions (like extended rigid bodies, or 'turning on' curvature and seeing where objects in formerly flat space are now located) are blatantly incompatible with the theory of GR itself.