How Do FRW Coordinates Transform SR Effects in Cosmology?

[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.