How Do FRW Coordinates Transform SR Effects in Cosmology?

[nutgeb]

In hypothetical empty space, Bob stays home and Alice flies radially away from home at .9c. In the flat Minkowski coordinates, each determines that the other's clock is running slower than their own. Each determines that the radial distance between them is Lorentz-contracted.

Now consider the same scenario, except we change to expanding space in FRW coordinates, with vanishingly small mass density. Now the space has negative (hyperbolic) curvature. Let's say that Alice is very far away, her large recession velocity happens to be exactly comoving with local fundamental observers in the Hubble flow. Each will determine that the other's clock is running at the same rate as their own, i.e. at cosmological proper time. Each determines that the radial proper distance between them is not Lorentz-contracted.

It appears to me that the effect of transforming the scenario from Minkowski to expanding FRW coordinates is to: (a) change from flat space to hyperbolic spatial curvature; (b) eliminate Lorentz contraction of the space between them (by exactly offsetting it with hyperbolic spatial curvature), and (c) eliminate time dilation between them (by hyperbolically stretching the time "axis".

Mathematically, it appears that the difference between Minkowski and FRW coordinates is that the FRW metric exactly transforms away (or offsets) the effects of SR as between fundamental comovers. Presumably that transformation can be easily demonstrated mathematically, but I haven't done it yet. Anyone know of such a mathematical demonstration?

 

[George Jones]

In hypothetical empty space, Bob stays home and Alice flies radially away from home at .9c. In the flat Minkowski coordinates, each determines that the other's clock is running slower than their own. Each determines that the radial distance between them is Lorentz-contracted.

Now consider the same scenario, except we change to expanding space in FRW coordinates, with vanishingly small mass density. Now the space has negative (hyperbolic) curvature. Let's say that Alice is very far away, her large recession velocity happens to be exactly comoving with local fundamental observers in the Hubble flow. Each will determine that the other's clock is running at the same rate as their own, i.e. at cosmological proper time. Each determines that the radial proper distance between them is not Lorentz-contracted.

Because the above doesn't contain any mathematics, I'm not sure what to make of it. For me, words help to explain mathematics, but, also, mathematics helps to explain words.

Setting aside the question of cosmological redshift, it appears to me that the effect of transforming the scenario from Minkowski to expanding FRW coordinates is to: (a) change from flat space to hyperbolic spatial curvature;

Yes.

(b) eliminate Lorentz contraction of the space between them (by exactly offsetting it with hyperbolic spatial curvature), and (c) eliminate time dilation between them (by hyperbolically stretching the time "axis".

Not sure.

Mathematically, it appears that the difference between Minkowski and FRW coordinates is that the FRW metric exactly transforms away (or offsets) the effects of SR as between fundamental comovers. Presumably that transformation can be easily demonstrated mathematically, but I haven't done it yet. Anyone know of such a mathematical demonstration?

Again, I'm not sure, but do you mean this transformation,

https://www.physicsforums.com/showthread.php?p=1757634#post1757634?

 

[nutgeb]

Oops, now that I wrote the post, I can see I got part of it wrong. The empty FRW metric exactly compensates for the SR time dilation of fundamental comovers, but it overcompensates for the SR spatial Lorentz contraction. The negatively curved space of the empty FRW model is "Lorentz dilated", in that the radius of a sphere is lengthened compared to its circumference (i.e., radial distances are dilated). This is a normal attribute of negatively curved space.

I expect the change in spatial curvature arising from transforming from the Minkowski metric to the empty FRW metric is exactly equal to the square of the Lorentz contraction that SR would otherwise imply for fundamental comovers in the empty FRW model.

I think this is straightforward math, but I haven't seen it done.

 

[nutgeb]

For me, words help to explain mathematics, but, also, mathematics helps to explain words.

Agreed, that's why I'm asking whether anyone has seen this math done.

Again, I'm not sure, but do you mean this transformation,
https://www.physicsforums.com/showthread.php?p=1757634#post1757634?

Yes, I think so.

 

[George Jones]

I just ran into the American Journal of Physics paper "Interpretation of the Cosmological Metric,"

http://arxiv.org/abs/0803.2701.

From its abstract:

"We present a particular Robertson–Walker metric (an empty universe metric) for which a coordinate transformation shows that none of these interpretation necessarily holds."

 

[nutgeb]

I just ran into the American Journal of Physics paper "Interpretation of the Cosmological Metric,"

From its abstract:

"We present a particular Robertson–Walker metric (an empty universe metric) for which a coordinate transformation shows that none of these interpretation necessarily holds."

Thanks George, I had read that article a while ago but forgot about it. It's helpful, especially because it demonstrates so much math.

Unlike some other sources, they seem comfortable with the idea that SR can be applied (with appropriate adjustments) in a universe with non-zero gravitational density. I think that's likely to be true.

Sadly, here's yet another paper which asserts that the SR Doppler shift (perhaps together with GR time dilation) should integrate over an infinite series of reference frames to equal the cosmological redshift, but is unable to provide a mathematical proof (beyond the infinitesimal local frame). I don't understand why the mathematics and heuristics for translating between SR and FRW frames remain incomplete after so many years of study. The equations are not all that complicated.

If SR and FRW in fact provide different but completely covariant interpretations of the same "real" observations, then why is the FRW version generally treated as the "correct" portrayal while the SR version is so often said to be "wrong" or "inapplicable"?

 

[JesseM]

A while ago on this thread pervect gave me a link to this set of quiz solutions by Alan Guth where he talks about how an expanding empty "Milne universe" (which I gather is just the limit of a FRW universe as density goes to zero) is equivalent to Minkowski spacetime:

The Milne model describes an empty universe: it is open (k = −1) and has no matter or radiation in it (\rho = 0). Its scale factor grows linearly with time, since there's no matter to slow down the Hubble expansion. (One normally includes "test" particles in the description of the Milne universe, so that we can talk about their motion. But the mass of these test particles is taken to be arbitrarily small, so we completely ignore any gravitational eld that they might produce.)

As an interesting aside, we might ask why the Milne model has k = −1. Since there is no matter, there shouldn't be any general relativity effects, and so we would ordinarily expect that the metric should be the normal, flat, Minkowski special relativity metric. Why is this space hyperbolic instead?

The answer is an illustration of the subtleties that can arise in changing coordinate systems. In fact, the metric of the Milne universe can be viewed as either a flat, Minkowski metric, or as the negatively curved metric of an open universe, depending on what coordinate system one uses. If one uses coordinates for time and space as they would be measured by a single inertial observer, then one finds a Minkowski metric; in this way of describing the model, it is clear that special relativity is sufficient, and general relativity plays no role. In this coordinate system all the test particles start at the origin at time t = 0, and they move outward from the origin at speeds ranging from zero, up to (but not including) the speed of light.

On the other hand, we can describe the same universe in a way that treats all the test particles on an equal footing. In this description we define time not as it would be measured by a single observer, but instead we define the time at each location as the time that would be measured by observers riding with the test particles at that location. This definition is what we have been calling "cosmic time" in our description of cosmology. One can also introduce a comoving spatial coordinate system that expands with the motion of the particles. With a particular definition of these spatial coordinates, one can show that the metric is precisely that of an open Robertson-Walker universe with R(t) = t.

The derivation is left as an exercise for the curious student. You should find that the normal special-relativistic time dilation and Lorentz contraction formulas, when applied to the velocities of a Hubble expansion to construct the comoving coordinate system, introduce the negative curvature to the metric.

 

[sylas]

If SR and FRW in fact provide different but completely covariant interpretations of the same "real" observations, then why is the FRW version generally treated as the "correct" portrayal while the SR version is so often said to be "wrong" or "inapplicable"?

The paper repeatedly emphasizes that there's one case where the FRW model corresponds to an SR interpretation -- the empty universe model -- and it repeatedly points out that case is inapplicable to the actual universe we live in.

Or that's what it looks like to me...

 

[nutgeb]

A while ago on this thread pervect gave me a link to this set of quiz solutions by Alan Guth where he talks about how an expanding empty "Milne universe" (which I gather is just the limit of a FRW universe as density goes to zero) is equivalent to Minkowski spacetime:

Thanks for the quote Jesse.

Guth makes the same point as Cook & Burns and others, which is that spatial curvature is inherently flat from the perspective of the "rigid observer" at rest at the Milne origin, while the curvature is inherently negative when viewed from the frame of all fundamental comovers in the Milne expansion. I have two conceptual problems with this distinction.

First, the "rigid observer" is in fact just one of the fundamental comovers. Test particles depart from the Milne origin at every speed from zero to (approaching) c. The rigid observer is simply that particular fundamental comover whose recession speed happens to be zero. In which case, how can it be meaningful to say that his reference frame is different from that of all the other comovers? That he sees Lorentz contraction and time dilation which none of the other comovers does?

Second, since all of the comovers see various other comovers moving away from them at a full range of different recession velocities, how is it meaningful to say that there is a single, unique "comoving frame" in this SR model in which none of the comovers observe each other to be Lorentz-contracted or time-dilated? It is impossible in SR for all observers to consider themselves to all be both at rest and in relativistic motion relative to each other, all within a single shared reference frame. From an SR perspective, the comoving Milne reference frame seems physically absurd and fictional. From an FRW perspective, one might claim the opposite, but none of us has ever had the privilege of experiencing an FRW comoving frame in our quasi-local neighborhood, where we can actually exchange light signals and test the synchronization of comoving clocks. It seems impossible that we could ever construct a properly functional "clockwork" toy model of a shared comoving Milne frame in the physical world accessible to us, even with maximum recession velocities << c. (Even setting gravitational issues aside.) The practical problem is that we need to arrange for some negatively curved space in order to conduct a physical experiment in it.

At the end of the day, it is the fortuitous imposition of negative spatial curvature in empty space that enables a shared comoving reference frame; not the choice of a particular metric or reference frame per se. The negative curvature of space exactly offsets the effects of Lorentz-contraction and time dilation, since expansion velocity is defined to be a hyperbolic function of distance in all homogeneous expanding models. And somewhat counterintuitively, starting from a foundation of negative spatial curvature in empty space seems to be what enables us to avoid time dilation among comovers even in the gravitating FRW model at critical density, even though gravity causes the spatial curvature itself to flatten out.

 

[atyy]

http://books.google.com/books?id=LkEhsgmP4vEC&printsec=frontcover#PPA360,M1

 

[JesseM]

Thanks for the quote Jesse.

Guth points to the same point as Cook & Burns and others, which is that spatial curvature is inherently flat from the perspective of the "rigid observer" at rest at the Milne origin, while the curvature is inherently negative when viewed from the frame of all fundamental comovers in the Milne expansion.

However, in post #71 of that thread George Jones also said that although the spatial curvature can be negative or flat depending on the choice of coordinate systems, the spacetime curvature is zero in both cases:

Milne's universe is just an interesting coordinate system on a proper subset of Minkowski spacetime that"splits" the subset into time and space. In this splitting, the spatial curvature for space at any "instant" of time is negative, but spacetime curvature is zero, as it must be. It is impossible to transform zero spacetime curvature into non-zero spacetime curvature by a change of coordinates.

First, the "rigid observer" is in fact just one of the fundamental comovers. Test particles depart from the Milne origin at every speed from zero to (approaching) c. The rigid observer is simply that particular fundamental comover whose recession speed happens to be zero. In which case, how can it be meaningful to say that his reference frame is different from that of all the other comovers?

Where did you get that from Guth's quote? Pretty sure he was saying that for any of the inertial test particles, you can define an inertial frame in which that particle is at rest and the metric is just the Minkowski metric.

Second, since all of the comovers see various other comovers moving away from them at a full range of different recession velocities, how is it meaningful to say that there is a single, unique "comoving frame" in this SR model in which none of the comovers observe each other to be Lorentz-contracted or time-dilated? It is impossible in SR for all observers to consider themselves to all be both at rest and in relativistic motion relative to each other, all within a single shared reference frame.

In an inertial frame this is impossible, but there's no law that says you can't use non-inertial coordinate systems in SR (see discussions here and here for example), as long as you understand that the equations for the laws of physics will look different in this frame than they do in inertial frames. The "comoving frame" seems to be a non-inertial coordinate system that is specifically defined so that each particle has a constant position coordinate, and so that at any given time coordinate each particle's proper time will be the same (with the zero of the proper time being when they were all at the initial position that they then moved away from in different directions)

 

[nutgeb]

Where did you get that from Guth's quote? Pretty sure he was saying that for any of the inertial test particles, you can define an inertial frame in which that particle is at rest and the metric is just the Minkowski metric.

Your missing my point. Guth draws a distinction between a frame that's valid for the single observer, and another frame that's valid for all comoving observers. He implies (and most authors state explicitly) that the single observer frame is centered on the origin. This is done for ease of analysis. My point is simply that the observer at the origin is indistinguishable from all other comoving observers. The choice of which comover will be placed at the coordinate origin is entirely arbitrary; each comover probably considers themselves to be the origin in their private coordinate system.

In an inertial frame this is impossible, but there's no law that says you can't use non-inertial coordinate systems in SR (see discussions here and here for example), as long as you understand that the equations for the laws of physics will look different in this frame than they do in inertial frames. The "comoving frame" seems to be a non-inertial coordinate system that is specifically defined so that each particle has a constant position coordinate, and so that at any given time coordinate each particle's proper time will be the same (with the zero of the proper time being when they were all at the initial position that they then moved away from in different directions)

Yes, as I said, it is specifically the imposition of a foundation of negative curvature for empty space which enables a single shared reference frame for all comovers in any homogeneous expanding metric. Without that foundation, it is also impossible to model a single shared reference frame for all comovers in a flat gravitating matter-only FRW model. Otherwise SR time dilation as between comovers would be inevitable.

That's why we can never construct a clockwork toy model of any homogeneous matter distribution that functions with a shared reference frame for comovers. Starting with empty flat space and adding gravitating particles won't work. You need to start with negatively curved empty space and then add gravitating particles.

Edit: Actually the recipe for the universe construction project begins with flat, static, empty space. Comovers are stationary, so there are no Lorentz effects. Then you add expansion motion, which introduces Lorentz effects between homogeneous nongravitating comovers. You then eliminate the Lorentz effects by adding negative curvature. (Note that it this stage, Lorentz transformations occur only with respect to peculiar velocities, not with respect to proper velocities per se.) Then you add gravity to the homogeneous comovers, which flattens out the spatial curvature but does not cause gravitational time dilation. Bake in CMB oven for 13.7 Gy and serve cold.

It does seem artificial, however, to just stir in some negative curvature as if it is an inevitable consequence of adding expansion motion. One can't just "choose" to have spatial curvature because one finds it convenient. Perhaps the "natural" condition of empty space is negative curvature, even if there is no expansion motion.

 

[JesseM]

Your missing my point. Guth draws a distinction between a frame that's valid for the single observer,

What do you mean "valid"? Guth didn't use that word. If by "valid" you just mean "doesn't lead to any incorrect predictions about coordinate-invariant physical facts", then all coordinate systems are equally valid, even non-inertial ones (provided you adjust the equations for the laws of physics to fit the non-inertial coordinate system, and don't mistakenly think you can still use the same equations from inertial frames).

He implies (and most authors state explicitly) that the single observer frame is centered on the origin.

Since all the observers move inertially, all will remain centered on the origin (i.e. the position where all the observers were at the start before they departed in different directions) in their own inertial rest frame. So I don't know what you mean by "the single observer frame", Guth never suggests that any particular observer be singled out as special, he's saying you can define an inertial frame for any single observer.

It does seem artificial to just stir in some negative curvature as if it is an inevitable consequence of adding expansion motion. One can't just "choose" to have spatial curvature because one finds it convenient. Perhaps the "natural" condition of empty space is negative curvature, even if there is no expansion motion.

Did you read the quote from George Jones I posted? I'm pretty sure he's saying that "spatial curvature" is a totally coordinate-dependent notion, it has no more objective physical reality than simultaneity (both depend on exactly how you choose to slice up 4D spacetime into a stack of 3D slices). It's only spacetime curvature which is genuinely physical, I think.

 

[nutgeb]

What do you mean "valid"? Guth didn't use that word. If by "valid" you just mean "doesn't lead to any incorrect predictions about coordinate-invariant physical facts", then all coordinate systems are equally valid, even non-inertial ones (provided you adjust the equations for the laws of physics to fit the non-inertial coordinate system, and don't mistakenly think you can still use the same equations from inertial frames).

Since all the observers move inertially, all will remain centered on the origin (i.e. the position where all the observers were at the start before they departed in different directions) in their own inertial rest frame. So I don't know what you mean by "the single observer frame", Guth never suggests that any particular observer be singled out as special, he's saying you can define an inertial frame for any single observer.

Well it isn't really clear what Guth is saying about the single observer. Some other authors make the distinction more clearly than this brief passage from him does. Anyway, it seems that you and I are in agreement that the origin observer is also a comover, so let's not argue.

Did you read the quote from George Jones I posted? I'm pretty sure he's saying that "spatial curvature" is a totally coordinate-dependent notion, it has no more objective physical reality than simultaneity (both depend on exactly how you choose to slice up 4D spacetime into a stack of 3D slices). It's only spacetime curvature which is genuinely physical, I think.

Yes I read everything.

Does it matter whether spatial curvature is "real", whatever that means?

I think it's circular to argue that spatial curvature is coordinate dependent. One could just as well turn that statement around and argue that the mechanics of particular coordinate systems are dependent on the specific kind of underlying spatial curvature they assume. For example the empty FRW metric assumes that empty space has underlying negative curvature. FRW can't assume anything else for empty space, or it couldn't accurately model homogeneity. In the Milne metric, one can assume either flat or negatively curved space. One must assume negatively curved space in order to achieve homogeneity. That automatically requires the Milne metric to mathematically become the empty FRW metric.

As I understand it, the RW metric is the one and only solution for dynamic homogeneous, isotropic space, with or without regard to the Friedmann equations and the Einstein Field Equations. If the homogeneity we observe is "real", then we cannot generate mathematical predictions that are consistent with cosmological observations if we try to use a metric that specifies a different spatial curvature than FRW does. Maybe homogeneity too isn't real, and it's just the result of a random choice of metric. But if so, then when we observe the cosmos, why are our eyes attuned the predictions of one random metric and not to the others?

 

[JesseM]

Yes I read everything.

Does it matter whether spatial curvature is "real", whatever that means?

It matters in relativity to distinguish coordinate-dependent quantities from coordinate-independent ones.

I think it's circular to argue that spatial curvature is coordinate dependent.

Do you think it's circular to argue that simultaneity or velocity or the rate a clock is ticking are coordinate dependent?

One could just as well turn that statement around and argue that the mechanics of particular coordinate systems are dependent on the specific kind of underlying spatial curvature they assume.

I don't understand what you mean by the "mechanics" of a coordinate system. And would you say the mechanics of a coordinate system are dependent on the specific ways they assign simultaneity to distant events, or the specific ways they assign velocities to different objects?

For example the empty FRW metric assumes that empty space has underlying negative curvature. FRW can't assume anything else for empty space, or it couldn't accurately model homogeneity. In the Milne metric, one can assume either flat or negatively curved space. One must assume negatively curved space in order to achieve homogeneity. That automatically requires the Milne metric to mathematically become the empty FRW metric.

As I understand it, the RW metric is the one and only solution for dynamic homogeneous, isotropic space, with or without regard to the Friedmann equations and the Einstein Field Equations. If the homogeneity we observe is "real", then we cannot generate mathematical predictions that are consistent with cosmological observations if we try to use a metric that specifies a different spatial curvature than FRW does. Maybe homogeneity too isn't real, and it's just the result of a random choice of metric. But if so, then when we observe the cosmos, why are our eyes attuned the predictions of one random metric and not to the others?

Are you not distinguishing between the geometry of spacetime and way the metric is expressed in a particular coordinate system on that spacetime geometry? As an analogy, if we're dealing with curved 2D surfaces the curvature can also be described entirely by a metric tailored to a particular coordinate system drawn on that surface, so if we pick a 2D surface such as a sphere there are an infinite number of possible coordinate systems that could be placed on the sphere and thus an infinite number of possible ways of writing down the metric, one for each coordinate system. But each of these metrics would define the same unique geometry (you could use the metric to determine the geometric length of every possible path on the sphere, and if you translate any given path into each coordinate system, the metric for that coordinate system will give the correct length), so I'm pretty sure they form a sort of equivalence class, and they'd all be different from the equivalence class of metrics associated with different coordinate systems on some other 2D surface like an ovoid.

I believe it's exactly the same with 4D spacetime (though harder to visualize!) For any given metric, it should be a member of an equivalence class of the metrics for all possible coordinate systems on a given spacetime geometry, all of which are different from metrics on any other spacetime geometry. So it may well be true that the geometries defined by the FRW metric have some unique properties, like the property that it is possible to foliate them into a stack of spacelike surfaces such that for any given surface, what is seen by observers at each event on that surface would be identical everywhere (homogeneous and isotropic). So certainly if you are given such a geometry, it is most "natural" to use a coordinate system where each such surface of homogeneity and isotropy is also a surface of constant time coordinate, and where the density defined in terms of volumes of space using the space coordinate is uniform in each spacelike surface; I think this would give you the coordinate system assumed in the FRW metric. But there'd be nothing stopping you from defining a different type of coordinate system on the same spacetime, which would mean the form of the metric would look differently expressed in this coordinate system, but it would still be the same spacetime geometry, and all coordinate-independent statements about what is seen by observers would remain the same (like the fact that for any event on an observer's worldline, you can find a spacelike surface including that event such that any other observer in the surface would see the same thing in all directions, even if this surface is not a surface of constant t relative to the coordinate system).

Also, although there may be a unique choice of coordinate system for any FRW universe with a nonzero mass/energy density such that all observers in a surface of constant t see the same thing, wouldn't this break down when you reach the point of exactly zero density? After all, observers in a surface of constant t in some other type of coordinate system like an inertial frame will also see exactly the same thing (total emptiness in all directions).

By the same token, according to Birkhoff's Theorem, the Schwarzschild metric is the one and only solution for a spherically symmetrical mass distribution in flat background spacetime.

But the spacetime geometry is exactly the same in some other coordinate system where the metric is expressed differently, like Eddington-Finkelstein coordinates or Kruskal-Szekeres coordinates. If you use the corresponding metric for any of these coordinate systems, you'll always get exactly the same predictions about what is seen by physical observers in this spacetime.

One can always specify an arbitrary change in the geodesic, the rest frame, the location of the coordinate origin, the units of measurement, etc., but one is still required to apply a particular metric in order to make predictions that are consistent with observations.

That's only true if you are using "metric" to refer to the spacetime geometry rather than to the way a metric is actually written down relative to a particular coordinate system on that spacetime.

 

[Vanadium 50]

Each will determine that the other's clock is running at the same rate as their own, i.e. at cosmological proper time.

How exactly do they do this? This is a key question in deciding what exactly they infer about each others clocks.

 

[nutgeb]

Do you think it's circular to argue that simultaneity or velocity or the rate a clock is ticking are coordinate dependent?

Well I don't see value in arguing about it, but I would say that one could just as well rephrase the question as: In what circumstances does a particular metric inherently mandate time dilation and lack of simultaneity between reference frames?

Are you not distinguishing between the geometry of spacetime and way the metric is expressed in a particular coordinate system on that spacetime geometry? ... For any given metric, it should be a member of an equivalence class of the metrics for all possible coordinate systems on a given spacetime geometry, all of which are different from metrics on any other spacetime geometry. ... But there'd be nothing stopping you from defining a different type of coordinate system on the same spacetime, which would mean the form of the metric would look differently expressed in this coordinate system, but it would still be the same spacetime geometry, and all coordinate-independent statements about what is seen by observers would remain the same (like the fact that for any event on an observer's worldline, you can find a spacelike surface including that event such that any other observer in the surface would see the same thing in all directions, even if this surface is not a surface of constant t relative to the coordinate system).

Yes you've put your finger on exactly the distinction I was making. I think at any effect (e.g. spatial curvature or time dilation) which is mandated by a particular metric (or class of equivalent metrics) in a particular scenario is "real" in the sense that we don't have to option to avoid that effect by arbitrarily selecting a different, non-equivalent metric. But as you say we are free to adopt any reasonable coordinate system for writing down the required metric, and we should expect the metric's predictions to be invariant or covariant as between those different coordinate systems.

Also, although there may be a unique choice of coordinate system for any FRW universe with a nonzero mass/energy density such that all observers in a surface of constant t see the same thing, wouldn't this break down when you reach the point of exactly zero density? After all, observers in a surface of constant t in some other type of coordinate system like an inertial frame will also see exactly the same thing (total emptiness in all directions).

Yes, as I understand it the FRW metric technically can't be applied to a completely empty universe. That's why we use terminology such as "vanishingly small density". It makes me wonder whether a completely empty Milne model with negative spatial curvature is a fictitious, physically unreal model. More on that in a separate post.

But the spacetime geometry is exactly the same in some other coordinate system where the metric is expressed differently, like Eddington-Finkelstein coordinates or Kruskal-Szekeres coordinates. If you use the corresponding metric for any of these coordinate systems, you'll always get exactly the same predictions about what is seen by physical observers in this spacetime.

Yes, you may have noticed that I edited the Schwarzschild paragraph out of my post shortly after submitting it, because I anticipated that you would point to these alternative coordinates. But I think you agree that a metric written in these alternative coordinates is in the same equivalence class as the Schwarzschild metric, so this is what you describe as a change of coordinate system rather than a true change of metric. So we all agree. Although I haven't verified the point, I believe your statement that Eddington-Finkelstein and Kruskal-Szerkeres will generate the same predictions of spatial curvature and time dilation in the same physical scenarios; but as you know this equivalence is slightly qualified by the fact that these alternative coordinate systems avoid certain singularities where the straight Schwarzschild coordinates will "blow up."

 

[nutgeb]

How exactly do they do this? This is a key question in deciding what exactly they infer about each others clocks.

In his textbook "Cosmological Physics" Prof Peacock suggests:

"The [cosmological time] coordinate is useful globally rather than locally because the clocks can be synchronized by the exchange of light signals between observers, who agree to set their clocks to a standard time when e.g. the universal homogeneous density reaches some given value."

Obviously this method would take a very long time, and one questions how accurately observers on different galaxies can locally measure the average homogeneous density (by observing the CMB, etc.) But it's the principle that it can be done which is important, not the "how to" details.

 

[nutgeb]

I said:

Actually the recipe for the universe construction project begins with flat, static, empty space. Comovers are stationary, so there are no Lorentz effects. Then you add expansion motion, which introduces Lorentz effects between homogeneous nongravitating comovers. You then eliminate the Lorentz effects by adding negative curvature. (Note that it this stage, Lorentz transformations occur only with respect to peculiar velocities, not with respect to proper velocities per se.) Then you add gravity to the homogeneous comovers, which flattens out the spatial curvature but does not cause gravitational time dilation. Bake in CMB oven for 13.7 Gy and serve cold.

It does seem artificial, however, to just stir in some negative curvature as if it is an inevitable consequence of adding expansion motion. One can't just "choose" to have spatial curvature because one finds it convenient. Perhaps the "natural" condition of empty space is negative curvature, even if there is no expansion motion.

Continuing down this path:

In the Milne model, the degree of negative curvature (i.e., the radius of curvature) must be finely tuned in the initial conditions, such that the amount that space "bends" as a function of distance is proportional to the increasing velocity as a function of distance. This enables the Lorentz contraction and time dilation to be exactly offset by the curvature. Presumably a Milne model can select between an infinite choice of settings for proper velocity at a given proper distance, as long as the linear velocity-distance Hubble law is satisfied. Is it reasonable to expect empty space to accommodate such infinite flexibility in the scale factor, while spontaneously generating the corresponding needed amount of negative curvature? Maybe this is an inherent attribute of empty space, but it seems unlikely. Since the comoving Milne test particles are massless, how would empty space "detect" (I'm anthropomorphizing here) the underlying scale factor of the particle distribution in order to adjust its curvature accordingly? What radius of curvature would characterize a static set of massless particles? And how can mere motion through space, without mass, cause space itself to bend?

It seems more likely that if one tried to build a toy Milne model in empty space, space would not cooperate in spontaneously supplying the desired negative curvature. In which case, the Milne model with negative curvature may be an unphysical fiction.

It also is counterintuitive that merely sprinkling a "vanishingly small" amount of massive particles into the mix would suffice to cause space to curl tightly into the maximum possible degree of global negative curvature. Adding mass causes the spatial curvature to become more positive. Therefore it is contradictory that sprinkling the first few grains of mass into an otherwise empty space would cause maximal negative curvature. This makes me wonder whether there is any physical effect that could be plausibly described as the cause of negative curvature.

It is interesting that this (seeming) paradox can be avoided if there is Lambda equal to the cosmological constant. The cosmological constant happens to be characterized by its own mass-energy (gravity) which is exactly in balance with its negative pressure, such that it causes a spherical region of otherwise empty space to expand at exactly the escape velocity of its mass-energy. By that means, the cosmological constant automatically offsets the negative spatial curvature that empty space would otherwise require. Note that the cosmological constant doesn't eliminate the need for the effect caused by the underlying negative spatial curvature. A cosmological constant added to otherwise flat space would not itself offset the SR time dilation resulting from recession velocities. (As explained in my other thread, in the Schwarzschild metric both SR velocity and mass act in the same direction to increase the time dilation; neither one is capable of reducing it). The negative curvature is needed first, in order to to offset SR time dilation; then when the cosmological constant is added to that mix, its gravity and acceleration effects combine to flatten out the spatial geometry without introducing any gravitational time dilation. Underlying negative curvature is required, but is always offset by the cosmological constant. In other words, (and rather obviously), if there is a cosmological constant, then the condition of "naked" negative curvature can never exist; it must be "clothed" in the curvature-flattening cosmological constant.

The cosmological constant affects the spatial curvature in the same way as matter, except that matter must rely on finely tuned initial conditions to ensure that its recession velocity equals its escape velocity. By contrast, over time the cosmological constant tends to automatically readjust the balance between mass-energy and the cosmic recession velocity, in the direction of the balance needed for spatial flatness.

 

[George Jones]

In his textbook "Cosmological Physics" Prof Peacock suggests:

"The [cosmological time] coordinate is useful globally rather than locally because the clocks can be synchronized by the exchange of light signals between observers, who agree to set their clocks to a standard time when e.g. the universal homogeneous density reaches some given value."

Obviously this method would take a very long time, and one questions how accurately observers on different galaxies can locally measure the average homogeneous density (by observing the CMB, etc.) But it's the principle that it can be done which is important, not the "how to" details.

But this thread is about an empty FRW model, and this method clearly doesn't work for such a model.

 

[nutgeb]

But this thread is about an empty FRW model, and this method clearly doesn't work for such a model.

Well of course the word "empty" in the title of the thread really means vanishingly empty, not completely empty.

Just because the receding particles are massless doesn't mean that we have defined them to be invisible or not emitting massless "test light". It seems to me that the massless model is so abstract that one can arbitrarily change the characteristics to make it either practical or impractical for massless observers to synchronize their massless clocks.

 

[JesseM]

Yes you've put your finger on exactly the distinction I was making. I think at any effect (e.g. spatial curvature or time dilation) which is mandated by a particular metric (or class of equivalent metrics) in a particular scenario is "real" in the sense that we don't have to option to avoid that effect by arbitrarily selecting a different, non-equivalent metric.

But my point was that we should really differentiate between features which are mandated by "a particular metric" and features which are mandated by a "class of equivalent metrics" (equivalent in the sense that they describe the same spacetime geometry). Given any particular metric there will be some facts about simultaneity and spatial curvature for that metric, but then if you look at the class of equivalent metrics to which it belongs, you can find other metrics in the same class which give different answers about simultaneity and spatial curvature. Agreed? In the case of simultaneity this is pretty easy to see intuitively, since given any 4D spacetime geometry there are many possible ways to slice it into a stack of 3D surfaces, and for each of these ways you can define a coordinate system where each slice is a surface of constant t (and then find the correct form of the metric in that coordinate system which describes the underlying geometry).

 

[George Jones]

Well of course the word "empty" in the title of the thread really means vanishingly empty, not completely empty.

Just because the receding particles are massless doesn't mean that we have defined them to be invisible or not emitting massless "test light". It seems to me that the massless model is so abstract that one can arbitrarily change the characteristics to make it either practical or impractical for massless observers to synchronize their massless clocks.

In the empty FRW universe, the energy/mass density is zero at all times, but, I suppose that number density could be used.

Why not just use identical massless clocks that are all set to zero at the "Big Bang" event? Then, the hyperbolic spatial FRW surfaces are hypersurfaces of constant t as registered by these clocks.

 

[nutgeb]

Why not just use identical massless clocks that are all set to zero at the "Big Bang" event? Then, the hyperbolic spatial FRW surfaces are hypersurfaces of constant t as registered by these clocks.

That works for me.

 

[nutgeb]

But my point was that we should really differentiate between features which are mandated by "a particular metric" and features which are mandated by a "class of equivalent metrics" (equivalent in the sense that they describe the same spacetime geometry). Given any particular metric there will be some facts about simultaneity and spatial curvature for that metric, but then if you look at the class of equivalent metrics to which it belongs, you can find other metrics in the same class which give different answers about simultaneity and spatial curvature. Agreed? In the case of simultaneity this is pretty easy to see intuitively, since given any 4D spacetime geometry there are many possible ways to slice it into a stack of 3D surfaces, and for each of these ways you can define a coordinate system where each slice is a surface of constant t (and then find the correct form of the metric in that coordinate system which describes the underlying geometry).

Maybe it would help if you give a specific example involving spatial curvature.

 

[JesseM]

Maybe it would help if you give a specific example involving spatial curvature.

I'm just basing my comments about spatial curvature on what George Jones said in the quote from the old thread I posted; the Milne metric and the Minkowski metric are part of the same equivalence class since they both correspond to the same spacetime geometry (zero spacetime curvature everywhere), but he said the spatial curvature was negative in the first and zero in the second.

I would also speculate that if you picked two different metrics in the same equivalence class for some other spacetime geometry, like the Eddington-Finkelstein metric vs. the Kruskal-Szekeres metric for a nonrotating black hole, then the spatial curvature in slices of constant t would probably be different too. But I don't know how spatial curvature is calculated in GR (in fact I know little about the detailed math of GR) so I can't verify this.

 

[George Jones]

Milne spacetime and Minkowski spacetime use exactly the same spactime metric, but components of this metric are expressed with respect to two different coordinate systems. The components of all tensors, including the curvature tensor, in the two coordinate systems are related in the standard way, so that all components zero in one coordinate system iff all components zero in the other coordinate system.

Let t be time in an inertial coordinate system and T be FRW time.

Consider a surface N of constant t and a surface N' of constant T. Restricting the spacetime metric g to N results in a metric h defined on N, and restricting the spacetime metric g to N' results in a metric h' defined on N'. Each of (N , h) and (N' , h') is a 3-dimensional differentiable manifold with metric. (Intrinsic) Curvature tensors for N and N' can be calculated from the metrics h and h' in the standard way. The curvature tensor for (N , h) is congruently zero, while the curvature tensor for (N' , h') is non-zero.

This is covered in Chapter 3, Hypersurfaces, of Poisson's book, A Relativist's Toolkit.

 

[JesseM]

Milne spacetime and Minkowski spacetime use exactly the same spactime metric, but components of this metric are expressed with respect to two different coordinate systems.

When you say "same spacetime metric", should I assume you're using the word "metric" to refer to the underlying spacetime geometry as opposed to the way the the metric is actually written in a particular coordinate system? For example, would you say the Schwarzschild metric, the Eddington-Finkelstein metric, and the Kruskal-Szekeres metric all just represent a single metric expressed in different coordinate systems? I was distinguishing between "metrics" as actual equations giving the curvature in a particular coordinate system and an "equivalence class of metrics" that all corresponded to the same spacetime geometry, but written differently because they were based on different coordinate systems on that geometry.

 

[Vanadium 50]

Why not just use identical massless clocks that are all set to zero at the "Big Bang" event? Then, the hyperbolic spatial FRW surfaces are hypersurfaces of constant t as registered by these clocks.

If they are massless, they aren't comoving. They are moving at c and the clocks continue to read zero.

If they are light but massive and co-moving, does an observer riding with a clock agree that the other clocks are synchronized? I believe not.

 

[nutgeb]

I said:

In the Milne model, the degree of negative curvature (i.e., the radius of curvature) must be finely tuned in the initial conditions, such that the amount that space "bends" as a function of distance is proportional to the increasing velocity as a function of distance. ... Is it reasonable to expect empty space to accommodate such infinite flexibility in the scale factor, while spontaneously generating the corresponding needed amount of negative curvature?

I realize that my posts on this subject have wandered around, but I think I’ve hashed through this enough to come to some conclusions. I had started with the assumption (or hope) that Birkhoff’s Theorem could be combined with SR to find equivalence between a snapshot of the FRW model and the Schwarzschild static model. I will describe why I no longer think that’s true. The best way to sort this problem out is to contrast the FRW metric under the ‘expanding space’ paradigm with the Schwarzschild metric under the “kinematic” paradigm, with Lambda=0. First I will set out the RW line element (which is the relevant part of the FRW metric):

ds^{2} = -dt^{2} + a^{2}(t) \left[ dx^{2} + \left\{sin^{2} \chi, \chi^{2}, sinh^{2} \chi \right\} \left( d\theta^{2} + sin^{2 \theta} d\phi ^{2} \right) \right] \\ \left\{ closed, flat, open \right\}

FRW metric with 'expanding space' paradigm: In this paradigm, empty, static space must start out with an intrinsic, maximal negative curvature. Then the 'expansion of space' causes space itself to automatically stretch exactly in proportion with the growing scale factor, causing the negative curvature to progressively flatten out, i.e. become less curved per unit of proper distance. In this respect the model does not require fine tuning of the initial conditions.

Comoving observers remain "stationary" while the space between them expands. Since they are stationary, there are no SR velocity-related effects at all. Instead, in FRW there is negative spatial curvature. In mathematical terms, the FRW metric simply inserts the hyperbolic sinh function in the space part of the metric, as can be seen in the RW line element. The metric seems to require that the underlying distribution of comovers would be hyperbolically contracted if empty space were flat. Such that when this hyperbolic distribution of comovers is viewed through hyperbolically curved space, it actually looks flat, in the sense that the distribution of comovers looks homogenous. This is nicely pictured in Figure 4.1 of Tamara Davis’ http://arxiv.org/abs/astro-ph/0402278v1" .

The intrinsic hyperbolic distribution of comovers required by the metric seems oddly out of place, because nothing is (or ever was) moving in the ‘expanding space’ model. What effect other than velocity would cause this exactly Lorentz-equivalent hyperbolic distribution? It is equally curious that FRW doesn’t apply the same requirement of intrinsic SR time dilation, which normally would accompany Lorentz-contraction as a velocity-induced effect. The FRW metric includes no time dilation element; the time part of the FRW metric can be seen to be linear. This allows comovers to share the same cosmological time even in an empty model.

Adding a critical density of matter in the FRW ‘expanding space’ paradigm introduces positive spatial curvature which exactly offsets the intrinsic negative curvature of empty space, so space becomes flat. The RW line element accomplishes this by simply deleting the sinh function from the space part of the metric. That’s fine, but one then wonders, what became of the original the distribution of comovers that would have been hyperbolically non-homogeneous if the space were intrinsically flat? In other words, how can the same distribution of comovers be homogeneous in negatively curved space and yet remain homogeneous when space becomes flat? I suppose one answer is that the expansion of space over time could (almost) accomplish that feat if it (almost) infinitely stretched out the scale factor. Inflation theory uses a similar explanation. I’ll adopt that answer for now, although it suggests that the FRW metric for an empty, expanding model permits a homogeneous distribution of comovers only at physical scales that are infinitesimal compared to the scale at which a homogeneous distribution is observed in flat space.

The FRW metric does not introduce any time contraction on account of the added matter, so comovers continue to share the same cosmological time and homogeneity.

Schwarzschild metric with ‘kinematic’ paradigm: In this paradigm, empty, static space starts out intrinsically flat. When recession velocity is introduced, comoving observers really are in motion relative to each other. This motion causes Lorentz-contraction and SR time dilation as between comovers. There is no underlying curvature to offset the Lorentz effects. The distribution of comovers (as viewed by each other) is hyperbolically non-homogeneous due to the Lorentz-contraction. Comovers do not share a common cosmological time, because time is dilated.

Adding a critical density of matter in this paradigm again introduces positive spatial curvature which exactly offsets the Lorentz-contraction of comovers. The Lorentz SR time dilation at escape velocity is:

d\tau = \gamma dt = \frac{dt}{\sqrt{1 - v^{2} /c^{2}}} = \frac{dt}{ \left( 1 - \frac{2M}{r} \right) ^{1/2} }

Although the spatial curvature theoretically has become locally positive everywhere (as measured by an observer at zero-density infinity,) it is measured to be locally flat as between all comovers, who see each other to be homogeneously distributed.

One could argue that adding matter introduces no gravitational time dilation or contraction in the Schwarzschild ‘kinematic’ model, but I now think the better answer is that it introduces the amount of gravitational time contraction specified by the Schwarzschild interior metric. In the spatially flat model expanding exactly at escape velocity, this amount of gravitational time contraction is never enough to mathematically entirely offset the SR time dilation. So an element of time dilation remains, and comovers still do not share a common cosmological time. (Note in passing that in a very over dense model in which expansion velocity is less than escape velocity, there is a balance point at which the Schwarzschild interior time contraction does match the SR time dilation, and comovers can share a common cosmological time, but this does not coincide with spatial flatness.)

. . . . . . . . . .

This comparison of the FRW ‘expanding space’ and Schwarzschild ‘kinematic’ paradigms shows that their treatment of the time element for comovers is different and irreconcilable. There is no direct Schwarzschild kinematic analogy for the relativistic time observed by FRW comovers. One must sadly conclude that while Birkhoff’s Theorem can model the spatial aspects of comovers in the FRW metric, it cannot be combined with SR Lorentz effects to model the time relationship between comovers.

This analysis reinforces the limitations on how SR Lorentz effects can be used within an FRW model. Since FRW comovers are not actually moving, SR doesn’t apply to their recession "motions", even by analogy. However, I see no reason why both Lorentz space and time effects can’t be applied validly to peculiar motions (i.e. the difference between proper motion and the local Hubble rate). That could explain why we can clearly observe many kinds of SR effects across distances far greater than the infinitesimal extent of a true "local" inertial reference frame, despite the fact that we are always immersed in the cosmic gravitational background.

This analysis also supports the proposition that SR and gravitational time dilation are not contributing causes of the cosmological redshift, because that would require net time dilation as between FRW comovers.

I don’t want to express any preference here for the FRW ‘expanding space’ paradigm or the Schwarzschild ‘kinematic’ paradigm. I’m just explaining why they seem to predict observational differences, due specifically to the incompatible ways in which they treat time dilation.

 

[nutgeb]

I've reposted the edited version of post #30 at 11:30PM (in Physics Forums relativistic time).

 

[nutgeb]

Edit note: Sorry, I made another dumb mistake, so I had to re-edit the description of the 'expanding space' paradigm under the FRW metric. I had said that in the empty model, the intrinsic spatial distribution of comovers was hyperbolic. That's wrong, the intrinsic distribution really is inverse-hyperbolic. Correcting that mistake didn't change my conclusions, but it changes part of the explanation.

I said:

In the Milne model, the degree of negative curvature (i.e., the radius of curvature) must be finely tuned in the initial conditions, such that the amount that space "bends" as a function of distance is proportional to the increasing velocity as a function of distance. ... Is it reasonable to expect empty space to accommodate such infinite flexibility in the scale factor, while spontaneously generating the corresponding needed amount of negative curvature?

I realize that my posts on this subject have wandered around, but I think I’ve hashed through this enough to come to some conclusions (which apparently are always subject to further correction!) I had started with the assumption (or hope) that Birkhoff’s Theorem could be combined with SR to find equivalence between a snapshot of the FRW model and the Schwarzschild static model. I will describe why I no longer think that’s true. The best way to sort this problem out is to contrast the FRW metric under the ‘expanding space’ paradigm with the Schwarzschild metric under the “kinematic” paradigm, with Lambda=0. First I will set out the RW line element (which is the relevant part of the FRW metric):

ds^{2} = -dt^{2} + a^{2}(t) \left[ dx^{2} + \left\{sin^{2} \chi, \chi^{2}, sinh^{2} \chi \right\} \left( d\theta^{2} + sin^{2 \theta} d\phi ^{2} \right) \right] \\ \left\{ closed, flat, open \right\}

FRW metric with 'expanding space' paradigm: In this paradigm, empty, static space must start out with an intrinsic, maximal negative curvature. Then the 'expansion of space' causes space itself to automatically stretch exactly in proportion with the growing scale factor, causing the negative curvature to progressively flatten out, i.e. become less curved per unit of proper distance. In this respect the model does not require fine tuning of the initial conditions.

Comoving observers remain "stationary" while the space between them expands. Since they are stationary, there are no SR velocity-related effects at all. Instead, in FRW there is negative spatial curvature. In mathematical terms, the FRW metric simply inserts the hyperbolic sinh function in the space part of the metric, as can be seen in the RW line element.

The metric seems to require that the underlying distribution of comovers would be hyperbolically dilated (the inverse of Lorentz contraction: radial distances are dilated) if empty space were flat. Such that when this inverse-hyperbolic distribution of comovers is viewed through hyperbolically curved space, it actually looks flat, in the sense that the distribution of comovers looks homogeneous. This is nicely pictured in Figure 4.1 of Tamara Davis’ http://arxiv.org/abs/astro-ph/0402278v1" . I haven't thought of any explanation why comovers should start out distributed through space in an inverse-hyperbolic pattern. What physical effect would cause that? Well, the purpose of the metric is to calculate what is required in order for the distribution of comovers to appear homogeneous regardless of matter content; the metric itself doesn't explain how the initial conditions might arise. This looks like an example of fine tuning.

The FRW metric includes no time dilation element; the time part of the RW line element is linear. This allows comovers to share the same cosmological time even in an empty model. This absence of intrinsic time dilation is somewhat ambiguously portrayed in Davis' Fig. 4.1. One might get the impression from the graph that as between comovers, time would be intrinsically Lorentz-contracted, and the empty-space metric then compensates for this by introducing an element of time dilation. However, it is clear that the RW line element itself contains no time-dilating component, so one must interpret Davis' circular T = constant lines as portraying only the inverse-hyperbolic spatial distribution of comovers.

Adding a critical density of matter in the FRW ‘expanding space’ paradigm introduces positive spatial curvature which exactly offsets the intrinsic negative curvature of empty space, so space becomes flat. The RW line element accomplishes this by simply deleting the sinh function from the space part of the metric. That’s fine, but one then wonders, what became of the original distribution of comovers that would have been inverse-hyperbolically non-homogeneous if the space were intrinsically flat? In other words, how can the same distribution of comovers be homogeneous in negatively curved space and yet remain homogeneous when space becomes flat? I suppose one answer is that the expansion of space over time could (almost) accomplish that feat if it (almost) infinitely stretched out the scale factor. Inflation theory uses a similar explanation. I’ll adopt that answer for now, although it suggests that the FRW metric for an empty, expanding model permits a homogeneous distribution of comovers only at physical scales that are infinitesimal compared to the scale at which a homogeneous distribution is observed in flat space.

The FRW metric does not introduce any time contraction on account of the added matter, so comovers continue to share the same cosmological time and homogeneity.

Schwarzschild metric with ‘kinematic’ paradigm: In this paradigm, empty, static space starts out intrinsically flat. When recession velocity is introduced, comoving observers really are in motion relative to each other. This motion causes Lorentz-contraction and SR time dilation as between comovers. There is no underlying curvature to offset the Lorentz effects. The distribution of comovers (as viewed by each other) is hyperbolically non-homogeneous due to the Lorentz-contraction. Comovers do not share a common cosmological time, because time is dilated.

Adding a critical density of matter in this paradigm again introduces positive spatial curvature which exactly offsets the Lorentz-contraction of comovers. The Lorentz SR time dilation at escape velocity is:

d\tau = \gamma dt = \frac{dt}{\sqrt{1 - v^{2} /c^{2}}} = \frac{dt}{ \left( 1 - \frac{2M}{r} \right) ^{1/2} }

Although the spatial curvature theoretically has become locally positive everywhere (as measured by an observer at zero-density infinity,) it is measured to be locally flat as between all comovers, who see each other to be homogeneously distributed.

One could argue that adding matter introduces no gravitational time dilation or contraction in the Schwarzschild ‘kinematic’ model, but I now think the better answer is that it introduces the amount of gravitational time contraction specified by the Schwarzschild interior metric. The interior metric applies because we are comparing the time dilation at the central mass to the time dilation at the radius of another comover. In the spatially flat model expanding exactly at escape velocity, this amount of gravitational time contraction is never enough to mathematically entirely offset the SR time dilation. So an element of time dilation remains, and comovers still do not share a common cosmological time. (Note in passing that in a very over dense model in which expansion velocity is less than escape velocity, there is a balance point at which the Schwarzschild interior time contraction does match the SR time dilation, and comovers can share a common cosmological time, but this does not coincide with spatial flatness.)

. . . . . . . . . .

This comparison of the FRW ‘expanding space’ and Schwarzschild ‘kinematic’ paradigms shows that their treatment of the time element for comovers is different and irreconcilable. There is no direct Schwarzschild kinematic analogy for the undilated relativistic time observed by FRW comovers. One must sadly conclude that while Birkhoff’s Theorem can model the spatial aspects of comovers in the FRW metric, it cannot be combined with SR Lorentz effects to model the time relationship between comovers.

This analysis reinforces the limitations on how SR Lorentz effects can be used within an FRW model. Since FRW comovers are not actually moving, SR doesn’t apply to their recession "motions", even by analogy. However, I see no reason why both Lorentz space and time effects can’t be applied validly to peculiar motions (i.e. the difference between proper motion and the local Hubble rate). That could explain why we can clearly observe many kinds of SR effects across distances far greater than the infinitesimal extent of a true "local" inertial reference frame, despite the fact that we are always immersed in the cosmic gravitational background.

This analysis also supports the proposition that SR and gravitational time dilation are not contributing causes of the cosmological redshift, because that would require net time dilation as between FRW comovers.

I don’t want to express any preference here for the FRW ‘expanding space’ paradigm or the Schwarzschild ‘kinematic’ paradigm. I’m just explaining why they seem to predict observational differences, due specifically to the incompatible ways in which they treat time dilation.

 

[nutgeb]

My analysis of the 'expanding space' paradigm under the FRW metric seems pretty straightforward. If this analysis is correct, then it should be impossible to determine directly by observation whether a homogeneous space is closed, flat or open. If the global geometry is closed, then the 'intrinsic' spatial distribution of comovers must be hyperbolic, such that when the distribution is viewed through positive spatial curvature by other comovers, it appears flat. If the global geometry is open, then the 'intrinsic' spatial distribution of comovers must be inverse-hyperbolic, such that when the distribution is viewed through negative spatial curvature by other comovers, it appears flat.

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. This effect would be exactly offset when viewed through positively curved space, which causes their apparent angular size to look larger at the same comoving coordinate. Therefore, the apparent angular diameters should be consistent with spatial flatness, regardless of the actual spatial curvature.

In light of the foregoing, I don't understand the effort being undertaken by the WMAP http://map.gsfc.nasa.gov/mission/sgoals_parameters_geom.html" to try to determine the the global spatial curvature (and thus the cosmic density parameter) by analyzing the apparent angular size of CMB features as compared to the calculated dimensions of the baryon acoustical peaks (oscillations) of the CMB. If the universe looks exactly homogeneous at large scales and the FRW metric is an accurate model, then the apparent angular size of CMB features should always look the same regardless of the spatial curvature. Conversely, if a discrepancy in apparent angular size is detected (after accounting for measurement errors), that would suggest a location-dependent departure from homogeneity, in violation of the cosmological principle.

I must be missing something here (?)

 

[nutgeb]

It is interesting to think of spatial curvature as a physical embodiment of the visual effect of 'perspective distortion'. Perspective distortion is what happens visually when you look through a telephoto or wide angle lens.

http://en.wikipedia.org/wiki/Perspective_distortion_(photography)" describes the telephoto effect:

"For example, you could have two identical stone statues one of which is 10 meters in front of you in absolute distance while the other is 1 meter beyond at 11 meters in front of you. The relative distance of the far object to the near object is a ratio of 11/10 or 10% farther away.

"If you walk toward the statues until you are only 1 meter away from the nearer one you will be two meters away from the other in absolute distance. However, it will now be twice as far from you as the other in relative distance, a ratio of 2/1 or 100% farther away. The nearer statue will have gotten bigger in your field of vision at a faster rate than the farther statue and the farther statue will appear to be twice as far away. This is normal perspective. No distortion is taking place.

"If, on the other hand, you remain 10 meters away from the nearer statue and take a photo with a telephoto lens (or crop and blow up a photo taken with a normal lens, or just look at it through a telescope) so that the statue that is 10 meters away appears as if it were only 1 meter away, the relative distance of the other statue to it will appear unchanged at a ratio of 11/10 or only 10% farther away. In other words, through a telephoto lens, when the nearer statue appears as though it were 1 meter away, the farther statue will appear to be only 1.1 meters (1 meter and 10 centimeters) away. The nearer and farther statues will have gotten bigger in your field of vision at the same rate. Perspective distortion is taking place. Two objects that are actually 1 meter apart appear to be only 10 centimeters apart."

It is helpful to look at the photos in the linked article.

Telephoto perspective distortion is the non-physical analog of positive spatial curvature. Wide angle perspective distortion is the non-physical analog of negative spatial curvature.

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. This redistribution of the objects also eliminates the relative increase in angular size of more distant objects. Thus the redistribution compensates for the entire distortion effect of positive spatial curvature (or telephoto distortion).

Consequently, if a distribution of objects appears homogeneous, there cannot be any angular size distortion. A homogeneous FRW model will always appear to be spatially flat, regardless of its true spatial curvature.

So again, I don't see how it is possible for the WMAP project to measure angular size distortion as a gauge of spatial curvature. Homogeneity and angular size distortion cannot co-exist.

 

[JesseM]

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.

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.

 

[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.