# SR & the empty FRW model

1. Jun 3, 2009

### 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?

Last edited: Jun 3, 2009
2. Jun 3, 2009

### George Jones

Staff Emeritus
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.
Yes.
Not sure.
Again, I'm not sure, but do you mean this transformation,

3. Jun 3, 2009

### 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.

Last edited: Jun 3, 2009
4. Jun 3, 2009

5. Jun 9, 2009

### George Jones

Staff Emeritus
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."

Last edited: Jun 9, 2009
6. Jun 9, 2009

### nutgeb

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"?

7. Jun 9, 2009

### 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:

Last edited: Jun 9, 2009
8. Jun 9, 2009

### sylas

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....

9. Jun 9, 2009

### nutgeb

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.

Last edited: Jun 9, 2009
10. Jun 9, 2009

### atyy

11. Jun 9, 2009

### JesseM

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:
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.
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)

12. Jun 9, 2009

### nutgeb

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.
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.

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13. Jun 9, 2009

### JesseM

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.
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.

14. Jun 10, 2009

### nutgeb

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.

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?

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15. Jun 10, 2009

### JesseM

It matters in relativity to distinguish coordinate-dependent quantities from coordinate-independent ones.
Do you think it's circular to argue that simultaneity or velocity or the rate a clock is ticking are coordinate dependent?
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?
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).
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.
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.

16. Jun 10, 2009

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

17. Jun 10, 2009

### nutgeb

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?
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.
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.
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."

18. Jun 10, 2009

### nutgeb

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.

19. Jun 10, 2009

### nutgeb

I said:
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.

Last edited: Jun 10, 2009
20. Jun 10, 2009

### George Jones

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