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.