The Mystery of Expanding Space: Uncovering the Truth Behind Dark Energy

[yogi]

Question for Ich: I have not had time to review all the prior posts in this thread - so maybe my intrusion has been already discussed and resolved - but if the redshift is a traditional Doppler affect, are we not going to get a much different picture of the universe than if it is treated as stretching of space space - in the latter case, z relates directly the difference in the two scale factors (now and at emission time) irrespective of how caused and independent of the velocity and acceleration profile - in Doppler - an accelerating universe is going to lead to a different size than a decelerating universe - and it would also seem that if we are dealing with pure ballistic or Doppler phenomena, the estimate of the present size of the Hubble sphere would be undervalued since we are witnessing red shift photons that were emitted long ago - and the universe would necessarily have changed during the travel time

 

[nutgeb]

Without specifying the procedure, "synchronization" is not defined and thus not a "real" aspect of physics. When you claim that fundamental observers are synchronized if you use a coordinate time that equals the proper time since the big bang, that's ok. And when I say that they are not synchronized if I use the standard procedure to establish synchronizity, that's also ok. The covariance principle surely applies here.
But it's not ok to pick one definition to establish synchronizity, and claim that procedures that give a different result are wrong. They aren't, they're simply different.

I think your words in bold are wrong, if you are saying that within a single coordinate system (such as FRW), you are allowed to treat the clocks of fundamental comoving observers running cosmological time as being unsynchronized merely because you selected an arbitrarily different synchronization test, such as SR time dilation alone.

When performing calculations using the FRW metric, the question of whether or not clocks of fundamental comoving observers are to be treated as synchronized with each other MUST be determined solely by measuring their proper time since the origin (or a mathematical equivalent of such proper time measurement). Any calculations performed within the FRW metric will be wrong if they depend on a determination that the clocks of fundamental comoving observers are unsynchronized.

I guess we must agree to disagree. Let's solicit opinions from other knowledgeable readers.

But there is also a simple formula in minkowski coordinates, namely the SR doppler formula.

I think you will agree that a homogeneous, isotropic (your definitions are ok) zero-gravity SR expansion cannot be mapped to Euclidian spatial coordinates because it requires globally hyperbolic spatial curvature. I.e. the Milne model.

A Minkowski spacetime diagram can portray a globally hyperbolic spatial curvature by replacing a straight-line axis with hyperbolic lines. So in that sense I suppose a Minkowski diagram can be used to map a homogeneous, isotropic expansion. Is it technically correct to say that a homogeneous, isotropic Milne spacetime is "flat" when the underlying spatial geometry is not flat?

 

[Ich]

Hi yogi,

if the redshift is a traditional Doppler affect, are we not going to get a much different picture of the universe than if it is treated as stretching of space space

The paper treats redshift effectively as a doppler effect in curved spacetime. As long as spacetime is flat, there's no problem with treating it globally as a traditional doppler efferct. In general FRW spacetimes, gravitational effects are left to be incorporated in the exact formulation of the calculation, which can be quite tedious. The authors do not bother with this "fine point". I tried to explain their appoach with an analogy in my last post.

Hi nutgeb,

Any calculations performed within the FRW metric will be wrong if they depend on a determination that the clocks of fundamental comoving observers are unsynchronized.

Before we agree to disagree, let me try to resolve a potential misunderstanding I believe to have spotted:
When you talk about "the metric", you always refer to a specific coordinate representation of it. It seems that you have the impression that this representation is the only possible one, and changing it would change the physics behind.
The metric is expressed as a tensor, and tensors are covariant, i.e. independent of the coordinates used. If I choose to use a different set of coordinates, I do not change anything about the physics. If I choose to use a certain set of coordinates that is valid only locally, there's nothing wrong with it either, as long as I also use it only locally.
There are two different meanings of "the metric". One refers to the covariant tensor as it is, a physical property of spacetime, the other refers to a specific coordinate representation. The latter is arbitrary, and chosen for convenience rather than physical reasons. One is free to choose arbitrary coordinates even if "the metric" (first meaning) is FRW.

I think you will agree that a homogeneous, isotropic (your definitions are ok) zero-gravity SR expansion cannot be mapped to Euclidian coordinates because it requires globally hyperbolic spatial curvature.

No, I don't agree. Spatial curvature is nothing physical, it is coordinate dependent. The word "foliation" is quite suggestive, you split the (invariant, physical) spacetime into arbytrary sheets that you call "space". In one case, you choose hyperbolic sheets, in the other flat ones. Sapcetime is the same.

Can a Minkowski spacetime diagram accurately and globally portray a hyperbolic spatial curvature?

Of course. You simply plot hyperbolae of constant cosmological time. They are hyperbolae in Minkowski coordinates, that's why the respective spacetime foliation is called hyperbolic.

 

[nutgeb]

There are two different meanings of "the metric". One refers to the covariant tensor as it is, a physical property of spacetime, the other refers to a specific coordinate representation. The latter is arbitrary, and chosen for convenience rather than physical reasons. One is free to choose arbitrary coordinates even if "the metric" (first meaning) is FRW.

Ich, I agree that there is a distinction between a "metric" and "coordinate representation" within a metric. I probably haven't been careful enough with my wording.

However, I don't think that distinction is the source of our disagreement. Even given an arbitrary choice of coordinate representation, I believe that any calculations performed using the FLR "metric" must remain mathematically consistent with the absolute requirement that proper time since the BB is synchronized as between fundamental comoving observers. This should be true, for example, whether one attaches labels using (a) comoving coordinates, (b) proper coordinates with any fundamental comoving observe at the origin and zero peculiar velocity, or (c) proper coordinates with non-zero peculiar velocity at the origin relative to fundamental comoving observers.

Spatial curvature is nothing physical, it is coordinate dependent. The word "foliation" is quite suggestive, you split the (invariant, physical) spacetime into arbytrary sheets that you call "space". In one case, you choose hyperbolic sheets, in the other flat ones. Sapcetime is the same.

I mis-phrased my statement by using the word "mapped." I meant only that the geometry of hyperbolically curved space is non-Euclidian. I agree that the spatial curvature can be portrayed on Minkowski foliations that are themselves hyperbolically curved, but not on flat foliations.

I agree that spatial curvature is coordinate dependent and is not physical.

 

[nutgeb]

It seems that the internal symmetries of ANY homogeneous, isotropic metric originating at a single point or singularity require the clocks of all fundamental comoving observers to be synchronized (in the sense of the proper time elapsed since the origin) regardless of the metric or coodinate system employed.

If they are unaccelerated then in their own reference frame each of their functions (proper time = proper velocity x proper distance from the origin) must be identical. If the are all subjected to the same acceleration, then in their own reference frame each of their functions (proper time = average velocity x proper distance from the origin) must be identical.

 

[Ich]

Hi nutgeb,

It seems that the internal symmetries of ANY homogeneous, isotropic metric originating at a single point or singularity require the clocks of all fundamental comoving observers to be synchronized (in the sense of the proper time elapsed since the origin) regardless of the metric or coodinate system employed.

Yes, there seems to be a symmetry in the universe called homogeneity of space. That means that there is a definition of space that can be used without change at any point, where every comoving observer has to be of the same age. That's why cosmological time is defined as the proper time of said observers. And that's why I said that FRW coordinates have the advantage to reflect that symmetry.
But if you use the word "synchronized" in the way you define it here, you must be aware that this is just a (your) definition.
There is a standard meaning of this word, where it is defined by the exchange of light pulses. Other words like "time dilatation", which we are discussing, are themselves defined via the standard definition. And still valid, no matter what other symmetries are present.

 

[nutgeb]

Proper time and proper distance are invariant under coordinate transformations.