1. Order of Kilopi
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I'm going to hazard that in the statement "cosmological redshift is due to changing curvature", the curvature that is changing can be a faux curvature, ala Rindler (or ala Born's rotating frame). The Riemann tensor, the "real curvature" is not fooled -- it is looking for what geodesics are doing, and it knows that Rindler and Born geodesics are really straight lines, relative to themselves.

The faux curvature can from the "silly ruler" effects -- indeed we can view Rindler or Born, certainly as a pure reference frame with no real observer to carry his ruler and clocks as a silly ruler and silly clock coordinate definition! That just hit me, BTW. Don't be silly and spin around in your swivel chair, and don't be silly and accelerate away......

-Richard

2. Originally Posted by publius
Well, if the Riemann curvature is very small [and this would indeed be a "remarkable" condition if this "dark energy"/cosmological constant is very nearly balancing out regular stress-energy], then we are very close to an SR universe on the grand scale!
I don't mean to lead you down the primrose path with my naive sense of curvature, I think the Riemann curvature is actually not small, but the comoving space is very flat, so all the curvature goes into the time dimension. So that's the part I get hazy on-- it seems like if the antigravity and gravity have averaged effects that are pretty compensating so far, which is what you need if they are both about the same order, then you shouldn't have much acceleration or deceleration, but still you apparently do have lots of curvature because the energy density is near the critical value. If it were all normal mass density, you'd still have a flat space, so time curvature, but the fact that roughly half is dark energy seems to mean that the radius of curvature has increased, on average since early times, more or less linearly with age. But that's still a lot of curvature, I just don't see why you need it to get cosmological redshifts. In fact, they way we keep getting order-unity parameters, suggests to me in a really hazy way that the redshift component that traces to changing curvature is perhaps about the same order of effect as the part that traces back to the initial condition of expansion.
No matter the "religion" of most of the circles of which you speak, if the Riemann curvature is small, they can't transform that sacrilege away.
True, but it probably isn't that small, because my naive idea that dark energy cancels curvature is probably wrong-- it just causes the radius of curvature to accelerate rather than decelerate. My guess is, the best I could hope for is that the SR-esque effects are about the same as the GR-esque ones, in our universe where all the parameters have been order unity on the average. But even that would be sacrilege, if true. I'm already in an unconventional dogmatic position because I like to think of the GR-esque component as a gravitational redshift due to the changing gravitational potential of the universe, but most people reserve the term "gravitational redshift" for the minor effects of local perturbations.

3. Originally Posted by publius
I'm going to hazard that in the statement "cosmological redshift is due to changing curvature", the curvature that is changing can be a faux curvature, ala Rindler (or ala Born's rotating frame).
I think it may actually be due to both types-- if the statement is indeed true, then perhaps it involves both a changing in faux curvature (an artifact of the initial condition) and the real curvature (due to the stretching radius of curvature of the real gravity). I think your faux curvature idea fills in the gap in my thinking about how SR-esque effects (or "silly shrinking rulers") can contribute to cosmological redshifts.
The faux curvature can from the "silly ruler" effects -- indeed we can view Rindler or Born, certainly as a pure reference frame with no real observer to carry his ruler and clocks as a silly ruler and silly clock coordinate definition!
Yes, these are the lines along which I'm thinking, that "real" effects come from gravity, but "faux" effects come from using a particular choice of observers (like the comoving frame), and then maybe there's a third category, "really faux" effects, that come from coordinates that can't possibly even correspond to measurements by local observers (these are generally global extrapolations of a single local reference frame, like time and radius switching places inside event horizons, or superluminal motion when you spin in your chair.)
Don't be silly and spin around in your swivel chair, and don't be silly and accelerate away......
Yes, I think that's the key physical insight. I generally prefer to think physically over mathematically, it gives me a more concrete sense that I understand, though in this case, it's more like mud than concrete. I really need to be straightened out on these issues, from a physical perspective.

4. Order of Kilopi
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Well, it may not be a primrose path. From what I understood of that "tidal force on a ruler in the void" calculation, if the "acceleration parameter" (or deceleration parameter) is zero, then the Riemann curvature is zero, and it doesn't matter what balance of real matter and cosmological constant make it so. If the expansion is a straight line, then the Riemann curvature is zero. So if we are close to that straight line, then the Riemann curvature is small.

I may have misunderstood "acceleration" or something, but that's the way I read it.

If we just happen to be close to a straight line now, but the real curve is something significantly different, then the Riemann curvature would be something, maybe just passing close to zero now. And if I've got that right, that would still mean the tidal force in the void is very small *now*, but would change and become significant later.

-Richard

5. Originally Posted by publius
Well, it may not be a primrose path. From what I understood of that "tidal force on a ruler in the void" calculation, if the "acceleration parameter" (or deceleration parameter) is zero, then the Riemann curvature is zero, and it doesn't matter what balance of real matter and cosmological constant make it so. If the expansion is a straight line, then the Riemann curvature is zero. So if we are close to that straight line, then the Riemann curvature is small.
Then perhaps they are red roses and not primroses! I've just heard from several places that redshifts either require "curvature in the time domain", or "changing curvature". Perhaps the Milne model is telling-- no gravity, but an expanding initial condition. The comoving space is negatively curved, since it is way less than critical density, and so to get zero net curvature you need compensating curvature in the time domain. The scale parameter grows linearly, so there is changing spatial curvature, so also changing temporal curvature. It is starting to sound to me like the correct statement is that redshifts require "changing curvature in the time domain of the comoving-frame coordinate system". It is just so hard to find people who really tell it to you in a way that's actually right!
If we just happen to be close to a straight line now, but the real curve is something significantly different, then the Riemann curvature would be something, maybe just passing close to zero now.
Yes, it is only flat on the average over most of the age of the universe-- early in the Big Bang, there would have been a lot of deceleration due to gravity. But we can only see after the CMB forms, and if you are talking quasars then it's later still, so it seems to me that quasar redshifts at least must be from a fairly low average Riemann curvature. But there's still a shift from deceleration to acceleration, so presumably the Riemann curvature is itself steadily changing. But I'm starting to feel skeptical that Riemann curvature is what gives redshifts-- I'm thinking it's purely the changing time curvature in comoving coordinates.
And if I've got that right, that would still mean the tidal force in the void is very small *now*, but would change and become significant later.
Oh definitely, in 100 billion years there'll be all kinds of acceleration due to dark energy (although those tidal effects require billions of light years to amount to much!).

6. Order of Kilopi
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Ken,

Could a Rindler analogy help? You and I are in a Rindler frame, but separated by some spatial distance. Let's say you are at our origin, and I'm behind you some distance, and feeling a greater force.

Now, I send a signal to you, and you see it redshifted. You send a signal to me and I see it blueshifted. Our explanation for that, in our Rindler coordinates is "gravitational", or for the purposes of this discussion, due to some sort of notion of curvature. And it's completely some apparent curvature of time with position. Nothing is a *function of time*, but time is certainly curving somehow in our coordinates.

So we say curvature is afoot and explains the redshift. We are stationary. Now, what about a Lorentz observer watching our little light show? He sees me gaining on you, and thinks there is relative between us. But that relative velocity, closing distance alone does not explain why you think I'm redshifted, nor why why I think your clock is faster than mine.

Here, I think the Rindler metric is the easiest way to see it. But of course the Lorentz observer can explain it. And that has to do with our proper accelerations. When you recieve my light, you are going faster than I was when I emitted it, I think. And when I receive your light, I'm going *even faster* that you were when you emitted it because I'm actually gaining on you. Throw the clock and ruler changes in there and everything is explained in agreement with the Rindler metric view.

And what makes the difference there? It's our proper acceleration and the differences in our proper acceleration, which is *curvature* of our world lines with respect to our (straight line) geodesics.

Behind you, I'm more curved than you are. So it looks this has to do with *distance* in some way (which is constant in our frame) as well as the curvature. So is there any similiarity between the redshift you see of me even though I'm gaining on you (from the POV of the Lorentz observer) and the cosmological redshift "changing curvature" thing?

-Richard

7. Originally Posted by publius
Now, I send a signal to you, and you see it redshifted. You send a signal to me and I see it blueshifted. Our explanation for that, in our Rindler coordinates is "gravitational", or for the purposes of this discussion, due to some sort of notion of curvature.
OK, we might say that time is flowing faster or slower for the other, a gradient in the rate of flow of time is another way to think of time curvature.
We are stationary. Now, what about a Lorentz observer watching our little light show? He sees me gaining on you, and thinks there is relative between us. But that relative velocity, closing distance alone does not explain why you think I'm redshifted, nor why why I think your clock is faster than mine.
Yes, you're right the big effect is what happens during the time of flight of the light.

And what makes the difference there? It's our proper acceleration and the differences in our proper acceleration, which is *curvature* of our world lines with respect to our (straight line) geodesics.
Yes, the coordinate change means we have swapped in some spatial curvature for some of the temporal curvature. I hadn't appreciated that-- spatial curvature creates "illusions" during flight, whereas temporal curvature speaks about differences in the flow of proper time. I'm starting to think that time-of-flight illusions should not be looked at as fundamentally different from "actual" time dilation and length contraction, because causally connected events could always be viewed by the same observer, and you can make the the distance and time elapsed between the events arbitrarily small with a 'fast' enough observer. In that frame, the "illusory" time-of-flight effects are mostly swapped for "real" time dilation differences!

Behind you, I'm more curved than you are. So it looks this has to do with *distance* in some way (which is constant in our frame) as well as the curvature. So is there any similiarity between the redshift you see of me even though I'm gaining on you (from the POV of the Lorentz observer) and the cosmological redshift "changing curvature" thing?
There might be a deep connection there, I still don't get much insight from the concept of curvature and whether or not it's changing. There's my fundamental quandary right there-- when is it curvature, and when is it changing curvature?

8. Reason
already explained in next post after quote

9. Hi Ken and Richard,
I was finding this discussion fascinating, even if a lot of it was "heading straight through to the keeper". In an attempt to revive it I'll throw in my 0.002 cents worth.

This statement
Originally Posted by Ken G
I think the Riemann curvature is actually not small, but the comoving space is very flat, so all the curvature goes into the time dimension.
and this statement
Originally Posted by Ken G
There's my fundamental quandary right there-- when is it curvature, and when is it changing curvature?
brought to mind Barbour's "Relativity without Relativity" paper. He uses a concept of constant mean curvature to foliate a Riemannian 3-space in order to get rid of the time dimension.

Anderson's thesis paper at arXiv:gr-qc/0409123 (2.5MB) reviews various 3-space theories (native and embedded) and has a section on extrinsic curvature and CMC which you might find useful (or might not ). It has a pretty good reference section at the least.

Anyway thanks for a interesting thread.
Murray

10. Thanks for that link, it would probably be an excellent way to get to the bottom of this curvature business. For some reason my eyes still sort of glaze over whenever curvature comes up.

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