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

[Old Smuggler]

Why not? Central to GR is that it is locally like SR.

True, but..

Hence two sufficiently close comoving observers can also be thought of as two observers moving apart in minkowski space.

This does not follow. Recall that the set of comoving observers taken at some instant of cosmic time makes up a 3D spatial hypersurface. Two neighbouring comoving observers can be thought of as two observers moving apart in Minkowski space-time only if this 3D hypersurface coincides with the counterpart 3D hypersurface in the tangent space-time in the limit when the distance between the neighbours goes to zero. This works well for open FRW models but not for closed (or flat) FRW models.

This means that for closed (or flat) FRW models, the cosmic redshift should be thought of as an effect coming from the curvature of space-time, with no "kinematic" contribution (somewhat similarly to the observed spectral shift observed between two hovering observers at different heights in Schwarzschild space-time). For open FRW models the cosmic redshift should be thought of as an effect coming from both curvature and "kinematic" contributions, but such that for small enough distances, the "kinematic" effect will always dominate. A discussion of this including calculations can be found in the newsgroup sci.astro.research about November 2004.

 

[Ich]

Hi again, Old Smuggler.

This works well for open FRW models but not for closed (or flat) FRW models.

This works well in all FRW spacetimes, as long as you look at first order effects only. Locally, spacetime is flat and expansion is the same as motion.

This means that for closed (or flat) FRW models, the cosmic redshift should be thought of as an effect coming from the curvature of space-time, with no "kinematic" contribution (somewhat similarly to the observed spectral shift observed between two hovering observers at different heights in Schwarzschild space-time).

Gravitation is second order in distance. Bunn and Hogg explicitly exclude second order effects.

Hi nutgeb,

At nearly infinitely small distances, the erroneous SR time dilation component is nearly infinitely small, which is acceptable for any single measurement. But if the individual observations of a nearly infinitely large number of adjacent observers are integrated mathematically, those nearly infinitely small errors will accumulate into one large, significant error.

No, the error vanishes in this limit. But I have a correction to my earlier post: It's not a sum of classical redshifts, but a combination, i.e. a multiplication. If we denote rapidity with w, the effect is not
$$f=f_0 \, (1+w)$$, but
$$f=f_0 \, e^w$$.

Regardless, they must agree in principle that all observed comoving events in the universe have the same duration and occur simultaneously, when corrected for light travel distance from the comoving emitter to each comoving observer (and correcting for any intervening inhomogeneities).

No. Events don't have a duration, and when they correct for light travel distance, they use the standard method and thus will not agree. That is the point I am talking about: cosmological coordinates are not Minkowski coordinates and if you treat them as Minkowski coordinates, you will lose the way.

 

[nutgeb]

No, the error vanishes in this limit.

I agree that at the extreme limit, the accumulated error of including SR time dilation in the computation goes to zero. Zero X infinity = zero.

But that just brings us full circle on the analysis of Bunn & Hogg. When the difference between using SR Doppler and classic Doppler goes to zero as accumulated SR time dilation goes to zero, then Bunn & Hogg's approach becomes the same as if we had just used an accumulation of classical Doppler redshifts in the first place. And since we know that the accumulation of classical redshifts alone over the full light path does not yield a correct number for the cosmological redshift, Bunn & Hogg's mathematical approach must be fundamentally invalid.

Added by edit: I expect that at the extreme limit, the accumulated classical Doppler shift would also go to zero, because the discrete Doppler shift measured by each adjacent observer becomes infintesimal and approaches zero. This doesn't seem helpful to generating a useful outcome. If the accumulated SR and classical Doppler shift both approach zero regardless of how large the total proper distance is, the redshift calculation always has a 0 result which clearly is invalid.

No. Events don't have a duration, and when they correct for light travel distance, they use the standard method and thus will not agree.

I used the term "duration" to mean the duration of a discrete process, such as the duration of a supernova as viewed by two distantly separated observers. I don't understand your statement that because the two comoving observers will use the "standard method" to correct for light travel distance they will disagree on the duration of a comoving supernova process. Both comoving observers' clocks are running identical cosmological time. And they each will measure light travel distance from the single distant emitter to themselves by using the standard cosmological redshift formula based on the expansion of the scale factor. Why will that cause them to disagree about the light travel distance and/or the duration of the supernova process?

 

[Chronos]

What 'cosmological time' are you referring to nutgeb?

 

[nutgeb]

What 'cosmological time' are you referring to nutgeb?

Here's the quick and dirty Wikipedia definition. I'll look around for a more explanatory reference.

"Cosmic time (also known as "time since the big bang") is the time coordinate commonly used in the Big Bang models of physical cosmology. It is defined for homogeneous, expanding universes as follows: Choose a time coordinate so that the universe has the same density everywhere at each moment in time (the fact that this is possible means that the universe is, by definition, homogeneous). Measure the passage of time using clocks moving with the Hubble flow. Choose the big bang singularity as the origin of the time coordinate.

Cosmic time is the standard time coordinate for specifying the Friedmann-Lemaître-Robertson-Walker solutions of Einstein's equations."

 

[marcus]

Yes I think Wikipedia is basically right. It is a common accessory to doing cosmology. The Friedman equations model is based on it. Hubble law assumes it. When you say the comoving distance now is proportional to the rate it is increasing now, all that is meaningful because you have a standard way to slice, a standard concept of simultaneity that everybody uses.

I wouldn't compete with Wiki but I think of cosmological time as the time told by the temperature of the CMB. Two observers who both measure the same CMB temp (as long as one is not significantly farther down in some gravity well, as long as they are roughly on the same level so to speak) are contemporaries. Pun.

 

[nutgeb]

I agree (despite the blushworthy pun) that any privileged comoving observers will measure the same CMB temp. Without any dipole.

Here's another description of cosmological time, from Cosmology Physics by John Peacock (1999) p.67:

"COSMOLOGICAL TIME The first point to note is that something suspiciously like a universal time exists in an isotropic universe. Consider a set of observers in different locations, all of whom are at rest with respect to the matter in their vicinity (these characters are usually termed "fundamental observers"). We can envisage them as each sitting on a different galaxy, and so receding from each other with the general expansion (although real galaxies have in addition random velocities of order 100 km/s and so are not strictly fundamental observers). We can define a global time coordinate t, which is the time measured by the clocks of these observers - i.e. t is the proper time measured by an observer at rest with respect to the local matter distribution. The 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. Using this time coordinate plus isotropy, we already have enough information to conclude that the metric must take the following form [omitted]. Here, we have used the equivalence principle to say that the proper time interval between two distant events would look locally like special relativity to a fundamental observer on the spot: for them c²dt² = c²dt² - dx² - dy² - dz² . Since we use the same time coordinate as they do, our only difficulty is in the spatial part of the metric: relating their dx etc. to spatial coordinates centred on us."

 

[Chronos]

That is exactly right, nutgeb. I think there may be some constants lurking in the background, but, I agree with you [and Peacock] in principle.

 

[Old Smuggler]

This works well in all FRW spacetimes, as long as you look at first order effects only. Locally, spacetime is flat and expansion is the same as motion.

You have been seriously mislead by a really bad paper (see below).

There is no way to foliate Minkowski space-time into flat hypersurfaces such that their evolution
describes an isotropic expansion. The only way to foliate (a subset of) Minkowski space-time to describe an isotropic expansion is via hypersurfaces with negative curvature; this is the empty FRW (Milne) model. (Minkowski space-time cannot be foliated into hypersurfaces with positive curvature.)

By a suitable (constant) scaling of the time coordinate in the Milne model, it is possible to set up an
initial-value problem such that the metric of space and the first derivative of the spatial metric
with respect to time (essentially the extrinsic curvature) are identical (at some initial hypersurface)
for the Milne model and any open FRW model. The differences in evolution then appear only via
the second derivative of the spatial metric with respect to time (essentially space-time curvature).
This shows that the expansion for open FRW models can be said to be "kinematic" for small
enough times/distances.

For a flat FRW model no such correspondence exists (since flat hypersurfaces are geodesically
embedded into Minkowski space-time, their extrinsic curvature vanishes, i.e., the first derivative of the spatial geometry with respect to time vanishes. No scaling of the time coordinate can change that). This means that the expansion described in a flat FRW model has none resemblance whatsoever to any counterpart in Minkowski space-time. Thus for a flat (and closed) FRW model, the expansion is purely a curvature effect.

Gravitation is second order in distance. Bunn and Hogg explicitly exclude second order effects.

That paper is built on a simple misunderstanding. They notice that the components of the Riemann curvature tensor is small enough to be neglected. This is true, but the curvature of space-time do not influence spectral shift calculations via the Riemann tensor, but via its effect on the connection
coefficients. This effect is of first order in general, and cannot be neglected.

Since the Bunn & Hogg paper is built on a false premise, their conclusions are wrong and misleading. IMO this is a crackpot paper of the worst kind; it contains a lot of bogus claims and very little sound science. Hopefully it will not pass peer review. However, it would seem that merely by
posting this paper on the preprint server, Bunn & Hogg have made laughing stock of themselves.

 

[Ich]

And since we know that the accumulation of classical redshifts alone over the full light path does not yield a correct number

Why do you think so? Of course it yields a correct number, it couldn't be otherwise. Why should the result be different if you check it at infinitely many points?

And they each will measure light travel distance from the single distant emitter to themselves by using the standard cosmological redshift formula based on the expansion of the scale factor.

Ah, I see where we came apart. The "standard method" to get the light travel distance is to measure it, not to infer it from redshift. Whatever, the result is that you divide the redshifted duration by the classical doppler effect, which leaves time dilation.

There is no way to foliate Minkowski space-time into flat hypersurfaces such that their evolution describes an isotropic expansion.

I don't need to find a global foliation - or the other way round, the standard foliation will do because it deviates from the local "tangent space" (flat spacetime approximation) only in second order. IOW, if I take any comoving observers 4-velocity as time and create an orthonormal basis, expansion looks like motion.
I don't know why we disagree here, basically Bunn and Hogg say that spacetime looks locally flat, and that is simply true. The deviations are of second order, one higher than the velocity - expansion equivalence.

They notice that the components of the Riemann curvature tensor is small enough to be neglected. This is true, but the curvature of space-time do not influence spectral shift calculations via the Riemann tensor, but via its effect on the connection
coefficients. This effect is of first order in general, and cannot be neglected.

Their procedure is essentially a parallel transport of the wave vector. They don't neglect the effect of the nontrivial connection, it is implicitly incorporated in the shift from one flat frame to the next - which they don't specify.
That's one of the point I'd criticize, if you want to get quantitative results from their approach, you're thrown back on "general spacetime geometry", which means that you have to calculate everything the tedious way where this explanation as doppler shift doesn't helt at all.
Another point is that they deny the importance of gravitational redshift. If you include second-order terms, you get a quantitatively useful approximation, and gravitational redshift comes into play. Of course it was there all the time, it's just that "spacetime curvature which we don't have to care about" now gets its name and can be included as a perturbation.
But that paper is far from crackpottery, Bunn and Hogg just have an onesided agenda which they try to push.

 

[Old Smuggler]

I don't need to find a global foliation - or the other way round, the standard foliation will do because it deviates from the local "tangent space" (flat spacetime approximation) only in second order. IOW, if I take any comoving observers 4-velocity as time and create an orthonormal basis, expansion looks like motion.

You are confused. When discussing space expansion in context of the FRW models, these models do not come as featureless 4D manifolds, but as manifolds foliated into a specific set of hypersurfaces. The evolution of these hypersurfaces in terms of a parameter (time) describes the expansion. Equivalently, one may view space-time as threaded by a specific family of observers moving orthogonally to the hyper-surfaces. Other foliations of the FRW models or the evolution of other families of observers with time are simply irrelevant when discussing space expansion in context of the FRW models. In particular, any model involving Minkowski space-time foliated into flat hyper-surfaces and
describing the "expansion" by means of receding test particles (such that their velocities mimic the Hubble Law as seen by a particular observer), has nothing to do with the FRW models whatsoever.

This means that no part of space expansion (for small distances/times) in a closed or flat FRW model is captured in each tangent space-time - in other words, that for these cases, the expansion cannot reasonably be interpreted as motion in flat space-time, not even locally.

I don't know why we disagree here, basically Bunn and Hogg say that spacetime looks locally flat, and that is simply true. The deviations are of second order, one higher than the velocity - expansion equivalence.

The important deviations are in the connection coefficients - not negligible in general.

Their procedure is essentially a parallel transport of the wave vector. They don't neglect the effect of the nontrivial connection, it is implicitly incorporated in the shift from one flat frame to the next - which they don't specify.

Consider any flat FRW model (use cartesian space coordinates). The non-zero connection coefficients all contain the first derivative of the scale factor with respect to time. Now take the corresponding connection coefficients of the tangent space-time (same foliation). Here all connection coefficients vanish. It should be clear that for this case, by parallel transporting vectors along a null curve, no contribution at all to the spectral shift comes from each flat frame. Trying to avoid this fact by claiming that "the effect of the non-trivial connection coefficients are incorporated in the shift from one flat frame to the next" is just mumbo-jumbo when it is not shown how this yields the same quantitative result as parallel transport with the full set of connection coefficients coming from curved space-time. Anyway, in absence of space-time curvature, there would be no need to shift between flat frames.

That's one of the point I'd criticize, if you want to get quantitative results from their approach, you're thrown back on "general spacetime geometry", which means that you have to calculate everything the tedious way where this explanation as doppler shift doesn't helt at all.
Another point is that they deny the importance of gravitational redshift. If you include second-order terms, you get a quantitatively useful approximation, and gravitational redshift comes into play. Of course it was there all the time, it's just that "spacetime curvature which we don't have to care about" now gets its name and can be included as a perturbation.
But that paper is far from crackpottery, Bunn and Hogg just have an onesided agenda which they try to push.

This paper has succeeded in utterly confusing the issue and fooling (at least one of) its readers.
That is the most dangerous form of crackpottery.

 

[Peter Watkins]

Marcus, and others, thank you for your time. Post #18 seems to have resolved the multiverse question! It states that the "einstein online" reference to all being compressed refers only to that part of the universe that we can see. The clear implication being that there is more that we can't see. If there were other expansions like ours, whether hundreds, thousands or billions, ie. the big bang occurring everywhere, presumably we would never see them due to the expansion of the space between us and them? This in turn means that we only have to think about our own little universe and forget questions about infinity. I assume that our omni-directional expansion would have produced a more or less spherical universlet. Whilst we are in the centre of our sphere of vision, it is doubtful that we are at the centre of our universe. This means that we could only be positioned between the centre and the outer "edge". Is it known approximately where, along this radial line, we are positioned?

 

[nutgeb]

Why do you think so? Of course it yields a correct number, it couldn't be otherwise. Why should the result be different if you check it at infinitely many points?

Ich, I'm not sure your comment relates to what I said, which is that equation for calculating classical Doppler shift does not even remotely yield a correct approximation of cosmological redshift, whether or not one uses the Bunn & Hogg approach of comparing the emitter's recession velocity at emission time with the observer's recession velocity now.

For example, here are a few selected cosmological redshifts, compared with the corresponding emitter velocity at emission and observer velocity now in units of c (as calculated with the Wright and Morgan cosmic calculators):

============================================

Cosmological Redshift (z+1): ...4...32...256

Emitter V @ emission: ...26.58c...8.13c...1.62c

Observer V now:.....3.26c...2.76c...1.53c

Calculated Doppler shift: ...7.7...13.32...23.97

============================================

Using the classical Doppler equation:

$$\frac{\lambda_{now}}{\lambda_{emit}} = \frac{c + V_{em}}{c-V_{now}}$$

the calculated classical Doppler shift is larger than the cosmological redshift at low z values and many times smaller than it at high z values. There's just no way to manipulate recession velocities to yield a classical Doppler shift even remotely as large as the cosmological redshift at (z+1) = 256.

The example of calculated Doppler shift in the table above divides both recession velocities by 2, assuming the rough approximation that the emitter and source respectively would be moving away from an imaginary midpoint between them at half of their relative total velocities.

 

[Chronos]

I agree with Old Smuggler on that point, nutgeb. Your math is inconsistent.

 

[Ich]

Other foliations of the FRW models or the evolution of other families of observers with time are simply irrelevant when discussing space expansion in context of the FRW models.

Sorry, I don't follow. You may use whatver coordinates you like, and if the authors choose to use local standard inertial frames, that's perfectly legitimate. And since we're discussing this paper, this approach is anything but irrelevant.

In particular, any model involving Minkowski space-time foliated into flat hyper-surfaces and describing the "expansion" by means of receding test particles (such that their velocities mimic the Hubble Law as seen by a particular observer), has nothing to do with the FRW models whatsoever.

This specific "model" is nothing but a differet coordinate representation of a specific (the empty) FRW solution. Nothing wrong with it.

This means that no part of space expansion (for small distances/times) in a closed or flat FRW model is captured in each tangent space-time - in other words, that for these cases, the expansion cannot reasonably be interpreted as motion in flat space-time, not even locally.

I don't know how you come to this conclusion. If we ignore second order effect, any spacetime can locally (and for a short time) be described as flat minkowski space with moving particles in it. That has nothing to do with space curvature of the original foliation, that's second order and irrelevant.
Hey, for 70 years, nobody knew wheter space is flat or positively or negatively curved. This is irrelevant for nearby redshift observations, we see galaxies moving away from us, and that's it. It's irritating that you seem to deny this fact, maybe I misunderstood you. When you say "locally", don't you mean also "for a short time"?

Trying to avoid this fact by claiming that "the effect of the non-trivial connection coefficients are incorporated in the shift from one flat frame to the next" is just mumbo-jumbo when it is not shown how this yields the same quantitative result as parallel transport with the full set of connection coefficients coming from curved space-time.

We both agree that parallel transporting the emitter velocity to a nearby absorber along a null curve gives the correct SR doppler shift. Actually, you teached me that.
We both agree that on small scales, for short time, there is a standard inertial frame that covers any smooth spacetime and is accurate to firat order.
We both agree that parallel transport along arbitrary paths leaves a vector unchanged (again, to first order).
Which means that, in this frame, the emitter has some definite velocity relative to the observer, and that this velocity gives the correct SR doppler shift. The classical doppler will do also, because we're ignoring second order effects.

Anyway, in absence of space-time curvature, there would be no need to shift between flat frames.

Of course you have to boost from one frame to the next, if you use Bunn and Hogg's procedure, where the local observers are at rest in the respective inertial frame. Those small dv 's add up to the accurate rapidity.

This paper has succeeded in utterly confusing the issue and fooling (at least one of) its readers.

Agreed, but until now you haven't convinced me that I am this reader.

 

[Ich]

Ich, I'm not sure your comment relates to what I said, which is that equation for calculating classical Doppler shift does not even remotely yield a correct approximation of cosmological redshift, whether or not one uses the Bunn & Hogg approach of comparing the emitter's recession velocity at emission time with the observer's recession velocity now.

Hmm, but you know that Bunn and Hogg are not comparing those velocities? You already stated this view, and I already said that you must be misunderstanding what they do.

For example, here are a few selected cosmological redshifts, compared with the corresponding emitter velocity at emission and observer velocity now in units of c (as calculated with the Wright and Morgan cosmic calculators):

These velocities are not velocities. They are fancy numbers. If you plug them in any formula except some very specific ones, they give nonsense.
Not because the formula is "not applicable" for some mystic, space-stretching reason, but because the formula expects a velocity as input.

"proper distance" in cosmology is not "proper distance" of SR.
If you could stop the expansion, the distance between object would not equal their cosmological proper distance.
Accordingly, the derivation of cosmological proper distance with respect to cosmological time is not a regular velocity, it is merely a coordinate velocity.

 

[nutgeb]

Hmm, but you know that Bunn and Hogg are not comparing those velocities?

As far as I can tell, Bunn & Hogg don't actually calculate any numerical values. What they suggest should be calculated is an integration of small proper distances in nearly flat local inertial frames. They would not suggest calculating a global velocity using SR because of course there is no global SR inertial frame in a gravitating FRW model.

Since, unlike SR Doppler redshift, classical Doppler shift is a linear equation, it seems to me that integrating a nearly infinite series of classical Doppler shifts is not going to yield a fundamentally different numerical redshift value than is obtained by calculating a single global classical Doppler shift.

"proper distance" in cosmology is not "proper distance" of SR.
If you could stop the expansion, the distance between object would not equal their cosmological proper distance.

Interesting statement. What would you calculate to be the difference in the numerical value of the two kinds of proper distance for an object which emitted light at (z+1)=256 which we are receiving now?

Accordingly, the derivation of cosmological proper distance with respect to cosmological time is not a regular velocity, it is merely a coordinate velocity.

If you intend to compute the numerical value of the difference between "cosmological proper distance" and "SR proper distance" (and the corresponding recession velocity differential) without reference to cosmological time or any other time coordinate system, be my guest. Show me a numerical value that isn't associated with a time coordinate (or with a cosmic density value, which is merely a surrogate for cosmological time in the FRW model).

 

[Ich]

What they suggest should be calculated is an integration of small proper distances in nearly flat local inertial frames.

No, they don't suggest that you actually calculate redshift that way. That'd be pretty messy, and we know the result from the much simpler calculation in FRW-coordinates, where you can exploit the symmetries of a cosmological spacetime. And it's not an integration of small proper distances, it's an integration (actually an infinite product, not an infinite sum) of small redshifts. But no need to calculate, just imagine the process to get a different view on cosmological redshift.

Since, unlike SR Doppler redshift, classical Doppler shift is a linear equation, it seems to me that integrating a nearly infinite series of classical Doppler shifts is not going to yield a fundamentally different numerical redshift value than is obtained by calculating a single global classical Doppler shift.

No. Classical Doppler shift depends on absolute speed, or your speed through the medium. Here, for each state of motion, you are supposed to be at rest wrt "the medium". It's a numerical approximation, valid at each point, not a statement about the true nature of doppler shifts.

Interesting statement. What would you calculate to be the difference in the numerical value of the two kinds of proper distance for an object which emitted light at (z+1)=256 which we are receiving now?

Take an empty spacetime (age: 13.7 GY), as there exists an relatively easy unambiguous answer.
Flat speed now: 0.999969
cosmological speed now: 255
Flat distance now: 13.69958 GLY
cosmological distance now: 3493.5 GLY
If you're going to stop the expansion now (cosmological time), the flat distance (now the only sensible one) would be 10^110 GLY. Pretty far away, but that's the distance you have to travel if you want to reach this observer. The real distance, so to speak.

Don't get me wrong, I'm not proposing that I have some magical new coordinates that must be used instead of the usual ones. I merely point out that cosmological coordinates are just coordinates, admittedly very usual and in a sense preferred ones, but coordinates. If you can't plug those values in the standard equations, well, that's not nescessarily because relativity doesn't work anymore. Those coordinates give "nonsensical" answers even if SR works perfectly fine.
Of course cosmological coordinates are not nonsensical, but they are in no way Minkowski coordinates, and if they behave strangely - no, that's not necessarily because of different physics, it#s because coordinates do whatever you (or they) want.
I'm so insisting because I experienced that even experts often fail to distinguish between coordinates and physics. Just have a look at the http://arxiv.org/abs/astro-ph/0310808" , a famous one, a good one, and often cited. Section 4.2: horribly uninformed and wrong. These are supposed to be educational papers.

If you intend to compute the numerical value of the difference between "cosmological proper distance" and "SR proper distance" (and the corresponding recession velocity differential) without reference to cosmological time or any other time coordinate system, be my guest.

I do explicitly not intend so, except for special cases. And I do not propose a different better set of coordinates. As I said, I just want to point out that physics is coordinate independent, and if you can gain insight from a different set of coordinates: do so!

 

[Old Smuggler]

Sorry, I don't follow. You may use whatver coordinates you like, and if the authors choose to use local standard inertial frames, that's perfectly legitimate. And since we're discussing this paper, this approach is anything but irrelevant.

It's irrelevant because it confuses the issue (see below). The issue we are discussing is whether or not spectral shifts observed between comoving observers in any FRW model can reasonably be interpreted as due to motion in flat space-time for small distances/times.

This specific "model" is nothing but a differet coordinate representation of a specific (the empty) FRW solution. Nothing wrong with it.

Yes, for the empty FRW model that representation is equivalent - but it gives you the false impression
that you can use such an representation approximately for all FRW models to correctly decide the issue we are discussing.

I don't know how you come to this conclusion. If we ignore second order effect, any spacetime can locally (and for a short time) be described as flat minkowski space with moving particles in it. That has nothing to do with space curvature of the original foliation, that's second order and irrelevant.

Space curvature is not so important in itself, but in context of the FRW models, it is useful since it immediately tells
you whether or not the expansion includes an element of "motion". This is precisely because the empty FRW model shows the only possible
foliation of Minkowski space-time representing isotropic expansion, and that the geometry of the hypersurfaces is hyperbolic.

Another important point about keeping the original foliation is that this makes it easy to identify the comoving observers since those observers move orthogonally to these hypersurfaces. However, if you try to represent any FRW model as Minkowski space-time foliated by flat hypersurfaces, the world lines of the particles representing the expansion will not in general coincide with the comoving observers' world lines.
Here is where you go awry.

Take a flat FRW model as an example. Here the original foliation is the same as for Minkowski space-time. The comoving observers in the flat FRW model move orthogonally to the flat hypersurfaces. But the particles in the Minkowski representation
do not. This means that these particles do not represent the comoving observers - this is a set of
different observers irrelevant to the issue we are discussing.

Hey, for 70 years, nobody knew wheter space is flat or positively or negatively curved. This is irrelevant for nearby redshift observations, we see galaxies moving away from us, and that's it. It's irritating that you seem to deny this fact, maybe I misunderstood you. When you say "locally", don't you mean also "for a short time"?

All I am saying is that any spectral shift can reasonably be interpreted as a Doppler shift in flat space-time only if this shift is also present in the tangent space-time. That is, take the 4-velocity of
the emitter and parallel transport it along the null curve to a nearby receiver. Calculate the spectral shift. Do the same procedure in the tangent space-time. If the spectral shifts coincide to the relevant
accuracy, the shift can reasonably be interpreted as a Doppler shift in flat space-time. If not, it cannot.

We both agree that parallel transporting the emitter velocity to a nearby absorber along a null curve gives the correct SR doppler shift. Actually, you teached me that.

Yes, but this yields the generalized Doppler shift. It does not imply that the generalized Doppler shift can always reasonably be interpreted as due to motion in flat space-time.

We both agree that on small scales, for short time, there is a standard inertial frame that covers any smooth spacetime and is accurate to firat order.

Sorry, but this is too vague in the context of the issue we are discussing. Please clarify.

We both agree that parallel transport along arbitrary paths leaves a vector unchanged (again, to first order).

I cannot see that this is relevant for the issue we are discussing.

Which means that, in this frame, the emitter has some definite velocity relative to the observer, and that this velocity gives the correct SR doppler shift. The classical doppler will do also, because we're ignoring second order effects.

No. Contributions to the generalized Doppler shift come both from motion and from curvature effects. You cannot eliminate curvature effects the way you think, because they act via the connection
coefficients and thus are non-negligible in general. Besides, there is the problem of correctly representing the comoving observers in the tangent space-time mentioned above.

Again an illustrating example is a FRW model with flat space sections. What you really do here, is to transform the space-time curvature of the FRW model into a velocity field in Minkowski space-time. That might not be so bad, but when you then claim that the space-time curvature of the FRW model were negligible to begin with ("of higher order"), and that the corresponding spectral shift must be interpreted as due to motion in flat space-time, it is just crazy.

Your comment on the classical Doppler effect is irrelevant.

Of course you have to boost from one frame to the next, if you use Bunn and Hogg's procedure, where the local observers are at rest in the respective inertial frame. Those small dv 's add up to the accurate rapidity.

Rather than parallel transporting the emitter's 4-velocity along the null curve to the receiver in one
go, one may indeed do the transport via many intervening comoving observers. But this does not
change anything - as long as each observed frequency is passed along, the total generalized Doppler effect is unaffected, and so is its interpretation.

Agreed, but until now you haven't convinced me that I am this reader.

I'm not out to convince anybody of anything - that is a waste of time in my experience.
However, if you can convince yourself, that's another matter. Tomorrow seems to be an extremely
appropriate day for it.

 

[nutgeb]

But no need to calculate, just imagine the process to get a different view on cosmological redshift.

No, I don't want to just "imagine the process" Bunn & Hogg use, because I'm convinced it's wrong. It is wrong to use accumulated SR Doppler shifts to calculate cosmological redshift, because SR Doppler shift includes an element of SR time dilation which has no place in the light path between a clock-synchronized emitter and observer in the FRW model. It is also wrong to use accumulated non-SR classical Doppler shifts for the same purpose, because this approach cannot be demonstrated to yield even a remotely correct numerical result.

Classical Doppler shift depends on absolute speed, or your speed through the medium. Here, for each state of motion, you are supposed to be at rest wrt "the medium".

If one is at rest wrt the medium in each adjacent observer's small local space along a light path, then each such observer must measure 0 classical Doppler shift. An infinite accumulation of 0's is 0, clearly an invalid result. That approach is a dead end.

Take an empty spacetime (age: 13.7 GY), as there exists an relatively easy unambiguous answer.
Flat speed now: 0.999969
cosmological speed now: 255
Flat distance now: 13.69958 GLY
cosmological distance now: 3493.5 GLY
If you're going to stop the expansion now (cosmological time), the flat distance (now the only sensible one) would be 10^110 GLY.

I don't follow what your definition of "cosmological speed" and "cosmological distance" are in this context. Your empty spacetime has no gravity, so no two observers should be able to have recession velocities > c with respect to each other, regardless of how you parse it.

Just have a look at the http://arxiv.org/abs/astro-ph/0310808" , a famous one, a good one, and often cited. Section 4.2: horribly uninformed and wrong. These are supposed to be educational papers.

What specifically is your problem with the cited passage?

 

[Ich]

No, I don't want to ...etc.

You're going in circles. Comoving observers still are not synchronized, time dilatation still can be ignored, and the approch with classical redshifts still yields correct results, as sketched in https://www.physicsforums.com/showpost.php?p=2135265&postcount=37".

If one is at rest wrt the medium in each adjacent observer's small local space along a light path, then each such observer must measure 0 classical Doppler shift. An infinite accumulation of 0's is 0, clearly an invalid result. That approach is a dead end.

That was hard work, quoting out of context to make sure you can deliberately misunderstand the rest. Or did you really not understand what I was saying? If so, I apologize.

I don't follow what your definition of "cosmological speed" and "cosmological distance" are in this context.

Today, I can't follow neither, because I forgot the logarithm yesterday. The (hopefully) correct values are:
Flat speed now: 0.999969
cosmological speed now: 5.545
Flat distance now: 13.69958 GLY
cosmological distance now: 76 GLY
If you're going to stop the expansion now (cosmological time), the flat distance (now the only sensible one) would be 1754 GLY.

Your empty spacetime has no gravity, so no two observers should be able to have recession velocities > c with respect to each other, regardless of how you parse it.

You're getting close. Go to Ned Wrights calculator and enter zero density everywhere. See what happens, and click on "comoving radial distance" for an explanation.

What specifically is your problem with the cited passage?

Davis & Lineweaver fail to correctly state the SR-only case. That's obviously the empty universe with H~1/t (not H=const.), and it's excluded by SN data alone by 3 standard deviations, not 28 or whatever they claim.

 

[Ich]

The issue we are discussing is whether or not spectral shifts observed between comoving observers in any FRW model can reasonably be interpreted as due to motion in flat space-time for small distances/times.

Exactly.

This specific "model" is nothing but a differet coordinate representation of a specific (the empty) FRW solution. Nothing wrong with it.

Yes, for the empty FRW model that representation is equivalent - but it gives you the false impression
that you can use such an representation approximately for all FRW models to correctly decide the issue we are discussing.

Actually, I can use the model to first order precision. I'll explain the procedure later.

However, if you try to represent any FRW model as Minkowski space-time foliated by flat hypersurfaces, the world lines of the particles representing the expansion will not in general coincide with the comoving observers' world lines.

This is a point where I brought confusion into the discussion by referring to "flat" approximations. The approximation I have in mind neglects second order terms and thus describes a flat spacetime. But that does not mean that it uses a foliation where space is flat, it simply is insensitive to curvature.

All I am saying is that any spectral shift can reasonably be interpreted as a Doppler shift in flat space-time only if this shift is also present in the tangent space-time. That is, take the 4-velocity of
the emitter and parallel transport it along the null curve to a nearby receiver. Calculate the spectral shift. Do the same procedure in the tangent space-time. If the spectral shifts coincide to the relevant
accuracy, the shift can reasonably be interpreted as a Doppler shift in flat space-time. If not, it cannot.

I agree.

Again an illustrating example is a FRW model with flat space sections. What you really do here, is to transform the space-time curvature of the FRW model into a velocity field in Minkowski space-time.

No, I transform explicitly time-dependent coordinates to coordinates with the standard minkowski interval. That has nothing to do with spacetime curvature. It's just a coordinate transformation accompanied by a linearization. In the new coordinates, comoving observers have a definite velocity. If this had to do with curvature, the linearization you eliminate these velocities.
Specificly, here's the procedure:
I start with the FRW metric ds² = dt² - a(t)²dr². Depending on the details of the spacetime and the transformation I use, the other two space dimensions deviate from flat space in second order. that doesn't bother me, I'm after first order effects only.
Now, at a specific epoch t0, I can linearize the funktion a(t) by setting a(t)=const. * (t-t0'), where
$$(t_0-t_0')=1/H_0=a/ \dot a$$
and the constant ensures that a(t0)=1.
Now that a(t) is linear, I can get rid of it by the same transformations that bring the empty FRW coordinates to Minkowski coordinates, i.e.
$$t_{FRW} = \sqrt{t_{mink}^2 - x^2}$$
$$r = 1/H_0 \tanh^{-1}(x/t_{mink})$$
In these standard coordinates, comoving observers have the claimed velocities. That works because these velocities are proportional to $$\dot a$$ and independent of $$\ddot a$$. They are not a curvature effect.

 

[Old Smuggler]

Specificly, here's the procedure:
I start with the FRW metric ds² = dt² - a(t)²dr². Depending on the details of the spacetime and the transformation I use, the other two space dimensions deviate from flat space in second order. that doesn't bother me, I'm after first order effects only.
Now, at a specific epoch t0, I can linearize the funktion a(t) by setting a(t)=const. * (t-t0'), where
$$(t_0-t_0')=1/H_0=a/ \dot a$$
and the constant ensures that a(t0)=1.
Now that a(t) is linear, I can get rid of it by the same transformations that bring the empty FRW coordinates to Minkowski coordinates, i.e.
$$t_{FRW} = \sqrt{t_{mink}^2 - x^2}$$
$$r = 1/H_0 \tanh^{-1}(x/t_{mink})$$
In these standard coordinates, comoving observers have the claimed velocities. That works because these velocities are proportional to $$\dot a$$ and independent of $$\ddot a$$. They are not a curvature effect.

You have assumed that the effects of curvature must be in $$\ddot a$$ or higher order terms
and that $$\dot a$$ is always independent of curvature effects. That assumption is quite wrong.
The point is that the affine connection is curved in general. This means that the Riemann tensor is non-zero, but also that there are curvature effects via non-zero connection coefficients.
For a flat connection, the non-zero connection coefficients can be eliminated via a suitable coordinate
transformation. That cannot be done (globally) in a curved manifold.

To illustrate this point; for a flat FRW model the time-dependent connection coefficients are proportional to $$\dot a$$. The only difference between the line elements of Minkowski space-time on the one hand and of the flat FRW model on the other (using standard coordinates), is the presence of a time-dependent scale factor. Yet the latter line element yields non-zero connection coefficients proportional to $$\dot a$$. The coordinate systems used are the same, so the non-zero connection coefficients cannot be blamed on a coordinate effect in flat space-time. The only reasonable explanation is that the non-zero connection coefficients (and thus $$\dot a$$) comes from curvature. This means that your assumption that $$\dot a$$ is independent of curvature effects is incorrect.

If you do not agree with this, we should agree to disagree.

I see that I wrote somewhere that the interpretation of spectral shifts in a non-empty, open FRW
model can always be interpreted as motion in flat space-time for small times/distances. In light of
my subsequent posts this view would be wrong - for a non-empty open FRW model the spectral shift
should be interpreted as a mix of curvature effects and velocity in flat space-time. Also some
comments on the initial-value problem for open FRW models were a bit misleading. Otherwise,
what I have written in this discussion should be reasonably correct (except some minor nitpicks).

Anyway, since it is now quite clear at what points we disagree, we should round off this discussion.

By the way, it's All April Fool's Day today. Do you consider yourself fooled?

 

[sylas]

Davis & Lineweaver fail to correctly state the SR-only case. That's obviously the empty universe with H~1/t (not H=const.), and it's excluded by SN data alone by 3 standard deviations, not 28 or whatever they claim.

Say what? The SR only case is that redshifts are a function of velocity using the Doppler shift. A major difference between SR models of the redshift and expansion models is that an assumption of homogeneity has different implications for observations from deep space.

I suspect you are mixing up definitions of H. In conventional GR based cosmology, H can be defined as (da/dt)/a where a is the scale factor (wrt to present) and t is a proper time co-ordinate. H~1/t is the nice simple solution you get for an empty universe using GR.

In SR, you can't define H in terms of a scale factor, because SR doesn't have expansion of space. However, you can assume that particles with recession velocity v are at a distance proportional to v. That is, we all started out close together and have been moving apart at constant velocities. The particle with redshift z has recession velocity [(1+z)^2-1]/[(1+z)^2+1].c, and it is at distance given by recession velocity by some constant time T by its recession velocity, on the assumption that everything started out from close together; that is, v is proportional to distance. H can be defined as the relationship between distance and v.

Now of course, under this assumption, the value of "H" for an observer at different times is proportional to 1/t. We can't test that, because we can't take observations billions of years apart in time.

However -- and THIS is the key point you seem to be missing -- H is defined here as a common feature of all observations, no matter how distant they may be. In GR the function H is a function of proper time, and so when you look into deep space you are seeing things when H was larger. Given information about time between events in deep space (SN data, for example) you can put strong constraints on the development of the scale factor over time. That is, there is a function from z to age, and from age to the scale factor, and from that to a value for H which was in play at the time the photon left whatever we are observing.

But in the SR model, H is a description of the observation, and it is identical for every particle we observe. When we look at distant particles, we are looking back in time, but the H is the same for all those particles. THAT's what is meant by constant H, I am pretty sure.

Davis and Lineweaver is excellent as an educational tutorial, helping to clear up all kinds of common popular misconceptions. It's perfectly normal to think they've done something wrong; and this is precisely because they tackle popular and entrenched misconceptions. If you think that they have made a mistake, you are probably in a good position to be learning something.

Cheers -- Sylas

 

[Ich]

You have assumed that the effects of curvature must be in$$\ddot a$$ or higher order terms
and that $$\dot a$$is always independent of curvature effects. That assumption is quite wrong.

No. I approximated the scalar function a(t) by its tangent at the point of interest, therefore there are no first order deviations. That's not an assumption, that's basic calculus. All deviations are of second order in cosmological time, therefore at most second order also in private time and private space.

The point is that the affine connection is curved in general. This means that the Riemann tensor is non-zero, but also that there are curvature effects via non-zero connection coefficients.

That's not a point, as these are second order contribuions.

For a flat connection, the non-zero connection coefficients can be eliminated via a suitable coordinate
transformation. That cannot be done (globally) in a curved manifold.

So what? It can be done locally, and that's what we are talking about. More to the point, I actually showed how it is done locally, so unless you're objecting to specific points in the transformation, there's no use telling me that curved space is not globally flat. I know this.
But you should know also that, in suitable coordinates, spacetime can be, locally and to first order, approximated by flat minkowski spacetime with zero connection coefficients (to be sure: first order). You simply have to find the correct local tranformation, and then show that lines of constant r have the appropriate velocity in these coordinates. That's what I have done, maybe you should try also.

The only difference between the line elements of Minkowski space-time on the one hand and of the flat FRW model on the other (using standard coordinates), is the presence of a time-dependent scale factor. Yet the latter line element yields non-zero connection coefficients proportional to LaTeX Code: \\dot a . The coordinate systems used are the same, so the non-zero connection coefficients cannot be blamed on a coordinate effect in flat space-time.

Wow, the line element is different, but the coordinates are the same. Now that's interesting.
And what does "flat FRW model" mean? The empty one? One with flat space?

If you do not agree with this, we should agree to disagree.

Yeah, I also get more and more the impression that this discussions makes no sense.

I see that I wrote somewhere that the interpretation of spectral shifts in a non-empty, open FRW
model can always be interpreted as motion in flat space-time for small times/distances. In light of
my subsequent posts this view would be wrong

In light of your previous writings, which were consistent with basic premises of GR (equivalence principle), observations (we actually see nearby galaxies receding), and the paper we're discussing, you should re-think this statement:

what I have written in this discussion should be reasonably correct (except some minor nitpicks).

because now you're struggling with all three points.

By the way, it's All April Fool's Day today. Do you consider yourself fooled?

I admit, I had the impression already back in March. Someone knowledgeable denying either the existence of local inertial frames or Hubble's law (you see, it's a first order effect in the local frame), maybe there's some spacetime fooliation going on.
Anyway, it was fun.

cheers
Ich

 

[Ich]

I suspect you are mixing up definitions of H. In conventional GR based cosmology, H can be defined as (da/dt)/a where a is the scale factor (wrt to present) and t is a proper time co-ordinate. H~1/t is the nice simple solution you get for an empty universe using GR.

With all respect, I suspect that you're missing a crucial point: The empty expanding universe is a valid FRW solution, with $$\Omega_{\Lambda} = \Omega_M = 0$$, and it's only 3 sigma away from LCDM. Empty spacetime is flat. SR can handle a flat spacetime, you simply have to use a different set of coordinates. Predicted observations, such as redshift of test particles, are independent of the choice of coordinates.
Maybe you want to read what http://www.astro.ucla.edu/~wright/cosmo_02.htm#MD" has to say, or you want to convince yourself.
Start with FRW coordinates (a(T)=T, T0=age of the universe)
$$ds^2=dT^2-T^2dr^2$$
and apply the transformations
$$T = \sqrt{t^2 - x^2}$$
$$r = T_0 \tanh^{-1}(x/t)$$
You'll get
$$ds^2=dt^2-dx^2$$
and you can perform the necessary calculations (redshift, luminositiy distance, angular size distance...) purely in SR.

It's perfectly normal to think they've done something wrong; and this is precisely because they tackle popular and entrenched misconceptions. If you think that they have made a mistake, you are probably in a good position to be learning something.

Ha, that's what I'm telling crackpots all along.
Pease understand that I'm not trying to sell a pet theory of mine. Davis&Lineweavers' analysis contradicts http://books.google.de/books?id=e-w...universe"&as_brr=3&ei=ta7USYHbHYGuzATAmuTeAg", you can convince yourself if you're familiar with th idea of a metric, you can read what other authorities in the field have to say. Or you can take the fact that even Old Smuggler, who disagrees generally with everything I say, agrees with me as evidence with the status of a mathematical proof.
Really, I'm not doing original research here, that chapter is simply wrong.

 

[sylas]

With all respect, I suspect that you're missing a crucial point: The empty expanding universe is a valid FRW solution, with $$\Omega_{\Lambda} = \Omega_M = 0$$, and it's only 3 sigma away from LCDM. Empty spacetime is flat. SR can handle a flat spacetime, you simply have to use a different set of coordinates. Predicted observations, such as redshift of test particles, are independent of the choice of coordinates.

That is not what is meant by an SR model. I do know about the FRW solutions.

The SR model described in Davis and Lineweaver is the model obtained by taking redshift as due to motions in a simple non-expanding space, and calculated as Doppler shift.

That's DIFFERENT from the FRW solution with an empty universe.

There's no error in the Davis and Lineweaver paper on this point, because they are quite clear on what they mean by SR model. It's not just taking an FRW solution and applying SR. It's taking redshift as being a Doppler shift in non-expanding space.

The luminosity distance with z arising from Doppler shifts for particles receding with at uniform velocity from a common origin event is different from that in the empty FRW model.

Cheers -- Sylas

 

[Wallace]

I hope I can contribute here. I think you (sylas and ich) are both basically right.

The empty FRW universe is indeed only 'ruled out' at 3 sigma, but as sylas suggests this is not the model D&L mean by saying 'SR model', they are referring to a particular assumption, valid at low redshift, that gives a bogus result at high redshift.

The point that leads to disagreement is actually a bit subtle. In post #61 ich makes a conformal tranformation between the empty FRW metric and a Minkowski like metric. This is all well and good, however this is only valid radially. If you put the angular terms back into the first line you will see that your transformation does not return a fully conformally Minkowski metric. This means that you cannot use this to determine either the angular diameter or luminosities distances. You need to do a more complex fully conformal transformation in order to do this.

Some technical details of this can be found http://adsabs.harvard.edu/abs/2007MNRAS.381L..50L".

I*think* that the error in the SR model the D&L discuss is that if you work through the details, you can see that that way we define distance in the SR model violates simultaneity, which is why it is okay for small distances but gets worse and worse the further you go.

So yes, a *correct* SR model is identical to an empty FRW universe and to work out the relationship between the FRW co-ordinates and the co-ordinates of this model you need to do the fully conformal transfomation, but D&L are talking about a model that, due to the misidentification of the meaning of co-ordinates, is only a low redshift approximation.

I hope that helps!

 

[Ich]

The SR model described in Davis and Lineweaver is the model obtained by taking redshift as due to motions in a simple non-expanding space, and calculated as Doppler shift.

No, the SR model they use is, frankly, BS. Read this:

However, since SR does not provide a technique for incorporating
acceleration into our calculations for the expansion of the Universe, the best we
can do is assume that the recession velocity, and thus Hubble’s constant, are approximately
the same at the time of emission as they are now6.

A non accelerating universe has $$\dot a = const.$$, thus H=1/t. Constant H is the de Sitter universe, nothing to do with SR, and the extreme case of an accelerating universe.

There's no error in the Davis and Lineweaver paper on this point, because they are quite clear on what they mean by SR model. It's not just taking an FRW solution and applying SR. It's taking redshift as being a Doppler shift in non-expanding space.

Please try to understand my point: in the empty universe, the only difference between "expanding space" and "constant velocity in non-expanding space" is a coordinate transformation. That's nothing more than just taking an FRW solution and applying SR.

The luminosity distance with z arising from Doppler shifts for particles receding with at uniform velocity from a common origin event is different from that in the empty FRW model.

No, if "uniform velocity" means what it should for particles receding from a common origin.
Now this is your claim, please back it up with calculations. You are probably in a good position to be learning something.

 

[Wallace]

A non accelerating universe has $$\dot a = const.$$, thus H=1/t. Constant H is the de Sitter universe, nothing to do with SR, and the extreme case of an accelerating universe.

Hmm, good point. I guess the best we can say then is that D&L introduce a very bad model and then demonstrate that it doesn't fit the data. I'm not sure that they intended it to be a 'correct' model in the sense of it correctly using relativity (SR and GR are of course identical if the universe is empty), I think they were trying to show that a mugs 'SR' model doesn't work, but maybe it was a bit too muggy.

I really think the points of agreement are much more than those of disagreement here, stemming from maybe some loose terminology. I think we can all agree that the 23 sigma model from D&L is not a 'correct' SR model. The disagreement appears to be just how incorrect it is, yes?

 

[Ich]

In post #61 ich makes a conformal tranformation between the empty FRW metric and a Minkowski like metric.

Stop, no! I just match coordinates locally to first order, and drop all the higher order terms. It's neither valid radially nor in the transverse directions, if you're looking at higher orders.
What I'm doing here is an exact coordinate transformation. The angular directions (hyperbolic to flat space) transform correctly, no need to bend the laws of physics. We're talking about a flat spacetime in both cases.

This means that you cannot use this to determine either the angular diameter or luminosities distances.

Of course you can. The hyperbolic space in FRW coordinates stems solely from the definition of the radial coordinate as being measured by comoving observers. If you "fix" that, everything is ok again.

but D&L are talking about a model that, due to the misidentification of the meaning of co-ordinates, is only a low redshift approximation.

Yes, they talk about the wrong model and therefore come to wrong conclusions. I think this is most clearly seen in the passage I quoted before, where they identify "not accelerated" with "constant H", which is bogus.

 

[Wallace]

Alright, I don't want to introduce additional disagreement. As you say, minkowski space and an empty FRW metric are both flat space-times (they have a vanishing Ricci scalar). You can transform between these two co-ordinate systems, without being forced to be vaild only to a given order, via a fully conformal transformation.

Of course you can. The hyperbolic space in FRW coordinates stems solely from the definition of the radial coordinate as being measured by comoving observers. If you "fix" that, everything is ok again.

Right, this 'fixing' is exactly what the transformation does.

I get what you are saying, any co-ordinate transformation is exact, so if your original space-time is flat the transformed one is as well. Just pointing out that the one you suggest doesn't work, on it's own to relate FRW co-moving co-ordinates to their Minkowski counterparts. Clearly you agree with this point, it just wasn't clear to me what you were demonstrating with it original, but now I see.

 

[Ich]

I guess the best we can say then is that D&L introduce a very bad model and then demonstrate that it doesn't fit the data.

Yes, it's a strawman.

I'm not sure that they intended it to be a 'correct' model

They say "the best we can do", so I'd say that they simply didn't know better. I'm convinced that this section would look quite different if they'd write it today.

The disagreement appears to be just how incorrect it is, yes?

Of course. D&L claim incorrectly that the "Doppler/SR interpretation" is ruled out by 23 sigma by SNIa observations alone, I (we) say it's ruled out by ~3 sigma. Taking other observations into account, I think we're rather getting back to 23 sigma.

 

[Old Smuggler]

For the benefit of readers who may possibly fall for the misunderstandings Ich seems to be promoting,
I will contribute with one last post in this discussion.

No. I approximated the scalar function a(t) by its tangent at the point of interest, therefore there are no first order deviations. That's not an assumption, that's basic calculus. All deviations are of second order in cosmological time, therefore at most second order also in private time and private space.

Your "linearisation procedure" is, with a suitable choice of constants, equivalent to expanding a(t) as a truncated Taylor series around some arbitrary time t_0. That is, you set a(t) = a(t_0) + $$\dot a$$(t_0)(t-t_0) and neglect higher order terms. But then you assume that no space-time curvature effects are included since the series is terminated after the linear term. This is not necessarily true since curvature effects
may be included into $$\dot a$$(t_0) (as is the case, in general). This simple misunderstanding may be appropriately called "the Bunn/Hogg fallacy", and you have endorsed it.

So what? It can be done locally, and that's what we are talking about. More to the point, I actually showed how it is done locally, so unless you're objecting to specific points in the transformation, there's no use telling me that curved space is not globally flat. I know this.

The problem is that you do not do what you think you do. Take again the FRW model with flat space
sections. The non-zero connection coefficients are proportional to $$\dot a$$, as usual. But here, since
we cannot perform any relevant coordinate transformation in order to change the connection coefficients
(the coordinates already have the standard form), the correct flat space-time approximation is to neglect $$\dot a$$ altogether. On the other hand, for a non-empty, open FRW model where
the line element is expressed in comoving coordinates, a coordinate transformation to standard
coordinates will change the connection coefficients, but not get rid of them altogether. What is left
should be due to curvature and must be neglected in the correct flat space-time approximation. It
is only for the empty FRW model a coordinate transformation from comoving to standard coordinates
can completely get rid of all the connection coefficients.

On the other hand, approximating a(t) as a Taylor series to first order the way you do, is effectively to include all the crucial effects of the connection coefficients (expressed in comoving coordinates) at the time t_0, since the relevant connection coefficients expressed in comoving coordinates are always proportional to $$\dot a$$. After making a local transformation to standard coordinates, the resulting non-zero velocity field is then just an expression of the fact that the connection coefficients (expressed in comoving coordinates) at the time t_0, are proportional to $$\dot a$$(t_0). You have absolutely no guarantee that these connection coefficients do not include some effects of curvature so that this procedure yields the correct flat space-time approximation for the issue we were discussing. In fact, it fails.

Wow, the line element is different, but the coordinates are the same. Now that's interesting.

You think so? Of course you can keep the coordinate system and change the metric as long as
the coordinate system covers the relevant part of the manifold. That is basic differential geometry.
You should try to learn it some time.

And what does "flat FRW model" mean? The empty one? One with flat space?

I have consistently used "flat FRW model' to mean the FRW model with flat space sections.

That concludes all I have to say in this discussion. You are on your own now. Good luck.

 

[sylas]

No, if "uniform velocity" means what it should for particles receding from a common origin.
Now this is your claim, please back it up with calculations. You are probably in a good position to be learning something.

There's no question about that! I'm sure most of you guys here know more than I do about GR, and metrics and tensors. I learn a lot by trying to work through these kinds of problems.

In any case, I'll go away and try my own analysis, and report back.

Cheers -- Sylas