How to prove the stretching of space

[Naty1]

Here is an interesting 8 page paper I stumbled across in my notes:

Expanding Space: the Root of all Evil?

http://arxiv.org/PS_cache/arxiv/pdf/0707/0707.0380v1.pdf

Several of these 'issues' have been discussed in other threads...I found these are insightful:...the expansion of space is neither more nor less than the increase over time of the distance between observers at rest with respect to the cosmic fluid in terms of the
FRW metric. With this metric... the density and pressures of cosmological fluids must change over cosmic time, and it is this change that represents the basic property of an expanding (or contracting) universe.

The proper time for …..privileged observers at rest with regards to the cosmic fluid ticks at the same rate as cosmic time and hence the watches of all privileged observers are synchronised.

In an expanding universe, the change of the metric implies that the physical distance between any two privileged [comoving] observers increases with time... The Hubble flow is then viewed as a purely kinematical phenomenon —
objects recede because they have been given an initial velocity proportional to distance.

the velocity of [a] particle due its motion relative to the Hubble flow (or equivalently the homogeneous fluid defining the FRW metric) must be less than the speed of light; its velocity due to the increase of the scale factor is not restricted in this way…..

cosmological redshift is not, as is often implied, a gradual process caused by the stretching of the space a photon is traveling through. Rather cosmological redshift is caused by the photon being observed in a different frame to that which it is emitted.

 

[timmdeeg]

Here is an interesting 8 page paper I stumbled across in my notes:

Expanding Space: the Root of all Evil?

http://arxiv.org/PS_cache/arxiv/pdf/0707/0707.0380v1.pdf

Thank you, this paper is very interesting and worthwile to be read. It moreover shows the controversy between cosmologists regarding thought experiments (one of my questions),

Expanding Space: the Root of all Evil? page 2: To illustrate how short this pragamatic formalism falls being platitude, one need no further than Abramowicz et al. (2006), in which a thought experiment of laser ranging in an FRW Universe is proposed to 'prove' that space must expand. This is sensibly refuted by Chodorowski (2006b), but followed by a spurious counter-claim that such a refutation likewise proves space does not expand.

which makes it not easier to improve one's own understanding.

To me this statement

Expanding Space: the Root of all Evil? page 2: The expansion of space is no more extant than magnetic fields are and exists only as a tool for understanding the unambiguous predictions of GR, not a force-like term in a dynamical equation.

sounds very agreeable.

Expanding Space: the Root of all Evil? page 7: The key is to make it clear that the cosmological redshift is not, as is often implied, a gradual process caused by the stretching of the space a photon is traveling through. Rather cosmological redshift is caused by the photon being observed in a different frame to that which it is emitted. In this way it is not as dissimilar to a Doppler shift as is often implied. The difference between frames relates to a changing background metric rather than a different velocity.

Is this proposal in accordance with the parallel transport of the 4-velocity vector?

And how about this thought experiment: Supposed the universe doesn't expand at the time of emission and absorption but expands during the photon's travelling. What kind of shift if any will be measured?

 

[Chalnoth]

And how about this thought experiment: Supposed the universe doesn't expand at the time of emission and absorption but expands during the photon's travelling. What kind of shift if any will be measured?

The observed redshift will be equal to the total amount of expansion between the emission and absorption of the photon, regardless of what the rate of that expansion was at different times.

 

[Old Smuggler]

Sorry for the late reply (no internet connection for the last week).

If I understood you correctly, the redshift observed between FOs depends in the non-empty RW model on whether these are closed, flat, or open and on the space-time curvature and thus not on the choice of coordinates (i).

The redshift depends only on the 4-velocities of the FOs and on the space-time geometry. (This is most easily seen by using said procedure of parallel-transport.) On the other hand, the INTERPRETATION of the redshift depends on the spatial geometry, since the spatial geometry is crucial for determining how well a flat space-time connection approximates the curved space-time connection.

Would you please specify in which cases the redshift is purely gravitational and gravitational/kinematic respectively, including the Lambda-CDM model, the universe in which we live.

As far as the Lambda-CDM model is based on RW-models, the properties of the RW-models apply (see below). If inhomogenities are taken into account, the effects of these come in addition.

In RW-models with flat or spherical space sections, the redshift is entirely due to the non-flat connection and thus indirectly to space-time curvature (i.e., "gravitational"). (See, e.g., arXiv:0911.1205.) For RW-models with hyperbolic space sections things are more complicated, and some part of the redshift is "kinematic" (meaning that some part of the redshift survives even if one replaces the curved space-time metric with a flat one). To decide how much of the redshift is "kinematic", a recipe for spectral shift split-up into "kinematic" and "gravitational" parts is necessary (this can be done unambiguously, at least for small distances).

 

[Naty1]

Would you please specify in which cases the redshift is purely gravitational and gravitational/kinematic respectively, including the Lambda-CDM model, the universe in which we live.

As I understand the consensus from earlier discussions on this subject, such a split in our universe, represented by the Lambda-CDM model, over cosmological distances is arbitrary.

 

[timmdeeg]

The observed redshift will be equal to the total amount of expansion between the emission and absorption of the photon, regardless of what the rate of that expansion was at different times.

Thanks for this clear and unambiguous answer.

 

[timmdeeg]

The redshift depends only on the 4-velocities of the FOs and on the space-time geometry. (This is most easily seen by using said procedure of parallel-transport.) On the other hand, the INTERPRETATION of the redshift depends on the spatial geometry, since the spatial geometry is crucial for determining how well a flat space-time connection approximates the curved space-time connection.
.
In RW-models with flat or spherical space sections, the redshift is entirely due to the non-flat connection and thus indirectly to space-time curvature (i.e., "gravitational"). (See, e.g., arXiv:0911.1205.)

These authors argue that in order to interpret the cosmological redshift in terms of a Doppler effect in non-expanding Minkowskian space-time the observer would have to move away from himself and thus claim (spatial curvature >= 0) "The Doppler interpretation is clearly self-contradictory (page 5). But this is relativised later (page 6):
"Hence, ironically in the context of the recent debate, parallel-transport of four-velocities along photons path can allow cosmological redshifts to be interpretet as a relativistic Doppler effect without the contradiction presented here, provided that the concept of expanding space is added to the Minkowski space-time ... and provided that the velocity is thought of as being tied to a path and not as a global concept."

But irrespective of such an ambiguous debate I have a problem to understand the cosmological redshift in the sense of a purely gravitational shift. It is quite clear that a photon looses energy und thus becomes redshifted as it climbs out of a gravitational field or in other words as it moves away from a mass (i). In contrast the photon traveling through homogeneous space doesn't move away from a gravitational center, but undergoes a redshift (= looses energy) as well. How shall I understand this (obvious?) discrepancy? You mentioned already the dependence on spatial geometrie ... . Is there any explanation besides the stretched wavelenght picture as simpel as (i)?

 

[Old Smuggler]

These authors argue that in order to interpret the cosmological redshift in terms of a Doppler effect in non-expanding Minkowskian space-time the observer would have to move away from himself and thus claim (spatial curvature >= 0) "The Doppler interpretation is clearly self-contradictory (page 5). But this is relativised later (page 6):
"Hence, ironically in the context of the recent debate, parallel-transport of four-velocities along photons path can allow cosmological redshifts to be interpretet as a relativistic Doppler effect without the contradiction presented here, provided that the concept of expanding space is added to the Minkowski space-time ... and provided that the velocity is thought of as being tied to a path and not as a global concept."

Yes, cosmological redshifts can always be interpreted as Doppler shifts in CURVED space-time. However, they cannot in general be interpreted as Doppler shifts in FLAT space-time, and it is the latter meaning that is usually understood with "kinematic" redshift.

But irrespective of such an ambiguous debate I have a problem to understand the cosmological redshift in the sense of a purely gravitational shift. It is quite clear that a photon looses energy und thus becomes redshifted as it climbs out of a gravitational field or in other words as it moves away from a mass (i). In contrast the photon traveling through homogeneous space doesn't move away from a gravitational center, but undergoes a redshift (= looses energy) as well. How shall I understand this (obvious?) discrepancy? You mentioned already the dependence on spatial geometrie ... . Is there any explanation besides the stretched wavelenght picture as simpel as (i)?

There is no obvious intuitive picture to decide the question of "kinematic" versus "gravitational" interpretations, I'm afraid. (If there were, this question would not have been debated so vigourously in the literature.) However, as I have mentioned earlier, there exists a general procedure to decide the matter for small distances, and for arbitrary space-times. That is, choose a pair of fixed ("close") observers with given world lines. Calculate spectral shifts obtained by exchanging photons between these observers. Then replace the space-time geometry in the relevant region with flat space-time (holding the chosen world lines and the coordinate system fixed). Calculate spectral shifts again, but now with the flat space-time geometry. If the latter calculation yields no spectral shifts at all, the spectral shifts obtained in the first calculation must be entirely due to space-time curvature, i.e., "gravitational".

For example, in the Schwarzschild metric, the chosen observers defining gravitational spectral shifts are observers with fixed spatial Schwarzschild coordinates. The flat space-time limit of this metric is obtained by setting the mass M=0. Now it is rather obvious that there is no spectral shift between the chosen observers in the Schwarzscild metric with M=0, so the spectral shift obtained when M is nonzero must be purely gravitational. A similar situation to that of the Schwarzschild metric occurs for RW-models with flat or spherical space sections, so the spectral shifts obtained between the FOs in these models must also be purely gravitational.

 

[Chalnoth]

Yes, cosmological redshifts can always be interpreted as Doppler shifts in CURVED space-time. However, they cannot in general be interpreted as Doppler shifts in FLAT space-time, and it is the latter meaning that is usually understood with "kinematic" redshift.

I don't understand what you mean. In flat space-time, there is no curvature, and thus in general you don't expect there to be any gravitational redshift at all, meaning that any observed redshift would be purely kinematic (of course, you might still be able to impose what looks like gravitational redshift with an appropriate coordinate choice, such as Milne coordinates).

Either way, though, our space-time does have a definite degree of overall curvature, as it must due to the fact that our universe is not empty (more pedantically-stated, the average energy density of our universe is non-zero).

Regardless of the overall curvature, however, the amount of the redshift that is attributed to gravitation and the amount attribute to motion of the emitter or observer is still arbitrary. Some choices may seem more or less natural to some people, but many choices are possible in any event.

 

[Naty1]

Chalnoth:

Regardless of the overall curvature, however, the amount of the redshift that is attributed to gravitation and the amount attribute to motion of the emitter or observer is still arbitrary. Some choices may seem more or less natural to some people, but many choices are possible in any event.

That seemed to be the conclusion from another discussion on this topic, with some insights that may be of interest:

[Note, especially the change in scale factor and,in Schwarzschild coordinates, the change in velocity, comments.]https://www.physicsforums.com/showthr...nt+flow&page=4

edit: oops, that link no longer works?[In the great 2007 thread Wallace, Chronos and Oldman take a different view than expressed here [and there] by Marcus...you can read the posts from the 40's thru 50's and see the pros and cons.]

I do think it is better to think of (photons) as being redshifted by being observed in a different frame ...Now as t ticks along, the scale factor a(t) increases. Therefore two observers who are both at rest wrt to the CMB, but who have different times t will therefore be in different frames (have different metrics). This is what leads to photons being redshifted when observed and emitted at different times.

I tend to agree, photons are not redshifted by traveling through the universe, they are redshifted only because they are observed in a different frame from which they were emitted.

Marcus: # 48] I am not comfortable with that because among other things I see cosmologists doing inventories of the energy density which are implicitly estimated IN A CMB FRAME...

These 'conflicting' viewpoints stem from this as explained by Chalnoth elsewhere:

" … You get some total redshift for faraway objects due to cosmological expansion. How much of that redshift is due to the Doppler shift# and how much is due to the expansion between us and the far away object is completely arbitrary."

# Doppler shift is based on [relative velocity] frame based differences, not expansion, Hence photon frequency and wavelength can be viewed as fixed just like in a static Spacetime.. Doppler shift is a particular explanation of redshift, with a particular formula.Marcus:
Don’t think of the redshift as a Doppler [relative velocity] effect. It is not the result of some particular speed. The formula involves the entire [varying] factor by which distances have been expanded during the whole time the light has been traveling.

PeterDonis: The law governing the relationship of emitted to observed photon energies (or frequencies) is general and applies in any spacetime. The 4-momentum of the photon gets determined at the emitter; then it gets parallel transported along the photon's worldline from emitter to observer; then you contract that 4-momentum with the observer's 4-velocity to get the observed energy (or frequency if you throw in a factor of Planck's constant). That "parallel transport" process is actually where the "redshift" occurs in an expanding universe; the expansion alters the 4-momentum of the photon as it travels (or at least that's one way of looking at it), whereas in a static universe the photon's 4-momentum would "stay the same" as it traveled.

There's another complication here, btw; what about the gravitational redshift of photons in Schwarzschild spacetime? Here the "change" with changing radius is actually in the 4-velocity of the observer; the photon's 4-momentum stays the same, but the 4-velocities of "hovering" observers are different at different radii, so they contract differently with the constant photon 4-momentum.

PAllen:

Redshift is a measured shift in received frequency versus emitted frequency. Doppler [shift] refers to one of two formulas (pre-relativistic; relativistic) for relating redshift to velocity. Doppler shift is a particular explanation of redshift, with a particular formula. It is not a measure of redshift.

Where the speeds of source and the receiver relative to the medium are lower than the velocity of waves in the medium, the classical Doppler shift formula; in cosmology, where we deal with lightspeed 'c' and recessional 'velocities' greater than 'c' we need the relativistic version of the formula. [Doppler is like a radar speed trap: The radar signal goes out and returns and keeps the same 'color', but we record the difference in wavelength as a speed measure.]

Cosmological redshift is typically considered distinct from Doppler redshift because it is a relation between distance and redshift rather than speed and redshift, under the assumption that both source and target are motionless relative to center of mass of the local matter (here, local is quite large - galaxy or galaxy cluster).

 

[Chalnoth]

https://www.physicsforums.com/showthr...nt+flow&page=4

edit: oops, that link no longer works?

You copied and pasted the shortened display text. Try right clicking and copying the link itself. Easier still if you copy the link of the post itself (which can be found by clicking the post number next to the post).

 

[Old Smuggler]

I don't understand what you mean. In flat space-time, there is no curvature, and thus in general you don't expect there to be any gravitational redshift at all, meaning that any observed redshift would be purely kinematic (of course, you might still be able to impose what looks like gravitational redshift with an appropriate coordinate choice, such as Milne coordinates).

You misunderstand. If the redshift in a general RW-model were purely "kinematic", the procedure described in #43 would yield the same redshift for small enough distances both for the curved space-time geometry and for flat space-time. Since this does not happen in general, the nature of the redshift in a general RW-model cannot be interpreted as purely kinematic.

The empty RW-model is an exceptional case since the space-time geometry is flat, so in this case, the observed redshift would be purely kinematic. But this does not apply to a general RW-model where the space-time geometry is not flat. The interpretation of the redshift in a general RW-model depends on the spatial geometry. (Only if the spatial geometry is hyperbolic there will be a non-zero "kinematic" contribution to the redshift.)

Regardless of the overall curvature, however, the amount of the redshift that is attributed to gravitation and the amount attribute to motion of the emitter or observer is still arbitrary. Some choices may seem more or less natural to some people, but many choices are possible in any event.

This may seem reasonable, but a proper mathematical analysis shows that it is simply not true. For example, no "kinematic" interpretation is consistent with the fact that the procedure described in #43 yields no cosmic redhifts between FOs for e.g., an arbitrary RW-model with flat space sections if the space-time geometry is replaced with flat space-time. Please do this (simple) calculation to convince yourself.

 

[Chalnoth]

This may seem reasonable, but a proper mathematical analysis shows that it is simply not true. For example, no "kinematic" interpretation is consistent with the fact that the procedure described in #43 yields no cosmic redhifts between FOs for e.g., an arbitrary RW-model with flat space sections if the space-time geometry is replaced with flat space-time. Please do this (simple) calculation to convince yourself.

I think the problem is that the procedure in #43 is still an arbitrary way of distinguishing between gravitational redshift and kinematic redshift. And I'm not sure it works in any event, because the relative velocity of two objects separated by some distance is arbitrary. If I select some coordinates with an interpretation of velocity which precisely gives the relative velocity between two objects in FRW space-time which would correspond to a Doppler shift, and then replace the space-time with flat space-time in those same coordinates, I'll have nothing but a Doppler shift.

 

[Old Smuggler]

I think the problem is that the procedure in #43 is still an arbitrary way of distinguishing between gravitational redshift and kinematic redshift. And I'm not sure it works in any event, because the relative velocity of two objects separated by some distance is arbitrary.

Nothing is arbitrary with the procedure described in #43. That is, the world lines of the FOs and their 4-velocities are not arbitrary and neither are the null curves.
(For sufficiently small distances the effects of geodesic deviation can be neglected, so the world lines of the FOs are still geodesics and the null curves are still null when replacing the curved space-time metric with a flat space-time metric.) Since the redshift is obtained by parallel-transporting the 4-velocity of the emitter along a null curve to the observer, this shows that the redshift obtained using the procedure described in #43 is unambiguous, only depending on the space-time geometry. Thus, changing the space-time geometry from curved to flat will in general change the redshift, so it cannot be interpreted as purely kinematic. Any concept of "relative velocity of two objects separated by some distance" is not part of the procedure; this is irrelevant since the coordinate-free concept of parallel-transport makes it unnecessary.

If I select some coordinates with an interpretation of velocity which precisely gives the relative velocity between two objects in FRW space-time which would correspond to a Doppler shift, and then replace the space-time with flat space-time in those same coordinates, I'll have nothing but a Doppler shift.

Whatever it is you are thinking of here, it would not correspond to selecting fixed observers (the FOs) and then changing the space-time geometry so the argument is quite irrelevant.

 

[Chalnoth]

Since the redshift is obtained by parallel-transporting the 4-velocity of the emitter along a null curve to the observer, this shows that the redshift obtained using the procedure described in #43 is unambiguous, only depending on the space-time geometry.

Ahh, okay, I missed that bit. This does seem like a somewhat-reasonable way of distinguishing between redshift and Doppler shift, as it is sort of a means of estimating the space-time curvature along the path of the photon. But as you mention, it's not the only way, so I still think it's worth keeping in mind that the distinction between gravitational and kinematic redshift isn't completely cut-and-dried. Some prescriptions are easier to interpret than others, of course.

 

[Murdstone]

I enjoyed this tread and thank the diligence of all contributors. What I found the most stimulating was Naty1’s inclusion of the link to Expanding Space: the Root of all Evil?

In this article it mentions that space might not be stretching but that additional space is being created.

The author implies that stretching and creating are equivalent. I find it more satisfying to consider the processes different.

We are still back to square one – what is space? However, now attention is focused on how space is created rather than just assuming that it is there.

 

[timmdeeg]

Yes, cosmological redshifts can always be interpreted as Doppler shifts in CURVED space-time. However, they cannot in general be interpreted as Doppler shifts in FLAT space-time, and it is the latter meaning that is usually understood with "kinematic" redshift.

Does it mean that in curved space-time a given cosmological redshift may be interpreted as a non-kinematic Doppler shift or equivalently as a gravitational shift as well? To me 'non-kinematic Doppler shift' sounds a bit contradictory.

For example, in the Schwarzschild metric, the chosen observers defining gravitational spectral shifts are observers with fixed spatial Schwarzschild coordinates. The flat space-time limit of this metric is obtained by setting the mass M=0. Now it is rather obvious that there is no spectral shift between the chosen observers in the Schwarzscild metric with M=0, so the spectral shift obtained when M is nonzero must be purely gravitational. A similar situation to that of the Schwarzschild metric occurs for RW-models with flat or spherical space sections, so the spectral shifts obtained between the FOs in these models must also be purely gravitational.

Okay, this clarifies that the notion of a 'climbing photon' is avoidable, which is very helpful, thanks.

 

[Old Smuggler]

Does it mean that in curved space-time a given cosmological redshift may be interpreted as a non-kinematic Doppler shift or equivalently as a gravitational shift as well? To me 'non-kinematic Doppler shift' sounds a bit contradictory.

A Doppler shift in curved space-time only means that the special-relativistic Doppler formula can be used after the described procedure of parallel-transport. Any further interpretation is not included. In particular, a Doppler shift in curved space-time has nothing to do with any possible split-up into "gravitational" plus "kinematic" parts (or any other way of decomposing the redshift for that matter). A "non-kinematic" Doppler shift is meaningless in this context.

 

[azzkika]

If space really does expand how do we know the expansion is uniform? If space expansion is variable then redshift readings cannot be relied on as an accurate measurement of distance/recessional velocity.

 

[Chalnoth]

If space really does expand how do we know the expansion is uniform? If space expansion is variable then redshift readings cannot be relied on as an accurate measurement of distance/recessional velocity.

The expansion is the same in every direction we look. Whether it is uniform in distance as well as direction is a bit harder to determine, but would appear in the data as a strongly-varying expansion rate with distance, which we just don't see.

 

[timmdeeg]

A Doppler shift in curved space-time only means that the special-relativistic Doppler formula can be used after the described procedure of parallel-transport. Any further interpretation is not included. In particular, a Doppler shift in curved space-time has nothing to do with any possible split-up into "gravitational" plus "kinematic" parts (or any other way of decomposing the redshift for that matter).

Ah, so one can say it's formally correct to use the Doppler formula in this context, but the statement "the most natural interpretation of the redshift is as a Doppler shift" (Bunn & Hogg 2009) goes too far.

In the example in #43 you mentioned "observers with fixed spatial Schwarzschild coordinates". Instead one could think of observers falling freely one after another on a radial path. Wouldn't then the similarity with FOs in curved RW models be even closer? And if yes, wouldn't they similarly wonder whether their "space" expands and how to interpret the redshift, as they recognize their moving away from each other, the faster the farther? Sorry, this may be quite silly, thanks for your patience.

 

[Chalnoth]

Ah, so one can say it's formally correct to use the Doppler formula in this context, but the statement "the most natural interpretation of the redshift is as a Doppler shift" (Bunn & Hogg 2009) goes too far.

The problem is that everybody's idea of "most natural" is different.

 

[timmdeeg]

The problem is that everybody's idea of "most natural" is different.

Yes, hence it becomes easily a semantic discussion. But the gap beween "most natural" and "meaningless" is remarkable. It is also possible that B&H don't share the latter classification.

 

[Old Smuggler]

Ah, so one can say it's formally correct to use the Doppler formula in this context, but the statement "the most natural interpretation of the redshift is as a Doppler shift" (Bunn & Hogg 2009) goes too far.

The problem is B&H's claim that it is "most natural" to interpret the cosmic redshift as a Doppler shift in FLAT space-time for sufficiently small distances. But as I have argued in this
thread this claim is false; with a few exceptions said interpretation is in general simply inconsistent with the geometry of the RW-models.

So in my opinion the B&H paper is an extremely lousy paper; just about everything in that paper is wrong or misleading and it should never have been published (the reputation of the American Journal of Physics has been tainted by accepting it). The paper is a prime example of how bad things may turn out when trying to do physics by gut feeling. Furthermore, based on the reception of the paper, B&H have not only succeded in fooling themselves, but have apparently done a good job of fooling some other professionals as well. This is what I find the most remarkable about the paper.

In the example in #43 you mentioned "observers with fixed spatial Schwarzschild coordinates". Instead one could think of observers falling freely one after another on a radial path. Wouldn't then the similarity with FOs in curved RW models be even closer? And if yes, wouldn't they similarly wonder whether their "space" expands and how to interpret the redshift, as they recognize their moving away from each other, the faster the farther? Sorry, this may be quite silly, thanks for your patience.

The similarity between the Schwarzschild metric and the RW-models I was trying to illustrate in #43 was the effect of space-time curvature on the spectral shift in the two cases, not any similarity regarding "space expansion".

 

[timmdeeg]

The problem is B&H's claim that it is "most natural" to interpret the cosmic redshift as a Doppler shift in FLAT space-time for sufficiently small distances. But as I have argued in this thread this claim is false; with a few exceptions said interpretation is in general simply inconsistent with the geometry of the RW-models.
.
Furthermore, based on the reception of the paper, B&H have not only succeded in fooling themselves, but have apparently done a good job of fooling some other professionals as well. This is what I find the most remarkable about the paper.

Chodorowski (Abstract, 2011) then seemingly has criticised B&H for false reasons by stating: "We find that the resulting relation between the transported velocity and the redshift of arriving photons is not given by a relativistic Doppler formula. Instead, for small redshifts it coincides with the well known non-relativistic decomposition of the redshift into a Doppler (kinematic) component and a gravitational one." But, as you have pointed out, such a decomposition is mathematically inconsistent.

The similarity between the Schwarzschild metric and the RW-models I was trying to illustrate in #43 was the effect of space-time curvature on the spectral shift in the two cases, not any similarity regarding "space expansion".

That was just the perhaps crazy idea to compare the cosmological tidal stretching with that happening to freely falling observers towards a mass. Things move away from each other with increasing acceleration in either case. The freely falling observers (not knowing about the mass) develop models just as the FOs do too. I wonder whether the phenomenon tidal stretching should be subject of interpretation, e.g. as expansion of space (not truly physical, meaning not measurable) in both cases.

 

[Old Smuggler]

Chodorowski (Abstract, 2011) then seemingly has criticised B&H for false reasons by stating: "We find that the resulting relation between the transported velocity and the redshift of arriving photons is not given by a relativistic Doppler formula. Instead, for small redshifts it coincides with the well known non-relativistic decomposition of the redshift into a Doppler (kinematic) component and a gravitational one." But, as you have pointed out, such a decomposition is mathematically inconsistent.

This non-relativistic decomposition is arrived at by using a Newtonian approximation to calculate the "gravitational" contribution. But this approach is misguided, and inconsistent with a relativistic approach.

The cited statement should not be taken as a criticism of B&H, since the parallel-transport procedure Chodorowski performs is different from that referred to in B&H. That is, Chodorowski defines a "recession velocity" as the 3-velocity obtained by parallel-transporting the 4-velocity of the emitting FO along a space-like geodesic rather than along a null geodesic. This is a perfectly valid procedure to do mathematically. But then he defines the redshift obtained from this "recession velocity" as the "kinematic" part of the redshift and the remainder part of the redshift is defined as "gravitational". These definitions are very misleading, since (for small distances) the definition of "kinematic redshift" does not correspond to the redshift obtained by using SR, and the definition of "gravitational redshift" does not correspond to the effects of space-time curvature. Chodorowski should have named his spectral shift split-up in some other way, reflecting the procedure on which it is based.

 

[George Jones]

That is, Chodorowski defines a "recession velocity" as the 3-velocity obtained by parallel-transporting the 4-velocity of the emitting FO along a space-like geodesic rather than along a null geodesic.

Chodorowski writes

In practice, however, as a 'preferred' path one can choose a geodesic connecting the galaxy an the observer.

Some care is need with the terminology. The path chosen is spacelike, but it is not a spacetime geodesic, it is spatial geodesic.

Consider an analogy: a portion of a great circle is geodesic on the 2-dimensional surface of the Earth, but it is not geodesic in 3-dimensional space.

Similarly, the path chosen is a geodesic for the 3-dimensional spatial hypersurface that represents all of space for one instant of cosmic time, but it is not a geodesic for 4-dimensional spacetime.

 

[Naty1]

Two issues of interest:

I just came across the following in my notes..and had forgotten about the concept:

http://www.astro.ucla.edu/~wright/cosmo_02.htm

To say the Universe is homogeneous means that any measurable property of the Universe is the same everywhere. This is only approximately true, but it appears to be an excellent approximation when one averages over large regions…. The homogeneity of the Universe must be defined on a surface of constant proper time since the Big Bang. [Because the density of the universe was different a billion years ago than today.]

So it seems we may not have such as widely varied choice of space-times as is often implied??. Some do seem 'natural' given this criteria as a starting point!

GeorgeJones:

Consider an analogy: a portion of a great circle is geodesic on the 2-dimensional surface of the Earth, but it is not geodesic in 3-dimensional space.

This seems really interesting...sometime ago there was a discussion about weaknesses in the balloon analogy...I think it may have been when phinds was developing an explanation of how far the balloon analogy should be taken.
I suggested the great circle on a balloon was not the same path as that used in coordinate and proper distance measurements...and consequently there were severe weakness in the balloon analogy regarding cosmological distances...

I like to use the Wikipedia illustration to 'picture' this for myself:

http://en.wikipedia.org/wiki/Metric..._two_points_measured_if_space_is_expanding.3F

...we can single out two distances which appear to be physically meaningful: the distance between the Earth and the quasar when the light was emitted [red curve], and the distance between them in the present era [orange curve] (taking a slice of the cone along the dimension that we've declared to be the spatial dimension). The former distance is about 4 billion light years, much smaller than ct because the universe expanded as the light traveled the distance, the light had to "run against the treadmill" and therefore went farther than the initial separation between the Earth and the quasar. The latter distance (shown by the orange line) is about 28 billion light years, much larger than ct. If expansion could be instantaneously stopped today, it would take 28 billion years for light to travel between the Earth and the quasar while if the expansion had stopped at the earlier time, it would have taken only 4 billion years.

Anyway, how do we describe the 'curve' in space at a constant coordinate time...[this is the 'path' of parallel-transport of four-velocities along which the FLRW metric is calculated].

I've heard it described as a 'straight line...as when laying rulers end to end' [for a proper distance 'measure' ] and I believe also as a space-like geodesic and a space-time geodesic...

Seems like the proper description is 'a geodesic in three dimensional space'...

edit: Found this in my notes:
Wallace: {commenting on weakness in balloon and raisin bread analogy}
The rate of expansion [velocity] is unimportant; It is the rate of acceleration of the expansion [a’[t] that tells you what happens. So in a contracting universe a distant particle could move away, or in an expanding universe a distant particle could come toward you. You don't intuitively expect this behavior if you think of the universe as the model loaf of rising bread filled with raisins! [or the balloon analogy]

Source not recorded:

A curve of constant cosmological time [along which we would like to measure a proper distance’ ] connecting two points in a FRW [model] universe is not a "straight line", i.e. it is not a geodesic.

But it IS the Hubble ‘distance’ calculated distance.

So what, then is the 'Hubble curve' over which distance is calculated called?

 

[timmdeeg]

The cited statement should not be taken as a criticism of B&H, since the parallel-transport procedure Chodorowski performs is different from that referred to in B&H. That is, Chodorowski defines a "recession velocity" as the 3-velocity obtained by parallel-transporting the 4-velocity of the emitting FO along a space-like geodesic rather than along a null geodesic. This is a perfectly valid procedure to do mathematically. But then he defines the redshift obtained from this "recession velocity" as the "kinematic" part of the redshift and the remainder part of the redshift is defined as "gravitational". These definitions are very misleading, since (for small distances) the definition of "kinematic redshift" does not correspond to the redshift obtained by using SR, and the definition of "gravitational redshift" does not correspond to the effects of space-time curvature. Chodorowski should have named his spectral shift split-up in some other way, reflecting the procedure on which it is based.

Okay, thanks, you gave me a better understanding of Chodorowski's paper.

And I want to thank all participants of this thread for helpful comments.

So, in simple words, one should use the notion of 'expanding space' with some care, as the 'stretching or the creation of space' is not measurable, or perhaps better per se is physically not meaningful.
Then the demystfied version could be the conclusion, "that the expansion of space is neither more or less than the increase over time of the distance between observers at rest with respect to the cosmic fluid", referring to the author's of the paper Expanding Space: The Root of all Evil?. The knowledge of the increasing distances results from the cosmological redshifts, which depend only on the space-time curvature. I hope, that's correct so far.

 

[Naty1]

timmdeeg

This issue is not one easy to describe in a few sentences. And summarizing long discussions about this issue as understandings and explanations evolve is also not so easy.

Timmdeeg

So, in simple words, one should use the notion of 'expanding space' with some care, as the 'stretching or the creation of space' is not measurable, or perhaps better per se is physically not meaningful.

I heartily disagree! It is, in principle observable, as post #2 shows clearly.

The issue is what does the observation [measurement] mean? How do we interpret observed redshift, exactly as posted by Chalnoth, post #2.
Further,if you conclude the effect is 'physically not meaningful', how do you explain that CMBR radiation emitted at almost 3,000 K is today observed at about 2.7K? If this loss of energy had NOT occurred, there would have been no evolution of the universe as we observe it...no stars, no galaxies, no us...also recall I quoted Chalnoth in post #33 integrated Sachs=Wolf effect as experimental evidence.

..."that the expansion of space is neither more or less than the increase over time of the distance between observers at rest with respect to the cosmic fluid",

This gets a LOT closer, but the overall quote I posted is more complete

...the expansion of space is neither more nor less than the increase over time of the distance between observers at rest with respect to the cosmic fluid in terms of the FRW metric. With this metric... the density and pressures of cosmological fluids must change over cosmic time, and it is this change that represents the basic property of an expanding (or contracting) universe.

The rest of this is important because the explanations discussed here are based on the assumptions and conventions in the FLRW cosmological model...and are unique to that model...assumptions like an isotropic and homogeneous universe, conventions like co-moving observers [at rest wrsp to the CMBR]...

For example as bapowell posted in #3 with the FLRW model the wavelength of a cosmological photon [or a mass particle] λ(t) varies according to the scale factor a[t]. In this model, they go together.

I don't even prefer the wording of the quote I posted...

"... the density and pressures of cosmological fluids must change over cosmic time, and it is this change that represents the basic property of an expanding (or contracting) universe..."

I would have said something like

"...it is this change [from general relativity] that causes [or 'powers' or 'determines']...the basic expanding universe...that is, Einstein's equations relate the evolution the scale factor to the changes in pressure and energy density of the matter in the universe.

 

[timmdeeg]

Naty1,
I welcome you criticism.

This issue is not one easy to describe in a few sentences. And summarizing long discussions about this issue as understandings and explanations evolve is also not so easy.

So, in simple words, one should use the notion of 'expanding space' with some care, as the 'stretching or the creation of space' is not measurable, or perhaps better per se is physically not meaningful.

I heartily disagree! It is, in principle observable, as post #2 shows clearly.

The issue is what does the observation [measurement] mean? How do we interpret observed redshift, exactly as posted by Chalnoth, post #2.
.
Further,if you conclude the effect is 'physically not meaningful', how do you explain that CMBR radiation emitted at almost 3,000 K is today observed at about 2.7K?

I think, it might be helpful to distinguish between 'truely physical', 'measurable', 'interpretation', 'not measurable' and 'correct description' in order to better clarfy the issue.

Would you agree with that:

Truely physical: the cosmological redshift, increasing distances between FOs.

Measurable: The redshift. It yields information about the increase of the scale-factor between emission and absorption and thus about increasing distances.

Interpretation : the redshift can be interpreted as due to the stretching of space or as due to the motion of galaxies, #2. Furthermore, the interpretation of the redshift depends "on the spatial geometrie", #39.

Stretching of space not measurable: a thought experiment may result in increasing distances, but these again can be interptreted in this or that way, #16.
So, the experiment doesn't prove the stretching of space (or the creation of space, resp.). If I claim that I have measured the stretching of space, you could say, no, you have measured just motion.
Especially here I ask for any differing opinions.

Correct description: the interpretations are "correct descriptions" of a "real physical phenomenon", #31. And "as space expands, the wavelenth must increase", #3, is to my understanding also covered under correct description.
So, "correct description" and "real phenomenon" don't have the same meaning.

I agree, without further explanation the wording "physically not meaningful" gives rise to misunderstanding. Perhaps "not truly physical" or "not a real physical phenomenon", would make more sense.

 

[ConformalGrpOp]

I appreciate the contributions that have been made to this thread. I found it helpful and stimulating, especially in confirming my view that a large percentage of cosmologists misconceive the concept of expanding space, space stretching, etc.

One question I do have, which I believe is subsumed within the subject matter of this thread, (and perhaps many others), is the following:

What is the experimental basis for the assumption that the frequency/wavelength of light remains constant when traveling over cosmologically relevant distances? Are there any plans for conducting an experiment to verify the behavior of light at such distances?

My sense is that there really is no experimental data on this issue and that astronomers, astrophysicists and cosmologists all rely on evidence produced from local experiments for the interpretation of data obtained from light received from distant sources.

It just seems to me that until such an experiment is conducted, much of the interpretation of what we observe from sources throughout the universe remains open to controversy. Is there any reason why the scientific community would not be interested in the results of such an experiment?

[if this deserves its own thread, I trust someone will pursue that]

 

[timmdeeg]

What is the experimental basis for the assumption that the frequency/wavelength of light remains constant when traveling over cosmologically relevant distances? Are there any plans for conducting an experiment to verify the behavior of light at such distances?

There is perhaps a misunderstanding.
Nobody assumes that. Since Hubble we know about the redshift (i.e. the non constancy of the wavelength) of distant galaxies.

 

[ConformalGrpOp]

Tim, I believe it is assumed that a fundamental property of EM is that the wavelength of light emitted from a distant source propagating along an unobstructed, field free path of fixed distance to an observer with no relative motion with respect to the source will exhibit no spectral shift.

In other words, it is assumed that the "doppler" and "cosmological" red shifts are not the result of the "non constancy of the wavelength", but a result of the relative motion of the observer with respect to the source (doppler), and the additional distance the light traveled from the time it was emitted to the time it was observed due to the expansion of the universe (cosmological), which looks like the same thing insofar as the expansion imparts a de facto relative motion between the source and the observer. In both cases, it is assumed that the wavelength of the light remains unchanged from the point of emission to the point of observation, and it is only the relative motion of the observer with respect to the source that results in the observer detecting a shift in the wavelength. At least, that is my understanding of the current state of knowledge with respect to the behavior of light. [Note: Some theorists contend that "space stretching", a term I believe is a misnomer, imparts a stretching of the wavelength analogous to the stretching of a dot on the surface of an inflating balloon. But, most knowledgeable scientists would view this conception as little more than a second rate, and certainly misleading pedagogical device. The more accurate explanation is that as the universe expands, the successive photons being emitted from the source travel further than the preceding ones, and this, like the doppler effect, accounts for the observed spectral shift.]

I am inclined to view the problem slightly differently; that Maxwell's equations admit a solution that permits light to "travel" within a metric that is not locally Minkowskian. Therefore, not inconsistent with Milne's view, the observed spectral shift, or perhaps a component of it, is quite likely to be the result of an intrinsic characteristic of light which is not velocity dependent, but rather is time/distance dependent.

Verification of the behavior of light over distances where Hubble's law becomes relevant is within our ability to ascertain by experiment, a circumstance wasnt even a fathomable possibility at the time when Eddington and others announced that the relation between red shift and distance codified in Hubble's law provided compelling evidence that the universe was expanding as predicted by the de Sitter and LaMaitre models.

I don't know of any experiments such as those hypothesized by Poincare, Milne and Whitrow (doing away with rigid rods and synched clocks, and using calibrated light signals), that have been proposed or carried out to specifically test the behavior of light over cosmologically relevant distances.

One might ponder the idea of a massively scaled up version of the Michelson-Morley experiment, (or its modern analog), and wonder would the experiment still yield a null result?

From a historical standpoint, its a bit intriguing to think about what might have happened if the late 19th Century experimentalists had access to the technology necessary to perform such a scaled up version of the M-M experiment and the experiment had yielded a positive result! Would Einstein have been able to successfully convince all those who were held to the theory that light could only propagate if there existed an omnipresent aether that: "NO! The positive results demonstrate a previously unknown property of EM allowed by Maxwell's equations, not proof of our motion through the aether!" ? (Rendering the theory of aether obsolete is, in my view, one of the most significant advances resulting from Einstein's theory of Special Relativity; which is a bit ironic for Einstein's SR was developed out of, and informed by the work of Lorentz and Poincare in attempting to explain the results of the M-M experiment with respect to the aether). Anyway, just a thought. Thanks for taking the time to respond.

 

[Naty1]

Interpretation : the redshift can be interpreted as due to the stretching of space or as due to the motion of galaxies, #2. Furthermore, the interpretation of the redshift depends "on the spatial geometrie", #39.

That captures the alternative perspectives for me...