Insights The Balloon Analogy .... the Good, the Bad, and the Ugly - Comments

[PeterDonis]

the difference in the space station and the sun is that the space station is not accelerating away from the earth. Due to red shift we know that we are accelerating away from other stars. Right?

Not in the sense I have been using the term "acceleration". You will notice that I have tried to say "proper acceleration", to make it clear that I am talking about acceleration that is actually felt--or, in the case of the space station and the sun and distant galaxies, not felt. All of those objects are in free fall, feeling zero acceleration.

The kind of "acceleration" you are referring to when you say that the sun is accelerating away from other stars is more precisely called "coordinate acceleration". (Actually, the red shifts you are referring to are from other galaxies, not other stars; we can't see individual stars at the distances at which the cosmological redshift becomes measurable. So it's more correct to say that our galaxy as a whole sees other galaxies' light as redshifted.) The key point about coordinate acceleration is that you can change it by changing coordinates--i.e., by a mathematical abstraction that doesn't change anything physical. So if you are trying to understand the actual physics going on, coordinate acceleration is the wrong thing to focus on.

red shift tells us that the universe is accelerating.

No, they don't. Red shifts, in and of themselves, only tell us that the universe is expanding. They do not tell us that the expansion is accelerating. For that, we need not only the observation that there are red shifts, but detailed correlations between red shifts and other observations, like the brightness and angular size of distant galaxies.

Also, once again, when we say the universe's expansion is "accelerating", we mean this in the sense of coordinate acceleration, not proper acceleration. All of the galaxies have zero proper acceleration. See below.

Therefore the sun can not be in a balanced free fall because it is accelerating.

You are confusing coordinate acceleration and proper acceleration here. The sun has zero proper acceleration, and that means there is no net force acting on the sun. The sun does have nonzero coordinate acceleration in certain coordinates, but coordinate acceleration does not tell you whether there is an unbalanced net force. Only proper acceleration does. As above, all of the galaxies have zero proper acceleration, so there is no unbalanced net force acting on any of them.

 

[1oldman2]

As a "layman"(barely) I find the balloon analogy and related conversations very useful and interesting in regards to my understanding of the principles discussed.I must add at this point that I'm no scholar, we won't be collaborating on any papers and the Nobel prize is safe from me. in fact if intelligence is relative then compared to yours mine would be measured on the Planck scale.My point is when someone takes the time to develop good analogies such as the balloon one here then a much broader segment of society is able to learn these concepts and hopefully get interested and excited about the sciences in general. Good job ! looking forward to more writing in this manner.

 

[RUTA]

So is there a valid feature of the Balloon analogy that actually contradicts with Doppler recession?

Narlikar once told me you can derive the cosmological redshift using the Doppler shift between local frames, integrated from emission event to reception event. I've always meant to check that, but never got around to it ...

 

[marcus]

Narlikar once told me you can derive the cosmological redshift using the Doppler shift between local frames, integrated from emission event to reception event. I've always meant to check that, but never got around to it ...

http://arxiv.org/abs/0808.1081
The kinematic origin of the cosmological redshift
Emory F. Bunn, David W. Hogg
(Submitted on 7 Aug 2008 (v1), last revised 14 Apr 2009 (this version, v2))
A common belief about big-bang cosmology is that the cosmological redshift cannot be properly viewed as a Doppler shift (that is, as evidence for a recession velocity), but must be viewed in terms of the stretching of space. We argue that, contrary to this view, the most natural interpretation of the redshift is as a Doppler shift, or rather as the accumulation of many infinitesimal Doppler shifts. The stretching-of-space interpretation obscures a central idea of relativity, namely that it is always valid to choose a coordinate system that is locally Minkowskian. We show that an observed frequency shift in any spacetime can be interpreted either as a kinematic (Doppler) shift or a gravitational shift by imagining a suitable family of observers along the photon's path. In the context of the expanding universe the kinematic interpretation corresponds to a family of comoving observers and hence is more natural.

 

[RUTA]

That looks familiar, marcus. I think we've had this exchange before :-)

 

[RUTA]

I finally took the time to derive the cosmological redshift z+1 = \frac{a_o}{a_e} using the accumulated nonrelativistic redshifts along the path of the light, as in Eqs (4) & (5) of http://arxiv.org/pdf/0808.1081v2.pdf. It's straightforward and nicely shows how the local flat frames can be pieced together along the light path in the global curved spacetime. I also liked the authors' derivation of their Eq (6) \sqrt{\frac{c + v_{rel}}{c - v_{rel}}} = \frac{a_o}{a_e} using the SR velocity addition equation in local flat frames along the light path, i.e., parallel transport. Very cool. I will add the contents of this paper to my GR course.

That being said, I don't agree with their view that v_{rel} constitutes a "natural" notion of velocity for cosmology. "... the velocity of the galaxy at the time of light emission relative to the observer at the present time" is about as unnatural as I can imagine haha. The natural notion is clearly v = Hr, i.e., proper time rate of change of proper distance, which holds not only locally, they use it to get both Eqs (5) and (6), but globally in the "expanding bread" or "stretching rubber sheet" analogies. That view is very Newtonian, so it's very intuitive and as Rindler once told me, "It's not just an analogy, it's exact!" But, it is certainly the case that there is no unique way to define velocity between objects that don't share a local flat frame in curved spacetime, so to each his own :-)

Thanks again for sharing that article, marcus. Hopefully, now that I've gone through the derivation myself, I won't keep requiring you to post it :-)

 

[RUTA]

Someone who wishes to remain anonymous sent me the following link: http://arxiv.org/abs/1111.6704. In this paper, Dag Østvang (O) shows that Bunn & Hogg's (B&H's) claim that a pair of co-moving observers (COs) who have small enough Hubble velocities (v = Hr) relative to each can be considered to occupy the same flat spacetime frame is false. O shows specifically that in the flat or closed RW models, it is not possible to have H nonzero if the curvature is zero. Thus, O concludes that the redshift caused by v = Hr is 100% attributable to spacetime curvature, so it is best understood as a "gravitational redshift" not a "kinematic redshift" as B&H claim. Further, one cannot use the SR Doppler formula between observers with very small v = Hr, since the SR Doppler formula can only be applied between observers who share the same flat frame and any nonzero v = Hr entails nonzero spacetime curvature. O does agree that the SR Doppler formula can be used on the parallel transported 4-velocity of the emitter at emission, which is the Nalikar-Synge computation. He also eschews the notion of "expanding space," so I don't think O would disagree with B&H's conclusion that the redshift is a "Doppler shift" not an "expanding space shift." But, O's analysis refutes B&H's argument using SR with small v = Hr between COs.

None of this causes me to change what I have my students compute as far as kinematics in RW cosmology (although I will definitely add this material to my GR course). I have them compute recession velocity of and proper distance to the source at time of reception and time of emission as a function of z in the Einstein-deSitter model. r = 3ct_o \left(1 - \frac{1}{\sqrt{1 + z}}\right), v = 2c \left(1 - \frac{1}{\sqrt{1 + z}}\right), v_e = v \sqrt{1+z}, r_e = \frac{r}{1+z}, and t_e = \frac{t_o}{\left(1+z\right)^{1.5}}. Then we add \Lambda and obtain \dot{a} = H_o \sqrt{\frac{\Omega _m}{a} + \Omega _\Lambda a^2} with \Omega _m + \Omega _\Lambda = 1 which I have them use to verify \frac{t_e}{t_o} = 0.0366 for z = 9.6, \Omega _m = 0.3 and \Omega _\Lambda = 0.7 found in http://arxiv.org/abs/1204.2305. Then I have them obtain \ddot{a} = 0 at z = 0.671 \left(\frac{t_e}{t_o} = 0.544 \right) with \Omega _m = 0.3 for comparison with http://www.ptep-online.com/index_files/2012/PP-29-02.PDF .

 

[Jorrie]

But, O's analysis refutes B&H's argument using SR with small v = Hr between COs.

I've always been skeptical of the B&H argument, since it would imply that a string of COs could not all be at the same cosmological time. It may not be a sufficient argument, but to me it made SR redshift as an interpretation for cosmological redshift uncomfortable, to say the least.

 

[timmdeeg]

I've always been skeptical of the B&H argument, since it would imply that a string of COs could not all be at the same cosmological time. It may not be a sufficient argument, but to me it made SR redshift as an interpretation for cosmological redshift uncomfortable, to say the least.

Peacock and Chodorowski argue that interpreting the cosmological redshift one should take gravity into account. And I think this makes sense because in curved space-time the redshift isn't purely dopplerian per se. But at the end, as we talk about interpretations the answer to this question is not wright or wrong but might be rather a matter of personal taste.
http://arxiv.org/abs/0809.4573
http://arxiv.org/abs/0911.3536