Relativistic Doppler and Hubble's law

[giulio_hep]

I have a quick question about the Special Relativity. Non-relativistically, we expect no doppler shift in the wavelength of the emitted light if the source is moving at right angles to the line of sight to the observer. However there is a transverse doppler shift even in this case, caused by time dilation of the source. Now, correct me if I'm wrong, but the mechanism for the cosmological redshift is not the same as the above relativistic shift. I'm confused because I often find the two associated together, like in this answer of anna v. On the opposite edge, I would reply that relativistic doppler effect is a more simple subject of a PHY206 (special relativity) while cosmological redshifts are discussed in PHY314 (relativity and cosmology) and PHY306 (introduction to cosmology). I think that the analogy between the Hubble expansion and a simple recession would be justified only if the scale factor would be increasing linearly: is it correct?

 

[mfb]

the mechanism for the cosmological redshift is not the same as the above relativistic shift

Correct.
I don't see the transverse Doppler effect mentioned in the linked page.

For small distances, it is possible to consider the objects as moving away in space instead of an expansion of space. For large distances this does not work any more, at least not with "realistic" speed values (there is always some specific speed that gives the correct redshift of course).

 

[PAllen]

It is certainly true that red shifts of distant objects in a realistic model of the universe do not correspond to any model using special relativity (in flat spacetime). However, it is completely up to interpretation whether or nor cosmological redshifts are Doppler generalized to curved spacetime versus something else altogether (e.g. expanding space).

To quickly justify that cosmological redshift can be treated as Doppler in curved spacetime, note that the correct prediction, in all cases, arises by parallel transporting the emitter 4-velocity along the light path to the receiver, and then using pure SR Doppler in a local frame at the receiver. Curvature is involved both in determining what the light path is, and what the behavior of parallel transport is. However, while curvature is involved in this formulation, nothing about expansion of space or recession rate is involved.

 

[giulio_hep]

Correct.
I don't see the transverse Doppler effect mentioned in the linked page.

For small distances, it is possible to consider the objects as moving away in space instead of an expansion of space. For large distances this does not work any more, at least not with "realistic" speed values (there is always some specific speed that gives the correct redshift of course).

Another link/example here

 

[bcrowell]

Now, correct me if I'm wrong, but the mechanism for the cosmological redshift is not the same as the above relativistic shift. I'm confused because I often find the two associated together, like in this answer of anna v.

Anna V's answer is incorrect. Note that the question is marked as a duplicate. The one that it duplicates has a correct answer by Ted Bunn, who is a professional relativist. PF also has a FAQ that answers the question about energy conservation posed in the physics.SE thread that you linked: https://www.physicsforums.com/threads/what-is-the-total-mass-energy-of-the-universe.506985/ .

It is neither true nor false that cosmological Doppler shifts are the same phenomenon as special-relativistic Doppler shifts. To test whether they were the same, we would need to be able to define, in an unambiguous way, the velocity vector of galaxy A relative to cosmologically distant galaxy B. If we could do that, we could then plug that velocity vector into the equation for the special-relativistic Doppler shift and see whether it agreed with observation. But GR doesn't provide any unambiguous way of defining the relative velocities of distant objects.

 

[giulio_hep]

It is certainly true that red shifts of distant objects in a realistic model of the universe do not correspond to any model using special relativity (in flat spacetime). However, it is completely up to interpretation whether or nor cosmological redshifts are Doppler generalized to curved spacetime versus something else altogether (e.g. expanding space).

To quickly justify that cosmological redshift can be treated as Doppler in curved spacetime, note that the correct prediction, in all cases, arises by parallel transporting the emitter 4-velocity along the light path to the receiver, and then using pure SR Doppler in a local frame at the receiver. Curvature is involved both in determining what the light path is, and what the behavior of parallel transport is. However, while curvature is involved in this formulation, nothing about expansion of space or recession rate is involved.

They claim that the resulting relation between the transported velocity and the redshift of arriving photons is not given by a relativistic Doppler formula

 

[giulio_hep]

Anna V's answer is incorrect. Note that the question is marked as a duplicate. The one that it duplicates has a correct answer by Ted Bunn, who is a professional relativist. PF also has a FAQ that answers the question about energy conservation posed in the physics.SE thread that you linked: https://www.physicsforums.com/threads/what-is-the-total-mass-energy-of-the-universe.506985/ .

Thanks, that's what I suspected.

 

[bcrowell]

They claim that the resulting relation between the transported velocity and the redshift of arriving photons is not given by a relativistic Doppler formula

Note the following from the abstract: "Last but not least, we show that the so-called proper recession velocities of galaxies, commonly used in cosmology, are in fact radial components of the galaxies’ four-velocity vectors. As such, they can indeed attain superluminal values, but should not be regarded as real velocities."

This confirms what we told you in #3 and #5, that there is no yes/no answer to your question.

 

[giulio_hep]

Note the following from the abstract: "Last but not least, we show that the so-called proper recession velocities of galaxies, commonly used in cosmology, are in fact radial components of the galaxies’ four-velocity vectors. As such, they can indeed attain superluminal values, but should not be regarded as real velocities."

This confirms what we told you in #3 and #5, that there is no yes/no answer to your question.

Ok, fair enough.

 

[PAllen]

They claim that the resulting relation between the transported velocity and the redshift of arriving photons is not given by a relativistic Doppler formula

They choose not to transport it on light path, but instead on a spatial hypersurface of their choosing. That is why I used the word 'could be interpreted'. I don't need to add anything to Bcrowell's #5. This paper together with Bunn's shows the fundamental ambiguity of both velocity and distance of 'far away' objects in GR.

 

[PAllen]

Note the following from the abstract: "Last but not least, we show that the so-called proper recession velocities of galaxies, commonly used in cosmology, are in fact radial components of the galaxies’ four-velocity vectors. As such, they can indeed attain superluminal values, but should not be regarded as real velocities."

This confirms what we told you in #3 and #5, that there is no yes/no answer to your question.

Interesting and relates to a point I"ve made recently. Spatial component of a 4-velocity is (in some local frame) gamma(v)*v , where v is the speed of the 4-velocity in that local frame. That is exactly a celerity. If you divide by the time component in any such local basis, you get the relative velocity. Of course, more simply, parallel transport always preserved the timelike character of 4-velocity, so you always end up with with (totally ambiguous) relative velocity (<c).

 

[giulio_hep]

Parallel transport, shape space, holonomy... The energy–momentum method is applied to dynamic problems in many fields, including chemistry, quantum and classical physics, and engineering. If we want to find the shortest path with respect to a metric induced by the function we wish to minimize, it turns out that the solution is closely related to Wong’s equations that describe the motion of a colored particle in a Yang–Mills field... (an old problematic discussion with a reference to parallel transport in relativistic string theory here )

Since special relativity and quantum mechanics were combined, we know that, because of the equivalence of mass and energy (E2 = p2 + m2) and the uncertainty principle (∆E∆t ∼ ∆p∆x ∼ h), the number of particles is not fixed but subject to quantum fluctuations.
Also from http://math.ucr.edu/home/baez/physics/Relativity/GR/hubble.html
The Doppler explanation fails for very large redshifts, for then we must consider how Hubble's "constant" changes over the course of the journey.
And from http://alemanow.narod.ru/hubbles.htm
After putting Hubble's law into its quantum form vn = nH0 it becomes apparent that the cosmological redshift of the photon's frequency has a quantum nature ... If the frequency decreases by the Hubble constant with each new period, then such process presents wave energy dissipation and not the Doppler effect.