# Cosmological redshift as kinematical Doppler effect

## Main Question or Discussion Point

In http://arxiv.org/abs/0808.1081 / Am.J.Phys.77:688-694,2009 Bunn and Hogg explain how cosmological redshift can be interpreted as accumulation of infinitesimal Doppler shifts. This suggested to invert z = z(v) and interpret v(z) as relative velocity of two objects with redshift z.

It seems that this view has become popular over the last years, however I still have doubts whether this interpretation does make much sense.

Relative velocities as used in the Doppler shift formulas must be defined w.r.t. a common (i.e. globally defined) inertial frame - which does not exist. So using such a v(z) still means that v(z) is not a relative velocity (even so it has been constructed from infinitesimal relative velocities).

In an expanding universe there may exist objects P and Q which cannot be connected by a light-like geodesic. It may even be the case that there are objects for which such a geodesic will never exist. So for these objects there will never be a redshift nor a velocity v(z) defined a la Bunn and Hogg. However these objects somehow "recede" from each other which can be understood using the picture of expanding space, but not a picture relying on a geodesic that does not exist by construction. So the kinematical interpretation breaks down for cosmological horizons.

However it seems that all this could be a rather academic discussion resulting from "intuitive pictures" instead of well-defined math only ...

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Dale
Mentor
The paper is a purely semantic discussion about popular and "hand waving" terms. I wouldn't lose any sleep about it.

In general, I don't like to attempt to decompose Doppler shifts into gravitational and kinematic parts, but that in itself is a personal preference.

The paper is a purely semantic discussion about popular and "hand waving" terms. I wouldn't lose any sleep about it.
Fine, thanks ;-)

In general, I don't like to attempt to decompose Doppler shifts into gravitational and kinematic parts, but that in itself is a personal preference.
I think you can't. The most general expression I am aware of contains the tangent vectors to the light-like geodesic and the sender's / receiver's local 4-velocity such that you can't decompose anything in a reasonable and unambiguous way.

PAllen
2019 Award
Well, the position outlined in that paper was pushed originally by J.L. Synge in 1960. It can be stated rigorously as follows:

What general procedures are there for determining redshift between arbitrary source and arbitrary receiver in an arbitrary manifold with Minkowskian metric? Synge showed the following works in all cases:

Parallel transport the source 4-velocity along the the null geodesic connecting it to the receiver, and then apply SR redshift in any local frame of the receiver, using the transported 4-velocity and the receiver 4-velocity.

Note, this works in any combination of gravitational sources and cosmological solutions. There are a couple of other equivalent universal formulations. Any such universal formulation shows, at minimum, that there is no objective basis to separating redshift into components due gravitational fields, cosmological expansion, and relative motion. This particular one makes it look like kinematic Doppler with the most natural possible way to compare velocities at a distance in GR.

As to sources that are causally disconnected, this seems like a complete red herring to me. There can be no redshift if A and B are causally disconnected. That is as totally absurd as asking what's the redshift factor between a source inside an event horizon and one outside?

A totally separate question is whether there is expansion or loss of energy in the universe. These, to me, are completely independent from whether and how you decompose redshift. As for expansion, to me the way to demonstrate that is simply to show that any foliation that manifests the global symmetry (especially isotropy) of the cosmology, will show growing proper distance (computed via the foliation) between comoving world lines. As for energy, I find it more objective to argue that total energy is undefined rather than decreasing.

vanhees71
Gold Member
2019 Award
I think that's just semantics. What really counts in the FLRW cosmological standard model are the measurable coordinate independent quantities to determine the scale parameter $a(t)$ like the luminosity-distance to redshift relationship of supernovae. Here you have a clearly defined observable with a clear interpretation within FLRW spacetime, and this is a gravitational redshift. This is already clear that at large distances and high redshift from a naive analysis in terms of the Doppler-shift picture you'd conclude faster-than-light recession speeds, which doesn't make too much sense as a physical relative speed between source and observer. For not too far objects in certain coordinates you may interpret the Hubble redshift as a Doppler effect, but that's approximate and not applicable to far-distant large-redshift objects. A nice review is given in the usenet FAQ by Baez et al:

http://www.edu-observatory.org/physics-faq/Relativity/GR/hubble.html

PAllen
2019 Award
I think that's just semantics. What really counts in the FLRW cosmological standard model are the measurable coordinate independent quantities to determine the scale parameter $a(t)$ like the luminosity-distance to redshift relationship of supernovae. Here you have a clearly defined observable with a clear interpretation within FLRW spacetime, and this is a gravitational redshift. This is already clear that at large distances and high redshift from a naive analysis in terms of the Doppler-shift picture you'd conclude faster-than-light recession speeds, which doesn't make too much sense as a physical relative speed between source and observer. For not too far objects in certain coordinates you may interpret the Hubble redshift as a Doppler effect, but that's approximate and not applicable to far-distant large-redshift objects. A nice review is given in the usenet FAQ by Baez et al:

http://www.edu-observatory.org/physics-faq/Relativity/GR/hubble.html
Synge's proof is universal, for all pseudo-riemannian manifolds. The idea that the redhsift is superluminal comes from comparing apples to oranges - treating growth of proper distance in a standard foliation as if it were a relative velocity. To me, that is methodologically absurd on its face for GR. Note that you can construct foliations in flat Minkowski space that show growth of proper distance between timelike world lines exceeding c by a factor greater than 2.

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As to sources that are causally disconnected, this seems like a complete red herring to me. There can be no redshift if A and B are causally disconnected.
That was not my intention.

The idea is that if there's redshift z then one could calculate some v(z) - an idea which I don't like.

In a homogeneous and isotropic, expanding universe one can say - in some sense - that arbitrary points do move away from each other; something you can't say by looking at geodesics b/c they may not even exist. So the picture of expanding space seems to be more general than the geodesics.

PAllen
2019 Award
That was not my intention.

The idea is that if there's redshift z then one could calculate some v(z) - an idea which I don't like.

In a homogeneous and isotropic, expanding universe one can say - in some sense - that arbitrary points do move away from each other; something you can't say by looking at geodesics b/c they may not even exist. So the picture of expanding space seems to be more general than the geodesics.
Expansion I agree with, and explained how it flows from any isotropic foliation of the cosmology. What I have a problem with is treating a recession velocity (growth of proper distance along a foliation, between two world lines) as a relative velocity, and wondering how it can be superluminal. Even in SR in standard coordinates you have up to 2c growth in separation between world lines.

Parallel transport shows the obvious in GR: that relative velocity is path dependent, yet < c for any path over which you transport, just like SR.

My feeling on redshift is that there is simply one type of redshift in GR, that is related to SR redshift, but affected by curvature - the same as you can say about most anything going from SR to GR. Thus, arguing that some redshift is kinematic versus gravitational is what I find meaningless. There is just redshift as a function two world lines and an emission event in a given manifold (universe). Any decomposition is coordinate dependent, even in the simplest, case - spherically symmetric, static solution.

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Thx PAllen, very clear and helpful - as usual ;-)

PAllen
2019 Award
Just thought I'd add another couple of thoughts here. I generally subscribe to the notion there is one form of redshift in GR that is a generalization of SR Doppler to curvature, and that factoring into types is useful only as a calculation convenience (e.g. where you can set up gravitational potential, including SR cases of pseudogravity, you can reduce computation considerably).

Yet, another line of argument is to look at symmetry vs. asymmetry in frequency shifts. Then you might say pure kinematic Doppler is when the shifts are symmetric; and pure gravitational (or pseudo-gravity in SR) shifts are when it is asymmetric, and inverse. [The frame dependence of this leaves me, personally, unattached to the idea this has much meaning; any of the gravitational cases become primarily or exclusively kinematic viewed from an appropriate inertial frame.]

Be that as it may, this point of view certainly supports cosmological redshifts as kinematic since they are symmetric.

Expansion I agree with, and explained how it flows from any isotropic foliation of the cosmology. What I have a problem with is treating a recession velocity (growth of proper distance along a foliation, between two world lines) as a relative velocity, and wondering how it can be superluminal. Even in SR in standard coordinates you have up to 2c growth in separation between world lines.

Parallel transport shows the obvious in GR: that relative velocity is path dependent, yet < c for any path over which you transport, just like SR.

My feeling on redshift is that there is simply one type of redshift in GR, that is related to SR redshift, but affected by curvature - the same as you can say about most anything going from SR to GR. Thus, arguing that some redshift is kinematic versus gravitational is what I find meaningless. There is just redshift as a function two world lines and an emission event in a given manifold (universe). Any decomposition is coordinate dependent, even in the simplest, case - spherically symmetric, static solution.
This is a topic that has been occupying my thoughts recently as a result of reading H.P Robertson's 1949 paper Postulate and Observation in the Special Theory of Relativity (http://authors.library.caltech.edu/11476/1/ROBrmp49.pdf), in the context of Bun and Hogg's paper cited in TS's initial post, so its somewhat fortuitous that I came across this thread.

The question of whether it is possible (or even meaningful) to distinquish between Doppler (SR) vs Cosmological (GR) interpretation of the Hubble redshift at first glance, indeed, may not amount to anything more than a question of semantics (though we recognize the conundrums raised by the SR view as vanhees and PAllen point out).

Now, I am probably conceptually off base here, but what brought this issue to mind is the point that Robertson makes in his concluding remarks that the Michelson-Morley, Kennedy-Thorndike and the Ives-Stillwell experiments prove that:

"The kinematics im kleinen of physical space-time is thus found to be governed by the Minkowski metric, whose motions are the Lorentz transformations, the background upon which the special theory of relativity and its later extension to the general theory are based. " (Isnt it amazing what can be proved when you perform an experiment and absolutely nothing happens??)

So, the question this raised in my mind is what kind of experiment could we design that could distinguish between a Doppler shift vs a Cosmological shift? That is to say, in light of J.L. Synge's analysis, what, if anything, might we be able to learn by sending a space probe with suitable electronics capable of receiving and retransmitting calibrated radar signals back to earth from a distance where the Hubble shift would be detectable?

How far out would the probe have to be sent before the EM signals sent to and returned by the probe would detect an effect of the presence of CDM and DE and be taken up (naively stated), in the "Hubble flow"? Would such an experiment be able to verify whether the "kinematics" im grossen of "physical space-time" is governed by the Minkowski metric? Or would there be no such observable effect at all? (Which is what I suppose Synge's analysis teaches).

PAllen
2019 Award
This is a topic that has been occupying my thoughts recently as a result of reading H.P Robertson's 1949 paper Postulate and Observation in the Special Theory of Relativity (http://authors.library.caltech.edu/11476/1/ROBrmp49.pdf), in the context of Bun and Hogg's paper cited in TS's initial post, so its somewhat fortuitous that I came across this thread.

The question of whether it is possible (or even meaningful) to distinquish between Doppler (SR) vs Cosmological (GR) interpretation of the Hubble redshift at first glance, indeed, may not amount to anything more than a question of semantics (though we recognize the conundrums raised by the SR view as vanhees and PAllen point out).

Now, I am probably conceptually off base here, but what brought this issue to mind is the point that Robertson makes in his concluding remarks that the Michelson-Morley, Kennedy-Thorndike and the Ives-Stillwell experiments prove that:

"The kinematics im kleinen of physical space-time is thus found to be governed by the Minkowski metric, whose motions are the Lorentz transformations, the background upon which the special theory of relativity and its later extension to the general theory are based. " (Isnt it amazing what can be proved when you perform an experiment and absolutely nothing happens??)

So, the question this raised in my mind is what kind of experiment could we design that could distinguish between a Doppler shift vs a Cosmological shift? That is to say, in light of J.L. Synge's analysis, what, if anything, might we be able to learn by sending a space probe with suitable electronics capable of receiving and retransmitting calibrated radar signals back to earth from a distance where the Hubble shift would be detectable?

How far out would the probe have to be sent before the EM signals sent to and returned by the probe would detect an effect of the presence of CDM and DE and be taken up (naively stated), in the "Hubble flow"? Would such an experiment be able to verify whether the "kinematics" im grossen of "physical space-time" is governed by the Minkowski metric? Or would there be no such observable effect at all? (Which is what I suppose Synge's analysis teaches).
Any valid method for computing red shift in GR (I know of 3 universal methods that don't factor red shift into 'types'; Synge's is just one) must predict the same thing. You would find redshift that cannot be accounted for assuming global Minkowski metric. Right around the earth, you can find redshift that is incompatible with the pure SR red shift: measure altitude dependent red shift on opposite sides of the earth. There is no possible explanation of this using SR. Yet this is perfectly predicted, for example, by Synge's method. What Synge argued (and I think there is no substance in the pro or con of this argument) is that all red shift can be viewed as due relative motion of world lines; NOT that red shift does not distinguish Minkowski metric from general metric. Curvature plays a fundamental role in what it means to try to talk about relative velocity. You must go from unique path independent relative velocity, to path dependent relative velocity, that is sensitive to the path used for comparison and the intervening curvature (including its dynamics in cosmology). Synge simply proved that if you define relative velocity for Doppler as the parallel transport of emitter 4-velocity over the light path connecting emitter and target, then this specific choice used with SR formula in the local frame of the target universally gives the same answer as any other valid method in GR.

That is understandable insofar as I think I had trouble stating what I have been trying to figure out. I will come back to the principal question I have been thinking about but I wanted to reference Kaiser's arXiv preprint which I think is highly topical to this thread. It can be found at http://arxiv.org/pdf/1312.1190.pdf. Its an extraordinarily well written paper.

PAllen