I Cosmological Redshift and Expansion

[Jaime Rudas]

TL;DR Summary

Can cosmological redshift be attributed to expansion?

Citing this paper as a source, Wikipedia says regarding cosmological redshift:

Many popular accounts attribute the cosmological redshift to the expansion of space. This can be misleading because the expansion of space is only a coordinate choice. The most natural interpretation of the cosmological redshift is that it is a Doppler shift.

Since I'm one of those who has attributed cosmological redshift to expansion, I'd like to know if this is really considered a misleading statement.

 

[PeterDonis]

Since I'm one of those who has attributed cosmological redshift to expansion, I'd like to know if this is really considered a misleading statement.

It's an attempt to be able to have one's cake and eat it too--to justify using the term "Doppler shift" while at the same time acknowledging that the curvature of the spacetime, which is the reason why so many cosmologists object to using the term "Doppler shift" (since that term implies an analysis using flat spacetime), is in fact significant.

IMO the best solution to such problems is to stop worrying about words and focus on the physics. The physics is simple: the redshift of a distant object tells you by how much the scale factor has increased between the time the light you're seeing from that object was emitted, and now. So, for example, if you see a distant object with a redshift $z = 1$, the scale factor has increased by a factor of 2 ($1 + z$) since that light was emitted.

It's true that the easiest way to understand the scale factor is as a measure of "expansion of space", which does indeed depend on adopting a specific coordinate chart (standard FLRW coordinates). However, it's worth noting that the observed redshift $z$ itself is an invariant, and therefore the expansion factor $1 + z$ is also an invariant. That means it's telling you about actual physics, not just coordinates. And one thing that actual physics is not is a standard Doppler shift, which is a measure of relative speed (in flat spacetime), not expansion factor.

 

[Bandersnatch]

Didn't @Orodruin write an insight article showing how (iirc) both the cosmological redshift and the Doppler shift are coordinate dependent statements, and equivalent in this context?

 

[Ibix]

There's a school of thought that invariants, which direct observations necessarily are, cannot be explained by non-invariants. If I have an observation, the physics behind it cannot depend on choices I personally made.

In that case, then, the observed redshift cannot be explained by "the expansion of space" because how you define space is a matter of choice. If I pick some coordinate system with its spatial planes not parallel to the usual FLRW ones then I don't have a uniformly expanding space - but my telescopes still show the same spectra in all directions, which I can't attribute to expanding space. However, $a(t)$ can be determined from the expansion scalar of the co-moving congruence and, as Peter noted, $1+z=a_\mathrm{now}/a_\mathrm{then}$ so an explanation in terms of the scale factor is still valid in this view.

It does often help to think in terms of invariants - it's harder to be misled by things that are artifacts of your coordinate system. But it must be said that in the case of FLRW spacetime everybody always uses the same spatial slices, if not always the same coordinates, so it's probably not as important as (e.g.) Schwarzschild spacetime where over-reliance on Schwarzschild coordinates leads to a lot of the common popsci errors.

 

[Ibix]

Didn't @Orodruin write an insight article showing how (iirc) both the cosmological redshift and the Doppler shift are coordinate dependent statements, and equivalent in this context?

He did: https://www.physicsforums.com/insights/coordinate-dependent-statements-expanding-universe/. The reference he uses is Bunn and Hogg, which is the paper cited in the OP.

 

[PAllen]

I would phrase it as that the red shift between bodies that see isotropy is invariant, but treating as generalized Doppler versus expansion of space is a choice, similar to coordinate choice. Note, that one must, in GR, adopt some generalization of Doppler to cover spectral shifts between arbitrary bodies in arbitrary GR solutions. Having done so, cosmological red shift is nothing other than this generalized Doppler between comoving world lines in FLRW solutions.

The choice is evident in flat spacetime in that Milne coordinates demonstrate cosmological redshift via apparently expanding space between comoving worldlines, that is nothing but Doppler in standard coordinates.

Similarly, gravitational redshift may also be viewed as Doppler in a free fall frame.

Similarly, muons created in the upper atmosphere reaching the ground may be viewed as due to time dilation or length contraction,

 

[PeterDonis]

gravitational redshift may also be viewed as Doppler in a free fall frame.

Only if the two observers are close enough that a single local inertial frame can cover them both. But in many cases of gravitational redshift, that is not the case.

You can adopt the "generalized Doppler" approach to cover gravitational redshift between observers that are too far apart to both be covered by a single local inertial frame. But IMO, as I said in post #2, that's an attempt to have one's cake and eat it too. If spacetime isn't flat, it isn't flat, and "Doppler shift" is a term that is supposed to refer to something that happens in flat spacetime.

 

[PAllen]

that's an attempt to have one's cake and eat it too. If spacetime isn't flat, it isn't flat, and "Doppler shift" is a term that is supposed to refer to something that happens in flat spacetime.

That is very much a matter of opinion. IMO, Doppler refers to the fact that the relative motion of emitter and receiver determines spectral shifts, and in GR this implies picking a specific way to compare at a distance - and the most obvious way is correct - parallel transport along the light path. The only difference from SR is that relative motion comparison is path independent.

 

[PeterDonis]

Doppler refers to the fact that the relative motion of emitter and receiver determines spectral shifts

Yes, and the "relative motion" in question only has meaning in flat spacetime, or within a single local inertial frame in a curved spacetime (i.e., in a small enough patch that the curvature can be ignored). Other than that, there is simply no invariant corresponding to "relative motion". Trying to generalize "Doppler shift" to cover such cases means giving up the very meaning of "Doppler shift" that you describe.

Also, as I pointed out in post #2, the cosmological redshift actually tells you how the scale factor has increased, which is not an indication of "relative motion".

 

[PeterDonis]

parallel transport along the light path. The only difference from SR is that relative motion comparison is path independent.

No, that's not the only difference from SR. The other difference from SR is that, in SR, given two inertial observers, parallel transport along a light path between the two gives the same "relative motion" as comparison along a line of simultaneity in any inertial frame. To put it another way, the "relative motion" between two inertial observers is constant with respect to time--and that holds whether "time" means coordinate time in any inertial frame, or proper time along either observer's worldline.

In FLRW spacetime, however, that is not the case. Parallel transporting along a light path between two comoving observers mixes apples and oranges: the 4-velocity of the emitter at the time of emission, and the 4-velocity of the receiver at the time of reception. And the "relative motion" between these two changes with time--and that's true whether "time" means FLRW coordinate time or proper time along either observer's worldline. So you can't just blithely interpret this as "Doppler shift due to relative motion" the way you do in flat spacetime: things are changing underneath the light as it travels, so to speak, which is not the case in flat spacetime, and those changes undermine the interpretation.

And, of course, that undermining becomes clearer when you realize that the parallel transport comparison you describe actually tells you how much the scale factor has increased, which, as I pointed out in my previous post, is not a measure of "relative motion".

 

[PAllen]

Yes, and the "relative motion" in question only has meaning in flat spacetime, or within a single local inertial frame in a curved spacetime (i.e., in a small enough patch that the curvature can be ignored). Other than that, there is simply no invariant corresponding to "relative motion". Trying to generalize "Doppler shift" to cover such cases means giving up the very meaning of "Doppler shift" that you describe.

I disagree, as follows. It is necessary generalize Doppler because for any given detector at a reception event, every different state of motion of emitter at an emission event produces a different spectral shift. You must handle this, and parallel transport on the only physically relevant path - the path of the light itself - provides the answer. In, particular, parallel transport of the detector 4 velocity back along the light path to emission event determines which state of emitter motion has no spectral shift.

 

[PeterDonis]

It is necessary generalize Doppler because for any given detector at a reception event, every different state of motion of emitter at an emission event produces a different spectral shift.

You don't have to "generalize Doppler" to understand that. Parallel transport along the light path is the physics, yes, but calling it "generalized Doppler" is not. That's a matter of words, and IMO they're not appropriate words, but of course opinions will differ on that. But you don't need to make any such choice of words at all to do the physics.

 

[Ibix]

Aren't we just taking the inner product of the wave four-vector with the four-velocities of the emitter and receiver? Are we parallel transporting anything here? (Apart from the light's wave vector, of course.)

 

[PAllen]

..

And, of course, that undermining becomes clearer when you realize that the parallel transport comparison you describe actually tells you how much the scale factor has increased, which, as I pointed out in my previous post, is not a measure of "relative motion".

Only if you are comparing comoving world lines.

 

[PAllen]

Aren't we just taking the inner product of the wave four-vector with the four-velocities of the emitter and receiver? Are we parallel transporting anything here? (Apart from the light's wave vector, of course.)

Right, that is another formulation that is simpler for SR Doppler as well. But if you transport the emitter 4 velocity as well as the light wave vector ( or 4 momentum), you see that the shift between emission and reception is the same as SR Doppler between the transported emitter motion and the detector motion.

 

[PeterDonis]

Only if you are comparing comoving world lines.

Which is the assumption behind the "cosmological redshift" anyway. If we're talking about non-comoving objects, we're off the topic of this thread.

That said, the comments I made about "generalized Doppler" not being appropriate words apply to any worldlines in any curved spacetime, not just comoving worldlines in FLRW spacetime.

 

[PeterDonis]

Aren't we just taking the inner product of the wave four-vector with the four-velocities of the emitter and receiver?

That amounts to the same thing, because the wave 4-vector at the emission event is determined by the emitter's 4-velocity, and to take its inner product with the receiver's 4-velocity, you have to parallel transport the wave 4-vector along the light path.

Are we parallel transporting anything here?

Yes. See above. Whether you call it parallel transporting the wave 4-vector or the emitter's 4-velocity is a case of six of one vs. half a dozen of the other. But you have to parallel transport something from the emission event to the reception event in order to know the proper 4-vector to use to take the inner product with the receiver's 4-velocity.

 

[PAllen]

Which is the assumption behind the "cosmological redshift" anyway. If we're talking about non-comoving objects, we're off the topic of this thread.

That said, the comments I made about "generalized Doppler" not being appropriate words apply to any worldlines in any curved spacetime, not just comoving worldlines in FLRW spacetime.

And what would call the phenomenon that light received from some physical emission process can be anything, depending on the state of motion of the emitter? To me, Doppler is the most appropriate name for this fact.

 

[PeterDonis]

what would call the phenomenon that light received from some physical emission process can be anything, depending on the state of motion of the emitter?

It doesn't just depend on the 4-velocity of the emitter. It also depends on the spacetime geometry between the emission and reception events, and the 4-velocity of the receiver.

Calling it "Doppler" includes the first (and the last, if you unpack the SR analysis fully), but not the middle of those three things. I don't know that there is a simple ordinary language term that does include the middle thing, but after all, that's why we don't do physics in ordinary language, we do it in math. The math is not in dispute here, only what to call it in ordinary language--but we don't have to call it anything in ordinary language to get the right answer. That last statement is my basic response to questions of the kind raised in this thread.

 

[PAllen]

It doesn't just depend on the 4-velocity of the emitter. It also depends on the spacetime geometry between the emission and reception events, and the 4-velocity of the receiver.

Calling it "Doppler" includes the first (and the last, if you unpack the SR analysis fully), but not the middle of those three things. I don't know that there is a simple ordinary language term that does include the middle thing, but after all, that's why we don't do physics in ordinary language, we do it in math. The math is not in dispute here, only what to call it in ordinary language--but we don't have to call it anything in ordinary language to get the right answer. That last statement is my basic response to questions of the kind raised in this thread.

And to me, it is perfectly intuitive and appropriate to call the 'middle' generalized Doppler - that is Doppler generalized for curved spacetime. The curvature determines how to relate the states of motion between emission and reception.

 

[PeterDonis]

And to me, it is perfectly intuitive and appropriate to call the 'middle' generalized Doppler - that is Doppler generalized for curved spacetime. The curvature determines how to relate the states of motion between emission and reception.

Then we have a difference of opinion--to me, the curvature means we can't relate the states of motion between emission and reception, at least not in any way that makes an interpretation as "relative motion" valid.

But, as I've said, this is only a difference of opinion about words, not about physics. And for that matter, so is disagreement about whether "expansion of space" is a suitable ordinary language description.