# Concerning the origin of cosmological redshift

I stumbled across this paper, which is published in MNRAS, that wants to show that the cosmological redshift is not due to some "stretching of space," by showing that this interpretation is coordinate dependent.

The background is that cosmological redshift in a FLRW metric is given by $$1 + z = \frac{a(t_O)}{a(t_S)},$$ where $t_O$ is the cosmological time of observation, and $t_S$ is the time of emission. This lends itself to the interpretation that as the space "scales up" or "stretches" the light is redshifted.

However, it was my belief that anyone who actually does general relativity knows that this is a faulty picture, which is at best used to explain the phenomenon pictorially. For let $u^a$ be the fundamental observer velocity field, then in a FLRW universe we have $$u_{a;b} = \frac{1}{3}h_{ab}\theta,$$ where $h_{ab}$ is the induced metric on 3-space and $\theta = u^a{}_{;a} = 3(\dot{a}/a)$ is the expansion parameter of the fluid, i.e. its divergence (here overdot notation indicated the derivative along $u^a$). More generally $$u_{a;b} = \dot{u}_au_b + \Theta_{ab} + \omega_{ab},$$ where $\dot{u}_a$ is the acceleration of the fluid, $\Theta_{ab} = h_{a}{}^ch_b{}^du_{(c;d)}$ is the deformation tensor, and $\omega_{ab} = h_a{}^ch_b{}^du_{[c;d]}$ is the vorticity tensor. If we let $k^a = \omega(u^a + e^a) = \frac{d}{d\lambda}$ be the light ray vector, for some unit spacelike vector $e^a$, orthogonal to $u^a$, then $$k^a\omega_{|a} = k^ak^bu_{a;b} = \omega^2(e^a\dot{u}_a + e^ae^b\Theta_{ab}).$$ Introducing a characteristic length scale $\ell$ that satisfies $\dot{\ell}/\ell = -e^ae^b\Theta_{ab}$, we find that if we assume homogeneity and vanishing acceleration we have $$\frac{d\omega}{\omega\,d\lambda} = -\frac{d\ell}{\ell\, d\lambda}.$$ But $\Theta^a{}_a = \theta = 3(\dot{a}/a)$, where $a$ is the (multiplicative) average $\ell$ over the different directions. In FLRW we have local isotropy so that $a = \ell$, and the result follows by integration.

To my mind, this derivation clearly shows that the factor $a(t_O)/a(t_S)$ arises from the divergence of the observer velocity field, i.e. that it is a purely kinematic effect, and should be interpreted thus. In fact, I feel that the "stretching of space" can only be given meaning as precisely the expansions of the normal velocity field. Therefore, I thought it commonly understood that the cosmological redshift is due to doppler shift, and that any other description is merely a re-casting into different language (i.e. that it makes no claim of different physics). The above linked paper seems to disagree concerning how well accepted this is, in that it depends on the coordinate system, and this answer (and references therein) seems to agree.

So my question is: am I incorrect in that my short derivation shows that the redshift is entirely due to kinematic effects (regardless of coordinate system) and that intepretation of it as being due to stretching of space is merely recasting the same fact into different language?

• Without the technical part I also thought like you. The Doppler effect as a merely kinematic aspect. Nov 16, 2017 at 20:25

You write cosmological redshift "arises from the divergence of the observer velocity field". But, is this divergence due to kinematics or the stretching of space??

I intend this question as rhetorical, and it seems to me both are valid interpretations. In my Master's thesis (MacLaurin 2015) I argued that one could interpret redshifts (such as cosmological redshift) as any combination of Doppler and gravitational redshift one chooses. The concepts had been proposed by others: to interpret as a gravitational shift, set up a line of observers along a photon worldline, with observers remaining at constant distance from one another. Take a setup with Rindler-like acceleration. For a Doppler shift, set up observers which are all in freefall, then by the equivalence principle they might interpret and redshift as kinematic, since they feel no force from resisting gravity. I implemented these ideas with some precise examples. Others have parallel-propagated photon 4-vectors and then argued for a purely kinematic/Doppler interpretation. (The usual statements that cosmological redshifts are not Doppler really only apply to a naive application of a special relativity formula.)

Your derivation seems a nice one. A PhD thesis (Davis 2004) has a similar conclusion, that cosmological redshifts are locally Doppler shifts.