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While reading Sean Carroll's book on General Relativity, I understood that the concept of velocity is ill-defined over large distances in arbitrarily curved manifolds, like the one used to describe our expanding universe. If this is the case, is it possible to find the velocity of a celestial object using Doppler Redshift? I know this cannot be done for cosmological redshift since it's conceptually different. But since velocity itself is ill-defined, can we do the same for Doppler shift from, say, a star very far away?

Additionally, it would be nice if someone could also explain how to distinguish between the types of redshifts, observationally. Thank you very much!

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For distant objects in a curved spacetime, you can define their velocities by means of parallel transport. Choose a spacetime path between yourself and the object, and drag its 4-velocity vector along that path, while preserving the direction of the vector during each infinitesimal step along the path.

The outcome from this process depends on the path that you choose. This is why velocities of distant objects are said to be ill defined. However, if you receive light from the distant object and interpret its frequency shift as a relativistic Doppler shift, that tells you the object's velocity based on parallel transport along the spacetime path of the light. Note that this is true whether you regard the frequency shift as kinematic (Doppler), gravitational, "cosmological", or whatever. There is no fundamental difference between these frequency shifts -- they are all the same phenomenon viewed from different perspectives.

In practice, for a distant galaxy, it is often convenient to decompose the redshift into a cosmological contribution and a peculiar contribution. The cosmological contribution tells you the distance of the object, in accordance with Hubble's law. The peculiar contribution tells you the motion of the object with respect to its local environment. For an individual galaxy, you can only do this decomposition if you have an independent distance measurement (e.g., a supernova that happened in the galaxy, from which you can infer distance from brightness). However, for a collection of galaxies, you can do it statistically.

For example, if you naively interpret all redshifts as cosmological, then you would obtain a highly anisotropic galaxy distribution, because the galaxy peculiar velocities are distorting their inferred positions along the line of sight, and not in any other direction. However, we expect the galaxy distribution to be statistically isotropic. Thus, you can decompose the statistics of your naively inferred galaxy distribution into isotropic and anisotropic components, and the latter can be re-interpreted as coming entirely from peculiar motion of the galaxies. This is the basic idea behind using redshift-space distortions as a cosmological probe.

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  • $\begingroup$ Sorry, I could get back to this only now. So, the 'velocity' of the object is measured along the path in spacetime through which light travels, right? Also, could you elaborate on the statement "There is no fundamental difference between these frequency shifts -- they are all the same phenomenon viewed from different perspectives?" Because I thought we could not associate a galaxy's velocity with cosmological redshift, as sometimes it exceeds the speed of light, and also since velocity is an ill-defined concept in spacetime, as you have said in the answer. $\endgroup$
    – V Govind
    Commented 6 hours ago
  • $\begingroup$ There are no faster-than-light velocities in cosmic expansion (see e.g. physics.stackexchange.com/a/820618/180843). You can associate a galaxy's velocity with the cosmological redshift if you use the correct velocity (arxiv.org/abs/0808.1081 discusses this point in detail), which is basically what I said here with the point about parallel transport. I'm not sure it's accurate to say this is the velocity "measured along the path . . ." because I don't know what that means. $\endgroup$
    – Sten
    Commented 3 hours ago
  • $\begingroup$ But to make a more concrete picture, you can imagine a photon passing a series of observers along its path to you. Take the number of observers to be large, so sequential observers are close enough in space that you can uniquely define the relative velocity between them. For each observer, you can use that relative velocity to calculate the frequency shift relative to what the previous observer sees. The total redshift is just the composition of all of those shifts. $\endgroup$
    – Sten
    Commented 3 hours ago

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