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**a special set of freely-falling observer**s' . I have the following definitions:

**Comoving Frame**: "defined at a time t is the

*inertial*frame in which the accelerated observer is instantaneously at rest at t=t

_{0}. (Thus the term 'comoving frame' actually refers to a different frame for each t)". has

##dx^i =0 ##.

**I'm unsure which 'special set' they are**- I believe a freely-falling observe is one that follows the geodesics establised by the space-time curvature of any bodies whose path it may across. In addition to following these geodesics, I believe it will have motion due to the expansion of space-time.

And I believe a comoving observer moves with the expansion of the universe, and has ##x^{i}## a constant. So that any relative motion between 2 comoving observers is solely due to the expansion of space-time itself. I'm really struggling to tie this with the definition of a freely-falling observer , if we have a different frame for each t, then in each frame the observer would only have motion due to the expansion of space and would not follow a geodesic - but once you piece all frames together they would follow the geodesics?

**Also just to clarify some definitions,**I have peculiar velocity - the velocity of an object as measured by a comoving observer. Am I correct in thinking apparent velocity = peculiar velocity + velocity due to expansion of space-time.Thanks very much !