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I am trying to understand a little general relativity using the book "Spacetime and geometry An introduction to general relativity" by Sean Carroll. I finished my Masters Degree in physics 10 years ago, but I am no expert. I (think I) understand that you can't put up a global inertial coordinatsystem and talk about relative velocity in curved spacetime. On page 345-346 of the book he derives the Hubble law. He says that for small distances we can use the instantanious distance given by the scalefactor times the comoving coordinate. If you differentiate that you get the Hubble law. I understand that the Hubble law is only valid for small distances in a curved spacetime. But we think that space is flat. My question is then if the Hubble law then still is only valid for small distances?

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As far as I understand, "instantaneous" distance here means just proper distance, and the Hubble law v = H(t)*D(t), where v is recession velocity and D is the proper distance, holds for any universe (curved or flat) and for all distances. The problem is that proper distance is not measurable, so other definitions are used (e.g., luminosity distance, angular distance), which behave differently with increasing redshift, but are all close to each other at low redshifts (~<0.8). I suggest reading this paper, in particular, Appendix A.

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Given the scale factor $a(t)$, the proper distance $D(t)$ for objects following the Hubble flow evolves according to $$ \frac{D(t+\Delta t)}{D(t)} = \frac{a(t+\Delta t)}{a(t)} $$ It follows that $$ v_r(t) = \dot D(t) = \frac{\dot a(t)}{a(t)} D(t) \equiv H(t)\, D(t) $$ That's valid for all times and at all distances, but of course only instantaneously at a given cosmological time.

If $H$ varies slowly compared to the time scales involved, we may use the approximation $$ v_r \approx H_0 D $$ but in principle, we need to take into account the variation of $H$ between emission and absorption of any specific signal.

Also note that while observation indicates that space (ie the homogeneous spatial slices at constant cosmological time) is approximately flat, spacetime is still curved.

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