Regarding the expansion of the Universe, Wikipedia states:
The expansion of the universe is the increase in distance between any two given gravitationally unbound parts of the observable universe with time. … As the spatial part of the universe's spacetime metric increases in scale, objects become more distant from one another.
This can be modeled to first order with the FLRW metric:
$$g(\mathbf{x},t) = \begin{pmatrix} -1 & 0 & 0 & 0\\ & a(t) & 0 & 0\\ && a(t) & 0\\ &&& a(t) \end{pmatrix}.$$
But isn’t the partitioning of 4D spacetime into “space” and “time” observer-dependent? How can “space” be expanding over “time” when not all observers agree on space & time axes?
I suspect that the statement being made by FLRW must be true only in one particular frame of reference. If so, which one? And what does that statement translate to when boosted to a different frame? I’m having trouble imagining what “expansion of the time-coordinate” would mean, since the expansion of space is defined with respect to time.
