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## Summary:

- How should I write the transformation matrix for an expanding space?

## Main Question or Discussion Point

Hello. I am confused with this matter that how should we write the transformation matrix for an expanding space. consider a spacetime that is expading with a constant rate of a. now normally we scale the coordinates for expansion which makes the transformation matrix like this:

\begin{pmatrix} -1 & 0 & 0 & 0\\ 0 & a & 0 & 0 \\ 0 & 0 & a & 0 \\ 0 & 0 & 0 & a\end{pmatrix}

but there's one thing I can't figure out. If we take one point in this space and attribute a frame of reference to it, this tranformation matrix will scale it up to the rate of expansion, yet that frame also moves and I think it should also translate while scaling. because when expanding, distance between points increase, and increasing distance means the origin of the frame also changes because it is farther from any point before. which means the frame actually moved. so does that mean matrix would be like this?

\begin{pmatrix} -1 & 0 & 0 & a\\ 0 & a & 0 & a \\ 0 & 0 & a & a \\ 0 & 0 & 0 & a\end{pmatrix}

\begin{pmatrix} -1 & 0 & 0 & 0\\ 0 & a & 0 & 0 \\ 0 & 0 & a & 0 \\ 0 & 0 & 0 & a\end{pmatrix}

but there's one thing I can't figure out. If we take one point in this space and attribute a frame of reference to it, this tranformation matrix will scale it up to the rate of expansion, yet that frame also moves and I think it should also translate while scaling. because when expanding, distance between points increase, and increasing distance means the origin of the frame also changes because it is farther from any point before. which means the frame actually moved. so does that mean matrix would be like this?

\begin{pmatrix} -1 & 0 & 0 & a\\ 0 & a & 0 & a \\ 0 & 0 & a & a \\ 0 & 0 & 0 & a\end{pmatrix}