Heaven knows why I sometimes get confused by the simplest things.
In this case, the conversion between two commonly used cosmological coordinate systems: Comoving coordinates vs. coordinates that are, well, not comoving, in which cosmic expansion is ascribed to time dilation effects instead.
In the standard coordinates that are used to describe the homogeneous, isotropic universe of the Friedmann–Lemaître–Robertson–Walker (FLRW) metric, the metric is given by
$$ds^2=dt^2-a^2dR^2,$$
where \(a=a(t)\) is a function of the time coordinate, and \(R\) represents the triplet of spatial coordinates: e.g., \(dR^2=dx^2+dy^2+dz^2.\)
I want to transform this using \(R’=aR,\) i.e., transform away the time-dependent coefficient in front of the spatial term in the metric. The confusion comes because for some reason, I always manage to convince myself that I also have to make the simultaneous replacement \(t’=a^{-1}dt.\)
I do not. This is nonsense. I just need to introduce \(dR’\). The rest then presents itself automatically:
$$\begin{align*}
R’&=aR,\\
dR&=d(a^{-1}R’)=-a^{-2}\dot{a}R’dt+a^{-1}dR’,\\
ds^2&=dt^2-a^2[-a^{-2}\dot{a}R’dt+a^{-1}dR’]^2\\
&=(1-a^{-2}\dot{a}^2{R’}^2)dt^2+2a^{-1}\dot{a}R’dtdR’-d{R’}^2\\
&=(1-H^2{R’}^2)dt^2+2HR’dtdR’-d{R’}^2,
\end{align*}$$
where \(H=\dot{a}/a\) as usual.
OK, now that I recorded this here in my blog for posterity, perhaps the next time I need it, I’ll remember where to find it. For instance, the next time I manage to stumble upon one of my old Quora answers that, for five and a half years, advertised my stupidity to the world by presenting an incorrect answer on this topic.
This, incidentally, would serve as a suitable coordinate system representing the reference frame of an observer at the origin. It also demonstrates that such an observer sees an apparent horizon, the cosmological horizon, given by \(1-H^2{R’}^2=0,\), i.e., \(R’=H^{-1},\) the distance characterized by the inverse of the Hubble parameter.
