Friedmann-Robertson-Walker (FRW) metric and scale factor confusion

Question

I'm confused about the different ways of writing the Friedmann-Robertson-Walker (FRW) metric using the normalised and non-normalised scale factor. Peacock, for example, (see equation 3.13) gives $$c^{2}\,\mathrm d\tau^{2}=c^{2}\,\mathrm dt^{2}-R^{2}\left(t\right)\left(\frac{\mathrm dr^{2}}{1-kr^{2}}+r^{2}\,\mathrm d\psi^{2}\right),$$ where $R\left(t\right)$ is the non-normalised scale factor and $k=0,\,\pm1$.

He then gives an alternative form using $a$, the dimensionless, normalised scale factor, where $a\left(t\right)\equiv R\left(t\right)/R_{0}$ so $$c^{2}\,\mathrm d\tau^{2}=c^{2}\,\mathrm dt^{2}-a^{2}\left(t\right)\left(\frac{\mathrm dr^{2}}{1-k\left(Ar\right)^{2}}+r^{2}\,\mathrm d\psi^{2}\right),$$ where $A=1/R_{0}$. This surely means that the $kA^2$ term no longer equals 0, +1 or -1. However, and this is what loses me, I've also seen (equation 1, here: https://ned.ipac.caltech.edu/level5/Sept02/Reid/Reid2.html) the metric as $$c^{2}\,\mathrm d\tau^{2}=c^{2}\,\mathrm dt^{2}-a^{2}\left(t\right)\left(\frac{\mathrm dr^{2}}{1-kr^{2}}+r^{2}\,\mathrm d\psi^{2}\right),$$ where $a(t)$ is dimensionless and $k$ again equals 0, +1 or -1. What am I misunderstanding? Thank you.

ADDITIONAL EDIT

I can get from the first to the second metric by making the substitutions $R\rightarrow R/R_{0}$, $r\rightarrow rR_{0}$, $k\rightarrow k/R_{0}^{2}$. But I can't see how I can get to the third metric, where not only does (presumably - it's described as being "dimensionless") $a\left(t\right)\equiv R\left(t\right)/R_{0}$ but also $k$ still equals 0, +1 or -1.

Answer 1

In the first expression:

• $R$ has dimensions of length
• $r$ and $k$ have no dimensions
• $k$ is scaled so it's $-1$, $0$, or $1$

To go from the first expression to the second expression, we define $$r' = r R_0, \quad k' = k / R_0^2, \quad a(t) = R(t) / R_0$$ where I'm denoting the new variables with a prime just to be extra clear. Then

• $r'$ has dimensions of length and $k'$ has dimensions of inverse length squared
• $a$ has no dimensions

These are the two main conventions for the FRW metric. Your third example is a bit unusual, because it tries to set $k'$ to $-1$, $0$, or $1$, which doesn't make any sense because $k'$ has dimensions.

However, $k'$ is a constant, so we can use it to define our unit of length. Taking positive curvature for concreteness, this means that rather than measuring lengths in meters, we measure in a unit system where the numeric value of $k'$ is $1 \, (\text{length unit})^{-2}$. Another way of saying this is that this example is setting $k' = 1$ in the same sense that you may set $c = 1$ in special relativity. You can always do this by defining your unit system appropriately, but then naive dimensional analysis will stop working, and you'll have to put the $k'$'s and $c$'s back in at the end to get valid numbers out.