## Friday 4 December 2020

### Robertson-Walker metrics

In section 8.2 we meet what Carroll calls the Robertson-Walker metrics:$${ds}^2=-{dt}^2+R^2\left(t\right)\left[\frac{{d\bar{r}}^2}{1-k{\bar{r}}^2}+{\bar{r}}^2{d\Omega}^2\right]=-{dt}^2+a^2\left(t\right)\left[\frac{{\rm dr}^2}{1-\kappa r^2}+r^2{d\Omega}^2\right]$$The second version is Carroll's preferred form - 'flouting' conventional wisdom.

Four people found the equations in various teams. They are Alexander Friedmann (Russian), Georges LemaĆ®tre (Belgian), Howard Robertson (USian) and Arthur Walker (British) and the metrics are often named after one or some or all. Carroll favours the English speakers.

As usual I check Carroll's equations and (not so usual) have three minor complaints.
1) When he gives the Christoffel symbols at his equation 8.44 the third line is$$\Gamma_{01}^1=\Gamma_{02}^2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \Gamma_{03}^3=\frac{\dot{a}}{a}$$there should be an $=$ sign between $\Gamma_{02}^2$ and $\Gamma_{03}^3$!
2) Just after equation 8.35 he says that a flat 3-manifold could be described by 'a more complicated manifold such as the three-torus $S^1\times S^1\times S^1$'. How can a torus, something generated by circles, be flat? I think he should have said a flat thee-torus. It can be generate from three circles but not as simply as the usual doughnut which is not flat.
3) Before that, just after equation 8.29, he says $k$ sets the curvature and therefore the size of the spatial surfaces. One might think that $k=0$ sets the size as infinite (flat curvature) but the aforementioned flat three-torus is a counter example with $k=0$ and finite size. So $k=0$ does not set the size as you might have expected. Similar arguments apply when $k<0$.

I learned a few other things too: I suspected that if the metric is diagonal then the Ricci tensor must be diagonal. Not true! But on the plus side I found Win's supercharged formula for calculating Ricci tensor components in that case. It avoids the need to calculate Christoffel and Riemann components. I also learned a bit more about smooth isometric embedding.

I also noticed something slightly mysterious that Carroll does not draw to our attention: In the closed universe case (finite size?) the radial coordinate $\bar{r}$ is constrained by ${\bar{r}}^2<1$. So the universe might be even more closed than we thought.

Read it all at Commentary 8.2 Robertson-Walker Metrics.pdf (7 pages + 5 of calculations)