The metric of any two-dimensional rotationally-symmetric curved space can be written in terms of polar coordinates, $(r, \theta)$, with $0 \leqslant \theta<2 \pi, r \geqslant 0$, as

$$d s^{2}=e^{2 \phi}\left(d r^{2}+r^{2} d \theta^{2}\right)$$

where $\phi=\phi(r)$. Show that the Christoffel symbols $\Gamma_{r \theta}^{r}, \Gamma_{r r}^{\theta}$ and $\Gamma_{\theta \theta}^{\theta}$ are each zero, and compute $\Gamma_{r r}^{r}, \Gamma_{\theta \theta}^{r}$ and $\Gamma_{r \theta}^{\theta}=\Gamma_{\theta r}^{\theta}$.

The Ricci tensor is defined by

$$R_{a b}=\Gamma_{a b, c}^{c}-\Gamma_{a c, b}^{c}+\Gamma_{c d}^{c} \Gamma_{a b}^{d}-\Gamma_{a c}^{d} \Gamma_{b d}^{c}$$

where a comma here denotes partial derivative. Prove that $R_{r \theta}=0$ and that

$$R_{r r}=-\phi^{\prime \prime}-\frac{\phi^{\prime}}{r}, \quad R_{\theta \theta}=r^{2} R_{r r}$$

Suppose now that, in this space, the Ricci scalar takes the constant value $-2$. Find a differential equation for $\phi(r)$.

By a suitable coordinate transformation $r \rightarrow \chi(r), \theta$ unchanged, this space of constant Ricci scalar can be described by the metric

$$d s^{2}=d \chi^{2}+\sinh ^{2} \chi d \theta^{2}$$

From this coordinate transformation, find $\cosh \chi$ and $\sinh \chi$ in terms of $r$. Deduce that

$$e^{\phi(r)}=\frac{2 A}{1-A^{2} r^{2}},$$

where $0 \leqslant A r<1$, and $A$ is a positive constant.

[You may use

$$\left.\int \frac{d \chi}{\sinh \chi}=\frac{1}{2} \log (\cosh \chi-1)-\frac{1}{2} \log (\cosh \chi+1)+\text { constant } .\right]$$