(i) The metric of any two-dimensional curved space, rotationally symmetric about a point $P$, can by suitable choice of coordinates be written locally in the form

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

where $r=0$ at $P, r>0$ away from $P$, and $0 \leqslant \theta<2 \pi$. Labelling the coordinates as $\left(x^{1}, x^{2}\right)=(r, \theta)$, show that the Christoffel symbols $\Gamma_{12}^{1}, \Gamma_{11}^{2}$ and $\Gamma_{22}^{2}$ are each zero, and compute the non-zero Christoffel symbols $\Gamma_{11}^{1}, \Gamma_{22}^{1}$ and $\Gamma_{12}^{2}=\Gamma_{21}^{2}$.

The Ricci tensor $R_{a b}(a, b=1,2)$ 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 denotes a partial derivative. Show that $R_{12}=0$ and that

$$R_{11}=-\phi^{\prime \prime}-r^{-1} \phi^{\prime}, \quad R_{22}=r^{2} R_{11}$$

(ii) Suppose further that, in a neighbourhood of $P$, the Ricci scalar $R$ takes the constant value $-2$. Find a second order differential equation, which you should denote by $(*)$, for $\phi(r)$.

This space of constant Ricci scalar can, by a suitable coordinate transformation $r \rightarrow \chi(r)$, leaving $\theta$ invariant, be written locally as

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

By studying this coordinate transformation, or otherwise, find $\cosh \chi$ and $\sinh \chi$ in terms of $r$ (up to a constant of integration). Deduce that

$$e^{\phi(r)}=\frac{2 A}{\left(1-A^{2} r^{2}\right)} \quad, \quad(0 \leqslant A r<1)$$

where $\mathrm{A}$ is a positive constant and verify that your equation $(*)$ for $\phi$ holds.

[Note that

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

Part II