(i) Using the condition that the metric tensor $g_{a b}$ is covariantly constant, derive an expression for the Christoffel symbol $\Gamma_{b c}^{a}=\Gamma_{c b}^{a}$.

(ii) Show that

$$\Gamma_{b a}^{a}=\frac{1}{2} g^{a c} g_{a c, b}$$

Hence establish the covariant divergence formula

$$V_{; a}^{a}=\frac{1}{\sqrt{-g}} \frac{\partial}{\partial x^{a}}\left(\sqrt{-g} V^{a}\right)$$

where $g$ is the determinant of the metric tensor.

[It may be assumed that $\partial_{a}(\log \operatorname{det} M)=\operatorname{trace}\left(M^{-1} \partial_{a} M\right)$ for any invertible matrix $M$ ].

(iii) The Kerr-Newman metric, describing the spacetime outside a rotating black hole of mass $M$, charge $Q$ and angular momentum per unit mass $a$, is given in appropriate units by

$$\begin{aligned} d s^{2}=&-\left(d t-a \sin ^{2} \theta d \phi\right)^{2} \frac{\Delta}{\rho^{2}} \\ &+\left(\left(r^{2}+a^{2}\right) d \phi-a d t\right)^{2} \frac{\sin ^{2} \theta}{\rho^{2}}+\left(\frac{d r^{2}}{\Delta}+d \theta^{2}\right) \rho^{2} \end{aligned}$$

where $\rho^{2}=r^{2}+a^{2} \cos ^{2} \theta$ and $\Delta=r^{2}-2 M r+a^{2}+Q^{2}$. Explain why this metric is stationary, and make a choice of one of the parameters which reduces it to a static metric.

Show that, in the static metric obtained, the equation

$$\left(g^{a b} \Phi_{, b}\right)_{; a}=0$$

for a function $\Phi=\Phi(t, r)$ admits solutions of the form

$$\Phi=\sin (\omega t) R(r)$$

where $\omega$ is constant and $R(r)$ satisfies an ordinary differential equation which should be found.