(a) Consider the spherically symmetric spacetime metric

$$\tag{†} d s^{2}=-\lambda^{2} d t^{2}+\mu^{2} d r^{2}+r^{2} d \theta^{2}+r^{2} \sin ^{2} \theta d \phi^{2},$$

where $\lambda$ and $\mu$ are functions of $t$ and $r$. Use the Euler-Lagrange equations for the geodesics of the spacetime to compute all non-vanishing Christoffel symbols for this metric.

(b) Consider the static limit of the line element $(†)$ where $\lambda$ and $\mu$ are functions of the radius $r$ only, and let the matter coupled to gravity be a spherically symmetric fluid with energy momentum tensor

$$T^{\mu \nu}=(\rho+P) u^{\mu} u^{\nu}+P g^{\mu \nu}, \quad u^{\mu}=\left[\lambda^{-1}, 0,0,0\right]$$

where the pressure $P$ and energy density $\rho$ are also functions of the radius $r$. For these Tolman-Oppenheimer-Volkoff stellar models, the Einstein and matter equations $G_{\mu \nu}=8 \pi T_{\mu \nu}$ and $\nabla_{\mu} T_{\nu}^{\mu}=0$ reduce to

$$\begin{aligned} \frac{\partial_{r} \lambda}{\lambda} &=\frac{\mu^{2}-1}{2 r}+4 \pi r \mu^{2} P \\ \partial_{r} m &=4 \pi r^{2} \rho, \quad \text { where } \quad m(r)=\frac{r}{2}\left(1-\frac{1}{\mu^{2}}\right) \\ \partial_{r} P &=-(\rho+P)\left(\frac{\mu^{2}-1}{2 r}+4 \pi r \mu^{2} P\right) \end{aligned}$$

Consider now a constant density solution to the above Einstein and matter equations, where $\rho$ takes the non-zero constant value $\rho_{0}$ out to a radius $R$ and $\rho=0$ for $r>R$. Show that for such a star,

$$\partial_{r} P=\frac{4 \pi r}{1-\frac{8}{3} \pi \rho_{0} r^{2}}\left(P+\frac{1}{3} \rho_{0}\right)\left(P+\rho_{0}\right)$$

and that the pressure at the centre of the star is

$$P(0)=-\rho_{0} \frac{1-\sqrt{1-2 M / R}}{3 \sqrt{1-2 M / R}-1}, \quad \text { with } \quad M=\frac{4}{3} \pi \rho_{0} R^{3}$$

Show that $P(0)$ diverges if $M=4 R / 9 . \quad$ [Hint: at the surface of the star the pressure vanishes: $P(R)=0 .]$