Submit only four solutions. If more solutions are submitted only the first four will be marked.
Christoffel symbol:
$$\Gamma_{a b}^{c}=\frac{1}{2} g^{c d}\left(g_{d a, b}+g_{b d, a}-g_{a b, d}\right)$$
Geodesics parameterised by $\lambda:$
$$\frac{d^{2} X^{a}}{d \lambda^{2}}+\Gamma_{b c}^{a} \frac{d X^{b}}{d \lambda} \frac{d X^{c}}{d \lambda}=0$$
Riemann curvature tensor:
\begin{aligned} &R_{m n r}{ }^{s}=\partial_{n} \Gamma_{m r}^{s}-\partial_{m} \Gamma_{n r}^{s}+\Gamma_{m r}^{a} \Gamma_{a n}^{s}-\Gamma_{n r}^{a} \Gamma_{a m}^{s}, \ &R_{\text {abcd }}=-R_{\text {bacd }}, \quad R_{\text {abcd }}=-R_{\text {abdc }}, \quad R_{\text {abcd }}=R_{\text {cdab }}, \ &R_{\text {dabc }}+R_{\text {dcab }}+R_{\text {dbca }}=0 . \end{aligned}
Ricci tensor and Ricci scalar:
$$R_{m r}=R_{m n r}{ }^{n}, \quad R=g^{m r} R_{m r} .$$
Geodesic deviation equation:
$$\frac{D^{2} N^{a}}{D \lambda^{2}}=\left(R_{i j c}{ }^{a} T^{j} T^{c}\right) N^{i} .$$
Schwarzschild-de Sitter line element:
$$d s^{2}=-\left(1-\frac{2 m}{r}-\frac{\Lambda}{3} r^{2}\right) d t^{2}+\left(1-\frac{2 m}{r}-\frac{\Lambda}{3} r^{2}\right)^{-1} d r^{2}+r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)$$

Problem 1.

$\mathrm{A}$ (a) Let $A_{j}$ be a covariant vector. Show explicitly how $\partial_{i} A_{j}$ transforms under coordinate transformations.
(b) Now discuss the transformation properties of $\partial_{i} A_{j}-\partial_{j} A_{i}$ under coordinate transformations.
(c) Derive the identity
$$\nabla_{a} \nabla_{b} W^{c d}-\nabla_{b} \nabla_{a} W^{c d}+R_{a b s}{ }^{c} W^{s d}+R_{a b s}{ }^{d} W^{c s}=0$$
using the definition of the Riemann curvature tensor.
$\mathrm{B}$ (d) You are given a covariant derivative operator $\nabla_{a}$ and the quantity $\nabla_{a} \Gamma_{j k}^{i}$ Discuss whether this object is a tensor.
(e) Consider the conformal transformation $\tilde{g}{a b}=\Omega^{2}\left(X^{i}\right) g{a b}$ where $\Omega$ is an arbitrary function of the coordinates. Compute the Christoffel symbol $\tilde{\Gamma}{j k}^{i}$ and write it as the sum of $\Gamma{j k}^{i}$ and a new term. Show explicitly that the new term is symmetric in the lower indices.

Problem 2.

A You are given the Euclidean space $\mathbb{E}^{3}$ with Cartesian coordinates $X^{i}={x, y, z}$ and standard line element $d s^{2}=d x^{2}+d y^{2}+d z^{2}$. You are given the following surface: $x=$ $(1+(v / 2) \cos (u / 2)) \cos (u), y=(1+(v / 2) \cos (u / 2)) \sin (u)$ and $z=(v / 2) \sin (u / 2)$
(a) Compute the induced metric in coordinates $Y^{i}={u, v}, i=1,2 .$ It is of the form
$$d s^{2}=f(u, v) d u^{2}+\frac{1}{4} d v^{2}$$
State the function $f$ explicitly.
(b) Find $\Gamma_{11}^{1}, \Gamma_{12}^{1}$ and $\Gamma_{11}^{2}$. The other Christoffel symbol components vanish.
$\mathrm{B}$ (c) Show that all non-vanishing Riemann tensor components $R_{i j k l}$ are of the form $\beta / f$. Determine the constant $\beta$.
(d) Using cylindrical coordinates ${\rho, \varphi, z}$ show that the surface can be written as $\rho-1=z / \tan (\varphi / 2) .$

Problem 3.

A (a) Let $M$ be a symmetric $3 \times 3$ matrix. Show that $M:=M-\operatorname{Itr} M / 3$ is trace-free, here I stands for the identity matrix.
(b) Now consider an $n$ -dimensional manifold with metric tensor $g_{i j}$ and Ricci tensor $R_{i j}$. Let $Z_{i j}:=R_{i j}-g_{i j} R / \alpha$ where $\alpha$ is a constant. Determine the value of $\alpha$ if $Z_{i j}$ is assumed to be trace-free.
(c) Write the 4-dimensional Einstein field equations with cosmological constant without matter $\left(T_{i j}=0\right)$ in two separate equations: a trace-free equation and a trace equation. You should use the tensor $Z_{i j}$ for the trace-free equation.
B Your are given a vector $v_{a}$ that satisfies $\nabla_{a} v_{b}+\nabla_{b} v_{a}=0$
(d) Show that $\nabla_{a} \nabla_{b} v_{c}=R_{c b a}{ }^{s} v_{s}$
(e) Show that $v^{a} \nabla_{a} R=0$

Problem 4.

A The charged generalisation of the Schwarzschild solution is known as the ReissnerNordström solution. It is given by
$$d s^{2}=-\left(1-\frac{2 M}{r}+\frac{Q^{2}}{r^{2}}\right) d t^{2}+\left(1-\frac{2 M}{r}+\frac{Q^{2}}{r^{2}}\right)^{-1} d r^{2}+r^{2} d \Omega^{2}$$
where $M$ is the mass parameter and $Q$ is the charge. We assume $r>0$.
(a) This line element has up to two possible coordinate singularities $r_{+}$ and $r_{-}$, we will call these horizons. Find explicit expressions for $r_{\pm}$ in terms of $M$ and $Q$. State the necessary condition for the existence of two distinct horizons, one horizon and no horizon.
(b) Assuming $\theta=\pi / 2$ throughout, state the remaining geodesic equations and interpret the two conserved quantities which will appear.
$\mathrm{B}$ (c) Consider the trajectory of a photon (massless particle) and find possible radii such that this photon has an exactly circular orbit. This is known as the photon sphere.
(d) Discuss the condition under which such orbits can exist. What is the maximum possible number of such orbits for the Reissner-Nordström solutions?
(e) Assuming that there are two horizons $r_{+}$ and $r_{-}$, how many photon spheres are present in the Reissner-Nordström manifold? Carefully justify your answer.