• # Paper 1, Section II, C

The Weyl tensor $C_{\alpha \beta \gamma \delta}$ may be defined (in $n=4$ spacetime dimensions) as

$C_{\alpha \beta \gamma \delta}=R_{\alpha \beta \gamma \delta}-\frac{1}{2}\left(g_{\alpha \gamma} R_{\beta \delta}+g_{\beta \delta} R_{\alpha \gamma}-g_{\alpha \delta} R_{\beta \gamma}-g_{\beta \gamma} R_{\alpha \delta}\right)+\frac{1}{6}\left(g_{\alpha \gamma} g_{\beta \delta}-g_{\alpha \delta} g_{\beta \gamma}\right) R$

where $R_{\alpha \beta \gamma \delta}$ is the Riemann tensor, $R_{\alpha \beta}$ is the Ricci tensor and $R$ is the Ricci scalar.

(a) Show that $C_{\beta \alpha \delta}^{\alpha}=0$ and deduce that all other contractions vanish.

(b) A conformally flat metric takes the form

$g_{\alpha \beta}=e^{2 \omega} \eta_{\alpha \beta},$

where $\eta_{\alpha \beta}$ is the Minkowski metric and $\omega$ is a scalar function. Calculate the Weyl tensor at a given point $p$. [You may assume that $\partial_{\alpha} \omega=0$ at $p$.]

(c) The Schwarzschild metric outside a spherically symmetric mass (such as the Sun, Earth or Moon) is

$d s^{2}=-\left(1-\frac{2 M}{r}\right) d t^{2}+\left(1-\frac{2 M}{r}\right)^{-1} d r^{2}+r^{2} d \Omega^{2}$

(i) Calculate the leading-order contribution to the Weyl component $C_{t r t r}$ valid at large distances, $r \gg 2 M$, beyond the central spherical mass.

(ii) What physical phenomenon, known from ancient times, can be attributed to this component of the Weyl tensor at the location of the Earth? [This is after subtracting off the Earth's own gravitational field, and neglecting the Earth's motion within the solar system.] Briefly explain why your answer is consistent with the Einstein equivalence principle.

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• # Paper 2, Section II, 38C

Consider the following metric for a 3-dimensional, static and rotationally symmetric Lorentzian manifold:

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

(a) Write down a Lagrangian $\mathcal{L}$ for arbitrary geodesics in this metric, if the geodesic is affinely parameterized with respect to $\lambda$. What condition may be imposed to distinguish spacelike, timelike, and null geodesics?

(b) Find the three constants of motion for any geodesic.

(c) Two observation stations are sitting at radii $r=R$ and $r=2 R$ respectively, and at the same angular coordinate. Each is accelerating so as to remain stationary with respect to time translations. At $t=0$ a photon is emitted from the naked singularity at $r=0$.

(i) At what time $t_{1}$ does the photon reach the inner station?

(ii) Express the frequency $\nu_{2}$ of the photon at the outer station in terms of the frequency $\nu_{1}$ at the inner station. Explain whether the photon is redshifted or blueshifted as it travels.

(d) Consider a complete (i.e. infinite in both directions) spacelike geodesic on a constant- $t$ slice with impact parameter $b=r_{\min }>0$. What is the angle $\Delta \theta$ between the two asymptotes of the geodesic at $r=\infty$ ? [You need not be concerned with the sign of $\Delta \theta$ or the periodicity of the $\theta$ coordinate.]

[Hint: You may find integration by substitution useful.]

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• # Paper 3, Section II, 37C

(a) Determine the signature of the metric tensor $g_{\mu \nu}$ given by

$g_{\mu \nu}=\left(\begin{array}{cccc} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{array}\right)$

Is it Riemannian, Lorentzian, or neither?

(b) Consider a stationary black hole with the Schwarzschild metric:

$d s^{2}=-\left(1-\frac{2 M}{r}\right) d t^{2}+\left(1-\frac{2 M}{r}\right)^{-1} d r^{2}+r^{2} d \Omega^{2}$

These coordinates break down at the horizon $r=2 M$. By making a change of coordinates, show that this metric can be converted to infalling Eddington-Finkelstein coordinates.

(c) A spherically symmetric, narrow pulse of radiation with total energy $E$ falls radially inwards at the speed of light from infinity, towards the origin of a spherically symmetric spacetime that is otherwise empty. Assume that the radial width $\lambda$ of the pulse is very small compared to the energy $(\lambda \ll E)$, and the pulse can therefore be treated as instantaneous.

(i) Write down a metric for the region outside the pulse, which is free from coordinate singularities. Briefly justify your answer. For what range of coordinates is this metric valid?

(ii) Write down a metric for the region inside the pulse. Briefly justify your answer. For what range of coordinates is this metric valid?

(iii) What is the final state of the system?

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• # Paper 4 , Section II, $37 \mathrm{C}$

(a) A flat $(k=0)$, isotropic and homogeneous universe has metric $g_{\alpha \beta}$ given by

$d s^{2}=-d t^{2}+a^{2}(t)\left(d x^{2}+d y^{2}+d z^{2}\right)$

(i) Show that the non-vanishing Christoffel symbols and Ricci tensor components are

$\Gamma_{i i}^{0}=a \dot{a}, \quad \Gamma_{0 i}^{i}=\Gamma_{i 0}^{i}=\frac{\dot{a}}{a}, \quad R_{00}=-3 \frac{\ddot{a}}{a}, \quad R_{i i}=a \ddot{a}+2 \dot{a}^{2}$

where dots are time derivatives and $i \in\{1,2,3\}$ (no summation assumed).

(ii) Derive the first-order Friedmann equation from the Einstein equations, $G_{\alpha \beta}+\Lambda g_{\alpha \beta}=8 \pi T_{\alpha \beta} .$

(b) Consider a flat universe described by ( $\dagger$ ) with $\Lambda=0$ in which late-time acceleration is driven by "phantom" dark energy obeying an equation of state with pressure $P_{\mathrm{ph}}=w \rho_{\mathrm{ph}}$, where $w<-1$ and the energy density $\rho_{\mathrm{ph}}>0$. The remaining matter is dust, so we have $\rho=\rho_{\mathrm{ph}}+\rho_{\mathrm{dust}}$ with each component separately obeying $\dot{\rho}=-3 \frac{\dot{a}}{a}(\rho+P)$.

(i) Calculate an approximate solution for the scale factor $a(t)$ that is valid at late times. Show that the asymptotic behaviour is given by a Big Rip, that is, a singularity in which $a \rightarrow \infty$ at some finite time $t^{*}$.

(ii) Sketch a diagram of the scale factor $a$ as a function of $t$ for a convenient choice of $w$, ensuring that it includes (1) the Big Bang, (2) matter domination, (3) phantom-energy domination, and (4) the Big Rip. Label these epochs and mark them on the axes.

(iii) Most reasonable classical matter fields obey the null energy condition, which states that the energy-momentum tensor everywhere satisfies $T_{\alpha \beta} V^{\alpha} V^{\beta} \geqslant 0$ for any null vector $V^{\alpha}$. Determine if this applies to phantom energy.

[The energy-momentum tensor for a perfect fluid is $\left.T_{\alpha \beta}=(\rho+P) u_{\alpha} u_{\beta}+P g_{\alpha \beta}\right]$

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• # Paper 1, Section II, 38D

Let $(\mathcal{M}, \boldsymbol{g})$ be a four-dimensional manifold with metric $g_{\alpha \beta}$ of Lorentzian signature.

The Riemann tensor $\boldsymbol{R}$ is defined through its action on three vector fields $\boldsymbol{X}, \boldsymbol{V}, \boldsymbol{W}$ by

$\boldsymbol{R}(\boldsymbol{X}, \boldsymbol{V}) \boldsymbol{W}=\nabla_{\boldsymbol{X}} \nabla_{\boldsymbol{V}} \boldsymbol{W}-\nabla_{\boldsymbol{V}} \nabla_{\boldsymbol{X}} \boldsymbol{W}-\nabla_{[\boldsymbol{X}, \boldsymbol{V}]} \boldsymbol{W}$

and the Ricci identity is given by

$\nabla_{\alpha} \nabla_{\beta} V^{\gamma}-\nabla_{\beta} \nabla_{\alpha} V^{\gamma}=R_{\rho \alpha \beta}^{\gamma} V^{\rho} .$

(i) Show that for two arbitrary vector fields $\boldsymbol{V}, \boldsymbol{W}$, the commutator obeys

$[\boldsymbol{V}, \boldsymbol{W}]^{\alpha}=V^{\mu} \nabla_{\mu} W^{\alpha}-W^{\mu} \nabla_{\mu} V^{\alpha}$

(ii) Let $\gamma: I \times I^{\prime} \rightarrow \mathcal{M}, \quad I, I^{\prime} \subset \mathbb{R},(s, t) \mapsto \gamma(s, t)$ be a one-parameter family of affinely parametrized geodesics. Let $\boldsymbol{T}$ be the tangent vector to the geodesic $\gamma(s=$ const, $t)$ and $\boldsymbol{S}$ be the tangent vector to the curves $\gamma(s, t=$ const $)$. Derive the equation for geodesic deviation,

$\nabla_{T} \nabla_{T} \boldsymbol{S}=\boldsymbol{R}(\boldsymbol{T}, \boldsymbol{S}) \boldsymbol{T}$

(iii) Let $X^{\alpha}$ be a unit timelike vector field $\left(X^{\mu} X_{\mu}=-1\right)$ that satisfies the geodesic equation $\nabla_{\boldsymbol{X}} \boldsymbol{X}=0$ at every point of $\mathcal{M}$. Define

$\begin{array}{ll} B_{\alpha \beta}:=\nabla_{\beta} X_{\alpha}, & h_{\alpha \beta}:=g_{\alpha \beta}+X_{\alpha} X_{\beta}, \\ \Theta:=B^{\alpha \beta} h_{\alpha \beta}, \quad \sigma_{\alpha \beta}:=B_{(\alpha \beta)}-\frac{1}{3} \Theta h_{\alpha \beta}, & \omega_{\alpha \beta}:=B_{[\alpha \beta]} . \end{array}$

Show that

$\begin{gathered} B_{\alpha \beta} X^{\alpha}=B_{\alpha \beta} X^{\beta}=h_{\alpha \beta} X^{\alpha}=h_{\alpha \beta} X^{\beta}=0 \\ B_{\alpha \beta}=\frac{1}{3} \Theta h_{\alpha \beta}+\sigma_{\alpha \beta}+\omega_{\alpha \beta}, \quad g^{\alpha \beta} \sigma_{\alpha \beta}=0 \end{gathered}$

(iv) Let $\boldsymbol{S}$ denote the geodesic deviation vector, as defined in (ii), of the family of geodesics defined by the vector field $X^{\alpha}$. Show that $\boldsymbol{S}$ satisfies

$X^{\mu} \nabla_{\mu} S^{\alpha}=B_{\mu}^{\alpha} S^{\mu}$

(v) Show that

$X^{\mu} \nabla_{\mu} B_{\alpha \beta}=-B_{\beta}^{\mu} B_{\alpha \mu}+R_{\mu \beta \alpha}{ }^{\nu} X^{\mu} X_{\nu}$

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• # Paper 2, Section II, $37 \mathrm{D}$

The Schwarzschild metric is given by

$d s^{2}=-\left(1-\frac{2 M}{r}\right) d t^{2}+\left(1-\frac{2 M}{r}\right)^{-1} d r^{2}+r^{2} d \theta^{2}+r^{2} \sin ^{2} \theta d \phi^{2}$

(i) Show that geodesics in the Schwarzschild spacetime obey the equation

$\frac{1}{2} \dot{r}^{2}+V(r)=\frac{1}{2} E^{2}, \quad \text { where } V(r)=\frac{1}{2}\left(1-\frac{2 M}{r}\right)\left(\frac{L^{2}}{r^{2}}-Q\right)$

where $E, L, Q$ are constants and the dot denotes differentiation with respect to a suitably chosen affine parameter $\lambda$.

(ii) Consider the following three observers located in one and the same plane in the Schwarzschild spacetime which also passes through the centre of the black hole:

• Observer $\mathcal{O}_{1}$ is on board a spacecraft (to be modeled as a pointlike object moving on a geodesic) on a circular orbit of radius $r>3 M$ around the central mass $M$.

• Observer $\mathcal{O}_{2}$ starts at the same position as $\mathcal{O}_{1}$ but, instead of orbiting, stays fixed at the initial coordinate position by using rocket propulsion to counteract the gravitational pull.

• Observer $\mathcal{O}_{3}$ is also located at a fixed position but at large distance $r \rightarrow \infty$ from the central mass and is assumed to be able to see $\mathcal{O}_{1}$ whenever the two are at the same azimuthal angle $\phi$.

Show that the proper time intervals $\Delta \tau_{1}, \Delta \tau_{2}, \Delta \tau_{3}$, that are measured by the three observers during the completion of one full orbit of observer $\mathcal{O}_{1}$, are given by

$\Delta \tau_{i}=2 \pi \sqrt{\frac{r^{2}\left(r-\alpha_{i} M\right)}{M}}, \quad i=1,2,3$

where $\alpha_{1}, \alpha_{2}$ and $\alpha_{3}$ are numerical constants that you should determine.

(iii) Briefly interpret the result by arranging the $\Delta \tau_{i}$ in ascending order.

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• # Paper 3, Section II, 37D

(a) Let $(\mathcal{M}, \boldsymbol{g})$ be a four-dimensional spacetime and let $\boldsymbol{T}$ denote the rank $\left(\begin{array}{l}1 \\ 1\end{array}\right)$ tensor defined by

$\boldsymbol{T}: \mathcal{T}_{p}^{*}(\mathcal{M}) \times \mathcal{T}_{p}(\mathcal{M}) \rightarrow \mathbb{R}, \quad(\boldsymbol{\eta}, \boldsymbol{V}) \mapsto \boldsymbol{\eta}(\boldsymbol{V}), \quad \forall \boldsymbol{\eta} \in \mathcal{T}_{p}^{*}(\mathcal{M}), \quad \boldsymbol{V} \in \mathcal{T}_{p}(\mathcal{M})$

Determine the components of the tensor $\boldsymbol{T}$ and use the general law for the transformation of tensor components under a change of coordinates to show that the components of $\boldsymbol{T}$ are the same in any coordinate system.

(b) In Cartesian coordinates $(t, x, y, z)$ the Minkowski metric is given by

$d s^{2}=-d t^{2}+d x^{2}+d y^{2}+d z^{2} .$

Spheroidal coordinates $(r, \theta, \phi)$ are defined through

\begin{aligned} x &=\sqrt{r^{2}+a^{2}} \sin \theta \cos \phi \\ y &=\sqrt{r^{2}+a^{2}} \sin \theta \sin \phi \\ z &=r \cos \theta \end{aligned}

where $a \geqslant 0$ is a real constant.

(i) Show that the Minkowski metric in coordinates $(t, r, \theta, \phi)$ is given by

$d s^{2}=-d t^{2}+\frac{r^{2}+a^{2} \cos ^{2} \theta}{r^{2}+a^{2}} d r^{2}+\left(r^{2}+a^{2} \cos ^{2} \theta\right) d \theta^{2}+\left(r^{2}+a^{2}\right) \sin ^{2} \theta d \phi^{2}$

(ii) Transform the metric ( $\dagger$ ) to null coordinates given by $u=t-r, R=r$ and show that $\partial / \partial R$ is not a null vector field for $a>0$.

(iii) Determine a new azimuthal angle $\varphi=\phi-F(R)$ such that in the new coordinate system $(u, R, \theta, \varphi)$, the vector field $\partial / \partial R$ is null for any $a \geqslant 0$. Write down the Minkowski metric in this new coordinate system.

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• # Paper 4 , Section II, 37D

In linearized general relativity, we consider spacetime metrics that are perturbatively close to Minkowski, $g_{\mu \nu}=\eta_{\mu \nu}+h_{\mu \nu}$, where $\eta_{\mu \nu}=\operatorname{diag}(-1,1,1,1)$ and $h_{\mu \nu}=\mathcal{O}(\epsilon) \ll 1$. In the Lorenz gauge, the Einstein tensor, at linear order, is given by

$\tag{†} G_{\mu \nu}=-\frac{1}{2} \square \bar{h}_{\mu \nu}, \quad \bar{h}_{\mu \nu}=h_{\mu \nu}-\frac{1}{2} \eta_{\mu \nu} h$

where $\square=\eta^{\mu \nu} \partial_{\mu} \partial_{\nu}$ and $h=\eta^{\mu \nu} h_{\mu \nu}$.

(i) Show that the (fully nonlinear) Einstein equations $G_{\alpha \beta}=8 \pi T_{\alpha \beta}$ can be equivalently written in terms of the Ricci tensor $R_{\alpha \beta}$ as

$R_{\alpha \beta}=8 \pi\left(T_{\alpha \beta}-\frac{1}{2} g_{\alpha \beta} T\right), \quad T=g^{\mu \nu} T_{\mu \nu}$

Show likewise that equation $(†)$ can be written as

$\square h_{\mu \nu}=-16 \pi\left(T_{\mu \nu}-\frac{1}{2} \eta_{\mu \nu} T\right)$

(ii) In the Newtonian limit we consider matter sources with small velocities $v \ll 1$ such that time derivatives $\partial / \partial t \sim v \partial / \partial x^{i}$ can be neglected relative to spatial derivatives, and the only non-negligible component of the energy-momentum tensor is the energy density $T_{00}=\rho$. Show that in this limit, we recover from equation $(*)$ the Poisson equation $\vec{\nabla}^{2} \Phi=4 \pi \rho$ of Newtonian gravity if we identify $h_{00}=-2 \Phi$.

(iii) A point particle of mass $M$ is modelled by the energy density $\rho=M \delta(r)$. Derive the Newtonian potential $\Phi$ for this point particle by solving the Poisson equation.

[You can assume the solution of $\vec{\nabla}^{2} \varphi=f(\boldsymbol{r})$ is $\varphi(\boldsymbol{r})=-\int \frac{f\left(\boldsymbol{r}^{\prime}\right)}{4 \pi\left|\boldsymbol{r}-\boldsymbol{r}^{\prime}\right|} d^{3} r^{\prime} .$ ]

(iv) Now consider the Einstein equations with a small positive cosmological constant, $G_{\alpha \beta}+\Lambda g_{\alpha \beta}=8 \pi T_{\alpha \beta}, \Lambda=\mathcal{O}(\epsilon)>0$. Repeat the steps of questions (i)-(iii), again identifying $h_{00}=-2 \Phi$, to show that the Newtonian limit is now described by the Poisson equation $\vec{\nabla}^{2} \Phi=4 \pi \rho-\Lambda$, and that a solution for the potential of a point particle is given by

$\Phi=-\frac{M}{r}-B r^{2}$

where $B$ is a constant you should determine. Briefly discuss the effect of the $B r^{2}$ term and determine for which range of the radius $r$ the weak-field limit is a justified approximation. [Hint: Absorb the term $\Lambda g_{\alpha \beta}$ as part of the energy-momentum tensor. Note also that in spherical symmetry $\vec{\nabla}^{2} f=\frac{1}{r} \frac{\partial^{2}}{\partial r^{2}}(r f)$.]

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• # Paper 1, Section II, D

Let $(\mathcal{M}, \boldsymbol{g})$ be a spacetime and $\boldsymbol{\Gamma}$ the Levi-Civita connection of the metric $\boldsymbol{g}$. The Riemann tensor of this spacetime is given in terms of the connection by

$R_{\rho \alpha \beta}^{\gamma}=\partial_{\alpha} \Gamma_{\rho \beta}^{\gamma}-\partial_{\beta} \Gamma_{\rho \alpha}^{\gamma}+\Gamma_{\rho \beta}^{\mu} \Gamma_{\mu \alpha}^{\gamma}-\Gamma_{\rho \alpha}^{\mu} \Gamma_{\mu \beta}^{\gamma}$

The contracted Bianchi identities ensure that the Einstein tensor satisfies

$\nabla^{\mu} G_{\mu \nu}=0$

(a) Show that the Riemann tensor obeys the symmetry

$R_{\rho \alpha \beta}^{\mu}+R_{\beta \rho \alpha}^{\mu}+R_{\alpha \beta \rho}^{\mu}=0 .$

(b) Show that a vector field $V^{\alpha}$ satisfies the Ricci identity

$2 \nabla_{[\alpha} \nabla_{\beta]} V^{\gamma}=\nabla_{\alpha} \nabla_{\beta} V^{\gamma}-\nabla_{\beta} \nabla_{\alpha} V^{\gamma}=R_{\rho \alpha \beta}^{\gamma} V^{\rho}$

Calculate the analogous expression for a rank $\left(\begin{array}{l}2 \\ 0\end{array}\right)$ tensor $T^{\mu \nu}$, i.e. calculate $\nabla_{[\alpha} \nabla_{\beta]} T^{\mu \nu}$ in terms of the Riemann tensor.

(c) Let $K^{\alpha}$ be a vector that satisfies the Killing equation

$\nabla_{\alpha} K_{\beta}+\nabla_{\beta} K_{\alpha}=0$

Use the symmetry relation of part (a) to show that

$\begin{gathered} \nabla_{\nu} \nabla_{\mu} K^{\alpha}=R_{\mu \nu \beta}^{\alpha} K^{\beta} \\ \nabla^{\mu} \nabla_{\mu} K^{\alpha}=-R_{\beta}^{\alpha} K^{\beta} \end{gathered}$

where $R_{\alpha \beta}$ is the Ricci tensor.

(d) Show that

$K^{\alpha} \nabla_{\alpha} R=2 \nabla^{[\mu} \nabla^{\lambda]} \nabla_{[\mu} K_{\lambda]},$

and use the result of part (b) to show that the right hand side evaluates to zero, hence showing that $K^{\alpha} \nabla_{\alpha} R=0$.

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• # Paper 2, Section II, D

Consider the spacetime metric

$d s^{2}=-f(r) d t^{2}+\frac{1}{f(r)} d r^{2}+r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right), \quad \text { with } \quad f(r)=1-\frac{2 m}{r}-H^{2} r^{2}$

where $H>0$ and $m>0$ are constants.

(a) Write down the Lagrangian for geodesics in this spacetime, determine three independent constants of motion and show that geodesics obey the equation

$\dot{r}^{2}+V(r)=E^{2}$

where $E$ is constant, the overdot denotes differentiation with respect to an affine parameter and $V(r)$ is a potential function to be determined.

(b) Sketch the potential $V(r)$ for the case of null geodesics, find any circular null geodesics of this spacetime, and determine whether they are stable or unstable.

(c) Show that $f(r)$ has two positive roots $r_{-}$and $r_{+}$if $m H<1 / \sqrt{27}$ and that these satisfy the relation $r_{-}<1 /(\sqrt{3} H).

(d) Describe in one sentence the physical significance of those points where $f(r)=0$.

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• # Paper 3, Section II, D

(a) Let $\mathcal{M}$ be a manifold with coordinates $x^{\mu}$. The commutator of two vector fields $\boldsymbol{V}$ and $\boldsymbol{W}$ is defined as

$[\boldsymbol{V}, \boldsymbol{W}]^{\alpha}=V^{\nu} \partial_{\nu} W^{\alpha}-W^{\nu} \partial_{\nu} V^{\alpha}$

(i) Show that $[\boldsymbol{V}, \boldsymbol{W}]$ transforms like a vector field under a change of coordinates from $x^{\mu}$ to $\tilde{x}^{\mu}$.

(ii) Show that the commutator of any two basis vectors vanishes, i.e.

$\left[\frac{\partial}{\partial x^{\alpha}}, \frac{\partial}{\partial x^{\beta}}\right]=0$

(iii) Show that if $\boldsymbol{V}$ and $\boldsymbol{W}$ are linear combinations (not necessarily with constant coefficients) of $n$ vector fields $\boldsymbol{Z}_{(a)}, a=1, \ldots, n$ that all commute with one another, then the commutator $[\boldsymbol{V}, \boldsymbol{W}]$ is a linear combination of the same $n$ fields $Z_{(a)}$.

[You may use without proof the following relations which hold for any vector fields $\boldsymbol{V}_{1}, \boldsymbol{V}_{2}, \boldsymbol{V}_{3}$ and any function $f$ :

\begin{aligned} {\left[\boldsymbol{V}_{1}, \boldsymbol{V}_{2}\right] } &=-\left[\boldsymbol{V}_{2}, \boldsymbol{V}_{1}\right] \\ {\left[\boldsymbol{V}_{1}, \boldsymbol{V}_{2}+\boldsymbol{V}_{3}\right] } &=\left[\boldsymbol{V}_{1}, \boldsymbol{V}_{2}\right]+\left[\boldsymbol{V}_{1}, \boldsymbol{V}_{3}\right] \\ {\left[\boldsymbol{V}_{1}, f \boldsymbol{V}_{2}\right] } &=f\left[\boldsymbol{V}_{1}, \boldsymbol{V}_{2}\right]+\boldsymbol{V}_{1}(f) \boldsymbol{V}_{2} \end{aligned}

but you should clearly indicate each time relation $(1),(2)$, or (3) is used.]

(b) Consider the 2-dimensional manifold $\mathbb{R}^{2}$ with Cartesian coordinates $\left(x^{1}, x^{2}\right)=$ $(x, y)$ carrying the Euclidean metric $g_{\alpha \beta}=\delta_{\alpha \beta}$.

(i) Express the coordinate basis vectors $\partial_{r}$ and $\partial_{\theta}$, where $r$ and $\theta$ denote the usual polar coordinates, in terms of their Cartesian counterparts.

(ii) Define the unit vectors

$\hat{\boldsymbol{r}}=\frac{\partial_{r}}{\left\|\partial_{r}\right\|}, \quad \hat{\boldsymbol{\theta}}=\frac{\partial_{\theta}}{\left\|\partial_{\theta}\right\|}$

and show that $(\hat{\boldsymbol{r}}, \hat{\boldsymbol{\theta}})$ are not a coordinate basis, i.e. there exist no coordinates $z^{\alpha}$ such that $\hat{\boldsymbol{r}}=\partial / \partial z^{1}$ and $\hat{\boldsymbol{\theta}}=\partial / \partial z^{2}$.

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• # Paper 4, Section II, D

(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: $P(R)=0 .]$

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• # Paper 1, Section II, 37E

Consider the de Sitter metric

$d s^{2}=-d t^{2}+e^{2 H t}\left(d x^{2}+d y^{2}+d z^{2}\right)$

where $H>0$ is a constant.

(a) Write down the Lagrangian governing the geodesics of this metric. Use the Euler-Lagrange equations to determine all non-vanishing Christoffel symbols.

(b) Let $\mathcal{C}$ be a timelike geodesic parametrized by proper time $\tau$ with initial conditions at $\tau=0$,

$t=0, \quad x=y=z=0, \quad \dot{x}=v_{0}>0, \quad \dot{y}=\dot{z}=0,$

where the dot denotes differentiation with respect to $\tau$ and $v_{0}$ is a constant. Assuming both $t$ and $\tau$ to be future oriented, show that at $\tau=0$,

$\dot{t}=\sqrt{1+v_{0}^{2}}$

(c) Find a relation between $\tau$ and $t$ along the geodesic of part (b) and show that $t \rightarrow-\infty$ for a finite value of $\tau$. [You may use without proof that

$\left.\int \frac{1}{\sqrt{1+a e^{-b u}}} d u=\frac{1}{b} \ln \frac{\sqrt{1+a e^{-b u}}+1}{\sqrt{1+a e^{-b u}}-1}+\text { constant }, \quad a, b>0 .\right]$

(d) Briefly interpret this result.

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• # Paper 2, Section II, E

The Friedmann equations and the conservation of energy-momentum for a spatially homogeneous and isotropic universe are given by:

$3 \frac{\dot{a}^{2}+k}{a^{2}}-\Lambda=8 \pi \rho, \quad \frac{2 a \ddot{a}+\dot{a}^{2}+k}{a^{2}}-\Lambda=-8 \pi P, \quad \dot{\rho}=-3 \frac{\dot{a}}{a}(P+\rho),$

where $a$ is the scale factor, $\rho$ the energy density, $P$ the pressure, $\Lambda$ the cosmological constant and $k=+1,0,-1$.

(a) Show that for an equation of state $P=w \rho, w=$ constant, the energy density obeys $\rho=\frac{3 \mu}{8 \pi} a^{-3(1+w)}$, for some constant $\mu$.

(b) Consider the case of a matter dominated universe, $w=0$, with $\Lambda=0$. Write the equation of motion for the scale factor $a$ in the form of an effective potential equation,

$\dot{a}^{2}+V(a)=C$

where you should determine the constant $C$ and the potential $V(a)$. Sketch the potential $V(a)$ together with the possible values of $C$ and qualitatively discuss the long-term dynamics of an initially small and expanding universe for the cases $k=+1,0,-1$.

(c) Repeat the analysis of part (b), again assuming $w=0$, for the cases:

(i) $\Lambda>0, k=-1$,

(ii) $\Lambda<0, k=0$,

(iii) $\Lambda>0, k=1$.

Discuss all qualitatively different possibilities for the dynamics of the universe in each case.

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• # Paper 3, Section II, E

The Schwarzschild metric in isotropic coordinates $\bar{x}^{\bar{\alpha}}=(\bar{t}, \bar{x}, \bar{y}, \bar{z}), \bar{\alpha}=0, \ldots, 3$, is given by:

$d s^{2}=\bar{g}_{\bar{\alpha} \bar{\beta}} d \bar{x}^{\bar{\alpha}} d \bar{x}^{\bar{\beta}}=-\frac{(1-A)^{2}}{(1+A)^{2}} d \bar{t}^{2}+(1+A)^{4}\left(d \bar{x}^{2}+d \bar{y}^{2}+d \bar{z}^{2}\right)$

where

$A=\frac{m}{2 \bar{r}}, \quad \bar{r}=\sqrt{\bar{x}^{2}+\bar{y}^{2}+\bar{z}^{2}}$

and $m$ is the mass of the black hole.

(a) Let $x^{\mu}=(t, x, y, z), \mu=0, \ldots, 3$, denote a coordinate system related to $\bar{x}^{\bar{\alpha}}$ by

$\bar{t}=\gamma(t-v x), \quad \bar{x}=\gamma(x-v t), \quad \bar{y}=y, \quad \bar{z}=z,$

where $\gamma=1 / \sqrt{1-v^{2}}$ and $-1. Write down the transformation matrix $\partial \bar{x}^{\bar{\alpha}} / \partial x^{\mu}$, briefly explain its physical meaning and show that the inverse transformation is of the same form, but with $v \rightarrow-v$.

(b) Using the coordinate transformation matrix of part (a), or otherwise, show that the components $g_{\mu \nu}$ of the metric in coordinates $x^{\mu}$ are given by

$d s^{2}=g_{\mu \nu} d x^{\mu} d x^{\nu}=f(A)\left(-d t^{2}+d x^{2}+d y^{2}+d z^{2}\right)+\gamma^{2} g(A)(d t-v d x)^{2}$

where $f$ and $g$ are functions of $A$ that you should determine. You should also express $A$ in terms of the coordinates $(t, x, y, z)$.

(c) Consider the limit $v \rightarrow 1$ with $p=m \gamma$ held constant. Show that for points $x \neq t$ the function $A \rightarrow 0$, while $\gamma^{2} A$ tends to a finite value, which you should determine. Hence determine the metric components $g_{\mu \nu}$ at points $x \neq t$ in this limit.

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• # Paper 4, Section II, E

(a) In the Newtonian weak-field limit, we can write the spacetime metric in the form

$d s^{2}=-(1+2 \Phi) d t^{2}+(1-2 \Phi) \delta_{i j} d x^{i} d x^{j},$

where $\delta_{i j} d x^{i} d x^{j}=d x^{2}+d y^{2}+d z^{2}$ and the potential $\Phi(t, x, y, z)$, as well as the velocity $v$ of particles moving in the gravitational field are assumed to be small, i.e.,

$\Phi, \partial_{t} \Phi, \partial_{x^{i}} \Phi, v^{2} \ll 1$

Use the geodesic equation for this metric to derive the equation of motion for a massive point particle in the Newtonian limit.

(b) The far-field limit of the Schwarzschild metric is a special case of (*) given, in spherical coordinates, by

$d s^{2}=-\left(1-\frac{2 M}{r}\right) d t^{2}+\left(1+\frac{2 M}{r}\right)\left(d r^{2}+r^{2} d \theta^{2}+r^{2} \sin ^{2} \theta d \varphi^{2}\right)$

where now $M / r \ll 1$. For the following questions, state your results to first order in $M / r$, i.e. neglecting terms of $\mathcal{O}\left((M / r)^{2}\right)$.

(i) Let $r_{1}, r_{2} \gg M$. Calculate the proper length $S$ along the radial curve from $r_{1}$ to $r_{2}$ at fixed $t, \theta, \varphi$.

(ii) Consider a massless particle moving radially from $r=r_{1}$ to $r=r_{2}$. According to an observer at rest at $r_{2}$, what time $T$ elapses during this motion?

(iii) The effective velocity of the particle as seen by the observer at $r_{2}$ is defined as $v_{\text {eff }}:=S / T$. Evaluate $v_{\text {eff }}$ and then take the limit of this result as $r_{1} \rightarrow r_{2}$. Briefly discuss the value of $v_{\text {eff }}$ in this limit.

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• # Paper 1, Section II, D

A static black hole in a five-dimensional spacetime is described by the metric

$d s^{2}=-\left(1-\frac{\mu}{r^{2}}\right) d t^{2}+\left(1-\frac{\mu}{r^{2}}\right)^{-1} d r^{2}+r^{2}\left[d \psi^{2}+\sin ^{2} \psi\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)\right]$

where $\mu>0$ is a constant.

A geodesic lies in the plane $\theta=\psi=\pi / 2$ and has affine parameter $\lambda$. Show that

$E=\left(1-\frac{\mu}{r^{2}}\right) \frac{d t}{d \lambda} \quad \text { and } \quad L=r^{2} \frac{d \phi}{d \lambda}$

are both constants of motion. Write down a third constant of motion.

Show that timelike and null geodesics satisfy the equation

$\frac{1}{2}\left(\frac{d r}{d \lambda}\right)^{2}+V(r)=\frac{1}{2} E^{2}$

for some potential $V(r)$ which you should determine.

Circular geodesics satisfy the equation $V^{\prime}(r)=0$. Calculate the values of $r$ for which circular null geodesics exist and for which circular timelike geodesics exist. Which are stable and which are unstable? Briefly describe how this compares to circular geodesics in the four-dimensional Schwarzschild geometry.

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• # Paper 2, Section II, D

(a) The Friedmann-Robertson-Walker metric is given by

$d s^{2}=-d t^{2}+a^{2}(t)\left[\frac{d r^{2}}{1-k r^{2}}+r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)\right]$

where $k=-1,0,+1$ and $a(t)$ is the scale factor.

For $k=+1$, show that this metric can be written in the form

$d s^{2}=-d t^{2}+\gamma_{i j} d x^{i} d x^{j}=-d t^{2}+a^{2}(t)\left[d \chi^{2}+\sin ^{2} \chi\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)\right]$

Calculate the equatorial circumference $(\theta=\pi / 2)$ of the submanifold defined by constant $t$ and $\chi$.

Calculate the proper volume, defined by $\int \sqrt{\operatorname{det} \gamma} d^{3} x$, of the hypersurface defined by constant $t$.

(b) The Friedmann equations are

\begin{aligned} &3\left(\frac{\dot{a}^{2}+k}{a^{2}}\right)-\Lambda=8 \pi \rho, \\ &\frac{2 a \ddot{a}+\dot{a}^{2}+k}{a^{2}}-\Lambda=-8 \pi P, \end{aligned}

where $\rho(t)$ is the energy density, $P(t)$ is the pressure, $\Lambda$ is the cosmological constant and dot denotes $d / d t$.

The Einstein static universe has vanishing pressure, $P(t)=0$. Determine $a, k$ and $\Lambda$ as a function of the density $\rho$.

The Einstein static universe with $a=a_{0}$ and $\rho=\rho_{0}$ is perturbed by radiation such that

$a=a_{0}+\delta a(t), \quad \rho=\rho_{0}+\delta \rho(t), \quad P=\frac{1}{3} \delta \rho(t)$

where $\delta a \ll a_{0}$ and $\delta \rho \ll \rho_{0}$. Show that the Einstein static universe is unstable to this perturbation.

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• # Paper 3, Section II, D

Let $\mathcal{M}$ be a two-dimensional manifold with metric $\boldsymbol{g}$ of signature $-+$.

(i) Let $p \in \mathcal{M}$. Use normal coordinates at the point $p$ to show that one can choose two null vectors $\mathbf{V}, \mathbf{W}$ that form a basis of the vector space $\mathcal{T}_{p}(\mathcal{M})$.

(ii) Consider the interval $I \subset \mathbb{R}$. Let $\gamma: I \rightarrow \mathcal{M}$ be a null curve through $p$ and $\mathbf{U} \neq 0$ be the tangent vector to $\gamma$ at $p$. Show that the vector $\mathbf{U}$ is either parallel to $\mathbf{V}$ or parallel to $\mathbf{W}$.

(iii) Show that every null curve in $\mathcal{M}$ is a null geodesic.

[Hint: You may wish to consider the acceleration $a^{\alpha}=U^{\beta} \nabla_{\beta} U^{\alpha}$.]

(iv) By providing an example, show that not every null curve in four-dimensional Minkowski spacetime is a null geodesic.

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• # Paper 4, Section II, D

(a) In the transverse traceless gauge, a plane gravitational wave propagating in the $z$ direction is described by a perturbation $h_{\alpha \beta}$ of the Minkowski metric $\eta_{\alpha \beta}=$ $\operatorname{diag}(-1,1,1,1)$ in Cartesian coordinates $x^{\alpha}=(t, x, y, z)$, where

$h_{\alpha \beta}=H_{\alpha \beta} e^{i k_{\mu} x^{\mu}}, \quad \text { where } \quad k^{\mu}=\omega(1,0,0,1)$

and $H_{\alpha \beta}$ is a constant matrix. Spacetime indices in this question are raised or lowered with the Minkowski metric.

The energy-momentum tensor of a gravitational wave is defined to be

$\tau_{\mu \nu}=\frac{1}{32 \pi}\left(\partial_{\mu} h^{\alpha \beta}\right)\left(\partial_{\nu} h_{\alpha \beta}\right)$

Show that $\partial^{\nu} \tau_{\mu \nu}=\frac{1}{2} \partial_{\mu} \tau_{\nu}^{\nu}$ and hence, or otherwise, show that energy and momentum are conserved.

(b) A point mass $m$ undergoes harmonic motion along the $z$-axis with frequency $\omega$ and amplitude $L$. Compute the energy flux emitted in gravitational radiation.

[Hint: The quadrupole formula for time-averaged energy flux radiated in gravitational waves is

\left\langle\frac{d E}{d t}\right\rangle=\frac{1}{5}\left\langle\dddot{Q}_{i j} \dddot{Q}_{i j}\right\rangle

where $Q_{i j}$ is the reduced quadrupole tensor.]

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• # Paper 1, Section II, D

Consider a family of geodesics with $s$ an affine parameter and $V^{a}$ the tangent vector on each curve. The equation of geodesic deviation for a vector field $W^{a}$ is

$\frac{D^{2} W^{a}}{D s^{2}}=R_{b c d}^{a} V^{b} V^{c} W^{d}$

where $\frac{D}{D s}$ denotes the directional covariant derivative $V^{b} \nabla_{b}$.

(i) Show that if

$V^{b} \frac{\partial W^{a}}{\partial x^{b}}=W^{b} \frac{\partial V^{a}}{\partial x^{b}}$

then $W^{a}$ satisfies $(*)$.

(ii) Show that $V^{a}$ and $s V^{a}$ satisfy $(*)$.

(iii) Show that if $W^{a}$ is a Killing vector field, meaning that $\nabla_{b} W_{a}+\nabla_{a} W_{b}=0$, then $W^{a}$ satisfies $(*)$.

(iv) Show that if $W^{a}=w U^{a}$ satisfies $(*)$, where $w$ is a scalar field and $U^{a}$ is a time-like unit vector field, then

$\begin{gathered} \frac{d^{2} w}{d s^{2}}=\left(\Omega^{2}-K\right) w \\ -\frac{D U^{a}}{D s} \frac{D U_{a}}{D s} \quad \text { and } \quad K=R_{a b c d} U^{a} V^{b} V^{c} U^{d} \end{gathered}$

\begin{aligned} & \text { where } \quad \Omega^{2}=-\frac{D U^{a}}{D s} \frac{D U_{a}}{D s} \text { and } \quad K=R_{a b c d} U^{a} V^{b} V^{c} U^{d} \text {. } \end{aligned}

[You may use: $\nabla_{b} \nabla_{c} X^{a}-\nabla_{c} \nabla_{b} X^{a}=R_{d b c}^{a} X^{d}$ for any vector field $\left.X^{a} .\right]$

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• # Paper 2, Section II, D

The Kasner (vacuum) cosmological model is defined by the line element

$d s^{2}=-c^{2} d t^{2}+t^{2 p_{1}} d x^{2}+t^{2 p_{2}} d y^{2}+t^{2 p_{3}} d z^{2} \quad \text { with } \quad t>0$

where $p_{1}, p_{2}, p_{3}$ are constants with $p_{1}+p_{2}+p_{3}=p_{1}^{2}+p_{2}^{2}+p_{3}^{2}=1$ and $0. Show that $p_{2} p_{3}<0$.

Write down four equations that determine the null geodesics of the Kasner model.

If $k^{a}$ is the tangent vector to the trajectory of a photon and $u^{a}$ is the four-velocity of a comoving observer (i.e., an observer at rest in the $(t, x, y, z)$ coordinate system above), what is the physical interpretation of $k_{a} u^{a}$ ?

Let $O$ be a comoving observer at the origin, $x=y=z=0$, and let $S$ be a comoving source of photons located on one of the spatial coordinate axes.

(i) Show that photons emitted by $S$ and observed by $O$ can be either redshifted or blue-shifted, depending on the location of $S$.

(ii) Given any fixed time $t=T$, show that there are locations for $S$ on each coordinate axis from which no photons reach $O$ for $t \leqslant T$.

Now suppose that $p_{1}=1$ and $p_{2}=p_{3}=0$. Does the property in (ii) still hold?

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• # Paper 3, Section II, D

For a spacetime that is nearly flat, the metric $g_{a b}$ can be expressed in the form

$g_{a b}=\eta_{a b}+h_{a b}$

where $\eta_{a b}$ is a flat metric (not necessarily diagonal) with constant components, and the components of $h_{a b}$ and their derivatives are small. Show that

$2 R_{b d} \approx h_{d}^{a}, b a+h_{b}^{a}, d a-h_{a, b d}^{a}-h_{b d, a c} \eta^{a c}$

where indices are raised and lowered using $\eta_{a b}$.

[You may assume that $\left.R_{b c d}^{a}=\Gamma_{b d, c}^{a}-\Gamma_{b c, d}^{a}+\Gamma_{c e}^{a} \Gamma_{d b}^{e}-\Gamma_{d e}^{a} \Gamma_{c b}^{e} .\right]$