General Relativity

2021

• Paper 1, Section II, C

  General Relativity | Part II, 2021

  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

  General Relativity | Part II, 2021

  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

  General Relativity | Part II, 2021

  (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, $37 \mathrm{C}$

  General Relativity | Part II, 2021

  (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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2020

• Paper 1, Section II, 38D

  General Relativity | Part II, 2020

  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, $37 \mathrm{D}$

  General Relativity | Part II, 2020

  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

  General Relativity | Part II, 2020

  (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

  General Relativity | Part II, 2020

  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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2019

• Paper 1, Section II, D

  General Relativity | Part II, 2019

  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

  General Relativity | Part II, 2019

  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 $r_{-}<1 /(\sqrt{3} H)<r_{+}$.

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

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• Paper 3, Section II, D

  General Relativity | Part II, 2019

  (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

  General Relativity | Part II, 2019

  (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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2018

• Paper 1, Section II, 37E

  General Relativity | Part II, 2018

  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

  General Relativity | Part II, 2018

  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

  General Relativity | Part II, 2018

  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 $-1<v<1$. 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

  General Relativity | Part II, 2018

  (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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2017

• Paper 1, Section II, D

  General Relativity | Part II, 2017

  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

  General Relativity | Part II, 2017

  (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

  General Relativity | Part II, 2017

  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

  General Relativity | Part II, 2017

  (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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2016

• Paper 1, Section II, D

  General Relativity | Part II, 2016

  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

  General Relativity | Part II, 2016

  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 $0<p_{1}<1$. 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

  General Relativity | Part II, 2016

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

  For the line element

  $$d s^{2}=2 d u d v+d x^{2}+d y^{2}+H(u, x, y) d u^{2},$$

  where $H$ and its derivatives are small, show that the linearised vacuum field equations reduce to $\nabla^{2} H=0$, where $\nabla^{2}$ is the two-dimensional Laplacian operator in $x$ and $y$.

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• Paper 4, Section II, D

  General Relativity | Part II, 2016

  A spherically symmetric static spacetime has metric

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

  where $-\infty<t<\infty, r \geqslant 0, b$ is a positive constant, and units such that $c=1$ are used.

  (a) Explain why a time-like geodesic may be assumed, without loss of generality, to lie in the equatorial plane $\theta=\pi / 2$. For such a geodesic, show that the quantities

  $$E=\left(1+r^{2} / b^{2}\right) \dot{t} \quad \text { and } \quad h=r^{2} \dot{\phi}$$

  are constants of the motion, where a dot denotes differentiation with respect to proper time, $\tau$. Hence find a first-order differential equation for $r(\tau)$.

  (b) Consider a massive particle fired from the origin, $r=0$. Show that the particle will return to the origin and find the proper time taken.

  (c) Show that circular orbits $r=a$ are possible for any $a>0$ and determine whether such orbits are stable. Show that on any such orbit a clock measures coordinate time.

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2015

• Paper 1, Section II, 35D

  General Relativity | Part II, 2015

  A vector field $\xi^{a}$ is said to be a conformal Killing vector field of the metric $g_{a b}$ if

  $$\xi_{(a ; b)}=\frac{1}{2} \psi g_{a b}$$

  for some scalar field $\psi$. It is a Killing vector field if $\psi=0$.

  (a) Show that $(*)$ is equivalent to

  $$\xi^{c} g_{a b, c}+\xi_{, a}^{c} g_{b c}+\xi_{, b}^{c} g_{a c}=\psi g_{a b}$$

  (b) Show that if $\xi^{a}$ is a conformal Killing vector field of the metric $g_{a b}$, then $\xi^{a}$ is a Killing vector field of the metric $\mathrm{e}^{2 \phi} g_{a b}$, where $\phi$ is any function that obeys

  $$2 \xi^{c} \phi_{, c}+\psi=0 .$$

  (c) Use part (b) to find an example of a metric with coordinates $t, x, y$ and $z$ (where $t>0)$ for which $(t, x, y, z)$ are the contravariant components of a Killing vector field. [Hint: You may wish to start by considering what happens in Minkowski space.]

  comment

• Paper 2, Section II, D

  General Relativity | Part II, 2015

  (a) The Schwarzschild metric is

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

  (in units for which the speed of light $c=1$ ). Show that a timelike geodesic in the equatorial plane obeys

  $$\frac{1}{2} \dot{r}^{2}+V(r)=\frac{1}{2} E^{2}$$

  where

  $$2 V(r)=\left(1-\frac{r_{s}}{r}\right)\left(1+\frac{h^{2}}{r^{2}}\right)$$

  and $E$ and $h$ are constants.

  (b) For a circular orbit of radius $r$, show that

  $$h^{2}=\frac{r^{2} r_{s}}{2 r-3 r_{s}} .$$

  Given that the orbit is stable, show that $r>3 r_{s}$.

  (c) Alice lives on a small planet that is in a stable circular orbit of radius $r$ around a (non-rotating) black hole of radius $r_{s}$. Bob lives on a spacecraft in deep space far from the black hole and at rest relative to it. Bob is ageing $k$ times faster than Alice. Find an expression for $k^{2}$ in terms of $r$ and $r_{s}$ and show that $k<\sqrt{2}$.

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• Paper 3, Section II, D

  General Relativity | Part II, 2015

  Let $\Gamma_{b c}^{a}$ be the Levi-Civita connection and $R_{b c d}^{a}$ the Riemann tensor corresponding to a metric $g_{a b}(x)$, and let $\widetilde{\Gamma}_{b c}^{a}$ be the Levi-Civita connection and $\widetilde{R}_{b c d}^{a}$ the Riemann tensor corresponding to a metric $\tilde{g}_{a b}(x)$. Let $T_{b c}^{a}=\widetilde{\Gamma}_{b c}^{a}-\Gamma_{b c}^{a}$.

  (a) Show that $T_{b c}^{a}$ is a tensor.

  (b) Using local inertial coordinates for the metric $g_{a b}$, or otherwise, show that

  $$\widetilde{R}_{b c d}^{a}-R_{b c d}^{a}=2 T_{b[d ; c]}^{a}-2 T_{e[d}^{a} T_{c] b}^{e}$$

  holds in all coordinate systems, where the semi-colon denotes covariant differentiation using the connection $\Gamma_{b c}^{a}$. [You may assume that $R_{b c d}^{a}=2 \Gamma_{b[d, c]}^{a}-2 \Gamma_{e[d}^{a} \Gamma_{c] b}^{e}$.]

  (c) In the case that $T_{b c}^{a}=\ell^{a} g_{b c}$ for some vector field $\ell^{a}$, show that $\widetilde{R}_{b d}=R_{b d}$ if and only if

  $$\ell_{b ; d}+\ell_{b} \ell_{d}=0$$

  (d) Using the result that $\ell_{[a ; b]}=0$ if and only if $\ell_{a}=\phi_{, a}$ for some scalar field $\phi$, show that the condition on $\ell_{a}$ in part (c) can be written as

  $$k_{a ; b}=0$$

  for a certain covector field $k_{a}$, which you should define.

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• Paper 4, Section II, D

  General Relativity | Part II, 2015

  In static spherically symmetric coordinates, the metric $g_{a b}$ for de Sitter space is given by

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

  where $d \Omega^{2}=d \theta^{2}+\sin ^{2} \theta d \phi^{2}$ and $a$ is a constant.

  (a) Let $u=t-a \tanh ^{-1}(r / a)$ for $r \leqslant a$. Use the $(u, r, \theta, \phi)$ coordinates to show that the surface $r=a$ is non-singular. Is $r=0$ a space-time singularity?

  (b) Show that the vector field $g^{a b} u_{, a}$ is null.

  (c) Show that the radial null geodesics must obey either

  $$\frac{d u}{d r}=0 \quad \text { or } \quad \frac{d u}{d r}=-\frac{2}{1-r^{2} / a^{2}}$$

  Which of these families of geodesics is outgoing $(d r / d t>0) ?$

  Sketch these geodesics in the $(u, r)$ plane for $0 \leqslant r \leqslant a$, where the $r$-axis is horizontal and lines of constant $u$ are inclined at $45^{\circ}$ to the horizontal.

  (d) Show, by giving an explicit example, that an observer moving on a timelike geodesic starting at $r=0$ can cross the surface $r=a$ within a finite proper time.

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2014

• Paper 1, Section II, 37E

  General Relativity | Part II, 2014

  For a timelike geodesic in the equatorial plane $\left(\theta=\frac{1}{2} \pi\right)$ of the Schwarzschild spacetime with line element

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

  derive the equation

  $$\frac{1}{2} \dot{r}^{2}+V(r)=\frac{1}{2}(E / c)^{2}$$

  where

  $$\frac{2 V(r)}{c^{2}}=1-\frac{r_{s}}{r}+\frac{h^{2}}{c^{2} r^{2}}-\frac{h^{2} r_{s}}{c^{2} r^{3}}$$

  and $h$ and $E$ are constants. The dot denotes the derivative with respect to an affine parameter $\tau$ satisfying $c^{2} d \tau^{2}=-d s^{2}$.

  Given that there is a stable circular orbit at $r=R$, show that

  $$\frac{h^{2}}{c^{2}}=\frac{R^{2} \epsilon}{2-3 \epsilon}$$

  where $\epsilon=r_{s} / R$.

  Compute $\Omega$, the orbital angular frequency (with respect to $\tau$ ).

  Show that the angular frequency $\omega$ of small radial perturbations is given by

  $$\frac{\omega^{2} R^{2}}{c^{2}}=\frac{\epsilon(1-3 \epsilon)}{2-3 \epsilon}$$

  Deduce that the rate of precession of the perihelion of the Earth's orbit, $\Omega-\omega$, is approximately $3 \Omega^{3} T^{2}$, where $T$ is the time taken for light to travel from the Sun to the Earth. [You should assume that the Earth's orbit is approximately circular, with $r_{s} / R \ll 1$ and $\left.E \simeq c^{2} .\right]$

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• Paper 2, Section II, E

  General Relativity | Part II, 2014

  Show how the geodesic equations and hence the Christoffel symbols $\Gamma^{a} b c$ can be obtained from a Lagrangian.

  In units with $c=1$, the FLRW spacetime line element is

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

  Show that $\Gamma_{01}^{1}=\dot{a} / a$.

  You are given that, for the above metric, $G_{0}{ }^{0}=-3 \dot{a}^{2} / a^{2}$ and $G_{1}^{1}=-2 \ddot{a} / a-\dot{a}^{2} / a^{2}$, where $G_{a}^{b}$ is the Einstein tensor, which is diagonal. Verify by direct calculation that $\nabla_{b} G_{a}^{b}=0$.

  Solve the vacuum Einstein equations in the presence of a cosmological constant to determine the form of $a(t)$.

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• Paper 3, Section II, $37 \mathrm{E}$

  General Relativity | Part II, 2014

  The vector field $V^{a}$ is the normalised $\left(V_{a} V^{a}=-c^{2}\right)$ tangent to a congruence of timelike geodesics, and $B_{a b}=\nabla_{b} V_{a}$.

  Show that:

  (i) $V^{a} B_{a b}=V^{b} B_{a b}=0$;

  (ii) $V^{c} \nabla_{c} B_{a b}=-B_{b}^{c} B_{a c}-R_{a c b}^{d} V^{c} V_{d}$.

  [You may use the Ricci identity $\nabla_{c} \nabla_{b} X_{a}=\nabla_{b} \nabla_{c} X_{a}-R_{a c b}^{d} X_{d}$.]

  Now assume that $B_{a b}$ is symmetric and let $\theta=B_{a}{ }^{a}$. By writing $B_{a b}=\widetilde{B}_{a b}+\frac{1}{4} \theta g_{a b}$, or otherwise, show that

  $$\frac{d \theta}{d \tau} \leqslant-\frac{1}{4} \theta^{2}-R_{00}$$

  where $R_{00}=R_{a b} V^{a} V^{b}$ and $\frac{d \theta}{d \tau} \equiv V^{a} \nabla_{a} \theta$. [You may use without proof the result that \left.\widetilde{B}_{a b} \widetilde{B}^{a b} \geqslant 0 .\right]

  Assume, in addition, that the stress-energy tensor $T_{a b}$ takes the perfect-fluid form $\left(\rho+p / c^{2}\right) V_{a} V_{b}+p g_{a b}$ and that $\rho c^{2}+3 p>0$. Show that

  $$\frac{d \theta^{-1}}{d \tau}>\frac{1}{4}$$

  and deduce that, if $\theta(0)<0$, then $|\theta(\tau)|$ will become unbounded for some value of $\tau$ less than $4 /|\theta(0)|$.

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• Paper 4, Section II, E

  General Relativity | Part II, 2014

  A plane-wave spacetime has line element

  $$d s^{2}=H d u^{2}+2 d u d v+d x^{2}+d y^{2}$$

  where $H=x^{2}-y^{2}$. Show that the line element is unchanged by the coordinate transformation

  $$u=\bar{u}, \quad v=\bar{v}+\bar{x} e^{\bar{u}}-\frac{1}{2} e^{2 \bar{u}}, \quad x=\bar{x}-e^{\bar{u}}, \quad y=\bar{y}$$

  Show more generally that the line element is unchanged by coordinate transformations of the form

  $$u=\bar{u}+a, \quad v=\bar{v}+b \bar{x}+c, \quad x=\bar{x}+p, \quad y=\bar{y}$$

  where $a, b, c$ and $p$ are functions of $\bar{u}$, which you should determine and which depend in total on four parameters (arbitrary constants of integration).

  Deduce (without further calculation) that the line element is unchanged by a 6parameter family of coordinate transformations, of which a 5 -parameter family leave invariant the surfaces $u=$ constant.

  For a general coordinate transformation $x^{a}=x^{a}\left(\bar{x}^{b}\right)$, give an expression for the transformed Ricci tensor $\bar{R}_{c d}$ in terms of the Ricci tensor $R_{a b}$ and the transformation matrices $\frac{\partial x^{a}}{\partial \bar{x}^{c}}$. Calculate $\bar{R}_{\bar{x} \bar{x}}$ when the transformation is given by $(*)$ and deduce that $R_{v v}=R_{v x}$

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2013

• Paper 1, Section II, 37D

  General Relativity | Part II, 2013

  The curve $\gamma, x^{a}=x^{a}(\lambda)$, is a geodesic with affine parameter $\lambda$. Write down the geodesic equation satisfied by $x^{a}(\lambda)$.

  Suppose the parameter is changed to $\mu(\lambda)$, where $d \mu / d \lambda>0$. Obtain the corresponding equation and find the condition for $\mu$ to be affine. Deduce that, whatever parametrization $\nu$ is used along the curve $\gamma$, the tangent vector $K^{a}$ to $\gamma$ satisfies

  $$\left(\nabla_{\nu} K\right)^{[a} K^{b]}=0 \text {. }$$

  Now consider a spacetime with metric $g_{a b}$, and conformal transformation

  $$\tilde{g}_{a b}=\Omega^{2}\left(x^{c}\right) g_{a b}$$

  The curve $\gamma$ is a geodesic of the metric connection of $g_{a b}$. What further restriction has to be placed on $\gamma$ so that it is also a geodesic of the metric connection of $\tilde{g}_{a b}$ ? Justify your answer.

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• Paper 2, Section II, D

  General Relativity | Part II, 2013

  A spacetime contains a one-parameter family of geodesics $x^{a}=x^{a}(\lambda, \mu)$, where $\lambda$ is a parameter along each geodesic, and $\mu$ labels the geodesics. The tangent to the geodesics is $T^{a}=\partial x^{a} / \partial \lambda$, and $N^{a}=\partial x^{a} / \partial \mu$ is a connecting vector. Prove that

  $$\nabla_{\mu} T^{a}=\nabla_{\lambda} N^{a}$$

  and hence derive the equation of geodesic deviation:

  $$\nabla_{\lambda}^{2} N^{a}+R_{b c d}^{a} T^{b} N^{c} T^{d}=0$$

  [You may assume $R_{b c d}^{a}=-R_{b d c}^{a}$ and the Ricci identity in the form

  $$\left.\left(\nabla_{\lambda} \nabla_{\mu}-\nabla_{\mu} \nabla_{\lambda}\right) T^{a}=R_{b c d}^{a} T^{b} T^{c} N^{d}\right]$$

  Consider the two-dimensional space consisting of the sphere of radius $r$ with line element

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

  Show that one may choose $T^{a}=(1,0), N^{a}=(0,1)$, and that

  $$\nabla_{\theta} N^{a}=\cot \theta N^{a} .$$

  Hence show that $R=2 / r^{2}$, using the geodesic deviation equation and the identity in any two-dimensional space

  $$R_{b c d}^{a}=\frac{1}{2} R\left(\delta_{c}^{a} g_{b d}-\delta_{d}^{a} g_{b c}\right)$$

  where $R$ is the Ricci scalar.

  Verify your answer by direct computation of $R$.

  [You may assume that the only non-zero connection components are

  $$\Gamma_{\phi \theta}^{\phi}=\Gamma_{\theta \phi}^{\phi}=\cot \theta$$

  and

  $$\Gamma_{\phi \phi}^{\theta}=-\sin \theta \cos \theta .$$

  You may also use the definition

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

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• Paper 3, Section II, D

  General Relativity | Part II, 2013

  The Schwarzschild metric for a spherically symmetric black hole is given by

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

  where we have taken units in which we set $G=c=1$. Consider a photon moving within the equatorial plane $\theta=\frac{\pi}{2}$, along a path $x^{a}(\lambda)$ with affine parameter $\lambda$. Using a variational principle with Lagrangian

  $$L=g_{a b} \frac{d x^{a}}{d \lambda} \frac{d x^{b}}{d \lambda}$$

  or otherwise, show that

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

  for constants $E$ and $h$. Deduce that

  $$\left(\frac{d r}{d \lambda}\right)^{2}=E^{2}-\frac{h^{2}}{r^{2}}\left(1-\frac{2 M}{r}\right)$$

  Assume now that the photon approaches from infinity. Show that the impact parameter (distance of closest approach) is given by

  $$b=\frac{h}{E}$$

  Denote the right hand side of equation $(*)$ as $f(r)$. By sketching $f(r)$ in each of the cases below, or otherwise, show that:

  (a) if $b^{2}>27 M^{2}$, the photon is deflected but not captured by the black hole;

  (b) if $b^{2}<27 M^{2}$, the photon is captured;

  (c) if $b^{2}=27 M^{2}$, the photon orbit has a particular form, which should be described.

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• Paper 4, Section II, D

  General Relativity | Part II, 2013

  Consider the metric describing the interior of a star,

  $$d s^{2}=-e^{2 \alpha(r)} d t^{2}+e^{2 \beta(r)} d r^{2}+r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)$$

  defined for $0 \leqslant r \leqslant r_{0}$ by

  $$e^{\alpha(r)}=\frac{3}{2} e^{-\beta_{0}}-\frac{1}{2} e^{-\beta(r)},$$

  with

  $$e^{-2 \beta(r)}=1-A r^{2}$$

  Here $A=2 M / r_{0}^{3}$, where $M$ is the mass of the star, $\beta_{0}=\beta\left(r_{0}\right)$, and we have taken units in which we have set $G=c=1$.

  (i) The star is made of a perfect fluid with energy-momentum tensor

  $$T_{a b}=(p+\rho) u_{a} u_{b}+p g_{a b}$$

  Here $u^{a}$ is the 4-velocity of the fluid which is at rest, the density $\rho$ is constant throughout the star $\left(0 \leqslant r \leqslant r_{0}\right)$ and the pressure $p=p(r)$ depends only on the radial coordinate. Write down the Einstein field equations and show that they may be written as

  $$R_{a b}=8 \pi(p+\rho) u_{a} u_{b}+4 \pi(\rho-p) g_{a b}$$

  (ii) Using the formulae given below, or otherwise, show that for $0 \leqslant r \leqslant r_{0}$, one has

  $$\begin{aligned} &4 \pi(\rho+p)=\frac{\left(\alpha^{\prime}+\beta^{\prime}\right)}{r} e^{-2 \beta(r)} \\ &4 \pi(\rho-p)=\left(\frac{\beta^{\prime}-\alpha^{\prime}}{r}-\frac{1}{r^{2}}\right) e^{-2 \beta(r)}+\frac{1}{r^{2}} \end{aligned}$$

  where primes denote differentiation with respect to $r$. Hence show that

  $$\rho=\frac{3 A}{8 \pi} \quad, \quad p(r)=\frac{3 A}{8 \pi}\left(\frac{e^{-\beta(r)}-e^{-\beta_{0}}}{3 e^{-\beta_{0}}-e^{-\beta(r)}}\right)$$

  [The non-zero components of the Ricci tensor are

  $$\begin{aligned} &R_{00}=e^{2 \alpha-2 \beta}\left(\alpha^{\prime \prime}-\alpha^{\prime} \beta^{\prime}+\alpha^{\prime 2}+\frac{2 \alpha^{\prime}}{r}\right) \\ &R_{11}=-\alpha^{\prime \prime}+\alpha^{\prime} \beta^{\prime}-\alpha^{\prime 2}+\frac{2 \beta^{\prime}}{r} \\ &R_{22}=1+e^{-2 \beta}\left[\left(\beta^{\prime}-\alpha^{\prime}\right) r-1\right] \\ &R_{33}=\sin ^{2} \theta R_{22} \end{aligned}$$

  Note that

  $$\left.\alpha^{\prime}=\frac{1}{2} A r e^{\beta-\alpha} \quad, \quad \beta^{\prime}=A r e^{2 \beta} .\right]$$

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2012

• Paper 1, Section II, B

  General Relativity | Part II, 2012

  (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.

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• Paper 2, Section II, B

  General Relativity | Part II, 2012

  The metric of any two-dimensional rotationally-symmetric curved space can be written in terms of polar coordinates, $(r, \theta)$, with $0 \leqslant \theta<2 \pi, r \geqslant 0$, as

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

  where $\phi=\phi(r)$. Show that the Christoffel symbols $\Gamma_{r \theta}^{r}, \Gamma_{r r}^{\theta}$ and $\Gamma_{\theta \theta}^{\theta}$ are each zero, and compute $\Gamma_{r r}^{r}, \Gamma_{\theta \theta}^{r}$ and $\Gamma_{r \theta}^{\theta}=\Gamma_{\theta r}^{\theta}$.

  The Ricci tensor 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 here denotes partial derivative. Prove that $R_{r \theta}=0$ and that

  $$R_{r r}=-\phi^{\prime \prime}-\frac{\phi^{\prime}}{r}, \quad R_{\theta \theta}=r^{2} R_{r r}$$

  Suppose now that, in this space, the Ricci scalar takes the constant value $-2$. Find a differential equation for $\phi(r)$.

  By a suitable coordinate transformation $r \rightarrow \chi(r), \theta$ unchanged, this space of constant Ricci scalar can be described by the metric

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

  From this coordinate transformation, find $\cosh \chi$ and $\sinh \chi$ in terms of $r$. Deduce that

  $$e^{\phi(r)}=\frac{2 A}{1-A^{2} r^{2}},$$

  where $0 \leqslant A r<1$, and $A$ is a positive constant.

  [You may use

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

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• Paper 3, Section II, 37B

  General Relativity | Part II, 2012

  (i) The Schwarzschild metric is given by

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

  Consider a time-like geodesic $x^{a}(\tau)$, where $\tau$ is the proper time, lying in the plane $\theta=\pi / 2$. Use the Lagrangian $L=g_{a b} \dot{x}^{a} \dot{x}^{b}$ to derive the equations governing the geodesic, showing that

  $$r^{2} \dot{\phi}=h$$

  with $h$ constant, and hence demonstrate that

  $$\frac{d^{2} u}{d \phi^{2}}+u=\frac{M}{h^{2}}+3 M u^{2}$$

  where $u=1 / r$. State which term in this equation makes it different from an analogous equation in Newtonian theory.

  (ii) Now consider Kruskal coordinates, in which the Schwarzschild $t$ and $r$ are replaced by $U$ and $V$, defined for $r>2 M$ by

  $$\begin{aligned} &U \equiv\left(\frac{r}{2 M}-1\right)^{1 / 2} e^{r /(4 M)} \cosh \left(\frac{t}{4 M}\right) \\ &V \equiv\left(\frac{r}{2 M}-1\right)^{1 / 2} e^{r /(4 M)} \sinh \left(\frac{t}{4 M}\right) \end{aligned}$$

  and for $r<2 M$ by

  $$\begin{aligned} &U \equiv\left(1-\frac{r}{2 M}\right)^{1 / 2} e^{r /(4 M)} \sinh \left(\frac{t}{4 M}\right) \\ &V \equiv\left(1-\frac{r}{2 M}\right)^{1 / 2} e^{r /(4 M)} \cosh \left(\frac{t}{4 M}\right) \end{aligned}$$

  Given that the metric in these coordinates is

  $$d s^{2}=\frac{32 M^{3}}{r} e^{-r /(2 M)}\left(-d V^{2}+d U^{2}\right)+r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right),$$

  where $r=r(U, V)$ is defined implicitly by

  $$\left(\frac{r}{2 M}-1\right) e^{r /(2 M)}=U^{2}-V^{2}$$

  sketch the Kruskal diagram, indicating the positions of the singularity at $r=0$, the event horizon at $r=2 M$, and general lines of constant $r$ and of constant $t$.

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• Paper 4, Section II, B

  General Relativity | Part II, 2012

  The metric for a homogenous isotropic universe, in comoving coordinates, can be written as

  $$d s^{2}=-d t^{2}+a^{2}\left\{d r^{2}+f^{2}\left[d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right]\right\}$$

  where $a=a(t)$ and $f=f(r)$ are some functions.

  Write down expressions for the Hubble parameter $H$ and the deceleration parameter $q$ in terms of $a(\eta)$ and $h \equiv d \log a / d \eta$, where $\eta$ is conformal time, defined by $d \eta=a^{-1} d t$.

  The universe is composed of a perfect fluid of density $\rho$ and pressure $p=(\gamma-1) \rho$, where $\gamma$ is a constant. Defining $\Omega=\rho / \rho_{c}$, where $\rho_{c}=3 H^{2} / 8 \pi G$, show that

  $$\frac{k}{h^{2}}=\Omega-1, \quad q=\alpha \Omega, \quad \frac{d \Omega}{d \eta}=2 q h(\Omega-1)$$

  where $k$ is the curvature parameter $(k=+1,0$ or $-1)$ and $\alpha \equiv \frac{1}{2}(3 \gamma-2)$. Hence deduce that

  $$\frac{d \Omega}{d a}=\frac{2 \alpha}{a} \Omega(\Omega-1)$$

  and

  $$\Omega=\frac{1}{1-A a^{2 \alpha}}$$

  where $A$ is a constant. Given that $A=\frac{k}{2 G M}$, sketch curves of $\Omega$ against $a$ in the case when $\gamma>2 / 3$.

  [You may assume an Einstein equation, for the given metric, in the form

  $$\frac{h^{2}}{a^{2}}+\frac{k}{a^{2}}=\frac{8}{3} \pi G \rho$$

  and the energy conservation equation

  $$\left.\frac{d \rho}{d t}+3 H(\rho+p)=0 .\right]$$

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2011

• Paper 1, Section II, 37D

  General Relativity | Part II, 2011

  Consider a metric of the form

  $$d s^{2}=-2 d u d v+d x^{2}+d y^{2}-2 H(u, x, y) d u^{2} .$$

  Let $x^{a}(\lambda)$ describe an affinely-parametrised geodesic, where $x^{a} \equiv\left(x^{1}, x^{2}, x^{3}, x^{4}\right)=$ $(u, v, x, y)$. Write down explicitly the Lagrangian

  $$L=g_{a b} \dot{x}^{a} \dot{x}^{b},$$

  with $\dot{x}^{a}=d x^{a} / d \lambda$, using the given metric. Hence derive the four geodesic equations. In particular, show that

  $$\ddot{v}+2\left(\frac{\partial H}{\partial x} \dot{x}+\frac{\partial H}{\partial y} \dot{y}\right) \dot{u}+\frac{\partial H}{\partial u} \dot{u}^{2}=0$$

  By comparing these equations with the standard form of the geodesic equation, show that $\Gamma_{13}^{2}=\partial H / \partial x$ and derive the other Christoffel symbols.

  The Ricci tensor, $R_{a b}$, is defined by

  $$R_{a b}=\Gamma_{a b, d}^{d}-\Gamma_{a d, b}^{d}+\Gamma_{d f}^{d} \Gamma_{b a}^{f}-\Gamma_{b f}^{d} \Gamma_{d a}^{f}$$

  By considering the case $a=1, b=1$, show that the vacuum Einstein field equations imply

  $$\frac{\partial^{2} H}{\partial x^{2}}+\frac{\partial^{2} H}{\partial y^{2}}=0$$

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• Paper 2, Section II, 36D

  General Relativity | Part II, 2011

  The curvature tensor $R_{b c d}^{a}$ satisfies

  $$V_{a ; b c}-V_{a ; c b}=V_{e} R_{a b c}^{e}$$

  for any covariant vector field $V_{a}$. Hence express $R_{a b c}^{e}$ in terms of the Christoffel symbols and their derivatives. Show that

  $$R_{a b c}^{e}=-R_{a c b}^{e}$$

  Further, by setting $V_{a}=\partial \phi / \partial x^{a}$, deduce that

  $$R_{a b c}^{e}+R_{c a b}^{e}+R_{b c a}^{e}=0 .$$

  Using local inertial coordinates or otherwise, obtain the Bianchi identities.

  Define the Ricci tensor in terms of the curvature tensor and show that it is symmetric. [You may assume that $R_{a b c d}=-R_{b a c d}$.] Write down the contracted Bianchi identities.

  In certain spacetimes of dimension $n \geqslant 2, R_{a b c d}$ takes the form

  $$R_{a b c d}=K\left(g_{a c} g_{b d}-g_{a d} g_{b c}\right)$$

  Obtain the Ricci tensor and curvature scalar. Deduce, under some restriction on $n$ which should be stated, that $K$ is a constant.

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• Paper 4, Section II, D

  General Relativity | Part II, 2011

  The metric of the Schwarzschild solution is

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

  Show that, for an incoming radial light ray, the quantity

  $$v=t+r+2 M \log \left|\frac{r}{2 M}-1\right|$$

  is constant.

  Express $d s^{2}$ in terms of $r, v, \theta$ and $\phi$. Determine the light-cone structure in these coordinates, and use this to discuss the nature of the apparent singularity at $r=2 M$.

  An observer is falling radially inwards in the region $r<2 M$. Assuming that the metric for $r<2 M$ is again given by $(*)$, obtain a bound for $d \tau$, where $\tau$ is the proper time of the observer, in terms of $d r$. Hence, or otherwise, determine the maximum proper time that can elapse between the events at which the observer crosses $r=2 M$ and is torn apart at $r=0$.

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2010

• Paper 1, Section II, B

  General Relativity | Part II, 2010

  Consider a spacetime $\mathcal{M}$ with a metric $g_{a b}\left(x^{c}\right)$ and a corresponding connection $\Gamma_{b c}^{a}$. Write down the differential equation satisfied by a geodesic $x^{a}(\lambda)$, where $\lambda$ is an affine parameter.

  Show how the requirement that

  $$\delta \int g_{a b}\left(x^{c}\right) \frac{d}{d \lambda} x^{a}(\lambda) \frac{d}{d \lambda} x^{b}(\lambda) d \lambda=0$$

  where $\delta$ denotes variation of the path, gives the geodesic equation and determines $\Gamma_{b c}^{a}$.

  Show that the timelike geodesics for the 2 -manifold with line element

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

  are given by

  $$t^{2}=x^{2}+\alpha x+\beta$$

  where $\alpha$ and $\beta$ are constants.

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• Paper 2, Section II, B

  General Relativity | Part II, 2010

  A vector field $k^{a}$ which satisfies

  $$k_{a ; b}+k_{b ; a}=0$$

  is called a Killing vector field. Prove that $k^{a}$ is a Killing vector field if and only if

  $$k^{c} g_{a b, c}+k_{, b}^{c} g_{a c}+k_{, a}^{c} g_{b c}=0$$

  Prove also that if $V^{a}$ satisfies

  $$V_{; b}^{a} V^{b}=0$$

  then

  $$\left(V^{a} k_{a}\right)_{, b} V^{b}=0$$

  for any Killing vector field $k^{a}$.

  In the two-dimensional space-time with coordinates $x^{a}=(u, v)$ and line element

  $$d s^{2}=-d u^{2}+u^{2} d v^{2}$$

  verify that $(0,1), e^{-v}\left(1, u^{-1}\right)$ and $e^{v}\left(-1, u^{-1}\right)$ are Killing vector fields. Show, by using $(*)$ with $V^{a}$ the tangent vector to a geodesic, that geodesics in this space-time are given by

  $$\alpha e^{v}+\beta e^{-v}=2 \gamma u^{-1}$$

  where $\alpha, \beta$ and $\gamma$ are arbitrary real constants.

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• Paper 4, Section II, B

  General Relativity | Part II, 2010

  The Schwarzschild line element is given by

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

  where $F=1-r_{s} / r$ and $r_{s}$ is the Schwarzschild radius. Obtain the equation of geodesic motion of photons moving in the equatorial plane, $\theta=\pi / 2$, in the form

  $$\left(\frac{d r}{d \tau}\right)^{2}=E^{2}-\frac{h^{2} F}{r^{2}}$$

  where $\tau$ is proper time, and $E$ and $h$ are constants whose physical significance should be indicated briefly.

  Defining $u=1 / r$ show that light rays are determined by

  $$\left(\frac{d u}{d \phi}\right)^{2}=\left(\frac{1}{b}\right)^{2}-u^{2}+r_{s} u^{3}$$

  where $b=h / E$ and $r_{s}$ may be taken to be small. Show that, to zeroth order in $r_{s}$, a light ray is a straight line passing at distance $b$ from the origin. Show that, to first order in $r_{s}$, the light ray is deflected through an angle $2 r_{s} / b$. Comment briefly on some observational evidence for the result.

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2009

• Paper 1, Section II, D

  General Relativity | Part II, 2009

  Write down the differential equations governing geodesic curves $x^{a}(\lambda)$ both when $\lambda$ is an affine parameter and when it is a general one.

  A conformal transformation of a spacetime is given by

  $$g_{a b} \rightarrow \tilde{g}_{a b}=\Omega^{2}(x) g_{a b}$$

  Obtain a formula for the new Christoffel symbols $\tilde{\Gamma}_{b c}^{a}$ in terms of the old ones and the derivatives of $\Omega$. Hence show that null geodesics in the metric $g_{a b}$ are also geodesic in the metric $\tilde{g}_{a b}$.

  Show that the Riemann tensor has only one independent component in two dimensions, and hence derive

  $$R=2 \operatorname{det}\left(g^{a b}\right) R_{0101}$$

  where $R$ is the Ricci scalar.

  It is given that in a 2-dimensional spacetime $M, R$ transforms as

  $$R \rightarrow \tilde{R}=\Omega^{-2}(R-2 \square \log \Omega)$$

  where $\square \phi=g^{a b} \nabla_{a} \nabla_{b} \phi$. Assuming that the equation $\square \phi=\rho(x)$ can always be solved, show that $\Omega$ can be chosen to set $\tilde{g}$ to be the metric of 2-dimensional Minkowski spacetime. Hence show that all null curves in $M$ are geodesic.

  Discuss the null geodesics if the line element of $M$ is

  $$d s^{2}=-t^{-1} d t^{2}+t d \theta^{2}$$

  where $t \in(-\infty, 0)$ or $(0, \infty)$ and $\theta \in[0,2 \pi]$.

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• Paper 2, Section II, D

  General Relativity | Part II, 2009

  A spacetime has line element

  $$d s^{2}=-d t^{2}+t^{2 p_{1}} d x_{1}^{2}+t^{2 p_{2}} d x_{2}^{2}+t^{2 p_{3}} d x_{3}^{2},$$

  where $p_{1}, p_{2}$ and $p_{3}$ are constants. Calculate the Christoffel symbols.

  Find the constraints on $p_{1}, p_{2}$ and $p_{3}$ for this spacetime to be a solution of the vacuum Einstein equations with zero cosmological constant. For which values is the spacetime flat?

  Show that it is not possible to have all of $p_{1}, p_{2}$ and $p_{3}$ strictly positive, so that if they are all non-zero, the spacetime expands in at least one direction and contracts in at least one direction.

  [The Riemann tensor is given in terms of the Christoffel symbols by

  $$\left.R_{b c d}^{a}=\Gamma_{d b, c}^{a}-\Gamma_{c b, d}^{a}+\Gamma_{c f}^{a} \Gamma_{d b}^{f}-\Gamma_{d f}^{a} \Gamma_{c b}^{f}\right]$$

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• Paper 4, Section II, D

  General Relativity | Part II, 2009

  The Schwarzschild metric is given by

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

  where $M$ is the mass in gravitational units. By using the radial component of the geodesic equations, or otherwise, show for a particle moving on a geodesic in the equatorial plane $\theta=\pi / 2$ with $r$ constant that

  $$\left(\frac{d \phi}{d t}\right)^{2}=\frac{M}{r^{3}}$$

  Show that such an orbit is stable for $r>6 M$.

  An astronaut circles the Earth freely for a long time on a circular orbit of radius $R$, while the astronaut's twin remains motionless on Earth, which is assumed to be spherical, with radius $R_{0}$, and non-rotating. Show that, on returning to Earth, the astronaut will be younger than the twin only if $2 R<3 R_{0}$.

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2008

• 1.II.35E

  General Relativity | Part II, 2008

  For the metric

  $$\mathrm{d} s^{2}=\frac{1}{r^{2}}\left(-\mathrm{d} t^{2}+\mathrm{d} r^{2}\right), \quad r \geqslant 0,$$

  obtain the geodesic equations of motion. For a massive particle show that

  $$\left(\frac{\mathrm{d} r}{\mathrm{~d} t}\right)^{2}=1-\frac{1}{k^{2} r^{2}}$$

  for some constant $k$. Show that the particle moves on trajectories

  $$r^{2}-t^{2}=\frac{1}{k^{2}}, \quad k r=\sec \tau, \quad k t=\tan \tau$$

  where $\tau$ is the proper time, if the origins of $t, \tau$ are chosen appropriately.

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• 2.II.35E

  General Relativity | Part II, 2008

  Let $x^{a}(\lambda)$ be a path $P$ with tangent vector $T^{a}=\frac{d}{d \lambda} x^{a}(\lambda)$. For vectors $X^{a}(x(\lambda))$ and $Y^{a}(x(\lambda))$ defined on $P$ let

  $$\nabla_{T} X^{a}=\frac{d}{d \lambda} X^{a}+\Gamma_{b c}^{a}(x(\lambda)) X^{b} T^{c}$$

  where $\Gamma_{b c}^{a}(x)$ is the metric connection for a metric $g_{a b}(x) . \nabla_{T} Y^{a}$ is defined similarly. Suppose $P$ is geodesic and $\lambda$ is an affine parameter. Explain why $\nabla_{T} T^{a}=0$. Show that if $\nabla_{T} X^{a}=\nabla_{T} Y^{a}=0$ then $g_{a b}(x(\lambda)) X^{a}(x(\lambda)) Y^{b}(x(\lambda))$ is constant along $P$.

  If $x^{a}(\lambda, \mu)$ is a family of geodesics which depend on $\mu$, let $S^{a}=\frac{\partial}{\partial \mu} x^{a}$ and define

  $$\nabla_{S} X^{a}=\frac{\partial}{\partial \mu} X^{a}+\Gamma_{b c}^{a}(x(\lambda)) X^{b} S^{c}$$

  Show that $\nabla_{T} S^{a}=\nabla_{S} T^{a}$ and obtain

  $$\nabla_{T}{ }^{2} S^{a} \equiv \nabla_{T}\left(\nabla_{T} S^{a}\right)=R_{b c d}^{a} T^{b} T^{c} S^{d}$$

  What is the physical relevance of this equation in general relativity? Describe briefly how this is relevant for an observer moving under gravity.

  [You may assume $\left[\nabla_{T}, \nabla_{S}\right] X^{a}=R_{b c d}^{a} X^{b} T^{c} S^{d}$.]

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• 4.II.36E

  General Relativity | Part II, 2008

  A solution of the Einstein equations is given by the metric

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

  For an incoming light ray, with constant $\theta, \phi$, show that

  $$t=v-r-2 M \log \left|\frac{r}{2 M}-1\right|,$$

  for some fixed $v$ and find a similar solution for an outgoing light ray. For the outgoing case, assuming $r>2 M$, show that in the far past $\frac{r}{2 M}-1 \propto \exp \left(\frac{t}{2 M}\right)$ and in the far future $r \sim t$.

  Obtain the transformed metric after the change of variables $(t, r, \theta, \phi) \rightarrow(v, r, \theta, \phi)$. With coordinates $\hat{t}=v-r, r$ sketch, for fixed $\theta, \phi$, the trajectories followed by light rays. What is the significance of the line $r=2 M$ ?

  Show that, whatever path an observer with initial $r=r_{0}<2 M$ takes, he must reach $r=0$ in a finite proper time.

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2007

• 1.II.35A

  General Relativity | Part II, 2007

  Starting from the Riemann tensor for a metric $g_{a b}$, define the Ricci tensor $R_{a b}$ and the scalar curvature $R$.

  The Riemann tensor obeys

  $$\nabla_{e} R_{a b c d}+\nabla_{c} R_{a b d e}+\nabla_{d} R_{a b e c}=0$$

  Deduce that

  $$\nabla^{a} R_{a b}=\frac{1}{2} \nabla_{b} R$$

  Write down Einstein's field equations in the presence of a matter source, with energymomentum tensor $T_{a b}$. How is the relation $(*)$ important for the consistency of Einstein's equations?

  Show that, for a scalar function $\phi$, one has

  $$\nabla^{2} \nabla_{a} \phi=\nabla_{a} \nabla^{2} \phi+R_{a b} \nabla^{b} \phi .$$

  Assume that

  $$R_{a b}=\nabla_{a} \nabla_{b} \phi$$

  for a scalar field $\phi$. Show that the quantity

  $$R+\nabla^{a} \phi \nabla_{a} \phi$$

  is then a constant.

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• 2.II.35A

  General Relativity | Part II, 2007

  The symbol $\nabla_{a}$ denotes the covariant derivative defined by the Christoffel connection $\Gamma_{b c}^{a}$ for a metric $g_{a b}$. Explain briefly why

  $$\begin{gathered} \left(\nabla_{a} \nabla_{b}-\nabla_{b} \nabla_{a}\right) \phi=0 \\ \left(\nabla_{a} \nabla_{b}-\nabla_{b} \nabla_{a}\right) v_{c} \neq 0 \end{gathered}$$

  in general, where $\phi$ is a scalar field and $v_{c}$ is a covariant vector field.

  A Killing vector field $v_{a}$ satisfies the equation

  $$S_{a b} \equiv \nabla_{a} v_{b}+\nabla_{b} v_{a}=0$$

  By considering the quantity $\nabla_{a} S_{b c}+\nabla_{b} S_{a c}-\nabla_{c} S_{a b}$, show that

  $$\nabla_{a} \nabla_{b} v_{c}=-R_{a b c}^{d} v_{d}$$

  Find all Killing vector fields $v_{a}$ in the case of flat Minkowski space-time.

  For a metric of the form

  $$\mathrm{d} s^{2}=-f(\mathbf{x}) \mathrm{d} t^{2}+g_{i j}(\mathbf{x}) \mathrm{d} x^{i} \mathrm{~d} x^{j}, \quad i, j=1,2,3$$

  where $\mathbf{x}$ denotes the coordinates $x^{i}$, show that $\Gamma_{00}^{0}=\Gamma_{i j}^{0}=0$ and that $\Gamma_{0 i}^{0}=\Gamma_{i 0}^{0}=$ $\frac{1}{2}\left(\partial_{i} f\right) / f$. Deduce that the vector field $v^{a}=(1,0,0,0)$ is a Killing vector field.

  [You may assume the standard symmetries of the Riemann tensor.]

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• 4.II.36A

  General Relativity | Part II, 2007

  Consider a particle on a trajectory $x^{a}(\lambda)$. Show that the geodesic equations, with affine parameter $\lambda$, coincide with the variational equations obtained by varying the integral

  $$I=\int_{\lambda_{0}}^{\lambda_{1}} g_{a b}(x) \frac{\mathrm{d} x^{a}}{\mathrm{~d} \lambda} \frac{\mathrm{d} x^{b}}{\mathrm{~d} \lambda} \mathrm{d} \lambda$$

  the end-points being fixed.

  In the case that $f(r)=1-2 G M u$, show that the space-time metric is given in the form

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

  for a certain function $f(r)$. Assuming the particle motion takes place in the plane $\theta=\frac{\pi}{2}$ show that

  $$\frac{\mathrm{d} \phi}{\mathrm{d} \lambda}=\frac{h}{r^{2}}, \quad \frac{\mathrm{d} t}{\mathrm{~d} \lambda}=\frac{E}{f(r)},$$

  for $h, E$ constants. Writing $u=1 / r$, obtain the equation

  $$\left(\frac{\mathrm{d} u}{\mathrm{~d} \phi}\right)^{2}+f(r) u^{2}=-\frac{k}{h^{2}} f(r)+\frac{E^{2}}{h^{2}}$$

  where $k$ can be chosen to be 1 or 0 , according to whether the particle is massive or massless. In the case that $f(r)=1-G M u$, show that

  $$\frac{\mathrm{d}^{2} u}{\mathrm{~d} \phi^{2}}+u=k \frac{G M}{h^{2}}+3 G M u^{2}$$

  In the massive case, show that there is an approximate solution of the form

  $$u=\frac{1}{\ell}(1+e \cos (\alpha \phi)),$$

  where

  $$1-\alpha=\frac{3 G M}{\ell} .$$

  What is the interpretation of this solution?

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2006

• 1.II.35A

  General Relativity | Part II, 2006

  Let $\phi(x)$ be a scalar field and $\nabla_{a}$ denote the Levi-Civita covariant derivative operator of a metric tensor $g_{a b}$. Show that

  $$\nabla_{a} \nabla_{b} \phi=\nabla_{b} \nabla_{a} \phi$$

  If the Ricci tensor, $R_{a b}$, of the metric $g_{a b}$ satisfies

  $$R_{a b}=\partial_{a} \phi \partial_{b} \phi,$$

  find the energy momentum tensor $T_{a b}$ and use the contracted Bianchi identity to show that, if $\partial_{a} \phi \neq 0$, then

  $$\nabla_{a} \nabla^{a} \phi=0$$

  Show further that $(*)$ implies

  $$\partial_{a}\left(\sqrt{-g} g^{a b} \partial_{b} \phi\right)=0$$

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• 2.II.35A

  General Relativity | Part II, 2006

  The Schwarzschild metric is

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

  Writing $u=1 / r$, obtain the equation

  $$\frac{d^{2} u}{d \phi^{2}}+u=3 M u^{2}$$

  determining the spatial orbit of a null (massless) particle moving in the equatorial plane $\theta=\pi / 2$.

  Verify that two solutions of $(*)$ are

  $$\begin{aligned} \text { (i) } u &=\frac{1}{3 M}, \quad \text { and } \\ \text { (ii) } \quad u &=\frac{1}{3 M}-\frac{1}{M} \frac{1}{\cosh \phi+1} . \end{aligned}$$

  What is the significance of solution (i)? Sketch solution (ii) and describe its relation to solution (i).

  Show that, near $\phi=\cosh ^{-1} 2$, one may approximate the solution (ii) by

  $$r \sin \left(\phi-\cosh ^{-1} 2\right) \approx \sqrt{27} M$$

  and hence obtain the impact parameter.

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• 4.II.36A

  General Relativity | Part II, 2006

  What are local inertial co-ordinates? What is their physical significance and how are they related to the equivalence principle?

  If $V_{a}$ are the components of a covariant vector field, show that

  $$\partial_{a} V_{b}-\partial_{b} V_{a}$$

  are the components of an anti-symmetric second rank covariant tensor field.

  If $K^{a}$ are the components of a contravariant vector field and $g_{a b}$ the components of a metric tensor, let

  $$Q_{a b}=K^{c} \partial_{c} g_{a b}+g_{a c} \partial_{b} K^{c}+g_{c b} \partial_{a} K^{c}$$

  Show that

  $$Q_{a b}=2 \nabla_{(a} K_{b)},$$

  where $K_{a}=g_{a b} K^{b}$, and $\nabla_{a}$ is the Levi-Civita covariant derivative operator of the metric $g_{a b}$.

  In a particular co-ordinate system $\left(x^{1}, x^{2}, x^{3}, x^{4}\right)$, it is given that $K^{a}=(0,0,0,1)$, $Q_{a b}=0$. Deduce that, in this co-ordinate system, the metric tensor $g_{a b}$ is independent of the co-ordinate $x^{4}$. Hence show that

  $$\nabla_{a} K_{b}=\frac{1}{2}\left(\partial_{a} K_{b}-\partial_{b} K_{a}\right)$$

  and that

  $$E=-K_{a} \frac{d x^{a}}{d \lambda}$$

  is constant along every geodesic $x^{a}(\lambda)$ in every co-ordinate system.

  What further conditions must one impose on $K^{a}$ and $d x^{a} / d \lambda$ to ensure that the metric is stationary and that $E$ is proportional to the energy of a particle moving along the geodesic?

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2005

• 1.II.35C

  General Relativity | Part II, 2005

  Suppose $(x(\tau), t(\tau))$ is a timelike geodesic of the metric

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

  where $\tau$ is proper time along the world line. Show that $d t / d \tau=E /\left(1+x^{2}\right)$, where $E>1$ is a constant whose physical significance should be stated. Setting $a^{2}=E^{2}-1$, show that

  $$d \tau=\frac{d x}{\sqrt{a^{2}-x^{2}}}, \quad d t=\frac{E d x}{\left(1+x^{2}\right) \sqrt{a^{2}-x^{2}}} .$$

  Deduce that $x$ is a periodic function of proper time $\tau$ with period $2 \pi$. Sketch $d x / d \tau$ as a function of $x$ and superpose on this a sketch of $d x / d t$ as a function of $x$. Given the identity

  $$\int_{-a}^{a} \frac{E d x}{\left(1+x^{2}\right) \sqrt{a^{2}-x^{2}}}=\pi$$

  deduce that $x$ is also a periodic function of $t$ with period $2 \pi$.

  Next consider the family of metrics

  $$d s^{2}=\frac{[1+f(x)]^{2} d x^{2}}{1+x^{2}}-\left(1+x^{2}\right) d t^{2},$$

  where $f$ is an odd function of $x, f(-x)=-f(x)$, and $|f(x)|<1$ for all $x$. Derive expressions analogous to $(*)$ above. Deduce that $x$ is a periodic function of $\tau$ and also that $x$ is a periodic function of $t$. What are the periods?

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• 2.II.35C

  General Relativity | Part II, 2005

  State without proof the properties of local inertial coordinates $x^{a}$ centred on an arbitrary spacetime event $p$. Explain their physical significance.

  Obtain an expression for $\partial_{a} \Gamma_{b}{ }^{c}$ at $p$ in inertial coordinates. Use it to derive the formula

  $$R_{a b c d}=\frac{1}{2}\left(\partial_{b} \partial_{c} g_{a d}+\partial_{a} \partial_{d} g_{b c}-\partial_{b} \partial_{d} g_{a c}-\partial_{a} \partial_{c} g_{b d}\right)$$

  for the components of the Riemann tensor at $p$ in local inertial coordinates. Hence deduce that at any point in any chart $R_{a b c d}=R_{c d a b}$.

  Consider the metric

  $$d s^{2}=\frac{\eta_{a b} d x^{a} d x^{b}}{\left(1+L^{-2} \eta_{a b} x^{a} x^{b}\right)^{2}}$$

  where $\eta_{a b}=\operatorname{diag}(1,1,1,-1)$ is the Minkowski metric tensor and $L$ is a constant. Compute the Ricci scalar $R=R_{a b}^{a b}$ at the origin $x^{a}=0$.

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• 4.II.36C

  General Relativity | Part II, 2005

  State clearly, but do not prove, Birkhoff's Theorem about spherically symmetric spacetimes. Let $(r, \theta, \phi)$ be standard spherical polar coordinates and define $F(r)=$ $1-2 M / r$, where $M$ is a constant. Consider the metric

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

  Explain carefully why this is appropriate for the region outside a spherically symmetric star that is collapsing to form a black hole.

  By considering radially infalling timelike geodesics $x^{a}=(r(\tau), 0,0, t(\tau))$, where $\tau$ is proper time along the curve, show that a freely falling observer will reach the event horizon after a finite proper time. Show also that a distant observer would see the horizon crossing only after an infinite time.

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2004

• A1.15 B1.24

  General Relativity | Part II, 2004

  (i) What is an affine parameter $\lambda$ of a timelike or null geodesic? Prove that for a timelike geodesic one may take $\lambda$ to be proper time $\tau$. The metric

  $$d s^{2}=-d t^{2}+a^{2}(t) d \mathbf{x}^{2}$$

  with $\dot{a}(t)>0$ represents an expanding universe. Calculate the Christoffel symbols.

  (ii) Obtain the law of spatial momentum conservation for a particle of rest mass $m$ in the form

  $$m a^{2} \frac{d \mathbf{x}}{d \tau}=\mathbf{p}=\text { constant }$$

  Assuming that the energy $E=m d t / d \tau$, derive an expression for $E$ in terms of $m, \mathbf{p}$ and $a(t)$ and show that the energy is not conserved but rather that it decreases with time. In particular, show that if the particle is moving extremely relativistically then the energy decreases as $a^{-1}(t)$, and if it is moving non-relativistically then the kinetic energy, $E-m$, decreases as $a^{-2}(t)$.

  Show that the frequency $\omega_{e}$ of a photon emitted at time $t_{e}$ will be observed at time $t_{o}$ to have frequency

  $$\omega_{o}=\omega_{e} \frac{a\left(t_{e}\right)}{a\left(t_{o}\right)}$$

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• A2.15 B2.24

  General Relativity | Part II, 2004

  (i) State and prove Birkhoff's theorem.

  (ii) Derive the Schwarzschild metric and discuss its relevance to the problem of gravitational collapse and the formation of black holes.

  [Hint: You may assume that the metric takes the form

  $$d s^{2}=-e^{\nu(r, t)} d t^{2}+e^{\lambda(r, t)} d r^{2}+r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)$$

  and that the non-vanishing components of the Einstein tensor are given by

  $$\begin{aligned} & G_{t t}=\frac{e^{2 \nu+\lambda}}{r^{2}}\left(-1+e^{\lambda}+r \lambda^{\prime}\right), \quad G_{r t}=e^{(\nu+\lambda) / 2} \frac{\dot{\lambda}}{r}, \quad G_{r r}=\frac{e^{\lambda}}{r^{2}}\left(1-e^{-\lambda}+r \nu^{\prime}\right), \\ & G_{\theta \theta}=\frac{1}{4} r^{2} e^{-\lambda}\left[2 \nu^{\prime \prime}+\left(\nu^{\prime}\right)^{2}+\frac{2}{r}\left(\nu^{\prime}-\lambda^{\prime}\right)-\nu^{\prime} \lambda^{\prime}\right]-\frac{1}{4} r^{2} e^{-\nu}\left[2 \ddot{\lambda}+(\dot{\lambda})^{2}-\dot{\lambda} \dot{\nu}\right] \\ =&\left.G_{r t} \text { and } G_{\phi \phi}=\sin ^{2} \theta G_{\theta \theta} .\right] \end{aligned}$$

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• A4.17 B4.25

  General Relativity | Part II, 2004

  Starting from the Ricci identity

  $$V_{a ; b ; c}-V_{a ; c ; b}=V_{e} R_{a b c}^{e}$$

  give an expression for the curvature tensor $R_{a b c}^{e}$ of the Levi-Civita connection in terms of the Christoffel symbols and their partial derivatives. Using local inertial coordinates, or otherwise, establish that

  $$R_{a b c}^{e}+R_{b c a}^{e}+R_{c a b}^{e}=0 .$$

  A vector field with components $V^{a}$ satisfies

  $$V_{a ; b}+V_{b ; a}=0$$

  Show, using equation $(*)$ that

  $$V_{a ; b ; c}=V_{e} R_{c b a}^{e}$$

  and hence that

  $$V_{a ; b} ; b+R_{a}{ }^{c} V_{c}=0,$$

  where $R_{a b}$ is the Ricci tensor. Show that equation $(* *)$ may be written as

  $$\left(\partial_{c} g_{a b}\right) V^{c}+g_{c b} \partial_{a} V^{c}+g_{a c} \partial_{b} V^{c}=0$$

  If the metric is taken to be the Schwarzschild metric

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

  show that $V^{a}=\delta^{a}{ }_{0}$ is a solution of $(* * *)$. Calculate $V^{a} ; a$.

  Electromagnetism can be described by a vector potential $A_{a}$ and a Maxwell field tensor $F_{a b}$ satisfying

  $$F_{a b}=A_{b ; a}-A_{a ; b} \quad \text { and } \quad F_{a b} ; b=0 .$$

  The divergence of $A_{a}$ is arbitrary and we may choose $A_{a} ; a=0$. With this choice show that in a general spacetime

  $$A_{a ; b}{ }^{; b}-R_{a}{ }^{c} A_{c}=0 .$$

  Hence show that in the Schwarzschild spacetime a tensor field whose only non-trivial components are $F_{t r}=-F_{r t}=Q / r^{2}$, where $Q$ is a constant, satisfies the field equations $(* * * *)$.

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2003

• A1.15 B1.24

  General Relativity | Part II, 2003

  (i) The worldline $x^{a}(\lambda)$ of a massive particle moving in a spacetime with metric $g_{a b}$ obeys the geodesic equation

  $$\frac{d^{2} x^{a}}{d \tau^{2}}+\Gamma^a_{bc} \frac{d x^{b}}{d \tau} \frac{d x^{c}}{d \tau}=0$$

  where $\tau$ is the particle's proper time and $\Gamma^a_{bc}$ are the Christoffel symbols; these are the equations of motion for the Lagrangian

  $$L_{1}=-m \sqrt{-g_{a b} \dot{x}^{a} \dot{x}^{b}}$$

  where $m$ is the particle's mass, and $\dot{x}^{a}=d x^{a} / d \lambda$. Why is the choice of worldline parameter $\lambda$ irrelevant? Among all possible worldlines passing through points $A$ and $B$, why is $x^{a}(\lambda)$ the one that extremizes the proper time elapsed between $A$ and $B$ ?

  Explain how the equations of motion for a massive particle may be obtained from the alternative Lagrangian

  $$L_{2}=\frac{1}{2} g_{a b} \dot{x}^{a} \dot{x}^{b} .$$

  What can you conclude from the fact that $L_{2}$ has no explicit dependence on $\lambda$ ? How are the equations of motion for a massless particle obtained from $L_{2}$ ?

  (ii) A photon moves in the Schwarzschild metric

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

  Given that the motion is confined to the plane $\theta=\pi / 2$, obtain the radial equation

  $$\left(\frac{d r}{d \lambda}\right)^{2}=E^{2}-\frac{h^{2}}{r^{2}}\left(1-\frac{2 M}{r}\right)$$

  where $E$ and $h$ are constants, the physical meaning of which should be stated.

  Setting $u=1 / r$, obtain the equation

  $$\frac{d^{2} u}{d \phi^{2}}+u=3 M u^{2}$$

  Using the approximate solution

  $$u=\frac{1}{b} \sin \phi+\frac{M}{2 b^{2}}(3+\cos 2 \phi)+\ldots$$

  obtain the standard formula for the deflection of light passing far from a body of mass $M$ with impact parameter $b$. Reinstate factors of $G$ and $c$ to give your result in physical units.

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• A2.15 B2.23

  General Relativity | Part II, 2003

  (i) What is a "stationary" metric? What distinguishes a stationary metric from a "static" metric?

  A Killing vector field $K^{a}$ of a metric $g_{a b}$ satisfies

  $$K_{a ; b}+K_{b ; a}=0$$

  Show that this is equivalent to

  $$g_{a b, c} K^{c}+g_{a c} K_{, b}^{c}+g_{c b} K_{, a}^{c}=0$$

  Hence show that a constant vector field $K^{a}$ with one non-zero component, $K^{4}$ say, is a Killing vector field if $g_{a b}$ is independent of $x^{4}$.

  (ii) Given that $K^{a}$ is a Killing vector field, show that $K_{a} u^{a}$ is constant along the geodesic worldline of a massive particle with 4-velocity $u^{a}$. Hence find the energy $\varepsilon$ of a particle of unit mass moving in a static spacetime with metric

  $$d s^{2}=h_{i j} d x^{i} d x^{j}-e^{2 U} d t^{2}$$

  where $h_{i j}$ and $U$ are functions only of the space coordinates $x^{i}$. By considering a particle with speed small compared with that of light, and given that $U \ll 1$, show that $h_{i j}=\delta_{i j}$ to lowest order in the Newtonian approximation, and that $U$ is the Newtonian potential.

  A metric admits an antisymmetric tensor $Y_{a b}$ satisfying

  $$Y_{a b ; c}+Y_{a c ; b}=0$$

  Given a geodesic $x^{a}(\lambda)$, let $s_{a}=Y_{a b} \dot{x}^{b}$. Show that $s_{a}$ is parallelly propagated along the geodesic, and that it is orthogonal to the tangent vector of the geodesic. Hence show that the scalar

  $$\phi=s^{a} s_{a}$$

  is constant along the geodesic.

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• A4.17 B4.25

  General Relativity | Part II, 2003

  What are "inertial coordinates" and what is their physical significance? [A proof of the existence of inertial coordinates is not required.] Let $O$ be the origin of inertial coordinates and let $\left.R_{a b c d}\right|_{O}$ be the curvature tensor at $O$ (with all indices lowered). Show that $\left.R_{a b c d}\right|_{O}$ can be expressed entirely in terms of second partial derivatives of the metric $g_{a b}$, evaluated at $O$. Use this expression to deduce that (a) $R_{a b c d}=-R_{b a c d}$ (b) $R_{a b c d}=R_{c d a b}$ (c) $R_{a[b c d]}=0$.

  Starting from the expression for $R_{b c d}^{a}$ in terms of the Christoffel symbols, show (again by using inertial coordinates) that

  $$R_{a b[c d ; e]}=0$$

  Obtain the contracted Bianchi identities and explain why the Einstein equations take the form

  $$R_{a b}-\frac{1}{2} R g_{a b}=8 \pi T_{a b}-\Lambda g_{a b},$$

  where $T_{a b}$ is the energy-momentum tensor of the matter and $\Lambda$ is an arbitrary constant.

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2002

• A1.15 B1.24

  General Relativity | Part II, 2002

  (i) Given a covariant vector field $V_{a}$, define the curvature tensor $R_{b c d}^{a}$ by

  $$V_{a ; b c}-V_{a ; c b}=V_{e} R_{a b c}^{e}$$

  Express $R_{a b c}^{e}$ in terms of the Christoffel symbols and their derivatives. Show that

  $$R_{a b c}^{e}=-R_{a c b}^{e}$$

  Further, by setting $V_{a}=\partial \phi / \partial x^{a}$, deduce that

  $$R_{a b c}^{e}+R_{c a b}^{e}+R_{b c a}^{e}=0 .$$

  (ii) Write down an expression similar to (*) given in Part (i) for the quantity

  $$g_{a b ; c d}-g_{a b ; d c}$$

  and hence show that

  $$R_{e a b c}=-R_{a e b c} .$$

  Define the Ricci tensor, show that it is symmetric and write down the contracted Bianchi identities.

  In certain spacetimes of dimension $n \geq 2, R_{a b c d}$ takes the form

  $$R_{a b c d}=K\left(x^{e}\right)\left[g_{a c} g_{b d}-g_{a d} g_{b c}\right]$$

  Obtain the Ricci tensor and Ricci scalar. Deduce that $K$ is a constant in such spacetimes if the dimension $n$ is greater than 2 .

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• A2.15 B2.23

  General Relativity | Part II, 2002

  (i) Consider the line element describing the interior of a star,

  $$d s^{2}=e^{2 \alpha(r)} d r^{2}+r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)-e^{2 \gamma(r)} d t^{2},$$

  defined for $0 \leq r \leq r_{0}$ by

  $$e^{-2 \alpha(r)}=1-A r^{2}$$

  and

  $$e^{\gamma(r)}=\frac{3}{2} e^{-\alpha_{0}}-\frac{1}{2} e^{-\alpha(r)}$$

  Here $A=2 M / r_{0}^{3}, M$ is the mass of the star, and $\alpha_{0}$ is defined to be $\alpha\left(r_{0}\right)$.

  The star is made of a perfect fluid with energy-momentum tensor

  $$T_{a b}=(p+\rho) u_{a} u_{b}+p g_{a b} .$$

  Here $u^{a}$ is the 4 -velocity of the fluid which is at rest, the density $\rho$ is constant throughout the star $\left(0 \leq r \leq r_{0}\right)$ and the pressure $p=p(r)$ depends only on the radial coordinate. Write down the Einstein field equations and show that (in geometrical units with $G=c=1$ ) they may equivalently be written as

  $$R_{a b}=8 \pi(p+\rho) u_{a} u_{b}+4 \pi(p-\rho) g_{a b} .$$

  (ii) Using the formulae below, or otherwise, show that for $0 \leq r \leq r_{0}$ one has

  $$\rho=\frac{3 A}{8 \pi}, \quad p(r)=\frac{3 A}{8 \pi}\left(\frac{e^{-\alpha(r)}-e^{-\alpha_{0}}}{3 e^{-\alpha_{0}}-e^{-\alpha(r)}}\right)$$

  [The non-zero components of the Ricci tensor are:

  $$\begin{gathered} R_{11}=-\gamma^{\prime \prime}+\alpha^{\prime} \gamma^{\prime}-\gamma^{\prime 2}+\frac{2 \alpha^{\prime}}{r}, \quad R_{22}=e^{-2 \alpha}\left[\left(\alpha^{\prime}-\gamma^{\prime}\right) r-1\right]+1 \\ R_{33}=\sin ^{2} \theta R_{22}, \quad R_{44}=e^{2 \gamma-2 \alpha}\left[\gamma^{\prime \prime}-\alpha^{\prime} \gamma^{\prime}+\gamma^{\prime 2}+\frac{2 \gamma^{\prime}}{r}\right] \end{gathered}$$

  Note that

  $$\alpha^{\prime}=A r e^{2 \alpha}, \quad \gamma^{\prime}=\frac{1}{2} A r e^{\alpha-\gamma}, \quad \gamma^{\prime \prime}=\frac{1}{2} A e^{\alpha-\gamma}+\frac{1}{2} A^{2} r^{2} e^{3 \alpha-\gamma}-\frac{1}{4} A^{2} r^{2} e^{2 \alpha-2 \gamma} \text { ] ] }$$

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• A4.17 B4.25

  General Relativity | Part II, 2002

  With respect to the Schwarzschild coordinates $(r, \theta, \phi, t)$, the Schwarzschild geometry is given by

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

  where $r_{s}=2 M$ is the Schwarzschild radius and $M$ is the Schwarzschild mass. Show that, by a suitable choice of $(\theta, \phi)$, the general geodesic can regarded as moving in the equatorial plane $\theta=\pi / 2$. Obtain the equations governing timelike and null geodesics in terms of $u(\phi)$, where $u=1 / r$.

  Discuss light bending and perihelion precession in the solar system.

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2001

• A1.15 B1.24

  General Relativity | Part II, 2001

  (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

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• A2.15 B2.23

  General Relativity | Part II, 2001

  (i) Show that the geodesic equation follows from a variational principle with Lagrangian

  $$L=g_{a b} \dot{x}^{a} \dot{x}^{b}$$

  where the path of the particle is $x^{a}(\lambda)$, and $\lambda$ is an affine parameter along that path.

  (ii) The Schwarzschild metric is given by

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

  Consider a photon which moves within the equatorial plane $\theta=\frac{\pi}{2}$. Using the above Lagrangian, or otherwise, show that

  $$\left(1-\frac{2 M}{r}\right)\left(\frac{d t}{d \lambda}\right)=E, \text { and } r^{2}\left(\frac{d \phi}{d \lambda}\right)=h$$

  for constants $E$ and $h$. Deduce that

  $$\left(\frac{d r}{d \lambda}\right)^{2}=E^{2}-\frac{h^{2}}{r^{2}}\left(1-\frac{2 M}{r}\right)$$

  Assume further that the photon approaches from infinity. Show that the impact parameter $b$ is given by

  $$b=\frac{h}{E}$$

  By considering the equation $(*)$, or otherwise

  (a) show that, if $b^{2}>27 M^{2}$, the photon is deflected but not captured by the black hole;

  (b) show that, if $b^{2}<27 M^{2}$, the photon is captured;

  (c) describe, with justification, the qualitative form of the photon's orbit in the case $b^{2}=27 M^{2}$.

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• A4.17 B4.25

  General Relativity | Part II, 2001

  Discuss how Einstein's theory of gravitation reduces to Newton's in the limit of weak fields. Your answer should include discussion of: (a) the field equations; (b) the motion of a point particle; (c) the motion of a pressureless fluid.

  [The metric in a weak gravitational field, with Newtonian potential $\phi$, may be taken as

  $$d s^{2}=d x^{2}+d y^{2}+d z^{2}-(1+2 \phi) d t^{2} .$$

  The Riemann tensor is

  $$\left.R_{b c d}^{a}=\Gamma_{b d, c}^{a}-\Gamma_{b c, d}^{a}+\Gamma_{c f}^{a} \Gamma_{b d}^{f}-\Gamma_{d f}^{a} \Gamma_{b c}^{f}\right]$$

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