A First Course in General Relativity is in my opinion the best book for those who are learning GR for the first time. It does not pretend to be mathematically rigorous by definining manifolds, a common attempt that invariably distracts the student away from the underlying physics, and is often utterly wrong when the author is hardly a mathematician him- or herself. The book also covers everything essential for the student to understand what GR is really about, and offers a lot of interesting problems that are hard yet solvable. The only prerequisite for this book is calculus and Newtonian mechanics, although the student should be very good at mathematical reasoning.

Despite all these advantages, the book is not without errors, some of which are rather serious and themselves very good reminders that one must be very careful with the implicit assumptions valid only in Euclidean geometry. Certain parts deserve more explanation than it is easy to show that … as one could in fact easily “prove” the right result in the wrong way. Also, the incompleteness of the official solution manual makes the book less approachable for students without easy access to a professor.

For these reasons, I have decided to publish my own versions of Errata, Notes, and Solutions for this wonderful book. I hope the reader will find them helpful.

Errata

Notes

Solutions

Physics

Chapter 1

Convert the following to units in which \( c=1 \), expressing everything in terms of m and kg. Too trivial.  Convert the following from natural units (\( c = 1 \)) to SI units. Too trivial. Draw the \( t \) and \( x \) axes of the spacetime coordinates of an observer \( \mathcal{O} \) and then draw... Too trivial. Write out all the terms of the following sums, substituting the coordinate names \( (t, x, y, z) \) for \( (x^0, x^1, x^2, x^3) \).
Physics

Chapter 2

Given the numbers \( \{A^0 = 5, A^1 = 0, A^2 = -1, A^3 = -6\} \), \( \{B_0 = 0, B_1 = -2, B_2 = 4, B_3 = 0\} \), \( \{C_{00} = 1, C_{01} = 0, C_{02} = 2, C_{03} = 3, C_{30} = -1, C_{10} = 5, C_{11} = -2, C_{12} = -2, C_{13} = 0, C_{21} = 5, C_{22} = 2, C_{23} = -2, C_{20} = 4, C_{31} = -1, C_{32} = -3, C_{33} = 0\} \),
Physics

Chapter 3

Given an arbitrary set of numbers \( \{ M_{\alpha \beta}; \alpha = 0, \ldots, 3; \beta = 0, \ldots, 3 \} \) and two arbitrary sets of vector components \( \{ A^\mu, \mu = 0, \ldots, 3 \} \) and \( \{ B^\nu, \nu = 0, \ldots, 3 \} \), show that the two expressions \[ M_{\alpha \beta} A^\alpha B^\beta := \sum_{\alpha = 0}^{3} \sum_{\beta = 0}^{3} M_{\alpha \beta} A^\alpha B^\beta \]
Physics

Chapter 4

Comment on whether the continuum approximation is likely to apply to the following physical systems: (a) planetary motions in the solar system; (b) lava flow from a volcano; (c) traffic on a major road at rush hour; (d) traffic at an intersection controlled by stop signs for each incoming road; (e) plasma dynamics. (a) No; (b) yes; (c) yes; (d) no; (e) no (See the Wikipedia page about plasma).
Physics

Chapter 5

Repeat the argument that led to Eq. (5.1) under more realistic assumptions: suppose a fraction ε of the kinetic energy of the mass at the bottom can be converted into a photon and sent back up, the remaining energy staying at ground level in a useful form. Devise a perpetual motion engine if Eq. (5.1) is violated. Trivial. Explain why a uniform external gravitational field would raise no tides on Earth.
Physics

Chapter 6

Decide if the following sets are manifolds and say why. If there are exceptional points at which the sets are not manifolds, give them: phase space of Hamiltonian mechanics, the space of the canonical coordinates and momenta \( p_i \) and \( q^i \); Yes. the interior of a circle of unit radius in two-dimensional Euclidean space; Yes. the set of permutations of \( n \) objects;
Physics

Chapter 7

If Eq. (7.3) were the correct generalization of Eq. (7.1) to a curved spacetime, how would you interpret it? What would happen to the number of particles in a comoving volume of the fluid, as time evolves? In principle, can we distinguish experimentally between Eqs. (7.2) and (7.3)? The discussion under Eq. (7.3) is clear enough. It should be very easy, in principle, to distinguish experimentally between Eqs. (7.2) and (7.
Physics

Chapter 8

Show that Eq. (8.2) is a solution of Eq. (8.1) by the following method. Assume the point particle to be at the origin, \( r = 0 \), and to produce a spherically symmetric field. Then use Gauss’ law on a sphere of radius \( r \) to conclude \[ \frac{\mathrm{d} \phi}{\mathrm{d} r} = G m / r^2. \] Deduce Eq. (8.2) from this. (Consider the behavior at infinity.)
Physics

Errata

Page 2: It should be \( a' = \mathrm{d} v'/ \mathrm{d} t \) instead of \( a' - \mathrm{d} v'/\mathrm{d} t \). Page 20: The first (i.e. from \( \mathcal{A} \) to \( \mathcal{B} \)) should be (i.e. from \( \mathcal{A} \) to \( \mathcal{E} \)). Page 30: In Exer. 12b, it should be \( (\Delta s^2)_{\mathcal{AE}} = (1 - v^2) (\Delta s^2)_{\mathcal{AB}} \) instead of \( (\Delta s^2)_{\mathcal{AC}} = (1 - v^2) (\Delta s^2)_{\mathcal{AB}} \).
Physics

Notes

Page 160: To derive Eq. (6.68) from Eq. (6.67), simply notice that \( g_{\alpha \lambda} g^{\lambda \sigma} = {\delta^\sigma}_\alpha \). Page 160: In Eq. (6.72), it is important to understand that \( {V^\mu}_{; \beta} \) in \( \nabla_\alpha ({V^\mu}_{; \beta}) \) is not a number. The whole \( \nabla_\alpha ({V^\mu}_{; \beta}) \) denotes \( {(\boldsymbol{\nabla}(\boldsymbol{\nabla} \vec{V}))_{\alpha \beta}}^\mu \). Page 176: For Eq. (7.14), an underlying assumption is that \( \phi_{, 0} = O(\phi) \), i.