## Differential Forms and the Geometry of General Relativity — Errata

Last update: 5/28/21

• Page 17, Equation (2.33): The implicitly defined derivative of $\phat$ is correct; there is no $\rhat$ term when working intrinsically on the sphere. However, the derivation of this result requires machinery we don't yet have; see Section 17.6 and Problem 2 in Section 17.11. Alternatively, the same result can be obtained extrinsically, working in 3-dimensional Euclidean space. In this case, $d\phat$ does contain an $\rhat$ term, which however drops out of Equation (2.35), leading to the same conclusion.
• Page 30, line 8: "with respect to $r$" should be "with respect to $\tau$".
• Page 87: The treatment of number density is inconsistent. The easiest fix is to regard $\ell$ as the distance between particles (as stated), in which case the number density should not contain $N$. Thus, Equation (8.2) should read \begin{equation} n = \frac{1}{\ell} \end{equation}
• Page 105, Equation (9.11): The second basis 1-form should be $\sigma^\phi$. The equation should therefore read: \begin{equation} \Omega^t{}_\phi = \frac{\ddot{a}}{a}\sigma^t\wedge \sigma^\phi \end{equation}
• Page 111, Equations (9.42) and (9.43): The units should be $\frac{\mathrm{g}}{\mathrm{cm}^3}$ in both cases.
• Page 144, Equation (13.6): The second pair of $u$s in the third line should be replaced by $v$s. That line of the equation should read \begin{equation} = \left( \Partial{f}{u} \Partial{u}{x} + \Partial{f}{v} \Partial{v}{x} + ... \right) \,dx + ... \end{equation}
• Page 179, Equation (15.31): The single instance of $dx^J\wedge dx^J$ should be $dx^I\wedge dx^J$. That line of the equation should read \begin{equation} = (h\,df+f\,dh) \wedge dx^I \wedge dx^J \end{equation}
• Page 185, Equation (15.87): $h_u$, $h_v$, $h_w$ should all be $f$. The full equation should read \begin{equation} df = \Partial{f}{u}\,du + \Partial{f}{v}\,dv + \Partial{f}{w}\,dw = \grad f\cdot d\rr \end{equation}
• Page 203, Figure 17.1: The caption should read: "The polar basis vectors $\rhat$ and $\phat$ at three nearby points."
• Page 204, Equations (17.4) and (17.5): There should be no factors of $r$ on the right-hand side. The full equations should read \begin{align} \rhat &= \cos\phi\,\xhat + \sin\phi\,\yhat \\ \phat &= -\sin\phi\,\xhat + \cos\phi\,\yhat \end{align}
• Page 210, line 5: $\sigma_m$ should be $\sigma_p$.
• Page 223: $4\times4$ should really be $N\times N$ (7 occurrences), where $N$ is the dimension of the space.
• Page 235, Equation (19.4): $\hat{e}^i$ should be $\hat{e}_i$.
• Page 242, Equation (19.59): The numerator should be $\ell\cot\theta$, rather than $\ell$. The full equation should read \begin{equation} \phi = \mp\arcsin\left( \frac{\ell\cot\theta}{\sqrt{r^2-\ell^2}} \right) + \hbox{constant} \end{equation}
• Page 242, just after Equation (19.61): $y=\sin\theta\sin\phi$ should be $y=r\sin\theta\sin\phi$.
• Page 250, last paragraph: "simply connected" should be replaced by "contractible".
(The same change should be made in the parenthetical remark that follows, whose first sentence should be deleted.)
• Page 258, Equation (A.29): $\Omega^i{}_j$ should be $\Omega^k{}_i$.
• Page 260, Equations (A.52) and (A.53): The middle expressions are missing a factor of $\frac12$. The full equations should read \begin{align} \Omega^t{}_\theta &= d\omega^t{}_\theta + \omega^t{}_m\wedge\omega^m{}_\theta = - \frac{f'}{2}\sqrt{f}\,dt\wedge d\theta = -\frac{f'}{2r} \sigma^t\wedge\sigma^\theta \\ \Omega^t{}_\phi &= d\omega^t{}_\phi + \omega^t{}_m\wedge\omega^m{}_\phi = - \frac{f'}{2}\sqrt{f}\,\sin\theta\,dt\wedge d\phi = -\frac{f'}{2r} \sigma^t\wedge\sigma^\phi \\ \end{align}
• Page 261, Equation (A.61): The initial minus sign should be deleted. The full equation should read \begin{equation} R^\phi{}_{\theta\phi\theta} = \frac{1-f}{r^2} \end{equation}
• Page 264, just after Equation (A.97): $s=-1$ should be $s=1$.
• Pages 269–271, §A.9: $\kappa$ should be $k$ throughout this section (17 occurances).
• Page 272, Equation (A.155): The final minus sign should be a plus sign. The full equation should read: \begin{align} ds^2 &= -A^2\,dt^2 + B^2 dr^2 + 2 C\,dt\,dr \nonumber\\ &= - \left(A\,dt - \frac{C}{A}\,dr\right)^2 + \left(B^2 + \frac{C^2}{A^2}\right)\,dr^2 \end{align} (The parenthetical comment after (A.157) is no longer necessary.)
• Page 275, Equation (A.182): Each factor of $q/r^2$ should be squared. The full equation should read: \begin{equation} 4\pi \left(T^i{}_j\right) = \frac12 \begin{pmatrix} -q^2/r^4 & 0 & 0 & 0 \\ 0 & -q^2/r^4 & 0 & 0 \\ 0 & 0 & q^2/r^4 & 0 \\ 0 & 0 & 0 & q^2/r^4 \end{pmatrix} \end{equation}