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
3dimensional 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 1form 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 righthand 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{1f}{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}
Tevian Dray