I think the confusion with the Length Contraction item in the table (in section 2.1.1)
lies in a subtle switch of situation between his
"introduction of Length Contraction in section 1.3.3"
and his
"derivation of the Lorentz Transformation in section 2.1.1".
(I think spacetime diagrams and additional notation would have helped distinguish the situations. Symbolic equations are occasionally not enough.)
When Morin introduces Length Contraction in section 1.3.3,
person-B (GROUND) is measuring the apparent-length of train-A (MOVING-TRAIN) that has rest length $L_A$.
His analysis determines that person-B measures
$$L_{B}=\frac{L_{A}}{\gamma}.\qquad(1.19)$$
At the expense of extra notation, this might be better expressed (using "wrt" = "with respect to") as
$$L_{A,wrtB}=\frac{L_{A,wrtA}}{\gamma}.$$
Then, Morin summarizes the result as
$$L_{observed}=\frac{L_{proper}}{\gamma}.\qquad(1.20)$$
On a spacetime diagram drawn by person-B (GROUND),
we draw the blue parallel timelines (worldlines) of the front and back of train-A. Note
- OQ (the proper length of train-A) and
- OM (person-B's measured-length of train-A).
Note that MQ (along the timeline of the front of train-A) is orthogonal to OQ
(along the spaceline $t'=0$). So this is a Minkowski-right triangle $OQM$ with right-angle at
$Q$, where $\theta$ is the equal to the rapidity
($v_{B,wrt A}=\tanh\theta$ and $\gamma=\cosh\theta$).
In this triangle, $OM$ is the hypotenuse (since it is opposite the right-angle)
and $OQ$ is the adjacent-side.
So, $\displaystyle\gamma=\cosh\theta=\frac{ADJ}{HYP}=\frac{OQ}{OM}=\frac{L_{A,wrtA}}{L_{B,wrtA}}$ which can be written as
\begin{align}
(HYP)&=\frac{(ADJ)}{\cosh\theta}\\
OM&=\frac{OQ}{\cosh\theta}=\frac{OQ}{\gamma}\qquad(1.19, 1.20)
\end{align}
Now, when deriving the Lorentz transformation in 2.1.1,
Consider a reference frame $S'$ moving relative to another frame $S$, as shown in Fig. 2.1.
Let the constant relative speed of the frames be $v$
....
Our goal is to look at two events and relate the $\Delta x$ and $\Delta t$ in $S$
to the $\Delta x'$ and $\Delta t'$ in $S'$.
In this section, Morin sets up the form of the Lorentz transformation with
$\Delta x =A \Delta x' +B\Delta t'$.
Morin wants to use "Length Contraction" to obtain the coefficient-$A=\gamma$
by setting up $\Delta x= \gamma \Delta x'$.
From the table, choose two events with $\Delta t'=0$ (so we don't need to worry about the so-far unknown coefficient-$B$):
events $O$ and $Q$, which are simultaneous according to person-A (MOVING-TRAIN)
$$\Delta t'_{OQ}=(t'_Q-t'_O)=0.$$
Now, here is the important part.
For this derivation, Morin wants (according to his goal)
$$\Delta x_{OQ}=(x_Q-x_O)\qquad\mbox{ in terms of $\Delta x'_{OQ}=(x'_Q-x'_O)$ } $$
but this does not directly involve $\Delta x_{OM}=(x_M-x_O)$ from 1.3.3!
Instead, it involves a different pair ($O$ and $N$) of simultaneous events
according to person-B (GROUND), where
$$ \Delta x_{OQ}=\Delta x_{ON}.$$
That is to say, use the event $N$ so that $x_Q = x_N$.
(What is special about event $M$ is that
$x’_Q=x’_M$, which doesn’t help us with $x_Q$ like $N$ does.)
On a spacetime diagram drawn by person-B (GROUND),
consider a differently-sized train at rest according to person-B (GROUND, WITH REST-TRAIN)
that person-A (MOVING-TRAIN) measures using events $O$ and $Q$.
Draw red parallels to B's timeline through event-O and through event-Q.
Note:
- ON (the proper-length of this differently-sized train-B),
where $N$
is the event on the front parallel that person-B says simultaneous with $O$.
- OQ (person-A's measured-length of this differently-sized train-B).
Note that $ONQ$ is a Minkowski-right triangle, with right-angle at $N$ (so $OQ$ is the hypotenuse)
and the same $\theta$ as before.
Thus (following the ideas of above),
\begin{align}
(HYP)&=\frac{(ADJ)}{\cosh\theta}\\
OQ&=\frac{ON}{\cosh\theta}=\frac{ON}{\gamma}
\end{align}
Compare the role of $OQ$ here and in (1.19) above.
Expressing these results in the $\Delta x$-notations (needed for 2.1.1)
from 1.3.3,
\begin{align}
(HYP)&=\frac{(ADJ)}{\cosh\theta}\\
OM&=\frac{OQ}{\cosh\theta}=\frac{OQ}{\gamma} \qquad(1.19,1.20)\\
\Delta x_{OM}&=\frac{\Delta x'_{OQ}}{\cosh\theta}=\frac{\Delta x'_{OQ}}{\gamma} \qquad(1.19,1.20 \mbox{ where $OQ$ is the proper-length of train-A})
\end{align}
from 2.1.1,
\begin{align}
(HYP)&=\frac{(ADJ)}{\cosh\theta}\\
OQ&=\frac{ON}{\cosh\theta}=\frac{ON}{\gamma}\\
\Delta x'_{OQ}&=\frac{\Delta x_{ON}}{\cosh\theta}=\frac{\Delta x_{ON}}{\gamma}=\frac{\Delta x_{OQ}}{\gamma}
\qquad(1.20 \mbox{ where $OQ$ is the observed-length of train-B})
\end{align}
so that $\Delta x_{OQ} = \gamma \Delta x'_{OQ}$,
implying that coefficient-$A$ is equal to $\gamma$.
Again,
I think spacetime diagrams and additional notation would help distinguish the situations.
