Muon travel distance vs Atmosphere Thickness?

[DrGreg]

The Lorentz transformation is$$\begin{align}
x' &= \gamma (x - vt) \\
t' &= \gamma ( t - vx/c^2),
\end{align}$$ and the inverse transformation is$$\begin{align}
x &= \gamma (x' + vt') \\
t &= \gamma ( t' + vx'/c^2).
\end{align}$$So, from (1), $x' = \gamma x$ is true only when $t = 0$, and, from (3), $x' = x / \gamma$ is true only when $t' = 0$.

So you need to look through the argument to see whether there is an implicit or explicit assumption that either $t = 0$ or $t' = 0$.

 

[ersmith]

I have no doubt that we can solve for $L' = \frac{L}{\gamma}$. The question is: what is the mistake in his manipulation of the same equation such that he obtains $L' = \gamma L$ ?

What it amounts to is that he identified the events $(0, L)$ in the Earth frame with $(0, L')$ in the muon frame. These are not the same spacetime events. This is obscured by his poor choice of coordinate systems and muddled notation.