Update study notes

parts/special-relativity.tex

@@ -531,18 +531,15 @@ \subsubsection{\hfil \S3. Lorentz Transformation - Quantifying Non-Simultaneity
 Eq.~\ref{eq:transformation},~\ref{eq:x-transformation},~\ref{eq:y-transformation},~\ref{eq:z-transformation}:
 
 \begin{align}
-\tau &= \varphi(v)\left[ t - \frac{v}{V^2 - v^2} (x - vt) \right] \\
-     &= \varphi(v)\left( \frac{V^2t - v^2t - vx + v^2t}{V^2 - v^2} \right) \\
-     &= \varphi(v)\left( \frac{V^2t - vx}{V^2 - v^2} \right) \\
-     &= \varphi(v)\frac{V^2}{V^2 - v^2}\left( t - \frac{v}{V^2}x \right)
+&\tau &= \varphi(v)\left[ t - \frac{v}{V^2 - v^2} (x - vt) \right] \\
+      &= \varphi(v)\left( \frac{V^2t - v^2t - vx + v^2t}{V^2 - v^2} \right) \\
+      &= \varphi(v)\left( \frac{V^2t - vx}{V^2 - v^2} \right) \\
+      &= \varphi(v)\frac{V^2}{V^2 - v^2}\left( t - \frac{v}{V^2}x \right) \\
+\xi   &= \varphi(v)\left( \frac{V^2}{V^2 - v^2} \right) (x - vt) \\
+\eta  &= \varphi(v)\frac{V}{\sqrt{V^2 - v^2}}y \\
+\zeta &= \varphi(v)\frac{V}{\sqrt{V^2 - v^2}}z \\
 \end{align}
 
-\begin{equation}
-    \xi = \varphi(v)\left( \frac{V^2}{V^2 - v^2} \right) (x - vt)
-\end{equation}
-
-
-
 It does not lose generality to say if $k$ is moving relative to $K$ at speed $v$ then $K$ is moving relative to $k$ at
 speed $-v$. Directly measuring coordinates in $K$ is effectively the same thing as transforming $(x, y, z, t)$ in $K$
 to $(\xi, \eta, \zeta, \tau)$ and then back to $(x, y, z, t)$ so we have along the x-axis: