With the value for \( \gamma \) found in Example 1-2, Equation 1-13 can be written in a somewhat simpler form and with it the complete Lorentz transformation becomes
\[
\begin{array}{ll}
x^{\prime}=\gamma(x-v t) & y^{\prime}=y \\
t^{\prime}=\gamma\left(t-\frac{v x}{c^{2}}\right) & z^{\prime}=z
\end{array}
\]
1-18
and the inverse
\[
\begin{array}{ll}
x=\gamma\left(x^{\prime}+v t^{\prime}\right) & y=y^{\prime} \\
t=\gamma\left(t^{\prime}+\frac{v x^{\prime}}{c^{2}}\right) & z=z^{\prime}
\end{array}
\]
with
\[
\gamma=\frac{1}{\sqrt{1-\beta^{2}}}
\]