 Solution

The one-dimensional Lorentz transformation givesHence

As $|\beta |\to 0$, $\gamma =1+O({\beta }^2)$. The spatial transformation is therefore $x^{\prime }=x-ut+O({\beta }^2)$, whileUnder the stated condition $|x|<c|t|$, the fractional correction $|ux|/(c^2|t|)$ is smaller than $|\beta |$ and tends to zero. Thus the Lorentz transformation has the nonrelativistic limit given by the Galilean transformation.