# Galilean space

The space-time of classical mechanics according to Galilei–Newton, in which the distance between two events taking place at the points $M_{1}$ and $M_{2}$ at different moments of time $t_{1}$ and $t_{2}$ respectively is taken to be the time interval $|t_{1} - t_{2}|$, while if these events take place at the same time, it is then considered to be the Euclidean distance between the points $M_{1}$ and $M_{2}$. For an $n$-dimensional Galilean space, the distance is defined as follows: $$d(\mathbf{x},\mathbf{y}) \stackrel{\text{df}}{=} \begin{cases} |x^{1} - y^{1}| & \text{if  x^{1} \neq y^{1} }; \\\\ \displaystyle \sqrt{\sum_{i = 2}^{n} (x^{i} - y^{i})^{2}} & \text{if  x^{1} = y^{1} }. \end{cases}$$
A Galilean space is a semi-pseudo-Euclidean space of nullity $1$; it may be considered as the limit case of a pseudo-Euclidean space in which the isotropic cone degenerates to a plane. This limit transition corresponds to the limit transition from the special theory of relativity to classical mechanics.