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.

References

[1]	B.A. Rozenfel’d, “Non-Euclidean spaces”, Moscow (1969). (In Russian)
[2]	R. Penrose, “Structure of space-time”, C.M. DeWitt (ed.), J.A. Wheeler (ed.), Batelle Rencontres 1967 Lectures in Math. Physics, Benjamin (1968), pp. 121–235 (Chapt. VII).