Groups of spacetime transformations and symmetries of four-dimensional spacetime. I

The relativistic 4-interval $(X-X_0)^2=s_0^2$ is interpreted as the 4-hyperboloid of the radius $s_0$ with the center at $X_0^{\mu }$, composed by the particles isotropically radiated from its center with rapidities $0<\beta \le 1$ and whose position in the 4d space–time is fixed at the same moment of the proper time $s_0/c$. Thus, the 4-hyperboloid can be considered as the model of an isotropically radiating source and all the transformations of space–time variables that leave its equation invariant have the physical sence and determine the symmetry properties of the 4d space–time. They compose the group of motions of the rotating 4-hyperboloid. Under the constant radius $s_0=\operatorname {const}$ the configuration space is the 8-dimensional bundle $\mathcal R(1.3)=\mathcal R(1.3)\otimes \Phi (1.3)$ with the minimal group of motions: $\mathcal K=\mathcal P\otimes O(1.3)$. It is shown that the known groups $\mathcal P$ and $O(1.3)$ defined only on the base $\mathcal R(1.3)$ and on the fiber $\Phi (1.3)$ of the space $\mathcal R(1.3)$ respectively and the corresponding symmetry properties of the 4d space–time are not complete. The group $\mathcal K$ extends the isotropy properties of the 4d space–time to moving frameworks. The space–time transformation group is constructed for the case of $N$ bundles. The new interpretation of the 4-interval has to regard the radius $s_0$ as the variable. The groups of motions of 4-hyperboloid with the varying radius are constructed in the second part of the work. They introduce new symmetry properties of the 4d space–time.