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

Abstract

The relativistic 4-interval (X-X ₍₀₎ ²=s ⁽⁰⁾₂ is interpreted as a 4-hyperboloid of radiuss ₍₀₎ and center at the pointX ⁽⁰⁾_μ that is formed by particles radiated isotropically from its center with velocities 0<β≤1 whose positions in 4d spacetime are fixed at a proper times _(0)/c that is the same for all of them. Therefore, the 4-hyperboloid can be regarded as a mathematical model of an isotropically radiating source, and all transformations of the spacetime variables that leave its equation invariant have a physical meaning and determine the symmetry properties of 4d spacetime. These transformations form the group of motions of a rotating 4-hyperboloid. For constant radiuss ₍₀₎=const, its configuration space is the 8-dimensional bundleR(1,3)=R(1,3) ⊗Φ(1,3), and the minimal group of motions isK=P ⊗O(1,3). It is shown that the well-known groupsP andO(1,3) are defined, respectively, only on the baseR(1,3) and only on the fiber Φ(1,3) of the spaceR(1,3) and that the symmetry properties of 4d spacetime introduced by them are incomplete. The groupK extends the isotropy property of 4d spacetime to moving frames of reference. The group of spacetime transformations is extended to the case ofN bundles. It is shown that the new interpretation of the 4-interval makes it necessary to assume that the radiuss ₍₀₎ is variable. The groups of motion of a 4-hyperboloid of variable radius are constructed in the second part of the paper. They introduce new symmetry properties of 4d spacetime.