Unit-free transformations of linear relativity

Linear relativistic transformations of special relativity considered in [A. Einstein, Zur Elektrodynamik der bewegter Körper, Ann. Phys. 17 (1905) 891–921] are analyzed and their representation through unit-free parameters is obtained in the form: τ/t=ξ/x′=γ(p)=[1+(p/V)2]0.5∈[1,∞),η/y=ζ/z=1τ/t=ξ/x′=γ(p)=[1+(p/V)2]0.5∈[1,∞),η/y=ζ/z=1, where x′=x−vtx′=x−vt, the relative velocity v>0v>0 of a moving frame (k)(k) with respect to a still frame (K)(K) is constant and directed along 0x∈(K)0x∈(K), the value p=dξ/dt=−βv=const, with ββ being the Einstein calibration factor, and the distance ξ(t)ξ(t) to the origin of (k)(k), a moving body, is being measured by radar from a still point x∈(K)x∈(K), on Earth, with respect to its natural time tt. A new relativistic invariant is derived which admits trigonometric representation and can be extended onto nonlinear relativity with variable velocities. The unit-free form of relativistic transformations demonstrates that special relativity considerations are valid with respect to any available information transmitting signal, thus extending special relativity onto all interacting processes linked by such signals. Multi-relativistic situations are discussed in relation to the structure of observed   time ττ. Natural time delays in transmission of information by physical processes are considered in relation to Minkowski’s 4D space–time geometry that becomes an affinely connected space–time structure with affinors being conditioned on the actually interacting physical processes. The results open new avenues for further research in relativity theory and its applications.