### Closed Isometric Linear Transformations of Complex Spacetime endowed with Euclidean, or Lorentz, or generally Isotropic Metric

#### Abstract

This paper is the first in a series of papers showing that Newtonian physics and Einsteinian relativity theory can be unified, by using a Generalized Real Boost (GRB), which expresses both the Galilean Transformation (GT) and the Lorentz Boost. Here, it is proved that the Closed Linear Transformations (CLTs) in Spacetime (ST) correlating frames having parallel spatial axes, are expressed via a 4x4 matrix Λ I , which contains complex Cartesian Coordinates (CCs) of the velocity of one Observer / Frame (O/F) wrt another. In the case of generalized Special Relativity (SR), the inertial Os/Fs are related via isotropic ST endowed with constant real metric, which yields the constant characteristic parameter ω I that is contained in the CLT and GRB of the specific SR. If ω I is imaginary number, the ST can only be described by using complex CCs and there

exists real Universal Speed (c I ). The specific value ω I =±i gives the Lorentzian-Einsteinian versions of CLT and GRB in ST endowed with metric: -g I00 η and c I =c, where i; c; g I00 ; η are the imaginary unit; speed of light in vacuum; time-coefficient of metric; Lorentz metric, respectively. If ω I is real number, the corresponding ST can be described by using real CCs, but does not exist c I . The specific value ω I =0 gives GT with infinite c I . GT is also the reduction of the CLT and GRB, if one O/F has small velocity wrt another.

The results may be applied to any ST endowed with isotropic metric, whose elements (four-vectors) have spatial part (vector) that is element of the ordinary Euclidean space.

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