### Euclidean Closed Linear Transformations of Complex Spacetime and generally of Complex Spaces of dimension four endowed with the Same or Different Metric

Spyridon Vossos, Elias Vossos
2016 Journal of Physics, Conference Series
Relativity Theory and the corresponding Relativistic Quantum Mechanics are the fundamental theories of physics. Special Relativity (SR) relates the frames of Relativistic Inertial observers (RIOs), through Linear Spacetime Transformation (LSTT) of linear spacetime. Classic Special Relativity uses real spacetime endowed with Lorentz metric and the frames of two RIOs with parallel spatial axes are always related through Lorentz Boost (LB). This cancels the transitive attribute in parallelism,
more » ... in parallelism, when three RIOs are related, because LB is not closed transformation, causing Thomas Rotation. In this presentation, we consider closed LSTT of Complex Spacetime, so there is no necessity for spatial axes rotation and all the frames are chosen having parallel spatial axes. The solution is expressed by a 4x4 matrix (Λ) containing components of the complex velocity of one Observer wrt another and two functions depended by the metric of Spacetime. Demanding isometric transformation, it emerges a class of metrics that are in accordance with the closed LSTT and the transformation matrix contains one parameter ω depended by the metric of Spacetime. In case that we relate RIOs with steady metric, it emerges one steady number (ω I ) depended by the metric of Spacetime of the specific SR. If ω I is an imaginary number, the elements of the Λ are complex numbers, so the corresponding spacetime is necessarily complex and there exists real Universal Speed (U I ). The specific value ω I =±i gives Vossos transformation (VT) endowed with Lorentz metric (for g ii =1) of complex spacetime and invariant spacetime interval (or equivalently invariant speed of light in vacuum), which produce the theory of Euclidean Complex Relativistic Mechanics (ECRMs). If ω I is a real number (ω I ≠0) the elements of the Λ are real numbers, so the corresponding spacetime is real, but there exist imaginary U I . The specific value ω I =0 gives Galileo Transformation (GT) with the invariant time, in which any other closed LSTT is reduced, if one RIO has small velocity wrt another RIO. Thus, we have infinite number of closed LSTTs, each one with the corresponding SR theory. In case that we relate accelerated