On a complex spacetime frame: new foundation for physics and mathematics

Abstract

This paper provides a derivation of a unit vector specifying the temporal axis of four-dimensional space-time frame as an imaginary axis in the direction of light or general wave propagation. The basic elements of the resulting complex four-dimensional spacetime frame are complex four-vectors expressed in standard form as four-component quantities specified by four unit vectors, three along space axes and one along the imaginary temporal axis. Consistent mathematical operations with the complex four-vectors have been developed, which provide extensions of standard vector analysis theorems to complex four-dimensional spacetime. The general orientation of the temporal unit vector relative to all the three mutually perpendicular spatial unit vectors leads to appropriate modifications of fundamental results describing the invariant length of the spacetime event interval, time dilation, mass increase with speed and relativistic energy conservation law. Contravariant and covariant forms have been defined, providing appropriate definition of the invariant length of a complex four-vector and appropriate definitions of complex tensors within the complex four-dimensional spacetime frame, thus setting a new geometrical framework for physics and mathematics.