Abstract
This paper provides derivation of some basic identities for complex four-component vectors defined in a complex four-dimensional spacetime frame specified by an imaginary temporal axis. The resulting four-vector identities take exactly the same forms of the standard vector identities established in the familiar three-dimensional space, thereby confirming the consistency of the definition of the complex four-vectors and their mathematical operations in the general complex spacetime frame. Contravariant and covariant forms have been defined, providing appropriate definitions of complex tensors, which point to the possibility of reformulating differential geometry within a spacetime frame.
Highlights
In a recent derivation [1], the present author identified the unit wave vector kto be the temporal unit vector within four-dimensional spacetime frame
In developing the mathematical operations in general form, we shall take the temporal unit vector kto be of general orientation relative to the spatial unit vectors x, y, z, satisfying the conditions in Equation (1)
To complete the mathematical formalism within complex four-dimensional spacetime frame, we introduce contravariant and covariant forms, which are useful in carrying out general mathematical operations with four-vectors
Summary
In a recent derivation [1], the present author identified the unit wave vector kto be the temporal unit vector within four-dimensional spacetime frame. We observe that the concept of imaginary temporal axis developed here, represents a rediscovery of the idea of imaginary time first introduced independently by Poincare [2], Lorentz [3] and Einstein [4] in their original theories of electrodynamics or special relativity in a four-dimensional spacetime frame. These authors did not identify the temporal unit vector and could not completely specify the complex spacetime frame and develop the full mathematical operations using complex four-vectors in the manner presented in this paper
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