Complex Spacetime Frame: Four-Vector Identities and Tensors

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.

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.

KEYWORDS

Complex Spacetime Frame, Four-Vector Identities, Contravariant and Covariant Forms, Complex Tensors

Complex Spacetime Frame, Four-Vector Identities, Contravariant and Covariant Forms, Complex Tensors

Cite this paper

Omolo, J. (2014) Complex Spacetime Frame: Four-Vector Identities and Tensors.*Advances in Pure Mathematics*, **4**, 567-579. doi: 10.4236/apm.2014.411065.

Omolo, J. (2014) Complex Spacetime Frame: Four-Vector Identities and Tensors.

References

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[10] Gibbons, G.W., Gomis, J. and Pope, C.N. (2007) General Very Special Relativity in Finsler Geometry. Physical Review D, 76, Article ID: 081701. http://dx.doi.org/10.1103/PhysRevD.76.081701

[1] Akeyo Omolo, J. (2014) On a Derivation of the Temporal Unit Vector of Space-Time Frame. Journal of Applied Mathematics and Physics, in Press.

[2] Poincare, H. (1906) Sur la dynamique de l’e’lectron (On the Dynamics of the Electron). Rendiconti del Circolo Matematico di Palermo, 21, 129.

[3] Lorentz, H.A. (1904) Electromagnetic Phenomena in a System Moving with Any Velocity Smaller than That of Light. Proceedings of the Royal Netherlands Academy of Arts and Sciences, 6, 809

[4] Einstein, A. (2003) The Meaning of Relativity. Routledge Classics, London and New York.

[5] Charbonneau, P. (2013) Solar and Stellar Dynamics. Saas-Fee Advanced Course, 39, 215.

[6] Arfken, G.B. and Weber, H.J. (1995) Mathematical Methods for Physicists. Academic Press Inc., San Diego.

[7] Landau, L.D. and Lifshitz, E.M. (1975) The Classical Theory of Fields. Pergamon Press Ltd., Oxford.

[8] Dirac, P.A.M. (1975) General Theory of Relativity. John Wiley and Sons Inc., New York.

[9] Bogoslovsky, G.Y. (2007) Some Physical Displays of the Space Anisotropy Relevant to the Feasibility of Its Being Detected at a Laboratory. arXiv:0706.2621 [gr-qc]

[10] Gibbons, G.W., Gomis, J. and Pope, C.N. (2007) General Very Special Relativity in Finsler Geometry. Physical Review D, 76, Article ID: 081701. http://dx.doi.org/10.1103/PhysRevD.76.081701