Complex Riemannian Geometry—Bianchi Identities and Einstein Tensor

ABSTRACT

Riemannian geometry has proved itself to be a useful model of the gravitational phenomena in the universe, but generalizations of it to include other forces have so far not been successful. Here we explore an extension of Riemannian geometry using a complex Hermitian metric tensor. We find that the standard electromagnetic field naturally appears along with two additional fields, which act as mass and charge sources. A first paper set up the basic geometry and derived the Christoffel symbols plus the E&M field equation. This paper continues development with the generalized Riemann curvature tensor, Bianchi identities and the Einstein tensor, laying the basis for field equations. A final paper will then present the field equations.

Riemannian geometry has proved itself to be a useful model of the gravitational phenomena in the universe, but generalizations of it to include other forces have so far not been successful. Here we explore an extension of Riemannian geometry using a complex Hermitian metric tensor. We find that the standard electromagnetic field naturally appears along with two additional fields, which act as mass and charge sources. A first paper set up the basic geometry and derived the Christoffel symbols plus the E&M field equation. This paper continues development with the generalized Riemann curvature tensor, Bianchi identities and the Einstein tensor, laying the basis for field equations. A final paper will then present the field equations.

KEYWORDS

Unified Field Theory, Gravity, Electromagnetism, Complex Riemannian Geometry, Riemannian Geometry

Unified Field Theory, Gravity, Electromagnetism, Complex Riemannian Geometry, Riemannian Geometry

Cite this paper

Hutchin, R. (2015) Complex Riemannian Geometry—Bianchi Identities and Einstein Tensor.*Journal of Modern Physics*, **6**, 1572-1585. doi: 10.4236/jmp.2015.611159.

Hutchin, R. (2015) Complex Riemannian Geometry—Bianchi Identities and Einstein Tensor.

References

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[2] Kaluza, T. (1921) Zum Unittsproblem in der Physik. 2nd Edition, Math. Phys, Berlin, 966-972.

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[4] Mie, G. (1912) Annals of Physics, 37, 511-534.

http://dx.doi.org/10.1002/andp.19123420306

[5] Reichenbächer, E. (1917) Annals of Physics, 52, 134-173.

http://dx.doi.org/10.1002/andp.19173570203

[6] Hutchin, R.A. (2015) Journal of Modern Physics, 6, 749-757.

http://dx.doi.org/10.4236/jmp.2015.66080

[7] Adler, R., Basin, M. and Schiffer, M. (1975) Introduction to General Relativity. McGraw Hill, New York.

[1] Weyl, H. (1918) Gravitation und Elektrizität. Sitzungsberichte der K öniglich Preussischen Akademie der Wissenschaften, 465.

[2] Kaluza, T. (1921) Zum Unittsproblem in der Physik. 2nd Edition, Math. Phys, Berlin, 966-972.

[3] Eddington, A.S. (1924) The Mathematical Theory of Relativity. 2nd Edition, Cambridge University Press, Cambridge.

[4] Mie, G. (1912) Annals of Physics, 37, 511-534.

http://dx.doi.org/10.1002/andp.19123420306

[5] Reichenbächer, E. (1917) Annals of Physics, 52, 134-173.

http://dx.doi.org/10.1002/andp.19173570203

[6] Hutchin, R.A. (2015) Journal of Modern Physics, 6, 749-757.

http://dx.doi.org/10.4236/jmp.2015.66080

[7] Adler, R., Basin, M. and Schiffer, M. (1975) Introduction to General Relativity. McGraw Hill, New York.