Share This Article:

Riemannian Space-Time, de Donder Conditions and Gravitational Field in Flat Space-Time

Full-Text HTML XML Download Download as PDF (Size:246KB) PP. 8-19
DOI: 10.4236/ijaa.2013.31002    3,378 Downloads   6,532 Views   Citations
Author(s)    Leave a comment

ABSTRACT

Let the coordinate system xi of flat space-time to absorb a second rank tensor field Φij of the flat space-time deforming into a Riemannian space-time, namely, the tensor field Φuv is regarded as a metric tensor with respect to the coordinate system xu. After done this, xu is not the coordinate system of flat space-time anymore, but is the coordinate system of the new Riemannian space-time. The inverse operation also can be done. According to these notions, the concepts of the absorption operation and the desorption operation are proposed. These notions are actually compatible with Einstein’s equivalence principle. By using these concepts, the relationships of the Riemannian space-time, the de Donder conditions and the gravitational field in flat space-time are analyzed and elaborated. The essential significance of the de Donder conditions (the harmonic conditions or gauge) is to desorb the tensor field of gravitation from the Riemannian space-time to the Minkowski space-time with the Cartesian coordinates. Einstein equations with de Donder conditions can be solved in flat space-time. Base on Fock’s works, the equations of gravitational field in flat space-time are obtained, and the tensor expression of the energy-momentum of gravitational field is found. They all satisfy the global Lorentz covariance.

Cite this paper

G. Liu, "Riemannian Space-Time, de Donder Conditions and Gravitational Field in Flat Space-Time," International Journal of Astronomy and Astrophysics, Vol. 3 No. 1, 2013, pp. 8-19. doi: 10.4236/ijaa.2013.31002.

References

[1] L. Smolin, “How far are we from the quantum theory of gravity?” Reports on Progress in Physics, Vol. 72, No. 12, 2009, Article ID: 126002. doi:10.1088/0034-4885/72/12/126002
[2] L. M. Butcher, A. N. Lasenby and M. P. Hobson, “Physical Significance of the Babak-Grishchuk Gravitational Energy-Momentum Tensor,” Physical Review D, Vol. 78, No. 6, 2008, Article ID: 064034. doi:10.1103/PhysRevD.78.064034
[3] N. Rosen, “General Relativity and Flat Space I,” Physical Review, Vol. 57, No. 2, 1940, pp. 147-150. doi:10.1103/PhysRev.57147
[4] R. Kraichnan, “Special-Relativistic Derivation of Generally Covariant Gravitation Theory,” Physical Review, Vol. 98, No. 4, 1955, pp. 1118-1122.. doi:10.1103/PhysRev.98.1118
[5] S. N. Gupta, “Einstein’s and Other Theories of Gravitation,” Reviews of Modern Physics, Vol. 29, No. 3, 1957, pp. 334-336. doi:10.1103/RevModPhys.29.334
[6] W. Thirring, “An Alternative Approach to the Theory of Gravitation,” Annals of Physics, Vol. 16, No. 1, 1961, pp. 96-117. doi:10.1016/0003-4916(61)90182-8
[7] S. Weinberg, “Derivation of Gauge Invariance and the Equivalence Principle from Lorentz Invariance of the S-Matrix,” Physics Letters, Vol. 9, No. 4, 1964, pp. 357-359. doi:10.1016/0031-9163(64)90396-8
[8] V. I. Ogievetsky and I. V. Polubarinov, “Interacting Field of Spin 2 and the Einstein Equations,” Annals of Physics, Vol. 35, No. 2, 1965, pp. 167-208. doi:10.1016/0003-4916(65)90077-1
[9] S. Deser, “Self-Interaction and Gauge Invariance,” General Relativity and Gravitation, Vol. 1, No. 1, 1970, pp. 9-18. doi:10.1007/BF00759198
[10] P. van Nieuwenhuizen, “On Ghost-Free Tensor Lagrangians and Linearizes Gravitation,” Nuclear Physics B, Vol. 60, 1973, pp. 478-492. doi:10.1016/0550-3213(73)90194-6
[11] T. M. Nieuwenhuizen, “Einsein vs Maxwell: Is Gravitation a Curvature of Space, a Field in Flat Space, or Both?” Europhysical Letters, Vol. 78, 2007, p. 10010.
[12] D. G. Boulware and S. Deser, “Classical General Relativity Derived from Quantum Gravity,” Annals of Physics, Vol. 89, No. 1, 1975, pp. 193-240. doi:10.1016/0003-4916(75)90302-4
[13] L. P. Grishchuk, A. N. Petrov and A. D. Popova, “Exact Theory of the (Einstein) Gravitational Field in an Arbitrary Background Space-Time,” Communications in Mathematical Physics, Vol. 94, No. 3, 1984, pp. 379-396. doi:10.1007/BF01224832
[14] A. A. Logunov and M. A. Mestvirishvili, “The Fundamental Principles of the Relativistic Theory of Gravitation,” Theoretical and Mathematical Physics, Vol. 86, No. 1, 1991, pp. 1-9. doi:10.1007/BF01018491
[15] M. Visser, “Mass for the Graviton,” General Relativity and Gravitation, Vol. 30, No. 12, 1998, pp. 1717-1728. doi:10.1023/A:1026611026766
[16] S. V. Babak and L. P. Grishchuk, “Energy-Momentum Tensor for the Gravitational Field,” Physical Review D, Vol. 61, No. 2, 1999, Article ID: 024038. doi:10.1103/PhysRevD.61.024038
[17] J. B. Pitts and W. C. Schieve, “Slightly Bimetric Gravitation,” General Relativity and Gravitation, Vol. 33, No. 8, 2001, pp. 1319-1350. doi:10.1023/A:1012005508094
[18] L. P. Grishchuk, “Some Uncomfortable Thoughts on the Nature of Gravity, Cosmology, and the Early Universe,” Space Science Reviews, Vol. 148, No. 1-4, 2009, pp. 315-328. doi:10.1007/s11214-009-9509-6
[19] E. R. Huggins, Ph.D. Thesis, California Institute of Technology, Pasadena, 1962.
[20] R. P. Feynman, F. Morinigo, W. Wagner and B. Hatfield, “Feynman Lectures on Gravitation,” Addison Wesley, Boston, 1995.
[21] A. A. Logunov, “The Relativistic Theory of Gravitation,” Nauka, Moscow, 2000.
[22] L. V. Verozub, “Gravitation as Field and Curvature,”
[23] T. Padmanabhan, “From Gravitons to Gravity: Myths and Reality,” International Journal of Modern Physics D, Vol. 17, No. 03n04, 2008, p. 367. doi:10.1142/S0218271808012085
[24] L. Liu, “General Relativity,” Advanced Education Publishing Company, Beijing, 1987, p. 133.
[25] R. V. Eotvos, D. Pekar and E. Fekete, Annals of Physics, Vol. 68, 1922, p. 11.
[26] P.-Y. Chou, “On the Physical Significance of Coordinates and the Solutions of the Field Equations in Einstein’s Theory of Gravitation,” Scientia Sinica (Series A), Vol. XXV, No. 6, 1982, pp. 628-643. http://www.cnki.com.cn/Article/CJFDTotal-JAXK198204005.htm http://www.emw21.com/CTS/Global%20Chinese/ZhouPY/ZhouPYFrame.htm
[27] T. de Donder, “La Gravifique Einsteinienne,” University of Michigan Library, Paris, 1921.
[28] V. Fock, “The Theory of Space Time and Gravitation,” Pergamon Press, New York, 1959, p. 175.
[29] Y.-S. Duan and J.-Y. Zhang, “On Fock’s Harmonic Conditions in General Relativity,” Acta Physica Sinica, Vol. 18, No. 4, 1962, p. 211. http://www.oaj.cas.cn/cn
[30] A. Einstein, Berl. Ber., 1915, p. 178.
[31] R. Tolman, “On the Use of the Energy-Momentum Principle in General Relativity,” Physical Review, Vol. 35, No. 8, 1930, pp. 875-895. doi:10.1103/PhysRev.35.875
[32] Y. A. Krutkov, “The Stress Tensor and the Solution of General Problems in the Static Theory of Elasticity,” Akad Nauk, U.S.S.R., 1949.

  
comments powered by Disqus

Copyright © 2018 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.