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

Author(s)
Gordon Liu

ABSTRACT

Let the coordinate system *x ^{i}* of flat space-time to absorb a second rank tensor field Φ

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.

G. Liu, "Riemannian Space-Time, de Donder Conditions and Gravitational Field in Flat Space-Time,"

References

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[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.

[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.