Riemannian Space-Time, de Donder Conditions and Gravitational Field in Flat Space-Time
- 1 Copernicus Institute for Physics and Astronomy, Toronto, Canada
Abstract
Let the coordinate system x i of flat space-time to absorb a second rank tensor field Φ ij of the flat space-time deforming into a Riemannian space-time, namely, the tensor field Φ uv is regarded as a metric tensor with respect to the coordinate system x u . After done this, x u 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.
- 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
- 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
- N. Rosen, “General Relativity and Flat Space I,” Physical Review, Vol. 57, No. 2, 1940, pp. 147-150. doi:10.1103/PhysRev.57147
- 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
- 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
- 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
- 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
- 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
- S. Deser, “Self-Interaction and Gauge Invariance,” General Relativity and Gravitation, Vol. 1, No. 1, 1970, pp. 9-18. doi:10.1007/BF00759198
- 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
- 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.
- 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
- 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
- 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