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

Gordon Liu

Copernicus Institute for Physics and Astronomy, Toronto, Canada.

DOI: 10.4236/ijaa.2013.31002                

Abstract

Let the coordinate system xⁱ 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.

Keywords

General Relativity; Gravitation; Riemannian Space-Time; Flat Space-Time; Einstein Equations; Harmonic Conditions; Energy-Momentum Tensor; Significance of the Coordinates; Gravitational Red-Shift

Conflicts of Interest

The authors declare no conflicts of interest.

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