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On energy-momentum tensors of gravitational field

A. I. Nikishov
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Abstract

Several energy-momentum "tensors" of gravitational field are considered and compared in the lowest approximation. Each of them together with energy-momentum tensor of point-like particles satisfies the conservation laws when equations of motion of particles are the same as in general relativity. It is shown that in Newtonian approximation the considered tensors differ one from the other in the way their energy density is distributed between energy density of interaction (nonzero only at locations of particles) and energy density of gravitational field. Starting from Lorentz invariance the Lagrangians for spin-2, mass-0 field are considered. They differ only by divergences. From these Lagrangians by Belinfante-Rosenfeld procedure the energy-momentum tensors are build. Using each of these tensors in 3-graviton vertex we obtain the corresponding metric of a Newtonian center in $G^2$ approximation. Only one of these ''field-theretical''tensors (namely the half sum of Thirring tensor and tensor obtained from Lagrangian given by Misner, Thorne and Wheeler) leads to correct value of the perihelion shift. This tensor does not coincide with Weinberg`s one (directly obtainable from Einstein equation) and gives metric of a spherical body differing (in space part of metric in the first nonlinear approximation) from Schwarzschild field in harmonic coordinates. As a result a relativistic particle in such field must move note according general relativity prescriptions. This approach puts the gravitational energy-momentum tensor on the same footing as any other energy-momentum tensor.

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A. I. Nikishov. 1999-12-09. On energy-momentum tensors of gravitational field. https://arxiv.org/abs/gr-qc/9912034

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