The Einstein–Maxwell-particle system in the York canonical basis of ADM tetrad gravity. Part 3. The post-minkowskian N-body problem, its post-newtonian limit in nonharmonic 3-orthogonal gauges and dark matter as an inertial effect ¹This paper is one of three companion papers published in the same issue of Can. J. Phys.

Abstract

We conclude the study of the post-minkowskian (PM) linearization of ADM tetrad gravity in the York canonical basis for asymptotically minkowskian space-times in the family of nonharmonic 3-orthogonal gauges parametrized by the York time 3 K. ; / (the inertial gauge variable, not existing in Newton gravity, describing the general relativistic remnant of the freedom in clock synchronization in the definition of the shape of the instantaneous 3-spaces as 3-submanifolds of space-time). As matter we consider only N scalar point particles with a Grassmann regularization of the self-energies and with an ultraviolet cutoff making possible the PM linearization and the evaluation of the PM solution for the gravitational field. We study in detail all the properties of these PM space-times emphasizing their dependence on the gauge variable 3 K .1/ D .1=4/ 3 K .1/ (the nonlocal York time): Riemann and Weyl tensors, 3-spaces, time-like and null geodesics, red-shift, and luminosity distance. Then we study the post-newtonian (PN) expansion of the PM equations of motion of the particles. We find that in the two-body case at the 0.5PN order there is a damping (or antidamping) term depending only on 3 K .1/ . This opens the possibility of explaining dark matter in Einstein theory as a relativistic inertial effect: the determination of 3 K .1/ from the masses and rotation curves of galaxies would give information on how to find a PM extension of the existing PN celestial frame used as an observational convention in the 4-dimensional description of stars and galaxies. Dark matter would describe the difference between the inertial and gravitational masses seen in the noneuclidean 3-spaces, without a violation of their equality in the 4-dimensional space-time as required by the equivalence principle. PACS Nos.: 04.20.Fy, 04.20.Ha, 04.30.-w, 04.40.Nr, 95.35.+dRésumé : Nous concluons l'étude de la linéarisation post-minkowskienne (PM) de la gravité tétrade de ADM dans la base canonique de York pour des espace-temps asymptotiquement minkowskiens dans la famille des jauges 3-orthogonales paramétrées par le temps de York 3 K (la variable de jauge inertielle, qui n'existe pas en gravité de Newton, décrivant ce qui reste, provenant de la relativité générale, de la liberté de synchronisation de l'horloge dans la définition du 3-espace instantané). La seule matière considérée ici est constituée de N particules scalaires ponctuelles avec une régularisation de Grassmann des self énergies et avec une coupure ultraviolette rendant possible la linéarisation PM et l'évaluation des solutions PM pour le champ de gravité. Nous étudions en détails toutes les propriétés de l'espace-temps PM, insistant sur leur dépendance sur la variable de jauge 3 K .1/ D 1=4 3 K .1/ (le temps de York non local) : les tenseurs de Riemann et de Weyl, les géodésiques de type temps (3D) et nulles, le déplacement vers le rouge et la distance de luminosité. Nous étudions alors l'expansion post-newtonienne (PN) des équations PM du mouvement des particules. Nous trouvons que dan...

From Relativistic Mechanics towards Relativistic Statistical Mechanics

“…In the family of globally-hyperbolic, asymptotically-Minkowskian space-times without super-translations discussed in [76][77][78][79][80][81][82][83], there is the possibility of defining the asymptotic ADM Poincaré generators at spatial infinity, where the instantaneous three-spaces Σ τ become Euclidean, for every kind of matter. The absence of super-translations implies that these three-spaces are relativistic non-inertial rest frames.…”

“…The four-coordinates σ A = (τ; σ r ) are named radar four-coordinates: they were introduced by Bondi [75] in general relativity and have been used for the definition of global non-inertial frames by means of 3 + 1 splittings of a certain family of Einstein space-times (see [76][77][78][79][80][81] and the reviews [82,83]). …”

“…Their relevance is due to the fact that they are the only global non-inertial frames allowed by the equivalence principle (forbidding the existence of global inertial frames) in the canonical metric and tetrad Einstein gravity with matter [79][80][81], in globally hyperbolic, asymptotically flat (asymptotically Minkowskian) space-times without super-translations, so as to have the asymptotic ADM Poincaré group becoming the standard Poincaré group of the matter in Minkowski space-time in the limit of the zero Newton constant.…”

“…96. See Appendix B of [35] for the Galilei generators of N scalar particles in Newtonian gravity and for the ADM Poincaré generators in the Post-Minkowskian approximation [79][80][81] ( x 1 , p 1 , ..., x N , p N , t) = 1. 102.…”

“…121. In Appendix A of [35] are the Poincaré generators for N charged scalar particles with the Coulomb plus Darwin potentials of [29], while in Appendix B is their expression for N scalar particles in the post-Minkowskian tetrad gravity of [79][80][81]. 122.…”