Post-Minkowskian Gravity: Dark Matter as a Relativistic Inertial Effect?

TL;DR

This paper reviews non-inertial frames in Minkowski space-time, explores their application to tetrad gravity, and proposes that dark matter could be a relativistic inertial effect related to gauge choices in gravity.

Contribution

It introduces a formalism linking inertial effects in general relativity to gauge variables, proposing a novel explanation for dark matter as a relativistic inertial effect.

Findings

Identification of gauge variables as inertial effects in gravity.

Definition of post-Minkowskian gravitational waves in non-Euclidean 3-space.

Hypothesis that dark matter may be a relativistic inertial effect from gauge choices.

Abstract

A review is given of the theory of non-inertial frames (with the associated inertial effects and the study of the non-relativistic limit) in Minkowski space-time, of parametrized Minkowski theories and of the rest-frame instant form of dynamics for isolated systems admitting a Lagrangian description. The relevance and gauge equivalence of the clock synchronization conventions for the identification of the instantaneous 3-spaces (Euclidean only in inertial frames) are described. Then this formalism is applied to tetrad gravity in globally hyperbolic, asymptotically Minkowskian space-times without super-translations, where the equivalence principle implies the absence of global inertial frames. The recently discovered York canonical basis, diagonalizing the York-Lichnerowicz approach, allows to identify the gauge variables (inertial effects in general relativity) and the tidal ones (the gravitational waves of the linearized theory) and to clarify the meaning of the Hamilton equations. The role of the gauge variable ${}^3K$, the trace of the extrinsic curvature of the non-Euclidean 3-space (the York time not existing in Newton theory), as a source of inertial effects is emphasized. After the presentation of preliminary results on the linearization of tetrad gravity in the family of non-harmonic 3-orthogonal gauges with a free value of ${}^3K$, we define post-Minkowskian gravitational waves (without post-Newtonian approximations on the matter sources) propagating in a non-Euclidean 3-space, emphasizing the non-graviton-like aspects of gravity. It is conjectured that dark matter may be explained as a relativistic inertial effect induced by ${}^3K$: it would simulate the need to choose a privileged gauge connected with the observational conventions for the description of matter.