The York map as a Shanmugadhasan canonical transformation in tetrad gravity and the role of non-inertial frames in the geometrical view of the gravitational field

TL;DR

This paper introduces a new parametrization in tetrad gravity that explicitly finds the York map, clarifies the role of gauge variables and non-inertial frames, and explores their implications for gravitational dynamics and potential dark matter effects.

Contribution

It presents an explicit York map in canonical tetrad gravity, analyzes gauge variables including non-inertial effects, and discusses their influence on gravitational dynamics and the possible emergence of dark matter as a gauge inertial effect.

Findings

Explicit form of the Hamilton equations for tidal degrees of freedom.

Existence of a generalized Gribov ambiguity in super-momentum constraints.

Dependence of gravitational dynamics on the gauge variable ${}^3K$ related to clock synchronization.

Abstract

A new parametrization of the 3-metric allows to find explicitly a York map in canonical ADM tetrad gravity, the two pairs of physical tidal degrees of freedom and 14 gauge variables. These gauge quantities (generalized inertial effects) are all configurational except the trace ${}^3K(\tau ,\vec \sigma)$ of the extrinsic curvature of the instantaneous 3-spaces $\Sigma_{\tau}$ (clock synchronization convention) of a non-inertial frame. The Dirac hamiltonian is the sum of the weak ADM energy $E_{ADM} = \int d^3\sigma {\cal E}_{ADM}(\tau ,\vec \sigma)$ (whose density is coordinate-dependent due to the inertial potentials) and of the first-class constraints. Then: i) The explicit form of the Hamilton equations for the two tidal degrees of freedom in an arbitrary gauge: a deterministic evolution can be defined only in a completely fixed gauge, i.e. in a non-inertial frame with its pattern of inertial forces. ii) A general solution of the super-momentum constraints, which shows the existence of a generalized Gribov ambiguity associated to the 3-diffeomorphism gauge group. It influences: a) the explicit form of the weak ADM energy and of the super-momentum constraint; b) the determination of the shift functions and then of the lapse one. iii) The dependence of the Hamilton equations for the two pairs of dynamical gravitational degrees of freedom (the generalized tidal effects) and for the matter, written in a completely fixed 3-orthogonal Schwinger time gauge, upon the gauge variable ${}^3K(\tau ,\vec \sigma)$, determining the convention of clock synchronization. Therefore it should be possible (for instance in the weak field limit but with relativistic motion) to try to check whether in Einstein's theory the {\it dark matter} is a gauge relativistic inertial effect induced by ${}^3K(\tau ,\vec \sigma)$.