The Chrono-Geometrical Structure of Special and General Relativity: A Re-Visitation of Canonical Geometrodynamics

Abstract

A modern re-visitation of the consequences of the lack of an intrinsic notion of instantaneous 3-space in relativistic theories leads to a reformulation of their kinematical basis emphasizing the role of non-inertial frames centered on an arbitrary accelerated observer. In special relativity the exigence of predictability implies the adoption of the 3+1 point of view, which leads to a well posed initial value problem for field equations in a framework where the change of the convention of synchronization of distant clocks is realized by means of a gauge transformation. This point of view is also at the heart of the canonical approach to metric and tetrad gravity in globally hyperbolic asymptotically flat space-times, where the use of Shanmugadhasan canonical transformations allows the separation of the physical degrees of freedom of the gravitational field (the tidal effects) from the arbitrary gauge variables. Since a global vision of the equivalence principle implies that only global non-inertial frames can exist in general relativity, the gauge variables are naturally interpreted as generalized relativistic inertial effects, which have to be fixed to get a deterministic evolution in a given non-inertial frame. As a consequence, in each Einstein's space-time in this class the whole chrono-geometrical structure, including also the clock synchronization convention, is dynamically determined and a new approach to the Hole Argument leads to the conclusion that "gravitational field" and "space-time" are two faces of the same entity. This view allows to get a classical scenario for the unification of the four interactions in a scheme suited to the description of the solar system or our galaxy with a deperametrization to special relativity and the subsequent possibility to take the non-relativistic limit.

Relativistic quantum mechanics and relativistic entanglement in the rest-frame instant form of dynamics

“…See Ref. [5] for the Møller non-covariance world-tube around the Fokker-Pryce 4-vector identified by these collective variables. Their transformation properties under Poincare' transformations (a, Λ) can be deduced from those for h, z and τ (see Appendix B of Ref.…”

“…[5], in each Lorentz frame one has different pseudo-worldlines describing R µ andx µ : the canonical 4-center of massx µ lies in between Y µ and R µ in every (non rest)-frame. As discussed in Subsection IIF of paper I, this leads to the existence of the Møller non-covariance world-tube, around the world-line Y µ of the covariant non-canonical Fokker-Pryce 4-center of inertia Y µ .…”

“…Then in Subsection B we study its non-relativistic limit. Finally in 5 In Refs. [12] a first attempt of quantization of relativistic mechanics in inertial and non-inertial frames (with the non-relativistic limit given in Ref.…”

“…[4][5][6][7] it was shown that it is possible to describe any isolated relativistic system (particles, fields, strings, fluids) admitting a Lagrangian formulation (allowing one to define the energy-momentum tensor of the system) in arbitrary non-inertial frames in Minkowski space-time by means of parametrized Minkowski theories. The transition among different non-inertial frames (with their different clock synchronization conventions identifying the instantaneous, in general non-Euclidean, 3-spaces) is performed by frame-preserving diffeomorphisms, i.e.…”

From Clock Synchronization to Dark Matter as a Relativistic Inertial Effect

“…If one restricts himself to inertial frames, one can define the inertial rest-frame instant form of dynamics for isolated systems by choosing the 3+1 splitting corresponding to the intrinsic inertial rest frame of the isolated system centered on an inertial observer: the instantaneous 3-spaces, named Wigner 3-spaces due to the fact that the 3-vectors inside them are Wigner spin-1 3-vectors [23,24], are orthogonal to the conserved 4-momentum P µ (assumed time-like, ǫ P 2 > 0) of the configuration.…”