After a review of regular Lagrangians, their Hamiltonian formulation, and the first Noether theorem, there is the exposition of theory of singular Lagrangians and of Dirac–Bergmann theory of first- and second-class constraints. Also, the gauge transformations of field theory and general relativity are analyzed.

There is a description of the 3+1 approach allowing definition of global non-inertial frames in Minkowski space-time. One gives a time-like observer and a nice foliation with 3-spaces (namely a clock synchronization convention). Then one introduces Lorentz scalar radar 4-coordinates: the time is an increasing function of the proper time of the observer and the 3-coordinates live in the instantaneous 3-spaces. The connection of the radar coordinates with the standard ones defines the four embedding functions describing the foliation with 3-spaces. Then there is the definition of parametrized Minkowski theories for every kind of matter admitting a Lagrangian description. The new Lagrangian is a function of the matter and of the embedding, but is singular so that the embedding variables are gauge variables. As a consequence, the transition from a non-inertial frame to either an inertial or non-inertial frame is a gauge transformation not changing the physics but only the inertial forces.

After a description of inertial and non-inertial frames in the Galilei space-time of non-relativistic Newtonian physics with a discussion of inertial forces, there is metrological definition of what is time and space in special relativity. Then there is a review of the standard 1+3 approach for the “local” description of non-inertial frames and of its limitations.

The following family of Einstein space-times allows the use of the 3+1 approach: (1) globally hyperbolic (this allows the ADM Hamiltonian formulation); (2) asymptotically Minkowskian at spatial infinity (all the 3-spaces approach parallel space-like hyper-planes); (3) without super-translations (at spatial infinity there is the asymptotic ADM Poincaré algebra needed for particle physics). It turns out that the asymptotic ADM Poincaré 4-momentum is orthogonal to the asymptotic hyper-planes. Therefore, the 3+1 approach allows describing the Hamiltonian formulation of metric gravity and of its first-class constraints in the family of the non-inertial rest-frames.

In this chapter there is the description of fields and fluids in the rest-frame instant form of dynamics with the definition of their Wigner-covariant degrees of freedom inside the Wigner 3-spaces after the decoupling of the external center of mass. This is done for the Klein–Gordon, electromagnetic, Dirac, and Yang–Mills fields. In the case of the electromagnetic field there is the identification of the Wigner-covariant Dirac observables. This procedure can be applied also to Yang–Mills fields, but to get the Dirac observables one needs the knowledge of an explicit solution of the Gauss’s law constraints. Then there is the description of relativistic fluids in this framework. In particular a definition of the relativistic micro-canonical ensemble in the Wigner 3-spaces is given and it is shown which equations have to be solved to get a consistent relativistic statistical mechanics.

The post-Minkowskian limit of ADM tetrad gravity in the 3-orthogonal gauges of the non-inertial rest-frames is defined with particles and the electromagnetic field as matter. Then, the post-Newtonian expansion of the post-Minkowskian linearization is studied. For binaries, the results are compatible with the standard one in harmonic gauges. However, there is the new result that a non-local version of the inertial gauge variable York time may explain many of the experimental data giving rise to the existence of dark matter, which would be reduced to a relativistic inertial effect to be treated by means of relativistic celestial metrology.

Given an isolated system of either free or interacting particles and the associated realization of the ten conserved Poincaré generators its total conserved time-like 4-momentum defines its inertial rest-frame as the 3+1 splitting whose space-like 3-spaces (named Wigner 3-spaces) are orthogonal to it and whose inertial observer is the Fokker–Pryce 4-center of inertia. There is a discussion of the problem of the relativistic center of mass based on the fact that the 4-center functions “only” of the Poincaré generators of the isolated system are the following three non-local quantities: the non-canonical covariant Fokker–Pryce 4-center of inertia, the canonical non-covariant Newton–Wigner 4-center of mass and the non-canonical non-covariant Mőller 4-center of energy. At the Hamiltonian level one is able to express the canonical world-lines of the particles and their momenta in terms of the Jacobi variables of the external Newton–Wigner center of mass (a non-local non-covariant non-measurable quantity) and of Wigner-covariant relative 3-coordinates and 3-momenta inside the Wigner 3-spaces. This solves the problem of the elimination of relative times in relativistic bound states and to formulate a consistent Wigner-covariant relativistic quantum mechanics of point particles. The non-relativistic limit gives the Hamilton–Jacobi description of the system after the separation of Newtonian center of mass. Finally there is the definition of the non-inertial rest-frames whose 3-spaces are orthogonal to the total 4-momentum of the isolated system at spatial infinity.