Hello,

John D. Norton (1993). General covariance and the foundations of general relativity: eight decades of dispute, Rep. Prog. Phys., 56, pp. 794, pp. 835-7.
"...the question of precisely what Einstein discovered remains unanswered, for we have no consensus over the exact nature of the theory's fondations. Is this the theory that extends the relativity of motion from inertial motion to accelerated motion, as Einstein contended ? Or is it just a theory that treats gravitation geometrically in the spacetime setting ? ... Of special importance for our purposes is that each frame of reference has a definite state of motion at each event of spacetime"

My document present the theory that extends the relativity of motion from inertial motion to accelerated motion and I hope you will try to learn more about its contents. It is built around the notion of observational frame of reference.

There are those who argue that a frame of reference is a coordinate system : they say that an observational frame of reference is a coordinate system. They claim that an entity continuously motionless in an observational frame of reference does not have a very specific trajectory in a coordinate system defined on a limited region of space-time.

There are those who think that a frame of reference is not a coordinate system. "...an observer in an observational frame of reference can choose to employ any coordinate system to describe observations made from that frame of reference. A change in the choice of this coordinate system does not change an observer's state of motion, and so does not entail a change in the observer's observational frame of reference". The notion of observational frame of reference has no meaning in general relativity.

If we know the path of an experimenter P in a cartesian coordinate system, we know its frame of reference only if we know the trajectories (in this coordinate system) of entities which are continuously stationary for P :
* In classical physics, all these trajectories are described by two functions velocity vectors : one translational and one rotational.
* In a Minkowski space, in some situations where P moves with constant velocity, it is argued that a particle which is continuously immobile for P moves with the same constant speed. But we can not say anything when the speed of P is not constant.

The mathematics of my document allow to clearly distinguish the concepts of "frame of reference" and "coordinate systems".

In the document I have isolated the three assumptions that are necessary and sufficient to construct the classical kinematics. They define the concepts of velocity vectors and their composition law. They postulate relationships between the choices that can carry out different experimenters.

These assumptions do not allow a covariant formulation of theory of electromagnetism, unlike the kinematics relativistic which is deducted from "postulat 1". This "postulat 1" summarizes interpretation of the special formulas of Lorentz made by Einstein and these Lorentz formulas have been introduced to make the wave equation invariant under a linear transformation of coordinates.

The solution of equation (11) must lead to new physics. This equation is the relativistic version of the purely classical that says that within a frame of reference, the movement of another frame of reference is described by two functions velocity vectors : one translational and one rotational.

To learn how to test the theory we must solve its equations (Equation 11). If you understand the construction of this equation, you will have a new look at the notions of "frames of reference" and "coordinate system".

The only unknown in equation (11) is the function f_{1} which depends on four real variables x_{1}, ... x_{4} (three spatial and one temporal). There is a notation that comes from equation (6) and f_ {1i} is the partial derivative of f_ {1} with respect to x_{i}.

In classical physics, if P moves in a cartesian coordinate system, the movement (in this coordinate system) of an entity that is continuously immobile in the observational frame of reference of P is described by only two functions velocity vectors: one translational and one rotational.

In this relativistic theory, the velocity components (in this coordinate system) of this entity come from equation (8) and depend only solutions of (11).

It is normal if the solutions of (11) are parameterized by at least an arbitrary function.

In the presentation of classical kinematics at the beginning of the paper, I introduce a new concept which becomes important in my theory : It is the operation that performs the linear combinations of vectors in an observational frame of reference.

Sincerely,