I would like to reproduce here part of an exchange with Willem Marinus de Muynck : 

In the objective of my project "Observational reference frames and electromagnetism" on ResearcGate, I wrote:

> 

It may be considered that the first part of this objective is already realized within the framework of classical kinematics, but this solution is incompatible with the second part of the objective and the properties of the fields generated by the electic charges in motion, even uniform translational motions in galilean reference frame. Furthermore, the Lorentz transformation is a partial solution to this first part of my objective but its usual generalization, I speak of the theory of general relativity, is no longer compatible. 

The kinematics that allows to realize the second part of my project is the one that I propose in my document and it is an extension of special relativity when we consider only the strict applications of the Lorentz transformations (without any extrapolation on the calculation of the proper time of a body which is accelerated in relation to a galilean reference frame). You wrote >. In fact my calculations were motivated by the belief that all the mathematical consequences of this idea of Einstein have not been established. Please see the first comment I made on my ResearcGate document.

In 2012, I obtained the general expression of the fields generated by an electric charge whose motion is arbitrary, in other words a generalization of the electromagnetic force of Lorentz which is a postulate in physics. The 2012 document that contains this is available in my project. 

Thus, on the basis of Einstein's interpretation of the Lorentz transformation, my preoccupation was to know whether one can consider that in a galilean reference frame:

(1) An electric point charge generates a static electric field when it is constantly at rest, a variable electric field and a variable magnetic field when it is in uniform motion (whatever its velocity), and more complex components when it is accelerated (possibly a component of a gravitational nature). 

(2) A gravitational point charge generates a static gravitational field when it is constantly at rest, and a component of magnetic nature when it is in uniform motion. 

You wrote: > 

Your question does not make sense in a curved spacetime of general relativity, and even in a flat spacetime it has meaning only if both observers have rigorously inertial motions. Indeed, the terms "being in motion" or "being constantly at rest" with respect to a reference object have no meaning in a spacetime that is not rigorously flat because the parallel transport of the four-velocity from the position of a body to the reference object will depend on the chosen path. 

In the framework of the kinematics that I propose, there is a family of three-dimensional Euclidean spaces which are in relative motions. If each of these observational reference frames uses a cartesian and rectangular coordinate system to describe the spatial positions, And a system of clocks constantly at rest and initialized according to the Einstein procedure (they will necessarily remain synchronized) to assign a date to each event, then the transformations between these defined four-dimensional coordinate systems are the ones I propose. 

Because I am not yet ready to study gravitation, I will answer your question in the case of a moving particle in a uniform electric field (which can be approximately realized between the plates of a capacitor).

(a) If the capacitor plates are constantly at rest in the study observational frame of reference, then we know how to calculate the trajectory of a body which is subjected to a constant force. 

(b) If the plates of the capacitors move with a constant velocity in the study observational frame of reference, then it is sufficient to apply the Lorentz transformation to the solution of (a). The Lorentz transformation is contained in those which I propose.

(c) If the motion of the capacitor plates is arbitrary in the study observational frame of reference, then the arbitrary transformations I have obtained must be applied. These arbitrary transformations are quantified in a certain sense (their dependence on the time parameter is not as arbitrary as the notions of instantaneous rotational and translational motions of classical kinematics).

Thank you,

Dear Rommel,

Newton then Mach and to a certain extent also Einstein formulated the "nihilist" hypothesys in the conception of what is *in between*  objects with mass (from Quarks to Quasars). Provided that it is not even well defined what objects are, Quantum electrodynamics which is the best tested theory so far, teaches us that  emptiness does not exist, in this view the Mach hypothesys is totally void of any meaning. Mach suggested that universe is determined by masses and their relations, QED and QCD teach us that governing dynamics are strictly related to an active Background, it is all the way around.

Newton looked for an absolute reference frame which he resonably found in the fixed stars, Mach did give  power to the stars to influence behavior of other "objects" We know that they are sufficiently far to nullify any gravitational or inertial influence.

Regarding that first paragraph you quote, I  would say that truth

is local, nothing to do with distant stars.

Dear Rommel,

Mach, as the other physics of early 1900 [including Einstein], though, had rather strange imaginations about Matter, space, time and spacetime notions/phenomena and his “conjecture” has no relations to the reality.

Material objects mutually interact, of course, but, for example, inertial and gravitational masses of concrete objects are just properties of this object and practically don’t depend on other objects.

The effects of other objects are very small, and are

– gravitational mass defects of bodies that compose some gravitationally interacting system, e.g. a planet system, but , for example in Sun system planets’ mass gravitational defects are near GMSmp/Rc2, where MS is the Sun/s mass, mp is a planet’s mass,  R is the distance “a planet - Sun”, i.e. they in very many orders are lesser then “own” [i.e. that are if there is no other objects]  planets’ masses in every case. It is quite evident that gravitational mass defects that are caused by gravity of some stars are practically zero.

-  the inertial masses are also a property of  concrete bodies that appear because of all/every particles that constitute a body are some 4D gyroscopes, which resist to attempts to change their 4D rotation rates, which [the rates] cannot by impacted practically  by “all rest matter in Matter”, excluding very small part of particles in Matter that were accelerated to near speed of light speeds.

Cheers

Thanks you Sergey.

This excellent article expands on Mach's principle in some amazing detail.

https://www.americanscientist.org/article/the-forgotten-mystery-of-inertia

Dear Mikhail Vasiliev,

Thank you for this link. I emphasize this remarkable passage: