In one of the letters written to the Infeld group in Warsaw Einstein wrote:
"A new manner of thinking is essential if humankind is to survive."

  

GALILEAN RELATIVITY AND NEWTON'S MODEL.

Pre Galilean, Aristotelian Physics considered immobility as steady
state of a body on which no force was acting. Under action of force
the body moved at speed proportional to the force. When force ceased
to act, the body returned to immobility. It may look strange to us,
but this model was universally accepted during 2000 years until
Galileo discovered that the steady state of a body is not immobility,
but uniform movement at constant (in value and direction) speed: 
A body on which no forces act moves at constant speed in an unchanged 
Euclidian direction. Galileo called such a body associated with 
concrete or virtual Observer an "Inertial Referential" (IR). This 
definition excludes absolute movement: all IR's move relatively to 
one another at constant speeds and none is in any way privileged.

Question arose if concept and laws describing mechanical and dynamic 
phenomena in some IR keep their validity transformed to other IR's 
and, if they do, than by being invariant or covariant. (A construct
is "invariant upon a transformation" when after being transformed it
stays unchanged; it is "covariant" when upon the transformation it
undergoes linear modification entirely determined by the 
transformation. This notion of "covariance" should not be confused
with "vector covariance" defined in "Vectors".) 
Invariant physical constructs are considered as absolute within the 
concerned Model. Relative constructs are Covariant upon the 
Transformation of the concerned Relativity Model. Galilean Relativity 
and Newton's Model are partially based upon an absolute construct, 
the Galilean #Space.

Galilean #Space is in contradiction to usual commonplaces, four
dimensional: an event is determined by 1 Time and 3 Space values.
It's indeed a TimeSpace, a 4D #Space encompassing two sub-#spaces of
our Raw View: 

-the 1 dimensional Time 
-the 3 dimensional euclidean Space. 

The confusion attributing "TimeSpace" to Einstein is due to Galilean 
Time and Space being incommensurable or Affine ("Space. Affine and 
Metric"), while Einstein's 4D continuum, as we shall see in the next 
chapter, is the Metric "LighttimeSpace" where the Galilean component 
"Time" is replaced by "Lighttime" having the measure of distance 
commensurable with the components of Space. 

Incommensurability of Galilean Time and Space excludes the notion of 
Time-Space Interval and, consequently, of Transformation involving 
the entire Galilean TimeSpace. On the other hand, constancy of speed 
among IR's excludes Rotation and we are left with two Translations,
one in Time and one in Space, which accounts for Galilean Relativity
being sometimes called "Translational Relativity" as opposed to the
"Rotational Relativity" of Einstein, which latter term we justify in 
the next chapter.

Given two IR's: R(O,X,Y,Z,T) and r(o,x,y,z,t) moving with respect to 
one another along the common axe X-x at speed V the Time and Space 
Translations from R to r are respectively:
t = T;   x = X + a + VT;                    [1]
where a is the distance of o, origine of r from O at T = 0.

[1] is called the Galilean Transformation. It supports invariant or
absolute Time and distance, i.e. Space. Indeed, distance DX in R
DX = X2 - X1, transforms into Dx in r: 
Dx = x2 -x1 = (X2 + a + VT) - (X1 + a + VT) = X2 - X1 = DX

Intervals of Time and Space are Invariant under the Galilean 
Transformation, or in epistemological terms, 
GALILEAN TIME AND SPACE ARE ABSOLUTE. 

A Detector moving along X/x at VDx with respect ot r moves at 
VDX = VDx + V with respect to R:

Speed is additively Covariant under Galilean Transformation, which,
after a few simple mathematical operations results in the rule:

MECHANICS AND DYNAMICS ARE COVARIANT UNDER GALILEAN TRANSFORMATION

Galilean Relativity reached its apogee in Newton's Model. In order to
investigate it from the point of view of RD we shall start by trying
to assign it to one of the basic classes: Continuous or Discrete.

At first glance the answer seems obvious: Discrete. At Newton's time
and for centuries afterwards the fabric of Universe was considered to 
be corpuscular. Newtonian #Space was compactly filled with corpuscules
acting directly on one another, like billiard balls thus supporting
Mechanics, Local Action and Causality. However, such corpuscules have
never been observed, nor was there even any hint to their conceptual
features such as size or mass. They have appearance of an ad hoc 
"crutch", of a hypothetical corpuscular "matter substance", introduced 
in order to formulate a microscopic model based intuitively upon 
macroscopic billiard. Now, hypothetical or not, this corpuscular 
"matter" accounted  well for the laws of Mechanics. 

However, serious  difficulties arose with Optics. In agreement with 
the postulate of discrete Universe, light was mapped as a flow of 
corpuscules thus introducing an additional ad hoc crutch, a 
"luminiferous" corpuscular substance. It supported rectilinear 
progression and reflection of light, as well as refraction (with help 
of hypothetical interactions between "material" and "luminiferous" 
substances).
In order to account for dispersion, or separation of white light into 
spectral components, it was necessary to split the "luminiferous"
substance into several substances, one  per colour, propagating for 
some reason each at different speed through various "material" 
substances. One accumulated uncountable ad hoc crutches, all 
hypothetical and not supported empirically. One grew a habit to add
for each new problem some new "substance" which indeed looks more like 
tautologies than solutions.

Finally, realizing that no conceivable new "substances" could possibly 
account for such phenomena as diffraction and interference patterns, 
physicists abandoned the corpuscular theory of light to the advantage 
of wave theory which, in conjunction with the discrete view of 
Universe, ushered in the Aether concept which dominated Physics till 
the beginning of the 20th century. The corpuscular theory fell into 
oblivion until it apparently resurrected in the Quantum Theory, at an 
incomparably higher level of complexity and in a form having nothing 
to do with rudimentary mechanistic thinking. 

Leaving for the moment Optics we shall move to Newton's Dynamics. 
It's based upon 3 Laws of Motion: 

1.In absence of force any object moves at constant speed.
2.Accelaration of an object is directly proportional to force acting
  on it and inversely to its mass.
3.For every action, there is an equal and opposite reaction.

Let's note that only 2. and 3. are original. 1. is a redefinition of
Galilean Inertial Referentials.

On the other hand, a fundamental concept is missing, that of kinetic
energy. It has been conceived and formulated at age of 16 by Pierrette 
Paulze - Lavoisier who, astounding as it may seem, fell into almost 
complete oblivion. Maybe not so astonishing after all: she was a woman; 
she did not belong to any Academy, none would accept a woman in 17th 
Century; and married to Lavoisier she was overshadowed by him. Not by 
his fault. She married him at age of 13 and from the first day became 
his full-fledged collaborator, which he emphasized at every occasion 
so that the rare cognoscenti who heard about her call them "Father and 
Mother of modern Chemistry". But there it is, hardly anybody heard 
about her.

Thus, if we consider Newton as founder of modern Physics and perhaps 
the greatest scientist of history, it's not due to his unconvincing 
Optics, nor to his Dynamics, certainly great, but incomplete and 
amended by a nearly unknown 16 years old girl. His outstanding 
significance is named "Gravitation", to which we shall further refer 
as to "Newton's Model".

It boils down to the following Law of Gravitation:

Attracting gravity force between two masses F(r)=G(m1*m2)/r^2 where 
G: gravity constant, m1,m2: respective masses, r: distance.

Let's start with our basic question: Is Newton's Model Continuous or
Discrete? On the face of it it seems Discrete, according to the
contemporary view of Universe and to his own explicit statements.
However, this assumption leads immediately to paradoxes:

The law of Gravitation assumes Action at Distance thus clearly
contradicting all principles of the corpuscular billiard-like view as 
well as the Local Action and Causality.   [3]

#Space (distance) determines the Gravity Force, without this Force 
impacting in any way the #Space, which contradicts the action/reaction
principle.    [4]

Paradox [3], Action at Distance in a Discrete Model, is not just a 
contradictory detail, but an essential inconsistency which casts doubt 
upon the very base of the theory. Consequently, question arises if
Newton's Model was not de facto Continuous, perhaps implicitly, 
against author's own explicit belief in the universally accepted 
discrete fabric of Universe. 

Indeed, taking the formula of gravitational Force: F(r)=G(m1*m2)/r^2, 
we may combine G*m1/r^2 into g(r)=G*m1/r^2 and define "g(r)" as
"Gravity Field in any point r of #Space".  

Then, F(r)=g(r)*m2 and replacing m2 with detector of mass m2=1, we get 
F(r)=g(r) - Gravity Force exerced by Field g(r) on a unit mass in 
every point r of #Space. 

Once we replace corpuscules acting paradoxically at distance, with 
Continuous Field acting locally in all points of #Space, Newton's 
Model becomes Continuous as well as consistent with Local Action and 
Causality; a pertinent foundation of modern Physics, precursor of 
Einstein's Relativity and of the Quantum Field Theory.

In this light, the paradox [4] involves Field rather than Force. 
Indeed, #Space being a Continuum and Force a point event, the 
original version of [4] simply does not make sense. In the Field 
version, on the contrary, both Field and #Space being Continua, 
it becomes a meaningful albeit yet unsolved inconsistency. 

Now, Field is a factual construct and #Space an abstraction, so by 
virtue of the Principle of Preponderance of Facts, it's Newtonian 
#Space that is primarily called into question by [4] and should be
reexamined in the first place. We shall see below that [4] has been 
indeed considered and solved in that sense by the General Relativity. 

Finally, let's recall that a major part in the acceptance of Newton's
ideas was due to Euler's re-writing of his entire family of ideas in 
the language of the calculus, which Newton had done so much to invent 
but largely left out of the Principia.

And who says calculus, talks about Continuity. This argument seals our 
conviction that Newton's Model was, at least implicitly, the first 
pioneering Continuous Field Model which founded and determined the 
subsequent progress of Physics till our own days.