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."

  

SPECIAL AND GENERAL RELATIVITY

Axioms, derivation and internals of Einstein's or Extended Relativity 
(ER) are shown in "Special Relativity" and "General Relativity". 
Here we shall look at it from outside, from the epistemological point 
of view concerning its context, foundations and impact.  

The impact of ER may be compared with that of Heliocentrism and
Evolution, similarly revolutionising the Weltanschaung and similarly 
attracting attacks of dogmatic obscurantists. At the time being they 
burned Giordano Bruno; at our own days the exorcists graduating from 
the Pontifical Regina Apostolorum University of Vatican, together with 
other Creationists and Intelligent Designers infiltrate "scientific" 
education and endeavor to resurrect geocentric cosmology founded in 
sacrosanct verbatim interpretation of Genesis. 

If by its impact ER is one of two or three most important Models, it 
stands alone by the simplicity of its epistemological foundation. 
The intellectually deepest, most revolutionary and complex Model is 
based on the simplest postulate ever formulated, that of Equivalence 
of Inertial and Gravitational Mass. It was not even totally new: 
equality of both Masses goes back to Galileo's Experiment, but has 
always been considered as coincidence. Replacing this Equality with 
Equivalence Einstein created new Physics, indeed new Science which 
implies in turn new Epistemology.

ER has of course several other postulated rules. We shall consider 
them in two steps of ER, the Special and the General Relativity, 
always from the epistemological view point and remembering that they 
all culminate in the Equivalence Postulate and its impact on Physics.

CONTEXT

The Michelson-Morley experiment sounded the knell of Mecanistic 
Dogmatism and of Aether phantasmagoric crutch. Physicists forgot to 
walk by their own means and Physics found itself void of foundations 
and thus meaningless. However, the void was not complete. Aether left 
in its ruins some most ingenious recipes hanging for the moment in 
emptiness, but ready for constructive use, once some firm ground found 
on which to lean. To mention two of them:

Maxwell described the structure of Electromagnetic Field, has shown 
that light is electromagnetic wave and determined analytically its 
speed C getting exact agreement with experimental data. The 
fundamental question "speed with respect to what?" stayed however 
unanswered and Maxwell's equations stayed a genial mathematical recipe 
void of physical sense.  Galilean Relativity suggested and Michelson-
Morley proved, that C is invariant with respect to all IR's (Inertial 
Referentials). But then, Galilean additive Transformation of speeds 
did not hold.

Lorentz conceived an ingenious recipe, called Lorentz Transformation 
(L-Tr) which replaced Galilean Transformation and supported the 
invariance of C. But he did it as last effort to save the Aether and 
with Aether's collapse L-Tr lost the physical foundation and sense.

In the meantime Mathematics made great progress. Minkowski, 
Lobatschevski, Rieman defined pseudo-euclidean, hyperbolic and 
parabolic #Spaces. Levi-Civita created Tensor Calculus, which allowed 
to precise metrics of non-euclidean #Spaces, to deal with their curved 
coordinates and to formulate physical laws in covariant form.

SPECIAL RELATIVITY (SR)

In this context, after the collapse of Aether, Einstein dared to throw 
away the crutches and to set out walking by himself. We shall try to 
retrace some of his most eventful walks. Having scrapped Aether and 
mechanistic dogmatism, Physic stayed without foundations and without 
a Model capable to represent Light (meant as Electromagnetic Field). 
The astounding effectivity of Newton's Model and Galilean Relativity
in representing mechanics and dynamics from local to cosmic scale 
pleaded for extending them over Electromagnetism and creating 
"Extended Relativity" (ER) rather than starting all from the scratch. 

This extention posed two essential problems:

1.With Aether we have lost the support of Field and were left with two 
apparently disparate Continua, Field and #Space without any physical 
relation between them. Therefore we had to look for some new base
which would reconcile Field and #Space without any intermediary agent. 
As we saw in "Galileo and Newton", Field is a factual construct and 
#Space an abstraction, so by virtue of the Principle of Preponderance 
of Facts, it's Newtonian #Space that had to be primarily reexamined.  
We shall start therefore by researching some new #Space which will 
hopefully found some Model supporting in IR's the covariance of 
Electromagnetism along with that of Mechanics and Dynamics. 

2.Incompatibility of the Galilean Transformation with invariance of C 
in IR's. We have seen above that, trying vainly to save Aether, 
Lorentz conceived his famous L-Tr which supports the invariance of C. 
However, deprived of foundation after Aether's collapse L-Tr stayed an 
ingenious mathematical recipe void of physical sense.            

Consequently our new MS, besides reconciling Field and #Space has to 
support some Lorentz-like, or if possible, the already existing L-Tr, 
providing it with physical foundation and sense.

The derivation of this MS called Special Relativity (SR) is described
with mathematical details in the chapter "Lorentz Transformations", 
which requires basic knowledge of Tensor Calculus and its application 
to Metric #Spaces. Here, we shall concentrate on the intuitive 
creativity in Einstein's train of thought, limiting mathematic tools 
to the Theorem of Pitagoras.

Let's start by recalling the concept of Elementary Displacement in 
the Euclidean #Space of the Galilean Relativity. Let RR be an IR 
with center in O and orthogonal coordinates X1,X2,X3.
 
RR: O,X1,X2,X3

For a point distant from O by Elementary Displacement dS we have by 
virtue of the Theorem of Pitagoras:
 
dS^2 = dX1^2 + dX2^2 + dX3^2
 
Now, let rr be another Inertial Referential rr: o,x1,x2,x3 with axes 
x1,x2,x3 respectively parallel to X1,X2,X3, moving with respect to RR 
along X1/x1 at speed V.
 
Galilean Transformation conserves the distance ("Galileo and Newton"), 
so that dX1 transforms from RR to rr invariantly: dx1 = dX1
Axes x2,x3 are not affected by the movement: dx2=dX2, dx3=dX3
and finally:
ds^2 = dx1^2 + dx2^2 + dx3^2 = dX1^2 + dX2^2 + dX3^2 = dS^2
 
ds^2 = dS^2: dS is invariant under Galilean Transformation
 
We shall call dS^2 = dX1^2 + dX2^2 + dX3^2 the "Quadratic Form"
characterizing Euclidean #Space and note that:
 
Quadratic Form of Euclidean #Space is invariant under the Galilean 
Transformation.

Derivation of SR Model #Space boils down to research of a Quadratic 
Form characterising a #Space replacing Galilean Transformation with 
another, supporting invariance of C, if possible with the already 
existing Lorentz Transformation.
 
We shall base this research upon comparison of sound wave and light 
wave emitted in the center O of RR.  Calling the sound speed U and 
supposing that sound emission starts at t=0, O will find itself at 
time T in the center of the sound sphere of radius R=UT. o of rr will 
find itself at T in the point seen from O as VT (V being the relative 
speed of rr with respect to RR), thus not in the center of the sphere, 
but at the distance (U-V)T from sphere's intersection point with x1, 
"heading" in the direction of V and at (U+V)T from the opposite 
"trailing" point. At the limit of V=U o will find itself confused with 
the "heading" point and over this limit, at V>U will become supersonic 
and find itself outside of the sphere, preceding the "heading" point. 
The speed of sound will appear to o as depending on its speed V with 
respect to the source O, slower in the V direction, faster in the 
opposite.

The essential difference between sound and light consists in light
speed C being, unlike that of sound, invariant with respect to V. In
other terms, both O and o will find themselves always in the center 
of the light sphere emitted in O. 

Radius R of the spere seen from RR will experience Lorentz Contraction
when seen from rr so that o will find itself in the middle of the 
sphere of radius r<R, smaller than the sphere seen by O. 
The clock in RR observed from rr will tick slower than observed from 
RR, so that o will count less ticks during light propagation. The 
sphere seen by o would have less time to extend, which accounts for 
its smaller dimension. At the limit, for V=C, the clock seen from rr
will stop ticking and the radius seen by o will be r=0. The spere 
seen from rr will stop to extend.

This construction exceeds human faculty of visualization but has been 
verified experimentally and if we want to investigate it, we must 
map it into some pertinent abstract Model #Space (MS). Let's look for 
a Quadratic Form defining this MS.                                              
 
Supposing now that the source at O of RR emits light, O will find 
itself after elementary time increment dT in the center of light 
sphere of radius dR = CdT. By virtue of the Pitagoras theorem for each 
point of the sphere we have:
dR^2 = dX1^2 + dX2^2 + dX3^2  or 
(CdT)^2 = dX1^2 + dX2^2 + dX3^2          [5]

Now, due to the invariance of C, o of rr will find itself also in the 
center of light sphere:
(Cdt)^2 = dx1^2 + dx2^2 + dx3^2          [6] 

Transfering CdT and Cdt to the right hand side of [5],[6]:
dX1^2+dX2^2+dX3^2-(CdT)^2 = dx1^2+dx2^2+dx3^2-(Cdt)^2 = 0

In order to maintain notation similar to that of the Quadratic Form of 
Euclidean #Space we shall write "-(CdT)^2" as "X4^2" and "-(Cdt)^2" as 
"x4^2" and say that the Quadratic Form of the SR #Space is invariantly 
equal zero:

dS^2=dX1^2+dX2^2+dX3^2+dX4^2 = 
ds^2=dx1^2+dx2^2+dx3^2+dx4^2=0     [7]

[7] defines the 4 dimensional SR #Space having 3 real coordinates 
x1,2,3 and one imaginary x4=sqrt(-(Cdt)^2) = iCdt  LightTime (LT) 
(where i=sqrt(-1)). Expressing these coordinates in units we get the 
"signature" of the #Space: 1,1,1,i

#Space of this signature is known as Minkowski #Space (#MinSp). The 
particular instance of #MinSp whose imaginary x4 is associated with 
physical LT and x1,2,3 with physical space may be called "SR #Space".

Let's note that SR #Space is Metric its all 4 dimensions including LT 
having the measure of distance. Thus it transforms covariantly in all 
4 dimensions. In other terms SR #Space, unlike the absolute Newtonian, 
is covariant , i.e. relative. under Lorentz Transformations. In 
particular, covariance of x4=-iCt, with invariant C implies covariance 
and relativity of time.

Now, a transformation may consist of translation, rotation or a 
combination of both. However, we have seen that translation would make 
C depending on the relative speed of source and observer, so it's 
excluded by the Axiom of C invariance and we are left with rotation: 
for rr moving along x1, the transformation consists in rotation in the 
plane x1 / x4 (or x1 / iCdt). Let's mention without going into details 
that x4 being imaginary, the plane is complex and our rotation is 
called "pseudo-rotation". This pseudo-rotation is equivalent with 
Lorentz Transformation (L-Tr) (see the corresponding chapter). 
SR is therefore sometimes called "Rotational Relativity" as compared 
with the Galilean "Translational Relativity".

SR marked the demise of dogmatic, noumenalistic Science and Philosophy
laying down the outset of axiomatic, rational Science and rational
phenomenalistic Weltanschaung. We shall return to the philosophical
impact in conclusions of this chapter, after having reviewed the
General Relativity. Now we shall briefly recollect some essential
findings and restrictions of SR. ("V" will mean the speed of observed 
Referential relatively to the observer.) 

Findings:
1.Phenomenal nature of Field observable as Force it exerts on 
  detectors.
2.Phenomenal Equivalence of Field and #Space resulting from dropping
  all physical supports of Field else than #Space itself.
3.Phenomenal nature of #Space; corollary of 2. 
4.Distance contraction and Moment dilation increasing with V.
5.Phenomenal Equivalence of Mass and Energy.
6.Increse of Mass (and Energy, corollary of 5.) with V.
7.Phenomenal Equivalence of Field, Energy (and Mass, corollary of 5).

Restrictions:
The French call SR "La Relativit� Restreinte" (Restricted Relativity),
which describes it quite well in our opinion: SR, while extended over
Electromagnetism, maintains the Galilean restriction to Inertial
Referentials. However, no strictly inertial Referential has ever been
observed and the Phenomenon #Space seems to encompass exclusively 
NIR's (Non Inertial Referentials). Still, many can be reasonably 
approximated as IR's and the SR may be considered as an acceptable, 
approximate Model of a vast range of Phenomena. Nevertheless, as one 
could not a priori except the possibility of other Phenomena being 
strictly incompatible with IR's, it appeared vital to extend the SR 
over NIR's, i.e. over the total Phenomenal #Space.

We shall now review this extension of the SR over NIR's, known as the 
General Relativity

GENERAL RELATIVITY (GR)

Special Relativity extended Galilean Relativity over Electromagnetism 
while preserving the Galilean validity scope, namely the Inertial 
Referentials (IR's). However, no IR has ever been observed and the 
Phenomenal #Space seems to encompass exclusively NIR's (Non Inertial 
Referentials) accelerating with respect to one another, or, more 
generally, experiencing inertial or gravitational Field ("Redefinition 
of NIR's" below). Some NIR's in which these effects are weak may be 
approximated as IR's, but no strict IR has ever been observed. 
Therefore, Einstein conceived the idea of General Relativity (GR) 
which would extend the scope of SR over NIR's, and hence over the 
entire Phenomenal #Space.

GR is founded upon two surprisingly simple principles; Reducibility 
to SR and Axiom of Equivalence.

-Reducibility to SR: Just as SR is an extension of Newton Model and 
reduces to it for slow speeds, GR was postulated as an extension of SR 
reducible to it locally for weak Field areas of NIR's.

-Axiom of Equivalence: Since Galileo's Experiment, gravitational and 
inertial Mass were known to be equal, but it has been attributed to 
coincidence. Einstein elevated this equality to equivalence.

Never, such a deep scientific, intellectual and philosophical 
revolution has been founded upon a simpler Postulate.

ROTATING DISK

And the actual derivation was just as simple: it boils down to a 
facile mental experiment known as "Rotating Disk" (see "Derivation of 
General Relativity 1 & 2" for details). Given on an equivalent 
stationary Disk a circle of radius R and circumference S, Euclidean 
geometry tells us that S = 2 pi R. Yet, on the Rotating Disk we'll
measure R and S with elementary physical rods ("Einstein's Covering
Principle" in Glossary). Upon R the rods are perpendicular to the 
sense of rotation, are not affected by it and R conserves its 
Euclidean length. However, upon S they move in the sense of the 
rotaion, and by virtue of Reducibility to SR experience the Lorentz
Contraction. S will need more contracted rods to be covered than in
Euclidean #Space, i.e. ils length will be measured as longer. On the 
Rotating Disk we have S > 2 pi R. This formula defines Lobachevsky or
hyperbolic curved #Space suggesting that rotation determines geometry 
and #Space.

And that's it, GR is derived. There may follow a lot of detailed 
mathematical expressions and physical interpretations, but they all 
rest on the "Rotating Disk".

FIELD and #SPACE

The NIR Rotating Disk suggested that rotation determines #Space. 
However, supposing that it rotates with respect to some IR, Relativity 
tells us that it is equivalent with this IR rotating with respect to 
our NIR. Now, IR's geometry is Euclidean and there is no reason for it
to change, just because some NIR rotates somewhere. In other terms 
the old Galilean definition of IR based upon relative movement is no
more sufficient for distinguishing IR's and NIR's and we have to look 
for some other criterion. Now, a detector placed on the circumference
S of the Rotating Disk experiences centrifugal force directed towards 
disk's exterior: our NIR is a seat of Inertial Field. On the contrary,
a detector placed in IR will detect no centrifugal force and no
Inertial Field.

REDEFINITION of IR and NIR

Thus Inertial Field is the criterion allowing to distinguish between 
IRs and NIRs and it is the Inertial Field engendered by rotation that 
determines #Space, rather than rotation itself. 
As, by virtue of the Equivalence Principle, Inertial and Gravitational
Fields are equivalent we may consider Field (indiscriminately Inertial 
or Gravitational),as the criterion allowing to distinguish between IRs 
and NIRs.

Finally, we may formulate the relation between Field and #Space as 
follows:

-Field determines #Space whose curvature increases with increasing 
 Field's strength.

-#Space determines Field whose strength is function of distance (from
 gravitational "mass" singularity, from rotation center, etc.)

-#Space and Field are phenomenally equivalent due to this mutual,
 reciprocal determination. This equivalence solved Newton's #Space-
 Field Paradox: Newtonian Field was determined by distance (#Space), 
 but had no impact on #Space, which contradicted the Action-Reaction 
 Principle. Einstein considered solving this Paradox as the most 
 gratifying of his achievements because he admired and venerated 
 Newton as his Master.

A FEW WORDS on NON-EUCLIDEAN #SPACES

We are all familiar with "S = 2 pi R" in the "flat" Euclidean #Space. 
The "S < 2 pi R" of Riemann or parabolic #Space may be easily 
visualized and verified with pieces of string and circles drawn on any 
sphere.

Euclidean #Space may be considered as a limit case of Riemannian sphere
getting "infinitely big", its radius tending to infinity and the 
curvature of the increasing sphere tending towards zero, or "flatness".

Hyperbolic #Space may be imagined as going past this limit, where our
surface gets "superflat", or its curvature gets negative.

PHILOSOPHICAL IMPLICATIONS OF EXTENDED RELATIVITY

SR marked the demise of dogmatic and the outset of axiomatic, rational
and falsifiable Science. One may object that its predecessor, the 
Aether Theory was already an Axiomatic Theory founded in the Axiom of
Aether.

It's true enough, but let's keep in mind that, besides its own 
"internal" Axioms, a Theory has usually other, "external" foundations.
When founded in external Theory, it accepts the external assertions 
as its own Axioms. Internal Axioms of the founded Theory must be 
compatible with the external assertions and often, as in the case 
of the Aether Theory, instantiate them.

Now, the Aether Theory, as the whole pre-Einsteinian Physics, was 
founded in dogmatic belief in Physics dealing directly with 
transcendental, noumenalistic "Reality", having the mechanistic 
structure of "billiard balls". Light has been considered as wave and 
in mechanistic world waves must have some corpuscular fluid support, 
like air or water. Light propagating through vacuum as well as through 
transparent "matter", its mechanistic fluid support had to be 
universally present.

Hence the Axiom of Aether, a universal mechanistic support of light,
which was not only compatible with the metaphysical mechanistic Dogma,
but actually instantiated it.

Consequently, Aether was a very particular concept, Axiom and Dogma at 
the same time. It was an Axiom within the Aether Theory, an ultimate 
premise of a perfectly legitimate inferencing structure exposed to 
inductive verification / falsification. Yet, it was also an instance 
of the mechanistic Dogma. As Axiom it has been falsified by the MM 
Experiment, but as Dogma it opposed unusual resistance to refutation. 
Greatest scientists endeavored to save it at the cost of superhuman 
efforts. Lorentz for instance derived his famous Transformation, a 
purely mathematical recipe compatible with the invariance of C, but 
void of physical sense in the mechanistic Newtonian #Space, affine in 
x1,2,3/t planes, ignoring in them the notion of distance and, 
consequently of any transformation. Facts contradicted Aether, but 
Lorentz believed in it and endeavored to save it at any price.      

One recalls Hegel who, informed that astronomical observations
contradict his aprioristic drivel, that facts disagree with his
views, answered: "So much the worse for the facts". 

However, Lorentz was not a foolish conceited metaphysical preacher,
but a great sincere scientist trapped by the universal belief in the 
apparently unique Foundation of Physics, the transcendental
mechanistic view.

It shows how deeply rooted and how difficult to eradicate is a Dogma,
once it takes over a domain of human cognition. 

It stresses also the courage it took to get rid of the Dogma with all
involved crutches and to set out walking by oneself over the swampy 
land lacking any founding bedrock. Physics deprived of mechanistic
foundation seemed to hang in the air; light waves propagating without
support defied the reason.

In that situation Einstein conceived new foundation of Physics 
providing the so far purely mathematical constructs Field and #Space 
with astounding physical, Phenomenal sense and equivalence. Physics
and, indeed Science, got new, purely Phenomenological Foundations.
All Concepts and Conceptual Systems fell from the inaccessible height
of the Transcendental Olympus of Apriori into the controllable domain
of empiric, phenomenal adequacy.

These new, Phenomenological Foundations of Science revolutionised the
Epistemology, which is not surprising, Epistemology dealing mainly, if
not exclusively with Foundations of Science and Cognition. However, 
their impact went still deeper, becoming a cornerstone of current
Ontology and Weltanschauung.

It's not by chance, that the vade mecum of current Ontology, Sartre's 
"L'�tre et le n�ant" carries the subtitle "Essai d'Ontologie 
Ph�nom�nologique", It's not by chance, that its first chapter "L'Id�e 
du Ph�nom�ne" starts with the phrase:

"La pens�e moderne a r�alis� un progr�s consid�rable en r�duisant 
l'existant � la s�rie des apparitions qui le manifestent." -
"The modern thought realized a considerable progress in reducing the 
existent to the series of appearances that manifest it".

This "modern thought" is that of Einstein's Extended Relativity.