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 VU 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 Relativité 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 Special 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'être et le néant" carries the subtitle "Essai d'Ontologie Phénoménologique", It's not by chance, that its first chapter "L'Idée du Phénomène" starts with the phrase:

"La pensée moderne a réalisé un progrès considérable en réduisant l'existant à la série 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 Relativity.