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PLAN OF 0.FIRST AND SECOND ENLIGHTENMENT
0a outline
FIRST ENLIGHTENMENT
0b galilean relativity and newton
0c descartes
0d aether and dogmatic thinking
SECOND ENLIGHTENMENT
0e einstein and extended relativity
0f kant and einstein
Og EPR Paradox
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0E.EINSTEIN AND EXTENDED RELATIVITY
NOTE: neologies and ambiguous terms clarified in GLOSSARY
are marked "[G]".
Axioms, derivation and internals of Einstein's Extended
Relativity (ER) are shown in
DB SPECIAL RELATIVITY
and
DC GENERAL RELATIVITY
Here we shall look at it from outside, from epistemological
point of view concerning its context, foundations and impact.
The impact of ER may be compared with that of Heliocentrism
and Evolution, similarly revolutionizing 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 Designists infiltrate
"scientific" education and endeavor to resurrect geocentric
cosmology founded on 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 of history is based on
the simplest postulate ever formulated, that of Equivalence
of Inertial and Gravitational Mass.
(Actually, "Equivalence" denotes here "Identity", but we
shall use the customary inexact term, always remembering
its rigorous meaning.)
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 implying
in turn new Ontology and Epistemology.
ER has of course several other postulated rules. We shall
consider them in two steps of ER, the Special and the General
Relativity.
CONTEXT
The Michelson-Morley experiment sounded the knell of
Mechanistic Dogmatism and of the Aether's phantasm. Physics,
deprived of its foundations lost momentarily its essential
meaning. 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 reestablished. To mention two of them:
1.Maxwell described the structure of electromagnetic Field,
has shown that light is electromagnetic wave and determined
analytically its speed C getting exact agreement with empiric
data. But speed with respect to what? Apparently, to Aether,
the unique absolute, omnipresent Referential, even if such
absolute Referential wrecks Galilean Relativity. But it was
Aether that got wrecked by the Michelson-Morley experiment.
2.Lorentz conceived an ingenious recipe, the Lorentz
Transformation (L-Tr) which replaced Galilean Transformation
and supported the invariance of C in Aether's context. It was
the last attempt to save the Aether and with it's failure
L-Tr lost its physical foundation and sense.
In the meantime Mathematics made great progress. Rieman and
Lobatschevski conceived parabolic and hyperbolic #Spaces[G].
Levi-Civita developed Tensor Calculus, which allowed to
define 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, unlike other physicists, who tried to mend
their collapsed dogmatic crutches, Einstein dared to throw
them away and to set out walking by himself. We shall try to
retrace some of his most eventful walks.
After the collapse of Aether and of the mechanistic Dogma,
Physic found itself devoid of foundations and lacking a Model
capable to represent Electromagnetism and Light. The
astounding effectivity of Newton's Model and Galilean
Relativity in representing mechanics from local to cosmic
scale pleaded for extending them over Electromagnetism and
creating an "Extended Relativity" (ER) rather than starting
all from the scratch.
This extension posed two essential problems:
1.With Aether Physics has lost the support of Field and was
left with two apparently disparate and unrelated Continua,
Field and #Space. One had therefore to look for some way of
reconciling them. Field being a factual construct and #Space
an abstraction, first to reexamine was Newtonian #Space, by
virtue of the Principle of Preponderance of Facts.
(see 0B GALILEAN RELATIVITY AND NEWTON'S MODEL )
The first step consisted therefore in researching some new
#Space capable of founding a Model supporting in IRs the
covariance of Electromagnetism alongside with that of
Mechanics.
2.Incompatibility of the Galilean Transformation with
invariance of C in IRs. 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 construct void of physical sense.
Consequently our new Abstract #Space (AS)[G], besides
reconciling Field and #Space has to support Transformation
similar to the L-Tr, or if possible identical with it, after
having provided it with physical foundation and sense.
The derivation of this AS called Special Relativity (SR) is
described with mathematical details in the chapter
DBB.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 mathematical tools to the Theorem of Pitagoras and
recalling that by virtue of L-Tr in an IR moving with
respect to an Observer, time appears to the Observer to slow
down and length of physical bodies to shrink in the direction
of the movement.
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 IR rr: o,x1,x2,x3 with axes x1,x2,x3
respectively parallel to X1,X2,X3, moving with respect to RR
in direction X1/x1 at speed V.
As distance is Invariant under Galilean Transformation
( 0B GALILEAN RELATIVITY AND NEWTON'S MODEL ),
dzX1 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 "Characteristic Quadratic Form" of Euclidean #Space
and note that:
Quadratic Form of Euclidean #Space is invariant under
the Galilean Transformation.
Derivation of SR Model boils down to research of a Quadratic
Form characterizing some SR #Space which would replace
Galilean Transformation with one supporting invariance of C,
if possible, with the already existing Lorentz Transformation.
We shall start this research comparing sound and light waves.
Let RR and rr referentials with observers O, o moving
relatively at Vs comparable with S the speed of sound.
At this speed Lorentz transformation is negligible and time
T is invariant for RR and rr. At To=0, O and o coincide and
O emits a sound signal propagating as a sphere. At time T,
O will find himself in the center of sound sphere of radius
ST. At Vs < o will "travel" inside the sphere and be
displaced by TVs from its center O. At Vs=S o will find
himself at the advancing surface of the sphere.
At Vs > S o will "overtake" the sound and find himself
supersonic, outside of the sphere, preceding its advance.
o will find that sphere surface advancing in the direction
of his Vs moves at decreased speed S-Vs and the opposite
surface recedes faster, at S+Vs. Speed of sound depends of
the speed of observer.
Now, let's replace sound with light propagating at C and the
relative speed Vs of RR and rr with subluminal speed Vl at
which Lorentz Transformations are significative. We have to
distinguish TRR, time of RR seen by O of RR and tRR, time of
RR seen by o of rr, the latter flowing slower: tRR < TRR.
Let's suppose that at tRRo=TRRo=0 both referentials RR and rr
coincide spacially and O emits a light signal propagating as
a sphere. O will find himself at his time TRR inside of light
sphere of radius RAD=C*TRR. What will observe o? As light,
unlike sound, is independent of observer's speed Vl, he will
find himself always in the middle of the light sphere.
However, he will see it at his tRR < TRR, of radius
rad=C*tRR, shorter than RAD observed by O: o will find
himself in the center of sphere smaller than one seen
by O. In other terms, o sees what O saw earlier: the same
sphere, but at earlier time, thus smaller. With increasing
Vl the radius of o's sphere will decrease and at the limit
Vl=C tRR will stop flowing and the sphere of o will contrtact
to a point. Actually, it will contract to a singularity
called photon, a "particle" of light, having no stationary
mass and moving by definition at C. It's a discrete "lump"
of continuous Field carrying energy of its wave oscillations
and studied by singularities oriented Quantum Field Theory.
Continuity/Discreteness, Field/Photon Dichotomy illustrates
the P-Equivalence of polar aspects of Phenomena: Phenomenon
"Light", inaccessible as such, is observable via its two
apparently contradictory, P-Equivalent aspects of continuous
field-wave and discrete photon.
This construction exceeds human imagination and Naive View,
but it has been verified experimentally and if we want to
investigate it, we must map it into some pertinent Abstract
#Space(AS). Let's look for a Quadratic Form defining such AS.
The source at O of RR emitting 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]
Transferring 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
Euclidean Quadratic Form 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 or 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 LightTime and x1,2,3 with
phenomenal space may be called "SR #Space".
Let's note that SR #Space is Metric as its all 4 dimensions
including LT have the measure of distance. Thus it transforms
covariantly in all 4 dimensions. In other terms SR #Space,
unlike the absolute Newtonian, is relative, being covariant
under L-Tr. 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. 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 amounts to rotation in the plane x1/x4 (or
x1/iCdt). Let's mention 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) (DBB.LORENTZ TRANSFORMATIONS).
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 Ontology. We shall return to its
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 Time dilation increasing with V.
5.Phenomenal Equivalence of Mass and Energy ( DBE.E=MC^2 ).
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: SR, while
extended over Electromagnetism, maintains the Galilean
restriction to Inertial Referentials (IRs). Yet, no strictly
inertial Referential has ever been observed and the
Phenomenal Cosmos seems to consist exclusively of NIRs
(Non Inertial Referentials). True, many can be approximated
as IRs and SR is an acceptable approximate Model of a vast
range of Phenomena. Nevertheless, as one could not a priori
except the possibility of some other Phenomena, incompatible
with IRs, it appeared vital to extend the SR over NIRs, i.e.
over the total Phenomenal Cosmos.
We shall now review this extension known as the General
Relativity.
GENERAL RELATIVITY (GR)
It may be said that the more complex a Model is in detail,
the simpler are its foundations. GR is founded on two
astoundingly simple principles: Reducibility to SR and Axiom
of Equivalence.
-Reducibility to SR: Just as SR is an extension of Newton's
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 NIRs.
-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 identity, not quite properly called
"Equivalence".
Never a deeper scientific, intellectual and philosophical
revolution has been founded on simpler Postulates.
ROTATING DISK
The actual derivation was just as simple as the Postulate:
it boils down to a simple mental experiment known as
"Rotating Disk" which we describe it very superfitially.
(Details in: DERIVATION OF GENERAL RELATIVITY, Steps 1, 2 )
Given on a stationary and rotating disk circles of radius R
and circumference S, Euclidean Geometry tells us that on the
stationary disk S=2 pi R. On the Rotating Disk we'll measure
R and S with elementary physical rods ("Einstein's Covering
Principle"[G]). Upon R the rods perpendicular to the rotation
are not affected by it and R conserves its Euclidean length.
However, upon S they move in rotation sense and experience
the Lorentz Contraction by virtue of Reducibility to SR.
Thus rotating S will need more contracted rods to be covered
than in Euclidean #Space and its length will be measured as
longer. On the Rotating Disk we have S > 2 pi R. This
defines Lobachevsky or hyperbolic curved #Space and suggests
that rotation determines geometry and #Space.
And that's it, GR is derived. There follows a lot of most
complex mathematical expressions, but they all rest on the
simple "Rotating Disk". The most complex Theory rests on
the simplest Postulate.
FIELD AND #SPACE
The NIR "Rotating Disk" suggests 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 and we have to look for some other criterion
to distinguish IRs and NIRs.
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 essential difference between IRs
and NIRs and #Space is determined by Inertial Field
engendered by rotation rather than by rotation itself. As,
by virtue of the Equivalence Principle, Inertial and
Gravitational Fields are equivalent we may consider Field
(indiscriminately Gravitational or Inertial), 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
Field's strength.
-#Space determines Field's strength as function of distance
(from gravitational "mass" singularity, from rotation
center, etc.)
-Due to this mutual, reciprocal determination #Space and
Field are PHENOMENALLY EQUIVALENT. Their P-Equivalence
solves 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 large", its radius
tending to infinity and the curvature of the increasing
sphere tending towards zero, or "flatness", giving
"S = 2 pi R".
Hyperbolic #Space may be imagined as going past this limit,
where our surface gets "superflat", or its curvature gets
negative, giving "S > 2 pi R".
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 on the Axiom of Aether.
It's true enough, but let's keep in mind that, besides its
own "intrinsic" Axioms, a Theory has usually other,
"external" foundations. When founded on an external Theory,
it accepts the external assertions as its own Axioms.
Intrinsic 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 on dogmatic belief that Physics deals 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
inferential 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 L-Tr, a mathematical construct compatible with the
invariance of C. However, it was void of physical sense in
the mechanistic Newtonian #Space, affine in space/time
planes, thus 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 observations contradict
his astronomical a priori drivel, that facts disagree with
his speculations, answered: "So much the worse for the facts".
However, Lorentz was not a conceited metaphysical preacher,
but a great sincere scientist trapped in the universal
belief in the apparently necessary 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 Science, got new, purely
Phenomenological Foundations.
All Concepts fell from the inaccessible heights of the
Transcendental Olympus of Apriori into the controllable
domain of empiric, phenomenal adequacy.
Just as the Scientific Revolution of the First Enlightenment,
Einstein's Extended Relativity revolutionized the Foundations
of Science which laid the cornerstone of the Second
Enlightenment and its Phenomenological Weltanschauung.
However, facing tenacious resistance of the still dominating
Noumenalism it could not rise far beyond this cornerstone.
While Popper's Epistemology and Heidegger/Sartre's Ontology
display some of its sprouts, they neglect several fundamental
concepts such as Phenomenal Equivalence, the base of
phenomenal Dialectic which iin our opinion incarnates the
Reason of the Second Enlightenment. Established Logic,
untouched by Enlightenment, is still entirely noumenalistic
and so are its explicit or implicit applications to human
and socio-political domains. The world is governed by inhuman
devastating Dogmatism calling in question the very survival
of the mankind. Einstein's call for Enlightenment, "A new
manner of thinking is essential if humankind is to survive"
gets every day more topical.
