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."

  

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Derivation of General Relativity, Steps 1, 2.
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A FEW TERMS.

FIELD: a mathematical term associating a potential
force vector with each point of MS #space. It's 
observable in PS not directly, but only via its 
manifestations - forces acting on physical bodies
which may be mapped into field in MS. For 
simplicity's sake we may say in metalanguage that 
field is "observed", however always keeping in mind 
the above rigorous implication.
In the present level of GR derivation we shall
disregard the electromagnetic field and consider 
only acceleration and gravity fields. 
In order to respect the term "inertial mass" of 
the Equivalence Principle we shall call call the
acceleration field "inertial field".
Equivalence Principle makes "inertial" and "gravity"
fields equivalent and indistinguishable by whatever 
experiment from inside of the involved referential. 
Thus, we may for convenience in derivation conceive
inertial and gravity fields, always keeping in
mind that we talk about two imaginary aspects of
a unique concept "field".

MASS: a mathematical coefficient of MS, non
observable in PS, but ordering in MS such
observations as force and acceleration into 
consistent patterns. In field formulas mass is
a singularity i.e. a point in which the formulas 
don't hold.

MATTER: a metalanguage concept not existing as
such in PS, nor in MS. In physical models 
singularities are not limited to a point, but
extend over neighboring areas where involved
equations cannot be solved. Such areas, where for 
instance field density exceeds some threshold. 
are for convenience sake called "matter", or the 
"black" area.

INERTIAL REFERENTIALS: 
Traditional Definition: set IR of referentials 
whose members move at constant speed with respect to
each other. We shall see that this definition is
not adequate when moving to GR and has to be
replaced by
GR Definition: set IR of referentials in which
no field is observed.

NON INERTIAL REFERENTIALS: set NIR of referentials
not belonging to IR, thus accelerating with respect
to any of IR, or set NIR of referentials in which
field is observed.

SCOPE OF SR: SR holds in IR. For the moment we 
don't know anything about NIR and in particular 
about relations between an IR and a NIR.

LOCAL INERTIAL REFERENTIAL (LIR): local elementary
referential within a NIR in which field disappears
eg. free falling box in the field of the NIR.

REFERENTIALS PARTICULAR TO GR DERIVATION:
I: an IR
F: a NIR with centriFugal field, observed from I
as rotating around center of I.
P: a NIR with centriPetal field.
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STEP 1.
Observer OF draws two circles of respectively
radii and circumferences RF1,SF1 and RF2,SF2, such
that between 0 and RF1 the field practically does
nor exist and at RF1 is strong enough to have all
eventual effects and not strong enough to fall into 
the "black" matter area.
OF accepts OI's view, considers SF1 and SF2 as
rotating with respect to the center of F and checks 
the tangential speed at RF1 to be negligible with
respect to C and that at RF2 not negligible.

Observer OI draws in I two circles RI1,SI1, 
RI2,SI2 exactly covering those of F. Then he makes
a straight physical unit rod UI1 short enough to 
cover a segment of SI1 with acceptable 
approximation. He measures his both radii and
circumferences with UI1 and finds 
SI1/RI1 = SI2/RI2 = 2pi.

Then he drops UI1 on F where it becomes UF1 
solidary with F and asks OF to make analogical 
measurements with its help, which OI can follow 
and observe. 

As long as UF1 covering a segment of SF1 or SF2
rotates with it, it stays a small NIR and we 
cannot say anything about it. So, OF cuts the
"string" attaching it to F's center and lets it
fly free at the tangential speed. Now, UF1 becomes
a LIR and all SR laws can be applied to it.

At RF1 its speed is too small to cause Lorentz
Contraction and measurements are identical in both
referentials: SF1=SI1 and SF1/RF1=2pi.
At RF2 however, its speed is sufficient to cause
Lorentz contraction, so that UF2SI2 and SF2/RF2>2pi.

Conclusion: Fast enough rotation changes F's
geometry to non-Euclidian.
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STEP 2.
Having complied with OI's wishes and with his view
OF proposes to carry out a symmetrical experiment:
While F appears rotating to OI solidary with I,
the inverse is just as true: to OF solidary with 
his referential I appears rotating and F stationary.
Thus OF expects symmetrical results, namely UI2
becoming shorter than UI1, SI2>SI1 and SI2/RI2>2pi.

However, these expectations prove false. 

Why? How do we know? How can they "prove"? 
- one may object: you talk as if you had some 
empiric facts proving false these expectations, 
but you have no facts, you are just carrying 
out a "mental experiment".

It's a tough objection, but a welcome one: it will
allow us to throw some light on the perhaps deepest
and most complex level of scientific inquiry:
factual certainty (see Crisis of Logic) within
mental experiments.

But as we said it's tough as well, so we shall ask 
the reader for a bit of patience and concentration
when considering the following lines.

It's true that within the SR phenomena observed
from one to another IR are mutually symmetrical:
To a fellow flying fast close to us the earth 
would look like a flat lens, but then to us this
fellow would look just as flat.

That holds among IR's, but there is no earthly 
reason to extend it beyond the SR and pretend 
that it holds between NIR's, or between an IR
and an NIR. 

Nor is there any to pretend that it does not hold.

We could rigorously chose one or the other option 
only in the light of empiric, experimental 
justification, but it seems that we cannot have
any within our mental experiment.

Cannot we? If it were true, if we could not be
empirical in mental experiments, no creativity, no
progress would exist in science. 
Indeed, in mental experiments we can check our 
assumptions against imaginary observations;
imaginary it's true, but encompassing our
knowledge of the investigated PS, the recollection
and the synthesis of all real experiments carried 
out there.

And in our particular case, real experience tells us
that PS or "universe" is overwhelmingly if not 
entirely non-inertial, that we are not even sure if 
any exact IR exists at all, that referentials which 
may be approximated as inertial are tiny islands 
within the non-inertial ocean and that this 
non-inertial vicinity does not upset at all their 
inertial character.

Thus, in our case the vicinity of NIR F does not
influence in any way the IR I which stays inertial,
independently of all neighboring NIR's.

Thus, the expectations of OF prove false. He may be 
solidary with his F and observe I as turning, but  
it has no bearing on I's inertial character and
euclidean geometry.

This statement has TREMENDOUS consequence: the 
traditional distinction between IR and NIR, the
criterion of movement, of constant speed against
acceleration is no more pertinent. From this
point of view F seen from I and I seen from F
are identical. The only difference between them
is FIELD which replaces from now on the traditional
criterion of movement. 

Conclusion: The criterion of movement, the 
traditional distinction between IR and NIR is no
more pertinent. The PERTINENT CRITERION IS FIELD.
A referential in which field is observed is NIR,
otherwise it is IR.

Corollary: FIELD DETERMINES GEOMETRY.
In presence of field geometry becomes 
non-Euclidian.
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