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PLAN OF DC GENERAL RELATIVITY
dca foundations of general relativity
dcb derivation of general relativity steps 1 and 2
dcc the centrifugal model step 3
dcd the centripetal model step 4
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Site Plan :
DCB DERIVATION OF GENERAL RELATIVITY, Steps 1, 2.
A FEW TERMS.
-SPACE (capitalized): abstract topological concept such as
Riemann SPACE, synonymous with Domain" or "Class", sharply
distinct from its unfortunate homonym, the "space" of
direct perception.
-AS (Abstract SPACE): Synonym of "Symbolism" ("STRUCTURES
OF MIND")
-PS (Phenomenon SPACE): Synonym of "Imagery" ("STRUCTURES
OF MIND")
-FIELD: a mathematical term associating a potential force
vector with each point of AS SPACE. It's observable in PS
not directly, but via its manifestations - forces acting
on physical bodies which may be mapped into Field in AS.
For simplicity's sake we may say in metalanguage that Field
is "observed", however always keeping in mind the above
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 interior
experiment in the involved Referential. Thus, we may talk
for convenience of derivation about inertial and gravity
Fields, always keeping in mind that we talk about two
P-Equivalent aspects of the unique Phenomenon "Field".
NOTE: By "P-Equivalence" (Phenomenal Equivalence) we mean
the relation among observable aspects of a not directly
observable phenomenon. For instance, continuous wave and
discrete photons are P-Equivalent aspects of the phenomenon
"light".
-MASS: a mathematical coefficient of AS, non observable in
PS, but helping to order in AS such observations as force
and acceleration into consistent patterns. In mathematical
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 AS. In physical models singularities are not limited
to a point, but extend over neighboring areas where Field
equations cannot be solved numerically. Such areas, where
for instance Field density exceeds some threshold. are for
convenience sake called in metalanguage "matter", or
"black" areas.
-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 in 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 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.
GR DERIVATION VIA MENTAL EXPERIMENT OF "ROTATING DISK".
Pertinent Referentials:
I: an IR
OI: Observer in the center of I solidary with I.
F: "Rotating Disk", a NIR with centriFugal Field, observed
from I as rotating around center of I, confused with the
center of F.
OF: Observer in the center of F solidary with F.
P: a NIR with centriPetal Field.
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 RF2 is
strong enough to have all observable effects, yet below the
threshold of 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 speeds at RF1 and at RF2 as
respectively negligible and effectual.
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,
in a way observable from I.
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 UF2<UF1 and SF2 needing more of
shorter units to be covered is measured as longer than SI2.
RF2 being perpendicular to the tangential speed stays
unaffected (in first approximation as we shall see) so that
SF2>SI2 and SF2/RF2>2pi.
Conclusion: Fast enough rotation changes F's SPACE to
non-Euclidean, namely to Hyperbolic or Lobatchevskian.
Note:
S = 2piR: Euclidean, "flat" SPACE,
S < 2piR: Riemannian, parabolic SPACE,
Positive curvature which may be visualized on a
sphere,
S > 2piR: Lobatchevskian or hyperbolic SPACE,
Negative curvature, impossible to visualize.
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 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 falsifying 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
problem of scientific inquiry: factual plausibility within
mental experiments. (see "ERN LOGIC")
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 observations of one from
another IR are 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 for 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 in
fact, 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 rigorous IR exists at all,
that Referentials approximated as inertial are tiny islands
within the non-inertial ocean and that this non-inertial
vicinity does not upset their inertial character at all.
So, 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.
Consequently, 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 SPACE.
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.
FIELD AND SPACE
Newton's Gravity Field:
Gravity Force exerted on a unit mass detector at distance r
from mass M F = -GM/r^2 may be expressed as gradient of
potential Field P(r)=GM/r.
In 0B GALILEAN RELATIVITY AND NEWTON'S MODEL
we discuss Newton's Paradoxes, in particular the Second
Newton's Paradox:
Gravity Field P(r)=GM/r is clearly determined by distance r,
thus by SPACE, but SPACE is presumed absolute and in no
way affected by Field, which clearly violates the
Reciprocity Principle (Action / Reaction).
Newton's Gravity generalized for a continuum point in
presence of ponderable "matter" in form of Poisson's
equation reiterates the Paradox:
lap(phi) = 4 pi K ro (Poisson's equation)
where
phi: scalar gravity field
lap: Laplace operator (divergence of the gradient of phi)
ro: density of ponderable "matter".
Indeed, density is a spatial concept associating some agent
(here an element of ponderable "matter") with a SPACE
element. This "material" SPACE element "generates" the
Field by determining its divergence without being
reciprocally affected in any way.
Resolution of Newton's Paradoxes had to wait till Einstein's
Relativity. The first, concerning the apparent action at
distance of gravity, contradicting the mechanistic dogma is
discussed in AXIOMS OF SPECIAL RELATIVITY and in
0B GALILEAN RELATIVITY AND NEWTON'S MODEL.
Here we shall concentrate on the solution of the Second
Paradox involved by the General Relativity.
Qualitative solution is immediately given by the Rotating
Disk: On the one hand Field increases with r, i.e. SPACE
(distance) apparently determines Field;
On the other hand the SPACE curvature (indicated by
circumference getting greater than 2 pi r) increases with
Field density, i.e. Field apparently determines SPACE.
Such a reciprocal apparent "determination" of two events,
each appearing in turn as cause and effect of the other,
should of course not be confused with causal, one-way
determination, but is the very essence of the P-Equivalence.
SPACE and Field are two P-Equivalent Aspects through which
manifests itself the most general domain of physical reality,
the Cosmos.
For readers familiar with Tensor Calculus we recall
Einstein's Field Equation, which may be considered as GR's
equivalent of Poisson's equation:
R(/m,/n)-0.5g(/m,/n)R=-mu*T(/m,/n) EFE(Einstein's Field Equ.)
where
R(/m,/n): contracted Riemann Tensor
g(/m,/n): Metric Tensor
R: scalar R=g(m/,n/)R(/m,/n)
T(/m,/n): energy tensor of the "matter"
mu: constant related to Newton's gravity constant.
Left side of EFE represents the Field OR SPACE in terms of
metric and curvature. This common representation shows the
homomorphism of both constructs which defines them as
P-Equivalent aspects of the Cosmos.
In spite of being abstract constructs of Mind, Field and
SPACE have full phenomenal sense secured by their
observable manifestations: force, speed; etc. for Field,
geometry for SPACE.
Right side of EFE represents the "singularity term"
replacing Poisson's "matter" with its relativistic
equivalent, viz. energy, encompassing that of gravity and
of electro-magnetism.
