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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 :
DCD THE CENTRIPETAL MODEL STEP 4
NOTE: neologies and ambiguous terms clarified in GLOSSARY
are marked "[G]".
The centripetal model consists of a disk with a mass
singularity in its center. As most if not all observable
centripetal mass referentials have a spin component, we
define our centripetal model by adding the central mass to
our previous centrifugal rotating disk. Strictly speaking
it's a mixed centrifugal-centripetal model. However, as we
shall concentrate here mainly on the centripetal effects
we shall call it "Centripetal Model with Spin".
The speed of our elementary detector at R VR will be the sum
of the opposed centriFugal speed VF and centriPetal speed VP
and will have the value of the absolute of their difference:
VR=mod(VF-VP)
As previously we set the angular speed W=1. Then, Centrifugal
force KF(Rt)=Vt^2/Rt=Rt*W^2 = Rt (as W=1).
Thus acceleration A(Rt)=K(Rt)=Rt.
Consequently Rt is a simple linear function of e^t.
We shall take as first approximation Rt=e^t.
Then, due to f'(e^t) = e^t:
Rt=VRt=ARt=e^t.
VF = R
On the other hand:
Newton's gravity: F=GMm/r^2=m(GM/r^2)=mg
(g: gravity acceleration at earth surface)
Setting M=1/G, m=1, the centripetal force KP(r):
KP(r) = 1/r^2
Free centripetal fall speed VP at R:
VP = int(1/r^2)[inf,R] = (0)-(-1/R) = 1/R:
VP = 1/R
and composed speed at R, VR:
VR = mod(VF-VP) = mod(R - 1/R)
Let R0 radius at which centrifugal and centripetal
forces cancel each other:
For R0, KF=KP: R0 = 1/R0^2
Thus R0=1
Detector (such as we defined in
DCC THE CENTRIFUGAL MODEL STEP 3 )
placed at R0 will turn arond M along a circular orbit of
radius R0.
In order to make our model more realistic, let's replace the
circular orbit with an ellipse obeying Keppler's laws, with
M in one of its foci.
The center of the ellipse will be at 1 and the focus carrying
M at 0. Thus the second focus will be at 2. We chose the
excentricity so that the periphelion (nearest point to M) is
at -0.0015 and, consequently, the aphelion (furthest point
from M) at 2.0015.
To resume:
Periphelion -0.0015
F1 0.0
Center 1.0
F2 2.0
Aphelion 2.0015
According to Keppler's laws tangential speeds at Periphelion
and Aphelion are inversly proportional to their radii and
with our unit conwentions are respectively: Vt(Per)=2.0011
and Vt(Aph)=0.0015.
As in DCC THE CENTRIFUGAL MODEL STEP 3
we shall perform simple calculations, remembering that for
reasons explained there, the results should be considered as
a rough approximation, valid only as qualitative indication.
We shall use following formulas:
gamma = 1 / sqrt(1 - (mod(V) / c)^2)
gamma D C A B (D=1/gamma)
VS= (Vt + Vc) / (1 + Vt * Vc / c^2)
VS P VS S Q R
dX1 = sqrt(dX2^2+dX3^2+dX4^2)
but dX4=dX2, thus
dX1 = sqrt(2*dX2^2 + dX3^2)
V V R P U S T
Note: "2*dX2^2" because we assume as in the centrifugal model
that dX4 = dX2.
Periphelion: Aphelion:
VF=R= 0.0015 2.0015
VP=1/R= 666.666666666667 0.499625281039221
VR=mod(VF-VP) 666.665166666667 1.50187471896078
A=VR/c 0.666665166666667 0.00150187471896078
B=A^2= 0.444442444446694 2.25562767145352e-06
C=1-B= 0.555557555553306 0.999997744372329
D=sqrtC= 0.745357334138 0.999998872185528
dX2=gammaR=1/D= 1.3416383715557 1.00000112781574
Vt= 2.0015 0.0015
Vc=VR*(gammaR-1)227.758401912906 1.69383795307362e-06
P=Vt+Vc= 229.759901912906 0.00150169383795307
Q=Vt*Vc= 455.858441428682 2.54075692961043e-09
R=Q/c^2= 0.00045585844142 2.54075692961043e-15
S=1+R= 1.00045585844143 1
VS=P/S= 229.655211646059 0.00150169383795307
A=VS/c= 0.229655211646059 1.50169383795307e-06
B=A^2= 0.052741516236196 2.25508438294623e-12
C=1-B= 0.947258483763804 0.999999999997745
D=sqrtC= 0.973272050232515 0.999999999998872
dX3=gammaS=1/D= 1.02746195142571 1.00000000000113
P.dX2= 1.3416383715557 1.00000112781574
R.2P^2= 3.59998704006126 2.0000045112655
S.dx3= 1.02746195142571 1.00000000000113
T.S^2= 1.05567806162753 1.00000000000226
U.R+T= 4.65566510168879 3.00000451126776
V.dX1=sqrt(U)= 2.15769902944984 1.73205210985922
Precession of the Periphelion.
Elementary distances covered by the detector at Periphelion
and Aphelion are respectively:
dX3(P)=1.027, dX3(A)=1 and their difference = 0.027.
This difference indicates that at Periphelion the detector
covers a longer elementary distance than at Aphelion. This
phenomenon is known as "Precession of the Periphelion" and
can be observed with respect to Mercury, whose Periphelion
is closer to the Sun than those of other planets. It means
in other terms that the whole elliptic orbit moves in the
direction of the rotation.
Bending of light
First approximation of the curved elementary geodesic may
be taken as: 1 - dx2/dx3, which gives
At Periphelion: 1 - 1.34 / 1.027 = 0.3
At Aphelion: 1 - 1 / 1 = 0
Close to M light is bent in the direction of M. Further away
the geometry becomes Euclidean and bending vanishes.
Einstein's Lense and Ring.
The following figure (very clumsy, as drawn with simple ASCII
means) illustrates the phenomenon of "Einstein's Lense",
caused by bending of light following curved geodesics close
to a mass.
i
i
-*-
*----------------------------*
*--------------------------------------------*
S--------------------------M-------------------------O
*--------------------------------------------*
*----------------------------*
-*-
i
i
S(ource) and O(bserver) are at opposite sides of M. Light
sphere originating in S would never reach O hidden by M, if
light traveled along straight lines.
However, due to the bending effect, light radius follows
the "*" trajectory and reaches O, who gets the impression
of light coming from image "i".
Our 2 dimensional figure shows two images "i", but it's easy
to see that in 3D the Observer will perceive a ring "i".
This phenomenon, the "Einstein's Ring" is due to M field
acting as a lense, the "Einstein's Lense".
Note: In order to make the clumsy figure slightly better
readable, the inside of the lense is filled with "-" .
Centripetal Event Horizon and Black (Matter) Hole.
For RH = 0.001 VP = 1000, i.e. c in our units, gamma
becomes infinite and all our formulas become meaningless.
RH represents the radius of the centripetal "Event Horizon",
the circular boundary of Field's singularity area, denoted
in metalanguage as "Black (matter) Hole".
(see Matter, Mass, Energy )
Comments
In function of R we have distinguished:
1.the central mathematical singularity at R=0,
2.the physical singularity area or "Black (matter) Hole"
within the centripetal "Event Horizon" RH=0.001,
3.Euclidean area adjacent to the no-field radius R0=1,
4.the area of centrifugal preponderance discussed in the
previous chapter, bounded mathematically by circumference
Sb of radius Rb and physically by the centrifugal Event
Horizon circumference Sh of radius Rh<Rb with the
"Black Matter" ring between Rh and Rb.
Yet, we have defined only two constructs:
A.CentriPetal Field (PF) with central singularity at M,
Following the common usage, we called it "Gravity Field",
which is not quite proper, as by virtue of the Equivalence
Principle, Gravity is within a Referential identical with
and undistinguishable from Inertia. The only specific
quality of PF is its centripetal direction.
B.CentriFugal Field (FF) of spin. Similarly to PF we followed
the common usage and called it improperly "Inertia Field",
while its only specific quality is the centrifugal direction.
Their density is P-Equivalent[G] to the curvature of #Space.
Consequently, the events 1-4 should be explainable in terms
of A-B.
1. Self-explanatory: singularity of PF (A.)
2. PF (A.) density is so high, #Space and light paths so
strongly bent that light will turn within the Black Hole
without the possibility to escape from it (See "Bending of
light" above). Thus, no event can be observed within the
Event Horizon.
3. Contracting PF and dilating FF cancel one another.
4. A transposition of 1. and 2. in which
-the centrifugal Inertia Field (FF) replaces the Centripetal
Gravity Field (PF),
-the boundary circumference Sb of radius Rb corresponds to
the central mathematical singularity of 1.
-the centrifugal Event Horizon Sh of radius Rh<Rb
corresponds to the centripetal Event Horizon.
-the "Black Matter" ring between Rh and Rb corresponds to
the centripetal Black Hole and the Inertia Field bends
there similarly the #Space preventing light from escaping
and events from being observable.
Disambiguation
Most publications, including Hawking's "A Brief History of
Time", profess that Einstein derived the General Relativity
(GR) by extending the Special Relativity (SR) over Gravity.
If he tried to do so, he would have sunk at the first step.
In fact, SR being restricted to Inertial Referentials (IR),
he extended it over the Non Inertial (NIR). This implied
accelerating (for instace rotating) NIRs, the involved
Inertial Field determining (bending) #Space and the
P-Equivalence of Field and #Space. Only afterwards the
"Equivalence Principle" identified Gravity with Inertia,
thus bringing it into the picture.
In most descriptions of Galaxies we find the structure of our
centripetal disk, however, usually buried under improper
terms. Most authors seem to forget, if not to ignore, that
GR boils down to two P-Equivalent constructs, "Field" and
"#Space", and describe the Extrinsicality in ambiguous terms, often
naive and misleading and always meaningless, unless reduced
explicitly to Field or #Space. Most frequent and muddling
terms are "Black Energy" and "Black Matter" implied by "Black
(matter) Hole" and appearing in Black Matter Ring". We shall
try to disambiguate them.
"Black Energy".
Gravity and Inertia are undistinguishable within the same
Referential, but in a Galaxy observed externally we
distinguish superposed contraction and dilation effects,
which we attribute respectively to the centripetal Field of
Gravity and to the centrifugal, Spin generated, Field of
Inertia. The mysterious "Black Energy", supposed to be
responsible for dilation makes the author apparently ignore
the fundamental GR concept of Non Inertial Referential and
muddles hopelessly the issue for the reader.
Let's note that "Black Energy" has several homonyms equally
muddling if not reduced to the centrifugal Field. The most
aberrant seems to be the "Cosmic Constant", when elevated
from the status of an abstract coefficient to the phenomenal
cause of the cosmic expansion.
"Black Matter"
"Matter"
A singular field is described by equations which lose
validity at its singularity points. That's mathematical. In
physical practice they lose validity in the neighborhood of
singularity where field density exceeds some value and which
we shall call "Singularity Area". Now, "Matter" is a
metalanguage term denoting for simplicity's sake SINGULARITY
AREAS of the four known fundamental Fields (and eventual
other ones not yet known). It does not appeare anywhere in
PS (Phenomenal #Space)[G], nor in any known AS (Abstract
#Space of a Model)[G]. Thus, it has no phenomenal, nor
abstract sense. Scientifically speaking it does not mean
anything at all.
In metalanguage of the General Relativity "Ponderable Matter"
denotes the energy tensor of a macroscopically considered
Field Singularity Area.
In the metalanguage of the Quantum Field Theory "Material
Particle" designates a microscopically considered Quantum
Singularity "packet" of Field.
"Black (Matter) Hole"
As we said above, the "Black (matter) Hole" is a centripetal
Field singularity area, where Field density, and the
P-Equivalent #Space curvature get so strong that light
signals following bent geodesics cannot escape from the area,
rendering unobservable the events within the Event Horizon.
"Black Hole" is a good Science Fiction term for #Space curved
beyond the light's non-escape threshold. Less good is talking
about "matter" or "masses" irremediably falling in the hole.
At least, as long as it tends to bestow on them phenomenal
quality and to make us forget that they designate Field
singularities.
"Black Matter Ring" and "Superluminal" Speed
"Black Matter Ring" (BMR) is an inverse map of the "Black
(Matter) Hole" and the previous paragraph applies to it
entirely.
Additionally, BMR is often associated with superluminal
speed apparently occurring at extremities of Galaxies.
One often reads in pseudo-scientific articles craving to show
that Einstein was wrong the story which may be resumed as
follows:
1.Galaxies are large enough for their rotation to engender
superluminal tangential speed at their extremities and such
speeds have been actually measured, which falsifies the rule
of unsurpassable C and consequently the whole Extended
Relativity.
2.Under the involved enormous centrifugal force the Galaxie
would fly apart if it were not "kept together" by some
compressing agent.
3.This compressing agent is the "Black Matter Ring" which
encircles the Galaxie beyond the centrifugal Event Horizon.
Let's return this complicated naive story to the simple frame
of the General Relativity, which, as we know, boils down to
two P-Equivalent constructs, Field and #Space:
1.While superluminal speed has indeed been measured at
Galaxies' extremities, it has been measured with external
clocks and distance units. Proper clocks and distance rods
of a rotating Referential are determined by #Space curvature
due to the centrifugal Field. The resulting tangential speed
tends asymptotically to C, without ever reaching it.
(Details in Appendix of DCC THE CENTRIFUGAL MODEL STEP 3 ).
Proper superluminal speed has never been observed.
2.Rotating Referential "keeps together" without any external
"compressors", due to its #Space curvature. As we can see
in the above link, dilation vanishes in comparison with the
Coriolis dominated rotation.
3.The idea of Galaxie kept together by a "Black Matter Ring"
in the manner of a barrel clamped with iron straps is rather
touching in its naivete. Galaxie is kept together by its
#Space curvature and the "Ring" enveloping the Event Horizon
is a singularity area bent beyond the treshold of expending,
of light escaping and events' observability. Thus, as any
Field singularity area it may be called "matter, as long as
one remembers its singularity gist and does not confuse it
with some Deus ex machina clamping strap.
Mass bending #Space
One often hears that that Mass bends #Space. Yet, some
pretend that Mass, being just an abstract coefficient cannot
bend anything and that the cause of bending is the Field.
But, others object that it's inversed, that it's #Space
curvature that causes the Field.
Sounds disturbing: Field as cause of curvature, and vice versa
looks like a vicious circle. Only very few dare to say that
causality does not apply here at all and that Field and #Space
are P-Equivalent "equally ranked" Aspects of "Cosmos", the most
general Phenomenon of "Extrinsicality", none causing the other.
Riemannian gravity SPACE.
In the "inertial" area outside of R0 the SPACE is hyperbolic,
as derived in the Rotating Disk. Estimating the circumference S
to radius R ratio we considered as first approximation that the
R, perpendicular to the tangential speed stays constant.
However, the field related SPACE curvature affects all dimensions,
so that R gets curved, thus longer. The exact effect may be only
represented with the Field Tensor, but R's extension is very small
compared with that of S and the "inertial area" stays hyperbolic.
However, in the "gravitational" area inside the R0, the tangential
speed and the extension of S decrease when approaching the center,
while the field density, the curvature and the extension of R
increase.
Thus, we get S < 2piR
Gravity preponderance is equivalent with Riemannian SPACE.
