Centripetal Model.
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, as previously:
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.
And 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) = (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 the Centrifugal
Model) 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 the centrifugal model we shall perform simple
calculations, remembering that for reasons
explained there the results should be considered as
a rough approximation, valid only as qualitative
indications.
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, or Einstein's Lense.
--------------------------------------
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 Euclidian 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 "-" .
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 "Event
Horizon", the sphere of singularity area of the field,
which we denote with the metalanguage term "matter".
(see " Matter, Mass, Energy" in the Depository
http://evans-experientialism.freewebspace.com/metanomskiindex.htm
)
Finally we have the central mathematical singularity
at R=0, the physical singularity area or "matter"
within RH=0.001, Euclidian area adjacent to the
no-field radius R0=1 and with further increasing R
the centrifugal model finishing at the centrifugal
Event Horizon Radius. |
Relativistic Dialectics