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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 :
DCC THE CENTRIFUGAL MODEL STEP 3
NOTE: neologies and ambiguous terms clarified in GLOSSARY
are marked "[G]".
In previous steps the observer OF ascertained the existence
of field in F and, looking at F from OI's point of view
considered it as acceleration field generated by rotation
of F. He noticed that F's geometry was non-Euclidean and
that its curvature (indicated by the ratio S/R) increased
with R, thus with the tangential speed, thus with the
strength of the acceleration Field.
These findings are essential, but much too vague to found
a Theory. In order to refine them we will look at F in more
detail considering it as stand alone referential. We may
therefore drop the indicator "F" and designate radius,
circumference, unit rod, etc. as Rn, Sn, Un, ...
We shall consider in F several circles Sn of radii Rn.
Inside of S1 field is negligible and geometry Euclidean.
Let's, as in step 1, make a straight unit rod of length U1
covering with acceptable approximation an element dS1 of S1.
Let's further construct an elementary physical spatial body
D (detector) in form of cube dX2*dX3*dX4=ds^3, where ds=dS1.
By physical we mean that D is capable to react to potential
forces of Field and in turn to create acceleration Field
when accelerated. (A rotating physical body generates for
instance Coriolis Field, etc.) D may be used as free falling
LIR (Local Inertial Referential), as a covering body of 3D
space patterns, etc.
Let's further consider OF free to observe F via instruments
of OI, seeing F as rotating and attributing its Field to
acceleration, or to use his own laboratory, observing F as
stationary and attributing its Field to gravity.
Then, let's associate F with an orthogonal coordinates system:
X1: lighttime ct
X2: a particular R
X3: orthogonal to X2 in the plane of S1
X4: orthogonal to the plane of S1.
This system will be cartesian within S1 and outside of it
will become Gaussian, embracing the #Space[G] curvature
generated by Field.
FREE FALLING D IN CENTRIFUGAL ACCELERATION FIELD.
Let:
t: time observed by OI.
Rt: R reached by D at t.
VRt: R component of free fall speed at t.
St: circumference of a circle of Rt (or "circle St").
W: constant angular speed of F = 1.
VSt: tangential speed at t
Then:
Centrifugal force K(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.
VRt causes Lorentz Contraction (LC) of D(Rt) thus dilating
and bending dRt. Rt as the whole dilates by factor of
integral of these dRt's from 0 to t.
VSt=Rt (as W=1) resulting in LC in X3 direction.
As rough approximation we shall assume the
curvature of dRt as inverse of its LC: dRt/dRIt.
(LC=1/gamma).
VX4t is local centrifugal effect caused by
curvature of dRt.
Finally curvature of Rt combined with VSt generates VCt
Coriolis component directed along tangential speed when D
moves along R towards increasing curvature. Thus in our case
VCt adds to VSt and contributes to LC(X3), in a decisive way
with increasing t.
Postulate of constant C requires: dX1^2 = dX2^2+dX3^2+dX4^2
Simple calculation gives:
______________________________________________
t__________11_____12______12.6____12.605_12.61
e^t_________59874_162754__296558__298045_299539
LC(X2(t))__0.9798_0.84____0.1510__0.1139_0.0554
dX2(t)_____1.0205_1.1904__6.6210__8.7738_18.046
dX3(t)_____1.0829_1.8341__86.675__152.96_650.30
dX4(t)_____1.0205_1.1904__6.6210__8.7738_18.046
dX1(t)_____1.8043_3.0878__87.180__153.02_650.80
V(X2(t))___33863___62745___22523___17089_460.29
V(X3(t))___69118__149385__298255__299800_299800
______________________________________________
DISCUSSION:
On a sphere we observe Coriolis forces oriented in the
direction of rotation when the detector moves from smaller
to greater curvature. Analogically, D falling free at Rt
will observe Coriolis force of magnitude growing with dSt/dRt
and oriented in the direction of rotation.
The original R direction of the free fall is bent towards
the direction of rotation. With LC tending towards 0 D's
trajectory tends towards S. At the limit D will rotate along
a boundary Sb of Rb which it will never cross. This purely
abstract consideration approximated by introduction of
"horizon radius" Rh<Rb in our case t approximately
greater than 12.6 . The region between Sh and Sb is
undetermined and represents the "Black Matter" ring.
(see "Disambiguation" of DCD THE CENTRIPETAL MODEL STEP 4 ).
Observable phenomena stop at the "Horizon Radius" Rh or at
the centrifugal "Event Horizon" circle Sh. Let's note that
dR (dX2) increases exponentially with t while the speed
along R after having reached a maximum decreases with t.
At the theoretical boundary Rb the speed along R becomes 0,
so that D will never cross it and will rotate at C along Sb.
But, as we said, this is purely abstract and physically
speaking D will touch the event horizon Sh at a very sharp
angle and disappear from observable PS.
Let's note that VS (V(X3(t))) increases when approaching the
event horizon. We shall recall it in the next, more realistic
centripetal model, trying to justify the precession of the
perihelion of a theoretical planet simulating Mercury.
In our example when we reach the event horizon, for
approximately t=12.6, the numerically most sensitive V(X3(t))
becomes close to C, but it's exact value is obscured by
round off error.
The present centrifugal model is not most pertinent from the
cosmological point of view, but is very much easier for
initial, intuitive considerations. That's why Einstein used
it as entry to GR.
In the next step we shall consider the more pertinent but
more complex centripetal model.
Before moving into it it seems advisable to have a clear
understanding of the present, centrifugal step, including
the details of the free fall table calculation shown in
Appendix.
COMMENTS
Calculations of STEP 3 are not precise and have only
indicative value guiding qualitative intuition. As such they
seem correct.
The qualitative statements that for the free falling LIR,
R and S tend to infinity, S much faster than R, and that
Time slows down to zero appear to be correct.
The whole train of thought can only give a rough qualitative
indication, indispensable for further mathematical refining,
but not precise as such.
The main reason is that we were taking 1)LC(Rt), deducing
from it the rotational 2)LC(St), the Coriolis 3)LC(St) and
4)LC(X4). Now, that's not exact:
1) cannot be taken separately, because it's affected
recursively by its effects 2,3,4. And, we should not forget,
affected also by LC(X1), the lighttime dimension.
To cut short a long story, we would have to have some
construct Lambda expressing all mutual actions of all
4 dimensions on one another i.e. having 16 dimensions
reduced by symmetry to 10, corresponding to the continuum
in which GR's curved 4D #Space is embedded.
Lambda presupposes a metric. Now, to the best of our
knowledge nobody determined a metric for centrifugal model,
not because it's more difficult than the centripetal, but
because it has not been considered interesting from the
cosmological point of view.
Such metric (eg. Schwarzschild's) exists for centripetal
Model, which we shall discuss it in the next chapter.
Appendix
Details of the free fall table calculation
Table:
t__________11_____12______12.6____12.605_12.61
e^t_________59874_162754__296558__298045_299539
LC(X2(t))__0.9798_0.84____0.1510__0.1139_0.0554
dX2(t)_____1.0205_1.1904__6.6210__8.7738_18.046
dX3(t)_____1.0829_1.8341__86.675__152.96_650.30
dX4(t)_____1.0205_1.1904__6.6210__8.7738_18.046
dX1(t)_____1.8043_3.0878__87.180__153.02_650.80
V(X2(t))___33863___62745___22523___17089_460.29
V(X3(t))___69118__149385__298255__299800_299800
Calculation for t=12.605:
Let:
t: time observed by OI.
Rt: R reached by D at t.
VRt: R component of free fall speed at t.
St: circumference of a circle of Rt (or circle St).
W: constant angular speed of F = 1.
VSt: tangential speed at t
Then:
Centrifugal force K(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.
VRt causes LC of D(Rt) thus dilating and bending
dRt. Rt as the whole dilates by factor of integral
of these dRt's from 0 to t.
VSt=Rt (as W=1) resulting in LC in X3 direction.
As rough approximation we shall assume the
curvature of dRt as inverse of its LC: dRt/dRIt.
(LC=1/gamma).
VX4t is local centrifugal effect caused by
curvature of dRt.
Finally curvature of Rt combined with VSt
generates VCt Coriolis component directed along
tangential speed when D moves along R towards
increasing curvature. Thus in our case VCt
cumulates relativistically with VSt and contributes
to LC(X3), in a decisive way with increasing t.
Postulate of constant C requires:
dX1^2 = dX2^2+dX3^2+dX4^2
Let's call v=VRt, a=ARt: Rt=v=a=e^t.
For the case t=12.605 we have:
A.t=12.605
B.v=e^t=298045.071292794
C.v/c=0.993483570975981
D.C^2 =0.987009605799187
E.1-D=0.0129903942008134
F.sqrt(E)=0.113975410509519
G.dX2=gamma=1/F=8.77382231421294
Relativistic cumulation of tangential
(=v) and Coriolis speed (=v), thus
relativistic v and v gives dX3:
H. 2v=c*C=596090.142585588
I. (v/c)^2+1=D+1=1.98700960579919
J. V=(v and v)=H/I=299993.588780784
K.V/c=0.999978629269279
L.K^2 =0.999957258995267
M.1-L=4.27410047333711e-05
N.sqrt(M)=0.00653766049389008
O.dX3=gamma=1/N=152.959916002762
dX1 = sqrt(dX2^2+dX3^2+dX4^2)
but dX4=dX2, thus
dX1 = sqrt(2*dX2^2+dX3^2)
P. dX2=8.77382231421294
R. 2P^2=17.5476446284259
S. dx3=152.959916002762
T. S^2=23396.7359035719
U. R+T=23414.2835482003
V. dX1=sqrt(U)=153.017265523209
SPEED considerations for t=12.605
RADIAL component:
v*dX2/dX1=O/V=17089.5388060197
TANGENTIAL component:
(v and v)*dX3/dX1=J*O/V=299880
Note: further digits are round-off errors. We
approach c and disappear in the "dark matter".
