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."

  

===================================================
Step 3, Centrifugal Model
===================================================
STEP 3.

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-Euclidian 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 Euclidian.

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 space 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
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 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.

VX4t is local centrifugal effect caused by Dt
/bin/bash: q: command not found
rough approximation we shall assume the 
curvature of dRt as inverse of its LC: dRt/dRIt.

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 originally 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 must be as usually approximated in 
Physics by introduction of "horizon radius" (Rh)sm(Rb) 
in our case t approximately greater than 12.6  .
The region between Sh and Sb is undetermined and
represents the physically unknown "black" matter
ring.
Observable phenomena stop at the "horizon radius" 
Rh or at the "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 will become 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 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 covered 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 into GR.

In the next step we shall consider the more 
pertinent but more difficult centripetal model.

Before moving into it let's discuss and clarify
as much as possible the present, centrifugal step.

===================================================
EXERCISE AND COMMENTS 
===================================================
EXERCISE

Let's repeat the reasoning of STEP 3 supposing that
F is stationary and the "black matter" ring 
between the Horizon Radius and the Boundary Radius
is rotating. Kind of rotating hollow body, doughnut
or torus. 

Have fun.
===================================================
COMMENTS

Calculations of STEP 3 are not precise and have
only indicative value guiding qualitative
intuition. As such they seem correct. They
indicate correctly that some U increases with V 
and does it more strongly with increasing W,
without precising the rate of increase of U.

The qualitative statements that for the free 
falling LIR, R and S tend to infinity, S much 
faster than R and 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 refinement, 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 been
considered less interesting from the cosmological 
point of view.

Such metric (eg. Schwarzschild's) exists for
centripetal models. We shall discuss it in the 
next chapter.

Now, centrifugal model is a map of the 
centripetal, so, if somebody has time to burn,
he may try, as second exercise, to transform
Schwarzschild's metric to the centrifugal
case. 
===================================================