Part
4 of "Cosmic Membrane Theory"
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.
8 Analysis and Conclusions (First version 05/1995, last update 07/2008)
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The 4-dimensional relativistic Cosmic Membrane
Theory is based on the imagination of an ether-filled 4-dimensional hyperspace S (or bulk space, as
sometimes called). In this hyperspace a 3-dimensional cosmic membrane expands
with high speed VE in the same manner as a balloon is blown up. This
membrane is our cosmos.
.
The author sees the fourth
spatial coordinate not as a trick to simlify mathematics. It is reality,
and we can measure the curvature of our 3D subspace.
.
The
author thinks the membrane consisting of tiny torus-shaped curls (German korns, grains) with diameters of Planck
length. The membrane is of tiny thickness in the fourth dimension and has a strong tension, but the fourth spatial
dimension of S is of the same kind as our
common three spatial dimensions x, y, z. Only the scale factor is unknown. The
motor of the expansion of our cosmos is the momentum of the mass of the
membrane
.
Gravitation is
not a direct force between masses. The membrane has nearly no resistance in the
etherwind of expansion, but matter inside the membrane is some disturbance and
causes resistance. The membrane deformes to a gravitational funnel. Another
(smaller) mass at the walls of the gravitational funnel is driven by the ether
force to the center of the funnel. That is what we call gravitation.
.
In the case of a single central load the memrane deformes with radial symmetry. The ordinary differential equation (ODE) of the deviation w(r) from its zero position is
.
(2.12)
.
Here w(r) is the depth of space in the fourth dimension. Positive w-direction is the
direction of the expansion of the cosmos. In all cases of weak curvature we may
neglect the small term w’2 and furthermore we may neglect the fine
difference between r and the true arc length inside the membrane. So we get the
simplified equation of curvature
.
w“(r)=‑2w‘(r)/r. (2.12
simplified)
.
Each
Newton's potential like function w(r)=Wo+C/r is a solution of the
simplified equation 2.12 . Since we often have to handle problems with
spherical symmetry (e.g. in our solar system), we write the solution of the
simplified equation 2.12 in the form of Eq 2.13a. R is the radius of the Sun, Wo
the depth of space at the edge of the
Sun. We define the mass-acceleration by
the ether wind (force for the unity of mass) just as Ae=gs/W’o,
where gs is the gravitational acceleration at the edge of Sun, W’o
the slope of the membrane at the edge of
Sun, g the gravitational constant.
.
, (2.13 a)
.
(2.13 b)
.
(2.14)
.
If
we do not have a central load, but a mass distribution of radial symmetry, we
get the differential equation (ODE)
.
. (3.5)
.
The
first term of the right-hand terms of the ODE 3.5 yields Newton’s gravitational
potential. The second term gives the influence of the ordinary matter.
.
Graviton: The Cosmic Membrane Theory does not use the concept graviton to transport gravitational
forces, and does not use this term, but the change of the position of a massive
body relatively to the membrane changes the curvature of the membrane. This
change propagates from korn to korn of the membrane, so as an acoustic
sound wave propagates from molecule to molecule of a gas. This mechanism of
propagation one can surely describe by the action of a graviton (gravitational
wave).
.
First estimation of the
Depth of Space Wo at the edge of the Sun:
To
calculate the exterior geometrical path lengthening dSE we solve the
integral eq. 3.6.
(3.6)
.
With
Feynman's value rEX=dSE=491 [m] and radius of sun R=6.958´108[m] we find a value of Wo= 1.432´106[m] or 1432 [km]. That is the depth of space at the edge
of Sun. The constants Ae , Fo und the ODE coefficient A
we can calculate now from the depth of space Wo:
Ether acceleration Ae=gs/W’o
= gsR/Wo =1.361´105[m/s2]
Membrane tension Fo=Mgs/(4pWo2)= 2.164´1019[N/m2]
ODE-coefficient A A=Ae/Fo=
6.289´10‑15[m2/kg].
.
Numerical calculations to curvature of space:
The
author performed in 1995-1996 numerical
calculations of the curvature of space to prove the correctness of the deduced
formulas. The first proof was done by a spatial grid made from regular
tetrahedras. Twenty regular tetrahedra form a regular icosahedra. The load of
the central point was directed in the fourth dimension. An improvement of the
icosahedra result was given by an enlarging of the number of points in the grid
and a better spherical shape. So the author used another grid forming a sphere.
The results showed convincing well the correctness of the formulas. The
calculated curvature of the grid yielded very exactly the 1/r-potential of
Newton's law of gravitation.
.
Speed of Gravity and gravitational waves:
Th.
Van Flandern (1998) supposes the propagation velocity of gravitation to be much
higher than the speed of light. The reason is the fact that the Earth
accelerates not towards the visible position of the Sun, but to its real
position. Inside the Cosmic Membrane Theory this contradiction is solvable.
Both directions - acceleration and path of light - are on one line. The reason
is the aberration that we do not see the Sun at its real place. The gravitation
has no aberration. It must not propagate, since it is already there in the form
of the gravitational funnel. Therefore, we do not have any reason to
suppose the propagation velocity of gravitation to be higher than the speed of
light c.
Gravitational waves may exist both, transversal
and longitudinal gravitational waves, comparable with gravity water waves and
water sound waves otherwise. The
amplitude of transversal gravitational waves is directed parallel to
w-direction. Here the membrane accelerates in w-direction, and we had to use
the unknown W-mass mW or the W-density rW. The only fact we can assume is that rW is much larger than the transversal inert
density r of the membrane. So, the
transversal waves are assumed to propagate much slower than longitudinal
gravitational waves. As final result for the total gravitational energy flow of
the system Sun-Earth we find
.
.
(6.2.14)
.
This
formula is equivalent to that given by Weinberg (1972) for the GT.
.
Momentum, Mass and Energy:
The
4-dimensional hyperspace S seems to be Newton's
absolute space. The confirmation of momentum is relative to this space, and not
relative to the other masses of the Universe, as Mach assumed, and also not
relative to the membrane. Heavy or gravitational mass is caused by the ether
wind force, and is proportional to the disturbance of the membrane by the
matter. Energy is the virial energy of the movement in the 4-dimensional
hyperspace as addition of the expansion speed VE and speed v inside
the membrane, and of the cyclic movement of the bricks of matter inside the
particles.
.
Special Relativity:
The
indroduction of a space filling resting stuff
leads necessarily to a special relativity, either based on the Lorentz
transformation or on an equivalent new transformation with cross and length
contraction. With both transformations one can explanate the change of mass and
time and the whole set of experiments, e.g. Thomas precession, the
interferometer experiments of Michelson-Morley, Fizeau (Fresnel's drag
coefficient), aberration and Airy's water experiment, the Trouton-Noble
condensor experiment, the Sagnac experiment with rotating interferometer, the
Kennedy-Thorndike long-time interferometer experiment.. The experiment of
Haefele and Keating needs additionally Einstein’s Principle of Equivalence. The
conservation laws of energy and momentum and Maxwell’s equations are valid.
Only the measured constancy of speed of
light is better to explanate with the new transformation.
×
Transformation with cross anf length contraction:
The
Lorentz transformation can not explanate exactly the constancy of speed of
light, as the SR and a lot of experiments demands. This fact is leading to the
new transformation with cross and length transformation. Some equations for
coordinates and time of points resting in a moved frame s' are (speed v directed in x-direction)
.
(4.2.8 a)
.
(4.2.8 b)
.
The
addition theorem for velocities is
.
, (4.2.18 b)
.
a
result, which is similar to the addition theorem of velocities in the special
theory of relativity.
.
Tidal and frequency effects:
The advance of time is a property of the
space, but no a dimension.
Kinematic Time-Dilation arises from any motion of a clock relative to CBR frame (Cosmic
Background Radiation). A moving clock runs more slowly. The formula is the same
as given by Lorentz or by Einstein
(4.7.1)
Gravitational Time-Dilation
of Clocks is an effect which is caused by a
gravitational field acting on the clock. The author has no objections to use
the explanation given by Einstein's GT for the gravitational time-dilation.
Gravitational Time Dilation
of Photons: Photons moving in a strong gravitational
field need more time than photons moving in field free space. In a week
gravitational field we find c(r)=co(1-2a/r) with 'a' the half
Schwarzschild radius of the central mass.
Doppler Effect is
an effect of first order in v/c. Since the light source or the receiver or both
are moving relative to the membrane (CBR frame) this effect is always connected
with the kinematic time dilation effect. For speeds v<<c we find the same
formulas as given in the case of the doppler effect of acoustic sound waves.
Gravitational Red Shift is an effect which is merged into the gravitational time dilation of clocks sometimes. This effects are
difficult to separate. The Cosmic Membrane Theory says: The frequency n does not change if the photon climbs upward, but the speed of light
changes with c(r)=co(1-rg/r)
and causes so a change of the wave length. Here co is the speed of
light outside of all gravitational fields. But what does energy and momentum?
If we use further the equation e=hn, we
have to assume that Planck's constant h depends on the gravitational field.
Red Shift caused by the
expansion of the Universe: The space stretches with the
expanding Universe, and stretches so a propagating photon. The effect is an
enlarging of the wave length.
×
Proofs of the Cosmic Membrane Theory:
The central assumption is the decrease of speed of light in the
gravitational funnel with
c(r)=co(1-2a/r) (5.1.1)
The Shapiro effect of signal retardation
by solar gravity is
(5.1.2)
The
result agrees with the General Relativity. The effect of higher order of the sun-near acceleration of speed of light
cr(r)=co(1+w’2/2) leads to a shortening of the
classical time dilation according to Eq. 5.1.4.
. (5.1.4)
Together
with the effect of the geometrical lengthening of the path both effects yield a
shortening of time dt=3pWo2R2/(16cy3), contrary to
signal retardation caused by the effect of first order. A regression analysis
of the 17 sun-near trajectory data of
Shapiro, Reasenberg et al. (1977) yields a value Wo=(1.204 ± 0,869) ´106 [m] for the depth of space at the edge of Sun.
×
The main effect of the solar gravitational deflection of
light or light bending is caused by the common gravitation together with
the effect of a decrease of the speed of light described above. The formula f(y)=-4a/y is the same as in the General Relativity. Additionally we find
three effects of the order 1/y4. Two of the three additional effects
of higher order cancel one another. The only remaining effect is the lateral
effect fC =‑3pWo2R2/(16y4)
of the centrifugal acceleration. The
data of the eclipses of Sobral 1919 (f=1.98±0.18“),
Principe 1919 (f=1.61±0.45“), Takegon 1929 (f=2.24±0.10“) and Timbuktu 1959 (f=2.17±0.34“)
yield the mean deflection angle Df=
2.093±0.193’’. The GR predicts f=4a/R=1.75“ for trajectories
grazing the Sun. We take the difference (2.093±0.193)’’-1,75’’=(0.342±0.193)’’
and compare it with the lateral effect fC. Thus, we get with y=R another estimate of the depth of space Wo=(1.168
± 0.659)´106 [m]. The
light deflection data quoted by Steven Weinberg (1972) deliver a weighted
average of Df= 1.92±0.18’’ and Wo=(0.823 ± 0.871)´106 [m].
.
Perihelion
Advance of Mercury: The perihelion Advance of elliptical orbits of
planets is caused by an additional relativistic term of Newton's
potential. The direct integration of the
equations of motion of a planet including the additional acceleration term ‑6gMa/r3 was done numerically for one revolution with sufficient
accuracy. The author got a value of 42.5“ per century and confirms so
sufficiently well the factor k=6 of the additional acceleration term. The hitherto
best experimental value is 43.20±0.20“.
An explanation of the factor k=6 and and therewith of the additional relativistic acceleration ‑6gM/r3 within the membrane theory we get from the relativistic
expression for the square of energy e2:
e2 = (mooco2)
2 + (pc) 2, (5.2.3)
and
from this the change of mass in the gravitational funnel,
m(r)=moo(1+3a/r). (5.2.6)
(Formula revised 11/2004)
The term 3a/r contains the kinematic dependence
of the mass on the speed, i.e. a/r, and the dependence of the mass on the
gravitational field, i.e. 2a/r.
.
If the NASA has neglected this acceleration, as
the author supposes, the parameter fit of the solar system parameters will be
only hold inside the planetary belt. As the spacecrafts Pioneer 10 and 11
leaved our solar system, the fit failed, and the implicitely added acceleration
of aA = (8 ± 1) ´10-10 [m/s2] became visible. The author
has written a small program in C, Pioneer.cpp. This program is calculating the
error made by the neglection of the Gerber-Einstein acceleration. By addition
of a "correction constant" of
CC = -10.5 ´10-10 [m/s2] the sum of error squares is
minimized inside the planetary belt (from Mercury to Uranus, or a distance
until 20 astronomical units). If one sums the errors until a distance of about
100 AU with additional measuring points, we had to reduce the correction
constant to a value of CC = -3.2 ´10-10 [m/s2]. We see, leaving the solar
system we had to diminish the falsely introduced correction. Between 20 AU and
100 AU by the amount of 7.3 ´10-10 [m/s2] getting a new value of CC =
-3.2 ´10-10 [m/s2]. The Gerber-Einstein acceleration does not influence
the rotation curves of the galaxies. Calculations with the program Darksim9.pas
show that the coefficient of the ODE term of curvature, C*w'2 with C=2.601´10-11 [1/m],
is to small. This value arises from the comparision of the integral of
the term C*w'2 multiplied by
the ether acceleration AE
compared with the Gerber-Einstein acceleration. A value of C=2.26´10-6 [1/m] is needed to yield a flat rotation curve with speed 245 km/s in
its flat part.
.
Particles:
The
basic brick of our world is a torus-shaped whirlpool (eye, in German Auge). The
finite extension of its inner channel causes a finite total ammount of the
outer flow. That means the flow decreases with a higher exponent of (1/r) than
3 and the force between two auges decreases with (1/r)6 at least.
.
The electron model: The basic idea is that the kinetic energy of
the drilling space surrounding a charge (a rotating pair of Auges) is
identically with the energy of the electric field. The membrane theory yields
the density of space. From this density and some assumptions about the angular
speed of the rotation we may calculate the energy. This energy depends on the
inner radius, where we are stopping the integration. If we compare the
integrated energy with the energy of the electric field of an electron, than
the stop radius of integration must be somewhat as the radius of the electron.
The calculation yields the results of the model in the first column. The known
values of literature are given in the second column:
J=5.269´10-35 [Js], JE =5.273´10-35 [Js], error=0.08%
m=9.307´10-24 [Am2], mB
=9.285´10-24 [Am2], error=0.24%
l=2.362´10-12 [m], lC = 2.426´10-12 [m], error=2.71%
RG=7.183´10-13 [m], RE=7.078´10-13 [m],
error=1.48%
Model parameters of the electron model are:
Great
half axis a =1.1210´10-13 [m],
Small
half axis b =0.6647´10-14 [m],
Radius
of auge RA =0.906·b
Ratio
of speeds VA/c = 1.860
Angular
frequency w = w(Z) / 2
with
angular frequency w(Z) =1.5527´1021
[1/s] of the zitterbewegung of
the electron, the spin angular momentum JE =5.273´10-35[Js], the Bohr's magneton of the electron mB =9.285´10-24 [Am2]
and the Compton wave length lC = 2.426´10-12
[m].
Cosmology:
There
is a significant difference between GR based cosmology and membrane Theory. In
all GR based models, e.g. the Einstein-Friedmann models or the
Friedmann-Lemaitre models, the expansion of space depends on mass or energy
parameters. The Friedmann equations suggest that baryonic mass, dark matter or
a cosmological constant (vacuum energy) can influence the expansion rate of our
universe. The Membrane Theory denies this imagination. The mass and the speed
of the membrane is so overwhelmingly greater than all baryonic mass, dark
matter and other forms of matter or energy inside our three spatial dimensions
that this forms of matter can not influence the speed of expansion in any way.
The Friedmann equations describe a three-dimensional cosmos, Membrane Theory a
four-dimensional one. In this context questions became obsolete concerning the
critical mass, W,
e.g., or the signature of the metric. The metric of our universe is spherical,
but nearly flat in the visible part because of the vast radius.
×
The
cosmological constant L is dominating the expansion
rate (Hubble constant) since 10 Billions of years (Blome and Priester 1991,
Priester 1995). This authors use the following numbers: Standardized
cosmological constant lo=Lc2/3Ho2=1.08, Hubble constant Ho=90
[km/sMpc]= 2.91210‑18[1/s],
radius of the universe R=36109[ly], =3.4081026[m].
Using the Friedman equation 7.1.1 we get the real cosmological acceleration aL by eq. 7.1.3.
(7.1.3)
With
the above value R=3.4081026[m] we find the value a L=‑3.1210‑9[m/s2].
The cosmic membrane model deals with two forces: The tensile force Fo=2.114´1019[N/m2]
of the membrane and derivated from it and from the radius of the universe the
pressure pL. From this we find
(7.1.5)
with
a valueof a M=‑0.79210‑9[m/s2].
There is the factor 4 between the both estimations of the acceleration. One should not over-evaluate the difference
between the two results, since one is calculated by Priester and Blome from
astronomical observations, the other from physical constants. But the values of
both accelerations are to high. Calculating the age TU of the universe
starting with an initial speed of expansion c=3´108[m/s], we get
not more than a value of TU=3´1017 [s] or 1010
years. This estimation is to short. The membrane model allows a higher speed VE
of the cosmic expansion than the speed of light c, since c holds only inside
the 3-d membrane. From this follows a much higher longitudinal inert mass rM=rW and additionally a higher speed. That means,
the radius R of the universe could be of another order of magnitude, and the
life-time of our universe may be much greater.
.
Dark
Matter denominates the phenomenon that the outer
stars of galaxies rotate with speeds which are too high and, therefore, not
explanable from the gravitational forces of the visible matter. Theory divides
Dark Matter in Cold and Hot Dark Matter.
The cosmic membrane theory says that dark matter is not really existent,
but it is an anomaly of the gravitational law. It is caused by effects of the
ether wind on a perturbated membrane. Two imaginable effects (two of some) the
author had selected. As seen in chapter 3, the korns (torus shaped curls) of the membrane experience a lateral
force Fec if the axes of the curls are not directed exactly parallel
to the ether wind, similarly to the buoyancy of a wing profile. This buoyancy
causes two other forces, Fa
and Fer. Fa is a direct component of the lateral force Fec,
and Fer is caused by a
thickening of the membrane stuff in the funnel by the hydrostatic pressure. We
get a new ODE for the curvature of space,
. 
(7.2.7)
The
third term is caused by the new force Fa and the fourth term by the new force Fer. With the depth of space Wo=1.432´106 [m] we get ODE-coefficient A with a value of A= 6.289´10‑15[m2/kg]. With this number, the ODE
7.2.7 has been solved numerically, seeking well performing values for C and D.
Values of C=1.909´10‑18[1/m]
and D=1.596´10‑26[1/m2]
yield a reasonable result (good looking rotation curve) with a dark-matter
coefficient of d=3 at a distance r=RG (radius of galaxy).
.
Although the Cosmic Membrane Theory does not
know the original frame dragging effect in the sense of Lense and Thirring, a
similar effect is thinkable. The membrane stuff is not a rigid body, not
crystal like, but is more a liquid with strong adhesion between the korns and
nearly frictionless. The word 'nearly' is essential here. If a heavy mass is
spinning over billions of years, one can imagine that it transmits a part of
its rotational energy to the
surrounding membrane and makes it rotate too.Since membrane stuff has a
density, and following inertial mass, we should await a centrifugal force,
because the trajectory of the membane particles is curved in the 4-dimensional
bulk space. This force is acting on the membrane, and causes a change of
curvature similarly to an additional load with dark matter.
The
rotation curve v(r) has not an even course for radii>0.3 Rg, as the example
of Begeman has, but v(r) is decreasing from 300 km/s to a value of about 220
km/s at radius Rg. That means that space drilling can not be the only source of
the dark matter effect.
The
choice of function w(r)= wo(Ro/r) modelling the decrease of
angular speed of rotation with increasing radius was controled by the aim to
get the ideal rotation. Other curve models, e.g., w(r)= wo(Ro/r)K with K>1 gave worse results, so that
K=1 was used. This function w(r)= wo(Ro/r) is adequate to a cylindrical flow with infinite
rotational axis. That is another weak point of the model. A spherical spinning
body as source for a pipe shaped flow pattern is difficulty to imagine. We had
to assume some decrease of angular
rotational speed propagating in direction of the rotational axis. But so we
lose the nice property of a curl free flow pattern.
.
Expansion of the Universe: An actual question of astrophysical relevance
is the discussed deceleration versus acceleration of the expansion of the
Universe. The Cosmic Membrane Theory yields here a "third way": The
real speed of expansion is constant or slows moderately, but because of an
acceleration of the velocity of light we get the impression of an accelerating
expansion of the Universe. One effect is that far light sources seem to be
fainter as expected from their redshift z.
The
modelling of the stretching membrane includes: A stretching membrane thins,
membrane tension F grows, membrane density r shrinks. This changes must
have some influence on the physical constants. We assume the total mass of the
membrane to be nearly constant and get the changing membrane load (load per
volume unit)
. (7.3.3)
Otherwise,
membrane tension F should linearly increase with an expanding balloon. That
follows from the assumed elastic behaviour. We find::
, (7.3.4)
and
from the wave equation c2=F/r we deduce the changing
speed of light
. (7.3.5)
Power
exponent P we had to find in the range P=0.5 (constant density r) to P=2 (case r=ro(R/Ro)3 ). The author has no physical argument to
chose the best value, but a good fit with the supernovae results we will find
with a value of P=0.5 only. Because of the direct equivalence of time ticks and
frequency, we find n(r)=no (1-a/r). Frequency no is the undisturbed frequency of the source far
away from each gravitational field. We suppose that both changes, the change of
c and n,
respectively, follow from a change of the properties of the membrane
surrounding the Sun. We get from this assumptions the changing time (frequency)
. (7.3.10)
Going
back in time the speed of light and the frequency of light sources decrease. In
the tough soup of the early Universe all things went slowly. What does our
length scale s? Our length scale, e.g. the meter,
is defined by a fixed number of wavelengths of
a certain type of light. We find
. (7.3.11)
Our
length scale s grows with the growing Universe. In the literature (e.g. Blome
et al. 2002) we find statements of the form "the distance between the galaxies grows with the expanding Universe".
The question, whether the galaxies grow too, is much more difficult. The author
thinks they do so, and eq. 7.3.11 is helpful here.
×
But
wat happens with a photon at his long travel through space and time? The common
assumption is that the photon is stretched as a drawing of an wave at the skin
of a balloon, which we blow up. The membrane model yields a second contibution
of stretching: If speed of light grows,
the wavelength grows too So, for the
membrane model redshift zm we find from the there-speed-and-frequency
transformation, back-speed transformation and back-stretching transformation
.
(7.3.12)
Eq. 7.3.12 differs somewhat from the common
redshift zF given by Friedman, i.e., zF=(Ro/R)‑1.
In the membrane model a given redshift zm yields a greater radius R
than Friedman's redshift zF does.
×
In
opposite to Friedman's model with constant speed of light the horizon grows
more slowly here. We will never see the whole Universe. We should remark two
important points:
·
The Universe seems to be flat for us, because
of the vast radius R. Expansion speed VE is assumed to be much
higher as speed of light.
·
The inflation of the Universe at the first
milliseconds after the Big Bang or Big Bounce (cf. Guth 1981) is not in
contradiction to Cosmic Membrane Theory. The first milliseconds of the
eigentime t had been very long compared with the milliseconds of an outer observer
with a recent time tick. In a short eigentime the radius R(t) grew with nearly
infinite speed. But that is relative. All processes went very slowly during the
real radius of the Universe was growing stadily with constant speed.
.
Now
back to the sopernavae candles. A calculation shows that using the above model
a far galaxy seems to glow fainter indeed than the distance calculated from
redshift prescribes.
×
The geodesic precession is a mathematical effect with no physical relevance. Astronomers are
using different reference frames. The GCRS (geocentric reference system) is a
non rotating reference system connected with the Earth. The BCRS (barycentric
reference system) is a non rotating reference system connected with the Sun. If
one goes from one system to the other, one had to consider the different time
flow influenced by Special and General Relativity.
×
The Lense-Thirring effect or frame dragging effect is
an effect of gravitational models using the graviton.
The graviton is thought to be a particle or wave transmitting the gravitational
force between distant masses with speed of light. The Cosmic Membrane Theory
does not know the point to point connection of particles by gravitons, and,
consequently, it does not know the Lense-Thirring effect in this form.
×
The
geodetic precession is the rotation of the spin
axis of a gyroscope in an isotropic gravitational field of a static mass
distribution. "Gravity Probe B - the 'gyroscope experiment' - was a NASA
space experiment in 2007 designed to measure the general relativistic effect
known as the dragging of inertial frames.
In Everitt (1991) we find the following formula of the geodetic
precession:
. (7.4.4)
The main effect causing the Geodetic Precession
is the decrease of speed in the gravitational funnel. The equation is c(r) = c0 ( 1-2a/r ). From this equation we
find by a conclusion by analogy the eq. 7.4.6.
.
v( r )
= vo ( 1 – 2a / r )
(7.4.6)
The gradient of the speed is dv/dr = vo 2a / r2 ,
and has the dimension of an angular speed. An integration over one year results
in the revolution angle W1 :
.
(7.4.7)
This angle is equivalent to 8.8
arcseconds. This is some more as the value of 6.6 arcseconds the
General Relativity is giving. The second effect of the membrane considered now will
give the correction. Both effects together yield the exact value of 6.6
arcseconds. Eq. 5.2.15 from section 5.2,
m(r) = moo (1 + 3a/r) = moo (1 + a/r + 2a/r) , says that mass will change with its
distance r from the gravitational center. Here is only the term 2a/r of
interest. It describes the change of mass caused by the changing properties of
the membrane in the gravitational funnel. In a position 1 a volume element of
the gyroscope moves away from the Earth on that side of the spinning top which
is seen by the viewer. On the backward side the volume elements are moving in
the direction towards the Earth. Supposing constant speed v, the rate of change
of mass, dm/dt, causes a change of the momentum v×dm/dt with the dimension of a
force. On the other side of the spinning top the rate of change of mass has the
opposite sign, and also the force. This pair of forces together with the lever
arms L will produce a torque. The torque does only depend on the angle q
now.
.
.
(7.4.11)
The precession per year is
.
.
(7.4.13)
.
This value is equivalent to 2.2
arcseconds. The total geodetic precession of one year is the difference
of W1 and
W2
.
.
.
(7.4.14)
.
Equation 7.4.14 is equivalent to Everitt’s
equation 7.4.4. This is the same value which is also given by the GR, about 6.6
arcseconds.
×
Change
of constants:
.
Mass
changes with the speed of the reference frame and with the gravitational field:
m(r)=moo(1+3a/r).
Velocity
of light changes with the gravitational field: c(r)=(1-2a/r). co is
the speed of light in a vacuum for r®¥, 2a
is the Schwarzschild radius of the Sun. The membrane gives a hypothetical
explanation for this effect using the hydrostatic pressure and the dispersion
equation n=1+4pN(e2/h)Sa2nk(nnk/(n2nk-n2))
.
Velocity
of light and with it the wave length, frequency of the source and with it the
emitted wave length change during expansion of the Universe: c(R)=co(R/Ro)0.5
.
Time
changes with the speed of the reference frame, with the gravitational field and
with expansion of the Universe
.
Planck's
constant changes with the gravitational field.
.
Length
scale changes with speed of a reference frame and with expansion of the
Universe
.
Further conclusions:
.
Spontaneous
creation of matter seems to be possible, because the resistance of the existing
matter inside the membrane is producing a great amount of energy This energy cames from the virial energy of
the expanding membrane.
.
The
radius of the universe may be much larger than 1026 [m] (assumed by
different authors), and the Universe is closed, i.e. there does not exist any
border and therefore no distance R. But there exists a radius R of the balloon.
.
The kinetic energy of the membrane stuff (sometimes
quintessence) is it what decreases during expansion. Most of this energy will
be stored in the tension of the membrane.
The existence of black holes is not in contradiction
to the cosmic membrane paradigm. On the contrary, the existence is well
imaginable using the membrane paradigm.
.
Cosmic membrane theory includes the possibility of a
membrane rupture comparable with the burst of a soap bubble. That will be the
end of this Universe, and mankind has
to seek a neighboured Universe to live in.
.
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of part 4 and end of script