Part one
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COSMIC MEMBRANE THEORY
OF GRAVITATION
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STEFAN M. VON WEBER
Department
of Mechanical and Environmental Engineering,
Fachhochschule Furtwangen, Germany, 78120 Furtwangen
×
ABSTRACT
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The proposed membrane theory of gravitation is of type Kaluza-Klein with
non-compactified fourth spatial dimension and delivers Newton’s law of
gravitation in a direct way and explains light bending, Shapiro effect and
perihelion advance of Mercury with the same accuracy as the GR. There is no
reason to believe that the speed of gravity is greater than the speed of light.
From the point of view of the proposed Cosmic Membrane Theory the General
Relativity of Albert Einstein is a projection of the 4-dimensional space into
the 4-dimensional spacetime. 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. Two new cold dark matter candidates (CDM)
are presented. It is a thickening of the cosmic membrane caused by ordinary
matter leading to a long-reaching violation of the 1/r-potential, and the
deformation of curvatre caused by the spinning center of a galaxy.
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Key words:
Kaluza-Klein, membrane, gravitation, relativity, dark matter
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The Content of this Paper
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------------------------
part 1 ----------------------------------------
1 Introduction
2 Newtons Law and the Cosmic Membrane
3 Differential Approach and Numerical Calculations to Curvature
4 Special Relativity
4.1 Longitudinal and Transversal Contraction
4.2 Coordinate and Time Transformation
---------------------------
part 2 ------------------------------------
4.3 Momentum, Mass and Energy
4.4 Membrane Theory and Basic Physical
Phenomenons
4.5 Experiments Concerning Special
Relativity
4.6 Apparent Constancy of Velocity of Light
4.7 Tidal and frequency effects
5 Classical Proofs of General
Relativity
5.1 Shapiro Effekt, Light Bending and Depth
of Space
5.2 Perihelion Advance of Mercury
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part 3 ------------------------------------
6 Novel Proofs of General Relativity
6.1 Electron and Space Torsion
6.2 Gravitational Waves
6.3 Anomaly of Pioneer 10 and 11 data
7 Cosmology
7.1 The Cosmological Constant and the
Expansion of the Universe
7.2 Dark Matter caused by the etherwind
7.3 Quintessence or Dark Matter and the Expansion
of the Universe
7.4 Dark Matter and Frame Dragging, Geodetic
Precession
8 Analysis and Conclusions
References
.
Link
back to the index:
home:
http://www.fh-furtwangen.de/~webers/index.htm
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Last
revision of this paper: 11/2006
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1 Indroduction
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(First version 03/1997, last update 05/2004) In 1919 two expeditions
were undertaken to Sobral and Principe to observe the solar eclipse with the
aim to proove the General Relativity of Albert Einstein (1916). One of the
initiators was A.S.Eddington. The vertical axis in Fig. 1 shows at that time observed angles (in
arcseconds) of light bending for sun-near trajectories (Mattig 1999). The
smooth line is the forecast of the General Relativity. The horizontal axis
shows the nearest distances y of the trajectories to the centre of the Sun
(scaled in radii of the Sun).
×
Fig.
1. - Light bending data 1919.
The
line is the forecast of General Relativity,
y is
the distance of trajectory in radii of Sun
×
Later
it was said Eddington had only measured what he had wished to see, and
moreover, the error of measurement had been greater than the effect he seeked.
Other measurements (in Takegon 1929, in Timbuktu 1959) confirmed Eddington, but
showed also that an additional amount of light bending is possible and likely
in the special case of sun-near trajectories, above the amount General
Relativity predicts.
The reason sun-near light bending
measurements are no longer carried out is the use of satellites and radio
signals of quasars. These measurements confirm the theoretical value f=4a/y of General Relativity with such an accuracy that sun-near
measurements are thought to be unnecessary. (2a is the Schwarzschild radius of the Sun, y the
perpendicular distance of the light trajectory from the centre of the Sun.)
What a pity! The HIPPARCOS data only covers values upto 45° from the Sun, e.g.
Robertson and Carter (1984) found sun-near deviations of the value f= ‑ 4a/y, too. They discussed the deviations, but took them as errors of
measurement or as influences of the
corona of the Sun. (To deletion of sun-near data see also Reasenberg, Shapiro
et al. 1979 , to variance of bending data see Hamana, Martel & Futamase
2000).
In 2002 Fomalont and Kopeikin measured the
light deflection in the gravitational field of the Jupiter. But nearest
distance was 14 times the Jovian radius (Fomalont & Kopeikin 2003), much to
far to detect effects of order 1/y4. C.R. Keeton and A. O. Petters
(2005, 2006) discuss the lensing of a quasar by the galactic black hole or
other black holes to find the signatures of the fourth spatial dimension. In
2003 Bertotti et al. measured with very high precission the frequency shift in
a radio signal transmitted by the Cassini spacecraft on its travel. The nearest
distance to sun of the signal trajectory was about 1.6 sun radii.
Unfortunately, for this conjunction point no frequency residuals are available.
Membrane Theory expects for this point of the trajectory a negative deviation
of about 3% from the General Relativity value according to the increase of the
radial component of light speed. The effect decreases with 1/y4,
where y is the nearest distance of the signal trajectory to the edge of sun.
The author is seeing here just the starting
point of a progress in the General Relativity theory. From point of view of the
Cosmic Membrane Theory the General Relativity of Albert Einstein is a special
projection of the 4-dimensional space into the 4-dimensional space-time.
×
In 1916 Albert Einstein had published his
General Relativity with the idea of the curved space-time. The fourth dimension
was born. In 1921 Kaluza (Kaluza 1921) extended the 4D space-time by the
addition of a fourthh spatial dimension and made so the first attempt to unify
electrodynamics and General Relativity, but the lack of physical evidence was
criticized by scientists at that time. In 1926 Oskar Klein (Klein 1926)
suggested that this extra dimension might be coiled into tiny, subatomic-scale
tubes of Planck length. That was the starting-point for leaving pure geometry
and entering subatomic strings and branes and to try the unification of quantum
theory with General Relativity. See e.g. Dirichlet-Branes by J. Polchinski
(1995), string and M-theory and supermembrane by Seiberg & Witten 1996,
Aldabe and Larsen 1996, Keyl 1997, Duff 1998, Duff, Liu and Sati 2000, Greene
2000, Assis et al. 2000, Furtado, Bezerra & Moraes 2000, Duff, Liu &
Sabra 2001, Khoury et al. 2001, Huey
and Lidsey 2002. To control the use of a lot of additional dimensions six
dimensions are curled into a Calabi-Yau manifold (c.f., e.g., Kachru, Laurence
and Silverstein (1998). Siegel (2006) denies additional, not compactified
spatial dimensions, because of a conflict with gravitational experiments.
.
The non-compactified additional fourth
dimension we find, e.g., in von Weber 1998, Randall-Sundrum 1999, Suntola
2003a. Non-compactified additional dimension Kaluza-Klein theory combined with
quantum theory we find in Wesson et al. 1996, Pavsic 1997, Randall-Sundrum
1999, Darabi, Sajko & Wesson 2000, Dvali, Gabadadze and Porrati (2000), Deffayet,
Landau, Raux, Zaldarriaga and Astier (2002), Louko and Wiltshire 2002, O'Callaghan,
Echaurren and Mills (2003). Spatial whirlpools in General Relativity were
investigated by Cartan in the 1920-th and then by Hehl et al. (Hehl 1973, Hehl,
Heyde, Kerlick and Nester 1976, Hehl and Obukhov 2001). Another strong impetus
came from the cosmology (cf. e.g. Turok & Hawking 1998, Turner & Tyson
1999). The big bang, inflation theory and cold dark matter are essential
mile-stones.
×
The idea that vacuum is more than empty
space goes back to Faraday and Maxwell, to Dirac, Casimir and de Sitter (cf.
Haisch & Rueda 2001), Brans and Dicke 1961. Ponce de Leon 1988 and Billyard
& Wesson 1994 connect GR, the fifth dimension and the de Sitter vacuum.
(Cf. also Haisch, Rueda & Dobyns 2001.) A class of Kaluza-Klein solutions
curved in 4D and flat in 5D, but without oscillatory term, gave Liu &
Wesson 1998, Liu & Mashhoon 2000. A 4D theory using higher terms of the GR
field equations is given by Lewis, Doran & Lasenby 2000. That idea will be
essential for the proof of the Cosmic Membrane Theory. W.G. Tifft (1997) and I.
Schmelzer (2000) remark that the cosmic frame appears to be fundamental.
×
The "membrane viewpoint" goes back
to the Znajek-Damour bubble-formalism and to the "membrane paradigm"
of Price and Thorne (Price & Thorne 1986). To quote them: "The membrane viewpoint has the goal
of providing astrophysicists with mental pictures physical intuition,
computational techniques, and other research tools." The notion domain wall is sometimes used
instead membrane (c.f., e.g., Huey
and Lidsey 2002, Gass and Mukherjee 1999). To quote Deffayet et. al. (2000); "
The brane-induced gravity models are a particular class of brane-world models,
which can be defined as models where our four dimensional (4D) universe is
considered to be a surface (called 'brane') embedded into a higher dimensional
bulk space-time. Brane-world models are inspired by superstring-M theory, and
can be regarded as some low energy effective models of more fundamental
underlying theories, but have also interest on their own in providing new
phenomenological ideas. We will only consider here the case where the bulk is
five dimensional (5D)".
×
Fig.
1.1 - Model of expanding cosmos
×
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 (brane, supermembrane, quantum vacuum, expanding
shell) expands with high speed VE in the same manner as a balloon is
blown up (Fig. 1.1). This membrane is our cosmos. (Do not take the Cosmic Membrane for a Domain Wall as described in Gass and
Mukherjee, 1999, but think a stuff with density, pressure or tension and
perhaps some velocity (Schmelzer 2000)). The author thinks the membrane
consisting of tiny torus-shaped curls (korns, grains) with diameters of Planck
length. In M-Theory we find similar constructions, e.g. the curled Calabi-Yau
spaces (cf. Kachru et al. 1998, Greene 2000), but in this theories the
existence of additional dimensions is taken as physical reality. Similiarly to
cosmic membrane theory the scalar field F in some quintessence
theories is a function of speed VE (e.g. Carroll 1998 or Albrecht et
al. 2001). But in the cosmic membrane theory the properties of the membrane
depend merely on the tension, i.e., they depend only in an indirect way on the
speed VE of expansion. Jimerson (1991) gave a similar model, but
without the ether flux.
×
The membrane is of tiny thickness in the
fourth dimension and has a strong tension (c.f. Aldabe and Larsen 1996, Battye
et al. 2005), 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. This model avoids the difficulties with
Maxwell’s equations (cf. Burgbacher, Lämmerzahl and Macias 1999). The size of
the curls hasn't changed from the Big Bang up to now (Kolb et. al. 1986) or has
changed (Schmelzer 2000). Here the author has no opinion. A five dimensional
space-time one may get also from the heterotic M-theory (cf. Witten 1996,
Horava and Witten 1996, Randall-Sundrum 1999, Dvali, Gabadadze and Porrat
(2000), Louko and Wiltshire 2002, Deffayet et al. (2002)). In the GR, time was
established instead of the fourth spatial dimension, but hitherto, the author
avoids its use as fifth dimension in the first instance. (Greene (2000) says
that the number of time dimensions is open.).
×
The motor of the expansion of our cosmos is
the momentum of the mass of the membrane (In this point there is a difference
between Schmelzer's General Ether Theory and Membrane Theory (Schmelzer 2000)).
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
(cf. Vuletic 1997). But this is not the return to the Steady State Universe
of Hoyle, Bondi and Gold (1948).
×
The Cosmic Membrane Theory is not a quantum
theory. But this fact does not mean that the author denies the existence of a
thing called graviton. 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. But that what
gravity means usually - the force between two massive bodies with a great
distance between them - is connected with the graviton indirectly only. In this
case the pure curvature of space, i.e. the slope of the membrane, together with
the force of the ether wind is the cause of gravity. In this sense the graviton
is only good for the micro-world (cf. Kogan 2001). The decission whether the
graviton is massless or massive is in this context not important, since a mass
has no influence on the large scale gravity (Kogan 2001).
×
The pro and cons to the vacuum (field,
fabric, zero-point energy, superfluid or membrane) and to the (a)ether fill
libraries. To quote here Einstein (1920, 1950), Ponce de Leon 1988, Billyard
& Wesson 1994, Weber 1995, Schmelzer 2000, Haisch & Rueda 2001, Haisch,
Rueda & Dobyns 2001, and to a serious aether discussion Prokhovnik 1993,
and the author remarks that membrane
stuff and aether are two different things. From point of view of Cosmic
Membrane Theory the General Relativity is in all static cases a projection of
the 4-dimensional space into the 4-dimensional space-time.
×
The Brans-Dicke relativistic theory of
gravitation (Brans and Dicke 1961) is founded on Mach's principle that the
phenomenon of inertia ought to arise from accelerations with respect to the
general mass distribution of the universe. To avoid long-range interactions,
Brans and Dicke suggest a scalar field f (cosmic field). Its field
strength has the dimension of G-1=c2/g, i.e. [Kg/m]. Field
strength f is
proportional to gravitational potential of the total mass MU of the
universe. We think a gas sphere with average density r=10-26[Kg/m3]. The radius of the Universe used by Brans and Dicke was about
R=1026[m]. The potential at the centre of the sphere is 
1026[Kg/m]. Because this value is not far from 
1027[Kg/m], Brans and Dicke stated f=G-1. The field equations for gravitation are now obtained by replacing G
with f-1 in Einstein's field equations and adding an
additional energy-momentum tensor Tfmn for the f-field
in the source of the gravitational field. The theory of Brans and Dicke is interesting
in that sense, that it is giving a deeper insight into the workshop of
gravitation compared with the GR of Einstein, and it is not far from membrane
theory. From point of view of the membrane theory now three remarks:
·
Field f has here properties of the membrane tension Fo, especially
its propotionality to G-1,
·
but the radius of the universe may be much
larger than 1026[Kg/m] (c.f. ch. 7.1), and
·
the Universe is closed, i.e. there does not
exist any border and therefore no distance R
×
The
class of Dvali-Gabadadze-Porrati brane worlds of 3D branes in 5D bulk
space-time yields Newton's law of gravity too, but only in the near surrounding
of a mass. For great distances r a potential arises which is descending with
1/r2.The supposition that gravity propagates through the bulk space
complicates the theory. The bulk metric (see Deffayet et al. 2002) can only be
an Euclidean metric. Correct is the idea of some "coupling between bulk space-time and brane localized matter
fields when the conformal invariance of the brane theory is broken". The
author calls this phenomenon etherwind.
×
It is interesting, that the biologist C.
Calvet (2000) finds a similar field (Background Field) by the means of pure logic. The features are similar to those of
the cosmic membrane, i.e. it may form a sphere in a hyperspace, and it has
tension by the action of strings, and last but not least, torsion yields
electromagnetic fields. Interesting also the idea that at the centre of the
Universum could exist a Primeordial Star as a remainder of the Big Bang
explosion. But from view of Cosmic Membrane Theory we would have no chance to
observe it.
×
The important questions of special
relativity will be discussed in an own chapter. The Lorentz transformation or a
similar transformation with length and cross contraction (Weber 1995) are
candidates. With both transformations one can declare the change of mass and
time and the whole set of experiments, e.g. Thomas precession, the experiments
of Michelson-Morley, Fizeau and Trouton-Noble. 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.
Gravitational waves are imaginable as transversal waves as well as longitudinal
waves. The existence follows from 3D analogy using the common wave equation. (Weinberg 1972, Fliessbach
1990, White et al. 1993). In the Cosmic Membrane
Theory the frame dragging (Thirring-Lense effect) is logically connected with
the electrostatic field of the electron. Starting from the depth of space at
the edge of Sun there exists a logical chain of calculations ending with the
Compton wave-length of the electron. It will be fitted with an accuracy of
0.8%.
×
Chapter 2 contains a geometrical derivation
of the curvature of the membrane for the special case of radial symmetry,
chapter 3 a differential approach. In chapter 4 it will be shown that the
existence of a space filling stuff (vacuum, superfluid or cosmic membrane) does
not contradict Special Relativity. Besides the Lorentz transformation the
author describes another possible transformation with length and cross
contraction, and well known experiments and their relevance to the membrane
theory will be discussed in brief. Chapter 5 deals with the classical proofs of
the General Relativity as Shapiro effect (travel time retardation), light
bending and the perihelion advance of mercury. The perihelion advance of
mercury is used to discuss some basic questions. Here we find the important new
constant moc2. Space drilling and gravitational waves are
novel proofs. Chapter 6 deals with them. Chapter 7 presents a new cold dark
matter candidate (CDM) and thoughts concerning some other cosmological questions.
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2 Newton’s Law and the Cosmic Membrane (First version 11/1995, last update 04/2005)
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In the past there were made several
attempts to explain Newton's law of gravitation. The General Relativity does
not derivate this law from basic principles, but uses it as limiting law to
find solutions of the field equations. All attempts to use quantum theory (cf.
e.g. Th. B. Andrews (2001)) had to define an unlimited speed of the wave
functions, because of the measurements of Tom Van Flandern (1998) concerning
the direction of gravity between earth and sun.
But Newton’s law we can derive from the
membrane hypothesis directly (Weber 1998). The expanding cosmic membrane is
exposed to the ether wind (Fig. 2.1). If the membrane and the ether wind are
exactly perpendicular then each korn
and so the membrane in all shall experience no or insignificant resistance. But
matter in the membrane resists and causes a deformation of the membrane. The resistance of the membrane is caused by
the tensile force Fo (connected over c2 with the density r of the membrane. In Turok
& Hawking (1988) Fo appears as four form field. Interesting here
is also the theory of Patricio Valdés Marin (2003). The author cites him:"
I suggest consequently that the cause of
universal gravitation is the expansion of the universe." O'Callaghan, Echaurren and Mills (2003) have
similar ideas. I quote them: "The
energy associated with these energy vortexes that is directed perpendicular to
the surface of three-dimensional space to become depressed with respect to the
fourth spatial dimension".
×
The hyperspace (or bulk space, as sometimes
called) S
with its four spatial orthogonal dimensions X, Y, Z, and W is supposed to be
flat (Liu & Wesson 1998, Darabi, Sajko & Wesson 2000, Schmelzer 2000,
Huey and Lidsey 2002, M. Grady 2002). So we have to deal with a simple
Euclidean metric gmn. A rest frame moving with the membrane shall have the orthogonal axes
x, y, z and w. The scale factors are unknown. They depend on the unknown
velocity VE of expansion. W is transformed to w by a Lorentz-like
formula w=(W-VE t)k. Factor k
contains the unknown scale and time transformation. In this rest frame the
Euclidean metric is simply
.
But x, y, z are not coordinates inside the
membrane. The simplicity of the 4D metric hides the complexity of 3D forms,
since there is no term containing the time as in the GR or in all other
Kaluza-Klein theories (cf. e.g. Ponce
×
×
Fig.2.1
- Deformation of memebrane by ether wind at place of matter
×
de Leon 1988). Furthermore, to
get a solution in our common coordinates, we had to perform some
transformations. In the membrane theory the velocity of light is a local
property of the membrane (Ellis et al. 2000). Time is a property of physical processes (Hawking 1988) and depends
on the local position at the membrane and on the motion relative to the
membrane. The absence of time in the metric means that the membrane theory in
this special form can deal with a static geometry only. The use of the
orthogonal "world coordinates" x, y, z, w we can compare with the
introduction of a hypersurface-orthogonal fiducial observer (FIDO) by Price and
Thorne 1986.
The author sees the fourth
spatial coordinate not as a trick to simlify mathematics (c.f. Durham 2000). It
is reality, and we can measure the curvature of our 3D subspace.
×
In
the case of a single central load the cosmic memrane (the 3D subspace) deformes
with radial symmetry. The ordinary differential equation (ODE) of the deviation
w(r) from its zero position we can deduce geometrically from the analogous case
of a 2-dimensional membrane in the 3-dimensional space x,y,z. (Weber 1998,
compare also Brill 2000). The membrane shall be ideal, i.e. it stretches linearly to any tension. Fig. 2.1 and 2.3
show the gravitational funnel surrounding a mass M. The funnel is of spherical
symmetry if the mass is a sphere and other masses are fare away. That means, we
can restrict us to the coordinates r and z. Our first aim is to find the
equation of curvature z"(r). Our way is purely gemetrical here. The
starting point is the equilibrium of forces for a small piece of the membrane
with the form of a saddle as shown in fig. 2.2. If the tension Fo of
the membrane is much greater as the additional tension caused by the load, then
the two pairs of forces Fr-Fr and Fz-Fz have got nearly identical amounts.
(Strictly speaking neither Fr is showing exactly in r-direction, but is laying
in the membrane, nor Fz is showing exactly in the z-direction.) As a
consequence the angle between the two forces Fr must be identically with the
angle between the two forces Fz, since the resultants must cancel one another.
From this follows the equality of the curvatures 1/R1 and 1/R2
and from this R1=R2.
×
Fig.
2.2: Forces on a piece of membrane
×
The
radius of curvature R1 of the membrane in r-direction (fig. 2.3) we
get from the deviation z(r) an its derivatives (Stoecker 1993).
(2.1)
R2
is the radius perpendicular to R1. We calculate R2 using
the imagination of a pipe with its axis in radial direction and a declination
according to the declination in r-direction of the membrane at point P. The
pipe touches the membrane in a small area surrounding point P (fig. 2.4).
×
Fig.
2.3 - Radius of curvature of the membrane in r-direction at point P
×
Fig.
2.4 - Radius of curvature R2 of the membrane perpendicular to R1
×
R2
we get from the vertex radius RE of curvature of the horizontal
ellipse of intersection (fig. 2.5). However, the radius RE is
identical with the radius r(P). r(P) is the radius of the circle surrounding the z-axis. With the small half-axis B=R2
and the great half-axis A=R2/sin(arctan(z’)),
, 
,
(2.2)
the
equation of the ellipse of intersection in the x-y-plane solved for x is
(2.3)
With
(2.4)
and
(2.5)
and
(2.6)
and y=0 the vertex radius of curvature RE
( identical to r(P) ) is
. (2.7)
×
Fig.
2.5.: Horizontal ellipse of intersection with Vertex Radius of curvature r(p)=RE
×
Eq.
2.2 inserted in Eq. 2.7 gives
(2.8)
According
to the equilibrium of forces on curved surfaces we set
. (2.9)
Since
RE=r, Eq. 2.8 solved for R2(=R1)
and inserted in Eq. 2.1 delivers the differential equation of the 2-dimensional
membrane deviation z(r)
(2.10)
To
find the 3-dimensional cosmic membrane deviation w(r) we take in account the
fact that the radius of curvature R1 will be compensated now by two perpendicular curvatures. Instead
of Eq. 2.9 we set
(2.11)
or R2=2
R1 . So we get the wished differential equation
.
(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.
Each function w(r)=Wo+C/r is a solution of the simplified equation
2.12 . The simlification of eq. 2.12 together with setting equivalent radius r
with the arc length makes the difference between our solution and an exact
solution of the field equations. Differentiation of w(r)=Wo+C/r
yields w‘(r)=‑C/r˛. Fig. 2.6 shows that force Fg is Fg=Ksin(a), if a is the angle of slope.
Considering only small angles it is sin(a)»tan(a)=w‘. Replacing w‘ by –C/r˛,
we get Fg=‑KC/r˛. That is Newton's Law of Gravitation in the case if two masses. K
is the ether force, C a free constant and Fg is the force of attraction.
×
Fig.
2.6: Parallelogram of forces
×
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 K 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)
×
The three classical proofs of the validity of a theory of gravitation are
explanation of perihelion advance of Mercury, of light bending and of Shapiro
effect of signal retardation by solar gravity. Novel proofs are the explanation
of the Lense-Thirring effect (or frame dragging) and the explanation of the
decrease of energy of pulsars orbiting each other based on the emission of
gravitational radiation. Here the author refers to the chapters 5 and 6.
×
3 Differential Approach and Numerical
Calculations to Curvature
×
(First
version 03/1997, last update 08/2006) We can find the differential equation of
the curvature of space in cases of radial symmetry in another way as shown in
chapter 2. We take a small sector of space with a small angle a<<1 starting at the centre of mass and opened outwards. On a
volume DV with length dr at a
distance r from the centre (see Fig. 3.1) the forces Fr, Fg
and Ft (see Fig. 3.2) are acting.
×
Fig.
3.1. - Sector of space with volume
element DV
×
Fc
is the tensile force of membrane acting in direction of the centre of the funnel. It is nearly identical with the
tensile force Fo at infinity distance from the centre. Fr
is the tensile force acting radially outwards and also nearly identical to the
tensile force Fo. Ft are the four tangential tensile
forces starting from the four lateral planes and also nearly of the value of Fo.
They have no w-component if we are considering a model with radial symmetry. Fg
is the gravitational force caused by the ether wind acting on the mass inside
the volume DV.
×
Fig.
3.2. - Forces acting on volume element DV
×
We
neglect changes of the x-y-z-components of the tensile force Fo of
the membrane and of the x-y-z-component of force Fec, which are all
small compared with Fo. Therefore, for the derivation of the ODE of
curvature, we use only the w-components of the forces acting on the volume
element DV. The components are:
.
(3.1)
.
(3.2)
.
(3.3)
.
Neglecting
small values the equilibrium of forces Frw+ Fcw + Fgw
=0 gives the ODE Eq. 3.4 for the depth of space w(r).
, (3.4)
or
with renamed coefficients
. (3.5)
The
first term of the right-hand terms of the ODE 3.4 (or 3.5, respectively) yields
Newton’s gravitational potential. The second term (with ether acceleration Ae)
gives the influence of the ordinary matter distributed under central symmetry
in the space.
×
A
first estimation of the depth of space Wo we find, and thus the
connection to the GR of A. Einstein (1916), if
we treat as equivalent formally Feynman’s radius of excess rEx=a/3=491[m]
to the geometrical path lengthening dSR from the edge of Sun to
its centre. (Feynman et al.1987). The
excess radius is calculated by Feynman for a sphere with constant density, but
this fact does not harm us. One can show (numerically e.g.) that the
geometrical path lengthening within and outside a sphere of constant density
have got equal amounts. The depth of space and from this the geometrical path
lengthening outside a sphere depends on the total mass of the sphere only, but
not on the interior density distribution.
To calculate the exterior geometrical path
lengthening dSE we solve the integral eq. 3.6. The lengthening Ddr of a piece dr of membran
is Ddr » dr(1+ w'2(r))1/2‑dr » dr w'2(r)/2.
With eq. 2.13b, i.e. w'(r)=WoR/r2, we find Ddr»(Wo2R2/r4)dr/2.
.
(3.6)
.
With
Feynman's value rEX=dSE=491 [m] and 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. To find numerically the interior geometrical path lengthening dSI we
calculate first the mean density of the Sun r= M/V= 1.991´1030[Kg]/((4p/3)(6.958´108[m])3)=
1.411 [Kg/m3]. With this value of r one can calculate an
estimation of the membrane slope w’(r) solving the difference equation
3.7.
.
(3.7)
.
Eq. 3.7 follows from fig. 3.3. The force Fg
=rAEdV is caused by the ether acceleration AE
acting on the volume element dV= r2dqdfdr
with density r at point of balance P. Fg is balanced by
the different directions of the membrane tension Fo inside and
outside the shell of thickness dr. The tension Fo acts on the area
of surface (r+dr)2dqdf outside with slope w'(r+dr) and on the area of
surface r2dqdf inside with slope w'(r). Since w' is small, we can
neglect all changes of the horizontal components of the membrane tension.
Considering only the w-components we find eq. 3.8 and from it eq. 3.7.
.
(3.8)
×
Fig. 3.3: Forces
at balance point P
×
One starts at r=R with a starting slope w'=Wo/R and goes step
for step to r=0 finding there w’(0)=0. Fo=MAe /(4pWoR) [N/m2]
is the membrane tension, Ae is the ether accelaration (Eq.
2.14). The path-lengthening dSI is given by equation 3.9. Varying Wo
and fitting so by iterative numerical integration the radius rEx=491
[m] to the value of dSI we get Wo = 1.432´106 [m] once more.
. (3.9)
The constants Ae , Fo
und the coefficient A we can now calculate from the depth of space Wo
at the edge of Sun. The following
numbers are calculated with the above found depth of space at the edge of Sun Wo=1.432´106 [m]. Ether acceleration is Ae=gs/W’o
= gsR/Wo =1.361´105[m/s2]
with gravitational acceleration gs=280.1[m/s2] at the
edge of Sun and radius of Sun R=6.958´108[m].
Tension or tensile force of the membrane is Fo=Mgs/(4pWo2)= 2.164´1019[N/m2]
with mass of Sun M=1.991´1030[kg].
ODE-coefficient A is A=Ae/Fo= 6.289´10‑15[m2/kg].
Of some interest may be the relation eq. 3.10 since the constant 4pg is often used in the
General Relativity. We find eq. 3.10 using the above relations for Fo and
AE and with the relation gs =gM/r2.
×
(3.10)
The relation Fo=Mgs/(4pWo2) for the membrane tension follows from fig.
3.4. The load L arises from the action of the ether wind on the total mass of
Sun. We find L=MAE or L= MgsR/Wo . With
w'=tan(a) » sin(a)= L/F we find F=L/w' and
with w'(R)=Wo/R we get F= MgsR2/Wo2.
Since F is the total amount of tensive force acting on the surface of the Sun,
we get the tension or force per unity of surface Fo by Fo=
F/A= F/(4pR2)= (MgsR2/Wo2)/
(4pR2)
or Fo=Mgs/(4pWo2).
Fig.
3.4: Membrane tension
×
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 (a fact which is confirmed by exact
astronomical observations), 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. Only small perturbations caused by the motion of the Sun around the
centre of mass propagate with finite speed and meet the orbit with an incorrect
angle. Therefore, we do not have any
reason to suppose the propagation velocity of gravitation to be higher than the
speed of light c.
×
The
author performed in 1995-1996 some numerical calculations of the curvature of
space to prove the correctness of the deduced formulas. The first calculation
was the simple curvature of a 2-dimensional membrane under central load in the
3-dimensional space (rubber sheet). The curvature of an ideal 2-dimensional
membrane is given by equ. 2.10. The author used a regular triangle with a grid
of smaller regular triangles. Six regular triangles form a regular hexagon, but
the author used only one and mirrored points near the boundaries. The solution
of the difference equations was found iteratively. The deviations of the
calculated curvature from the solution of equ. 2.10 are very small. Only at the
central point and near the boundary greater deviations arose. The reason is
that a hexagon is not a ideal circle (boundary effect) and that six connections
of the central point to its neighbours do not model well a continuous membrane.
The solution of the ODE equ. 2.10 the author calculated numerically by the Euler-Couchy
method. The initial values have been fitted iteratively to find the minimum
quadratic error.
×
The
next proof was done by a spatial grid made from a regular tetrahedra. Twenty
regular tetrahedra form a regular icosahedra. The similarity of an icosahedra
with a sphere is not very good, but the advantage is that the grid models a
dense package of spheres with 12 neighbours for each sphere. The load of the
central point was directed in the fourth dimension. Fig. 3.5 shows the used
grid for one tetrahedra. The grid does not show any displacement in the visible
three dimensions. But the calculated displacements in the w-dimension formed
nearly exactly the demanded 1/r-curve. Near the central point and at the
surface of the icosahedra the fit of both curves was not good, because of the
boundary effects of the grid.
×
Fig.
3.5: Grid in the shape of a tetrahedra
as
part of a regular icosahedra
×
Fig.
3.6: Deviations between calculated curvature
and
1/r-curve (Newtons potential)
×
Figure
3.6 shows the deviations of the grid results of curvature from the best fit of
the 1/r-potential. In the range of 0.4 to 0.8 of r/rmax the fit is very good. Here is r the distance of
the grid point from the centre. (see program Gravitat in http://www.fh-furtwangen.de/~webers/sourcode.htm
). The
program Gravitat the author has used
also to calculate the small changes of the r-coordinates of points near the
centre. This small displacements, which are decreasing with 1/r4,
play some role in the calculation of sun-near light bending. There the tension
of the membrane is increasing.
×
An improvement of the icosahedra result one
will find by an enlarging of the number of points in the grid and a better
spherical shape. So the author used another grid. Fig. 3.7 shows the grid
points forming a sphere. The grid points are calculated as the centres of dense
packed spheres (space-centered lattice of a dense package of spheres). It was
the same kind of grid structure as used in the icosahedra case. The load acted
only on the central point in the fourth dimension (w-dimension). Each point of
the grid has 12 neighbours (excepted boundary points).
×
Fig.
3.7: Spherical grid of a dense package of spheres.
Each
point marks the centre of a little sphere
×
Figure 3.8 shows in its upper part the
calculated 1/r-potential (Newtons potential). The boundary effect at the
central point is visible. Twelve connections are only an approximation of an
ideal 3-dimensional membrane. The lower part of fig. 3.8 shows the errors,
i.e., the deviations from the ideal 1/r-potential. In the range between 0.3 to
0.8 of r/rmax the error is very small. This grid the author also
used later to calculate dark matter effects in galaxy models (see chapter 7.2)
(see program Darksim3 in http://www.fh-furtwangen.de/~webers/sourcode.htm
).
×
Fig.
3.8 Calculated curvature from the spherical grid
and
deviations from the ideal 1/r-curve
×
Geometrical
derivation, differential derivation and numerical calculations show convincing
that a 3-dimensional membrane in 4 spatial dimensions delivers under central
load exactly such a curvature as needed for Newton's potential. This is a
strong argument for the assumption that our Universe is placed in or at such a
membrane - the cosmic membrane.
×
4 Special Relativity
×
The
indroduction of a space filling resting stuff (here called membrane) leads necessarily to a special relativity, either based
on the Lorentz transformation (c.f. Lorentz 1916) or on an equivalent
transformation. In 1887 Michelson and Morley published the results of their experiments conducted to measure the relative motion of
Earth and ether (cf. e.g. Michelson 1881). The negative result is known and led
to the special theory of relativity through the work of Voigt(1887), Lorentz
(1892), Poincaré and Einstein (Einstein 1905) .
×
4.1 Longitudinal and Transversal Contraction (Fist version 05/1977, last update 08/2002)
×
The Lorentz transformation of space and time
for the transition from one frame of reference to a frame of reference with
another speed is known since Lorentz (Einstein 1905). The formulas are
symmetrical. The transformation is in agreement with all experiments concerning
the Special Relativity, and therefore it is the first choice. But following
Hendrik Lorentz’ traces (cf. Prokhovnik 1993) the author (and other authors,
e.g. cf. Onoochin, 2004) has found still another transformation – so to speak
the second choice. The chapters 4.1 to 4.3 deal with this transformation.
×
Fig. 4.1.1 shows two particles A and B in rest and moving. After the time t the moved body A with
speed v in the rest frame s is no longer the centre of a field
of spherical symmetry (Fig. 4.1.2).
The body B with parallel motion diminishes its distance x to the body A to keep
its potential in the field of body A.
Otherwise the distance A-B’ in the
case of a moving pair would be greater than the distance A-B by reason of a finite speed of
propagation of the fields of forces. A cross contraction could solve the problem. The planes of constant
potential change from spheres in the case of rest to ellipsis in the case of
motion. All axes of the ellipsis are shortened. That the longitudinal
contraction is stronger than the perpendicular or cross contraction – this fact
follows from the Michelson-Morley experiments (see below).
Fig.
4.1.1. - Two particles in rest and moving with lines of constant potential
Fig.4.1.2 Field surrounding a moved point A
×
The velocity of waves (forces, fields) in our
3-dimensional world s (the membrane) is c. Now we consider photons in a cache
consisting of two mirrors. That is nearly equivalent to one of the arms of the
Michelson-Morley interferometer. If the cache is moving with a constant speed v (b=v/c) relatively to the
membrane, we may define parallel and perpendicular orientations of the cache
(see Fig. 4.1.3). The travel time of a photon there and back between the
mirrors a and b does not change after a rotation of the cache, since the number
of cycles remains unchanged.. Otherwise, we would observe in the interferometer
a change of the interference pattern. We postulate a contraction of length with
factor (1‑b2)1/2 during a rotation from perpendicular to
parallel orientation. Up to this point, the author agrees with H. Lorentz. The
lenght contraction during rotation explains the zero result of the
Michelson-Morley experiments.
×
Fig.
4.1.3. - Moving light cache perpendicular and parallel to v
×
But
from the perpendicular case we said above that there might be a physically
founded perpendicular or cross contraction. This cross contraction would get
the contraction factor (1‑b2)1/2 also. Together with the results
of the Michelson-Moorley experiments this assumption then gives a length contraction
of the value 1- b2.
×
For the most effects of time the real
physical foundation is still outstanding. Therefore, we indroduce the
speed-depending relativistic dilation of time as the experimental result of
Ives and Stilwell e.g. (1938), or
Otting respectively. In the case of the gravitational time effect, we follow
Albert Einstein and his Principle of
Equivalence (c.f. chapter 4.7) The time transformation for moving systems
without (or with weak) gravitational fields uses the common time dilation
according to Ives and Stilwell. This transformation is in agreement with the
Lorentz transformation and is used in the following chapters.
×
4.2 Coordinate and Time Transformation (First version 06/1991, last update 08/2002)
×
The chapter contains the formulas following from the special relativity
with both – cross and length contraction. First we consider the movement of a
signal in a moved frame of reference s‘.The
speed v of s‘ inside the rest frame s may be directed in negative z-direction. Let x‘ be a fixed distance in s‘. A light signal travels over the distance x‘, is reflected at the end
and returns to the starting point. The true path a-b in the rest frame s is shown in Fig. 4.2.1.
×
Fig.4.2.1. Light path in s, source and mirror resting in s‘
×
|
|
Equations
4.2.1 follow from Fig. 4.2.1.
, (4.2.1 a)
.
, (4.2.1 b)
.
. (4.1.2 c)
sina does not depend on the scales of the reference system, because it is a
dimensionless quotient. This is not valid for the coordinates of locus and time
and their derivatives.
×
All
longitudinal distances (with the same direction as speed v) in s' are contracted by the factor 1-b2, the transverse distances by (1- b2)1/2. Assuming, that v and x are parallel and that the
origins O and O' of the coordinate system s and
s' coincided at time t=0, for a fixed x' in s' has a length given by eq. 4.2.2 using s-scales.
(4.2.2)
An
observer in s' measures the distance x-vt
with his contracted scale. He gets a value enlarged by the factor 1/b2.
×
For
the construction of the time transformation we use the result of an experiment
performed in 1938 by Ives and Stievell as well as Otting. They investigated the
change of frequency of light emitted by canal rays in ray direction and in the
opposite direction. They found that the mean value of the two frequencies
follows
×
(4.2.3)
×
Here
v is the speed of the canal rays. v was measured by the simple Doppler
effect of first order. n is the frequency of the emitted line for v=0.We subtract from this result the amount of the common
longitudinal Doppler effect of second order with moved source and resting
observer and get
×
(4.2.4)
×
In
agreement with the special theory of relativity we extrapolate eq. 4.2.4 in
such a way, that the frequencies of periodical processes are diminished by the
factor b in a moved system. Our time
transformation for points resting in s' is
then
×
. (4.2.5)
×
We
get the inverse coordinate transformation by resolving eq. 4.2.5 into x and
replacing t by t‘/b in order to eq. 4.2.5.
×
(4.2.6)
×
Since
measurement of velocity in s' uses a scale which is
contracted by the factor b2 and additional the clocks in s'
have a factor b lower speed, a velocity v
in s will give
in s' the amount v '= v/b3. Introducing
this in eq.4.2.6 and with eq. 4.2.5 we
get
×
(4.2.7)
×
The
complete collection of transformation equations for coordinates and time of
points resting in s' is
(4.2.8 a)
.
(4.2.8 b)
.
(4.2.8 c)
.
(4.2.8 d)
.
(4.2.8 e)
.
(4.2.8)
×
Travels
a point P' in s' with velocity u' in the
direction of the positive x-coordinate during time t', it crosses through the
distance u't'. An observer in s has to subtract from the
coordinate x of P' in s the amount vt (the motion
of O' in s in the time t) and the
difference to multiply by the scale factor 1/b2 to fit the above value, i.e.
×
(4.2.9)
×
The
simple time transformation for the origin O' in s' is
t'= bt. But this transformation holds only in the
singular reference system s, since s is the only system in which we may install "master clocks" by
synchronizing them in the origin O and then moving them with "low"
speed u to their fixed position x.
Since the time dilatation diminishes in the case of small u by (1/2)(u2/c2),
but the time of transport increases only by 1/u, there remains a linear
decrease of dilatation usable for the transport of the clocks without
influencing the time flow.
×
In
the moved system s' this time invariant motion
is not possible. Each motion of the clock from the origin O' in s' to another point x' in s'
with a velocity v+u (in s-scales) leads to an
additional time dilatation
×
(4.2.10)
×
related
to a clock t in O of s. For u<<v we get
neglecting the term u2/c2 and by differentiation
×
(4.2.11)
×
For
small u we get neglecting 2uv/c2
(4.2.12)
and
with ut=x-vt
×
(4.2.13)
×
Denotes t(1-v2/c2)1/2
the time dilatation in O' of s' then the time t' in the
point x outside O' is
×
(4.2.14)
×
giving
(4.2.15)
×
a
result, which is nearly identical to the Lorentz transformation of time. See
also F. Selleri (2004). Going back to eq. 4.2.9 we get eq. 4.2.16.
(4.2.16)
×
and
with u=dx/dt and
×
(4.2.17)
×
we
get
, (4.2.18 a)
.
(4.2.18 b)
×
a
result, which is similar to the addition theorem of velocities in the special
theory of relativity. If one misses the Lorentz symmetry then compare
Schmelzer's (2000) discussion to this topic..
×
End of part 1.
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