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(The paper containes an error in the chapter "perihelion advance". Although the calculated result is true, the derivation is wrong. See Part 2 of "Cosmic Membrane Theory")

Membrane Theory of Gravity

.Stefan von Weber

Zusammenfassung

Im 4-dimensionalen Raum führt das Modell einer 3-dimensionalen elastischen Membran, die vom Ätherwind angeblasen wird, auf Newtons Gravitationspotential. Die Präzessionsbewegung der Merkurbahn kann allein durch die Massenzunahme gemäß der speziellen Relativitätstheorie erklärt werden.

Summary

In the 4-dimensional space the model of a 3-dimensional elastic membrane blown on by the etherwind leads to Newtons potential of gravity. The precession of the orbit of the Mercury one can calculate only by the increase of mass due to special relativity.

Key words: ether, relativity, gravitation

1. Introduction

The discovery of the cosmic background radiation by A.A.Penzias and R.W.Wilson lets revive the hypothesis of an etherfilled space. Hereby one may refer to papers of PROKHOVNIK [1]. The author showed that both, the modell of a resting ether and also the modell of a moving ether (etherwind), is in agreement with all known effects of special relativity (WEBER [2]). Such effects are for example the dilation of time, the contraction of length, the result of Michelson and Moorley, the addition of velocities, the increase of mass and the conservation laws of energy and momentum. Beyond this the 4-dimensional ether model explains the gravity in an illustrative manner. Fig. 1. shows a 3-dimensional picture of the 4-dimensional case. The time is not one of this dimensions. We assume that the 3-dimensional cosmos s expands with velocity VE (velocity of etherwind) as a hyperplane in the 4-dimensional space S.

Fig. 1.: Model of an expanding cosmos s

 

2. Newton’s Potential

We see our 3-dimensional world s as an elastic membrane stretched in the 4-dimensional space S. The etherwind passes the membrane with velocity -VE. Particles with mass are freely mobile within the membrane, but they have a higher resistance in the etherwind as the surrounding membrane. The membrane will be deformed.

Fig. 2.: Deformation of membrane s by the

etherwind at the place of a mass M

A 3-dimensional elastic membrane in the 4-dimensional space x,y,z,w with a load in w-direction deformes in radial symmetry. The differential equation of the deviation w(r) we can deduce from the analogous case of a 2-dimensional membrane in the 3-dimensional space x,y,z. The radius of curvature R1 of the membrane in r-direction we get from the deviation z(r) an its derivatives (STÖCKER [3]).

(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. 3.: Radius of curvature of the membrane in r-direction at point P

 

Fig. 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. However, the radius RE is identical with the radius r(P). r(P) is the radius of the circle surrounding the z-axis.

Fig. 5.: Horizontal ellipse of intersection with vertex radius

of curvature r(p)=RE

With the small half-axis B=R2 and the great half-axis A=R2/sin(arctan(z’)),

, B = R2 , (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)

Equ. (2.2) inserted in equ. (2.7) gives

. (2.8)

According to the equilibrium of forces on curved surfaces we set

R1=R2. (2.9)

Since RE=r, equ. (2.8) solved for R2(=R1) and inserted in equ. (2.1) delivers the differential equation of the 2-dimensional membrane derivation z(r)

. (2.10)

To find the 3-dimensional 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 equ. (2.9) we set

(2.11)

or R2=2 R1 . So we get the wished differential equation

. (2.12)

We neglect the term w’2 in the case of small deviations and therewith small w’ and get with two integration constants the solution

.. (2.13)

With the common commitments of a potential (negative and w=0 for r®¥ ) we write in the special case of spherical symmetry of the solar system

. (2.14)

Here Wo is the depth of space at the edge of the sun and R is the radius of the sun. Equ. (2.14) is identical to Newton’s potential neglecting the different commitment of the constant.

The classical proofs of any theory of gravity are ligth bending by the sun, travel time dilation of a radar echo, precession of the orbit of Mercury and the change of time. Light bending by the sun and travel time dilation of a radar echo lead to small differences (against the results of the General Relativity) for beam trajectories very near to the edge of the sun. Here the author has to do additional research work, especially to look for experimental data with sun-near trajectories. The nature of the dark matter is connected closely with this questions.The precession of the orbit of the Mercury is (by chapter 3) not an essential touchstone of any theory of gravity. Remains the change of time by a gravitational field. Here the author has to do some additional research work likewise. The relations between time and curvature of space are of another kind as in the General Relativity.

3. Precession of the Orbit of Mercury

In 1859 U.Leverrier and in 1882 Newcombs found both a precession of the orbit of the planet Mercury with the value 43’’ per century which they could not explain by the influence of other planets. In the Newtonian theory the orbits of planets are ellipses with fixed axes. This is only true for the potential 1/r and masses with absolut spherical symmetry. The center of mass of the 2-body-system is one of the two focal points of the ellipse. Possible disturbances of the exact elliptic orbit are caused by other planets, a departure from spherical symmetry of the sun, the finite speed of gravity waves and departures from the exact 1/r-potential. The disturbance of the spherical symmetry of the sun, it has a oblate form, is too small to be the reason of the precession (FLIESSBACH [4]). The influence of the other planets is eliminated already in the value Df=43" . Remains the effect of the finite speed c of gravity waves and the effect of virtual disturbances of the 1/r-potential.

Fig. 6.: The orbit of the Mercury and our coordinate system

We will see in the further text that the precession of the orbit of the planet Mercury is explicable with high accuracy by the relativistic increase of mass alone. Therefore the precession of the orbit is not an essential touchstone for the General Relativity or Membrane Theory or any other theory of gravity. If there exist further influences on the precession of the orbit, the sum must bee equal to zero as in General Relativity.

The trajectory of the Mercury is in the first order fixed by r+=70´109 [m] (largest distance from sun), r- =46´109 [m] (perihelion or smallest distance to sun) and the time of revolution T=87,969 [d] . One sidereal day is d=86164 [s]. The numerical eccentricity of the elliptic orbit of Mercury is e=(r+/r- - 1)/(r+/ r- +1) = 0,2069. We define the cartesian system of coordinates in such a way that the orbit lies in the x-y-plane, r- points to positive x-direction and the origin of coordinates coincides with the right focal point of the ellipse. In polar coordinates is (STÖCKER [3])

(3.1)

with

= 55,517´109 [m]. (3.2)

p is also called semilatus rectum or altitude over the focal point. We start with an exact elliptic orbit. The characteristical values (see also fig. 7a) are in the case of a constant increment df of angle f:

, (3.3)

,, (3.4)

, (3.5)

, (3.6)

v = ds / dt . (3.7)

Fig. 7.: a) Parameters of the orbit and b) displacements

Here vectors has been written with bold types, their absolute values with normal types. v- is the velocity of Mercury at point r- . The velocity v- was calculated numerically so that is fulfilled. The so calculated value is v- = 5,9309´104 [m/s]. ds is the arc element of the orbit to the differential of angle df. dr is the change of the absolute value of r. dt is the differential of time to arc element ds. a is the angle between v or ds and positive x-axis. v is the velocity of Mercury at point r of its orbit. The relativistic change of the mass of Mercury on its orbit causes a velocity which is too small. A little part of the potential F(r)= -GM/r is consumed by the increase of the mass. Because of

(3.8)

and v2/c2=a/r the square of velocity u2=(av2)/(2r) lacks. The gravitational acceleration GM/r2 cancels for velocity v the centrifugal acceleration exactly. The deficit u2 causes an additional acceleration

. (3.9)

The so caused additional speed implies together with two another velocities u2 and u3 a deviation with u=u1+u2+u3 between the real coordinates of the orbit and the exact elliptic orbit. (u2 and u3 are introduced below). This deviation implies a secondary effect, an additional acceleration

. (3.10)

Here is Dr=Ds cos(f-b) and b is the angle between Ds and the x-axis (fig. 7b). A positive Dr means for e. greater distance between Mercury and sun and therefore a smaller attraction force. In this case the additional acceleration is directed outside. This acceleration causes the additional speed . The additional speed u we had to add to speed v on the orbit and there it causes a change of the centrifugal force as tertiary effect. The so caused acceleration is

. (3.11)

Fig. 8.: Angles of precession Df

Here is g the angle between u and the x-axis (as illustration we can use fig. 7b and u instead of Ds and g instead of b). A3 gives integrated the additional speed u3. If we integrate numerically all equations parallel from f=0 to f=2p with the Euler-Cauchy method for e. and use stepwidth df=2p/40000, we get the speed component ux = -2,902´10 - 2 [m/s] and the deviation of orbit in y-direction Dsy=2,341´104 [m]. Fig 8. shows the angles. The angle Df1=-ux/v-=0,491´10-6 º 0,1009" is somewhat different from angle Df2=Dsy/r- =0,509´10-6º0,101050" . However, the average Df=0,1029" coincides rather exactly with Df=0,1038", a value given by FLIESSBACH [4] for the General Relativity. 415 revolutions per century of the Mercury round the sun give the total sum of 43’’ precession of the orbit.

References

[1] Prokhovnik,S.J.: The Physical Interpretation of Special Relativity - a Vindication of Hendrik Lorentz, Zeitschrift für Naturforschung , 48a, 925 (1993)

[2] Weber, S.v.: Newtons absoluter Raum, Forschungsbericht der Fachhochschule Furt-

wangen 1995, S.55-57.

[3] Stöcker,H.: Taschenbuch mathematischer Formeln, Verlag Harri Deutsch, 1993

[4] Fliessbach, T: Allgemeine Relativitätstheorie, Wissenschaftsverlag Mannheim 1990).

April 1998

Prof. Dr.sc.techn.Dr.rer.nat.Stefan von Weber,

FH Furtwangen, University of Applied Sciences,

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