Part 3 of membrane theory (last update 06/2008)

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6      Novel Proofs of General Relativity

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 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. The author has not found the equivalent of the Lense-Thirring effect in the Cosmic Membrane Theory hitherto, but another result of space torsion in chapter 6.1. The results to gravitational waves one may find in chapter 6.2.

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6.1   Electron and Space Torsion (First version 08/2001, last update 01/2005)

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  In this chapter the author will try to find some connection between the theory of gravitation (General Relativity or membrane theory, respectively) and Quantum Theory. The theory of gravitation handles the great distances and great masses, Quantum Theory the tiny distances and tiny quants of energy.

  The basic idea of this chapter is that the kinetic energy of the drilling space surrounding a charge is identically with the energy of the electric field. Opher (2003) discusses, e.g., the connection between angular momentum of heavy bodies and magnetic 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.

   In chapter 4.4 the author made the hypothesis that the basic brick of our world is a torus-shaped whirlpool (eye, in German Auge) (Fig. 4.4.1) moving with speed VE in direction of the W-axis of S. Two auges attract one another and form a rotating pair of auges (cf. fig.4.4.2). The rotation causes a whirlpool (space rotation) and is equivalent to an elementary charge e. This elementary charge e may be already identically with the electron or positron, but may be also somewhat in the sense of a quark. But here we suppress each doubt and use a simple model of the electron made from two auges only (fig. 6.1.1).

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  We consider the torsion of space in the environment of an electron. Starting with the estimation Wo = 1.433´106 [m] of depth of space at the edge of Sun we had found the tensile force Fo= 2.114´1019 [N/m2]. Using the wave equation (see also Carter 2004) we find r=Fo/c2=0,2352´103 [Kg/m3], the density of the membrane. The energy UE of the electrostatic field of an electron from distance RE to infinity is given by eq. 6.1.1.

                   (6.1.1)

Here qE is the charge of the electron and eo the electric absolute permittivity. Assuming speed VA for the drilling space at the distance of radius RE from the centre of the whirlpool and v(r)=VARE2/r2, the energy UM of the moving membrane outside RE is given by eq. 6.1.2.

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      (6.1.2)

We treat UE as equivalent to UM and find the eq. 6.1.3. It gives the relation between radius RE and density r. The drilling speed VA we suppose to be the speed of light c or to be greater. The analogon is the speed of sound waves in air. There the mean speed of the gas molecules is about 1.5 times graeter than the speed of sound.

                       (6.1.3)

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  Trying the same calculation for the momentum IM we get equ. 6.1.4.

                                   (6.1.4)

This integral does not deliver a finite value. But a consideration of the same integral in the 4-dimensional space with speed VE of the membrane will yield the finite value. The speed V in the 4-D space is the vectorial sum of v and VE according to equ. 6.1.5.

        (6.1.5)

Apart from an unknown scaling factor the second term of the Taylor series is yielding the same value as yielded by equ. 6.1.2.  Energy in the 3-D space is proportional to the absolute value of the 3-D component of the 4-D momentum. The first term of the Taylor series in equ. 6.1.5 is still yielding an infinite value. It means the infinite momentum of a infinite large piece of membrane. Replacing the infinite boundary of the integral 6.1.4 by the finite extension of the cosmos we get a finite value.

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To estmate RE and VA, we consider the model of the electron (fig. 6.1.1).

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Fig. 6.1.1: Auges model of electron

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In fig. 6.1.1 a and b are the half axes of the elliptic orbit of the drilling auges. RA is the radius of  an auge. VA is the averaged speed of the auges. The electron radius RE is the radius of that circle with the same perimeter as that of the ellipse with half axes a and b. Radius Rl describes the circle with diameter lC. lC is the Compton wave length of the electron. From Dirac's theory of the electron we know the angular frequency w(Z) =1.5527´1021 [1/s] of the zitterbewegung of the electron. Further we know the charge qE =1.602´10-19 [As], the rest mass mE=9.109´10-31[Kg], 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].

   Our model has two auges which we can not discriminate. That means we had to halve frequency w(Z) if we consider one auge. The angular frequency of a single auge on its elliptic orbit is w= w(Z) /2=7.7635´1020 [1/s].  To fit the constants JE, mB, lC of the electron and radius RE given by the density of the membrane, we use the model parameters a, b and RA. Angular frequency w and speed VA are not parameters, since w is given and VA is calculated from a,b and w.

The spin angular momentum of the electron JE = h/2 = h/4p we model with eq. 6.1.6, using a·b as an estimation of the square of the distance of the mass from the centre.

         (6.1.6)

Bohr's magneton of the electron is mB = qEh/(2mE) [Am2]. This formula we can rewrite in the form mB = (w(Z) /2)qE R2 if we use the known relation w(Z) =2mEc2/h and the relation R=c/ w(Z). R is some radius with no special meaning here. But now we can see that the electric current is given by I=wqE/2p [A], the area by A=pR2 [m2]. In analogy to the above formula we model Bohr's magneton by eq. 6.1.7. We remark on the addition of RA and a (RA and b, respectively) in formula 6.1.7  that many other authors suppose the charge in a greater distance from the centre, than the mass (cf. Joos 1989 e.g.). But we had also to remark that this part of the model is not a consequent membrane model in the sense that the charge is handled as some stuff  here. The membrane theory says that the charge is caused by the movement of the auges, but the author had not found the courage to model strictly this idea. So the model of the magneton  remains somewhat half silken.

      (6.1.7)

The Compton wave length is lC = h/(mEc). We rewrite this formula using once more the relation w(Z) =2mEc2/h and get lC = 2c/ w(Z). And with R=c/ w(Z)  we find lC =2R. We see that the Compton wave length is twice some radius R. So, in analogy, we model the Compton wave length lC by eq. 6.1.8. We remark on the addition of RA and a that there is no deeper sense than the best fit.

       (6.1.8)

The geometrical radius of the electron RG we model purely geometrically, indeed. We seek that circle with perimeter U of the elliptic orbit of the auges. So we find eq. 6.1.9.

              (6.1.9)

To calculate the perimeter U of an ellipse we had to solve an elliptic integral which was done numerically by the author. An approximative formula as U»p(1.5(a+b)-(ab)1/2) is only well for small excentricities of the ellipses or for a rough estimation. The geometrically found electron radius RG we compare with the radius RE calculated from the membrane density.

  The numerical fit yielded the numbers of table tab. 6.1.1. The calculated parameters are given in table tab. 6.1.2.

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Table 6.1.1: Model and theory of electron

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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%

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Table 6.1.2: Model parameters of the electron

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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

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  One may see that the constants of the electron are fitted with some accuracy. This is a first success. Nevertheless doubt remains. The most important point is the great excentricity of the orbit of the auges. The distance between them becomes too large, so that the consistency of the electron must be instable because of the 1/r6 statistic of the attractive force of the auges. Otherwise the excentricity is a niceful feature. If the electron is accelerated, its great half axis may start to turn. Perhaps one may explanate de Broglie's matter waves in this way. So we hope for the next generation of models.

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6.2   Gravitational Waves (First version 06/2002, last update 08/2002)

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Gravitational waves may arise from the movement of celestial bodies (Weinberg 1972, Fliessbach 1990) or from perturbations of relativistic stars, as coalescing neutron stars or supernovas (Allen et al. 1998). When an object's curvature varies rapidly it should emanate curvature ripples (gravitational waves) that propagate through the universe at the speed of light ... (Abramovici et al. 1992).

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From point of view of the Cosmic Membrane Theory there may exist both, transversal and longitudinal gravitational waves, comparable with gravity water waves and water sound waves otherwise (see also Battye et al. 2005). (Electro-magnetic waves are transversal waves too, but there the amplitide lays inside the x-y-z-space.)  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. In the rest of this chapter we investigate longitudinal waves only (c.f. Clayton and Moffat 2002, e.g.).

 

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The rotating system Sun-Earth makes a quadrupole, e.g., and is so the source of gravitational waves. To get an estimation of the total membrane energy flow FT caused by the gravitational waves of the system Sun-Earth, we take the point of balance PB at the distance RB= l/2p = cT/2p= 0.7529´1015 [m]  from the centre of mass of the system (COM) in the plane given by the orbit of the Earth (Weber 2002). T is half a year. The point of balance follows from Heinrich Hertz' solution of a radiating source and is exactly that point where the density of energy is balanced between close-environs term 1/r2 and distant-wave-area term 1/r. This distance is large enough to see the orbit of  Earth under a small angle. From mass of Earth ME=5.977´1024 [Kg], mass of Sun MS=1.991´1030 [Kg] and mean distance Earth-COM RE=1.496´1011 [m] we calculate the mean distance Sun-COM as RS=MERE/MS=4.491´105 [m]. The square of the distance RBE between the point of balance PB and Earth is given by eq. 6.2.1. using spherical coordinates. Fig. 6.2.1 illustrates the relations.

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Fig. 6.2.1: System Earth-Sun with point of balance PB

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(6.2.1)

In Euclidian coordinates x, y, z the square of the distance is given by eq. 6.2.2.

(6.2.2)

or

(6.2.3)

With cos(y-f)=cosycosf+sinysinf and with sin2a+cos2a=1 we get

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.          (6.2.4)

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By the same way we find the distance RBS between point of balance PB and the Sun. But here is fS=f+p and following from this cos(y-fS)=-cos(y-f). So we get:

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 .          (6.2.5)

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Newton's gravitational acceleration A at the point of balance PB is given by eq. 6.2.6 with gravitational constant g.

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                 (6.2.6)

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Expansion into a series of 1/R2BE and 1/R2BS yields eq. 6.2.7 because of 1/(1+e1+e2) » 1-e1-e2+e12 +2e1e2+e22  and neglecting members less than 10‑8. Here is  ½e1½»2RE/RB»10-4, ½e2½» R2E/R2B»10-8 and ½e12½»10-8. All other terms are less than 10-8.

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(6.2.7)

We calculate the difference of A(RB,q,y,f) and acceleration ACOM to find the yearly change DA(RB,q,y,f). Acceleration ACOM is that acceleration we would find if both masses, Sun and Earth, are united in the center of mass (COM). The formula is ACOM=g(ME+MS)/R2B. Because of MSRS=MERE the terms linear in RE/RB are canceled in the eq. 6.2.8. What remains, is:

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            (6.2.8)

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Since we are seeking a wave with amplitude and harmonic oscillations, we consider only the yearly deviations DA of DA from its mean value. Halving the first term in the brackets of eq. 6.2.8 and neglecting the term R2E/R2B as a constant bias, we find the amplitude DA of the changes of gravitation. The dependence on time is hidden in the term cos2(y-f) with f=wt.

.           (6.2.9)

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In the membrane model acceleration or gravitational force is connectet with the slope of the membrane. Because of A=AEw´ (acceleration = ether-acceleration ´ slope of membrane) we get the square of the quadrupole-caused membrane slope difference Dw´2 by eq. 6.2.10.

 .                (6.2.10)

The relative change of length b of a piece of membrane at the moment of maximum slope difference Dw´ is given by the relation 6.2.11.

                 (6.2.11)

With membrane tension Fo, which has the dimensions of a pressure, the change of energy DE of one unit of volume is bFo.  With speed c of gravitational waves we find the flow F per square unit of the surface and per unit of time by F=bFoc=(DA2/2A2E)Foc. Integrating F over the surface of the sphere with radius RB and calculating the temporal mean over one cycle of f (0 - 2p) we find the total energy flow FT given by eq. 6.2.12. An additional factor 2 arises from the fact that we have considered only the potential energy of the wave. Kinetic and potential energy of waves have the same amount. Another factor, 1/2, arises from the fact that at the point of balance the energy is balanced between close-environs term 1/r2 and distant-wave-area term 1/r. So we write

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(6.2.12)

With Fo/A2E=1/(4pg) we find

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          (6.2.13)

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or as final result for the total gravitational energy flow of the system Sun-Earth

.            (6.2.14)

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Weinberg (1972) or Fliessbach (1990), e.g., gives for the GR a value of FGR=195.4 [W]. He uses the eq. 6.2.15.

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                        (6.2.15)

To compare the two equations we introduce T2 by Kepler's law and find eq. 6.2.16.

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                                           (6.2.16)

With RB=cT/2p and from this c=2pRB/T and eq. 6.2.16 we find

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          (6.2.17)

or

 .            (6.2.18)

General Relativity and Membrane Theory are yielding here the same value.

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  For the experimental detection of gravitational waves there some projects existed and do exist. The first pioneer was Joseph Weber from Maryland University. He hung a massive aluminium cylinder in a vacuum chamber and insulated it from outer influences. Vibrations of the cylinder should signal gravitational waves. But the sensitivity of the system was not high enough. The signals found had other sources. Other teams continued Weber's work with cooling the cylinder by liquid helium, but without success also. Abramovici et al. (1992) describe a laser inferometer (LIGO). The authors say: LIGO offers an opportunity to bring nonlinear gravity, black holes, and the graviton out of their near isolation as theoretical constructs and into confrontation with experiment. For myself (the author of this paper), I do not believe in the graviton, but that gravitational wave detection will give us surely a lot of information about stars and black holes I do believe in. The theoretical sensitivity of the LIGO will be god enough to detect the gravitational waves of a supernova at a distance of 30 Mpc. There are at least three interferometers in construction: US, Italian-French, Britain-Germany, which will start with their work in 2004-2006 (state at June 2002).

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6.3   The Pioneer 10/11 Anomalous Acceleration  (First version 11/2005, last update 12/2005)

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In this chapter the author discusses the surprising anomalies of the Pioneer 10 and 11 acceleration data. We will see that presumably the NASA has neglected the Gerber-Einstein acceleration -gMs6a/r3, which causes, e.g., the perihel advance of the Mercury and the other planets. Because of the wonderful fit of the parameters of our solar system (mass of Sun, masses of planets, gravitational constant, distances, ... ) this lack was - not realized until now - I believe so .

 

Two spacecrafts of the NASA, the Pioneer 10 and Pioneer 11, were launched in 1972 and 1973 to leave our solar system, and to carry human technics to other solar systems. Underway, they informed us about the distant planets in our solar system. Now NASA lost contact with them, because of the vast distance. But before the lost of contact, the radio measurements of the speed of the spacecrafts during the last years of their existence seem to show a riddle: the acceleration of the spacecrafts is not so as expected. It seems that there is an unexpected contribution to the acceleration of about Da = (8 ± 1) ´10-10 [m/s2] directed towards the Sun.

 

   I copy here the abstract of J. Anderson et al. (2002): "Our previous analyses of radio Doppler and ranging data from distant spacecraft in the solar system indicated that an apparent anomalous acceleration is acting on Pioneer 10 and 11, with a magnitude a_P ~ 8 x 10^{-8} cm/s^2, directed towards the Sun (anderson,moriond). Much effort has been expended looking for possible systematic origins of the residuals, but none has been found. A detailed investigation of effects both external to and internal to the spacecraft, as well as those due to modeling and computational techniques, is provided. We also discuss the methods, theoretical models, and experimental techniques used to detect and study small forces acting on interplanetary spacecraft. These include the methods of radio Doppler data collection, data editing, and data reduction. There is now further data for the Pioneer 10 orbit determination. The extended Pioneer 10 data set spans 3 January 1987 to 22 July 1998. [For Pioneer 11 the shorter span goes from 5 January 1987 to the time of loss of coherent data on 1 October 1990.] With these data sets and more detailed studies of all the systematics, we now give a result, of a_P = (8.74 +/- 1.33) x 10^{-8} cm/s^2. (Annual/diurnal variations on top of a_P, that leave a_P unchanged, are also reported and discussed.) "

 

This number a = (8 ± 1) ´10-10 [m/s2] is currently discussed by the physical society: Is here the influence of a new, yet unknown physics, or is it only a technical problem? The enigma is not solved yet. The discussions go on, and vary from basic technical principles to extra spatial dimensions. Many physicists think, the answer can only be obtained after a new special mission is underway. These new probe could increase the precision, so that one could answer the question whether the anomaly is a result of a technical or systematic error (cf. Marmet 2005), the influence of new dark matter at the boundary of the Solar system (cf. e.g. Bertolami 2004) or a proof of new physics (e.g. Ranada 2005). A strange fact is that this new acceleration is about a=H0c. H0c is the Hubble constant multiplied by the speed of light. This fact is widely hailed by the modified Newtonian dynamics (MOND theory, see Milgrom 1983, 1993, Sanders and McGaugh 2002, Bekenstein and Sanders 2005). There the Hubble constant plays a central role. It seems to be the same acceleration as the critical acceleration relevant for this theory which replaces dark matter. Most physicists would of course be sceptical about the existence of this effect. A special "force acting on the spacecrafts" violates the equivalence principle.

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  Because the Cosmic Membrane theory deals with the Hubble constant also, we will cite Bekenstein and Sanders (2005): " MOND, invented by Milgrom, is a phenomenological scheme whose basic premise is that the visible matter distribution in a galaxy or cluster of galaxies alone determines its dynamics. MOND fits many observations surprisingly well. Could it be that there is no dark matter in these systems and we witness rather a violation of Newton's universal gravity law ? If so, Einstein's general relativity would also be violated. For long conceptual problems have prevented construction of a consistent relativistic substitute which does not obviously run afoul of the facts. Here I sketch TeVeS, a tensor-vector-scalar field theory which seems to fit the bill: it has no obvious conceptual problems and has a MOND and Newtonian limits under the proper circumstances. It also passes the elementary solar system tests of gravity theory."

The MOND theories modify the Newtonian laws in such a way that the inverse square law 1/r2 for gravity is replaced by the inverse distance law 1/r for objects whose acceleration is smaller than a critical value a0. This new law allows one to reproduce the observed results from a large number of galaxies without assuming any dark matter.

 

  The Cosmic Membrane theory has some points that could touch the anomalous acceleration matter:

·      Dark matter caused by the etherwind

·      Dependence of speed of light and time on gravity

·      Several effects caused by the expanding Universe

·      Effects by additional sun-near potentials

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Chapter 7.2 deals with dark matter caused by the ether wind. Using the coefficients from ODE 7.2.7 inside our solar system, we get a resulting additional acceleration which is 7 orders of magnitude to low. Therefore, this type of dark matter effect can not be involved in the anomalous acceleration of the pioneer 10 and 11 spacecrafts.

  The dependence of the speed of light and time on gravity is nearly the same in the GRT and in the Cosmic Membrane theory. Differences do exist for very sun-near positions only. Both effects have been considered by the NASA in the travel path computings of the spacecrafts, and so they can not be the source of the anomalous acceleration effect.

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The third point enumerates effects caused by the expanding Universe. A Hubble constant of H0=82 [Kms-1/Mparsec] is equivalent to a lifetime of  T=12´109 years in the Big Bang paradigm. The expanding membrane causes a linear increase of the membrane tension F0 (eq. 7.3.4), and so several effects, e.g.:

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·      the redshift of the radio waves

·      an additional enlarging of the distance r between spacecraft and receiver

·      a change in time (frequency of clocks)

·      a change in the speed of light

·      a change in gravitational constant g.

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The anomalous acceleration of about  aA = (8 ± 1) ´10-10 [m/s2] is equivalent to a lost of DkA=3.05´10-9 waves per second in each second, and accumulating to about 1.5 waves per second over 30 years in the radio band of n=2.29 MHz. This number  we can calculate by eq. 6.3.1.

.     (6.3.1)

This measured lost of waves we had now to compare with several effects enumerated above.

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The redshift effect DkRS waves per second is given in eq. 6.3.2 with speed v=12.000 [m/s] of the spacecraft, distance d  between spacecraft and Earth, frequency n=2.29´109 [Hz], and inverse Hubble constant H0-1= T=12´109 [years]  or  T=3.789´1017   seconds.

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.   (6.3.2)

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Faktor 1/2 arises by integration over the second. The term d/T we can neglect for distances near the solar system. We see, the effect of DkRS  is too small to explanate the anomalous acceleration.

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The additional enlarging of the distance d between spacecraft and Earth by the stretching membrane (expanding space) is leading to the same formula as above:

.   (6.3.3)

Term d/T we can neglect again. The effect is also too small to explanate the anomalous acceleration.

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Membrane theory postulates in chapter 7, eq. 7.3.10, that frequency  n of any clock changes with the enlarging radius R of the membrane balloon as n(R)=n0(R/R0)1/4 with some fixed radius R0. Then the change of frequency in Hz per second is given in eq. 6.3.4.

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   (6.3.4)

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Factor 1/4 arises by axpanding the root in a series. Term d/T is neglected. The effect is also too small to explanate the anomalous acceleration

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Membrane theory postulates in chapter 7, eq. 7.3.5, that speed of light changes with the enlarging radius R as c(R)=c0(R/R0)1/2. The effect of this change in the speed is given in waves per second by eq. 6.3.5. The travel time of the radio signal is d/c. In this time the radius R of our Universe changes by factor (1+d/cT), and c by the factor (1+d/2cT). The 2 in the denumerator arises because of the squareroot. Factor 1/2 arises by the integration over travel time. The Doppler effect 12000/c is subtracted also. Division by travel  time d/c is giving DkC, i.e. the increment per second.

,

and as a number,

.   (6.3.5)

Some small terms have been neglected in eq. 6.3.5.The effect is also too small to explanate the anomalous acceleration.

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The last effect considered here is the change of the gravitational constant g in Newton's law a=gM/r2. If the membrane stretches then the membrane tension Fo increases, and causes so a linear decrease of the slope w' of the membrane. The slope w' of the membrane is directly proportional to the gravitational constant g used in Newton's law. We find acceleration a= AEw'(r)= gM/r2 with mass M and distance r between mass and spacecraft. AE is the ether acceleration in negative w-direction caused by the ether wind of expansion, and w'(r) is the slope of the membrane in distance r from the central mass. The effect per second of the change of  g is  Dg= -g/T  or  Dg= -g/3.789´1017.  This effect is also too small to explanate the anomalous acceleration.

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We see that all effects caused by the expanding membrane fail here to explanate the anomalous acceleration of the two spacecrafts Pioneer 10 and 11, respectively. Now we consider effects of sun-near potentials. The Membrane theory postulates here two effects:

·      the Gerber-Einstein acceleration -gMs6a/r3

·      sun-near acceleration of light

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The sun-near acceleration of light is an effect discussed in chapter 5.1 (Shapiro Effect, Light Bending and Depth of Space). The effect decreases with 1/r4, and is very small. Calculations showed that this effect is too short and too small to cause the anomaly of the Pioneer data.

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The Gerber-Einstein acceleration -gMs6a/r3 is derived from the square of energy, e2, in chapter 5.2 (Perihelion Advance of Mercury). Ms is the mass of the Sun, g the gravitational constant, r the distance from Sun, and a thhe half of the Schwarzschild radius. The author has done in the past some effort to find another derivation of this acceleration. It uses the ether wind acting on a sloped membrane. The effort was not very successful. But this is not the question here. The acceleration -gMs6a/r3 explanates exactly the perihel advance of the mercury and of the other planets. 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 leaved our solar system, the fit failed, and the implicitely added acceleration of aA = (8 ± 1) ´10-10 [m/s2] became visible.

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  The author has written a small program in C, Pioneer.cpp. One can find it in the file Sourcode.htm. 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]. For a distance of 200 AU we get CC =  -1.9 ´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]., between 20 AU and 200 AU by an amount of  8.6 ´10-10 [m/s2], getting CC =  -1.9 ´10-10 [m/s2].

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We see that we can hide well the average anomalous additional acceleration of aA = (8 ± 1) ´10-10 [m/s2] by a single parameter fit. The parameter fit of the solar system uses a lot of parameters, and was done perfectly for the known solar system. But leaving the planetary belt the spacecrafts entered space with a bad fit, and exactly here the neglection of the Gerber-Einstein acceleration became visible.

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The Gerber-Einstein acceleration does not influence the rotation curves of the galaxies. Calculations with the program Darksim9.pas (also to find in Sourcode.htm) 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.

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Remains the question what role plays the mysterious acceleration H0c? In Milgrom (1993) we find a list of several numbers, all somewhat fitting more or less well the acceleration a0 = 1.2 ´10-10 [m/s2]. In chapter 7.2.1 the author considered models for dark matter, which are able to explanate the effects found in galaxies. There the acceleration H0c does not play any role until now, but we cannot see the future.

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7.     Cosmology

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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.

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7.1   The Cosmological Constant

(First version 04/2002, last update 08/2006)

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  The cosmological constant L is dominating the expansion rate (Hubble constant) since 10 Billions of years (Blome and Priester 1991, Priester 1995). Derivating the field equations of Einstein from first principles (e.g. from Hamiltonian principle) the L-term appears cogently. It has the character of an constant of integration. In connection with the quantum field theory we may interprete the L-term as energy density eL=rLc2=Lc4/8pG (c=speed of light, G=gravitational constant). Since this energy density had to obey the equation pL=-rLc2, one compares it with the energy density ev=rvc2 of the quantum vacuum.

  Quantum vacuum is a short denominator for the deepest energy level of all quantized fields of matter in the quantum field theory (Priester 1995). In this context the elementary particles are higher energy levels of the fields. This fact let us suppose that the quantum vacuum is something primary and the particles something secondary. The quantum vacuum could have existed already long before the formation of matter. So we could separate the formation of space and time from the formation of matter. The equations of Einstein permit solutions of a cosmos free from matter, filled only with the quantum vacuum. The formation of primordiale matter was then the consequence of a phase transition of the quantum vacuum. But this considerations demand to discriminate between the primordiale quantum vacuum, which was rich of energy until formation of matter, and the true quantum vacuum connected with the L-term (Priester 1995). The cosmological model of the Big Bounce (Blome and Priester, 1991) uses the cosmological term.

  In Priester (1995) we find the following numbers: Standardized cosmological constant lo=Lc2/3Ho2=1.08, Hubble constant Ho=90 [km/sMpc]=2.91210‑18[1/s] (a newer value of Ho=72 [km/sMpc] see Freedman et al. 2000),  radius of the universe R=36109[ly], =3.4081026[m]. Deviating the Friedman equation 7.1.1 with d/d(ct) and dividing the derivative then by 2dR/d(ct) we get the standardized acceleration of the radius R according to eq. 7.1.2.

.

  with           (7.1.1)

.

K, D, L and k are supposed to be constants.

.

                         (7.1.2)

.

The last term of the right side we name as standardized cosmological acceleration. Multiplying the standardized cosmological acceleration term –LR/3 by c2 and with the relation L=3loHo2/c2 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].

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The 4-dimensional 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. pL means the force acting on one unit of volume of the membrane (quantum vacuum) in negative w-direction. Figure 7.1.1 shows forces and areas in the cases 2d and 3d. In the case 2d a string surrounds a circle yielding a pressure p2d[N/m]=2Fo[N]/D[m]=Fo/R. In the case 3d the spherical elastic membrane is giving the pressure p3d[N/m2]= 2pRFo[N/m]/pR2= 2Fo/R.

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Fig. 7.1.1: a) Case 2d,   b) Case 3d

 

In the case 4d the tensile force is Fo[N/m2] and the pressure p4d is given by eq. 7.1.4.

.

                               (7.1.4)

.

The dimensions of p4d are [N/m3]. The density (mass of one unit of volume) of quantum vacuum (membrane stuff) is rM= Fo/c2= 235.2[Kg/m3]. That means, considering one unit of volume, the force p4d is acting on the mass rM. Equation 7.1.5 yields the acceleration aM. The value aM does not depend directly on any feature of the membrane, but we should remember that we are using the relation rM= Fo/c2 (see also Carter 2004). This relation says that gravitational waves propagate with the speed of light, a fact which is supposed, but experimentally not prooved till now. Another unsolved question is: Does membrane stuff consist of pure energy, or is there still another contribution to the density?

                              (7.1.5)

.

Using the same above value of R=3.408´1026[m] we find the value 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 and some Einstein-Friedmann equation, the other from physical constants under the membrane paradigma. But the values of both accelerations are to high. Calculating roughly the age TU of the universe in the frame of GR 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. We had now to discuss the consequences. The membrane model allows a higher speed, VE =2196 c2 (see chapter 3 in part 1) 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.

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7.2   Dark Matter caused by the etherwind (First version 05/1998, last update 08/2006)

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One of the most pressing issues in cosmology is the nature of the ubiquitous dark matter which pervades the Universe (Turner 1986, Overbye 2001, Feitzinger 2002). 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. It was postulated that at least 90% of the total mass of the universe are constituted by the dark matter. History (Markowitz 1996): In the early twentieth century Kapetyn and Jeans concluded that non-luminous matter is present. During the 1930s, Swiss astronomer Zwicky concluded that galaxies in the Coma cluster were moving too fast. In 1936 Smith found the same effect studying the Virgo cluster. In 1939 Babcock was attempting to measure the first time a rotational velocity curve. In the same year Rubin and Ford found the flat rotation curve of galaxy M31. The velocity leveled off at around 200 km/s.  During the 1940s and 1950s the Dutch astronomer Jan Oort studied the motions of stars normal to the plane of the galaxy disk. He found a Dark Matter coefficient of about 4. Since one would not cast doubt on Newton’s law of gravity, the idea was born of additional matter (dark matter or missing mass) inside the galaxies or clusters of galaxies. Its density is estimated to about r=5´10-28 [Kg/m3] (Turner 1986).

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Theory divides Dark Matter in Cold and Hot Dark Matter. Cold Dark Matter (CDM) is thought to be formed by MACHOs, i.e. small stars, dust clouds or black holes, or by WIMPS (Weakly Interacting Massive Particles, see e.g. Duerbeck 2002). Boehm et al. (2003) argue for light particles (1-100 MeV). These particles are generating positrons and electrons. If they collide, they generate gamma rays (511 keV).

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Some candidates of Hot Dark Matter (HDM) are heavy neutrinos, SUSY (Super Symmetry) particles as neutralinos or Axions (Kascer 1991, Pailer 1996, Turner 1986,Turner & Tyson 1999, Phoenix 1999). Dark Matter is playing an essential role in forming galaxies also (cf. e.g. Shapiro, Giroux and Babul 1994 or Silk 1989 or Blome et al. 2002). Fig. 7.1 shows a typical rotation curve of a galaxy (Begemannn 1989). The vertical axis gives the speed  v of the stars in km/s. The horizontal axis gives the distance of the stars from the centre of galaxy in kpc. Such dynamical studies were done also for gas and other clouds inside and outside of galaxies (LaRosa, Shore & Magnani 1999).

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Fig. 7.2.1. -  Rotation curve of galaxy

NGC 3198 from Begeman 1989

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The rotation curve following only from visible mass (called Newtonian curve, here not shown, but see e.g. Fig. 7.2.4.) would have a maximum at a distance of about 5 kpc. For greater distances, the Newtonian curve would descend rapidly. In reality observed rotation curves show a horizontal course for greater distances from the centre of the galaxy. The demanded amount of dark matter is estimated by numerous astronomers as 3 to10 times the mass of visible matter in galaxies, up to 300 times in clusters or superclusters.

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 Some properties of the dark matter (or halo-mass) are (Feitzinger 2002):

·      Dark matter is not a part of an uniform background mass distribution. It is concentrated near baryonic matter

·      At the outer parts of galaxies the density of dark matter is 100 to 100 times greater,  than the density of normal baryonic matter (e.g. gas or dust)

·      Dark matter has no dense points comparable with stars

·      The comparable shape of the rotation curves of different disc galaxies shows that both, luminous and dark matter, contribute to the radial mass distribution

·      Between luminous and dark matter is an relation, which depends on the type of the galaxy. Dark matter seems to know how much luminous matter is in a galaxis

·      Inside the optical radius of a galaxy the dark matter is about a half of the total mass

·      Dark matter we find also far outside the optical radius. So dark matter may have a percentage of up to 90%

·      There are indications to a baryonic nature of the dark matter.

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Some additional or differing features of dark matter we find in Blome, Hoell & Priester (2002):

·      The dark matter halo may reach up to 100 kpc outside (measured from centre)

·      The evolution of galaxy structures needed the dark matter percentage

·      Neutrinos with some rest mass can not be candidates of dark matter, because they move with relativistic speed, and so can not contribute to the forming of potential funnels as germ-cells of galaxy structures

·      Other candidates for dark matter are exotic particles. But hitherto one has nothing found.

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7.2.1 Dark Matter (caused by etherwind)

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The membrane theory delivers not only Newton’s potential, but it may deliver also a new model of the phenomenon of Dark Matter. In the membrane theory dark matter is not really existent and not generated in the early universe (Siegel 2006), but it is an anomaly of the gravitational law (for similar ideas cf. Kogan 2001). It is caused by effects of the ether wind on a perturbated membrane. Two imaginable effects (two of some) the author has 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 lateral force is in the first order proportional to the volume and proportional to the slope of the membrane, i.e. Fec=kecw’dV. The force pushes stuff in the central direction of the gravitational funnel, and so it causes an increase of the density r(r) of the membrane. The density of the membrane increases with 1+a/r according to the explanations of chapter 5 during approximation to the centre of the funnel. The increment a/r corresponds to the course of the depth of space w(r). Thus, by its enlarged resistance a thickened and inclined membrane may be the reason for a new force Fer=kerw÷w'÷dV. ker is a still  unknown coefficient. Here a new cold dark matter candidate (CDM) is presented. It is the thickening of the cosmic membrane caused by ordinary matter leading to a long-reaching violation of the 1/r-potential.

      We can find the differential equation of curvature of space in cases of a radial symmetry in a similar way as shown in chapter 3. Once more we take a small sector of space with a small angle g<<1 starting at the centre and opened outwards. On a section DV with length dr at a distance r from the centre (see Fig. 7.2.2) the forces Fr, Fg, Ft, Fec, Fecw and Fer (see Fig. 7.2.3) are acting.

Fig. 7.2.2 -  Sector of space with volume element DV

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Fc is the tensile force of membrane acting in direction of the centre of  the funnel. It is nearly identically with the tensile force Fo at infinity distance from the centre. Fr is the tensile force acting radially outwards and also nearly identically 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. Fe is the lateral force (buoyancy) of the membrane korns inside the volume DV and its norm is proportional to ÷w’÷, Fec is its component with the slope of the membrane, Fa= Fe-Fec is the difference and its norm is proportional to ÷w’2÷. Fec is directed to the centre (negative r-direction), Fa is directed to positive w-axis. Fer is an additional ether force caused by the increase of density of the inclined membrane inside the volume DV. It is supposed to be proportional to w(r) (or density) and proportional to ÷w’(r)÷ (or slope of the membrane), i.e. Fer » w(r)÷ w’(r)÷. The direction of Fer  is given by the sign of w. If depth of space w is positive, i.e. above the mean level, the membrane is thinned and its resistance is lower than the mean resistance. We count this case positive. In the other case we count negative.

 

Fig. 7.2.3. -  a) Forces acting on volume element DV

b) Forces acting on a korn (torus)

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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:

.

Frw = Fo g 2( w’ + dw’ ) ( r + dr ) 2         (7.2.1)

.

Fcw = - Fo g 2  w’ r 2                              (7.2.2)

.

Fgw = - Ae r(r) g 2 r 2 dr                        (7.2.3)

.

Fa = + ka w’2 g 2 r 2 dr.                           (7.2.4)

.

Fer = + ker w÷w'÷ g 2 r 2 dr.                     (7.2.5)

.

Neglecting small values the equilibrium of forces Frw+ Fcw + Fgw + Fa + Fer =0 gives the ODE Eq. 7.2.6 for the depth of space w(r).

,       (7.2.6)

.

or with renamed coefficients

.

.       (7.2.7)

.

The first term of the right-hand terms of the ODE 7.2.6 (or 7.2.7, 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. The third term is caused by the new force Fa  and the fourth term by the new force Fer  according to the density of membrane. Fo is the tensile force of the membrane.

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The constants Ae , Fo und the coefficient A we can calculate from the depth of space Wo at the edge of  Sun (see chapter 3) 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].

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Since the slope w’(r) of the membrane at the position r of a heavy mass is directly proportional to the gravitational force acting on the mass, we can define the ratio we’ / w’, i.e.

d(r) = we’(r) / w’(r)                               (7.2.8)

as the dark-matter coefficient d(r). we’(r) is the solution of the ODE 7.2.7 calculated with all right-hand terms, i.e. with the C- and D-members also. w’(r) is the solution calculated without the C- and D-members. A dark matter coefficient of the value d=4, e.g., means that the dark matter must have thrice the mass of the visible matter for explanation of the observed gravitational effect.

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  The author used a model of a spherical galaxy in several simulation calculations. This galaxy has got the fixed radius of 50,000 light years (Rg=4.733´1020[m]) and a density distribution of radial symmetry r(r)=rCO , modelling halo and centre (cf. Turner 1986 or Premadi, Martel and Matzner 1998). rCO is the density in the centre (bulge) of the radial distribution. The construction of the galaxy model was influenced by the papers of Turner (1986) and of Barnaby & Harley (1994). For calculation of masses of galactic nuclei see e.g. Fromerth & Melia (2001) or O’Neil, Bothun & Impey (2000), of stars from Einstein redshift see Sion et al. (1998). Parameter s (the sigma of the distribution) has got the value of s=0.20RG . The value of rCO =2.107´ 10‑20[kg/m3] gives a total mass of  5´1010 sun masses for this galaxy. With this numbers, the ODE 7.2.7 has been solved numerically. The author followed three principles during the simulation calculations. First: use only terms in the ODE, which can be declared by a physical model. Second : seek a solution which is stable for great values of r. Third: the so caused effects inside the solar system had to be undetectable small. Values of C=1.909´10‑18[1/m] and D=1.596´10‑26[1/m2] yield a reasonable result with a d-coefficient of d=3 at a distance r=RG. At this state of the theory the author has no idea how to get the values of the coefficients in a more theoretical way and not by ‘try and error’. The starting values of both we(r) and w(r) has been chosen so that the depth of space becomes about zero at infinite distance r.

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Fig.7.2.4. - Rotation curves ve(r) ¾ and v(r) ---

from r=0 to r=1.5 radii of galaxy

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  The program for the numerical solution of the ODE 7.2.7 delivered the rotation curves ve(r) and v(r) from Fig. 7.2.4. A star must have a determined speed orbiting the gravitational centre. ve(r) is the demanded orbital speed of a star in km/s about the centre of the galaxy with consideration of the dark matter, v(r) without consideration of the dark matter, i.e. the pure Newtonian solution. The distance r is scaled in parts of the radius Rg of the galaxy. Contrary to curve v(r), the decrease of the curve ve(r) behind the maximum is insignificant only. The Dark Matter area is greater than the visible part of the galaxy (Chengalur, Salpeter & Terzian 1994).  Thus, the curve ve(r) could resemble really observed rotation curves. Fig. 7.2.5 shows the course of the dark-matter coefficient d(r) with growing distance from the centre of galaxy. Up to a distance of value r»0.2 Rg, the dark-matter coefficient d(r) remains near the value of 1, i.e. in this range of the galaxy is no dark-matter effect visible. Starting at a distance r»0.35 Rg the curve d(r) is growing in this simulation calculation. The maximum we find outside the galaxy.

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Fig. 7.2.5. -  Course of the dark-matter coefficient

d(r) from r=0 to 1.5 radii of galaxy

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If a model does not have spherical symmetry (models of spiral or bar galaxies e.g.), we must solve a partial differential equation (PDE). The author used  a bar with a determined mass load as a model of a simple bar galaxy. The solution of the PDE has been programmed using a difference method for a section of space containing the bar. In such a method the mass load is defineable only at the grid points. The author used a point shaped central load LC together with a distributed load with the linear decrease L(r)=Lo(1-r/RG) from centre to the edges. Central load and the sum of the distributed loads were equal. The computational algorithm neglects small terms to save computer time. (Similar ideas one can find in Hamana, Martel & Futamase 2000.)

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Fig. 7.2.6 shows the rotation curves of the bar galaxy model. The Newtonian rotation curve v(r) is calculable with oscillating disturbances only. The reason is the grid structure of the distributed loads. ve(r) is defined by the difference method at the grid points only. The nearest point distance of the grid was 0.1 Rg. The total section of space used in the simulation had got a radius of  1.7 Rg. Greater space sections or finer grids lead rapidly to a great  expenditure. The course of the rotation curve ve(r) here is above the Newtonian curve v(r), too. The used C- and D-coefficients were C=0.00015 and D=0.000025. They are equivalent to C=2.577´10-18[1/m] and D=2.380´10-26[1/m2] of the ODE 7.2.7. One can compare them with the coefficients of the ODE by calibration with the same spheric model only. At r=Rg the bar model yielded a dark-matter coefficient d of  a value of  d=2.5.

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Fig. 7.2.6 -  Rotation curves ve(r) ¾ und v(r) --- of

the model of a bar galaxy with radius 10

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Terry Matilsky (Matilsky 2000) has found a similar solution without using a membrane model. I quote: "We propose an additional term in the classical gravitational force law, which is repelling in nature, and which may solve the dark matter problem. As an inverse cube field interaction, it operates over 4 real spatial dimensions and its effect on our observable 3-D space may account both for flat rotation curves and standard Newtonian dynamics at small radial distances."

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7.3  Expansion of the Universe (First version 02/2003, last update 02/2004)

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  Friedmann (1922) and Alpher, Bethe and Gamov (Alpher et al. 1948) were some of the first who investigated an expanding universe. An actual question of astrophysical relevance is the discussed deceleration versus acceleration of the expansion of the Universe (e.g., cf. Schmidt et al 1998, Perlmutter et al. 1999, Albrecht et al. 2001, Khoury et al. 2002, Suntola 2003 a). The Cosmic Membrane Theory yields here a "third way": The real speed of expansion 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.

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  "In 2001 A. Riess and M Livio of the Space Telescope Science Institute presented a 11´109 years old supernovae which is on the Hubble diagram half as bright as it should. The conclusion that the expansion of the universe is accelerating is based on the observation that Type Ia supernovae at redshifts greater than 0.5 are dimmer--and thus farther away--than their redshifts would suggest if the universe were coasting, or if expansion were slowing under the influence of gravity " [Overbye 2001].

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To quote Albrecht et al. (2001): "Recent observations of type Ia supernovae (SNe Ia) at high redshift indicate that the expansion of the Universe is accelerating [Perlmutter et al. (1999a), Riess et al. (1998a)] although concerns about systematic errors remain, these calibrated 'standard' candles appear fainter than would be expected if the expansion were slowing due to gravity. According to General Relativity, accelerated expansion requires a dominant component with effective negative pressure, w=p/r <0. Such a negative-pressure component is now generically termed Dark Energy; a cosmological constant L, with  pL= -rL, is the simplestbut not the only possibility.

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Another quotation from the same source: "With or without dark energy, a consistent description of the vacuum presents particle physics with a major challenge: the cosmological constant problem (see, e.g., Weinberg (1989), Carroll et al. (1992)]. The effective energy density of the vacuum - the cosmological constant - certainly satisfies WL < 1, which corresponds to a vacuum energy density rL=L/8pG < (0.003 eV)4. Within the context of quantum field theory, there is yet nounderstanding of why the vacuum energy density arising from zero-point fluctuatuions is not of order the Planck scale, M 4Pl , 120 orders of magnitude larger, or at least of order the supersymmetry breaking scale M 4 SYSY ~ TeV 4 , about 50 orders of magnitude larger. Within the context of classical field theory, there is no understanding of why vacuum energy density is not of the order of the scale of one of the vacuum condensates, such as M 4 GUT, M 4 SUSY,  MW 4 sin 4 (q ) / (4 p a) 2 ~ (175 GeV) 4 , or fp 4 ~ (100 MeV) 4 .

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The discrepancy between the supposed value of the standardized cosmological constant lo= Lc2/3Ho2 =1.08 (Priester 1995) and the demands of several theories is one of the most embarrassing problems for modern particle physics theory. A number of models, termed Quintessence models, in which the dark energy is associated with a scalar field and not with a fundamental cosmological constant have been discussed in the last years (e.g. Carroll 1998 or Albrecht et al. 2001). This models postulate that the vacuum energy of the Universe is zero or nearly zero. Dark Energy so has to be stored in the potential energy of a scalar field F.

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From point of view of the Membrane Theory the quintessence theories are very interesting. The scalar field F has some properties of the membrane tension Fo. But one problem arises immediately - the discussed acceleration of the expansion of the Universe. The Cosmic Membrane Theory model predicts a very faintly decelerated expansion (see chapter 7.1). A pressure or negative energy would mean an ether wind from inside the balloon, so that we had to assume an ether source there. We can not exclude this case, but it is difficulty to imagine. Reversely, the Cosmic Membrane Theory says that the ether rests, and the membrane moves very fast through the resting ether.

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The first question is how physical laws react to a changing Universe? There is a lot of literature concerning changes of physical constants (e.g. Webb et al. 1999, Ivanchik et al. 2002, Carlip 2002, , Capozziello 2002, Clayton and Moffat 2002, Suntola 2003 a). To quote Carlip: "Over the past few decades, there have been extensive searches for evidence of variation of fundamental "constants". Among the methods used have been astrophysical observations of the spectra of distant stars, searches for variations of planetary radii and moments of inertia, investigations of orbital evolution, searches for anomalous luminosities of faint stars, studies of abundance ratios of radiactive nuclides, and (for current variations) direct laboratory measurements.

    E.g., the time standard was set up with high precission. To quote Bize et al. (1999): "The achivment of high-precision Rb clocks also opens the way for several new experiments. As shown in [12], a search for a possible drift of the fine-structure constant can be done by comparision of the hyperfine energies of alkali atoms with different Z numbers. Interest in such a test has been renewed by recent developments of string theory that predict a possible drift of the fine-structure constant a with time [23]. We estimate that using Rb and Cs fountains, (da/dt)/a can be tested at a level of 10-16 per year, wich would represent a 100-fold improvement compared to the present best laboratory tests [12]."

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   A stretching membrane thins. Membrane tension Fo grows, membrane density r shrinks. This changes must have some influence on the physical constants. First we consider a uniformly expanding Universe with a constant speed of light. The space stretches with the expanding Universe. A photon with the emitted wave length lE has the obeyed wavelength lo. R(tE) shall be the radius of the Universe at emitting time tE, Ro the radius nowadays. As red shift z we spell the relative increase of the wavelength. One assumes lE=lo, i.e. the emitted wave length lE equals the measured wave length lo in the laboratory (e.g., cf. Blome et. al. 2002).

.

           (7.3.1)

.

Fig. 7.3.1 shows a 2-dimensional projection of the 5-d spacetime. The blue arcs show the surface of the balloon at different time points measured by their redshifts z. Time and radius are running upwards, starting at the Big Bang or Big Bounce.

Fig. 7.3.1: Light path in an uniformly ex-

panding Universe with constant speed

of light

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BB is the position of the Big Bang or Big Bounce. The sources of light are marked with S, the observer (or receiver) is R, red shift is Z. The light starts billions of years ago and moves from the boundaries of the visible part of the universe to our present position and present point of time. The light cone illustrates the space-time of that, what we can see. Each distance z is equivalent to some age of the object seen. (The graphic 7.3.1 and 7.3.2 use an expansion speed VE, which is of order of the speed of light, c. Otherwise, with VE >>c, the light cone would be too slim for a nice image.)

 

Since we had to suppose that a lot of fundamental physical constants vary in the course of the cosmological evolution, all conclusions from the simple red shift by the expanding Universe are worth to be considered carefully. The expanding membrane may change its properties, and therefore some other physical constants can vary, especially:

 

·      Velocity of light

·      Frequency of the source and local time

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We assume the total mass of the membrane to be nearly constant, although 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 surface of a 4d-balloon is

.     (7.3.2)

The radius R of the cosmic 4d-balloon grows nearly with R(t)=VEt. Time t shall be given by a clock with a tick dt of a second of the present time, and with no changes (4d-observer time). VE is the nearly constant speed of expansion of the balloon, and is assumed to be much more greater than speed of light c, i.e., VE>>c. While the skin of the balloon expands the load per area of the membrane decreases. ( In the case of a 3-dimensional membrane in a 4-dimensional space the load per area LA has the unit [kg/m3] ). From eq. 7.3.2 we find

.     (7.3.3)

Here LAO is the membrane load per area at present time. But how the density r does change? We do not know anything about the thickness or height hm of the membrane, i.e., the fourth dimension is unknown hitherto. A simple rubber membrane will not change dramatically its density if it is stretched. But if the membrane is a layer with thickness of one korn only, the density r will decrease with the same order of Ro/R as LA(R) does. We can check the limits of the power order only. The true order is in the range of 0 to 3. No change of density yields zero. Otherwise, membrane tension F should linearly increase with an expanding balloon. That follows from the assumed elastic behaviour (c.f. Capozziello 2002). We write:

.

.     (7.3.4)

.

Here Fo is the membrane tension at present time. From the wave equation c2=F/r we deduce

.

.       (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. Joao Maguelijo (2003) has similar ideas. From chapter 5.1 we know that the speed of light changes in a gravitational field, e.g., the field of the Sun. We found c(r)=co(1-2a/r). Here 2a is the Schwarzschild radius of the Sun, and r is the distance from the centre of the Sun. Another known relation (cf. Weinberg 1972, Fliessbach 1990, e.g.) is the change of time in a gravitational field. Equation 7.3.6 gives the change of the eigentime dt of a clock in the gravitational field of the Sun, e.g., compared with the time tick dt given by a clock outside the gravitational field.

.

      (7.3.6)

.

Here goo is one of the coefficients of the spacetime metric, g is the gravitational constant, MS the mass of Sun and r the distance from the centre of Sun. Equation 7.3.6 is experimentally well proved by different experiments, especially by the behaviour of clocks in the GPS satellites (Hatch 2000). 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. The quotient of the independent differentials (dc/dr)(dr/dr)dr and (dn/dr)(dr/dr)dr is at the present radius of  Ro of the Universe

.      (7.3.7)

.

If the change of speed of light c and the change of frequency n both depend on a change of the properties of the membrane, and those properties change with the cosmic radius R, we can write the independent differentials with eq. 7.3.5 and P=0.5 also in the form

.

.                           (7.3.8)

.

At point R=Ro we find with eq. 7.3.7 the relation

.

.      (7.3.9)

.

The simplest formula for n(R) yielding at point R=Ro this result is

.

.      (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 a wave at the skin of a balloon, which we blow up. The membrane model yields a second contibution: If  speed of light grows, the wavelength grows too (cf. chapter 4.7, Gravitational Red Shift ). This backward transformation, as described in ch. 4.7, does not include any frequency transformation. Here, in this model, both backward transformations are therefore of the form  lobs=lE(Ro/R)k, where k has different values for the stretching transformation (k=1) and for the speed-transformation (k=1/2).

  For a physical explanation of the speed-transformation of the wavelength we would need a good photon model. Although there are different models in the literature (e.g., cf. de Broglie 1943, Meyl 1990, Meno 1997, Lawler 1998, Krasnoholovets 2001 and 2002), the author is not able now to find the connection between photon models and membrane properties. 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. Armed with the formulas found above we have gotten a new model of the expanding Universe. An outer observer would see a nearly linearly expanding balloon (cf. Valdés Marin, 2003), starting at the Big Bang or Big Bounce. But our world is positioned at the skin of the balloon, and there time, speed of light and length scale change with the expanding balloon.

×

 

 Fig. 7.3.2: Light path in an Universe with

accelerating speed of light

×

Fig. 7.3.2 shows at the left side (blue) the redshifts zm and at the right side (brown) Friedmans redshift zF. The light cone (red) is still more curved now compared with the light cone from fig. 7.3.1. In opposite to Friedman's model with constant speed of light the horizon grows more slowly here. We will never see the whole Universe, althoug the farest galaxy hitherto (state 02/2003) with z=6.56 was found recently. We should remark two important points:

·      The Universe seems to be flat for us, because of the vast radius R (c.f. Capozziello 2002). Expansion speed VE is assumed to be much higher as speed of light. So the Cosmic Membrane Theory does not contradict de Bernadis et al. (2000), which had derivated from CBR data a flat, Euclidean metric (see also Raffelt 2006).

·      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, as fig. 7.3.3 shows. 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 (c.f. superluminal expansion in Capozziello 2002). But that is relative. All processes went very slowly during the real radius of the Universe was growing stadily with constant speed.

×

 

Fig. 7.3.3: Real radius R(t), redshift z(t) and

Friedman's radius RZ(t) over eigentime t

×

Fig. 7.3.3 shows that a redshift of z=10 adresses objects which emitted the light a long time after the Big Bang or Big Bounce. The radius of the Universe, R(t), starts with infinite slope and grows then nearly linearly. The curve RZ(t) gives the radius we get by RZ(t)=Ro/(z(t)+1). It is a straight line as expectet. Ro is here taken as 1. RZ is the radius which is commonly used in Friedman's model. The RZ(t)-curve does not show any inflation.

×

Now back to the supernavae candles. The author simplifies the calculation by the use of dimensionless values. All scales are scales of an recent observer. Recent time is to=1. Expansion speed VE and speed of light c are all set to 1. Recent radius of the Universe is so Ro=1. Since only ratios of surfaces are used the factor 4p is not needed.

×

First we consider a closed Fridman model and perform the following steps:

i) f1=1 shall be the measured light flux per unit of area from a nearby candle with redshift z1=0.05. We want to norm the candle in Friedman's model and calculate first the radius r1 of the candle using the equation r1= Ro / (z1+1) = 1/(0.05+1) = 0.9524.

ii) In the next step we calculate the radius y1 of the light sphere after expansion of the Universe to r=Ro. We find, e.g. in Blome et. al. 2002, the following equations:

.

        (7.3.13)

.

               (7.3.14)

.

Here c, the radial coordinate, is the angle between the world line of the source and the world line of a receiver positioned at the actual position of the photon. Distance y, the metric coordinate, is the distance between the actual position of the source and actual position of the photon. Time t is the Friedman time. In program expansi1.pas eq. 7.3.13 will be integrated numerically, and we get y1=0.04877. Now we can gauge the total light flux f of the candle with the value of f=f1y2=0.002379.

iii) Now we take a redshift z2=6 and estimate distance r2= Ro / (z2+1) = 1/(6+1) = 0.1429. From this radius we calculate using eqs. 7.3.13 and 7.3.14 the diameter y2 of the light sphere with a value of  y2=0.9305. With the total flux f of the candle (both candles are supposed to be of equal total flux) we get the lighting of an area with distance y2 from the source by f2=f/y22=0.002747.

 

×

Now we make a similar calculation using the assumptions of the cosmic membrane model:

i) Starting with z1=0.05 we get, using eq. 7.3.12, a value of r1=Ro/(z1+1)4/5=0.9617.

ii) We gauge the candle first calculating the diameter y1 of the light sphere. Since the membrane model uses 4 spatial coordinates and the time t of  the outer observer at position Ro, we can not use eq. 7.3.13. The tangential movement y of the photon is described by the ODE

.

.      (7.3.15)

.

The variable speed of light is c(R). Neglecting the deceleration of the expansion, R(t) follows the simple equation R(t)=VEt. The growth of y has two sources here - first the speed of light c(R) and second the stretching of the already covered distance y(t). By numerical integration we get a value of y1=0.03866, and from this value the total flux f with a value of f=f1*y12=0.001495.

iii) From redshift z2=6 we get r2=Ro/(z2+1)4/5=0.0.2108. Using ODE 7.3.15 we get the diameter of the light sphere y2 with a value of y2=1.0817. Lighting is f2=f/y22=0.001278. This is about one half of the value of f2 of the Friedman model. Tuomo Suntola (Suntola 2003 a) has found another solution of the supernovae issue, but also using varying speed of light.

×

The Cosmic Membrane Theory yields an alternative possibility of  an explanation of the assumed acceleration of the expansion of the Universe. That far supernavae of typ Ia seems to be fainter as assumed we could explanate by a changing speed of light and a changing frequency of the light sources depending on the tension of the cosmic membrane in four spatial dimensions. This explanation does not deny the existence of the dark energy, but the meaning is some different from the behaviour of the quintessence field. The kinetic energy of the expanding membrane could be this dark energy. The Cosmic Membrane Theory says also that the Universe must be nearly flat, since we can see only a tiny section of the surface of our balloon we live on. The inflation of the radius of the Universe at the first milliseconds after the Big Bang or Big Bounce we can explanate as a time effect: in the tough soup of the early membrane all processes went on very slowly. In Cosmic Membrane Theory inflation has only one phase starting together with the Big Bang. This is in contradiction to the supposition of several phases (cf., e.g., Taylor and Berera (2000)).

×

A. Starobinsky (2000) discusses the conclusions from the existence "of a positive and possibly varying Lambda-term". He makes three assumptions: The present Hubble constant Ho=50 kms-1Mpc-1 (Freedman et al. 200 are giving 72 kms-1Mpc-1 ) the present age of of the Universe to=10 Gy, and the energy density of the Lambda-term is non-negative (and will remain so for a period of about 22 Gy). Further assumptions of A. Starobinsky are: i) Dark energy (or quintessence) remains unclustered, but may change with time. ii) Loss of possibility to reach distant objects. iii) Probability of collision with a null singularity (super massive black hole). iiii) Formation of a classical space-like curvature singularity during expansion. v) Hitting a space-like singularity ib future due to quantum-gravitational effects. The author agrees to all assumptions and conclusions, but gives here some remarks: i) The kinetic energy of the membrane stuff (quintessence) is it what decreases. ii) Since cosmic membrane expansion VE>>c this assumption is immanently true. iii) The existence of black holes is not in contradiction to the cosmic membrane paradigm. iiii) Cosmic membrane theory includes the case of a membrane rupture comparable with the burst of a soap bubble. This is not the same what A. Starobinsky is assuming, but with similar effect. v) Superposition of gravitational waves may arise.

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7.4   Dark Matter and Frame Dragging, Geodetic Precession (First version 08/2004, last update 06/2008)

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Special and General Relativity are giving effects not known in the Newtonian mechanics. The geodetic effect, the Lense-Thirring effect (frame dragging) and the geodesic precession are interesting here.

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Geodesic Precession (First version 08/2004, last update)

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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 BCRS supposes time flow under the conditions zero speed and no gravity. The GCRS time flow tG is lowered by the kinematic time dilation t'=tB/(1- v2/c2) 0.5 due to Special Relativity and the gravitational time dilation t''=tB(1+a/r) due to General Relativity. Here tB is the time of the BCRS, v is the speed of Earth at its orbit, c the speed of light, "a" half  the Schwazschild radius of the Sun ( a=gMS/c2 ), g the gravitational constant, MS the mass of  Sun, r the distance Earth-Sun.. With the relation a/r= v2/c2 and the squareroot-series we find tG=tB(1+(3/2)(a/r)). With Earth's angular speed v/r and the surplus time tick of (3/2)(a/r) we find the geodesic precession (dw/dt)gds=3va/2r2 with a value of 1.9198"/century (Brumberg et al. 1992).

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Lense-Thirring Effect (First version 08/2004, last update)

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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. Especially, it means that each particle of  a first mass is connected via gravitons with each particle of  a second mass (see Fig. 7.4.1a). If one mass is rotating, the particles move, and following the gravitons have to transmit the new position with its changed gravitational force. Fig. 7.4.1b shows the edge of a rotating mass with  particles A and C. A' and C' are the same particles some time later on its new position. A and C sends gravitons to B. If the gravitons reach particle B, we find A at position A' and C at C'. The symmetric configuration A-B-C' causes the resultant force (red arrow), which is not centrally directed, but somewhat inclined. The calculations of Lense and Thirring and other authors show that we can describe this effect by field lines similarly to the field lines of a magnet, and they use the term gravitomagnetic field. The magnetic poles coincide with the two ends of the axis of rotation.

   Due to this theory a particle in free fall in direction to a spinning mass does not describe a straight line, but a slight spiral form (green trajectory in fig. 7.4.1b). Photons prapagate with rotation faster than counterwards. The whole space is moving like a fluid surrounding the spinning source.

×

Fig. 7.4.1: Lense-Thirring effect

×

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. A rotating spherical mass wraps the membrane in nearly the same kind as a non rotating spherical mass does. Small deviations arise from the oblate form of a roting and not rigid spherical mass. Only if rigid bodies are rotating (or coupled masses, e.g., double stars), as figure 7.4.1c is showing, then we will find a similar effect, i.e. similar to the original Lense-Thirring effect. The changes in the curvature of the membrane propagate with speed of light in the Cossmic Membrane model too. But this effect is of smaller order compared with the originally intended Lense-Thirring effect, because the deviations from the ideal spherical form are acting only.

×

Nevertheless, the author does not deny the possibility that any effect may exist causing a precession of the spinning axis of a gyroscope out of the orbital plane as the Lense-Thirring effect should do, but it will have only accidentally the same amount (see e.g. R. L. Collins 2004).

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Dark-Matter Effect by Spinning Membrane (First version 08/2004, last update 04/2005)

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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 (see also M.Grady (2002), or R. J. Cahill (2005)). 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. Unfortunately, the author sees no way to evaluate the amount of the rotation. The coupling between matter and membrane must be very small. Otherwise light would lose its energy or moving particles would stop.  

   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. Fig. 7.4.2a shows a space vortex surrounding the spinning mass m. Part b of the same fig. shows the effect of the centrifugal force on the curvature of the membane funnel. Because the angular speed of the outer spheres of the vortex is assumed to be smaller than near the spinning center, we will find a bell-shaped form in the case of rotating space instead of the 1/r form of the undistorted curvature.

×

Fig. 7.4.2: Frame dragging and change of

shape of curvature by centrifugal force

×

Fig. 7.4.3 shows the forces acting on a piece of the mebrane. The centrifugal force, Fc, is acting radially. Space tension Ft=Fo+DF acts tangentially to the membrane, i.e., it has the slope w' of the membrane at this point. DF is some enlarging of the natural space tension Fo. The resultant force Fw is directed in negative w-direction as an load by ordinary matter.

×

Fig. 7.4.3: Centrifugal force Fc, tension Ft

and resultant force Fw

×

The best method to approach the effect is simulation. It was done in two steps. The first model for simulation was the spherical model of a galaxy from chapter 7.2.1. It was used only to prove whether the estimated coefficient of the ODE is yielding acceptable results. The ODE 7.4.1 used for this rough estimation is similar to equ. 7.2.6.

.

        .          (7.4.1).

The first two right-hand terms are known from chapter 7.2.1. The third right-hand term has dimension [1/m] as all terms of the ODE. rM is the membrane density, w(r) is the angular speed of the membrane at distance r from the rotational axis. The centrifugal force per volume unit is then Fc(r)= rMw2(r)r. Multiplication by the slope w'(r) of the membrane is giving force Fw(r) per volume unit. Division by tension Fo yields the radius of curvature. This rough model has a serious disadvantage: No rotating galaxy will be of spherical shape. But to get a first indication of the order of the effect and what function w(r) to choose, this model is good enough.

    Our spinning body shall have a radius of  Ro=21.5 km and spins with angular speed wo=2p2200 [rad/s] (see e.g. Gondek-Rosinska et al. 2000). That means that the speed of the surface is near the speed of light. With membrane density  rM= Fo/c2= 241 [kg/m3], calculated from membrane tension Fo= 2.164´1019 [N/m2] and speed of light c=  2.99792 ´108 [m/s], and a function w(r)=  wo(Ro/r) modelling the decrease of angular speed of rotation with increasing radius we get the ODE equ. 7.4.1 rewritten as

.

        ,                (7.4.2).

 with the B-coefficient of the ODE as

.

        .                 (7.4.3)

.

Fig. 7.4.4 shows the rotation curve v(r). The speed is 280 km/s for radii greater 0.3 Rg. Rg is the radius of the model galaxy. The Newtonian rotation curve, calculated from ordinary matter distribution, decreases with increasing distance from the center. The dark matter coefficient for r=Rg was dmc=5.48.

×

Fig. 7.4.4: Radial rotation curves for the spheroidal

model. Curve vth(r) is the rotation curve without frame

dragging effect

×

        Next step of the simulation was the calibration of the difference method. The same spheroidal model was used and dmc(Rg) fitted to the above value of dmc=5.48. Now the disc model from chapter 7.2.1 was used with the calibrated B-coefficient (B=1.47 in the context of program DARKSIM7.PAS). The resulting rotation curves are given in Fig. 7.4.5

×

Fig. 7.4.5: Rotation curve v(r) caused by

frame dragging for a disc galaxy model and

Newtonian rotation curve vth(r) for visible

 matter

×

The rotation curve v(r) has not an even course for radii>0.3 Rg, as the example of Begeman has (see fig. 7.2.1), 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 curve from fig. 7.4.4. 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.

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Geodetic Precession (First version 11/2004, last update 06/2008)

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The geodetic precession is the rotation of the spin axis of a gyroscope in an isotropic gravitational field of a static mass distribution (Fließbach 1990). Fig. 7.4.6 shows a precessing gyroscope during its round trip around Earth.

Fig. 7.4.6: Precessing gyroscope orbiting the Earth.

×

We cite Clifford M. Will's (2002) clearly written paper: "Gravity Probe B - the 'gyroscope experiment' - is a NASA space experiment designed to measure the general relativistic effect known as the dragging of inertial frames. The experiment will place into Earth orbit a spacecraft containing four gyroscopes and a telescope (for details of the experiment see Everitt 1988 and http://einstein.stanford.edu).The gyroscopes are 4 cm diameter fused quartz or single crystal silicon balls, machined to be as spherical as possible (to tolerances better than one ppm), and coated with a thin film of niobium. At the low temperatures provided by a dewar of superfluid Helium surrounding the gyroscopes assembly, the niobium is superconducting, and each spinning gyroscope has a London magnetic moment parallel to its spin axis (stray magnetic fields, such as those due to the Earth, will have been supressed by many orders of magnitude before the niobium goes superconducting). The orientations of the gyros' magnetic moments are measured via changes in the magnetic flux through superconducting current loops that encircle each gyro, and that are attached to the gyroscope housing. The four gyros will initially have their spins alligned parallel to the symmetry axis of the spacecraft, which coincides with the optical axis of the telescope mounted  on the end of the spacecraft. In order to average out numerous unwanted torques on the gyros, the spacecraft will rotate about its symmetry axis at a rate of between one and 10 times per minute.

  According to general relativity, a perfect gyroscope in a orbit around the earth will precess relative to distant  inertial frames because of two effects. The first and most important, is the dragging of inertial frames, caused by the rotation of the Earth, also called the Lense-Thirring effect. Over a time of one year, for a polar orbit of about 640 km altitude, this causes the gyroscopes to precess in an East-West direction by around 42 milliarcseconds (mas). The second is the geodetic precession, caused by the curved spacetime around the Earth. This effect results in an annual precession in the North-South direction of about 6600 milliarcseconds.

  The reference direction to the 'distant stars' is provided by the onboard telescope, which is to be trained on a star IM Pegasus (HR 8703) in our galaxy. ..." (End of citation)

×

  In Everitt (1991) we find the following formula of the geodetic precession:

 

 .       (7.4.4)

.

A factor  1/c2  is omitted in Everitt’s formula by the chose of the coordinates. The change of the spin is perpendicular to the axis of the precession and perpendicular to the spin by

.

 .               (7.4.5)

.

Here in eq. 7.4.4,  WG is the angle of the geodetic precession, v is the speed of the spacecraft on its orbit. The geodetic effect of the Moon-Earth system was first time derivated by W. de Sitter (1916). Other derivations, most of them with the same result, one can find in Fliessbach (1990),  Ciufolini et al. (1995), Nordtvedt (1996, see also Nordtvedt 2003), Laemmerzahl und Neugebauer (2001), Ruggiero and Tartaglia (2002), Collins (2002), Will (2002). Collins uses only SR effects and predicts the same precession angles as GR does, but with opposite sign. The GR predicted sign of the geodetic effect we can see by the formulas 7.4.4 and 7.4.5.  Fig. 7.4.7 shows the vectorial construction.

Fig. 7.4.7: Vectorial construction of change dS

 of spin vector S

×

Here, the vector of precession WG shows in the same direction as the orbital rotation W0 with speed v.

.

(In 2004 the author has tried the first time to derivate a similar formula – at that time only from point of view of the special relativity of the membrane theory. He has found at that time only  -1/12  or  -8.333 % of the GR geodetic effect. Meanwhile, it is clear that special relativity effects cancel one another out, because these effects arise from torques caused by different weights of the half-spheres. We had also to delete this remaining 1/12 of the special relativity effect, because the center of mass changes always in that way that no torque will remain.)

.

The main effect causing the Geodetic Precession is the decrease of speed in the gravitational funnel, as it was discussed for the speed of light in section 5.1.  In this section we find eq. 5.1.1. The equation is   c(r) = c0 ( 1-2a/r ).   The decrease of speed of light is not a geometrical effect of the curvature of space, but an inner structural change of the membrane as discussed in section 5.1.  From eq. 5.1.1 we find by a conclusion by analogy the eq. 7.4.6.

.

v( r ) = vo ( 1 – 2a / r )                  (7.4.6)

.

Speed vo is here the speed of the center of mass of the gyroscope on its orbit. Due to eq. 7.4.6 those parts of the gyroscope are moving faster which are more distant from the Earth. The nearer parts move slower. Obviously, this effect doesn’t make any action of forces. Otherwise, we would have a precession perpendicular to the plane of the orbit. 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 of the spin axis of the gyroscope in the plane of the orbit, and in the same sense of revolution as the movement of the gyroscope on its orbit:

.

     (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. The term  a/r  is not relevant here, because it describes the change of mass caused by the increase of speed during a free fall from infinite distance to distance r in the direction of the center of gravity. The term  2a/r is important here. It describes the change of mass caused by the changing properties of the membrane in the gravitational funnel.

.

Fig. 7.4.8: The spinning gyroscope on its orbit

.

In 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. We make a mathematical statement for the distance of the volume element from the center of gravity

.

            (7.4.8)

.

Here  q  is the orbital angle,  R  is the distance between the center of mass of the Earth and the center of the gyroscope,  rV  is distance of a volume element from the spin axis, and  f  is the revolution angle of the gyroscope around its spin axis S. With angular frequency  w  of the gyroscope we get  f = w t . We differentiate the increase of mass   m(r) = moo (1 + 2a/r)  with respect to time t, and get

.

.       (7.4.9)

.

We think the spherical gyroscope divided into slices of thickness d perpendicular to the spin axis S. The density of the material is  r. Here, a volume element in cylindrical coordinates is  dV = rV df drV d.  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. That force arises from the membrane, and is acting in the direction opposite to the direction of speed v in the case of decreasing mass. On the other side of the slice 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. But if the gyroscope is in position 2 (see fig. 7.4.8) then the visible volume elements are moving in the direction towards the Earth, and their mass will increase. But at the same time, the speed v has changed its direction into the opposite direction. So, the torque is acting in the same direction as in position 1. Only in such positions the torque dD is zero when the spin axis is perpendicular to speed v. This behaviour is described by the factor cos2(q) in eq. 7.4.10.

  In fig. 7.4.9,  speed  v  is the speed of the gyroscope on its orbit,  w  the angular frequency of the spin rate of the gyroscope,  u  is the tangential relative speed of a volume element of the rotating slice,  f=wt  is the rotating angle of the slice,  uR  is the projection of  u  on the plane of the orbit,  L  is the lever arm lengths,  f  is the force acting on the volume element,  fS  is the projection of  f  on the direction of the spin axis,  d  is the thickness of the slice. Here, the plane of the orbit we can see only as line of intersection with the slice.

.

Fig. 7.4.9: Disc of the gyroscope

.

The lever arm  L  of the torque  dD=L×f  is depending on the angle  f  by  L = rV sin(f) . The same is holding for the projection  ux  of  u  on the plane of the orbit  (ux = u sin(f) ). The projection  fS  of the force  f  on the direction of the spin axis is depending on the angle q  of the orbit ( fS = f cos (q )  ). We replace the mass  moo  in eq. 5.2.15 by the mass of the volume element ( moo = r dV = r rV df drV d ), and pay attention to the fact that the torque is doubled because the volume element is mirrored along the spin axis. The mirrored element does experience a force of the same magnitude, but in the opposite direction. So we get the small torque

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     (7.4.10)

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Integration is performed in two steps, and only for a half slice. In the first step of integration angle  f  runs from 0 to p. The second integration goes outwards from the center of the slice, i.e.,  radius  rV  runs from 0  to  rG, the radius of the slice of the gyroscope. We get the full torque of the slice. The torque does only depend on the angle  q now.

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.     (7.4.11)

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The moment of inertia,  JS , of the slice with respect to its spin axis is  JS = (p d r rG 4 ) / 2.  If a torque  D  acts on a spinning top with angular frequency  w   and momentum of inertia  JS, and the torque acts orthogonally to the spin axis as here, then the precession of the gyroscope is

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.     (7.4.12)

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Considering the direction of the torque which one can find in fig. 7.4.9, the theory of spinning tops says that  W2  is growing with the opposite sign with respect to  W1, i.e., we need to subtract both angles. In fig. 7.4.8 the spin axis would turn clockwise. Integration of  over one year gives the correction angle W2. The additional factor  ½  arises by the integration of the term cos2(q), because the gyroscope and with it the orbital angle q make 5394 cycles per year.

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.     (7.4.13)

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This value is equivalent to  2.2  arcseconds. The total geodetic precession of one year is the difference of W1 and W2 .          

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.     (7.4.14)

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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.

 

 

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End of part 3