COSMIC MEMBRANE THEORY

Part 2 of membrane theory (last update 01/2005)

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4.3 Momentum, Mass and Energy (First version 06/1991, last update 12/2001)

All results of the chapters 4.3 to 4.5 one may show also only using the Lorentz transformation (see e.g. Lorenntz 1916). The choice of the transformation is here not relevant.

The increase in mass of a body with its velocity v results presumably from an increase in energy of the surrounding membrane (Haisch & Rueda 2001). The author does not agree Mach's principle that the inertia of a mass is caused by the distribution of all distant matter (c.f. Silk 1989, Herrmann 1999), but the exact physical explanation is still outstanding. Another unsolved question is, wether one should count twice the increase of mass, i.e. kinetic energy plus mass increase. The formula e_Ph=m_Phc²=hn of the energy of a photon initiates such thougths (c.f. Hatch 2000). Gedanken experiments, e.g. a collision of a photon with a black body, lead to the differential equation of increase of mass with the well known solution m(v)=m_o(1-b²)^-1/2 (see e.g. Weber 1995¹).

The increase in the mass of electrons experimentally investigated by Kauffmann (1900) is in good agreement with the formula

(4.3.1)

deduced by Lorentz from the length contraction.

Fig.4.3.1. Impact of a quant on a moving body m

One may derive eq. 4.3.1 from a thougt experiment similar to the thougt experiments of Tolman. A quant hn is travelling in s in the x-direction and collides with a black body of mass m and velocity v in x-direction, too. Now we replace the quant by a particle of mass m =hn/c² and speed c. s' is a frame moving with the center-of-mass of the particle-body system. The axes x and x' shall have the same direction. The speed u' of the particle in s' is

. (4.3.2)

Eq. 4.3.2 is deduced from eq. 4.2.18. With u=c we get

. (4.3.3)

Because of c'=c/(1-v²/c²)^3/2 (by change of x-scale and t-scale) is

. (4.3.4)

The particle (quant) m_q has in s' only 1-v²/c² of its momentum in the s-frame. Do we interprete this part as relevant to the part (hn/c)m·dv of the total differential

(4.3.5)

then because of the same direction of v and dv the part relevant to (hn/c)v·dm is equivalent to v²/c². The quotient of both parts is

. (4.3.6)

But this is the differential equation with the solution m(v)=m_o/(1-v²/c²)^1/2 .

Now we set the kinetic energy of a body with mass m in the 4-dimensional space in analogy to the special theory of relativity E=m·U², where U= V_E²+V². Here V_E is the speed of expansion (W-direction) and V is the speed of the body in the X-direction (in S-scales). We get the square E² using U_~C and m=m_o / (1-v²/c²)^1/2 as

E² = (m · U²)² _~ (m · C²)² = (m_o· C²)² + (m · V · C)²

(4.3.7)

Changing from S-scales to s-scales and with p=m·v we get the well known formula

e² = ( m_o· c² )² + (pc)² . (4.3.8)

The central elastic impact of two bodies with rest mass m_1,0and m_2,0 and the speeds v₁and v₂ before the impact we may calculate using a s'-frame with weighted speed

(4.3.9)

Using eq. 4.3.2 we transform the velocities from s-scale to s'-scale and after impact with eq. 4.2.18 back to s-scale. The two conservation laws

, (4.3.10)

(4.3.11)

are fulfilled in the same way using the addition of velocities according to the special theory of relativity, but the proof was done numerically only by the author.

4.4 Membrane Theory and Basic Physical Phenomenons

(First version 12/93, last update 09/2002) The imagination of a 4-dimensional etherfilled space is well suited to initiate new models of the basic physical phenomenons, but the author emphasizes that the following ideas are hypothetically only.

The basic brick of our world is a torus-shaped whirlpool (eye, in German Auge) (Fig. 4.4.1) moving with speed V_E in direction of the W-axis of S. The finite extension of its inner channel causes a finite total ammount of the outer flow. That means the flow decreases with a higher exponent of (1/r) than 3 and the force between two auges decreases with (1/r)⁶ at least.

Fig. 4.4.1. - The Auge - a torus-shaped whirlpool

Two auges attract one another and form a rotating pair of auges (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. Left and right rotation have the same probability (e.g. electron e_-, positron e₊), but in the 4-D-space we may define three different rotations, all three with right and left, supporting so the hypothesis of the quarks. Otherwise, the ALEPH detector of the Large Electron Positron Collider (LEP) of the CERN (Geneve) has picked up 18 unusual events. In each case the detector recorded four jets of mesons and similar particles spraying from high-energy collisions of electrons and antimatter positrons (Watson 1997). This would support the theory that electron and positron are made from two bricks.

The gradient of the space circulation is equivalent to the electric field. Whirlpools with the opposite rotation attract one another. The electric field of a resting charge has no rotation outside the centre, but relative movement against a body causes rotation in this point, the magnetic field.

Fig.4.4.2. Whirlpool formed by two rotating auges

4.5 Experiments Concerning Special Relativity

(First version 05/1977, last update 08/2002) In the past two centuries a lot of experiments had been performed concerning Special Relativity. The null results of the Michelson-Morley experiments is explained already by the deduction of transformation equations 4.2.8 (Weber 1995). Each transformation with quotient (1-b²)^1/2of length and cross contraction explaines the zero effect. The time transformation is not essential here, since interference of light waves takes place in one point and after the same run time inside both arms of the interferometer. That is valid for both transformations – Lorentz transformation and the transformation by eqs. 4.2.8, too.

Maxwell’s Equations (First version 01/2000, last update 02/2000) of electrodynamics are valid. In the case of a vacuum they are (vectors with an arrow above or in bold letters):

(4.5.1 a)

(4.5.1 b)

B [A/m] is the magnetic field strength, E [V/m] electric field strength. Electric absolute permittivity e_o and magnetic susceptibility m_o are constants. Their numerical value depends on the measuring system. We start with an electromagnetic plane wave. It propagates with angle sin(a)=v/c against the z-axis (Fig. 4.5.1). w is the angular frequency. Vector B points in the y-direction. Vector E is perpendicular to B and to the direction of propagation. Since l=2pc/w and l_z/cos(a)=2pc/(w(1‑b²)^1/2) and l_x/sin(a)=2pc²/(w v), we get B_y according to eqs. 4.5.2.

Fig. 4.5.1: Plane wave in s with wave length l

(4.5.2a)

(4.5.2b)

We get E_x and E_z according to eqs. 4.5.3 and 4.5.4 by formation of curl of B, use of Maxwell’s first equation in 4.5.1 and integration over time.

(4.5.3)

(4.5.4)

How does an observer in s’ see this wave? The difference (t-vx/c²) is transformed according to eq. 4.2.15 in t’(1‑b²)^1/2. For speed of light, we find eq. 4.5.5.

c’=c(1- b ²)^1/2(1- b²)^‑2/2=c(1- b²)^‑1/2 (4.5.5 a)

c=c’(1‑ b²)^1/2. (4.5.5 b)

This follows from a speed reduction according to the angle between the direction of propagation and the z-axis or the z’-axis, respectively, and from a speed acceleration according to scale transformations of the two cross dimensions.

Since space does not change itself, but the scale, the distance measured in the z’-direction is z’=z(1- b²)^‑1/2 or z=z’(1- b²)^1/2 . Speed v of the observer (resting in s’) has the component v²/c in the direction of the propagation of the wave. Therefore, the frequency of the observed plane wave is lowered, and we get w⁺=w(1- b²). Since the unit of time, the second, in s’ is longer than in s , time transformation yields w’=w(1- b²)^1/2 or w=w'(1- b²)^‑1/2 , respectively.

Magnetic amplitude A remains unchanged physically and adheres to the same scale, i.e. A’=A. This follows from constancy of charge, i.e. [As]=[A’s’]. The ampere must have the inverse correction factor in regard to the correction factor of second. Since B_y is perpendicular to v, scale corrections of magnetic field strength [A/m] cancel one another. We replace A, w, t-vx/c², z and c in Eq. 2.9 by A’, w’, t’, z’ und c’ together with their corrections described above and get so eq. 4.5.6. This is the equation of a plan wave in z’-direction.

(4.5.6)

Using Maxwell’s first equation in eqs. 4.5.1 and by integration over t’, we get the electric field strength in s’ according to eq. 4.5.7, but we had to define the magnetic susceptibility m'_o and electric absolute permittivity e'_o so that 1/c’²=m'_oe'_o holds.

(4.5.7)

The enlargment of the x’-component of E’ and the absence of a z’-component of E’, we can explain with Faraday’s law of induction. Motion of s’ in s induces a field strength with the same amount as E_z in eq. 4.5.4, but with the plus and minus sign reversed.

Fresnel's Drag Coefficient, Fizeau’s Experiment, Aberration and Airy's Experiment (First version 05/91, last update 08/2002) A. J. Fresnel (1788-1827) made in 1817 the forecast that light will experience an additional speed v(1‑1/n²) if it is traveling through a medium (e.g. water) moving with speed v. This formula was derived by Fresnel without experimental evidence. The term in the parantheses is known as Fresnel's drag coefficient (or convection coefficient). Here n is the refractive index of the medium. More than thirty years later, in 1851, Fizeau carried out the first experiment. It was repeated with higher accuracy in 1886 by A. Michelson and E. Morley and confirmed Fresnel's formula. Fizeaus's and Michelson's experiments used an interference pattern of two light beams. Both light pathes laid inside a water filled pipe of u-form. The water was moving through the pipe with a speed v of about v~7 m/s. One light beam had the same direction as the water flow, the other beam the opposite direction.

To explain Fresnel's formula without the use of "ether dragging" we could do in the membrane theory the same as in Special Relativity (SR). For the moved observer (resting in s’) the speed of light is near to u’=c/n. Neglecting the small terms with b², the first equation of eqs. 4.2.18a yields speed u according to eq. 4.5.8.

(4.5.8)

Neglecting the small term u’v²/c² and replacing u’ by c/n, we get u=u’+v(1-1/n²). But this solution of the problem is not satisfying from the physical point of view . In 1965 A. A. Penzias and R. W. Wilson had found the rest frame of the cosmos, the cosmic background radiation. The Earth moves with a speed u of about u~600 km/s in this frame. A relativistic time effect has only the magnitude of uv/c²= 6´10⁶´7/(3´10⁸)²» 0.5´10^‑9. But this effect is to small to explanate the dragging effect of »10^-8. Fresnel's drag coefficient and the aberration are effects of first order in v/c and so we may hope to find a classical derivation.

We follow in wide parts the paper of G. Antoni and U. Bartocci (1999) founded on Antoni 1953. First Antoni and Bartocci introduce the time delay Dt_d of a photon moving over the distance L in a medium with refractive index n:

Dt_d = nL/c - L/c = (n-1)L/c, (4.5.9)

and then Antoni and Bartocci make the natural assumption that such a delay is due to the contribution of many single delays, due to the total number NL of obstacles that light meets during its travel over the distance L. N is the number of obstacles per length unit, t is a single delay. Then is:

t = (n-1) L/(cNL) = (n-1)/cN. (4.5.10)

We suppose with Antoni and Bartocci now that in the given length L the water moves with uniform speed v (e.g. in the same sense of light). The physical phenomenon we are facing of in this case is: Light meets less obstacles during its travel through the moving water, say for a time Dt, according to N'=N(L-vDT). If the water is resting, light meets N obstacles per unit of length. If water moves with speed v then NvDt obstaceles disappear before light can meet them.

The following passage differs slightly from the paper of Antoni and Bartocci. The author will try to give here a physical model which can also be used later for the explanation of Airy's aberration experiment. The model says that an obstacle is an encapsulation of the photon over the single delay time t in a small volume of the medium. After the time t the photon continues its travel in its old direction. Fig. 4.5.2 illustrates this model.

Fig. 4.5.2: Light path with obstacles

If water moves with speed v then the obstacle volume has changed its position. The travel time Dt=Ln/c in resting water changes by the dragging effect of the moving water to the value Dt' of

Dt' = L/c + (n-1) (L-vDt')/c - (n-1) (L/c) (v/c). (4.5.11)

The first term is the travel time in air (or vacuum). The second term, in the same form given by Antoni and Bartocci, says that the number of obstacles decreases. If the photon reaches the end of the distance L the piece vDt' of water is disappeared. The third and last term says that the undisturbed beam has only to travel the distance L-vDt_d, since during the delay time Dt_d the encapsulated photon moves together with its water volume. With Dt' »Dt=Ln/c we get

Dt' = (L/c) [1 + n - 1 - (n-1)(vDt'/L) - (n-1)(vDt')/(nL) ] (4.5.12)

Dt' = (L/c) [n - (n-1) (Dt'/L) (v+v/n) ]. (4.5.13)

Because of c(v)=L/Dt' we find

c(v) = Lc / { Ln [ 1 - (1-1/n) (Dt'/L) v (1+1/n) ]} (4.5.14)

and with Dt' »Dt=Ln/c we get the final result

c(v) » (c/n) [1 + v (1-1/n²) (nL/cL) = c/n + v (1-1/n²). (4.5.15)

That is Fresnel's drag coefficient. Regardless of slightly differences with the paper of Antoni and Bartocci the author will quote here the referenced paper once more word for word: "Summing up the previous "logical" argumentation shows, in one more case, that completely different theories (actually, SR and a very "natural" ether theory) can give, sometimes, the same experimental previsions."

Stellar Aberration was first observed by the British astronomer J. Bradley in 1725. All fixed stars may be observed to describe a small ellipse in the sky, over the course of a year, if its ecliptic coordinates can be accurately measured. The classical explanation of this effect of order v/c compares the photon with a bullet flying through the laterally moving telescope. Fig. 4.5.3 illustrates the classical explanation. The speed of the telescope (of the Earth) in the Sun frame is v. In the Earth frame we find an angle q»v/c between light path and telescope axis.

Fig. 4.5.3: Stellar aberration

In 1871 Sir George Airy conducted an experiment in which the stellar aberration was measured by a water filled telescope. Airy found that the amount of aberration was the same as with an air filled telescope. Also in this case the SR uses the addition theorem of velocities and explanates so the the null result. We are trying to find the classical explanation using the above mentioned model of obstacles (Fig. 4.5.2). Since the speed v of the telescope is directed perpendicularly to the light beam, an obstacle does not change its vertical position during the delay time . Additionally, the number of obstacles per unit of length does not change, too. So, the speed of light inside the water in direction of the beam is not influenced. But what is the lateral drag coefficient?

During the time delay LNt with t=(n‑1)/(cN) the telescope of length L, together with its water within, moves laterally over the distance

vLNt = (v/c)(n-1)L (4.5.16)

with speed

vLNt/Dt=vLNt/(Ln/c)=(v/c)(n-1)L/(Ln/c)=v(n-1)/n=v(1-1/n).

(4.5.17)

The expression in parantheses, (1-1/n), is the lateral drag coefficient. Since the speed of light has in water a value of c/n instead of c, the lateral displacement D_o of the beam would be D_o=vDt=vL/(c/n)= (v/c)Ln at the bottom of the water filled telescope. But the lateral dragging of the beam reduces this displacement by D_v=Dtv(1-1/n)=(Ln/c)v(1-1/n)=(v/c)Ln(1-1/n). The final and real displacement D we find by subtraction, i.e.

D = D_o - Dv = (v/c)Ln - (v/c)Ln(1-1/n) = (v/c)Ln (1-1+1/n) = (v/c)L.

(4.5.18)

But D=(v/c)L is exactly the displacement, we find in the case of an air filled telescope. So the null result of Airy's experiment has a classical solution, too.

The atomic clock experiment of Haefele and Keating (First version 06/1992, last update 02/2000)

In 1971 Haefele and Keating let travel four atomic clocks round the Earth with normal airplanes. One pair of clocks were travelling westward, the other pair eastward. Upon return Haefele and Keating compared the times of the four clocks with clocks in the laboratory. The time differences they found were in agreement with the relativistic forecast. This experiment is a special case, since two time effects take part - the relativistic time dilation according to the motion and the time effect due to gravity following from Einstein’s Principle of Equivalence. Integral eq. 4.5.19 gives the relativistic time difference DT between a clock moving with speed v+u_i (v and u are vectors) and a clock resting in the membrane. c is the speed of light, v is the speed of the Earth in the membrane, u_i a velocity in that frame moving with the Earth but not rotating together with it. Subscript i may accept three values - „E“ (eastward), „W“ (westward“), „L“ (laboratory). a is the angle between u_i and v. T is the time of integration, i.e. the time of travelling around the Earth.

(4.5.19)

We model the trajectories of the clocks setting u=w_iR and a=w_it. Here w_i is the angular frequency and R the radius according to the degree of latitude. Then w_L=2p/d, w_O=w_L+2p/T and w_W=w_L‑2p/T hold. Here is d=24·3600 [s] the duration of one day and T=2pR/s the travel time. s is the speed of airplanes. The author used the values R=36000/2p [km] and s=900 [km/h] giving T=144000 [s]. We calculate the differences DT_W‑DT_L and DT_O‑DT_L, and then we add the effect DT_G=hg/c² due to gravity for altitude of flight h=10 [km]. g=9.81[m/s²] is the acceleration due to gravity. So we find the same results as found by Haefele and Keating. This result does not depend on the magnitude or direction of speed v. Small differences are caused by errors of the numerical integration. The author used the trapezoidal rule with 1,000,000 steps. v was varied from v=-500 to v=+500 km/s. The time effect of parallel movement in the membrane is nihilated by the formation of the difference. What remains is the relativistic effect. Compare also the nearly identical calculations of Franco Selleri (2004).

The Trouton-Noble (First version 02/2000, last update 02/2000) experiment: A parallel plate capacitor was suspended from a single line, thus allowing it to rotate freely around the line. It was expected that the translational motion of the Earth would result in a magnetic torque force on the charges resulting in the alignment of the plates parallel to the motion v of the Earth. The prerelativistic formulas had predicted different amounts of field energy for „parallel“ or „perpendicular“ alignment. No such effect was observed. Singal 1993 gives two explanations of the null results using the Lorentz-transformation, and beyond that an excellent review of references.

The author calculated numerically the electromagnetic field energy of a pair of charges in motion. Using transformation formulas 4.2.8 the integral eq. 4.5.20 is invariant. Change of b or change of alignment does not matter. There is no difference in energy between parallel or perpendicular alignment. As a side effect of the calculation Singal’s standing up for the common expression eq. 4.5.20 of field energy is confirmed once more. (Some authors plead in favour of difference E²-B² under the integral sign.)

(4.5.20)

In this case is vector E’= r_o /(8pr²) and vector B’= b(e_x ´ r_o)/(8pr²). Vector r, we get by back- transformation of vector r’ (Fig. 4.5.4) from moved frame s’ into rest frame s. e_x is the unit vector of x-direction, r_o’ and r_o are unit vectors, too.

Thus, the null results of the Trouton-Noble experiments had found their explanation. Extension from a pair of charges to a pair of parallel plates with area A, charge C and distance d will not cause any problems. All contractions due to a rotation of the capacitor will be compensated for by a higher density of charges and a higher field strength in direction of contraction. Magnetic field energy has the same behaviour as electric field energy and brings no unsymmetry in the system.

Fig. 4.5.4: Charge in s’ with field vectors

Thomas precession: (First version 02/2000, last update 02/2000)The spinning electron moves in an orbit in the electric field of the rest of the atom (Corben 1993). For an observer moving with the electron, the other charges appear as in motion, and by this motion the effect of a magnetic field is produced. The virtual magnetic field acting on the electron’s magnetic moment induces a torque. The torque makes the spinning particle precess, and the frequency of this precession is the frequency interval of the doublet level. If a is the acceleration and v is the speed of the electron in its orbit, we get the angular frequency of Thomas precession by w_T =[a´v]/(2c²). The factor 1/2 is called Thomas factor. Nonreletivistic calculation yields the double value, which is not in agreement with measurements of w_T. Furry 1955 e.g. gives a derivation of the formula using the Lorentz transformation ( x’=x-vt, t’=t-vx/c² ) in a series of infinitesimal small parts of the rotation of an electron in its orbit. Infinitesimal means that we may neglect terms with (v/c)². Transformation eqs. 4.2.8 yield the same infinitesimal transformation as derived by Furry. So Furry’s derivation of the Thomas factor remains valid here, too.

Sagnac effect (First version 01/2003, last update 01/2003) The Sagnac effect is an effect measured using a rotating interferometer. A light beam starting from the source is split up by a glass plate into two beams, which travel in different directions over the mirrors m1, m2 and m3 to the screen (detector, observer) and make there an interference pattern. Source, apparatus and screen are rotating together with uniform speed (see fig. 4.5.5). One observes a displacement of the interference fringe pattern on the screen if rotation changes. The Sagnac effect follows from the Doppler shift of the wave length (c.f. ch. 4.7). Following Joos (1989), e.g., we find the relative speeds of the different light beams

(4.5.21)

and

. (4.5.22)

Here w is the angular frequency of rotation, r is the radius from centre to any point of a beam, q the angle between beam and perpendicular to r, df the differential of the rotational angle, ds the differential of the light path and c the speed of light. The time difference of the running times is

, (4.5.23)

or because of v<<c

, (4.5.24)

where A is the area inside the square bordered by the light beams. The displacement DZ of the interference fringe pattern is then, with wave length l of the used source,

. (4.5.25)

(The expression "wA" should better be split in the expression "vd", where v is the rotational speed of the source and mirrors and d the distance over the round trip. This does better fit the physical sense.) A translational movement of the whole apparatus relative to the membrane (CBR frame) does not harm if one remembers the passages "Maxwell's equations" and "Haefele and Keating" above.

Fig. 4.5.5: Experiment of Sagnac

The Sagnac experiment is one of the cruxes in the discussions concerning the ether (aether). To quote here R. R. Hutch (2002): "Several types of modern gyroscopes function by using the Sagnac effect to measure rotation. Georges Sagnac performed the original experiment in 1913. He split a light beam into two parts, which traveled around the circumference of an area in opposite directions. He then measured the interference fringe effect when the two light beams were brought back together. He found that the fringe shift was a function of the rotational velocity. In other words, the speed of light relative to the rotating sensor was a function of whether the light beam traveled with or against the rotational velocity of the platform. The MLET (Modified Lorentzian Ether Transformation by R. R. Hatch) explanation of the Sagnac effect is again obvious. Simply stated, the motion of the detector (observer) has no effect on the speed of light; and therefore a non-isotropic light speed relative to the moving detector can be expected - which lets to the observed phenomenon.

4.6 Apparent Constancy of Velocity of Light (First version 08/2002, last update 03/2004)

Light does not have a constant velocity. It depends on the gravitation (cf. eq. 5.1.1). But beside this, a resting medium where the light is propagating in must cause special conditions for the propagation in moved frames of reference. A simple mind would await here some effects. But the constancy of all measurements of the speed of light in the constant gravitational field of the Earth is underpinned by a lot of experiments with high precission. But we should also remember that all measurements are using a closed light path with start and end at the same point, and all measurements are using clocks working due to the same principles and under the same conditions, as measurement of speed of light is using. So, e.g., the definition of the speed of light c=(distance light travels in any given time interval)/(the given time interval) was never realised so by any experiment (c.f. Durham 2000).

Fig. 4.6.1: Resting and moving frame

Fig. 4.6.1 shows the light path in a resting frame of reference. Source is at A or A₁ or A₂, respectively, mirrors at B and C. The moving frame has speed v. A light signal starting at point A₁ perpendicularly to v is mirrored by C and returns to the source at point A₂. A light beam parallel to v is mirrored at B and meets the source at point A₂. Measurement of speed of light and the Kennedy-Thorndike experiment are closely related. The Kennedy-Thorndike experiment is using an interferometer with arms of unequal length, and it was performed over half an year. It yields a null result only if speed of light is constant in all directions and at any time in the year. The Kennedy-Thorndike experiment yielded a null result indeed in the interference pattern shift, so, as the Michelson-Morley experiment had yielded. The theory has to explanate now, why the speed of light seems to be constant.

Contraction of length (or length and breadth) are essential for the null result of the original Michelson experiment with an interferometer using arms of equal length. Both, Lorentzian and Special Relativity, need additionally the time transformation for the null result of the Kennedy-Thorndike experiment. Both are using the kinematic time effect t'=t(1-b²)^-^1/2 .

Time is playing an essential role also in this theory . Here x, y, z and x’, y’, z’ are the coordinates of the resting and the moving frame, and x-axis is assumed to be parallel to v, without restriction of generality. Five effects do exist considering the conditions in the cross case:

· The acoustic Doppler effect of a moving source in a resting medium with l'=l_o(1-cos(a)v/c), where a is the angle between speed v of the frame and some chosen direction.

· The kinematic time effect, t'=t(1-b²)^-^1/2.

· The projection effect (1-b²)^1/2 if we consider the cross beam b_c' (beam A₁-C-A₂ in fig. 4.6.1) in the moved frame and the same beam b_c observed in the resting frame, where it has the angle v/c with the y axis or z axis, respectively.

· The cross contraction (1-b²)^1/2.

· The seeming decrease (1-b²)^1/2 of speed of light at beam b_c' seen in the moved frame.

Doppler effect and kinematic time effect cancel one another, since the Doppler effect is decreasing the wave lengt, but the kinematic time effect is increasing it with the same amount. Projection of the sloped beam b_c on the y axis to get the beam b_c' shortens the wavelength. So we would find more periods at the cross beam b_c', if there would not be the cross contraction. With cross contraction the number of periods remains unchanged. This considerations are sufficient to explanate the null result of the Michelson experiment and of the Kennedy-Thorndike experiment, performed at one moment. To explanate the missing summer-winter or day-night effect we had to consider that the seeming wavelength at beam b_c' is shortened by the factor (1-b²)^1/2, but the seeming speed of light is decreased also by the projection with angle v/c. So the number of incoming wave fronts remains unchanged also if speed v of the moved frame is changed. That means we will not find any summer-winter effect, also in the case of enequal arms of the interferometer, according to the null result of Kennedy-Thorndike.

In the parallel case we are considering the beam b_p (beam A₁-B- A₂ parallel to v in fig. 4.6.1). The Doppler effect is l₁'=l_o' (1-v/c), for those waves moving from A to B. The dash means that l_o' and n_o' have changed already by the kinematic time effect accordingly to the speed v of the moving frame. Since mirror B is moving also with speed v, the incomming waves have the frequency n'=(c-v)/ l₁' giving the old frequency c/l_o'=n_o'. The backward beam fom B to A has the Doppler effect l₂'=l_o'(1+v/c). We find anew the old frequency n'=(c+v)/ l₂ '=n_o' at mirror A. We see that the frequency of incoming wave fronts has not changed in the parallel case too. That means that any interference pattern will change whether by a rotation of the interferometer, nor by a change of speed v of the moving frame.

This theoreticall result is in full agreement with the experimental results. Although speed of light has changing amounts for different directions in a moving frame of reference, we will never find a change performing closed path measurements. Lorentzian ether theory is not complete, although it is yielding the correct null results in all cases too. This theory was established to explanate the result from basic assumptions, but it does not consider the Doppler effect. Special Relativity has no problems here since the SR is a phenomenological theory which is not laying claim to explanate the workshop of nature. But if research is climbing down to the bricks of space one may need the new transformation.

4.7 Tidal and frequency effects (First version 12/2002, last update 01/2003)

The advance of time is a property of the frame of reference and of the exact position in it where the clock is resting. Einstein's GR is showing here some tendency of mystification of time. Time is not a coordinate comparable with the spatial coordinates. Constructions as ct are meaningful only in the sense that our Universum expands into all four spatial directions. One of this directions is the fourth spatial coordinate w=V_Et. (Speed V_E is nearly C. C is the signal speed in the 4D ether.) Measuring of time has reached a high standard (cf. Bize et al. 19999), so that one can find small changes of time flow depending on different influences, e.g. depending on gravity.

In the past century several tidal effects were found and several effects concerning the frequency of a photon. This chapter is a collection and an evaluation of such effects:

· Kinematic Time-Dilation is an effect of the Special Relativity: Moving clocks run slower

· Gravitational Time-Dilation of Clocks: Clocks run slower in a gravitational field (some authors treat this effect wrongly as equivalent to the gravitational red shift)

· Gravitational Time-Dilation of Photons: Photons move slower in a gravitational field

· Doppler shift is the change of frequency in the case of moving source and/or receiver

· Gravitational Red Shift: A photon is red shifted if it climbs upward in a gravitational field (some authors treat this effect wrongly as equivalent to the gravitational time-dilation of clocks)

· Red Shift caused by the expansion of the universum

Kinematic Time-Dilation arises from any motion of a clock relative to CBR frame (Cosmic Background Radiation). A moving clock runs more slowly. The CBR frame is identical with the resting membrane. The formula is the same as given by Lorentz or by Einstein (c.f. chapter 4.2).

(4.7.1)

Here v is the speed of the clock in the CBR frame, c the speed of light in the absence of gravitational fields, t the time of a clock resting in the CBR frame and t' the eigen-time of the clock. The clocks had to be started from the same point in the membrane both of them with a starting value of t_o=t'_o=0.

This effect was proved very often by different methods (c.f. Ives and Stilwell 1938, Haefele and Keating 1972, comments c.f. R.A. Herrmann 1999, Hehl et al. 2001, R.R. Hatch 2002). First measurements were done by Ives and Stilwell in 1938. The most famous experiment was the "moving clock experiment" of Haefele and Keating in 1972, though still another effect was acting in this experiment.

The discussion to the Twin Paradox is directly related to the kinematic time dilation (c.f also chapter 5.5, passage Haefele and Keating). To quote here R. R. Hatch (2002), an expert in questions concerning clocks and their behaviour:

"Thoug entirely theoretical, the solution to the twin or clock paradox also reflects the different application of the Lorentz transformation in the two alternate theories (Lorentz and SRT - the author). Part of the problem with adressing the twin paradox of a travelling twin and a stay-at-home twin is that so many different mutually incompatible solutions are offered within SRT. However, as fare as I am aware, all the solutions claiming to be consistent with SRT involve changing inertial frames when the travelling twin turns around. The specific solution given by Ohanian [4] seems to be the most consistent with other applications of SRT (particularly with the Thomas precission above). Ohanian says that, when the twin turns around at the middle of his journey, he causes a hyperbolic rotation of the lines of simultaneity around his origin. Again, this involves magic because the position of any light signal in transit must suddenly adjust in both position and time to be consistent with this SRT solution. (This argument has been presented in greater detail in a prior paper [5].)

The solution to the twin paradox in MLET (Modified Lorentzian Ether Theory by R. R. Hatch) is the soul of simplicity. Pick any inertial frame you want for the twins and treat the frame as the absolute frame. Then stick with the frame as the isotropic-light-speed frame for the entire trip. Never change frames. Simply let each clock run at a rate consistent with its velocity in the chosen frame. The same observed slowing of the clock or decreased aging of the twin who makes the round trip relative to the stay-at-home twin will be observed independent of the chosen isotropic-light-speed frame."

Gravitational Time-Dilation of Clocks is an effect which is caused by a gravitational field acting on the clock. Robert A. Herrmann says in his paper "Gravitational Time-Dilation" in 1999, 'An Equivalence Principle appears to be necessary in every General Theory of Relativity (GR) derivation for expressions that predict gravitational time-dilation.' Following Herrmann, we could state that Einstein's conclusions were erroneous, but this would hold only in some philosophical sense. It is true that Einstein denyed the possibility to recognice an absolute movement or an absolute acceleration ( the CBR frame had been discovered after Einstein's death first), but the conclusions from a principle with a false foundation must not be inevitably false. So the author has no objections to use the explanation given by Einstein's GT for the gravitational time-dilation.

(4.7.2)

Eq. 4.7.2 describes the change of the eigen-time t of a clock in respect to another clock outside of all gravitational fields. The first clock is running in a gravitational field with a distance of r from a spherical mass with r_g as two times the Schwarzschildradius, or with escape-speed v, respectively. We find at earth-level dt_E=(1-7.1´10^-10)dt with respect to a clock at r®¥.

(4.7.3)

Eq. 4.7.3 describes the change of the time at earth-level (Earth radius r_E) in respect to the time at the higher level r_E+H above the earth (with H<<r_E). The gravitational earth acceleration is g. A good introduction to grvitational time-dilation is given, e.g., by Tycho Sleator from the New York University. He uses Einstein's famous elevator model for his gedanken experiment. At time t=t_o the elevator is in rest with respect to an inertial frame. A first observer rests in this frame (not in the elevator). This observer checks the frequency of time pulses of a clock at the bottom of the elevator to be 1 per second. Now the elevator is accelerating upward with acceleration g. Pulses of light are emitted upward from the bottom of the elevator with the frequency of the clock. By the time Dt=H/c the pulses reach the top of the elevator (where H is the height of the elevator), the velocity of the elevator has increased from V=0 to V= Dtg= gH/c. Because a second observer with the top of the elevator is moving away from the source at velocity V, this observer sees the frequency of the source shifted by the Doppler-shift:

(4.7.4)

(Note, that the above formula for the Doppler-shift is only correct for v<<c). By Einstein's principle of equivalence, this same result should occur for a stationary observer in a gravitational field. This observer states a decrease of the frequency of the clock by the factor 1-gH/c² compared with a clock not in any gravitational field. That is the gravitational time-dilation.

Several experiments had been done to prove the gravitational time-dilation. In 1960, e.g., Cranshaw, Schiffer and Egelstaff and in 1965 Pound and Snider used g-rays and the Mössbauer effect. But in this experiments the gravitational red shift and the gravitational time-dilation is involved, and we can not decide which effect causes the shift. The "fourth" test of GR by I. I. Shapiro (1964) should measure the "gravitational time-dilation", but what it had measured was the decrease of the velocity of light in a gravitational field. Citing once more R.A. Herrmann, we find, 'Experiments with flying atomic clocks, in aircraft and rockets, have a crucial conceptual advantage over the gamma-ray experiment, in that the former show in the most direct way that clocks in a gravitational potential run slower.' So we cite once more the "moving clock experiment" of Haefele and Keating in 1972, though still another effect was acting in this experiment. But in this case the influence of the second effect, i.e. the relativistic time dilation, is clearly separable. So, the gravitational time-dilation is experimentally proved.

Gravitational Time Dilation of Photons: Photons moving in a strong gravitational field need more time than photons moving in field free space. I.I. Shapiro et al. (1964, 1977) performed several experiments to measure the run time of radar signals at trajectories grazing the Sun. They used the Venus as reflector, later the Viking space probe. Shapiro found a signal retardation with the value predicted by A. Einstein. But the explanation given by the GR uses the gravitational time delay, although the gravitational field of the Earth does not change its strength, and following the clocks there run with uniform speed. Therefore the simplest explanation for the behaviour of the photons (to be late) here is to suppose a change of their speed. One can find in chapter 5.1. a discussion and the trial of an explanation, how the membrane acts on the photon and changes its speed. Furthermore, eq. 5.1.1 in chapter 5.1. allows to calculate the signal retardation with the same accuracy as the GR does.

Doppler Effect is an effect of first order in v/c. Since the light source or the receiver or both are moving relative to the membrane (CBR frame) this effect is always connected with the kinematic time dilation effect. For speeds v<<c we find the same formulas as given in the case of the doppler effect of acoustic sound waves. The relative change in the wave length l measured by the receiver is

, (4.7.5)

where c is the speed of light, v_R the speed of receiver, v_S the speed of source, both speeds here laying in the straight trajectory of the photon, and both speeds measured relative to the CBR frame with positive direction from receiver to source.

Gravitational Red Shift is an effect which is merged into the gravitational time dilation of clocks sometimes. Compare the discussion above connected with the experiment of Pound and Snider (1965). This effects are difficult to separate. Most of the literature does not see here a difference. A photon with starting frequency n and mass m=hn/c² loses at least the energy (m/2)v²= hnv²/(2c²) if it climbs up from the surface of a spherical mass with escape speed v to a distance of r®¥ (h is Planck's constant). If the photon climbs the height H with H<<r_m (r_m the radius of the spherical mass) the lost of energy is De= mgH= hngH/c². Starting from this equation GR and Membrane Theory use different equations.

The GR (see Weinberg 1972 e.g.) argues: Dn= De/h= hngH/(c²h)= ngH/c₂, or n-Dn=n(1-gH/c²). The frequency of the photon is red shifted. It is the same value as given by eq. 4.7.3 for the gravitational time dilation of clocks.

The Cosmic Membrane Theory says: The frequency n does not change if the photon climbs upward, but the speed of light changes with c(r)=c_o(1-r_g/r) (see eq. 5.1.1 in chapter 5.1) and causes so a change of the wave length. Here c_o is the speed of light outside of all gravitational fields. Why should frequency not change? Imagine a coherent light source at the surface of the massive body, e.g. a laser. There is no room for a change of frequency. If the laser radiates n cycles per day, the receiver had to detect n cycles too. A change of frequency of the individual photons without change of the frequency of the laser beam would be able only by a complicated continuous phase shift of the individual photons. But if the receiver is positioned at a field free place then there we will find no reason for such a phase shift.

The change of light speed causes a change of the starting wave length l with l+Dl=l(1+r_g/r). But what does energy and momentum? If we use further the equation e=hn, we have 'gotten an elk', since so energy would be a constant. To escape this trap we had to assume that Planck's constant h depends on the gravitational field (c.f. Webb 1999, Ivanchik et al. 2002, or Carlip 2002). So Planck's constant is in a 'good society':

· Mass changes with the gravitational field (see chapter 5.2)

· Velocity of light changes with the gravitational field (see chapter 5.1)

· Time changes with the gravitational field (see above)

· Planck's constant changes with the gravitational field.

Red Shift caused by the expansion of the Universe: The space stretches with the expanding Universe. A photon with the emitted wave length l_E has the obeyed wavelength l_o. R(t_E) shall be the radius of the Universum at emitting time t_E, R_o the radius nowadays. As red shift z we spell the relative increase of the wavelength. One assumes l_E=l_o, i.e. the emitted wave length l_E equals the measured wave length l_o in the laboratory (cf. Bergmann-Schaefer 2002 e.g.).

(4.7.6)

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, e.g.:

· Velocity of light and with it the wave length

· Frequency of the source and with it the emitted wave length.

In chapter 7.3 the author will try a discussion of problems connected with the expansion of the Universe.

5 Classical Proofs of General Relativity

The classical proofs of a theory of gravitation are Shapiro effect of signal retardation, light bending and perihelion advance of Mercury. In the chapters 5.1 and 5.2 the author will discuss these effects from the point of view of the Cosmic Membrane Theory.

5.1 Shapiro Effect, Light Bending and Depth of Space

(First version 12/96 , last update 04/2005) For explanation of light bending at the edge of Sun and Shapiro effect of signal retardation by solar gravity, we have to accept that the speed of light is a property of the membrane. The velocity depends on the distance r from the Sun according to Eq. 5.1.1. This topic is discussed very frequently in the physical community. The author cites Van Flandern 1993 or Ellis, Farakos, Mavromatos, Mitsou & Nanopoulos 2000, Haisch, Rueda & Dobyns 2001, Puthoff's polarizable vacuum approach (PV) 2002, Joao Magueijo 2003, Puthoff, Maccone & Davies 2004. Gass and Mukherjee (1999) discuss different types of vacua, Clayton and Moffat (2002) discuss vector field mediated models of dynamical light velocity. Opher (2003) discusses vacuum energy which is dependent on spatial position (Casimir effect).

(5.1.1)

c_o is the speed of light in a vacuum for r®¥, 2a is the Schwarzschild radius of the Sun. The membrane gives a hypothetical explanation for this effect. If the membrane is not perpendicular to the ether wind, e.g. in a gravitational funnel, then the ether wind is deflected by the particles (korns) of the membrane (Fig. 5.1.1). The deflection causes a counterforce F_ec at the membrane in a central direction. The author is imagining the korns of the membrane as small torus shaped curls with openings to the ether wind.

We assume F_ec to be proportional to the slope w’ of the membrane and proportional to the volume DV which flows through (in analogy to the buoyancy of a wing profile in aerodynamics). If the membrane is orientated parallel to the ether wind (the korns are affected at the side), it shall experience the same resistance A_er as common matter does. Here A_e is the ether acceleration and r the density of the membrane in respect to the x-, y- and z-coordinate (cf. Bahcall et al. 1999).

To calculate the density r of the membrane from the membrane tension F_o needs that we believe in the condition c²=F_o/r of the wave equation of the membrane (see e.g. Carter 2004). But that should not mean that the density of the membrane is caused only by pure energy. So we define an Inert Mass m_i or the density r_i, respectively, of the membrane stuff. Then we define the Energetic Mass m_e and the density r_e as the equivalent of the pure energy of the membrane tension. The sum is the Transversal Inert Mass m=m_i+m_e or transversial inert density r=r_i+r_e of the membrane. This transversal inert density r we had to use in all equations concerning x-, y- and z-coordinates. But there exists still the the w-coordinate. We assume that the law of change of mass of special relativity is valid in the 4-d ether space too. Transversal inert mass m is then m(V_E)= m_oi /(1-V_E²/C²)^1/2+ m_e, where m_oi is the mass of the acting ether building the membrane korns, C is the uppermost signal speed in the 4-d ether space and V_E the expansion speed of the membrane. The energetic mass m_e is assumed to have no rest mass. The longitudinal inert mass or W-Mass (seen in w-direction) is m_W=m_o/(1-V_E²/C²)^3/2+m_e (cf. Joos 1989, e.g.). The energetic mass m_e is here assumed to be independent of the direction, comparable with the mass of a photon. Because we have not an proper experiment hitherto, we are not able to estimate the mass ratio m_e/m_i. But since V_E is assumed to be of order C, the longitudinal inert mass m_W is assumed to be much greater than the transversal inert mass m (c.f., e.g., Huey and Lidsey 2002, where the density is with scalar field F).

Force F_ec is assumed to be orientated nearly perpendicular to w-direction. So in direction of membrane the shortened component F_em =‑A_er w’(1‑w’²)^1/2 does act. In the case of small amounts of w’, we set the squareroot equal to 1. The sum of forces from the outer edge of the gravitational funnel to distance r from the centre gives a hydrostatic pressure. This pressure we interpret as energy E_e per m³. The value E_e=A_erW_oR/r of the sum of forces we get by integration from r®¥ to r over r with w’(r)=W_oR/r² (see Eq.3.2). With A_e=gM/(W_oR) (see Eq. 3.2) we get the relation E_e=gMr/r. Relative energy-growth (or relative mass-growth respectively) is then d_r(r)=gM/(rc²). With a=gM/c² we get the relation d_r(r)=a/r. The density of the membrane grows inside the funnel as it approaches the centre as r(r)=r_o(1+a/r). r_o is the density outside the funnel.

Fig.5.1.1: Ether deflection force and counterforce

F_ec at torus shaped curls (korns) of membrane

The Dispersion Equation n=1+4pN(e²/h)Sa²_nk(n_nk/(n²_nk-n²)) (see. e.g. Joos 1989) gives the refractive index of a substance for frequency of light n. Here N is the number of atoms per unit of volume, e is the elementary charge, h Planck’s Constant, . The a_nk are transition probabilities of energy states, the n_nk are frequencies of spectral lines of atoms or molecules. The essential information here is the linear growth of the refractive index with growing number N of atoms per unit of volume (or growing density r of the substance). Between the velocity of light c_n in a medium with a refractive index n and the refractive index n itself the relation c_n=c_o/n holds. Here c_o is the vacuum velocity of light. As dependence of c_n of a change of density dr, we get the relation c_n=c_or/(n(r+dr)) with constant frequency n. Therefore, the speed of light diminishes with growing dr/r If we translate this known behaviour to the membrane, hypothetically, we find the half of the decrease of the speed of light according to Eq. 5.1.1 with the change of the density dr/r =a/r from the section above.

The Dispersion Equation yields the other half of the decrease of speed of light, too. The author was following here once more the General Relativity. There differentiation of a potential has some spatial components and a temporal component. By Dispersion Equation is n‑1»n_nk/(n_nk²-n²) or n‑1»1/(n_nk‑ n²/ n_nk), respectively. Here n is the refractive index, n_nk are Eigen-frequencies, and n is the frequency of light. If n_nk >> n²/ n_nk then n‑1»1/n_nk holds. If gravitation inside the funnel diminishes not only the frequencies of radiating atoms, but also the Eigen-frequencies n_nk of the membrane diminishes, then the refractive index of membrane will grow. Relative change of frequency (compared with frequency n for r®¥ ) in a gravitational field is Dn/n=‑a/r according to A. Einstein (see e.g. Schmutzer 1991). From this follows Dn»+a/r, and a second time the amount of speed of light is diminished by c_oa/r. Together with the effect of density growth of membrane in the funnel, we get Eq. 5.1.1.

The Shapiro effect of signal retardation by solar gravity is caused essentially by the decrease of speed of light c(r)=c_o(1-2a/r) in the gravitational funnel. The decrease of speed is v_d=2ca/r. We consider the light path in Fig. 5.1.2. The signal starts from the Earth, travels to the Venus (the reflector) and returns. Xe is the distance Earth-O, Re the distance Earth-Sun, Xr the distance Venus-O, Rr the distance Venus-Sun. y is the nearest distance of the signal trajectory to the Sun. Integration of deficit v_d over time (travel time of the signal at the path Xr from O to the Venus) gives the path difference compared with a signal without retardation in the gravitational funnel. With dt=dx/c and r²=x²+y² we get after integration dS=2a (ln(Xr+Rr) - ln(y)). The path O-Earth yields dS=2a (ln(Xe+Re) - ln(y)). Addition and division by c is giving the time retardation dt of one distance, e.g. Earth-Venus, according to Eq. 4.2.

Fig.5.1.2: Radar echo Earth-Venus-Earth

(5.1.2)

The result agrees with the General Relativity (e.g. see Weinberg 1972 or Fliessbach 1990 or cf. Bruckman & Esteban 1993 in the case of strong gravitational fields).

The Depth of space W_o at the edge of Sun is not deriveable from the effects of first order of light bending or signal retardation, since W_o does not appear here. Therefore, we must consider effects of higher order in 1/y (Lewis, Doran & Lasenby 2000). Simulation calculations of a 3-dimensional membrane with a central load in the 4-dimensional space yielded the formula dr/r=(1+w’²)^1/2 for the stretching of the membrane. It means, that the membrane is stretched in radial direction inside the gravitational funnel.. The tangential direction remains unchanged. The author supposes now that this stretching of the membrane causes a relative decrease of the density of the membrane, and thus by the Dispersion Equation it causes a relative increase of the speed of light according to dc/c=1 + w’²/2. (Other imaginable effects caused by a sloped membrane in the gravitational funnel were examined by the author, too.)

The experimental data by Shapiro et al. 1977 show a small tendency of reduction of the retardation in case of sun-near trajectories of radar signals. The influence of the solar corona has been eliminated in a large measure by the use of different wave lengthes, and furthermore, we had rather to assume a retardation by the corona effect. So the author explains this tendency by the sun-near acceleration of speed of light. This effect shortens the run time. Both other effects, reduction of the speed of light with -2a/r and geometrical lengthening of the path inside the gravitational funnel, cause signal retardation.

Geometrical lengthening of the path is the sum of all differential lengthenings dS=(dx²+dw²)^1/2. Eq. 5.1.3 gives the signal retardation dt_W of one part of the path (there or back). Fig. 5.1.3 shows the geometry.

. (5.1.3)

Fig. 5.1.3.: Path lengtening in x-w-plane

The effect of the sun-near acceleration of speed of light c_r(r)=c_o(1+w’²/2) leads to a shortening of time according to Eq. 5.1.4.

. (5.1.4)

Together with the effect of the geometrical lengthening of the path both effects yield a shortening of time dt=3pW_o²R²/(16cy³), contrary to signal retardation caused by the effect of first order. A regression analysis of the 17 sun-near trajectory data (Shapiro, Reasenberg et al. 1977) with weights 1/y³ yields a value W_o=(1.204 ± 0,869) ´10⁶[m] of the depth of space at the edge of Sun. Unfortunately the value is only reliable with t=1.38 and df=16 degrees of freedom.

The main effect f(y)=-4a/y of the solar gravitational deflection of light is caused by the common gravitation together with the effect of a decrease of the speed of light described above. This formula is the same as in the General Relativity (cf. Weinberg 1972, Premadi, Martel and Matzner 1998 or cf. Bruckman & Esteban 1993 in the case of strong gravitational fields). Here y is the nearest distance of the signal trajectory to the centre of the Sun, f the angle of deflection. Additionally we find three effects of the order 1/y⁴. All 5 effects of light deflection acting here are:

· G-effect or deflection caused by the y-component of gravitation

· B-effect or deflection caused by a brake-effect of decrease of the speed of light inside the gravitational funnel and by the x-component of gravitation

· C-effect or deflection caused by a lateral effect of the centrifugal force in the x‑w‑plane

· P-effect or deflection caused by the y-gradient of the lengthening of path

· A-effect or deflection caused by the sun-near acceleration of the speed of light

The G-effect: Common gravitational acceleration of the photon in y-direction is A_y=-(gM/r²)(y/r). With a=gM/c², dx=dt/c und r²=x²+y² the integration over time (from ‑¥ to ¥) yields the lateral speed v_y=‑2ac/y. Division by c gives the result f_G=-2a/y or 0.875“ angle of deflection in the direction of the Sun for a trajectory grazing the Sun (y=R, R is the radius of the Sun).

The B-effect: A photon is braked when entering the gravitational funnel. Fig. 5.1.4 shows the geometrical relations. Brake-acceleration is A_b=dc/dt. With dx=dt/c, r²=x²+y², c»c_o and c=c_o(1‑2a/r) from Eq. 5.1.1, we find A_b=2c²ax/r³. The x-component of gravitation is A_x=(‑gM/r²)(x/r), and with a=gM/c² we get A_x= -ac²x/r³. The sum of both accelerations A_bx=c²ax/r³ is negative at the entrance to the funnel (x<0), but positive at the exit. y-gradient of this acceleration is A_bxy=3c²axy/r⁵ [(m/s²)/m]. Integration over t=x/c (from ‑¥ to ¥) yields the difference of the speed of two trajectories of a distance of 1 meter. Repeated integration over t=x/c yields the difference of path at a fixed time and thus the angle of deflection f_B according to Eq. 5.1.5.

Fig. 5.1.4.: Brake-force and x-gravitation inside the funnel

. (5.1.5)

Together G- and B-effect yield f=‑4a/y.

The C-effect is shown in Fig. 5.1.5. Here k is the radius of curvature of the trajectory. With ¶²w/¶x²=1/k, simple centrifugal acceleration A_C= w²k= (c/k)²k= c²/k, r²=x²+y² and w(r)=‑W_oR/r we find Eq. 5.1.6 for centrifugal acceleration A_C.

. (5.1.6)

Fig. 5.1.5.: Centrifugal acceleration with lateral effect

The lateral effect is caused by the slope of the membrane w_y=¶w/¶y with respect to the x-y-plane. A_C is exactly inside the x-w-plane. A_C is positive for large values of x. Inside the funnel A_C is negative and, therefore, the component A_y is negative, too. With w_Y=W_oRy/r³ , A_Y=A_Zw_Y , v_Y=òA_Ydt , dt=dx/c and f_Z=v_Y/c we find Eq. 5.1.7 for the deflection angle f_C(cf. e.g. Integral Tables in Stöcker 1993). The deflection angle f_C is negative. That means that the trajectory is bent in the direction to the Sun.

. (5.1.7)

To evaluate the P-effect (caused by the lengthening of path inside the funnel) we start with the y-gradient of the differential path lengthening ¶S/¶y=¶((w’²/2)dx)/¶y =¶((W_o²R²/(2(x²+y²)²))dx)/¶y =‑2W_o²R²y/r³. Integration over x from -¥ to ¥ gives a path difference and, therefore, a deflection angle f_P= -6pW_o²R²/(8y⁴).

The A-effect of central acceleration c(r)=c_o(1+w’²/2) of the speed of light (caused by the stretching of the membrane inside the funnel) starts with acceleration of light A_A=dc/dt and dt=dx/c. We find A_A= -4c²W_o²R²x/(2r⁶). Entering the funnel (x<0) A_A is positive, leaving it A_A is negative. The y-gradient of A_A is A_Ay=12c²W_o²R²xy/r⁸. Twice integrated over t=x/c, we find the path difference of two pathes with a distance of 1 meter in y-direction at a fixed time, and, therefore, the deflection angle f_A= +6pW_o²R²/(8y⁴).

P-effect and A-effect cancel one another. The only remaining effect of higher order is the lateral effect f_C=‑3pW_o²R²/(16y⁴) of the centrifugal acceleration acting on the photon. Measurements of deflection angles of sun-near photon trajectories are only possible during eclipses of the Sun. But here an estimation of the influence of the solar corona is difficult to perform. Presumably, the corona causes a deflection to the Sun, too. Nevertheless, we will use the results of Sobral 1919 (f=1.98±0.18“), Principe 1919 (f=1.61±0.45“), Takegon 1929 (f=2.24±0.10“) and Timbuktu 1959 (f=2.17±0.34“) (data from Schmutzer 1991), weight them with the square of their inverse standard deviation, and so we may get the mean deflection angle Df= 2.093±0.193’’ of the above four eclipses of the Sun. The error of ±0.193’’ is the harmonic average of the single errors. The GR predicts f=4a/R=1.75“ for trajectories grazing the Sun. We take the difference (2.093±0.193)’’-1,75’’=(0.342±0.193)’’ and compare it with the second term of Eq. 5.1.7. Thus, we get with y=R another estimate of the depth of space W_o=(1.168 ± 0.659)´10⁶[m]. The uncertainty of this value is great, too. In addition, the observations of Robertson und Carter 1984 show an increase of the deflection angles in the case of sun-near signal trajectories, too.

The light deflection data quoted by Steven Weinberg (1972) deliver a weighted average of Df= 1.92±0.18’’ calculated from 6 eclipses of sun at 11 points on earth, some more as given by Schmutzer, but some of the points are identical. The results of the measurements are (May 29, 1919, Sobral, 1.98±0.16'' / Sept. 21, 1922, Australia, 1.61±0.40'' / 1.77±0.40'' / 1.79±0.19'' / 1.72±0.15'' / 1.82±0.20'' / May 9,1929, Sumatra, 2.24±0,10'' / June 19, 1936, U.S.S.R. 2.73±0.31'' / Japan, 1.70±0.21'' / May 20, 1947, Brazil, 2.01±0.27'' / Febr. 25, 1952, Sudan, 1.70±0.10''). We take the difference anew, here (1.92±0.18)’’-1,75’’=(0.17±0.18)’’ and compare it again with the second term of Eq. 5.1.7. Thus, we get with y=R the estimate of the depth of space W_o=(0.823 ± 0.871)´10⁶[m]. The uncertainty of this value is still greater compared with the result above.

5.2 Perihelion Advance of Mercury (First version 03/1997, last update 06/2008)

In 1859 U. Leverrier, and in 1982 Newcomb, have found independently of each other an advance of the perihelion of Mercury with a value of 43“ per century, which is not explanable from the influence of other planets. Meantime there exists measurements of such an advance of perihelion for all sun-near planets. Due to Newton’s theory the orbit of a planet is an ellipse with fixed axes starting at the centre of mass of the solar system. But this holds exactly only for an absolute spherical symmetry and the potential gM/r. The centre of mass coincides with one of the focal points of the elliptic trajectory. Possible perturbations of the elliptic trajectory may be caused by other planets, by deviations from the exact 1/r potential, by the finite propagation velocity of gravitation, by the oblateness of the Sun following from its rotation - in order to enumerate the most essential influences only. The influence of the other planets is already eliminated in the value 43“.

Weinberg 1972 or Fliessbach 1990, e.g., discuss the influence of the oblateness of the Sun and says that its influence may be neglected. The finite propagation velocity of gravitation was the base of the paper of Gerber, reprinted 1917. He had found already before Einstein the correct formula of the amount of the perihelion advance. (An elementary derivation of the Gerber-Einstein formula starting from a ‘Schwarzschild’ orbit see Cornbleet 1993. For orbits with great eccentricities see Shahid-Saless and Yeomans 1994.) But the author agrees with Einstein 1916 and Tom Van Flandern 1998 so far that the cause of the perihelion advance is not the finite propagation velocity of gravitation, but a deviation from the 1/r potential. From point of view of the GR and of the proposed membrane theory the perihelion advance is caused by an additional relativistic acceleration -6gMa/r³ added to the common gravitational acceleration ‑gM/r². Here g is the gravitational constant, M the mass of Sun, r the distance Sun-planet. The motion of the Sun around the common center of mass causes periodical changes of the gravitational potential propagating (probably) with speed of light. But the influence of these changes is so small that the Mercury will double its distance from the Sun not until after 10⁹ years.

Th. Van Flandern 1998 supposes the propagation velocity of gravitation to be much higher than the speed of light.The reason is the fact that the Earth accelerates not towards the visible position of the Sun (a fact which is confirmed by exact astronomical observations), but to its real position. Inside the membrane theory this contradiction is solvable. Both directions - acceleration and path of light - are on one line. The reason is the aberation that we do not see the Sun at its real place. The gravitation has no aberation. It must not propagate, since it is already there in the form of the gravitational funnel. Only small perturbations caused by the motion of the Sun around the center of mass propagate with finite speed and meet the orbit with an incorrect angle. Therefore, we do not have any reason to suppose the propagation velocity of gravitation to be higher than the speed of light c. Otherwise, Th. Van Flandern's exact measurements of the acceleration of Earth rule out that gravity is caused by gravitons with speed c. Each theory of gravitation must be geometrical, or it has to live with an infinite speed of the gravitons.

The author already delt with the perihelion advance in the paper WEBER 1998. Unfortunately, this part of the paper contains several errors, although the value of 43“ per century was computed. Meanwhile the theory had developed further, and the author makes now a new effort to explanate the perihelion advance from the point of view of membrane theory.

The direct integration of the equations of motion of a planet including an additional acceleration term ‑6gMa/r³ may be done numerically for one revolution with sufficient accuracy. The author used N=10⁹ steps of the Euler-Cauchy integration for one revolution. First, we had to integrate without the additional acceleration term to find the integration error ( or bias). The second integration including now the additional acceleration term then yields as difference the angle of perihelion advance of the Mercury, e.g., and it has here a value of 42.5“ and confirms so sufficiently well the factor k=6 of the additional acceleration term. The hitherto best experimental value is 43.20±0.20“ (Shapiro et al. 1972). The other parameters of the trajectory had values deviating from the starting values less than 10^‑9 after one integrated revolution. Variations of the starting values yielded different trajectories, but all of them confirmed the Einstein-Gerber formula.

How we can derive the additional acceleration term ‑6gMa/r³ from the GR? We start with the Einstein-Gerber formula for the perihelion advance per revolution dy=6pgM/(c²A(1‑e²)) =6pa/(A(1-e²))=6pa/p with p=A(1-e²). A is the great axis of the orbital ellipse, c is the speed of light, a is the half of the Schwarzschild radius, e is the numerical eccentricity and p is the semi-latus rectum (highness over the focal point) or radius of curvature at the perihel and aphel. The total energy of a planet of mass M is mv²/2‑gmM/r=Const (e.g. see Joos 1989). We try to express the additional potential term by an additional speed Dv. We get kgMa/(2r²) integrating from ¥ to r the additional acceleration -kgMa/r (with a still unknown factor k) We take this value as the half of the increase of the square of speed v·Dv or Dv= kgMa/(2r²v). The orbit of the Earth has a small eccentricity, e.g. Thus, we are allowed to set dy=6pa/R_E. R_E is the mean distance Earth-Sun. A second equation for the additional speed, here the speed of perihelion advance, is then Dv =dyR_E/T=6pa/T. T is the time for one revolution. In cases of planets with orbits of small eccentricity v=(gM/r)^1/2 is valid. Together with the first equation for Dv we find the new Eq. 5.2.1 for the additional speed Dv.

(5.2.1)

Comparing this value with the value yielded by the Einstein-Gerber formula, we find for factor k the value according to Eq. 5.2.2.

. (5.2.2)

Using Earth data or Venus data we find k=6, indeed. In the case of Mercury with its great orbital eccentricity we get k=5.64 (with r=p=55.517´10⁹[m] mean radius). But the numerical integration of the orbit yields the true value of the perihelion advance only with k=6, too.

For explanation of the factor k=6 and and therewith of the additional relativistic acceleration ‑6gM/r³ within the membrane theory, first we derive the relativistic expression for the square of energy e² (see chapter 4.3). We set the kinetic energy of a body with mass m in the 4-dimensional space in analogy to the special theory of relativity E=m·U², where U= V_E ²+V². Here V_E is the speed of expansion (W-direction) and V is the speed of the body in the X-direction (in S-scales). We get the square E² using U_~C and m=m_o / (1-v²/c²)^1/2 as E² = (m · U²)² _~ (m · C²)² = (m_o· C²)² + (m · V · C)² . Then we find with C»U and M=M_oo(1‑V²/C²)^‑1/2 the square of energy E²=(MU²)² » (MC²)² » (M_ooC²)² + (MVC)². After transition to our scales (using lower case characters) and with p=mv we get the known expression Eq. 5.2.3.

e² = (m_ooc_o²)² + (pc)². (5.2.3)

m_oo is the mass of a body at infinite distance from a gravitational centre and with velocity v=0. c_o is the speed of light at infinite distance from a graviutational centre. The use of an index is neccessary within the proposed membrane theory, since speed and mass changes with the gravitation. Differentiation of energy e yields the acceleration forces according to Eq. 52..4. Hereby, we use the relations c(r)=c_o(1-2a/r), m(r)»m_oo(1+Ka/r), v_f²=2gM/r, dc/dr=c_o2a/r², dm/dr = ‑m_ooKa/r², dv_f/dr= -gM/(r²v_f). v_f is the pure falling speed from infinite distance in direction of the Sun. The still unknown constant K determines the dependence on speed v=v_f and on distance r of the central mass m. We find

(5.2.4)

or neglecting terms with a²

. (5.2.5)

Differentiation is giving

. (5.2.7)

Writing

(5.2.8)

or with v_f²=2gM/r writing energy e as

, (5.2.9)

we get

(5.2.10)

and after multiplication of the brackets and neglecting terms with a²or terms with a higher power of r than 3 we get the equ. 5.2.11,

. (5.2.11)

The first right side term is the common gravitational force, the second and third terms are relativistic. Both forces, the common gravitational force and the relativistic additional force, are caused by the effect of the ether wind, i.e.

. (5.2.12)

Eqs. 5.2.11 and 5.2.12 are representing the same gravitational force. By setting them equal to each other, we find

, (5.2.13)

. (5.2.14)

The amount of constant K is K=3. (In a former derivation there was an error in the last calculation step from eq. 5.2.13 to eq. 5.2.14, giving the false value of 5). The dependence of mass m on the distance r to Sun is now given by Eq. 5.2.15 in the simple case of falling down.

m(r) = m_oo(1 + 3a/r) = m_oo(1 + a/r + 2a/r).

(5.2.15)

The term 3a/r contains the dependence of the mass on the speed, i.e. a/r, and the dependence of the mass on the gravitational field, i.e. 2a/r. The author is not able to deliver a physical explanation of Eq. 5.2.15 in the sense that we can derive it directly from the change of features of the membrane.

End of part 2.

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