Gravitational Mass Attraction Caused by Ultralong Wave Photons?

(Accepted for publication in “Advanced Studies in Theoretical Physics”, February 2008)

Adv. Studies Theor. Phys., Vol.2, 2008, no. 10, 491 – 505

http://www.m-hikari.com/astp/astp2008/astp9-12-2008/schultheissASTP9-12-2008.pdf

Christoph Schultheiss

In the frame of Special Relativity it can be demonstrated that the wavelength of a photon, which is confined between two perfectly reflecting moving mirrors, develops in congruence with the distance between the mirrors. The expansion of space after the big bang opens the possibility to deal with a flux of congruently expanded photons with ultra long wavelength (ULW). Each particle in space could undergo continuing collisions with ULW photons. This can lead to electric screening effects between particles proportional to their charge and consequently to virtual attraction forces, a force similar to gravitation. This idea is not new: In 1784 Le Sage already developed a theory of inward-directed pressure caused by a flux of hypothetic particles in space. For the case of extremely long wavelength photons – as Le Sage`s particle - it can be shown that relativity is approximately conserved. Movements in the frame of tiny fractions of wavelength (<< λ ~ 10²⁵ m) during short times (<< λ/c) will not cause Doppler shifts i.e., do not change the colour of the ULW spectrum. In the same way very small movements Δλ in comparison with the wavelength of ULW photons will also not cause an increased collision rate, since the collision probability underlies quantum mechanical uncertainty; the rate deviation in this case is proportional to Δλ/λ which is nearly zero. To simulate gravitation, a high flux of ULW photons in space is necessary corresponding to an energy density of 10⁶⁸ Joules/m³ (if results of the paper http://christoph-schultheiss.de/photonic_mass_model  are used). But in comparison with the zero point radiation which is assumed to have a total energy density of about 10¹¹⁴ Joules/m³ this can be considered as a small thermal fraction.

PACS numbers:  98.80.- k,  95.30.Sf,  03.30.+ p

Keywords: Cosmology, gravitation, Special Relativity

Author address:

Email: christoph.schultheiss@ihm.fzk.de

1.     Introduction

Currently, hypothetic dark energy is under discussion that could account for the phenomenon of accelerated expansion of the space [1]. Furthermore, the existence of gravitating dark matter is necessary to hold the gravitation law [2]. This paper should be understood as an approach to explain the expansion of the universe and the deviation of gravitational attraction in large mass collections like galaxies based on well-understood physical laws. 

The acceleration of expansion of universe reminds to a historic theory of gravitation developed by Le Sage and others in the late 17^th –century short after Newton’s detection of gravitation [3]. They understood gravitation as an inward-directed pressure on masses generated by friction of an isotropic flux of hypothetic particles. However, implications of the theory are deceleration of necessarily over light fast particles, corresponding heat development in masses by friction etc. so that discussions about this theory ended in 1920 with the upcoming of the General Relativity. In this paper it will be demonstrated that relativity and quantum mechanic facts of ultra long wavelength (ULW) photons as “hypothetic particles” will effort gravitational pressure without negative implication mentioned above.

It is necessary to recall Einstein’s argument [4] that electromagnetic fields are in reality swarms of photons that interact with charged particles by means of collisions [5]. Therefore, the exact description of collision processes is the Compton formalism. It is also necessary to recall the conclusion of Schrödinger that Compton shifts are in reality Doppler shifts i.e., changes of photon energy rely exclusively on Doppler transformation [6]. Within this framework, moving mirror processes are modeled in which a photon reflects back and forth between two mirrors, thus accelerating them and represent a compelling model of the photo effect. In this exact relativistic calculation, the light wave expands by multiple Doppler shifts proportional to the increasing distance between the mirrors (path-wave-congruence).

The idea presented here is that from the very first stage of the big bang, both the wavelength of light and (optically thick) matter [7] could have expanded in a congruent manner. Hence, space may be densely filled with photons of cosmic wavelength distribution, which interact with charges [8]. When such ULW photons penetrate large masses like planets, then a small fraction of flux can get lost by Compton scattering, comparable with radio long waves passing a dust particle. Thus, a mass located over the planet’s surface is more strongly impacted by ULW photons coming from the open sky than by ULW photons coming through the planet. Therefore, due to the photon momentum transfer, the mass is accelerated towards the planet’s surface, an effect similar to that of gravitation. However, the overall behavior of interaction between masses and ULW photons is an expansion or explosion driven by the pressure of ULW photon collisions.

2. Model of generation of ULW photons by means of moving mirrors and effect of congruence between path and wave.

    Let a moving mirror system consists of two mirrors. One can be fixed (m_∞) the other one is movable as shown in Fig.1. A photon moves back and forward in 180° reflections leading to an outward directed pressure. Under the action of a high number of reflections the photon energy shrinks, the kinetic energy of the mirror approaches hν and the mirror separation grows. The relationship between mirror separation and photon energy is the question and is the subject of the following investigation:

FIG.1 Fixed (S) and movable mirror (S´) with photon

The exact description of a 180° collision of a photon with a particle is the Compton formalism. Here the movable mirror mass as well as the photon changes energy and momentum during collision [9]. This changes cause the well known wavelength shifts for photons (Compton shift). Several authors [6, 9-13] concluded that these shifts are identical to Doppler shifts, that is, after the interaction the charge moves with the velocity β´= v`/c and the interacting photon, coming back from the moving mirror mass, becomes Doppler red-shifted corresponding to β´. This will be demonstrated next more detailed.

With the abbreviations: α = hν/mc², γ = (1-β²)^-½ and for an initial velocity β = 0, by mutual addition and subtraction of both energy and momentum equations:

α +1 =   α´+ γ´                                                           (1)

α + 0 = - α´+ β´γ´                                                       (2)

and the use of the defining equation: γ´+ β´γ´ = ( γ´ - β´γ´)^-1 the well known solutions: α´=α(1+2α)^-1 and γ´= 1 + 2α²(1 –+2α)^-1 result. Furthermore Eq.1 rewritten gives: γ´ = α - α´+1  and Eq.2 gives β´γ ´ = α + α´. By subtraction and addition we have γ´ - β´γ´= 1 - 2α´ and γ´ + β´γ´ = 1 + 2α. Because of the defining equation: γ´+ β´γ´ = ( γ´ - β´γ´)^-1 it follows that γ´- β´γ´ = 1 - 2α´= (1 + 2α)^-1 . A comparison with the solution for α´ (see above) gives:

γ´- β´γ´ = D´ = α´/α,                                                       (3)

which is the Doppler equation. For the case of an initial velocity analogous to Eq.1 and 2, the energy- and momentum laws have the form α + γ = α´ + γ´ and α + βγ = -α´+ β´γ´. With the abbreviated Doppler factor D = γ (1- β) the solution is [14]:

D D´ = α´/α                                                           (4)

Of course, each transformation during following reflections leads to additional Doppler terms in the product DD´D´´D´´´... of Eq.4 and so on. It will be shown next that multiple Doppler shifts lead to an expansion of the wavetrain c o n g r u e n t l y with the distance given by the equation of motion of the movable mirror. The Minkowski presentation of Fig. 2, left, shows the wavetrain path (shown as an arrow) in the fixed mirror system S and Fig.2, right, shows the wavetrain path in the movable mirror system S´. If the wavetrain arrow in one system is horizontally aligned (wavetrain head and tail arrive simultaneously), then in the other system the wavetrain arrow is slanted (the arrival times of wavetrain head and tail are not simultaneous). The coordinates of the laboratory system S in Fig.2 are transformed into the system S´ of the movable mass m by means of the Lorentz transformation ct´= γ (ct-βx) and x´= γ (x-βct). The transformation relates to the common origin, where the ct-axis meets the ct´-axis. This point is in the past at the virtual time ct₀ = -L/β. Therefore, before the transformation all coordinates in Fig.2 should be rewritten.

FIG. 2. Minkowski space-time presentation of 180°-Compton collisions of a photon between a fixed- (ct-axis) and a movable mirror (ct´-axis). The wavetrain arrow is alternately horizontal- (λ_0, λ₁ and λ₂ ) or slanted. The photon is assumed to be at the wave head.

The transformed coordinates are:

(0, L)´ → (0, L/β)´ =  [- γ L, γ L/β ]                                                (5)

(L, L)´→ (L, L/β)´ = [0, γ L/β (1- β² )]                                            (6)

(0, 2L)´ → (0, L/β·(1+β))´ =  [ -γ L (1 + β), γ L/β · (1 + β)] (7)

(x₁, ct₁)´ = (0, L/(1-β) , L/β·1/(1-β))´ = [0, γ L/β · (1+β)]                                           (8)

(x₂, ct₂)´ = (0, Lγ²(1+ β)², L/β · γ²(1+ β)²)´ = [0, γ L/β · (1 + β)²]                                         (9)

(0, ct₂)´ =  (0, L/β · γ² (1 + β)²)´ = [-Lγ³(1+ β)², L/β · γ³ (1 + β)²]                     (10)

This results in the following expressions for the wavetrain lengths:

λ₀ =   (L, L) – (0, L)  =         L                                                        (11)

     λ₁ = - (L, 2L)´ - (x₁, ct₁)´  =  LD^-1                                                                                

      λ₂ =   (x₂, ct₂) - (0, ct₂)   =   LD^-2                                                                                 

The scheme in Eq.11 demonstrates that just when the wave-head touches the opposite mirror, the mirror separation and the horizontally aligned wavetrain path are exactly equal in length. In the subsequent collisions (λ₃ etc.) we have to distinguish between Doppler factors with different values appearing in Eq.11 since the photon energy decreases.

Equation set 11 can be called “congruence equations”. With increasing mirror distance, the photon momentum decreases but will not vanish in this framework, even for cosmic separations. In this sense, the number of photons is conserved [15].

Note, that in this context a thermalization of ULW photon energy via an instantaneous photo effect process is irrelevant, since a further transfer of energy from ULW photons onto particles demands a continues expansion of the wave in the moving mirror system, respectively in space (see Fig,2) and calls for cosmic time scales.

Note, in the light of congruent expansion (Eq.11) a Photo Effect will have the following character:  For instance, visible light is “absorbed” by a black fabric which means the photon undergoes a number of (mainly) elastic Compton collisions with fabric electrons, loose energy due to Doppler shifts, and is converted into long wavelength infrared photons. The wrong picture is that the photon energy is convert into kinetic energy and then reemitted as infrared photons corresponding to the temperature of fabric. The correct picture is, one can follow an individual photon though processes as pointed above (conservation of the number of photons).

Note, inelastic collisions have the potential to absorb a photon within the short time interval λ/c. However, such considerations play no rule in ULW scenario where absorbing atoms or molecules with corresponding ULW-energy bands are not present.

Note, in the calculation of Fig.2, the photon is located at the wavetrain head. In reality, because of QM-uncertainty the photon collision can appear during the whole oscillation period [16]. This leads to QM-uncertainty in the mirror- and wave expansion, but in a statistical average the congruence between path and wave is true.

3.         Tiny movements in comparison with ULW wavelength and -oscillation time against the ULW photon spectrum and effect on relativity. Limitation of Doppler law in sub-wavelength processes

Figure 2 suggests that the change of wavelength in the Doppler process requires an expansion of the wavetrain between moving mirrors during an oscillation period. If the mirror movement is disturbed during the oscillation period by means of high frequent displacements caused by outer means, deviations from the Doppler process should be expected. Assume that at a time point ct = -T, just before the photon arrives at the point (L,L) in Fig.2 left, an external force starts to move the mirror outward (T is tiny in comparison to L). In a linear approximation the increase of separation as well as the increase of the wavetrain length is (for the case β << 1):

λ ≈ L(1- β· T/L )                                                             (12)

A derivation using the Lorentz transformations (not shown here) gives λ = L [1 - 2·β/(1-β) ·T/L]^½ . Equation 12 deviates from the Doppler shift expression because of the dominant term T/L, where L is in the order of 10²⁵ m. Even for particles moving with the speed of light, Eq.12 describes immeasurably small effects on λ. If, for instance, particles generated by a 10 km long linear accelerator have λ-values on the order of  β = 0,9999, then the factor in Eq.12 is in the order of 10^-21, which means if Eq.12 is considered as a Doppler equation a wavelength shift corresponding to a velocity of 3.10^-10 m/s can be expected. Therefore, Special Relativity is hardly affected by a ULW photon scenario. However, an increased collision rate because of the anisotropy of the ULW-spectrum during motion against would violate relativity. Since, on the other hand, it is uncertain whether the ULW photon is emitted at the head or at the tail of the wave, it is irrelevant (by the order L/T) whether the charge moved over the distance T during the ULW oscillation time L/c or is at rest. For Mercury with an orbital radius of T ≈ 10¹¹ and L = 10²⁶ the increase of the collision rate, proportional to T/L ≈ 10^-15, is tiny.

4 . 1     The appearance of an ULW photon flux in space.  

The critical point of the theory is whether an intense spectrum of ULW photons exists at all. This question couples back to the theory of big bang and its boundary conditions. The ULW scenario demands a very high initial number of photons. In the next chapter (Chap.4.2) indications are found that we have to deal with about 10¹¹⁹ photons/m³ respectively about 10¹⁹⁵ photons in space. If this is true, then in the early stage of big bang where space diameter and wavelengths are submicron, the total photon energy exceeds all values by many magnitudes which have been discussed up to now in big bang theories. On the other hand the big bang theory is lacking the zero-point radiation contribution (which exceeds the energy of hadrons - protons, electrons, neutrons etc. - roughly by a factor of 10⁹⁰), so that a decision pro or contra ULW spectrum is open.

4 . 2     Attraction on masses caused by screening effects within the ULW photon flux.  

Let j be the isotropic flux of ULW photons which penetrate the interaction sphere F_p of a charged particle (see Fig.3). Let us look for an estimation of F_p: Depending on the Thomson cross section

σ_p = π r_p² = 8/3·π r₀²· ω⁴/(ω²- ω_p²)² for elastic scattering of electromagnetic waves [17], Compton collisions between particle and photons take place. The radius r₀ = 10^-15 m is the classical electron radius and ω_p  is the “natural frequency” for the case that the frequencies of particle oscillations are higher than the frequency ω of the wave [18]. For ULW photons, one can assume that ω_p >> ω. Therefore, the ω-term in the cross section is ω⁴/ ω_p⁴ and very small in value [19]. Thus, the particle interaction radius derived from the above cross section is

r_p = 2(⅔)^½  r₀ ω²/ω_p²                                                        (13)

and the interaction sphere area is F_p = 4π r_p² = 32/3 ·π r₀² ω⁴/ω_p⁴ . The dimension of the flux j through F_p is the number of photons per unit time. A second particle located at the larger sphere area with radius R around the first particle (see Fig.3) reflects or scatters with the same probability σ_p a ULW photon approaching from the outside and prohibits a flight to and a reflection from the center particle. A virtual mutual central force between both particles appears under the assumption that after the reflection the photon is blocked from undergoing new Compton collisions as for example with charges at the outer sphere. It is true, that QM-uncertainty allows the spontaneous appearance of the photon at every path interval of the wavelength with the same probability [16]. However, within the ratio of planet dimension to the ULW-wavelength, which is in the order of 10^-19, the probability is practically zero.

FIG. 3. An isotropic flux j of penetrating ULW photons enters the interaction sphere F_p  of a charged particle located at the center of a larger sphere with radius R. A second charged particle located at the larger sphere with cross section σ_p screens photons from the center particle and virtual attraction forces will appear.

 Analytically it means before the isotropic photon flux j enters the particle interaction sphere F_p , it passes through the outer shell area F_S in a nearly normal direction because the quotient r_p/R is generally small (see Fig.2). The particle at the outer sphere scatters with the above mentioned Thomson cross section σ_p. Hence the flux j at the center particle is reduced by the quantity of scattered flux Δj a factor which is governed by the area factor σ_p/F_S :

Δj = j σ_p/F_S = ⅔· j · r_p²/R²                                          (14)

The reduced flux j-Δj appears at the screening particle site. In the opposite direction the flux j is unchanged. Thus, the center particle experiences an overshooting force caused by Δj via the screening particle.

To quantify the virtual attraction force one has to consider all types of scatterings from 180° reflections to small-angle scatterings. In the latter case nearly no momentum transfer from the photon to the particle occurs while in a 180° reflection, the (non-relativistic) momentum transfer is 2·h/λ. This will be taken into account by a correction factor ε. The force f on the center particle (charge) is:

                        f = Δj 2 ε h/λ                                                                (15)

f = ε 4/3 · j · r₀²/R² ·  ω⁴/ω_p⁴ ·  h/λ

If the nature of this force is gravitation, then j can be calculated by setting:

         f = f _GRAVITATION                              (16)

2ε 4/3 · j · r_p²/R² · h/λ  =  g · m_p²/R²

where g is the gravitational constant and the factor of 2 results from the fact that matter appears electrically neutral, i.e. in the form of positive-negative charge pairs. Effects coming from deviations from neutrality can be neglected because electrostatic forces are a factor of 10⁴⁰ stronger than gravitational forces. Hence, the force of ULW photons on neutral mass units counts double. The coupling strength in terms of q/m is 2 e/(m_p + m_e). A similar situation can be assumed with neutrons, where the magnetic moment [20] indicates negative and positive charges, which both participate in the interaction with ULW photons.

However, photon interactions with ULW photons look different. Following the QM uncertainty ΔE·Δt ≥ h, for the time interval Δt  photons decay into virtual electron-positron pairs with

ΔE = (m_e^- +  m_e⁺)·c² = e m_ec² and recombine afterwards in a process which takes place with a duty cycle of νΔt [16]. Collisions with ULW photons (photon-photon interaction) occur only during the uncertainty time interval Δt. The twin charges suddenly gain the deflection velocity v = g (νΔt)^-1 Δt . Thus the deflection d approaches double the Newtonian one [21]:

d = g (νΔt)^-1 Δt · t → g t²                                                   (17)

In the case of an anisotropic ULW photon flux the path of a photon is polygonal. An increased gravitational deflection of photons (by the factor of two) is also predicted by GR.

Finally from Eq.16 j takes the form:

J = ¾ ε^-1 g (λ m_p²)/(h r₀²) ·  ω_p⁴/ω⁴                                        (18)

To determine j numerically one has to give an estimate for ω_p; all the other parameters in Eq.18 are known and ε is about 0,3. With 2π/ω ≈ 10 billion years for ULW photons (age of the universe, diameter of universe in light years) and  2π/ω between 1 and 10,000,000 years for typical periodic motion of a charge collection like galaxies, Tab. 1 shows the flux j, the energy density Ε and the radius of the interaction sphere r_p for this cases. The latter one is small and could have values down to the Planck length. Although huge, the energy density Ε of the ULW flux in Tab.1 can be considered as a small thermal fraction upon the spectrum of the zero point radiation [22] which is assumed to have a total energy density of about 10¹¹⁴ Joules/m³ [23].

TABLE 1. Dependency of the ULW photon flux j, energy density Ε = jp/(4π r_p²), where p is the momentum of the ULW photon (p_ULW ~10^-60 kg m/s) and interaction sphere radius r_p as a function of disturbance frequency ω_p. A frequency of about 10 years is discussed below.

2π/ωp in years	j in photons/s	rp in m	Energy density ε in Joule/m^3	ULW-photons/m^3
10,000,000	10^27	10^-15	10^-4	10^47
1,000,000	10^35	10^-19	10^12	10^63
100,000	10^43	10^-23	10^28	10^79
10,000	10^51	10^-27	10^44	10^95
100	10^59	10^-31	10^60	10^111
10	10^63	10^-33	10^68	10^119
1	10^67	10^-35	10^76	10^127

Comments on this:

Table 1 offers a variety of disturbance frequencies, where 10,000,000 years corresponds approximately to the frequency of ULW photons and has maximum cross section (see Eq. 13, ω_p ~ ω), while the cross section in the one-year case shrinks down to nearly zero and extreme ULW photon flux densities are necessary to simulate gravitation.

An estimate for the real disturbance frequency could be the look to the single sided acceleration at the border of a black hole. If for instance a frequency corresponding to 100,000 years is chosen then an acceleration in the order of  b ~ 2 j p_ULW/m_p ~ 10¹⁰ m/s² result.

However if one takes the results of the paper http://www.christoph-schultheiss.de/photonic_mass_model into account, the disturbance frequency can be calculated to about 10 years. In this paper a huge external photon flux in space is necessary to stabilize a photonic mass, which consists of a photon confined between mass less mirrors. If a proton is simulated, then the cavity stabilisation formula is (see Eq.50): 3/2 · h/L⁴ · L/c ~ 10⁴. This has to be equal to: Photon flux j times momentum of a ULW photon (10^-60 kg m/s) i.e.  j · 10^-60. The comparison gives: j is about 10⁶⁴ photons per second though a proton. Such a value indicates that the cross section for gravitational electromagnetic scattering is in the order of the Planck length.

The zero-point-radiation energy density is estimated to be 10¹¹⁴ J/m³ (cut-up frequency is the Planck length). In comparison with that: if the disturbance frequency corresponds to 10 years then the thermal ULW-spectrum contributes with 10^-46 to the total zero point energy, which in relation is extremely small.

Charge objects with a lower periodic motion in comparison with ω_p will have an increased gravitation constant g and vice versa.

5.                     ULW-photons and the perihelion precession of planets

If a charge is in motion due to the relativistic length contraction, the interaction sphere F_p in Fig.2 is compressed in the flight direction (x-coordinate). The same elliptical compression happens for the larger sphere in Fig.2 as well as for the second charge with cross section σ_p which is located at the radius R around the centre charge.

The outer charge may move with the angular frequency ω around the larger sphere. Due to the length contraction the polar coordinates r(t) = r sin ωt and x(t) = r cos ωt changes into

y(t) = r(1-β² cos ω´t)^½ sin ωt  and  x(t) = r(1-β² cos ω´t)^½ cos ωt,                                         (19)

where ω´γ = ω. The x-axis is length contracted: (x´ = x γ^{-1 )} as well as the movement along this axis is slowed down (v´= v γ^-1 ).

Now, let a charge (or a planet) have a circular motion around a center charge (or a star) then a mixed situation appears since the charge in orbital motion is length contracted in flight direction (with respect to the center frozen at ωt = π/2) while the center charge at rest is not. Because of Eq.19 the acceleration ÿ in the direction of the center charge is:

ÿ = - R ω² (1+ β²)     or        Φ = M R² ω² (1+ β²)                                            (20)

The attraction force is increased by a term of the order of β² and is therefore identical to the second order approximation in β of the gravitational potential given in General Relativity [24]:

Φ ≈  - f · M/R · (1+ β²)                                                           (21)

Equations 20 and 21 describe the potential which are behind the motion of perihelion of elliptical planet orbital.

6. Conclusion and outlook

The model of mass attraction presented here is based on ULW photon-charge collisions as the main interaction process in space. Many questions arise and many aspects of the model have still to be investigated. The following substantial observations may support the ULW photon collision model:

The way the space expands could be a consequence of a universally present ULW photon pressure on charges in matter.

The increase of the Hubble constant at the outer space areas may indicate a single-sided photon pressure at the boundary.

The screening of penetrating ULW photons by outer regions of large mass accumulations such as disk-like galaxies may be the reason why the outer spirals rotate with abnormally high angular velocity while the inner one do not, a phenomenon which has been interpreted as an effect of gravitating dark matter.

The Pioneer anomaly (deceleration) may have the same back-ground [25].

Electrons experience the same force like protons although a 1836 times smaller mass.        

Acknowledgement

I would like to express my gratitude to A. Citron for helpful discussions about the interpretation of the path-wavelength congruence as well as E. Borie for critical reading of the draft.

References:

[1] K.C. Freeman, Astrophys. J. 160, 811 (1970)

[2] See, for example, D. Burstein, W.K. Ford, Jr.N. Tonnard, V. Rubin, Astrophy. J. 289, 81 (1985); C. Armendariz-Picon, V. Mukhanov, and Paul J. Steinhardt, Phys. Rev. Lett. 85, 4438 (2000)

[3] Aronson, S.,"The gravitational theory of Georges-Louis Le Sage", 1964, The Natural Philosopher 3, 51

[4] A. Einstein, Phys. Z. 22, 817 (1909) (in German)

[5] In 1909 Einstein demonstrated that during the interaction of gas atoms with electromagnetic waves (both thermal distributed) the gas cools down and the radiation heats up because of overshooting violet shifts (in comparison with red shifts). He suspected a principal failure of the field conception and concluded that the process is carried by gas atom-photon collisions.

[6] E. Schrödinger, Phys.. Z. 23, 301, (1922) (in German)

[7] To be distinguished from the Cosmic Background Microwave Radiation, which escapes about 4·10⁵ years after the big-bang. (see: R. Alpher et R. Herman, Nature 162, 774 (1948) ). In the time span between, which is subject of this paper, the space was hot and opaque, photons collided permanently with particles.

[8] The derivation is restricted to one dimension since conclusions with respect to momentum and energy do not demand higher dimensionality

[9] W. Cantor, Stroboscopy Letters 4 (3 & 4), 59  (1971)

[10] R. Kidd, J. Ardini, A. Anton, Am. J. Phys. 53 (7),  641 (1985)

[11] D. S. Lemons, Am. J. Phys. 59 (11), (1991)

[12] D. Wilkins, Am. J. Phys. 60 (3), 221 (1992)

[13] D. G. Ashwood, R. C. Jennison, J. Phys. A: Math. Nucl. Gen.7, No. 7, 803 (1994)

[14] This result recommends the following interpretation: The Compton effect can be considered as a two-step event. First step is the reflection of the photon, where both mass and photon change from the laboratory system S into the moved collided system S´. Then in a second step, the retransformation of the photon into the laboratory system takes place. In this step the Doppler red shift as formulated in Eq.3 occurs. This shift corresponds exactly to the velocity difference ß´ as energy and momentum law demand.

[15] The conservation of number of photons gives the certainty that ULW-photons can be generated (by any expanding systems) and will exist

[16] W. Heisenberg, "Physikalische Prinzipien der Quantentheorie” Hochschultaschenbücher, Band 1, p 20 (1958) (in German)

[17] See for instance: R. P. Feynman, Lectures on Physics, (California Institute of Technology 1965), Vol.1, p. 32-8

[18] See ref. [17] The ω/ω_p -factor is a measure for deviation from resonance and determines the decrease of energy transfer into the scattered charge

[19] Coherent interferences of neighbored mirrors with separations small in comparison with the wavelength may increase the reflection probability in square to their number.

[20] Particle Data Group, Rev. mod. Phys. 41, 109 (1969)

[21] A.J. Kox, M.J. Klein, R. Schulmann in “The Collected Papers of Albert Einstein” Princeton University Press, Vol.3, DOC 23, 497, footnote [11] (1996)

[22] H. B. G. Casimir, Koinkl. Ned. Akyd. Wetenschap. Proc. 51,793 (1948)

[23] see G.J.Maclay in http//www.quantumfields.com/ZPV.html

[24] R. Adler, M. Bazin and M. Schiffer, “Introduction to General Relativity”, McGraw-Hill, New York, p. 206 (1965)

[25] J.D. Anderson, P.A. Laign, E.L.Lau, A.S. Liu, M.M. Nieto, S.G. Tryshev, Pys. Rev. Lett. 81, Nr. 14, 2858 (1998) and http://www.physical-congress.spb.ru/english/cherep/doppler.asp

Impressum: Postal Address:         Dr. Christoph Schultheiss

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Last correction: March 18. 2008

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