quantumredshift

The following unpublished paper is being put here for purposes of reference only. It is currently being divided into several smaller papers in which certain parts are being re-worked and explained more fully. There are, in addition, parts of the paper below that have been superceded due to recent discoveries. These are marked with yellow highlights in this paper, so that it may be understood that this particular material has been superceded by current work.

A model is developed which accurately predicts observed redshift quantisations and explains periodicities of astronomical objects ranging from nearby galaxies to distant quasars. This is achieved by imposing a quantum condition on the second of the two equations governing the behaviour of the Bohr atom (the orbital energy equation). The outcome is that light emitted from distant galaxies should have a basic redshift quantisation of 2.671 km/s, which is in good agreement with Tifft's statistically treated observations that yield 2.665 km/s. Table 1 compares predicted and observed redshift quantisations.

This model offers a resolution to the missing mass problem for galaxy clusters, while the smoothness of the microwave background also becomes explicable. Black-hole radii should increase smoothly with time if no other effects operate concurrently.

The observed quantum shifts seem to derive from a changing energy density of the zero point fields (ZPF) that comprise the physical vacuum. This induces changes in some atomic quantities, including the speed of light, which are in accord with the observational data. These changes in the ZPF are traced to vacuum behaviour at the Planck length level.

1. INTRODUCTION

1.1. Initial Summary

In 1976, Tifft [1] presented the first in a series of papers that has continued for two decades relating to the redshift z of light from distant galaxies. His observations indicate that the redshift appears to be quantised in basic steps of 7.997/3 or 2.665 km/s [2] rather than being a smooth entity. So far there has been no generally accepted explanation for this basic quantisation. Also relevant to this paper are several other astronomical problems which science has been trying to deal with for the last twenty years: the smoothness of the microwave background, the missing mass in galaxy clusters, and problems associated with redshift periodicities.

A new cosmological model is presented here which indicates that redshifts should be quantised in steps of 2.671 km/s as atomic orbit energies undergo isotropic quantum increases with time. These progressive redshift quantum changes are linked with a smoothly changing zero-point energy (ZPE) of the physical vacuum. When a full quantum of additional energy becomes available to atoms from the zero-point fields (ZPF), then atomic orbits isotropically assume a higher energy state. On this model, the redshift data indicate that the ZPE has progressively increased with time. This increase is traced to the behaviour of the vacuum at the Planck length level and is discussed in the final sections of this paper.

Quantum	n = 1	n = 3	n = 6	n = 9	n = 14	n = 27	n = 54
PREDICTED	2.671	8.013	16.02	24.04	37.39	72.12	144.2
OBSERVED	2.665	7.997	15.99	24.15	37.5*	72.46	144.9

1. BASIC POSTULATE: Several astronomical enigmas may be open to resolution if a quantum condition is imposed upon the second of two equations governing the behaviour of the Bohr atom (the orbital energy equation). Since orbit energy is derived from particle potential energy, this procedure quantises atomic particle potential energy as well as orbital kinetic energy. This yields two Corollaries.

2. COROLLARY 1: Many atomic processes are quantised and quantum thresholds need to be attained before new energy levels can be accessed. However, any variation in the vacuum energy density should be smooth and isotropic. Therefore Corollary 1 states: Quantum changes in atomic behaviour will only occur when the smoothly increasing zero-point energy of the vacuum rises isotropically to an atomic quantum threshold. A quantum change in the energy of each atomic orbit will then occur.

3. COROLLARY 2: Atoms and atomic particles usually cannot access fractions of a full quantum of energy. Therefore Corollary 2 states: Within a quantum interval, atomic quantities that are affected by smooth changes in the properties of the physical vacuum will vary in such a way that energy is conserved.

4. NOTE: Two types of atomic behaviour are discussed here: the behaviour of atoms within each quantum interval, and the behaviour of atoms at the quantum jumps.

1.3. Defining the Problem

It is usually assumed that the redshift indicates universal expansion. If this is true, redshift measurements should show a smooth progression with distance. This does not seem to be happening, however. While quantisation of atomic processes has been accepted for the best part of a century, it is only in the last two decades that Tifft [1-5] and others have been pointing out that quantisation on an astronomical scale also appears to be occurring with redshifts from galaxies. This quantisation becomes apparent when redshifts of pairs of galaxies are compared, or when galaxy redshifts have the Doppler shift of our solar system motion removed [4, 6]. The question then arises why universal expansion should be quantised instead of being a smoothly occurring entity. One possible explanation is that the redshift is not really an artifact of expansion but is due to some other process.

1.4. Turning the Problem Around

The redshift, z, is defined as the change in wavelength, Dl, of a given spectral line compared with the laboratory standard for that line, divided by the laboratory standard wavelength, l, for that same spectral line. This is usually written as [7]

z = Dl /l (1)

Observations show that the more distant the astronomical object, the higher the redshift. However, as previously noted, Tifft’s observations indicate this shifting of wavelengths is not happening smoothly but in quantum jumps. Since similar quantum jumps are familiar to physicists dealing with atomic phenomena, the observed redshift quantisation might be explicable if all wavelengths from each atomic emitter undergo a series of discrete ‘jumps’ over time. If this alternative explanation is valid, then the observational evidence requires a series of isotropic decreases in all emitted wavelengths with time. For this to occur, the energy of each atomic orbit must increase in a series of discrete ‘jumps’.

This paper will use the Bohr atomic model for several reasons. First, as acknowledged by others, the Bohr theory of the atom gives results that are spectroscopically correct to a first approximation [8]. The more recent wave mechanical approach has simply confirmed and built upon the concepts that Bohr used [9]. Similarly, much of the recent work done on the zero-point energy (ZPE) and atoms in stochastic electro-dynamics (SED) has been largely formulated at the Bohr theory level. It has been stated that this approach permits both “intuitive insights and calculational ease” [10, 8].

From Bohr theory then, the total energy of an atomic orbit E_t, of radius r, is given by kinetic energy of the electron, E_k, plus the electrostatic potential energy of the electron in the field of the proton, E_p, so that [11, Eq. (4-18)],

E_t = E_k + E_p = [e²/(8per)] – [e²/(4per)] = - e²/(8per) (2)

Note that the kinetic energy of the electron E_k is equal in magnitude (but not sign) to the total orbit energy E_t. The negative sign in (2) reflects the fact that the electron is bound to the positive nucleus in the same way that the planets are bound to the sun. In order to avoid difficulties with negative quantities let E_t be re-defined at this juncture to represent the binding energy E₀ of the orbit in question. E₀ is then a positive quantity [12] and is equal in both magnitude and sign to the orbital kinetic energy E_k. Eq. (2) can now be specifically re-written as

When an electron falls into a given Bohr orbit from outside an atom, it releases a photon of light with energy E that is equal in magnitude (but not sign) to E_t in (2), and is equal in both sign and magnitude to E₀ and E_k from (3). Furthermore, this photon of energy E will have a wavelength l from the standard relationship E = hc/l where h is Planck’s constant and c is the velocity of light. The following equation can therefore be written for the energy of photons at the time of emission:

For the energy of a given Bohr orbit to undergo a series of discrete jumps as suggested above would require Eqs. (2) to (4) to have a quantum condition imposed. But the terms being quantised need to be determined.

At this point observations from geology impose a constraint. If the radii of atomic orbits r, and hence the radii of atoms, were to undergo a series of quantum expansions, dislocations in rock crystals would be evident, particularly in Precambrian strata. However, this is not observed. It must therefore be concluded that atomic orbit radii r, have not undergone any quantum changes. In the Friedmann model of an expanding cosmos, both the atoms and the space between them would expand uniformly so that no evidence of any change would be recorded geologically. However, a series of discrete, quantised changes in r over and above any universal expansion would be instantaneous and affect atoms only, not the space between them. Serious disruption to rock and crystal structure would then occur. The same argument holds whether the cosmos is expanding, contracting or static.

As a consequence of this observational constraint from geology, it becomes apparent that any quantum change in energy E_t, E₀, E_k or E implied for (2) to (4) cannot come from r, but must be in the e²/e ratio, which is common to both the proton and electron. A discrete orbit energy change thus requires a quantum change in particle energy itself via the e²/e ratio, which is maintained by virtual particles or the zero point fluctuations of the vacuum. This is discussed in detail later.

1.7. The Invariance of hc

There is a second important observational constraint in relation to (4). Experiments by Bahcall & Salpeter [13], Baum & Florentin-Nielsen [14], and Solheim, Barnes & Smith [15] indicate that the quantity hc is invariant over astronomical time. Indeed for distant astronomical objects, Noerdlinger found that: d[ln(hc)]/dz £ 3 ´ 10^–4 , where z is the redshift [16]. These cosmological results experimentally confirm that

Under these circumstances, when the result of (5) is applied to (4) it becomes apparent that any quantisation of energy E must be in the wavelengths l of the photons at the time of emission. It may therefore be concluded that a quantum change in atomic orbit energies and particle potential energies via the e²/e ratio will result in an observed quantisation of emitted wavelengths, l.

Mathematically, such a quantum change in atomic orbit energies is possible. There are two equations that govern atomic orbit energies. See for example Wehr and Richards [11, Eqs. (4-14) and (4-17)]. They are duplicated in French [17, Eqs. (5.1) and (5.3)]. The first of these equations specifies the angular momentum of atomic orbits in a hydrogen atom, which was quantised by the Bohr theory as

Here, m is the electron mass moving with velocity v, and n is the usual quantum integer. However, the second Bohr equation is of immediate interest as it contains the terms that appear in (2) to (4). It describes the kinetic energy of the electron in its orbit as

The results of (7) can now be added to (4) to give a complete set of terms involved in any discrete orbit energy change, namely

It therefore becomes apparent that any discrete change in orbit energy via the e²/e ratio, which brings about a discrete change in emitted wavelengths l, will also involve a change in the mass m of orbiting electrons and/or their velocity v. This follows since the mass of an electron is related to e² through the classical electron radius [18]. Changes in the electronic charge and/or the classical electron radius will inevitably affect the mass of the electron.

The next step is to note that, in the Bohr atom, all orbit energies are scaled according to the ground state orbit. Therefore, if the ground state orbit has an energy change, all other orbits will scale their energy proportionally. This means that wavelengths of emitted light will also be scaled in proportion to the energy of the ground state orbit of the atom. Now French points out in his derivation of the relevant equations that the energy of the ground state orbit can be written as [19]:

where R_¥ is the Rydberg constant for the infinite nucleus. Thus, for the ground state orbit, it follows that

where W is the Rydberg wavelength. Observationally, the incremental increase of redshift with distance indicates that the wavelengths of light emitted from successively more distant galaxies undergo a fractional increase. Therefore, in the case of the ground state orbit of the Bohr atom, the Rydberg wavelength W must change incrementally in steps of some set fraction of W, say W/z = R*. This means that W = z R*. Furthermore, the redshift wavelength increment Dl can then be defined as

Here, the term n is the new quantum integer, which fulfils a similar function as Bohr’s quantum number n. Note that if n decreases with time, it will mimic the behaviour of the redshift, which also decreases with time. Thus high redshift values from distant objects necessarily mean high values for n as well. From (11) it may be seen that R* is a specific fraction of the Rydberg wavelength. This specific fraction is given by the dimensionless number z which could be called the Rydberg quantum number. Analysis of the terms making up the Rydberg constant indicate that such a dimensionless number can indeed be obtained from R_¥ provided one reasonable assumption is made. This numerical value of z, the Rydberg quantum number, is derived by this and another method later in this paper.

To find the redshift quantum Dz from this approach, it need only be noted that, for the ground state orbit of the Bohr atom, the laboratory reference wavelength l will be equal to W. Using (11), the following equation can then be written:

Since Dz is a fixed dimensionless number equal to 1/z, it follows that the redshift z is proportional to n, which can be written as z µ n where the symbol (µ ) means “proportional to” throughout this paper. Furthermore, the redshift wavelength increment as given by (11) means that the energy of the first Bohr orbit given in (9) will increment in steps of DE such that

This procedure effectively quantises Bohr’s second equation given in (7) and (8). It then necessarily follows that there will be corresponding changes in both mv² and the e²/e ratio in (8). In practice this means that if a full quantum of energy becomes available to every proton and electron, each atomic orbit will then assume a proportionally higher energy state. This follows since all other orbits scale their energy in proportion to the ground state orbit, and the orbit radii remain fixed. Furthermore, in multiple proton atoms, the orbital energy levels will also change synchronously and instantaneously with Bohr hydrogen atoms. It is assumed that the quantum energy change in (13) occurs simultaneously for every proton and electron throughout the cosmos.

Consequently, the greater the distance we penetrate into the cosmos, and thus the further back in time we go, Eq. (13) indicates that n takes on successively higher quantum values resulting in corresponding quantum decreases in orbit energies. Furthermore, emitted wavelengths, such as l in (9), will become longer and hence redder in a series of Dl quantum steps governed by the value of n, since both R* in (11) and hc in (13) are invariant. This means that the observed change in wavelength, Dl, compared with l, the fixed laboratory standard on earth, will be directly proportional to n also.

This proposed new quantisation in proton/electron energy thereby supplies a potential solution to the problem. If particle energies have been increasing in discrete steps over time, then all atomic orbits would have had lower energy when photons were emitted in the past. There would then be a systematic and quantised redshift of received photon wavelengths compared with the laboratory standard. In the discussion associated with (20) and (21) it is also shown that emitted wavelengths remain unchanged in transit to earth.

The proposed jump in atomic orbit energy should occur isotropically throughout the cosmos as a result of the behaviour and properties of the vacuum. Currently there are two models for the physical vacuum: the quantum electro-dynamic (QED) model, and the stochastic electro-dynamic (SED) approach. Since they both give the same answers mathematically, the choice regarding their use becomes one of convenience. Although both are used here, the SED model frequently gives more visualisable results and avoids much of the esotericism associated with QED formalism.

Although the rise of the SED approach is more recent than the QED, thus making the QED model better established, the roots of the SED model go back to 1910, with Planck’s “second theory.” In this theory, Planck involved the zero-point field in an exploratory derivation of the formula for a blackbody spectrum [10]. In 1913, Einstein and Stern dealt with the interaction of matter and radiation using classical physics and a model of simple dipole oscillators representing charged particles [20]. They concluded that if the dipole oscillator had a zero-point energy for some reason, the result would be the Planck formula for the radiation spectrum without the need to invoke quantisation. By 1916, Nernst [21] had grasped the crucial significance for physics of such a universal zero-point field, but it was not until some comments by de Broglie [22] in 1962 that interest was revived in this. It resulted in several derivations of the formula for the black-body spectrum using classical physics and the zero-point fields (ZPF) without invoking quantisation [23]. In 1975, Boyer established that the fluctuations caused by the ZPF on the positions of particles were in exact agreement with Heisenberg’s uncertainty principle [24]. Using this approach, Planck’s constant h thereby becomes a measure of the strength of the ZPF, since the ZPF fluctuations provide an irreducible random noise at the atomic level, which is then interpreted as innate uncertainty [25, 10].

Reinforcing this approach outlined by Planck, Einstein, Stern, and Nernst, both Nelson in 1966 [26], and more recently Rueda [27], have demonstrated that an entirely classical derivation and interpretation of the Schroedinger equation can be obtained on the basis of ZPF induced Brownian motion. As a result of these and other more recent successes, it has been pointed out “The most optimistic outcome of the SED approach would be to demonstrate that classical physics plus a classical electromagnetic ZPF could successfully replicate all quantum phenomena” [10]. While SED formalism has been successful up to this point, many more years of work may be needed to fully achieve this goal.

Evidence for the existence of the ZPF comes from an effect on the electron known as the Zitterbewegung, or its “jitter motion” even at a temperature of absolute zero. There is also the Lamb shift of spectral lines, and the surface Casimir effect, whose magnitude was verified in a beautiful experiment by Lamoreaux [28]. In the case of closely spaced atoms or molecules, the all-pervasive ZPF result in short-range attractive Van der Waals forces [25, 10], which is merely the microscopic counterpart of the Casimir effect. Finally Sokolov has shown that the vacuum energy liberated by the volume Casimir effect during the collapse of dying stars may perhaps be the reason that they then go on to explode as supernovae [29]. All this observational evidence indicates that the ZPE is a physical reality, and is not simply a theoretical construct.

Indeed, both the QED and SED models come to the conclusion that, even at absolute zero, the physical vacuum has an inherent energy density. On the QED model this zero-point energy (ZPE) permits short-lived virtual particle pairs to form and annihilate. They flip in and out of existence on a time-scale governed by Planck’s constant h and so do not violate the Heisenberg uncertainty principle. According to the QED model, an atomic particle, even when alone, is emitting and absorbing these virtual particles from the vacuum. Consequently, a proton is considered to be the centre of continual activity; it is surrounded by a cloud of virtual particles with which it is interacting [30, 25].

In the SED approach, the vacuum at the atomic or sub-atomic level is considered to be comprised of a turbulent sea of randomly fluctuating electro-magnetic fields whose cut-off wavelength is the Planck length 1.61604 ´ 10^–33 cm. Pipkin and Ritter point out that “…the Planck length is a length at which the smoothness of space breaks down, and space assumes a granular structure” [31]. Wavelengths of the zero-point fields shorter than the Planck length would not be transmitted by the vacuum under these conditions. These zero-point fields (ZPF) are homogeneous and isotropic, and look the same to two observers no matter what their velocity or position is with respect to each other. Consequently the zero-point radiation (ZPR) is Lorentz invariant. In practical terms, this means that the intensity of the ZPR has a cubic frequency dependence up to the Planck length cutoff. Furthermore, in the SED approach the perturbations that result from the ZPF are the source of quantum indeterminacy. Planck’s constant h thereby becomes a measure of the strength of the ZPF as outlined above [25, 10].

The intrinsic energy inherent in the vacuum ZPF gives free-space its properties and governs the activity of virtual particles. Besides the ZPE, key vacuum properties include its permittivity and permeability. If any of these properties change isotropically, then atomic behaviour would also vary throughout the cosmos. For example, any increase in vacuum energy density implies there has been an increase in the ZPE. As a result, it might be expected that atomic particles would absorb more energy by the exchange process from the ZPE sea in which they are immersed, or the virtual particles with which they are interacting.

Since atomic orbit energies are quantised, it is reasonable to assume that a basic quantum of energy must become available from the vacuum before any change in orbit energy can occur. Although vacuum properties may be expected to vary smoothly, once these variations exceed a certain threshold, energy is accessed from the vacuum, and a quantum change then occurs in atomic behaviour as the e²/e ratio takes on a new value.

The way that the physical vacuum transfers its energy to the atom has an explanation on the SED approach that is useful here. Puthoff has demonstrated that there is a dynamic equilibrium between radiation emitted due to acceleration of the electron in its ground-state orbit, and radiation absorbed from the zero point fluctuations of the physical vacuum [32]. This dynamic equilibrium prevents the radiative collapse of the Bohr atom. The ZPE sea in which all atoms are immersed is made up of a complete spectrum of wavelengths down to the Planck length cutoff. An electron in a circular orbit comprises a waveform of a specific wavelength. That waveform is reinforced by energy of the same wavelength from the ZPF. All other ZPE wavelengths only act to give rise to the electron’s quantum indeterminacy.

Puthoff has explained it this way: “The circular motion [of an electron in its orbit] can be thought of as two harmonic oscillator motions at right angles and 90 degrees out of phase, superimposed. These two oscillators are driven by the resonant components of the ZPE just as you would keep a kid swinging on a swing by resonantly-timed pushes. The oscillator motion acts as a filter to select out the energy at the right frequency (around 450 angstroms wavelength for the hydrogen atom Bohr orbit ground state)” [33]. Consequently, it can be seen that energy is transferred from the ZPF to maintain electrons in their atomic orbits by this resonance mechanism. In fact, Puthoff has pointed out that the ZPE must be maintaining all atomic structures throughout the entire cosmos [32].

Puthoff also demonstrated [34, Eqs. (18) and (19)] that the power absorbed from the vacuum P_a was equal to power radiated P_r by the electron. Interestingly, while the other terms were unique to either P_a or P_r in Puthoff’s two equations, the e²/e ratio was common to both. These equations are important since the quantisation of the e²/e ratio, and hence orbit energy, may be derived from them. In turn, this means that the vacuum energy density must increase by a precise amount to meet the quantum condition.

Such an increase in orbit energy will necessarily result in shorter wavelengths for emitted light. Consequently, light from distant astronomical objects emitted before the quantum change will be redshifted compared with the laboratory standard on earth. However, changes in vacuum energy density inevitably mean changes in key properties of the physical vacuum. The reasons for these changes will be discussed later. The first task is to formalise the results of these changes mathematically.

In the SED approach adopted by Boyer [35], Puthoff [36, 37],and others, the ZPE of the “seething vacuum” is made up of electromagnetic fields. Since the vacuum is an isotropic, non-dispersive medium, the energy in all electromagnetic waves, including the ZPF, will be equally distributed between electric and magnetic fields. If the electric field strength is E, and the corresponding magnetic field strength is H, then the simplest electromagnetic waves have the form [38]: E_y = A₀ sin (wT – bx), while H_z = A₀ sin (wT – bx), where A₀ is the peak amplitude of the wave. These equations are therefore relevant to the electromagnetic waves of the ZPR. The standard equation for the energy density of such electromagnetic waves [39] will also be the equation for the energy density U of these zero-point fields, namely

Let us now assume that the electric and magnetic field strengths of the already existing photons making up the ZPF remain unchanged with time, which also implies an unchanged peak amplitude A₀. Then any increase in the energy density U of the ZPF will mean a proportional increase in e and m. This follows from (14) and can be expressed thus:

Indeed, Barton has shown that the vacuum Planck energy density U is directly related to both e and m [40]. Consequently, the zero-point energy density U in the physical vacuum would be expected to follow the same proportionality. Under these circumstances, with the vacuum permeability and permittivity directly proportional to the vacuum energy density, it follows that if the energy density of the ZPE ever increased, then there would be a proportional increase in the value of both the permeability and permittivity. In the SED approach, an increase in the energy density of the ZPE should signify that an isotropic increase in the total number of waves of any specified wavelength longer than the Planck length cutoff has occurred. In essence, this means that the cubic frequency distribution of the ZPE intensity is maintained, but this cubic frequency distribution is multiplied by a time-dependent factor, whose form can be discovered by redshift observations. It should be noted that, in the QED model, a higher energy density for the ZPE should imply more virtual particles per unit volume. The reason why the energy density of the ZPE increases with time is also deduced from the redshift data, and the derived equations appear in the closing section of this paper.

These results have several important consequences. As a result of (14) and (15), it follows from the definition of the intrinsic impedance of space, W, that

so W will always bear the value of 376.7 ohms. From (14) this reveals that, even with vacuum energy-density variations, the value of H will bear a constant ratio to E in plane waves, as required for a non-dispersive medium. Light-speed c is necessarily involved in this also, since the standard relationship reads [41]

When compared with (15), this result indicates that light-speed and vacuum energy density must obey the relationship

Since the vacuum permittivity and permeability are energy-density dependent, then Eqs. (14) to (19) show that light-speed c will also be affected by these changes in the physical vacuum.

This has been explored theoretically for the case of the reduced energy density of the vacuum between two Casimir plates. Scharnhorst [42] and Barton [40] linked this reduced energy density with a proportionally reduced permittivity and permeability of the vacuum and a higher light-speed that would result. This is explained in the QED model by the fact that a decrease in the energy density of the ZPE would also result in a decrease in the number of virtual particles per unit distance. Light photons are absorbed by these virtual particles and then re-emitted. As Barnett has pointed out, “this process makes a contribution to the permittivity of the vacuum (e ) and therefore to the speed of light” [43]. Consequently, if there is any decrease in the ZPE and the number of virtual particles per unit distance, then the speed of light will inevitably increase. Conversely, with any increase in the ZPE, and thus the number of virtual particles per unit distance, the speed of light will inevitably decrease.

In summarising these results, Barnett commented that “The vacuum is certainly a most mysterious and elusive object … The suggestion that [the] value of the speed of light is determined by its structure is worthy of serious investigation by theoretical physicists” [43]. These and other results were later generalised by Latorre et al. [44]. While some of the equations developed in those papers only applied under restricted conditions, the general principle was well established by Latorre. He concluded that if a vacuum had a lower energy density than the standard vacuum, then light-speed would be proportionally greater than the standard speed. Conversely, if a vacuum had a higher energy density than the standard vacuum, then light-speed would be proportionally less than the standard speed. These conclusions are in agreement with Eq. (19).

It might be objected that any such variation in the components of (18) is contrary to the theory of relativity. This objection is usually voiced because, according to relativity, free space should have the same properties to any observer in motion. Thus the individual values of both e and m should be the same for all inertial frames. The conclusion is that c must therefore be constant. However, these demands of relativity are still fulfilled if, at any instant, the individual values of e and m are, first, isotropic throughout the cosmos, and, second, vary slowly with respect to atomic processes. This second condition is explained further below. Therefore, if light-speed is varying, it must be doing so isotropically on a cosmological scale, and any such variation is slow with respect to atomic processes. These two conditions are maintained in this model.

Those who are accustomed to derive Maxwell’s equations from relativity may object that (18) is obtained on the assumption that c is a constant. However, as shown in Bleany & Bleany [41], (18) can be readily derived without any initial assumptions about the behaviour of c, e, or m. This is done by obtaining a set of four simultaneous partial differential equations based on (a) Gauss’s theorem applied to electrostatics; (b) Gauss’s theorem applied to magnetic fields; (c) Faraday’s and Lenz’s law of electromagnetic induction; and (d) Ampere’s law for magnetomotive force. As the resultant equations from (a) and (b) eventually become e div E = 0 and m div H = 0 for a vacuum, then it is apparent that they are independent of any variations in e and m. The equations that result from (c) and (d) eventually become curl E = - m (¶ H/¶ t) and curl H = e (¶ E/¶ t). Provided that m varies slowly with respect to H, and that e varies slowly with respect to E, the formulation is still valid. Similarly, the general wave equation which (a), (b), (c), & (d) reduce to has the form Ñ²A = (1/v²)(¶ ²A/¶ t²) where v² = 1/(me) = c² and A is some scalar or vector quantity. Again, this equation is valid for describing wave motion provided that v² varies slowly with respect to A. All that Maxwell’s equations require, therefore, is that e, m, and c vary slowly with respect to atomic processes. This illustrates the necessity of the second condition required by relativity as shown above. Since both conditions are fulfilled in the context of this model, (18) is still valid in a scenario where m, e, and c are varying.

Importantly, (19) indicates that light-speed is not quantised since the smoothly varying properties of a vacuum will result in both m and e smoothly varying and hence c also. Furthermore, as atomic behaviour is affected by these same vacuum properties, there should be synchronous variation with c and the relevant atomic quantities. If the redshift quantisation is indeed caused by increasing vacuum energy density, there should be experimental evidence for a decline in c and a synchronous variation of other associated atomic quantities within the quantum interval.

Observational evidence supports the possibility that light speed c may not be a constant. Some 40 articles about the matter appeared in the scientific literature from 1926 to 1944 alone. Some interesting points emerge from this literature. Despite a strong preference for the constancy of atomic quantities, Dorsey was reluctantly forced to admit: “As is well known to those acquainted with the several determinations of the velocity of light, the definitive values successively reported … have, in general, decreased monotonously from Cornu’s 300.4 megametres per second in 1874 to Anderson’s 299.776 in 1940…” [45]. Even Dorsey’s own re-working of the data could not avoid that conclusion.

However, the decline in the measured value of c was noticed much earlier. In 1886, Newcomb reluctantly concluded that the older results obtained around 1740 were in agreement with each other, but they indicated c was about 1% higher than in his own time [46]. In 1941 history repeated itself when Birge made a parallel statement while writing about the c values obtained by Newcomb, Michelson and others around 1880. Birge was forced to concede that: “…these older results are entirely consistent among themselves, but their average is nearly 100 km/s greater than that given by the eight more recent results” [47]. Each of these three eminent scientists held to a belief in the absolute constancy of c. This only makes their careful admission about experimentally declining c values all the more significant. The observational data that comprise this evidence for declining c values, termed c-decay here, was documented and discussed fully in the SRI Research Report “The Atomic Constants, Light, and Time” by Norman and Setterfield in August 1987, hereafter referred to as the Report [48].

The data obtained over the last 320 years do at least suggest a decay in c. Evidence for a decay trend exists within each measurement technique as well as overall. Furthermore, the measured values of the associated c-dependent atomic ‘constants’ change synchronously with c. The mass of data indicating such variation comprises some 638 values measured by 43 methods. They are detailed in the Report where 163 values of c were examined along with 475 measurements of 11 other atomic constants. Montgomery and Dolphin did further extensive statistical analyses on the data, the results of which supported the c-decay proposition [49]. Some similar conclusions about the behaviour of c and the atomic constants were arrived at by Troitskii using other methodologies [50].

Figure 1 plots a linear regression of the best 120 values of c by 17 methods as determined by the Montgomery and Dolphin statistical analysis. If the two high outliers are omitted, the linear regression reveals a decay in c of 3.15 km/s per year that has at least formal statistical significance. It serves to illustrate that a postulate of c-decay is an improvement over an assumption of a constant c value as residuals reduce markedly. Nevertheless, non-linear curve fits reduce residuals even further, indicating that any c-decay may well be non-linear. However, all questions on the statistical treatment of the c data are referred back to the authors of the Montgomery and Dolphin paper [49]. The purpose of the discussion here is to show that the experimental evidence does not contradict the model presented in this paper.

If c is indeed following a non-linear function, a tapering of any measured c-decay rate may be anticipated to lead to a flattening of the curve, even though c may still be varying. In fact the curve does seem to flatten after 1950. There are two reasons for this. The first has to do with the introduction of new timing devices. In 1949 the frequency-dependent ammonia-quartz clock became standard in many scientific laboratories [51]. Following that, atomic clocks were developed. By 1967, atomic clocks became uniformly adopted as timekeepers around the world. Methods that use atomic clocks to measure c will always fail to detect any changes, since their run-rate depends on atomic frequencies that vary directly as c varies. This follows as a consequence of (21) below and is explained more fully by the observationally based comments preceding it. Not surprisingly, the result was that the General Conference on Weights and Measures meeting in Paris, in October 1983, declared c an absolute constant [52].

Nevertheless, evidence presented in the next section implies that the c-decay curve continued to progressively flatten out until about 1980. This is confirmed by separate measurements of other c-dependent atomic constants that allow the behaviour of c to be indirectly monitored. These indirect data suggest that c may have started to rise marginally after 1980. This leads to the second reason for the character of the curve after 1950, which is linked with the behaviour of c itself. Observational evidence from astronomy and geology presented with Table 2 indicates that there is a naturally occurring oscillation, which has had the effect of flattening the curve to a broad plateau at certain points. This oscillation leaves a ripple superimposed on the general c-decay curve.

For the sake of uniformity in the rest of this presentation, the text of French [17] will be followed. Hereafter this will be designated ‘F’ with a page number.

Since atoms and light-speed appear to exhibit coordinated behaviour, it is logical to suggest that universal changes in the physical properties of the vacuum may be occurring. The possible reason for these changes is discussed after the redshift data has been considered, since these data provide the information needed about c and the energy density of the vacuum. The immediate task is to establish the extent of the coordinated behaviour between c and atomic phenomena.

Consider firstly an observational fact pointed out by Birge [53]. Light-speed had been measured as varying, but there was no observed change in wavelengths compared with the standard metre. Although Birge did not personally accept c variation, he admitted that this evidence left only one conclusion. Birge stated that “if the value of c … is actually changing with time, but the value of [wavelength] l in terms of the standard metre shows no corresponding change, then it necessarily follows that the value of every atomic frequency ... must be changing” [53]. This follows from the basic equation

Since wavelength l is unchanged in this case, then, as Birge noted, frequency f must obey the equation

This experimentally derived result will partly dictate the direction of the subsequent discussion. It means, for example, that wavelengths of light photons in transit are not altered by the processes being considered here.

In addition, because the velocities of particles in atomic orbits are directly related to the frequencies of light that they emit [F. 114], the run rate of any atomic clock will be a direct function of this frequency. In other words, when c is higher, atomic frequencies are also higher from (21). This means that atomic time intervals, t, are shorter as t is proportional to 1/c, so atomic clocks tick more rapidly. It can therefore be stated that ct = constant. By contrast, dynamical (or orbital) time, T, has units that are subdivisions of the earth's orbital period, which is essentially constant, and not affected by changing c, as shown independently below. Kovalevsky has pointed out that if the two clock rates were different, “then Planck's constant as well as atomic frequencies would drift” [54]. This can be seen by considering (5), (21)-(23), (29) and (36).

Such changes have been noted. For example during the interval 1955 to 1981 Van Flandern examined data from lunar laser ranging using atomic clocks and compared them with dynamical data. He concluded that: “the number of atomic seconds in a dynamical interval is becoming fewer. Presumably, if the result has any generality to it, this means that atomic phenomena are slowing down with respect to dynamical phenomena” [55]. Although controversial, these results nevertheless establish the general principle being outlined here. They also support the contention that c was flattening out to a minimum around 1980. Van Flandern also made one further point as a consequence of these results. He stated “Assumptions such as the constancy of the velocity of light … may be true only in one set of units (atomic or dynamical), but not the other” [55]. This is the kernel of what has already been said above. Since the run-rate of the atomic clock is proportional to c, it becomes apparent that c will always be a constant in terms of atomic time.

These results also led Van Flandern to make a perceptive observation. He stated that “… if the universe had constant linear dimensions in both dynamical and atomic units, the increase in redshift with distance (or equivalently, with lookback time) would imply an increase in c at past epochs, or that c was decreasing as time moves forward” [55]. The importance of the behaviour of the cosmos with time is discussed later in this paper.

4.5. The Isotropic Nature of Light-speed and Relativity

In the model presented here, light has the same speed at any instant throughout the entire cosmos. This is demanded if the vacuum energy density is smoothly and isotropically increasing with time throughout the universe. Experimentally, this constraint has been verified by Barnet, Davis, & Sanders [56]. They demonstrated that light from distant quasars arrived at our telescopes with the same velocity as light from local astronomical sources.

This constraint means that c would have the same value in all frames of reference throughout the cosmos at any instant. Furthermore, over a short dynamical time interval T, light-speed will be essentially constant. Therefore, the Lorentz transformations on which relativity is based will not be violated. But a further point can be made for atomic time intervals t. It was noted above that ct = constant. Now since the Lorentz transformations require the invariance of x² + y² + z² – c²t² it is usually assumed that c must be a constant [F. 154]. However, all that the transform really demands is that c²t², as a whole, be invariant without saying anything about the mutually cancelling variations in c and t. This condition is obviously met if atomic time t is used for any measurements. Since these same terms appear in the Schwarzschild metric, the same argument is valid there.

However, that is not all. Mermin [57] and Singh [58] have shown by an examination of the Lorentz transformation that relativity theory can be deduced without introducing c at all. Breitenberger also states “The special theory of relativity is shown to be independent of the assumption that the velocity of light, c, is a universal constant. … Existing theory-dependent arguments purporting to demonstrate the constancy of c are shown to be inadequate” [59]. In addition, at the time when c was measured as varying, Gheury de Bray drew attention to the fact that “Vrkljan has shown (Zeits. fur Phys., Vol.63, pp 688-691; 1930) that a decrease in the velocity of light is not in contradiction with the general theory of relativity” [60]. When all these facts are considered, it appears that light-speed declining isotropically throughout the cosmos poses little threat to the basis of relativity theory. Indeed, Magueijo recently pointed this out in Scientific American, saying, “…the urge to reconcile VSL [variable speed of light] to relativity is motivating much ongoing work… It now appears that the constancy of c is not so essential to relativity after all; the theory can be based on other postulates.” [J. Magueijo, “Plan B for the Cosmos,” Scientific American, January 2001, p.47].

The experimental evidence that demonstrates and confirms the invariance of hc, as noted in (5), has often been interpreted as setting limits on the variability of either h or c on a universal time scale. However, in each case an assumption is made about the constancy of the other term. The results that uphold (5) within the experimental limits say only that h must vary precisely as 1/c at all times. The results provide no information about any mutually cancelling variations in h and c individually.

The observational evidence documented in the Report showed that Planck’s constant h was independently measured as increasing over the same period when c was measured as declining. The data comprised 45 values of h measured by 7 methods over 64 years. Since the stringent astronomical data reveal hc as invariant, while the observational evidence indicated that h was increasing at the time when c was measured as decreasing, then the only possible conclusion from these data is that, at all times,

This result is further substantiated below in the preamble to (29) as well as in the Report. It can be explained via the SED approach since, as mentioned above, h is essentially a measure of the strength of the zero-point fields. This is confirmed by Haisch, Rueda and Puthoff: “In other words, in the SED perspective, h is an empirically determinable scale-factor for the strength of the ZPF” [10]. It follows then that if the energy density of the ZPE is increasing, so, in direct proportion, must h. The converse is that light-speed is inversely proportional to the energy density U of the ZPF so an increasing U causes a decreasing c. In other words, on the observational basis of (5) and (22), Eq. (19) can now be extended to read:

Thus Planck’s constant, as well as vacuum permittivity and permeability, are directly proportional to the energy density of the ZPF, while light-speed is inversely proportional to these quantities. Since U is a smoothly varying quantity, it follows that the other components of (23) will vary smoothly also.

4.7. Light-speed and the Early Cosmos

The issue of light-speed in the early cosmos is one that has received some recent attention. The Russian physicist V. S. Troitskii from the Radiophysical Research Institute in Gorky published a twenty-two page analysis in December 1987 regarding the problems cosmologists faced with the early universe [50]. Troitskii’s suggested solution involved the changing speed of light over the lifetime of the cosmos in association with some synchronously varying atomic constants. His solution included the idea that, at the origin of the cosmos, light might have travelled at 10¹⁰times its current speed. He also concluded that the cosmos was static and not expanding, in line with Van Flandern’s observation quoted above [55].

In 1993, J. W. Moffat of the University of Toronto, Canada, published two articles suggesting a high value for c during the earliest moments of the formation of the cosmos, following which it dropped rapidly to its present value [61, 62]. Then, in January 1999, a paper by Albrecht and Magueijo (hereafter referred to as the A-M paper), entitled “A Time Varying Speed Of Light As A Solution To Cosmological Puzzles” [63], received some prominence. These authors demonstrated that a number of serious problems facing cosmologists could be solved by a very high initial speed of light.

Like Moffat before them, Albrecht and Magueijo isolated their high initial light-speed, and its proposed dramatic drop to the current speed, to a very limited time during the early moments of the cosmos. However, John D. Barrow, of the University of Cambridge took this concept one step further in a paper published simultaneously with the A-M paper. He proposed that the speed of light has dropped from the initial value proposed by Albrecht and Magueijo (some 10⁶⁰ times its current speed) down to its current value over the lifetime of the universe [64].

The A-M and Barrow papers discuss a possible alternative to inflationary cosmology. Whereas inflationary models rely upon a modification of the matter content of the universe to drive the superluminal expansion, the A-M and Barrow papers rely on a change in the speed of light for the early cosmos. The models in both these papers seek to provide solutions to the horizon, flatness, and cosmological constant problems in the Standard Big Bang model of the universe. These results were summarised in one sentence by an editorial comment to another article by Barrow [65]. It read: “Call it heresy, but all the big cosmological problems will simply melt away, if you break one rule, says John D. Barrow – the rule that says the speed of light never varies.”

Importantly, both the Barrow and A-M papers use a principle of “minimal coupling” to determine the effects of changing c on other atomic quantities, and rarely rely on the observational evidence. By contrast, the model presented here is more experimentally oriented and primarily concentrates on the observational evidence of what happened to the cosmos after its origin and the ongoing behaviour of the atomic constants. Despite the different approaches, however, all these papers provide a framework for examining the effects of c-decay on the behaviour of the cosmos since its origin.

Throughout the rest of this paper c will denote the current value of light-speed, “c now”. The emitted value of light-speed at any given time will then be defined as c.

The theoretical and experimental evidence mentioned above has shown that light speed c will vary smoothly and inversely with vacuum energy density U as in (23). By contrast, atomic processes are often quantised. The photoelectric effect, atomic orbits, and wavelengths are each described by quantum integers and discrete energy values. This implies that a smoothly increasing vacuum property, such as U, e, or m must exceed a precise threshold value before the atom undergoes this new quantum energy change. This systematic difference between successive quantum threshold values will have been accompanied by a smooth change in the value of c by some set amount, Dc. Although light-speed is not quantised itself, this precise change in c between the quantum transitions may be referred to as a ± Dc variation. With each Dc change, the quantum number n changes by one. Thus the higher the value of c, the greater will be the value of n. Thus, at a given quantum jump n, c will be such that c = n(Dc) while at the precise moment of the next quantum jump c = (n + 1)(Dc). An equation can therefore be derived that links Dc, n, and c for distant astronomical objects. Since Dc is a fixed value, and the measurements of c indicate that light-speed itself is not quantised, it can be stated that, between jumps

where n is the quantum integer as before, and c is any value. This equation will hold at the quantum jump itself if at that point Y = 1. This will also mean that

Note that n(Dc) changes by quantum jumps so that Y is the value of a discontinuous function. Variations of c between the integer values of Dc will then produce the corresponding variation in the value of Y, which will thus vary within the range,

As c increases, instead of Y deviating from the interval of (26), n will shift to the next higher quantum integer. At that quantum jump, Y reverts to 1 again. In the limit, as n tends to infinity, Y approaches 1. Astronomically, the relative magnitudes of c and Dc give high values for n. Therefore, the following equation is a close approximation for distant astronomical objects:

(where the symbol » is used to mean “approximately equal to” throughout this paper.)

Accordingly, it can be stated that when a smoothly varying vacuum energy density exceeds a certain threshold value, atoms will undergo a quantum energy change given by the quantum integer n in (13). Furthermore, between successive quantum changes, there will also have been a precise ± Dc variation. It is only when the full variation of ± Dc is reached that atomic orbit energy jumps will occur, and, as a consequence, the energy of photons at emission.

5. THE TWO TYPES OF ATOMIC BEHAVIOUR CONSIDERED

By following through the implications of the above variations, it becomes apparent that two different effects on the atom need to be considered: the conservation of energy within a quantum interval, and then the results of the quantum jump.

First, because no extra energy from the ZPF is accessible to the atom until a full quantum threshold has been attained, atomic processes will proceed on the basis of the conservation of energy. There is no evidence that a quantum threshold has been exceeded recently. Consequently, observed atomic phenomena should be consistent with the conservation of energy within the quantum interval. Indeed, as detailed in the Report, the observed behaviour of the synchronously varying values of c and the associated atomic constants is only explicable if conservation of energy is occurring.

Second, the effect of each new quantum jump on the atom must be considered. In this case, a full quantum of additional energy from the vacuum becomes available to every atom in the cosmos each time the threshold is exceeded. Although this paper is primarily concerned with the second point, both these effects must be examined in turn.

Within a quantum interval between successive quantum jumps, energy will be conserved and atomic orbit energy levels remain unchanged. Many alternative scenarios had been explored in the time leading up to publication of the Report. However, the observational evidence of the behaviour of the various atomic quantities compelled the adoption of an approach based on energy conservation in the Report, and within the quantum interval in this paper. This approach, based on energy conservation, was the only one that agreed with the behaviour of all the data. This contrasts with the minimalist position adopted by both Barrow [64] and the A-M paper [63]. Within the quantum interval, then, particles of mass m or photons of wavelength l at the moment of emission will each have a constant energy E that will obey the equation

Now (20) and (21) show c and f vary proportionally, with l unchanged so that (28)

This is borne out by measured values of h and m throughout the 20^th century. Data details are recorded in the Report. Note that h is considered to be a measure of the strength of the ZPF in the SED approach [25, 10]. Therefore, the fact that h has actually been measured as increasing with time also implies that the strength of the ZPF has increased with time.

The next step is to note that the frequency of light emitted from an atom is directly related to v, the tangential velocity of particles in their orbits. Indeed, for the first Bohr orbit, “… the frequency of light emitted is identical with the frequency of revolution in the orbit, which corresponds exactly to the classical requirements” [F. 114]. It has already been noted in (21) that frequency of emitted light is proportional to light-speed. This statement by French therefore implies that, within a quantum interval with changing c, and hence f, the rate of revolution of the electron in its orbit will also be proportional to c. This becomes evident by comparing (6) with (21), (29) and (30) such that

In summary, because energy is conserved within the quantum interval, Eqs. (6), (7), (21) and (29) to (31) demand that, within each interval

In order to visualise what is happening both within the quantum interval and at the transition or jump, the behaviour of these and other quantities have been graphed in Figures 2 to 8.

Now while (14) to (20), as well as (5), (22), (23) and (29) remain valid for all values of c, including the quantum changes themselves, (28) and (30) to (32) only remain valid within the quantum interval. In other words, at the quantum jump E and m undergo an instantaneous, discrete, and predictable change in value as the additional energy becomes available to each atom from the vacuum.

Again, orbital energy is conserved within a quantum interval. Therefore, since r is invariant, these conditions require that for orbit energy E_k to be conserved in (7) and (8), then within the interval

Since e is proportional to vacuum energy density U from (23), a smoothly increasing energy density not only requires a smoothly increasing e, but (33) also requires that, within the quantum interval,

so that e² is changing synchronously with e. Consequently, since many atomic formulae include the ratio e²/e as a single identity, no change in e² or e can be measured separately. It should be noted that the fine structure constant, which is defined as a = e²/(2hce), is usually considered to be the measure of the electronic charge in “natural units” [66]. Since hc is invariant from (5) and (29), while e²/e is constant as given by (33), then a will not change within the quantum interval. In other words, the value of the electronic charge in natural units is constant between quantum jumps.

At the precise moment of the quantum jump, there will only be an infinitesimally small change in the energy density U of the ZPF. This also means from (23) that there will be essentially no change in the value of m, e, h, or c at the quantum jump, as these are all smoothly varying quantities. As a result, Eq. (6) for the old Bohr quantum condition reveals that, at the new quantum jump, the quantity nh/(2p ) must remain unchanged as both the Bohr quantum number n and h are unchanged. Therefore, the other side of the equation (mvr) also remains unchanged. In other words, at the new quantum jump, atomic angular momentum is conserved. Now since r is invariant, as already noted, then mv must also be a constant at the jump. But, as noted in (13), there is in fact a discrete instantaneous change in atomic orbit kinetic energy E_k at the new quantum jump. From (8) this means there must be an instantaneous change in mv². Consequently, from (6), m and v must change in such a way that

where the subscripts 1 and 2 refer again to the quantity before and after the jump with time increasing. Furthermore, as shown above, there will also be a discrete instantaneous change in wavelengths l emitted at the quantum jump. Since there is essentially no change in c at the quantum jump, it then follows from the basic equation c = f l that, for emitted frequencies at the new quantum change with time increasing,

Accordingly, since particle velocities, v, are directly proportional to frequency f, then, like f in (36), they must undergo an instantaneously discrete change at the quantum jump. Now the redshift observations demand that, with increasing time, the new wavelength l₂ is shorter than l_1. Consequently, from (36) it follows that f₂ is greater than f₁, while v₂ must be larger than v₁, and m₂ will be less than m₁. Therefore, from (35) and (36) it can be stated that

m₂v₂v₂/(m₁v₁v₁) = v₂/v₁= m₁/m₂= f₂ /f₁= l₁/l₂ = b (37)

where b > 1. In order to establish the composition of the term b in (37) it need only be pointed out that

l₂ = l₁ - Dl₁ therefore l₂/l₁ = (1 - Dl₁/l₁) = (1– z) (38)

where z is the redshift from (1). Now since l₁/l₂ is required in (37), which is the inverse of (38), and since z is very small, then to a close approximation it can be written that

Remembering that there is essentially no change in e at the quantum jump, a similar approach to (8) reveals that for orbit energy E₁ before the quantum jump and E₂ after the jump, with time increasing,

Therefore, from the definition of the fine structure constant given following (34), it follows that, at the quantum jump, with time increasing

It becomes apparent from (37) to (41) that the above changes in these physical quantities are themselves responsible for the observed quantised redshift. This argument may be turned around the other way; namely that the redshift of light from distant galaxies is in fact the prime evidence that these quantities are behaving this way.

The behaviour of electronic charge at the quantum jump may also be discerned, at least heuristically, by both the SED approach and the QED model. Of immediate interest here is the size of the electron. MacGregor has pointed out that there are seven possible values for this quantity [67]. They range from the Compton radius of 3.86151 ´ 10^–11 cm to those measured by high-energy scattering experiments that come out to something less than 10^–16 cm. The classical electron radius of 2.81785 ´ 10^–13 cm was obtained from calculations based on the mass being entirely electro-magnetic in origin [F. 66]. Interestingly, from the SED approach, Haisch, Rueda, & Puthoff point out that “one defensible interpretation is that the electron really is a point-like entity, smeared out to its quantum dimensions by the ZPF fluctuations” [10]. MacGregor initially emphasised that this “smearing out” of the electronic charge by the ZPF involves vacuum polarisation and the Zitterbewegung [67]. When Haisch, Rueda & Puthoff did the calculations for SED using these phenomena, the Compton radius for the electron was indeed obtained.

With this in mind, it might be anticipated, in the SED approach, that if the energy density of the ZPF increased, the “point-like entity” of the electron would be “smeared out” even more, thus appearing larger. This would follow since the Zitterbewegung would be more energetic, and vacuum polarization around charges would be more extensive. In other words, the spherical electron’s apparent radius and hence its area would be expected to increase at the quantum jump with increasing energy density of the ZPF. Importantly, the formula for the classical radius links the electron radius with the electronic charge and its mass-energy. A quantum change in radius therefore means a quantum change in charge, if other factors are equal. Therefore, at the quantum jump, when a full quantum of additional energy becomes available to the atom from the ZPE, the electron’s radius, and hence its area, would be expected to change and the electronic charge increase. This suggestion about the behaviour of the point-like spherical electron receives some backing by Boyer’s comment quoted in [68], namely that “the quantum zero-point force also expands the sphere”.

In the QED model there is a cloud of virtual particles around the “bare” electron interacting with it. When a full quantum increase in the vacuum energy density occurs, the strength of the charge increases. With a higher charge for the “point-like entity” of the electron, it would be expected that the size of the particle cloud would increase because of stronger vacuum polarisation and a more energetic Zitterbewegung. This larger cloud of virtual particles intimately associated with the “bare” electron would then give rise to an increase in the perceived radius of the “dressed” electron and its apparent area since both include the particle cloud. This inevitably means that the virtual particle cloud partially screens the full value of the “bare” charge. This has received some experimental backing. In the QED approach, Levine et al. have experimentally determined that the strength of the electronic charge is indeed modified by the cloud of virtual particles that shield the charge in the core [69]. As they probed deeper into this cloud, they found that the value of the electronic charge in natural units, namely a, increased from 1/137 to 1/128.5. Under these conditions Levine et al. concluded that the full value of the “bare” charge has yet to be determined [69].

However, as a result of (33), (34), (37) and (40), it is possible to examine the behaviour of the electron more precisely. It has already been noted that the fine structure constant is a measure of the electronic charge in natural units. By this criterion, there is no change in the electronic charge within the quantum interval, while (41) reveals an instantaneous change at the jump. There is another way of examining the electron behaviour through the classical electron radius r_c. This is defined [F. 67] as

Since e²/e and mc² are both constant in the interval, then from (42) it follows that r_c will also be constant in the interval. However, at the jump, it has been noted that both c and e are essentially constant. Therefore, as a result of (37) and (40) above, the following equation can be written:

r₂/r₁ = (e₂/e₁)²(m₁/m₂) = b ² = (1 + 2 z + z²) » (1 + 2 z) (43)

where r₁ and r₂ are the classical radii of the electron before and after the jump with time increasing. Thus the classical electron radius increases at the jump, with an associated increase in charge, in line with (41), and a decrease in electron mass.

The mass of atomic particles has entered the above discussion in several instances and (30) reveals that, within the quantum interval, m µ 1/c². Recent work in the SED approach supports this result. Note first that French points out [F. 66] “The very small value of the electron mass led physicists to suggest that the whole of this mass was electro-magnetic in origin.” Similarly, Ford states that “The mass of the electron can reasonably be attributed entirely to electromagnetic self interaction” [28]. Under these circumstances it would come as no surprise to find that the properties of the electron were governed by the electromagnetic ZPF of the vacuum. Indeed, using the SED approach, Haisch, Rueda & Puthoff (HRP) were able to show mathematically that “when an electromagnetically interacting particle is accelerated through the ZPF, a force is exerted on the charge; the force is directly proportional to the acceleration but acts in the direction to oppose it. In other words, the charge experiences an electromagnetic force as resistance to acceleration” [70]. Then, in 1997, HRP demonstrated that all “inertial mass may be due to a Lorenz force-like electro-magnetic interaction between charge at the quark or fundamental lepton level and the ZPF” [10, 148].

They went on to state that “If inertial mass m_i originates in ZPF-charge interactions, then, by the principle of equivalence, so must gravitational mass m_g.” They then noted that the ZPF causes the Zitterbewegung, where charged particles oscillate ultra-relativistically. Their calculations reveal that the energy of the Zitterbewegung can be equated to gravitational mass m_g. As Puthoff [36] stated “It is therefore simply a special case of the general proposition that the internal kinetic energy of a system contributes to the effective mass of that system” (see also Bohm [71]). It is then shown by HRP that the relationship between m_g and electrodynamic parameters is identical to the inertial mass m_i [10]. More recent work has confirmed this analysis and has extended it to include particle rest-masses as well [148].

It might therefore be anticipated that as the energy density of the ZPF increased, electron inertial mass, rest-mass, and gravitational mass would also increase. The reverse would also be true if the energy density of the ZPF decreased. In fact, HRP point out that, in a Casimir-type situation with light-speed perpendicular to the plates being greater than c, “inertial mass should also change [lessen] under such circumstances” [10]. This is in line with (30) and the experimental data listed in the Report. In their mathematical analysis HRP show that inertial mass (which corresponds to the “dressed” mass of the sub-atomic particle in question) is given by [10]

where w is the cutoff frequency of the electromagnetic ZPF which, on this present work, is proportional to c from (21). Therefore w ²/c² is constant within the interval. The definition of the damping constant G was given by Puthoff as [36]

The proportionality can be made because E is an energy term associated with the “bare” mass of the sub-atomic particle, and so is constant, as is e²/e. Now from (22) and (29) Planck’s constant h is proportional to 1/c in the interval. Therefore, it follows from (44) and (45) that

in the interval, just as derived in (30). So SED theory does not contradict the results obtained here.

The situation with regard to m_i at the quantum jump now needs to be considered. As noted above, there is essentially no change in the terms c, h, or e in (44) and (45) at the jump. Thus, from (44), it follows that, at the jump

As noted in (36) frequencies undergo a discrete instantaneous change at the jump. It can therefore be written that

From (47), this leaves the status of G at the jump to be considered. Examination of the definition of G in (45) reveals that it will be affected by the instantaneous discrete change in the value of e²in line with (40). However, G is a radiation damping constant that is directly related to the ZPF induced oscillation frequency of the electron. As a result, it will vary in inverse proportion with the radiation damping of this oscillatory motion. This radiation damping acts on the full area of the spherical electron, and the electron radius increases at the jump. Therefore it follows that G will change in inverse proportion to the square of the radius. When this effect from (43) is combined with the change in electronic charge from (40), then it transpires that

G₂/G₁ = [(e₂/e₁)²]/[(r₂/r₁)²] = b /b ⁴ = 1/b ³ (49)

which is in accord with (37). Although these results have been derived for the electron, they are also applicable for masses in general, since all matter is made up of quarks which each have charge and so will behave under the influence of the ZPF in basically the same way as electrons do. Thus SED formalism allows the reproduction of earlier mathematical results for the behaviour of masses, both in the quantum interval and at the jump.

There is one important conclusion from this treatment of sub-atomic particle behaviour at the jump. It was noted that the term E in (45) represented the “bare” mass-energy of the particle. This has already been determined as being constant in the quantum interval. This treatment at the jump reproduces the results obtained in (37) in a manner that indicates the “bare” mass-energy E of sub-atomic particles does not change at the jump either. Therefore, since c is essentially constant at the jump, and E is constant at the jump, then so is the “bare” mass, m₀. For the quantum jump, then, it can be written that

In other words, the “bare” mass-energy of subatomic particles is invariant at all times. Thus

Given the equivalence between gravitational mass m_g and inertial mass m_i, the “bare” mass m₀, though unobserved experimentally, is the quantity that could be said to be interacting at the level of the ZPF. In this context, then, G is the operative parameter of significance. Therefore, the changes in the gravitational and inertial masses at the jump come from the action of the ZPF on the changed charge and radius of the particle.

The behaviour of the Newtonian gravitational constant G with time must also be considered. In Section 4.4 and 4.5 above, and in the Report, the observational evidence suggests that dynamical time T runs at a constant rate, while atomic time intervals t are proportional to 1/c. It therefore follows that ct = constant as pointed out in the discussion following (21). Consequently the following equation can be written

and t is a dimensionless conversion factor. It is therefore possible to convert from atomic time t to dynamical or orbital time T using this factor t. Note that on this formulation t = 1 when c equals c, so that t is always in terms of the current value of c.

Obviously, all atomic constants are associated with the atom. Not so obviously, the Newtonian gravitational constant G is also associated with the atom, since it is needed to obtain the Planck mass, Planck length, and Planck time [72, 73]. For example, the Planck mass m_p is constructed as follows:

As Planck himself noted, this system of units is free from complications arising from elementary particle structure, atomic and molecular constitution, and problems associated with solid-state physics [29].

Since the Newtonian gravitational constant G is involved in the Planck length, mass and time, it too must be an atomic constant. As such it bears a proportionality G µ c⁴ as shown in the Report, and is revealed here by a comparison of (5) and (30) with (54). Similarly, the Sakharov-Puthoff derivation [36] has G = 2p ²c⁵/(hw_c²) in their SED approach, where w_c, the ZPE cutoff frequency, is itself proportional to c. Planck’s constant h, again, is proportional to 1/c. By comparison, Einstein [74] identified a gravitational constant g where g = 8p G/c² so that g µ c². However, while G is an atomic constant, it is also used in orbital mechanics on an astronomical scale. It is thus an important link between the atomic and astronomical domains, and, consequently, it then links atomic and orbital time.

Strictly speaking, then, the gravitational laws should be formulated in atomic time t and then converted to dynamical/orbital time T using (53). Basically, Newtonian mechanics allows us to write the following equation for orbiting astronomical bodies [75]

where R is the circular orbit radius, m is the mass of the planet (or any secondary body) with angular velocity w, and M is the mass of the sun (or any primary body). Note that the planet mass m drops out of the equation, since it appears on both sides of the equality. Since the mechanics are being formulated here in atomic time, the angular velocity of the planet about the sun is w = 2p /t. Substitution for w in (55) then gives

(GM)/R² = R(2p /t)² so that GM/(2p )² = R³/(t²) (56)

This can now be converted for astronomical bodies whose orbit period is required in dynamical time T by substituting t = T/t from (53) to obtain

which is the dynamical expression of Kepler’s Third Law. Here, G* is the gravitational ‘constant’ to be used in dynamical time when both c and t are varying. This dynamical gravitational constant is defined as

Since the Report concluded that G µ c⁴, then from (53) and (30) it follows that

within the quantum interval. As formulated here, within the current quantum interval, the dynamical gravitational constant G* still holds the same value and units as Newtonian G in atomic time, namely G* = 6.675 ´ 10^–8 (cm)³/[(gram)(sec)²], since the quantity t = 1 when c equals c, and t is dimensionless.

In a similar fashion, g, the acceleration due to gravity, may also be derived in terms of G* rather than G. The results are simply stated here. Let the earth’s mass be M and its radius be r. If there is a small mass m at the earth’s surface, then, from reference [75], the attractive force between the earth and mass m is given dynamically by

within the quantum interval on the basis of (59). Therefore, all orbital periods and gravitational accelerations will remain unchanged by c variation within the quantum interval as shown in (57) and (60). In addition, as the dynamical G*M and the Einsteinian g M appear as single identities in all our equations, no variation in G* and M separately or g and M separately can be found by experiment within any quantum interval.

As far as the quantum jump is concerned, consider the formulation of the “bare” mass m₀. According to Puthoff this can be written [36]

where m is the particle mass, which may be either m_i or m_gas discussed above. Now from (5) hc is invariant, and from (51) m₀ is constant at the jump. Therefore it follows that, at the quantum jump

Furthermore, since the jump occurs essentially instantaneously, and since c is unchanged, then from (53) it follows that t remains unchanged also. Combining this with the results from (59) means that, at the jump

Since it has been established already that the quantities G*m and g m are constant within the interval as well, then it follows that at all times

Accordingly, it can be stated that gravitational acceleration and orbital phenomena remain unaffected by changes in particle masses, the ZPF, and the associated atomic constants. The natural outcome of (64) on the SED approach is that particle mass and gravitation fields must be entirely related to the ZPF interactions with the charges on quarks and electrons. It has already been pointed out that the energy associated with the Zitterbewegung itself gives rise to gravitational mass, and that inertia is also an electromagnetic phenomenon involving interactions between charge and the ZPF. This has stood up well to analysis. Gravity must therefore be a related phenomenon. HRP [70, 10, 76] have worked with some success on this line of inquiry initiated by Sakharov [77] and later by Puthoff [36]. As a result, they have shown that the ZPE in and of itself does not generate a gravitational field; neither does a point particle “bare” mass such as described in (61).

Issues related to energy emission in its various forms are sometimes seen as potentially problematical for this model. Accordingly, it is appropriate to briefly examine these matters here. One overriding factor is discussed first, namely the energy density of emitted radiation, as distinct from the ZPE itself. A photon of light with wavelength l has an energy given by the standard relation hc/l. As noted in (5), (22) and (29) above, hc is invariant. Therefore, a photon of given wavelength l will have the same energy at all times irrespective of c variation. Consider a situation in which a laser emits photons with a wavelength l such that each photon has an energy of 1 erg. Furthermore, let this source be so manipulated that it consistently emits one photon per dynamical second. Let the experimental arrangement be such that this photon enters an extended cylinder of 1 square centimetre in cross section and c centimetres long. Then the rate of flow of electromagnetic energy through any cross section of the cylinder is 1 erg per second per square centimetre. This rate of flow of energy is described by the quantity S in the standard equation S = cr where r is the energy density of the radiation in ergs per cubic centimetre. If S is to remain constant, as our experimental arrangement ensured, then as c varies it necessarily requires the energy density of radiation r to follow the equation

Thus the energy density of electromagnetic radiation in the cylinder is proportional to 1/c.

This may be examined in another manner. The energy density r of electromagnetic radiation is also given by the standard equation in references [38, 39] and [F. 40]

The simplest electromagnetic wave is described by a sine function as in the preamble to (14). The peak amplitude A₀ of this wave is such that E² = ½A₀². This follows because the factor of ½ is the average of the square of the sine over all angles. Hence the energy density of an electromagnetic wave is related to its amplitude [38] as shown in (66). However, since the field strengths of both the electric field E, and the magnetic field H are constant with changing c, this means that the wave amplitudes A₀ are also unchanged. Therefore, since both e and m are proportional to 1/c from (15) and (19), it then follows that the energy density of the electromagnetic wave r is again proportional to 1/c as in (65).

If there has been an increase in e and m with time, which means a decrease in c, then the energy density of electromagnetic radiation would also increase with time. Therefore, the higher the value of c was in the past, the lower was the energy density of radiation. There are no quantum jumps in this process since both e and m are smoothly changing. Nevertheless, insofar as redshift and c are associated, this effect can be considered to be redshift dependent.

There are two main factors in determining the luminosity of a star in circumstances of varying light-speed. One of these is the rate of photon production, which depends on the rate of reaction for the nuclear fuel that powers the star. The other factor is the star’s opacity, which inhibits the transport of energy from the nuclear reaction centre of the star out to its exterior where the energy radiates into space. There are four key processes that will potentially contribute to a star’s opacity: free-free transitions, bound-free transitions, bound-bound transitions, and scattering by electrons. Importantly, Harwit notes that the bound-bound transitions play a negligible part in stellar interiors [78]. In addition, the statement has been made that for “typical stellar interior densities, electron scattering will predominate over bound-free and free-free absorption for temperatures in excess of, say, some 10⁷ °K” [79].

For the central area of stars, this leaves electron scattering of photons as the prime source of opacity. If n_e is the number of electrons in unit volume and r_m is the mass-density [80, 81], then the opacity resulting from scattering by free electrons k_e is given by [78, 82]

Here, s is the Thomson scattering cross-section where s = (8p /3)[e²/(e m_e c²)]² which is constant, as is n_e. However, r_m is proportional to 1/c² from (30), so the end result is that the opacity from electron scattering is proportional to c² as shown in (67). For some stars, the interior conditions are not favourable for the high energy Compton or inverse Compton effects to be significant [83]. Nevertheless, Cox and Giuli [84] quote a formula derived by Sampson for the Rosseland mean opacity for Compton scattering by free electrons in the non-degenerative limit. Since it differs from (67) only by a dimensionless number dependent upon temperature, it too shows a similar c² proportionality.

Harwit states “hydrogen and helium do not contribute significantly to the bound-free transitions” [85]. Instead, in the relevant equation, a factor Z is included which is the metal abundance expressed as a fraction of the total mass, which will be constant for all c. In the resultant equation that describes Kramer’s Law of Opacity for bound-free absorption, Harwit has detailed the terms that make up the numerical co-efficient [86]. When these are included, the following equation results:

k_bf = (2/3)Ö(2p /3)[e⁶h²r_m/(e ³cm_H²m_e^3/2)][Z(1+ X)/(kT)^7/2][kT/(hf)]³ <g_bf >/F µ c² (68)

Here, k is Boltzmann’s constant, which is invariant for changes in c. The terms m_H and m_e are the masses of the hydrogen atom and the electron respectively [87] and are proportional to 1/c² from (30). The term F contains correction factors and other factors independent of c, and <g_bf> is the dimensionless mean Gaunt factor that is of the order of unity. X is also a dimensionless number, and T is temperature. When the c-dependent terms are analysed, the result is that the bound-free opacity is proportional to c², the same as for electron scattering. In reference [86], figure 8.3 reveals that the stellar conditions allowing free-free transitions are restricted. Nevertheless Harwit repeats the above process for these free-free transitions. The resulting equation gives a result similar to (68), but an (X + Y) term is included in place of Z. Schwarzschild lists the equation in full as [88]:

k_ff=(2/3)Ö(2p /3)[e⁶h²r_m/(e ³cm_H²m_e^3/2k^7/2)](1/196.5)<g_ff>(X+Y)(1+X)(1/T^7/2) µ c²(69)

By comparison with (68) it can be seen that the same c² proportionality for the free-free opacity results. Therefore, even though bound-free and free-free absorptions play a lesser role to electron scattering in the considerations here, it nonetheless appears that all stellar opacities will be proportional to c². It can therefore be written that the average value for the opacity of the whole star is

The other factor affecting a star’s luminosity is its rate of burning nuclear fuel. Nuclear reactions are temperature sensitive. As a result, the proton-proton reactions dominate at lower stellar temperatures with the carbon cycle becoming prominent later [89]. The key reaction that is needed to get the proton-proton sequence started is given by H¹ + H¹ ® D² + e⁺ + n. This is a beta process with a mean reaction time for any given particle of 14 billion years [89]. However, this reaction rate should be proportional to c as indicated by the treatment of Swihart [90]. He states that the reaction rate per unit volume is given by

N_r = {4h²DE^1/6N₁N₂/ [M^3/2(kT)^2/3]}e ^{–1.89[(E/(kT)]^}^1/3 µ c (71)

Here the temperature is T, Boltzmann’s constant is k, and Planck’s constant is h. N₁ and N₂ are the numbers of interacting particles per unit volume, M is the reduced mass of the two reacting particles, D is the probability of reaction between nuclei, and E is the energy required to penetrate the Coulomb barrier. In this equation, the only two factors that are c-dependent are M and h. Their ratio, h²/M^3/2, is proportional to c. Stellar nuclear reactions are thereby proportional to c as shown in (71) and so will be the photon production rate.

When photon production rate and opacity are then considered together, the luminosity of a star under conditions of varying c should be established. The key formula for stellar luminosity is given by Chandrasekhar as [91]

Here G*M is defined in (59) and (64) and so is invariant with changing c, while M is the total mass of the star. The term g is a ratio of pressures while h is a ratio of energies, so both are thereby dimensionless and independent of c. Since k_A is proportional to c² from (70), it follows that the luminosity L_s in (72) is proportional to 1/c. This result, however, is for an unchanged nuclear reaction rate per unit volume. To accommodate a changing nuclear reaction rate with varying c, the c-dependency of N_r needs to be included. To do this the c-dependent dimensionless factor t from (53) will be employed again so that the luminosity equation reads:

There is an alternative formulation by Schwarzschild [92] that may also be relevant. He firstly defines the fraction of energy of the beam of light that is lost by scattering or absorption over a distance dr. This is given by the term k¢ multiplied by dr where k¢ bears the same proportionality as k_A. The equation for the luminosity at temperature T then becomes

Here the term a is given by the relationship a = 8p ⁵k⁴/[15(hc)³] which is constant as hc is constant and Boltzmann’s constant k is invariant. As in the case of (72), Eq. (74) is proportional to 1/c. But again that is for an unchanged reaction rate per unit volume. Therefore, as in (72), the c-dependent dimensionless number t needs to be included so that this luminosity equation will read for varying c:

As a consequence, stellar luminosities should remain essentially unaffected by conditions of varying c.

In a fashion similar to the analysis of (71), it can be shown that the equations governing the rate of radioactive decay each contain c-dependent atomic constants. These c-dependent constants all combine in such a way as to make the rate of radioactive decay processes proportional to c. This is fully discussed in the Report as well as an earlier paper [93]. In both papers, the analysis has shown that decay constants for alpha and beta decay, electron capture and spontaneous fission are all proportional to c. These results are simply summarised here by stating the result for all radioactive half-lives t_½ is such that

It has been shown that the range of alpha and beta particles remains unchanged with variations in c values [93]. The size of pleochroic halos will therefore remain unchanged as a consequence. However, with higher c values, x-rays and gamma rays will be redshifted at emission, and each photon will have a lower energy density as indicated in Eq. (65).

By contrast with the radioactive decay processes just discussed, those reactions needing slow neutrons, such as neutron-induced fission in uranium ores, may not be enhanced with higher c values. The reason is that this process is velocity dependent since the neutron is electrically neutral. Only slow neutrons are captured or absorbed, which is why these nuclides are called “1/v absorbers” [94]. Since neutron velocities v will be higher, proportional to c, it follows that the probability of capture or absorption for any given neutron should be retarded as a result. However, the flux of neutrons should also be higher. Analysis therefore suggests that the number of neutrons able to be captured or absorbed per unit time should be maintained at approximately the current rate.

It has been suggested, usually on the basis of collision theory, that chemical, and therefore biological, processes would be adversely affected by a changing speed of light. This is an important topic and can only be touched on briefly here. In most chemical reactions, a series of steps is involved, many of which occur rapidly. Indeed, if the old collision theory was truly correct, “all chemical reactions would be completed in a fraction of a second” [95]. It is now generally acknowledged that the collision theory “is of only limited usefulness” [96]. In actual fact, the overall reaction rate is generally governed by the slowest step in the reaction series, referred to as the “rate determining step” [97]. In this step it is common for an activated complex to be formed, which is then disrupted, yielding the products needed for the rest of the reaction series. Clearly, it is the physics of the formation of the activated complex that is of crucial importance here. The equation that describes this process [98] can be presented in the form

where k* is the reaction rate constant, and K^¹ is the equilibrium constant for the activated complex. The term A is the probability of reaction, or the chance of forming the activated complex. Since the rate determining step is a time dependent phenomenon, the chance of forming the activated complex in this step will also be time dependent.

From a consideration of masses and velocities starting with Eqs. (30) to (32), it can be shown that, at a constant temperature, the number of approaches per second by ions or charged molecules to a potential reactant is proportional to c. Let us designate this quantity by the symbol Y*. Because the velocity of these approaches is faster when c is higher, the time that each ion or charged molecule is in the vicinity of the potential reactant is therefore proportional to 1/c. Let us designate this quantity by the symbol X*. It can then be established that the chance of forming the activated complex, A, is equal to the number of approaches by ions to the reactant per second, Y*, multiplied by the time the ion spends in the vicinity of the reactant, X*. Thus we have the relationship that, for any given c value,

The chance of forming the activated complex at a given temperature is given by the equilibrium constant, K^¹, which describes the “equilibrium” between reactants and activated complexes. This remains constant within the quantum interval because the temperature and kinetic energy are constant. However, it is at this point that it becomes necessary to consider the effects at the quantum jump with time increasing. The basis of the effects can be found in Eq. (37) where it can be seen that the kinetic energy of the systems being considered here will increase by a factor b at the quantum jump. Note that in (78) any such b factors at the jump appear in both denominator and numerator and so cancel out leaving the result as stated. However, in the case of the equilibrium constant, the b factor, which indicates higher particle kinetic energies after the jump, means that the chance of forming the activated complex is greater after the jump than before. Therefore, the higher the redshift, the lower the chance of forming the activated complex. This means the reaction rate constant, k*, in (77) very gradually increases with time as c decays. These explanations will be expanded out in another forum.

7.1. Determining the Rydberg Quantum Number

In determining the basic redshift quantum nDz where n = 1, there are two factors involved. The first is the increase in available energy from the zero-point fields, and the second is the behaviour of the electronic charge when this energy becomes accessible at the quantum jump. The necessary equation, which describes the transference of energy from the vacuum ZPE to the electron, has already been derived by Puthoff [32, Eq. 16]. His analysis revealed that P_a (the power available for absorption from the random background zero-point field by a charged harmonic oscillator) is given by the expression

P_a = e²hw ³/(24p ²e m₀ c³)

(79)

However, in this equation the power available from all possible directions of propagation of the ZPF has been included, whereas on average only one third of the total energy of the field will be absorbed by such an oscillator [F. 82]. Therefore the utilisable power P_u available will be given by

The matter for investigation is the way the terms in (80) behave at the quantum jump. It has already been pointed out that h, c and e each remain essentially unchanged at the jump, as does the “bare” mass m₀ from (51). The oscillator resonance frequency is w = kc in Puthoff’s formulation where k is independent of c. Therefore, since c is a smoothly changing quantity, it will remain essentially unchanged at the quantum jump itself, and, consequently, so will w. These unchanging quantities therefore have no role to play in the quantum jump process here.

All the changes at the jump that have been considered in this paper are due to the change in the utilisable energy or power available from the ZPE. If E_u is the utilisable energy available for absorption from the ZPE, then E_u is proportional to P_u. Therefore, if the power maintaining the electron before the jump is P₁ and the power utilised after the jump is P₂ with time increasing, then it follows from (40) that

E₂/E₁ = (e₂ /e₁)² = b = (1 + z) = P₂ /P₁

(81)

As a consequence, (81) can be written as:

P₂ = P₁ + zP₁

(82)

Eq. (82) therefore indicates that the increase in P₁ at the quantum jump is given by the dimensionless fraction z. This requires z to be a dimensionless component of (80), that is 1/(72p ²).

However, since e² changes at the jump, the possibility of a dimensionless component of e² is also relevant. As shown above in (43) the electron radius increases at the jump. This means the area of the spherical point-like electron also increases. It is this surface area where the electron interacts with the ZPE, which in turn results in the increase in its charge. The electronic charge e is expressed in units of Coulombs, where one Coulomb is an Ampere-second. By definition, the Ampere is a force per unit length, which has the same dimensions as energy per unit area. If the electron’s surface area is represented by a, the following equation holds:

e = Coulombs = (energy/area) ´ time = (power/area) ´ time² = q/a (83)

where q contains proportionality factors. In this equation, the electron surface area a can be replaced by a = 4p r₀², where r₀ is the “bare” electron radius. The dimensionless component of e that emerges from this can be seen in the expression

where Q also contains proportionality factors. If the results of (84) are substituted in (80), then the utilisable power becomes

From (85), the full dimensionless component making up the value of the redshift quantum nDz when n = 1 can finally be written as:

[(1/4p )²] [1/(72p ²)] = 1/(1152p ⁴) = 1/112215 = 8.91144 ´ 10^{– 6} = Dz (86)

This procedure has also determined the Rydberg quantum number z since from (12) and (86) it follows that

There is a means whereby the numerical value of the dimensionless number z may be cross-checked. It has been pointed out in the preamble to (11) that, for a quantised redshift, the ground state orbit of the Bohr atom must have its emitted Rydberg wavelength W advance incrementally in steps of some set fraction of W, such as W/z = R*. In order to find the value of z by this method, it is important to examine the terms making up the value of W to discover their dimensionless components. From [F. 109] the definition of W coupled with (11) yields the following results:

It has been shown above that the change in wavelength occurs at the quantum jump. Furthermore, it has been pointed out that, at the jump, there is only an infinitesimally small change in e, c, and h. This leaves all the changes to occur in the denominator of (88) that combine to give the value of z. For this purpose it can be written that

It would be convenient to have the electron mass m in (89) in terms of electronic charge e and dimensionless numbers. In fact, the classical electron radius r_c defined in (42) directly links the electromagnetic mass of the electron with the electronic charge. From this expression, the electron mass m can be isolated as

However, the situation with regard to m here is not straightforward. To begin, French points out a very interesting anomaly [F. 63-67]. If the value of m is derived from the energy density of the electron’s magnetic field, a factor of 2/3 is required on the right hand side of (90). By contrast, French also noted that using the Poynting vector and the electron’s electric field, a value is obtained which includes a factor of Ö(8/3) on the right hand side of (90). Because of these conflicting results, French makes the comment that “…without some idea concerning the precise distribution of charge within the electron volume, we cannot be sure about numerical factors of order unity. It was therefore decided to define a so-called ‘classical electron radius’ r_c …” without any numerical coefficients at all. This problem has been discussed at some length by MacGregor who lists some six additional options [67]. However, if the Compton radius R_c = h/mc, as in Ford [99], is multiplied by the fine structure constant designated as a = e²/(2e hc), as noted in Ford [66], the result is a more precise expression for r_c. When the terms are re-arranged the expression for m becomes

The precision of (91) is now solely dependent upon the accuracy of the Compton radius. Since c and e take no part in the quantum jump, the following proportionality holds:

where (82) allowed a substitution for e in the second proportionality of (93). Furthermore, since the relationship r_c² = a/4p holds for the electron, then it follows that the dimensionless component of r_c that is relevant at the jump is given by

Then finally, if the dimensionless component of (94) is now substituted into (93), we find the value of z emerges as

This value of z in (95) differs from the value of 112215 determined in (87) by 0.3%. The difference probably originates in the precise correction needed for m, or in this case the Compton radius, as explained above and elaborated by MacGregor [67]. Because of this uncertainty about m and R_c and the more clean-cut nature of the result from (87), let us accept here the value of z, the Rydberg quantum number in that equation, namely 112215, as definitive.

These results allow the determination of the size of the redshift quantum jump. From Eq. (12) the following relationships hold:

In addition, when the approximation from (27) is included, the redshift may be expressed as

When n = 1 is substituted in (96), the basic redshift quantum jump Dz has been established. Now the redshift is often expressed as a Hubble velocity, V, by multiplying z by the current speed of light c. When this is done in (96), the quantum change in velocity emerges as

V = cz = (299792 km/s)(1)(8.91144 ´ 10^–6) = 2.671 km/s = c(Dz) ((98))

This compares favourably with Tifft’s basic value of 2.665 km/s. The process that Tifft used to come to this conclusion now claims our attention.

The initial redshift quantisation that alerted Tifft to the phenomena was 72.46 km/s in the Coma cluster [3]. It was later discovered that quantisation figures of up to 13 multiples of 72.46 km/s existed. Further work established a more basic quantisation at just half this figure, namely 36.2 km/s. This was subsequently supported by Guthrie & Napier who concluded that a quantisation at 37.6 km/s really did exist [100, 101]. This is very close to the 37.4 km/s that this model predicts. On continuing his investigation, Tifft announced in 1991 that these and other earlier redshift quantisation figures were numerically related to the 8.05 km/s quantisation [2]. Thus when multiplied by 9 the original 72.46 km/s was obtained, or when multiplied by 9/2 the 36.2 km/s value is obtained. However, after some statistical treatment of the observations, Tifft noted that the actual figure was 7.997 km/s rather than 8.05 km/s [2]. But even this was not the most basic result as data revealed an (8.05)/3 = 2.68 km/s quantisation which was even more fundamental. When the more statistically accurate 7.997 km/s replaces the suggested 8.05 km/s, the best observationally derived basic redshift quantum becomes (7.997)/3 = 2.665 km/s. This is close to the theoretical value obtained above of 2.671 km/s. Tifft’s primary values from observation [2] are compared with the theoretical prediction in Table 1 above.

Other quantum values can now be determined from Eqs. (96) to (98). These equations, in combination with (27) yield a Hubble velocity of

V = cz = n(2.671) km/s » (c/Dc)(2.671) km/s n = 1, 2, 3, ... ((99))

showing that the redshift value cz undergoes a discrete change when c changes by ± Dc. However, in astronomical work, observers are often dealing with redshift differences by comparing one redshift with another. If the model presented here is correct, it is the differences between quantum numbers that are being observed. Eq. (99) can then be amended to read:

Note that these measured differences in the cz values are here attributed to a changing ZPE and its effects on c and the associated atomic constants. Because these changes are isotropic throughout the cosmos, then (m - n) is generally a measure of distance, just as the value of n is when a given object is observed from earth. Eq. (100) can now be simplified by defining N as the quantum difference number given by (m - n) such that

One additional step is needed to generalise (99) to (101). Some observations involve whole sequences of redshifts with these N spacings. To accommodate this circumstance, a quantum sequence number L can be defined such that

Importantly, for N = 14, there exists a redshift sequence beginning with L = 1 with a spacing of 37.39 km/s. This value begins a quantisation or periodicity that has been noted throughout our Local Supercluster of galaxies by Tifft, as well as Guthrie & Napier. Tifft [2] assessed it to be 36.22 km/s, while Guthrie and Napier noted strong quantisation at 37.2 km/s [100]. Following that, a more exhaustive analysis by Guthrie & Napier yielded 37.6 ± 2 km/s [101]. They finalised their research by a study of 204 additional galaxies. The final plot of galaxy numbers against cz differences, out to a redshift velocity near 2500 km/s, resulted in a series of 65 regularly spaced peaks and troughs. This means the most distant galaxies surveyed in the Local Supercluster were about 125 million light years away. The power spectrum of galaxy redshifts revealed a strong signal at 37.5 km/s [102]. The value derived theoretically here (37.4 km/s) is very close to this observationally determined quantisation figure. In this regard, it might also be noted that much of Tifft’s later data came from 21-cm measurements at high signal-to-noise levels where random uncertainties were very small. Indeed, under these conditions, Lewis has claimed an accuracy in redshift measurements in excess of 0.1 km/s [

(103)].

A major conclusion from the quantised redshift is that the cosmos may not be expanding now. It is accepted here that the microwave background does point to an initial hot, superdense state that cooled by expansion. Even so, once the initial cosmological expansion had occurred, it appears that a position of cosmic stability may have been reached and maintained ever since. Narliker and Arp have shown that a static, matter-filled universe will remain stable and not collapse [

(104)

(104)]. The only condition that their field equations demand for this stability is that particle masses increase with time, which is the situation in this present model as demonstrated by (30) above.

There is a reason why the quantised redshift demands a static cosmos. In the case of Friedmann universes, it is generally accepted that the vacuum permittivity, e, is directly proportional to the Friedmann radius [

(105),

(106)]. This result is achieved by generalising Maxwell's equations to the coordinates of general relativity. It is therefore tempting to attribute the above changes in e (and therefore in c and the other atomic constants) to just such a variation in space-time geometry as an on-going effect of cosmological expansion. But an analysis of this conclusion reveals a problem.

The necessary analysis was performed by Sumner in 1994 [

(107)]. He points out that in an expanding Friedmann universe, atoms as well as photons are involved in the expansion. Thus, even though light photons in transit will be stretched (redshifted), the final observational result will still be a blue shift of light from distant galaxies due to the effects of expansion on the atom. Since a blue shift is not what is observed, a reasonable conclusion is that the cosmos is not expanding.

Even if one prefers to ignore Sumner’s analysis, a further problem emerges. Light photons in transit will be stretched (redshifted) if the Friedmann radius expands, or alternatively contracted (blue shifted) if the Friedmann radius becomes less. This would immediately obliterate or ‘smear out’ any sign of a precise redshift (or blue shift) quantisation: the photon z value at emission would alter in transit as the Friedmann radius changed. In other words, the very fact that this z quantisation exists at all necessarily implies that the Friedmann radius is fixed.

The fact that this z quantisation exists also implies that the redshift and its quantisation are not a velocity phenomena. Indeed, a large velocity component in cz values would destroy the quantisation effect. This point is emphasized by recent work on galaxy clusters which has revealed the significant information that in “the inner parts of the Virgo cluster, deeper in the potential well, [galaxies] were moving fast enough to wash out the [redshift] periodicity” [

(108)]. Thus, if the cosmos were expanding, this motion would also be expected to smear out the quantisation. As a consequence, it may be concluded that the cosmos has remained static after an initial period of expansion. This is not only in line with Narliker and Arp [104], but also with the treatment by Troitskii [50] and Van Flandern [55].

In relativity theory, the properties of the physical vacuum can be described by the action of a cosmological constant L. It might be thought, then, that any change in vacuum energy density might be due to the direct action of this cosmological constant. However, as Barrow and Magueijo [

(109)] point out “If L > 0, then cosmology faces a very serious fine-tuning problem…There is no theoretical motivation for a value of L of currently observable magnitude…” Furthermore, the problem of trying to incorporate L into ZPE theory has proved very difficult. In the first instance, Zeldovich pointed out that the numerical value obtained for L from any proposed vacuum and particle theory can disagree with observation by up to 10⁴⁶as observation suggests L » 10^–54 cm^–2 or lower [

(110)]. The same conclusion was reached via a different line of reasoning by Abbott [

(111)].

More recently, Greene noted that “…the cosmological constant can be interpreted as a kind of overall energy stored in the vacuum of space, and hence its value should be theoretically calculable and experimentally measurable. But, to date, such calculations and measurements lead to a colossal mismatch: Observations show that the cosmological constant is either zero (as Einstein ultimately suggested) or quite small; calculations [based on QED theory] indicate that quantum-mechanical fluctuations in the vacuum of empty space tend to generate a nonzero cosmological constant whose value is some 120 orders of magnitude larger than experiment allows!” [

(112)].

This might, perhaps, indicate that the cause of the problem exists in the QED approach, and that SED formalism has an assessment of the situation that is more in keeping with observation. This line of thinking seems to be confirmed by Haisch and Rueda who noted that [

(113)] “the ZPF cannot be the manifestation of a cosmological constant, L, or vice versa.” Indeed, they emphatically point out that “The ZPF is NOT a candidate source for a cosmological constant. The ZPF…can have nothing to do with L and is not, of itself, a source of gravitation…Gravitation is not caused by the mere presence of the ZPF, [but] rather by secondary motions of charged particles driven by the ZPF. In this view it is impossible for the ZPF to give rise to a cosmological constant” [113].

When all these comments are coupled with the statements by Narliker and Arp that L is not needed to maintain a static, matter filled cosmos [104], it is possible that the cosmological constant has no current role to play in any change in the energy density of the ZPE. Furthermore, if, as the quantised redshift suggests, the Friedmann radius of the cosmos is not currently changing, then this, too, is removed as an explanation for any change in vacuum properties as well. The interpretation of the most recent observations of distant Type Ia supernovae is that the cosmological constant is accelerating the expansion of the universe. However, these observations may have a different explanation as explained below.

Under these circumstances, any change in e, c, and/or the atomic constants through an increase in the energy density of the ZPF requires an explanation as to why the ZPE should increase with time. Importantly, Boyer [24] and Puthoff [32] demonstrated that the ZPE prevents the radiative collapse of the Bohr atom. Granted that point, a logical non sequitur then appears in another paper by Puthoff, where he looks upon particle motion and the ZPE as “...a self-regenerating cosmological feedback cycle” [37]. If that is true, it would be helpful to know how this cycle started after the Big Bang before the radiative collapse of all atoms in the cosmos occurred. This problem suggests that the ZPE may originate in a manner different from Puthoff’s preferred option of a feedback cycle. His alternative option, namely that the vacuum energy came from the conditions at the origin of the cosmos, may therefore be worthy of examination. Considerations that lead to a discussion of that option introduce the final aspect of this paper: the light-speed quantum and its distance relationships.

From (25) and (27) it can be seen that, for high values of n, the function Y will always be very nearly equal to 1, even within the quantum interval. With (25) and (27) in mind, the following relationship exists between n, z and Dc from (96) and (97):

z = n(8.9114 ´ 10^–6) = [c/(YDc)](8.9114 ´ 10^–6) = c/K (103)

In Eq. (103), K = YDc/(8.9114 ´ 10^–6). Since the value of z ranges beyond 6, which gives a value for n of about 690,000, it can be seen that n will be a large number for most astronomical objects. Since Y is then effectively equal to 1, the approximation in (27) essentially becomes an equality. Under these conditions, (103) can be written in its approximate form where Y = 1, which will hold for most practical purposes as

where k is a constant since Dc is a constant as outlined in the preamble to Eq. (25). Since the value of Dc will be in terms of the current value of c, namely c, it follows that the value of c obtained from (104) will also be in those terms. If the value of the quantum number n is required, a convenient form of (103) and (104) is as follows:

Eqs. (104) and (105) therefore express the relationship between redshifts, z, the quantum number, n, and the speed of light at the time of emission, as a multiple of c. Distance now needs to be tied into these equations.

Throughout this section, the light-year, LY, is often used. The other unit of distance is the mega-parsec, Mpc, equivalent to 3.26 million LY. The Hubble velocity V of an astronomical object is given by V = cz, so on the basis of Eq. (1) the redshift may be written as

Since (104) links z and light-speed at emission, (106) can now be converted to include these factors. To that end, (104) is multiplied by c to obtain a Hubble velocity in km/s.

cz = V = cDl/l = (299,792)[c/(Dc)](8.9114 ´ 10^–6) = [c/(Dc)](2.671) (107)

The conversion of z to a velocity as in (106) allowed the redshift Dl/l to be interpreted as an effect of a recessional velocity by the classical Doppler formula for light [

(114)]. This implied a linear redshift/distance relationship, a position which astronomers held up until 1953 at least [

(115)]. Later, distant astronomical objects were found with z ³ 1 which implied V ³ c in (106). Consequently, the full relativistic Doppler formula (as in [

(116)]) was applied and became the standard. That resulted in a non-linear redshift/distance relationship for objects beyond about z = 0.2, and an age for the cosmos that depended on our local value for the Hubble constant H₀ [

(117),

(118)].

However, the model in this paper suggests that some re-examination may be necessary. As shown by (96) to (98) the actual redshift of Dl/l may not be the result of cosmological expansion at all, but rather an expression of changing atomic orbit energies over time. This would negate the original assumption that Dl/l resulted from a Doppler velocity shift. Nevertheless, the convenience of calling V the Hubble velocity is retained here, but on the understanding that it may not be a true velocity at all [

(119)].

Despite these developments, the general form of the redshift/distance relationship is considered to be reasonably well established. On that basis, a graph of redshift z on the y-axis against distance in LY on the x-axis can be drawn as in [117] and [118]. The main point of contention is the precise distance scale that should be employed. To overcome this problem on the horizontal axis, the values there can be arranged to go from x = 0 near our own locality in space, to x = 1, the furthest distance in space, close to the origin of the cosmos. On this approach, the redshift/distance relationship may then be expressed as

The total distance in LY at x = 1 is then dependent upon the value of the Hubble constant H_o. Work on re-calibrating the distance scale using Cepheid variables and Type Ia supernovae resulted in a recent assessment by Burrows that H₀ ~ 64-69 km/s per Mpc which, he stated, is “amply within the error bars of competing techniques” [

(120)]. If the middle of this range is chosen, namely 66.5 km/s per Mpc, this gives a value for the furthest distance, at x = 1, of about 15 billion LY.

One of the issues discussed in this paper has been the relationship between the redshift, z, and the speed of light, c. This has resulted in Eq. (103) along with its close approximation in (104). This equation can be expressed in the alternative form

Therefore, the y-axis of the redshift graph can changed to read c instead of z if the scale is changed by the factor k. Furthermore, since increasing astronomical distance means that we are looking further back in time, there should also be a direct relationship between astronomical distance x in (108) and dynamical time T. Indeed, if dynamical time is expressed on the horizontal axis of a graph from T = 0 to T = 1 in the same way that the distance, x, is, then from (108) and (109) the equation for c behaviour would then become

Thus the curve for the standard redshift/distance relationship is the same as that for the relationship of light-speed/dynamical time, except that the resulting graph will have re-scaled axes. It should also be noted that, in a way that parallels the notation for x, the notation T = 1 represents dynamical time at the origin of the universe, while T = 0 is dynamical time close to the present. The result can be seen in Figure 8.

This may not be the end of the matter. Some geological and astronomical data discussed in Section 9.7 suggest that there may be a ripple or oscillation associated with the main decay pattern given by z in (108) and therefore c in (110). If this proves to be the case, and some evidence to support this possibility is presented below, it need not be a complete surprise. In many physical systems, the overall response to the processes acting, y(T), comprises two parts: a forced response, y_f(T), plus the natural, or free response, y_n(T). In the case under discussion here, it appears that there is a decaying response, y_f(T), as in (110), upon which is superimposed a resonating free response, y_n(T), which gives an oscillation or ripple on the main pattern. In such cases, the complete equation would be the sum of the two responses and so would read [

(121)]

The observational evidence presented in the Report, and subsequent analysis of atomic clock dates compared with actual historical dates, does seem to suggest that an oscillation may be imposed upon the main c-decay pattern in the way indicated by (111). These archaeological data do not stand alone. There is some astronomical evidence for an oscillatory natural response. Any such oscillation would leave a ripple on the final c-decay curve, which has the effect of flattening the curve at those points before it resumes its steep drop. Since the light-speed curve is directly related to redshift behaviour, these “flat points” in the c-decay curve would correspond to similar “flat points” in redshift behaviour. At these flat points the value of z should vary only slowly over a large distance. Consequently, we should see a significant number of galaxies appearing to congregate at or near any such key redshift values. This should be in contrast with a much smaller number of galaxies expected to populate those regions where the value of z changes significantly over a short distance.

This is in line with the observational evidence. The congregating of galaxies at these flat points has been termed redshift periodicities, which are different from redshift quantisations. Tifft’s redshift quantisations were usually obtained by comparing redshift differences between pairs of galaxies. In addition, Tifft claimed that he has discerned boundaries between quantum states within individual galaxies [1]. This reveals that redshift quantisation is not object dependent. By contrast, redshift periodicities are specifically related to the number of galaxies known to exist at a given z value, and as such are definitely an object dependent phenomenon.

It should be made clear that, if it withstands analysis, this whole periodicity effect would be a numerical artifact brought on by the flattening of the redshift or c-decay curve at specific points due to isotropic behaviour of the ZPE. Because light takes a finite time to travel, and has undergone the same velocity changes from objects that are equidistant from our planet, these redshift periodicities, as well as the quantisations discussed above, should occur in apparent concentric “shells” with the earth seeming to be at their focus. This is precisely the impression that this redshift congregating of galaxies creates [

(122)]. Because light-speed would undergo exactly the same changes for observers elsewhere in the universe, they would see exactly the same effect. Therefore, this phenomenon does not indicate a favoured or central position for our solar system.

After reviewing the observational history of the preferred redshifts at which distant objects appeared to congregate, Burbidge and Hewitt in 1990 noted that those redshifts were in agreement with the Karlsson formula and listed the data as follows [

(123)]

The Karlsson formula was thus verified and Burbidge and Hewitt quoted it as being

(1 + z₂)/(1 + z₁) = 1.2285 or (1 + z_n)/(1 + z₀) = 1.2285ⁿ (113)

Here, z₂ is the next higher redshift from z₁ and the analysis here has determined that 1.2285 is a better figure describing the observational data, compared with 1.227 in the Burbidge and Hewitt paper. These equations can be simplified since it can be established that z₀ = -0.13821. As a consequence (113) can be written as

These preferred redshifts were not just in narrow sectors, but occurred globally across the whole sky. A straightforward interpretation of these apparent periodicities is that the redshift (and hence light-speed) is itself going in a series of steps and stairs on a large scale. These data engender the suspicion that an oscillation could perhaps be superimposed on the main decay pattern in a way suggested by (111). If this is indeed the case, then the mid-points between the values in (112) should give the start of the steeply rising part of the steps and stairs pattern, namely at redshifts of

These considerations are important in another context. Just recently, four Type Ia supernovae have been examined as a result of photos taken by the Hubble Space Telescope. They were SN 1997ff at z = 1.7; SN 1997fg at z = 0.95; SN 1998ef at a value of z = 1.2; and SN 1999fv also at z = 1.2. In 1999, the supernovae at z = 0.95 and z = 1.2 attracted attention because they were 10% to 15% fainter, and hence further away than expected [124, 125, 126]. This led cosmologists to state that Einstein’s Cosmological constant must be operating to expand the cosmos progressively faster with time. However, it now turns out that the object SN 1997ff, the most distant of the four, is brighter than expected for its redshift value. This has elicited a further batch of explanations [127, 128].

In essence, the objects at z = 0.95 and z = 1.2 are systematically faint for their assumed redshift distance. By contrast, the object at z = 1.7 is brighter than expected for its assumed redshift distance. Notice that the object at z = 0.95 is at the middle of the flat part of the step according to the Burbidge and Hewitt analysis of (112). Consequently, for its redshift value, it will be further away than expected, and will therefore appear fainter. By contrast, the object at z = 1.7 is fully on the steeply rising part of the pattern. Because the redshift is changing rapidly over a short distance astronomically speaking, the object will be assumed to be further away than it actually is, and will thus appear to be brighter than anticipated. It would also appear that the precise redshift of the two supernovae listed at z = 1.2 may have been marginally overestimated, and instead may be around z = 1.19. They would then be right at the back of the step, just at the beginning of the steep climb as seen in (115), and hence fainter than anticipated.

These results do not depend entirely on the Burbidge and Hewitt paper. In 1992, Duari et al. examined 2164 objects with redshifts ranging out to z = 4.43 in a statistical analysis [129]. Their analysis eliminated some suspected periodicities as not statistically significant. Only two candidates were left, with one being “difficult to detect visually from the histogram” [129]. However, the other was mathematically precise at a confidence interval exceeding 99% in four statistical tests. When subjected to the comb-template test, the period of this candidate was so perfect that the formula could be specifically given as:

The basic period here of z = 0.0565 is equivalent to 6340 steps of Dz. Importantly, this smaller scale periodicity peaked on or very near each of the six redshift data periodicities noted by Burbidge and Hewitt [129]. The results from (116) therefore indicate that there is a significant but smaller scale ripple on the main pattern of (112). Since several modes of oscillation often accompany the fundamental vibration, these results lend weight to the possibility that such an oscillation may be superimposed on the main decay pattern as suggested by (111). Furthermore, they make it difficult to accept that the redshift is a measure of universal expansion.

There is another aspect of this redshift clustering of galaxies as given in (112) and (116) that may be important. The fact that radioactive decay rates are necessarily proportional to c has been outlined in section 6.3. This means that radiometric dates for geological phenomena should also be c-dependent in the same way that the redshift is. Indeed, the Report as well as (21) and (53) above show that the run rate of atomic clocks is linked with light-speed. Therefore geological data from the radiometric clock in atomic years should coincide with astronomical data from the redshift. In other words, at the “flat points” in the c-decay curve, greater numbers of galaxies should be counted for the corresponding redshifts, and there should also be a greater number of radiometric dates recorded over the atomic time-range covered by those “flat points.”

What is found to be the case in practice? Here is a comment on the geological effect by Read and Watson:

“In 1960 Gastil [130 this paper] published a histogram showing the time-distribution of mineral dates then available from all over the world and concluded that the pattern of peaks revealed by it suggested lengthy fluctuations in the earth’s orogenic history. This conclusion, based on a small sample, has been endorsed in a general way by several later authors who have been able to use a larger number of age-determinations. A diagram published more recently by Dearnley [131 this paper] shows a simplification that suggests that some of the gaps were due to lack of data. Nevertheless, several low points persist …” [132].

Read and Watson then show a histogram from 3400 determinations in which the number of age-determinations is plotted against the radiometric time-scale in millions of atomic years. The resulting geological histogram shows some 10 spikes superimposed on the overall trend.

The positioning of these geological peaks given by the atomic clock now needs to be assessed to see if there is any coincidence with similar peaks in the redshift. In order to do this, it is recalled that the slowing atomic clock ticks at a rate proportional to c. Atomic time elapsed is therefore given by the integral of the c decay curve from the present, where T = 0, back to the chosen time of T = T_c. But the value of T_c must first be determined for the chosen value of z, so that it can be substituted into the formula for the integral to obtain this elapsed atomic time. Because there is an equivalence between a given value for x in (108) and T in (110), the first step is to isolate x for the chosen value of z from (108). This is possible since it can be established that

Verification may be obtained by inserting this expression for x back in (108). Substitution for the chosen value of z will then give the chosen value of x, which also determines T_c. The time elapsed, t, on the atomic clock is then given by substituting T_c for T in the integral of (110) which is formulated as follows:

The final term of unity in (118) is included since this gives the value of t at T = 0. Its inclusion thereby allows a straight substitution for T_c to give t directly. This will allow the values of t in millions of atomic years to be determined from the peaks of redshift data in (116). These peak values of t from Duari et al. using 2164 objects are then compared with the position of the various spikes, in millions of years, that occur on the pattern of the histogram of geological data given in [132] using some 3,400 determinations. The results are recorded in Table II. 10⁶

100	x = 2; 125
300	x = 3; 270
450	x = 4; 450
600	x = 5; 670
950	x = 6; 925
?	x = 7; 1200
1400	x = 8; 1500
1800	x = 9; 1820
2100	x = 10; 2150
2500	x = 11; 2500
3000	x = 12; 2850
Uncertainty ± 50

The uncertainty of ± 50 million atomic years is the error in determining the precise position of the peaks, not the error in the radiometric measurements. As can be seen, the agreement between the two data sets in Table II is reasonably close, and an even closer match may be possible by refining the value of K. This reasonable agreement tends to confirm the general thesis that the behaviour of light speed is uniformly affecting both the redshift and radiometric data.

From the astronomical data, the range of the validity of the redshift and light-speed equations given by (108) and (110) may be determined. Since there are no significant blue-shifts in the astronomical data, it may be deduced that the redshift function in (108) stops at T = 0 and does not go into the negative region of the function. This in turn means that (110) is likewise only valid down to T = 0 for light-speed and does not go into the negative region either. However, the oscillations that the astronomical and geological data have revealed appear to be the natural or free response to the main decaying function. In the same way that a child’s swing continues to oscillate after the pushing has ceased, these oscillations might be expected to continue and dampen out after the main function has reached its zero point in the manner illustrated in reference [133]. This is the precise situation revealed by the archaeological data, the last 300 years of light-speed measurements, and the recent comparisons between the atomic and dynamical clocks. The data minimum, which appeared to be attained in 1980, may be another “flat point” from the effects of Eq. (116) following which the drop may be resumed in a manner illustrated in reference [134], or, alternatively, the function may rise somewhat after this minimum. Further data and analysis is needed to determine which of these oscillation scenarios is being followed.

If the behaviour of c after T = 0 is yet to be determined precisely, the redshift data reveal the nature of the curve between T = 1 and T = 0. When all the geological, astronomical and archaeological data are taken into consideration, a concordant value for K emerges. However, it is not necessarily unique, as other options are possible. This value for K then allows k to be determined, since (110) requires the time to be in dynamical years rather than a number between 0 and 1 as needed in (118). Then from (104) the value of c may be established for a given value of z. Finally, having found c, the value of Dc may be may be found from (109). Using this procedure, the option envisaged here indicates that

Galaxy velocity differences in clusters are measured by their cz values. However, the quantised redshift largely explains the changing cz values across the diameters of most clusters of galaxies, without the necessity of invoking high velocity movements. Indeed, a large actual velocity component in these cz values would destroy any quantisation effect. Under these circumstances, the real velocity component in these redshifts is probably small, as Arp has pointed out [135]. More recent work on galaxy clusters has revealed the significant information that in the center of the Virgo cluster, galaxies “were moving fast enough to wash out the [redshift] periodicity” [108]. Consequently, the necessity for additional mass to retain high-velocity galaxies within these clusters is precluded. This model therefore provides a possible solution to the problem of mass “missing” between galaxies in clusters. Note, however, that problem of mass missing within each individual galaxy remains unresolved by this treatment.

Another astronomical comment is also relevant here. The Schwarzschild radius of a black-hole is given by r_s = (2G*m)/c². As shown by the discussion around (59) to (64), G*m will remain invariant at all times. Consequently, the following equality holds at all times:

Several points can be made here. First, this equation reveals that with higher values of c, a smaller black-hole radius results for a given mass. Alternatively, with higher c values, a black-hole of any given radius must have been more massive than at present. Second, as c declines smoothly with the smooth increase in vacuum permeability and permittivity, there will be a smooth increase in r_s. This should allow black-holes to progressively engulf material from their accretion disks. In turn, this may keep feeding their high velocity jets [136].

In the 1987 Report as well as here, it was upheld that the value of c is isotropic throughout the cosmos. Note that this does not preclude local variations in c. It is possible that rapidly changing energetic processes may alter the physical properties of the vacuum in their vicinity. A local variation in c, z and n would then result. Indeed, a minority view among astronomers holds that the abnormally high z values of quasars do not relate to distance. If they do not, then it is feasible that some high z values may be caused by processes related to those envisaged by Arp, such as his “white-holes” and “little bangs” [137].

On this model, a high redshift inevitably implies a very high c value. Until recently, the highest value of the redshift for any galaxy was z = 5.34 [138], with a gamma-ray burster detected at z = 6.2 according to both Wijers [139] and Chown [140]. However, results released from galaxies in the Hubble Deep Field South NICMOS Field have broken all records. This Team analysis for the Hubble Space Telescope picked up 14 galaxies with redshifts between 5 and 10 with a further five candidates of redshift greater than 10 and less than 14 [141]. If the limit of z = 14 is chosen, then (109) reveals that c = 1.001´ 10⁸ c. In a similar way, the maximum value for c at the origin of the cosmos, c₀, is about

One very important effect should be noted that would occur with the initial c values of the order of 10¹¹c. It was suggested originally by Troitskii [50]. He pointed out that if c was of the order of 10¹⁰ c initially, this would allow rapid homogenisation of radiation at that epoch. This process in itself would largely explain the general uniformity of the microwave background without the necessity of secondary assumptions about matter distribution and galaxy formation. The A-M [63] and Barrow [64, 65] papers take a similar line, but suggest that c was about 10⁶⁰ c to begin with. By contrast, the redshift data presented here are in general agreement with Troitskii’s more conservative figure as about the maximum needed to account for all the observational evidence.

In Science Vol. 292, p.414 for 29 June 2001, Charles Seife reported on the results of an investigation by Lubin and Sandage slated to appear in Astronomical Journal. This study analysed space-based data from distant galaxies and applied the Tolman surface brightness test. In principle, this test can distinguish between expanding and static models of the universe. This is possible because an object at redshift z with radiation surface brightness I, as measured by an observer at rest at the object, will have an observed surface brightness integrated over all wavelengths in an expanding cosmos of I_o = I(1 + z) ^–⁴. The four powers of the equation’s expansion factor arise because light at reception has been affected in a manner proportional to z by four factors: one due to the effect of redshift on the energy of each photon, one from the effect of time dilation on the reception of photons, and two from aberration effects.

By contrast, the tired light hypothesis initially proposed by Fritz Zwicky predicts only a decreasing photon energy for a static cosmos. The equation for the observed surface brightness in that case becomes I_o = I(1 + z) ^–¹. Zwicky’s hypothesis has thereby been ruled out by the most recent results. In comparison, the model presented here has the following redshift dependent factors affecting brightness: one due to the redshift in energy, one due to a time dilation factor that is exhibited in Fig. 5 and Eq. (37), one due to the slower rate of reception of photons compared with that at emission, and one due to the lower energy density of photons at emission as discussed around Eq. (66). This gives a result of I_o = I(1 + z) ^–⁴. Consequently, this model is in complete accord with the results obtained by Lubin and Sandage.

Wesson [142] and others have shown that the vacuum could initially acquire an elastic tension as a result of the inflationary expansion of the cosmos [143, 144]. In a similar way, it is proposed here that an elasticity or vacuum elastic tension has originated as the cosmos was expanded out to its current static state. Like other forms of elastic tension, this cosmological expansion would have invested the vacuum with a potential energy per unit volume [145, 146]. In order to appreciate what is happening to the structure of the vacuum under these conditions, the statement of Pipkin and Ritter is again relevant, namely that “…the Planck length is a length at which the smoothness of space breaks down, and space assumes a granular structure” [31]. Furthermore, this granular structure of space can be considered to be made up of a seething sea of Planck particle pairs whose dimensions are equal to the Planck length (1.61604 ´ 10^–³³ cm), with a mass equivalent to the Planck mass (2.17657 ´ 10^–⁵ gm), and which flip in and out of existence over an interval equal to the Planck time (5.39053 ´ 10^–⁴⁴ sec).

It is at this level, then, that the vacuum was likely to respond with an increased separation and spin of the particle pairs as the cosmos was expanded. Such pairs will simultaneously have both electric and magnetic dipole moments due to charge separation and spin motion respectively. In turn, these dipole moments will give rise to the electric and magnetic fields, which comprise the ZPE. In that sense, then, the original expansion set the initial parameters governing the ZPE. However, once those parameters were set, the energy density of the ZPE would then depend on the number of Planck particle pairs that manifest per unit volume in any given dynamical interval. Anything that changes this number will also change the energy density of the ZPE, along with all the effects noted here. In this way, the structure and behaviour of the vacuum at the Planck particle level is determining all the observed effects at the atomic level.

An important factor in the discussion then becomes an interval known as the Planck time t_p, which is defined as [73]

This proportionality holds since we are talking of effects at the subatomic level and so G is proportional to c⁴ as outlined in the comments following (54). An alternative definition can be found in French [F. 193], which gives the result that

where E_p is the Planck energy and m_p is the Planck mass. As defined in (122) and (123), the Planck time interval therefore increases with the passing of dynamical time. Thus this slowing atomic clock ticks at a rate proportional to c as dynamical time progresses (rather like a cheap watch that slows as its spring unwinds). The function governing the rate of ticking is thereby the same as the function governing the behaviour of light-speed, namely Eq. (110).

The length of time that Planck particle pairs exist before annihilating is referred to as the Planck time interval t_p. However, as dynamical time progressed from the origin of the cosmos at T = 1 to a more recent epoch at T = 0, the Planck time interval increased in length since the rate of ticking of this atomic clock has slowed. This means that Planck particle pairs remain in existence for progressively longer periods before annihilating. This effectively means that, for any given dynamical interval, more particle pairs will be in existence per unit volume as dynamical time goes forward.

In order to illustrate this more effectively, consider a unit volume of space in which the conditions are such that a Planck particle pair manifests every dynamical second. Furthermore, let the Planck time interval also be one dynamical second. Thus, at any given observed interval of one dynamical second, only one particle pair will exist in that unit volume. Let the Planck time then be increased by a factor of 3, so that each particle pair exists for 3 dynamical seconds. Since other conditions remain unchanged, a new particle pair will still manifest every second. Thus 3 particle pairs will exist during any given dynamical second. First, there is the pair that originated at the beginning of that interval, just as the situation was before. Then there is also the pair that originated one second earlier, so that the observational interval is the middle second of their 3 second lifespan. Then in addition there is also the pair that originated two seconds earlier, so that the observational second is the 3^rd second of their existence.

It can thus be demonstrated that if t_p is increased by a factor N, there will be N times as many particle pairs per unit volume for a given dynamical interval. From the relationship in (122) and (123), it follows that at all times

Therefore, as time progresses, there is an effective increase in the number of Planck particle pairs per unit volume in a given dynamical interval proportional to 1/c. This means the strength of the ZPF will be increased such that the energy density U of the ZPE will also be proportional to 1/c as in (19). All the effects outlined in this paper then follow on as a consequence.

There might also be one final effect worth noting. A systematic increase in Planck particle numbers per unit volume making up the fabric of space might be expected to systematically lower the intrinsic temperature of the vacuum. This result might also be anticipated if the total radiation energy density of space is given by U* = A + B. Here, A is the cubic frequency distribution of the ZPE (which is independent of temperature), and B is the usual Planck term (in which temperature is variable) and which gives rise to the microwave background. If U* is constant in a static cosmos, then as the ZPE increases with time, the term A also increases. This means that B must decrease, so that the temperature of the microwave background must then decrease with time. This effect has recently been verified by the new VLT spectrum of the quasar PKS 1232+0815 whose light passed through a gaseous cloud in a galaxy of redshift z = 2.34 giving a wealth of spectral lines from a number of elements for analysis, including neutral carbon and molecular hydrogen. At that early epoch, the temperature of the microwave background was hotter than 6K and cooler than 14K [147].

The only remaining issue for discussion is the basic reason for the relationship in (124). Since c is dependent upon the ZPE as outlined above, it cannot be the independent variable here. In a similar way it can be argued that m_p and t_p are dependent upon other quantities for their formulation, as in (54) and (122) respectively. Indeed, the behaviour of the atomic clock itself has been shown to be ZPE dependent. On the SED approach, even G is a ZPE phenomenon, which removes it from contention here. In (122) and (54) this only leaves Planck’s constant h as the possible independent variable which is influencing vacuum behaviour at the Planck particle level. If this is the case, it means that, rather than the SED definition of h being a measure of the strength of the ZPE, h would actually become the cause of ZPE behaviour via its effects on Planck particles. In this case, an increase of quantum uncertainty with the passage of dynamical time would be an intrinsic feature of the cosmos. The inverse of this uncertainty must then behave in a Lorentzian fashion since (22), (108), (110) and (124) mean that

1/N µ 1/h µ c = k{[(1 + T)/Ö(1 – T²)] – 1} µ z ((125))

However, if some entirely independent reason is sought for the behaviour of h and N, it is possible that the cosmos might oscillate at its natural frequency on a small scale around its equilibrium position once its static state had been attained. Such an oscillation would affect the Planck particle numbers per unit volume and hence the ZPE and all associated quantities, including h. This may be the origin of the periodicity effects in (112) and (116). Such oscillations might be expected to be somewhat limited on a cosmic scale to avoid smearing out the quantisation, but large enough to account for the periodicities. As such, they would be a secondary rather than a primary cause for the observational data. All these factors thereby become a matter for ongoing research.

1. Both atomic orbit energy and particle potential energy are linked with the electromagnetic properties of the physical vacuum. According to stochastic electro-dynamics (SED), energy is transferred from the vacuum to electrons and quarks through the electromagnetic zero-point fluctuations.

2. The vacuum zero-point energy (ZPE) has been shown by Puthoff [32] and Boyer [24] to sustain sub-atomic particles in their orbits. Since the quantised redshift increases with distance, this may indicate that atomic orbits had lower energy in the past. This in turn may imply that the ZPE was itself lower in the past. The converse is that the vacuum energy density of the zero-point fields (ZPF) may be smoothly and isotropically increasing with time throughout the cosmos.

3. A smoothly increasing ZPE would be expected to induce smooth changes in some atomic quantities, including a smooth increase in Planck’s constant h, a slowing of atomic clocks, and a smooth decay in the speed of light. The redshift data indicates that c-decay is following a Lorentzian function with a natural ripple superimposed that results in some “flat points” on the curve. This ripple explains the large-scale clustering of galaxy redshifts in astronomy, and a concordant clustering of radiometric dates in geology, corresponding to the “flat points” shown in Table II.

4. Whenever the increasing vacuum energy density exceeds a quantum threshold, a full quantum of additional energy becomes accessible to all atomic orbits isotropically throughout the cosmos from the electromagnetic zero-point fields. All atomic orbits then take up a new quantum energy state throughout the cosmos.

5. From Corollary 2, energy is conserved in all atomic processes within each quantum interval since any variation in the vacuum energy density less than a full quantum change is inaccessible to the atom. The expected behaviour of atomic quantities is borne out by the observational data, which was listed and examined in detail in the 1987 Report [48].

6. The discrete increase in orbit energy at the quantum jump, with increasing time, causes a blue quantum shift in all emitted wavelengths since the radii of atomic orbits must remain invariant to accord with observational evidence. At the quantum jump, the angular momenta of all atomic orbits will remain unchanged.

7. A finite time is taken for light emitted by atomic processes to reach the observer after each quantum change. Consequently, the observed redshift will appear to be quantised in spherical shells centred about any observer anywhere in the universe. All objects that emit light within a given shell will have the same redshift. A discrete redshift change of 2.671 km/sec exists between adjacent shells.

8. Any change in the Friedmann radius of the cosmos would also change the wavelengths of light in transit. But any major change in radius would obliterate the wavelength quantisation effects. The very fact that this redshift quantisation exists at all necessarily implies that the Friedmann radius is substantially stable. Therefore, the conclusion follows that the cosmos has remained essentially static after the initial expansion. Evidence for this initial expansion comes from the microwave background radiation.

9. The velocities of galaxies within clusters are measured by their redshift z. However, the quantised redshift effect accounts for most Hubble velocities within clusters of galaxies. The actual velocity component in many galaxy redshifts may thus be small, except near cluster centres where higher galaxy velocities destroy the quantisation. The actual velocities of galaxies within clusters may not be high enough to require any “missing” mass to hold the clusters together. However, the problem of mass missing within each galaxy remains.

10. The orbit period of astronomical bodies and the acceleration due to gravity is invariant.

11. The Schwarzschild radius of a black-hole will smoothly increase with time if no other effects are operating. This should allow material from the accretion disk to be progressively engulfed, which may in turn activate the high velocity jets.

12. The isotropic assumptions for the cosmos in Corollaries 1 and 2 in no way preclude the properties of the physical vacuum varying locally, perhaps due to energetic processes such as Arp’s “white-holes” and “little bangs”. All the subsequent effects outlined in this paper would then occur in a local astronomical sense rather than cosmologically.

13. The reason for the progressive change in the strength of the ZPE may be traced to the behaviour of the vacuum at the Planck length level. On this model, ZPE originates with the Planck particle pairs that comprise the structure of the vacuum. The increase in the quantum uncertainty as dynamical time progressed effectively allowed more Planck particles to manifest per unit volume in a given dynamical interval. This meant the energy density of the ZPE also increased.

This model suggests that, following an initial rapid expansion event, the universe attained a maximum size, followed by an essentially static cosmos thereafter. A high initial value for light-speed allowed rapid homogenisation of the radiation resulting in a smooth microwave background. The response of the vacuum at the Planck length level gave rise to an increasing energy density for the ZPE, which had two effects. First, light-speed decayed in a manner concordant with the behaviour of redshift with distance. Second, and concurrently, atomic particle and orbit energies underwent a series of isotropic quantum increases, as more energy became progressively available to them from the vacuum. With increasing time, atoms emitted light that shifted in jumps towards the blue end of the spectrum. With increasing astronomical distance (looking back in time), the resulting redshift increased in cz quanta of 2.671 km/s in accord with Tifft’s observations.

My deep appreciation goes first to Helen (Fryman) Setterfield for her perceptive comments, her probing questions, her incredible wisdom, and much practical assistance. Thanks are also due to Trevor Norman, Lambert Dolphin, Michael Webb and Daniel Dzimano for their many important discussions. A debt of gratitude is owed to Bernard Brandstater, Walt Brown, Alan J. Rankin, Gerard Fothergill, and Graham Mortimer who kindly gave their time to review various drafts of this paper. High praise goes to John Heidenreich and Derek Miller who exercised their computer skills during many months. This paper is dedicated to my late mother Beth and patient sister Marilyn.

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1. INTRODUCTION

1.1. Initial Summary

1.3. Defining the Problem

1.4. Turning the Problem Around

The redshift, z, is defined as the change in wavelength, Dl, of a given spectral line compared with the laboratory standard for that line, divided by the laboratory standard wavelength, l, for that same spectral line. This is usually written as [7]

z = Dl /l (1)

1.7. The Invariance of hc

4.5. The Isotropic Nature of Light-speed and Relativity

4.7. Light-speed and the Early Cosmos

5. THE TWO TYPES OF ATOMIC BEHAVIOUR CONSIDERED

In order to visualise what is happening both within the quantum interval and at the transition or jump, the behaviour of these and other quantities have been graphed in Figures 2 to 8.

7.1. Determining the Rydberg Quantum Number

P_a = e²hw ³/(24p ²e m₀ c³) (79)

However, in this equation the power available from all possible directions of propagation of the ZPF has been included, whereas on average only one third of the total energy of the field will be absorbed by such an oscillator [F. 82]. Therefore the utilisable power P_u available will be given by

E₂/E₁ = (e₂ /e₁)² = b = (1 + z) = P₂ /P₁ (81)

As a consequence, (81) can be written as:

P₂ = P₁ + zP₁ (82)

1. INTRODUCTION

1.1. Initial Summary

1.3. Defining the Problem

1.4. Turning the Problem Around

The redshift, z, is defined as the change in wavelength, Dl, of a given spectral line compared with the laboratory standard for that line, divided by the laboratory standard wavelength, l, for that same spectral line. This is usually written as [7]

z = Dl /l (1)

1.7. The Invariance of hc

4.5. The Isotropic Nature of Light-speed and Relativity

4.7. Light-speed and the Early Cosmos

5. THE TWO TYPES OF ATOMIC BEHAVIOUR CONSIDERED

In order to visualise what is happening both within the quantum interval and at the transition or jump, the behaviour of these and other quantities have been graphed in Figures 2 to 8.

7.1. Determining the Rydberg Quantum Number

Pa = e2hw 3/(24p 2e m0 c3) (79)

However, in this equation the power available from all possible directions of propagation of the ZPF has been included, whereas on average only one third of the total energy of the field will be absorbed by such an oscillator [F. 82]. Therefore the utilisable power Pu available will be given by

E2 /E1 = (e2 /e1)2 = b = (1 + z) = P2 /P1 (81)

As a consequence, (81) can be written as:

P2 = P1 + zP1 (82)

P_a = e²hw ³/(24p ²e m₀ c³) (79)

However, in this equation the power available from all possible directions of propagation of the ZPF has been included, whereas on average only one third of the total energy of the field will be absorbed by such an oscillator [F. 82]. Therefore the utilisable power P_u available will be given by

E₂/E₁ = (e₂ /e₁)² = b = (1 + z) = P₂ /P₁ (81)

P₂ = P₁ + zP₁ (82)