ATOMIC QUANTUM STATES,

The following unpublished paper is being put here for purposes of reference only.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 There are, in addition, parts of the paper below that have been superceded due to recent discoveries.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are marked with yellow highlights in this paper, so that it may be understood that this particular material has been superceded by current work.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

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