It might be realised that one of the fundamental assumptions of a causal wave model is the need for a wave media, which can transport energy in 3D space as a function of time. However, the nature of this wave media has been described in many ways, i.e. as a distortion of space, as a fundamental electromagnetic field, as a multitude of different quantum fields or even possibly as a superfluid that can isolate angular momentum.
Note: To be honest, this discussion will not make any real reference to LaFreniere’s page ‘The Aether’ as it appears to be more focused on the historic description of the ‘Aether’ and the ‘virtual aether’ of his simulated space. Equally, this discussion prefers to use the term ‘ether’ or simply ‘space’ to describe the wave media under review, as it hopefully avoids some of the historic ambiguity surrounding the ‘luminiferous aether’.
This discussion will focus on the idea that space itself is the fundamental media of wave propagation. While accepting that this is a speculative idea, it would appear that even accepted science suggests that space can be subject to a relativistic ‘curvature’ as well as cosmological ‘expansion’. In this respect, the current discussion may be far more modest in its speculation as it may only imply a microscopically small quantum distortion of the fabric of space, sufficient to propagate waves. However, we will start by simply questioning whether space is a continuous or discrete wave media. Initially, most wave models might assume space to be a propagation media that is essentially continuous rather than discrete in nature. For example, in the macroscopic world, the propagation of 2D surface waves in the media of water appears continuous, although we know that this media is discrete in terms of its molecular structure. Equally, we understand that the propagation of 3D sound waves in the media of air also depends on a molecular granularity inherent within this media. Of course, in the case of molecules, these points of oscillations are also connected by bonds that represent an elasticity [k] within the media that allows energy to propagate.
But how might space have some form of granularity?
While there is no obvious granular structure of space, we might speculate that should any granularity exist, it may have to be associated with the most fundamental level of physical existence. In terms of the Planck scale, there is a minimum limit to length, i.e. the Planck length, which might then define a minimal Planck volume as the smallest unit of space – see discussion of Planck scale for more details. If this were the case, these ‘granules’ of space might be modelled as an interconnected set of harmonic oscillators, where the displacement amplitude [A] from some equilibrium state would be proportional to the potential energy being transported by a wave. While this granularity of space might be pure speculation, any wave model will require a causal mechanism by which scalar potential energy [Ep] can be propagated through space as a function of time. While both the LaFreniere and Wolff wave models make the assumption that the media of space is capable of propagating waves, there is an issue, which may require space to have some form of granularity, if it is to be resolved.
Note: Today, one of the biggest numerical conflicts in physics is the approximate 10120 difference between the critical energy density of the universe, i.e. 10-9 J/m3 obtained from general relativity (GR) and the energy density of the vacuum obtained from quantum mechanics (QM) and its related fields. Names such as vacuum energy, zero-point energy or even vacuum fluctuations are all used to represent an implied energy density of about 10113 J/m3. The 10120 difference between these two numbers is usually believed to require a mathematical resolution known as renormalisation, which it is assumed eliminates the implied energy density of the vacuum. Aspects of this issue have also been discussed under a second heading of renormalisation.
However, in the current speculative context of the wave models being discussed, space is assumed to be the media of wave propagation. From this perspective, the square of the wave amplitude [A2] would be proportional to the energy required to distort the medium of space at a given point. However, we might expand this idea based on the assumption that wave energy conforms to , which suggests a functional dependence on both amplitude [A] and frequency [f].
Note: In , we might see how the Planck constant [h] could obscure the need for any wave amplitude [A2] or the second component of frequency [f]. It might also be highlighted that the equation [E=hf] appears to provide no causal explanation of how energy [E] propagates or why it is a function of only frequency.
The first part of the expression, i.e. [kA2], is normally associated with Hooke’s law, where [k] is typically defined in terms of the elastic constant of a spring. However, for [k] to help define the energy as a function of an offset amplitude [A], we might initially associate [k] with a mass-frequency of oscillation, i.e. [mf2], where any numeric factors will be ignored for the purpose of this general outline. However, we might also assume that any space wave media model would have to operate on a scale that is much smaller than anything suggested by the Compton wavelength of an electron, e.g. [10-12 metres].
Note: Various estimates for the radius of an electron exist over 7 orders of magnitude from [10-11m] to [10-18m]. Therefore, as a starting point, we might recognise that the granularity of space would have to be much smaller than the electron radius. However, while the LaFreniere and Wolff models only address the issue of an electron wave structure, if a neutrino is also a ‘particle’ within the accepted model, which has a near-zero mass, we may need to consider the granularity of space existing on an equally smaller distance scale.
Based on the note above, the granularity of space might have to be conceived in terms of the smallest scale recognised by physics, i.e. the Planck scale. If so, the Planck volume would be the smallest 3D granularity of space based on the Planck length, i.e. (10-35m)3, where the energy of a harmonic oscillating granule of space would be a function of frequency [f] and amplitude [A], as defined in .
Note: As a slight aside at this point, if we assume the velocity [c] of any wave through this media to be constant in all reference frames, the relativistic effects of time dilation and length contraction may only be perceived at some higher frame of reference moving with a velocity [v] with respect to the wave media, even though this may not be realised.
What else might any wave model require of its wave media?
As outlined, we are proceeding on the assumption that the ‘granules’ of space can be modelled as an interconnected set of harmonic oscillators, where the displacement amplitude [A] from some equilibrium state is proportional to the potential energy being transported by the wave. So, while the granularity of space might be questionable, the idea is proceeding on the fundamental assumption that any wave model will require a causal mechanism by which scalar potential energy [Ep] can be propagated through space as a function of time.
Might we make some initial estimate about the size of these space ‘granules’?
In response to the question above, we might make a speculative ballpark calculation, although still anchored to the accepted idea of the Planck scale. We will start by defining a conceptual Planck particle as one that has the maximum energy density of a blackhole, but with a Schwarzschild radius [Rs] equal the Planck length [Lp]. At this stage, it is highlighted that there is no obvious physical justification of the Planck length, other than it is the minimal product of the three most fundamental constants, i.e. [c, h & G]. However, if the Planck length is the smallest length that has any physical meaning, we might consider the Planck volume to be the smallest granule of 3D space. Some attempt is now made to quantify this assumption.
Again, no justification of the Planck length will be made, but is used to help define the Planck mass [mP] and Planck energy [EP] associated with a black-hole with a radius defined by the Planck length [LP], although a factor of 2 is missing from the Schwarzschild radius equation. However, from these figures we might estimate one extreme of energy density [Eρ] by confining this energy within the smallest Planck volume [Vp] based on the Planck length.
From , we might see a correlation of the Planck energy-density being in the region of [10113 J/m3], which is in the same ballpark as the energy-density assumed by quantum theory. However, we might go one-step further in these ballpark estimates by trying to compare the Planck Energy [Ep] estimated above with the Planck energy equation [E=hf], where frequency is correlated to Planck Time [Tp].
Based on  and , it might be suggested that the Planck volume [Vp] may represent the smallest granularity of space and the largest possible energy [Ep] that can be supported within this wave media model. However, while this figure broadly aligns to the energy-density defined by quantum theory, it is not supported by general relativity and the observational estimates of cosmology, which differ by some 10122 based on a figure of 10-9J/m3 in comparison to the 10113J/m3 estimated in . So, while the figures in ,  and  are not necessarily supported by empirical verification, it might be highlighted that we are initially proceeding on the same mathematical assumptions as the quantum model.
What other assumptions might we make about space as a wave media?
Before we begin to consider this question, we might introduce a few fundamental concepts, first by introducing the idea of intensity [I] in two equally equivalent forms:
In terms of the speculative wave media model being outlined, we are assuming that wave energy has to be transported by the oscillations of the granules of space. If so, then the potential energy being transferred by the wave must have some correlation to the kinetic oscillating energy of these conceptual ‘granules’ of space, i.e. the displacement velocity [v], such that we might try to quantify the energy component in  as follows:
The first step in  possibly needs some explanation and justification. In terms of the conceptual oscillation of a ‘granule’ of space, the velocity [v] in  would relate to the velocity [v] of oscillation, not the wave velocity [c]. As such, [v] is a function of wave amplitude [A] and frequency [f] of oscillation. If so, we might now substitute  back into the energy form shown in , such that we might define the intensity of a wave with amplitude [A] and frequency [f].
Pursuing this line of speculation, we might continue by introducing the idea of a wave impedance [Z], which we shall initially describe as the ‘resistance’ of the media to a longitudinal wave motion in a 3D wave media. Based on the idea of specific acoustic impedance, we might also consider a relationship between pressure [P] and the velocity of a wave [c]. While this model is essentially an acoustic wave model, 3D sound waves are also longitudinal and non-dispersive, which propagate through a 3D media. In this context, we might extend the idea of impedance [Z] to the media of space.
Note: By way of a general description, when an oscillating source propagates waves into the media, the media opposes the immediate propagation velocity [c] of the wave in terms of an impedance [Z]. As such, the wave needs a pressure [P] or energy [E], which can be translated into a force/area or energy/volume. However, this description might also be related to the idea of stiffness [K], if defined in terms of a displacement [d] of the media when subject to a force [F].
As indicated, we might attempt to quantify another attribute of the wave media in terms of stiffness [K] as follows:
At this point, we might consider the possibility of combining the definitions of wave intensity [I] and wave impedance [Z] in  and  as follows, where any numeric values have been dropped in order to simply focus on the variables.
While recognising that we have probably push speculation beyond its limits, we might still consider some of the possible implications stemming from this conceptual wave media model. However, having reached this point, it is recognised that there are no obvious values to substitute into the equations outlined, such that we cannot really quantify any of the key attributes of the wave media, as defined. Of course, we might return to some of the initial Planck scale assumptions underpinning ,  and , which suggested a possible causal resolution of the renormalisation issue that divides quantum theory and general relativity. Given that this wave media model is predicated on Planck scale assumptions, we might relate the parameters in ,  and  to other Planck parameters starting with wave intensity [I] as defined in .
Here we see energy-density [E/V] defined in terms of the Planck energy [Ep] and Planck volume [Vp] multiplied by the wave propagation velocity [c]. While it is recognised that these figures might appear ridiculously large, such numbers are simply derived from the extremes of the Planck scale, which appear to be generally supported by quantum theory. We might now take the same approach with the impedance [Z] as defined in .
Finally, we will attempt to defined the stiffness [K] of the wave media as follows.
Note: Given the extremely large figure for stiffness [K] in , we might comment on the apparent contradiction of the wave media of space, which while appearing to be extraordinarily ‘stiff’ is often described as ephemeral in terms of any obvious physical ‘substance’. However, in the context of the WSE model, everything exists as a wave structure dependent on the attributes of a wave media, as generally defined in terms of intensity, impedance and stiffness. As such, these wave structures may be unaware of any underlying wave media attributes being an obstacle to propagation, as they are simply a prerequisite necessity. If so, the wave media might indeed be perceived as having no substance as it exists on a level that we have no direct perception.
Having quantify some highly speculative values for [I] and [Z], based on the Planck scale parameters, we might now transpose the basic formulation of [I] in  in terms of the two remaining unknown parameters, i.e. wave amplitude and frequency.
Clearly, we do not know the values of either the wave amplitude [A] or frequency [f], although we might initially consider the broad frequency spectrum of electromagnetic waves from gamma waves [300*1018Hz] through to the bottom end of radio waves [3*103Hz], such that we might consider the amplitude [A] suggested by these frequencies based on .
In , we first see an estimate of the distortion amplitude [A] of the space wave media for gamma frequencies, the scale of which we might initially entertain. However, the idea of a distortion of the space media being in the order of 10,000 metres for radio waves has to be questioned. If so, we need to consider the possibility that the propagation of an electromagnetic wave operates on an entirely different scale via a different mechanism. In this context, it might be highlighted that the wave model, as outlined, has only assumed that the standing waves that form ‘particles’ are predicated on longitudinal waves propagating through 3D space, which do not appear to directly explain transverse nature of EM waves. In terms of Maxwell’s equations, electromagnetic waves are assumed to be transverse waves, which propagate through an EM field having been sourced by an oscillating charged particle. While we will not focus on this issue at the moment, the diagram above might suggest how a standing matter wave, i.e. a vibrating charged particle, constructed of longitudinal waves, might generate a transverse wave on a completely different scale, if oscillating vertically about a point in space. If so, the frequencies used in  would not provide any indication of the wave amplitude of the underlying matter waves.
However, might we still consider  to determine frequency if we assume a matter wave amplitude?
Of course, in order to attempt any sort of answer to this question, we would require an amplitude [A], which for speculative purposes we might align to the Planck length as follows.
In essence, the answer in  simply turns out to be related to the inverse value of minimal Planck time. This is not really so surprising given the circular nature of so many of the Planck parameters based on only 3 constants, i.e. [c, G and h]. However, while we need to consider any estimate of the frequency associated with any fundamental matter waves with extreme caution, if not scepticism, we might consider some tangential arguments. Within Bohr’s atomic model, the first orbital is determined to have an energy [E=13.6eV], which if equated to the Planck energy equation [E=hf] would translate to a frequency [f= 2.42*1014 Hz] and a wavelength [λ=1.24*10-06m] based on [λ=c/f].
Note: It is recognised that the Bohr model was an initial and overly simplistic particle model. Equally, as already indicated, estimates for the radius of an electron exist over 7 orders of magnitude from [10-11m] to [10-18m], such that we should be cautious of any estimated figures associated with an electron.
The frequency suggested above for the electron also ignores any wave structure that might possibly be needed to support a ‘particle’ like the neutrino, which we might assume has a much smaller defined radius, which would have to be supported by the granularity of space. So, while the figure in  may be completely wrong, it might only be suggesting that the granularity of space, as a wave media, has to exist on the most fundamental level of existence, i.e. the Planck scale. Of course, if this were the case, we might then have to question whether the electron wave models of both LaFreniere and Wolff would need to be revised or depend on some form of sub-structure not yet considered.
Note: While this discussion has alluded to the possibility of some Planck scale distortion of space in order to support wave propagation, it does not necessarily explain the large-scale curvature of space-time assumed by general relativity or the wholesale expansion of space assumed by cosmology. However, the wave media model might be used to explain the Planck scale itself, if it represents the smallest units of distance and time, which are actually realised in terms of wavelength and frequency. Of course, we might recognise that the granularity suggested may only be another mathematical model of space, such that the physical reality of space may yet be shown to be essentially continuous, while still allowing Planck scale distortions.
In equation , there was a suggestion that a granule of space might be modelled as a harmonic oscillator based on Hooke’s law, as now repeated in .
However, there is an implication in  that may not be immediately obvious. If we equate the elasticity constant [k] to mass-frequency [mf2], where mass [m] might be re-interpreted as some form of energy concentration, i.e. [E=mc2], which differs from a point of undisturbed equilibrium, we might ask whether this granule of space would be subject to a gravitational force? This issue is raised because it might suggest that space can be subject to a large wholesale distortion around some large gravitational mass [M]. However, having simply raised this speculative idea, it will not be pursued at this stage.