The previous outline of the Lorentz Doppler effect, as forwarded by the MMW model, suggests support for time dilation. However, the initial evidence for this conclusion seems to rest on the propagation time as determined by observer-S in the stationary [S] frame is longer due to the change in geometry caused by velocity [β]. As outlined, the MMW model also forwards the idea of an active and reactive mass, which assumes that frequency [f] is associated with mass [m], slows because of time dilation.
Note: If we transpose the SR postulate associated with the constancy of the speed of light in all frames, [c=fλ], it suggests the wavelength relationship [λ=c/f=ct]. As such, any change in wavelength [λ] must infer a change in frequency [f] and therefore an associated change in time [t]. However, this transposition is predicated on the assumption that the relative velocity [c±v] can never be resolved within an inertial frame due to combinations of the normal and virtual Doppler effects. However, these Doppler effects also apply to any wave model that forwards the idea that a particle has a standing wave resonance structure that can be subject to a Standing Wave Compression.
We might initially anchor the discussion to the equation in , which is described by the MMW model as the all-azimuth Lorentz Doppler effect, although the notation on the wavelength symbols has been changed and explained below.
The wavelength [λS’] is associated with an oscillating source that exists in the moving [S’] frame to which we will simply assign a unity value, such that [λS’=1]. However, the wavelength [λSθ] that propagates out into a wave media has to be a function of both velocity [β] and angle [θ], which can only be measured by observer-S in the stationary [S] frame.
Note: At this point, it needs to be highlighted that we are pursuing the idea that length contraction may be linked to the Ivanov model of standing wave compression, which might then be superimposed onto the MMW wave model, where IN waves and OUT waves conceptually combine in superposition to create a standing wave – see Comparative Wave Models for a wider discussion of some of the problems with these models.
However, first, we might try to illustrate the results from  in the table below, where [β=0.5].
In the first highlighted column [λSθ], we see the results associated with  for a specified angle [θ], where we might recognize the values of [λ1] and [λ2] from the Configuration-1 discussion linked to the Lorentz transforms. However, in the current context, the values of [λ1] and [λ2] now reflect the wavelengths of the IN and OUT waves, as described above, which might form a standing wave [λSW] according to .
In the second highlighted column in the table above, we see the resulting compressed wavelength [λSW], as defined by , which appears to correspond to the length contraction as seen by the stationary observer-S, where [λSW=0.866] along the [x] axis that conforms to the Lorentz transforms. However, unlike the inference made by the MMW model, this discussion has made no direct reference to time dilation.
Note: What we might see in these results is the basis of a causal mechanism that may help explain the length contraction of a physical object made up of some complex hierarchy of standing wave resonances within the moving frame. As described, this length contraction would only apply to physical objects constructed from subatomic particles. i.e. wave-centres. As such, length contraction cannot be applied to the wave media itself, although wave models may require the fabric of space, possibly at the Planck scale, to be distorted by the amplitude of the waves in motion. If so, the measure of length of a physical object may have some correlation to the underlying wavelengths.
The previous discussion of Relative Geometry, specifically Configuration-2, suggested that more than one interpretation of the coordinate mappings between the moving [S’] and stationary [S] frames may be possible. As this has a bearing of how length contraction may occur, the following equation in  may be seen as an alternative to  linked to the Ivanov model, where the equation right is the simplified form for the forward and backward wavelengths to be discussed below.
As the descriptive details of the table below are essentially identical to the one just discussed, only the differences in values in the shaded columns will be highlighted.
However, if we now translate these wavelengths into the forward and backward waves for a moving [S’] frame, where [β=0.5] and [g=0.866], as per the earlier Lorentz Doppler effect example, the component wavelengths will change as shown in .
Again, we can use , but will now described the resulting standing wave as a ‘compressed’ waveform using  and .
Again, the Lorentz and Ivanov transforms might support a compressed standing wave, where the resulting compressed wavelength aligns to either [λF/B=gλ] or [λF/B=g2λ].
But which transform is right?
At this stage, the review is still looking for causal mechanisms that might physically explain why a physical object might contract in length, while in motion with velocity [β]. However, while both sets of transforms appear to be generally supportive of the idea of length contraction, the Lorentz transforms also require time dilation, which this review is still struggling to explain in terms of a causal mechanism. Of course, it might be highlighted that standing wave compression is an idea that was only proposed long after the publication of SR in 1904, which also makes reference to a wave propagation media that is essentially rejected by SR. However, if we were to question the assumption of SR from the perspective of verified physical science, beyond mathematical abstraction of any transform, we require causal mechanisms that explain all phenomena, including gravity, light, magnetic and electrostatic fields, nuclear forces, etc. While it is accepted that the wave models being referenced might not provide the actual causal explanation, if there is no wave propagation frame, do we have any causal description.
Note: As an initial summary, the MMW model proposes that a wave media does exist and possibly explains why the Michelson-Morley experiment failed to detect it. The model also proposes that all matter particles have an underlying standing wave structure, where various Doppler effects help explain length contraction and why mass [m] is subject to a relativistic effect. However, at this point, it is still unclear that we have a direct causal mechanism for time dilation.
If we were to accept the basis of this questioning, then we might
need to reconsider whether moving observer-S’ has a ‘distorted’
perception about its inertial status. In the context of a wave model,
there is only one stationary frame from which [β] can be determined,
i.e. the wave propagation frame. This position therefore must question
the idea that any inertial frame can simply assume itself to be stationary,
such that the Lorentz transforms can be applied to any other frame moving
relative to it.