Relativistic Factors

This section is an initial commentary on a paper entitled ‘Effect of Matter Distribution on Relativistic Time Dilation’, which is included as part of the description of the Cordus Conjecture. This paper considers the possibility of an extension of the Lorentz transforms and the nature of relativistic time dilation assumed to take place within the moving frame with respect to a stationary frame.  As this is an issue already discussed under the MMW wave model section of the Mysearch website, it might be compared against the Cordus model. In a wider context, the Lorentz transforms underpin the postulates of special relativity, which predict how an observer in one frame of reference will perceive motion in another, which it is generally assumed to require both length contraction and time dilation plus consideration of various Doppler effects. As an introduction of the Cordus model has already been outlined in previous discussions, this review shall start with the stated goal of the paper, which is to explain the Lorentz transform in terms of a non-local hidden-variable model, such that it might unify its ‘particule model’ with relativity. Since the goal of the Cordus model is to show a physical causality does exist, an alternative derivation might provide a better explanation of relativistic motion grounded in physical causality. One of the fundamental assumptions of the Cordus model is that the speed of light [c] in vacuum and the rate of time are inversely related to the fabric density.

Note: By way of general reference, the Cordus particule model covers both matter and photons that are assumed to be ‘energised’ at a given frequency that defines the particule type. This particule model then emits discrete forces, although the exact number and direction of these forces also depends on the type of particule being described. These forces are then assumed to propagate along ‘flux lines’ out into the space between particules. As a consequence, the vacuum of space is filled with a tangle of flux force lines that defines the functional nature of the fabric of space.

The Cordus model also assumes that the Lorentz transforms require modification to account for the fabric density, as it impacts the conventional concept of an inertial frame of reference. This idea then has implications on the definition of time dilation and the nature of any transverse Doppler Effect, when considered in terms of a particule model.

So, how might the Lorentz transforms be derived within this model?

The model starts by considering a mass particule [B], e.g. an electron, moving with constant velocity [vB] along the x-axis, as shown in the diagram below. It then assumes that particule [B] is emitting a discrete frequency, which propagates outwards with a velocity initially assumed to be the speed of light [c].

 

Note: The diagram above might immediately be questioned, if the inference of the red circle implies  the radius of the emission at time [t1]. The details of this configuration have been discussed in some detail under the heading of Ivanov Waves. So while the emission first sourced at time [t0] at [O] may have reached [Q], no emissions from [B]  having just reach [R] at time [t1] will have had time to propagate outwards.

It is stated that the derivation of the Lorentz transforms can be achieved by a geometric consideration of the effect of movement on the ‘flux tube of discrete forces’. Based on this model, particule [B] passes point [O] at time [t0] and emits a discrete field at this moment. After time [t1] this field emission moves out radially on the y-axis to a distance [ct1], i.e. point [Q]. During this same time [B] moves a distance [vBt1] to point [R] along the x-axis. [B] continues its field emission during this process. Of course, if [B] had been stationary at [R], its emission would have reached point [P] on the original vertical y-axis based on the propagation [ct1].

Note: While the description above might address the issue raised in the previous note, the statement: “the emission from [B] as it moves from [O] to [R] must have continuity of the flux tube rather than be broken” is not really understood, nor is the statement: “The emission from location [R] is not absent at [Q], but is instead stretched, hence redshifted”.

Based on the Doppler effect, the wavelength associated with emissions from [B] may be subject to redshift or blueshift depending on whether the receiver is behind or in front of [B]. We might also consider a number of other implications at this point. If [B] were stationary at [O] at time [t0], then its emission would have reached [Q] after time [t1] given a propagation velocity [c]. Alternatively, if [B] were stationary at [R] at time [t0], then its emission would have reached [P] in the previous diagram after time [t1] given a propagation velocity [c]. However, in the case where [B] is stationary, the wavelength would not have been ‘stretched’. If [B] were moving with velocity [vB] then it would reach [R] after time [t1]. However, in none of these scenarios does any emission from [B] at [R] have time to reach [Q] after time [t1]. As such, the formulation in [1.1] appears to have no basis in terms of emission propagation from [B], i.e. [RP], in time [t1] under any of the configurations suggested.

[1.1]          

Therefore, an alternative diagram is proposed to explain the revised form in [1.2], where the emission from [B] at [O] at time [t0] is seen propagating outwards with velocity [c] towards positions [Q] and [P], but where [P] is now defined as maintaining its vertical position above [B] at [R] at time [t1]. As such, we now perceive a revised geometry based on propagation velocity of the emission from [B] in time [t1], although it now requires the distance labels in [1.1] to be revised as follows.

[1.2]          

If we initially consider the revised result in [1.2] from the perspective of [B] being stationary, where [vB=0], then [RP] is equal to [OQ], i.e. ct1. Of course, by the postulates of special relativity, [B] might still consider itself as being stationary, as an inertial reference frame, but clearly from the perspective of the revised diagram, [B] is known to have velocity [vB]. Therefore, the vertical distance [RP] might now be interpreted as having been subject to a length contraction relative to [OQ]. Again, while we need to revised the distance labels, we might still arrive at an analogous form of the Lorentz factor [γ] shown in [1.3]

[1.3]          

While it is argued that [1.3] is generally compatible with the version in the Cordus paper, it is unclear that it depends on any of the assumptions of the Cordus particule model. Based on the revised diagram and the postulates of special relativity, [B] can consider itself to be an inertial reference frame, i.e. [vB=0], such that there is a discrepancy between the perception that distances [RP] and [OQ] are equivalent without knowledge of velocity [vB]. However, the postulates of special relativity create two problems that the Lorentz transforms sought to solve. First, there is the implied length contraction of [OQ] to [RP], which if assumed to be the light propagation distance in the inertial frame also requires a time dilation of time [t1], such that the speed of light [c] is maintained as invariant, which is also a postulate of special relativity.

But how might time dilation affect the measure of frequency of the emissions from [B]?

While the Doppler effect alone may cause a redshift or blueshift of wavelength depending on the position of the receiver with respect to a moving frequency source, time dilation is required to maintain the invariance of [c], which causes another effect on the perceived frequency of the source. Based on the accepted form of the Lorentz factor [γ] in [1.3], we can see that [γ] has a unity value when [vB=0], which will increase towards infinity as [vB] approaches [c]. However, a slowing of the rate of time in the moving frame also suggests that the frequency of any emission from [B] would also be a function of [γ], as per [1.4], although the subscript notation has been modified.

[1.4]          

The form of [1.4] has adopted the normal primed and unprimed notation to signified the relative value of frequency [f’] in the moving frame with respect to the frequency [f] in the stationary frame. However, again, it might be highlighted that the assumptions leading to time dilation are anchored to the postulates of special relativity without any obvious reference to any causal mechanisms assumed by the Cordus model

Note: If the assumptions of the Lorentz transforms are held true, then we might assume that they would apply to the Cordus particule model, i.e. the distance between the ‘reactive ends’ might be affected by length contraction. However, this assumption would only hold true if the axis of the particule model was orientated along the axis of motion. 

With the previous issues tabled, we will continue to pursue the idea that particle [B] is sourcing a discrete field with a given frequency, as described by the Cordus model. While it is not necessarily understood whether this really provides a causal explanation for this frequency, we shall assume that the particule emits an alternating ‘discrete force’ from one of its two ‘reactive ends’. This force is then described as being confined within a ‘flux tube’ that provides a rationale for the emissions from [O] and [R] being both stretched and synchronized. So, to reiterate some of the basic assumptions of the Cordus model, particule [B] has two reactive ends separated by a span, which is defined in terms of a fibril, although the physical structure and size of this span is unknown.

Note: The previous description has possibly given the impression that [B] is a single particule rather than an assembly of particles, such that there is only one field being emitted, rather than multiple overlapping fields. However, the model describes a single particule as being ‘coherent’ in the context that the concept of time is based on its own frequency cycle. In contrast, macroscopic assemblies of particules might generally be assumed to be ‘decoherent’ assemblies of multiple frequencies from different particule that overlap.

While we have initially assumed that the speed of light [c] is a constant, the Cordus model predicts that this speed of light is not universally constant, but may depend on the local ‘fabric density’. As previously outlined, the fabric-density appears to refer to the mesh of moving flux tubes that are assumed to exist in 3D space, where the density is determined by the ‘discrete forces’ emitted by all particules within some region of space-time, which ultimately explain all the different type of observed fields, e.g. electric, magnetic and gravitational. Based on this possibly flawed outline, it is assumed that the fabric-density is not always constant, such that the speed of light [c] may also be a variable under certain conditions, which is some function of the particule flux density in a given region of space-time.

Note: The Cordus model assumes that light, i.e. all EM waves, are photon particules, such that we need to understand the causal mechanism that explains why photons propagate at a certain speed through the fabric-density, which is a ‘tangle of discrete forces’ collectively created by all other particules, presumably both photon and matter. While the Cordus model alludes to an interaction between the photon flux lines and the sum total of the flux lines comprising the fabric-density, the exact causal mechanism is unclear at this point.

So, some attempt may be needed to clarify the scope of concepts being introduced in connection with the Cordus model. First, it is stated that time is described as an emergent property of matter, although the details behind this assumption are the subject of another paper – see next review section on emergent time. However, without going into these details at this stage, rate of time is assumed to exist on multiple levels, which is simply summarised in the note below.

Note: The perception that time exists at various spatial levels that define different coherent and decoherent boundaries, i.e. single electron particule, atom, molecules and organic cells. As such, time is initially correlated to the frequency of a single particule, but then aggregated as the structural complexity increases, which is then used to explain the irreversible nature of time, i.e. entropy, at the macroscopic level.

At the most fundamental level of a particule, time is dependent on frequency which corresponds to the rate at which the particule emits a discrete force. However, this frequency cycle also quantifies the ability of a particule to interact with its surroundings, i.e. only when it is energized. As such, we might perceive several ambiguities within this model when trying to quantify any measure of space-time, when defined in terms of the speed of light [c] and time [t], if both can be variables within different frames of reference. However, at this point, it is highlighted that the connectivity of any particule in space-time is still generally defined by its past light-cone with the caveat of the constancy of [c] and the somewhat contentious issue of superluminal communication that may exist between reactive ends connected by a fibril.

What else might be highlighted in terms of time and frequency?

While the Cordus model addresses this question in its own way, it is not unreasonable that this commentary offers up a different perspective, although possibly equally speculative.

Note: We might realise that the question above is a somewhat ‘chicken-and-egg’ issue, which might be worthy of some wider consideration. For example, from a human perspective, it is difficult to defined the idea of frequency other than as a cycle-count per unit time, even though this definition might be reversed, such that time is defined in terms of frequency. There is also a similar issue as to whether wavelength is a measure of space or vice versa. However, both of these issues are implicitly making reference to wave mechanisms, i.e. frequency and wavelength, which do not require any reference to matter particles, such that we might question the fundamental causal mechanism at work in the universe. We might take this line of tangential reasoning a step further by inquiring into the ‘substance’ of any fundamental particle. If we accept the basic inference of the energy equations of Einstein and Planck, as shown in [A.1] below, we might transpose the definition of mass [m] into either energy [E] or frequency [f] as follows:

[A.1]             

However, the scope of [A.1] possibly returns us to the quantum idea of the mass of a point-particule, which in any physical reality has to occupy some finite volume of space, such that it defines an energy-density. Of course, the idea of energy [E] can be an elusive concept, but one that might also be rationalised in terms of a wave model. In this context, energy is a scalar quantity that corresponds to some displacement within a wave media, i.e. potential energy, which then seeks to return to some equilibrium state. Again, it might be suggested that a wave mechanism might provide the most fundamental explanation of how this might be achieved, where excess energy is propagated away through the wave media with some unit velocity [c=1], which we might generalise in terms of the following equation.

[A.2]             

In terms of known causal mechanisms, the basic relationship in [A.2] can be explained in terms of a displacement of a wave media, i.e. energy [E], where frequency [f] is associated with some physical oscillation and propagation velocity [c] is an attributes of the wave media, such that wavelength [λ] becomes a resulting function. In this context, we might return to the issue as to whether frequency and wavelength are any more fundamental than time and space. For while frequency and wavelength might be described as the most fundamental causal mechanisms in a wave model, they do not necessarily replace the concept of time and space. For the wave mechanisms outlined still require the spatial distribution implied by a wave media, while the process of energy dissipation still requires time, irrespective of the frequency and propagation velocity of the waves. So, while the Cordus model might define a fabric-density in terms of a ‘tangle of discrete forces’, the flux lines associated with these forces still require the concept of a spatial distribution between particule, which propagate as a function of time.

If we return to the description of the Cordus model, the frequency of the particule is said to define its local rate of time, but where this frequency is also affected by the ‘resistance’ of the surrounding fabric-density into which the particules is emitting its own discrete force lines. As outlined, the fabric-density is not necessarily uniform and can encompass a range of field types, e.g. electric, magnetic and gravitational. In this context, any field emissions being sourced by a particule may also affect the fabric-density, which can also be compounded by the velocity and acceleration of the particule. Collectively, all these factors lead to the assumption that the frequency of the particule field emission can be slowed, such that the local measure of time might be subject to a ‘dilation’.

But how is this idea incorporated into the Lorentz transforms?

A variable [φ] is now introduced an associated with the fabric density, which reflects the gradient of the fabric density, which may be linked to the expansion of the universe and the assumption of a non-homogenous spatial distribution of matter. This latter assumption differs from the standard cosmological model, which assumes the universe to be homogeneous, at least, on a cosmological scale. Another difference in respect to the standard model, e.g. relativity, is that the Cordus model assumes that two inertial frames of reference are only equivalent if the fabric density is also equal. However, in terms of the previous example of particule [B], it was generally assumed that the fabric density [φ] was constant. However, in the case of a non-homogeneous fabric density, particule [B] will start in one fabric density and move into another fabric density. If this is the case, the frequency associated with particule [B] changes, as a function inversely proportional to fabric density, as indicated in [2.1].

[2.1]          

Initially, we might consider a general case, where particule [B] is moving with a non-relativistic velocity [vB1], when positioned in a fabric density [φ1] with frequency [fB1]. It then moves into a different fabric density [φ2], such that the frequency becomes [fB2], as characterised in [2.2].

[2.2]          

Again, the Cordus model assumes that frequency, at the fundamental level, defines the rate of time for the particule, though this discussion has questioned aspects of this assumption. However, according to the Cordus model, as particule [B] moves into a region of higher fabric density, e.g. [φ2], then the fabric resistance to the emission of discrete forces increases, such that emission frequency decreases. As a consequence, all processes associated with [B] are subject to a slower rate of time.

Note: Previously, it was argued that frequency is measured in terms of time, not vice versa. However, if all processes built upon particules are subject to a lower frequency, then it is possible that they would also experience a slower rate of time. In this context, we might see a general correlation with accepted relativistic models, where time dilation is a description of time ticking slower, such that it would correspond to a higher fabric density.

An extrapolation of [2.1] suggests that there would be a similar effect on the external measure of velocity, as it is also a function of time, as characterised in [2.3]. However, the causal mechanism by which the velocity [v] of a particule, i.e. single or multiple assembly, is affected is not understood in terms of normal kinematics laws.

[2.3]          

So, as described, a change in the fabric density [φ2] can cause an internal change in the frequency of emissions produced by the particule, which also affects the fabric density. If we are to correlate this effect with relativity, both special and general, then time dilation must be associated with a significant increase in the fabric density, which must be non-linear in order to conform to the Lorentz [γ] factor defined in [1.3]. We might also assume that the change defined by [2.2] and [2.3] would only be perceived by an external observer, who is positioned in some region of space-time, where the fabric density [φ1] is unaffected.

Note: There is some confusion in the suggestion of [2.3] when considered in terms of special relativity. In this context, velocity as determined by the moving and stationary observers is normally invariant and reconciled by time dilation and length contraction as shown in [A.3], where the primed notation reflects the measure of time and length in the moving frame and the unprimed notation is associated with the stationary frame

[A.3]             

It is also highlighted that special relativity allows the assignment of the unprimed and primed frames to be reversed, if both are inertial frames. However, it is unclear that this reversal could take place in the Cordus model as it is not obvious how the fabric density could simply be reversed.

In the description, so far, the inference has always been that particule [B] is a matter particle of some description, e.g. an electron. However, the Cordus model has a somewhat similar particule model for a photon of light, which it is assumed can only propagate through space at the speed of light [c], although this velocity is also inversely proportional to the fabric density [φ], such that [2.3] is extended to [2.4].

[2.4]          

Note: The form of [2.4] has been extended to highlight the implication that if the fabric density [φ2] increases above [φ1], for any reason, the speed of light [c2] falls relative to [c1].

Clearly, the idea that the speed of light in vacuum is not a universal constant is a major departure from mainstream models. While we will need to come back to the implications stemming from [2.4], the focus will now jump ahead of the detail in the paper to consider the issue of gravitational time dilation on matter particules. We might introduce the approach of the Cordus model by considering a matter particule, e.g. [B], moving away from some much larger gravitational mass [M]. In this context, it is assumed that [B] experiences a decrease in the surrounding fabric density [φ] as a function of an increasing radial distance from mass [M], where its velocity [v] is normally described as the escape velocity as per [2.9]

[2.9]          

As the issue of general relativity has been previously discussed, reference can be made via the previous link, although it is highlighted that this discussion was in the context of a wave model. Based on a somewhat classical derivation of the Schwarzschild metric, an equivalent form of the relativistic [γ] factor, as first shown in [1.3], is revised as per [2.10].

[2.10]       

Note: The final equation above has adopted the definition of the Schwarzschild radius [Rs], which is defined as the radius at which the escape velocity [v], shown in [2.9], equals the speed of light [c], which is also the definition of an event horizon of a blackhole. However, the Schwarzschild radius [Rs] can be calculated for any mass [M], although in most cases it will be far smaller than the physical radius of mass [M].

While the form of [2.9] and [2.10] might be considered conformant to the standard model, the Cordus model does not require either [c], or [G], to be universal constants, which will have impact on subsequent assumptions. Again, the initial inference of general relativity in connection with particule [B] is that we are discussing matter interaction with a gravitational field. Of course, general relativity also has some predictions about the gravitational effects on light, either as an EM wave or photon. However, at this point, the Cordus paper makes some statements that are not understood, which are first paraphrased and then discussed.

Gravitational redshift of photons: The previous case was for matter particule experiencing a changing gravitational field, where the frequency of the particule increases as it moves outward, i.e. away from mass [M], which is reflected in a changing rate of time. However, a different behaviour arises when the particule is a photon, and in this case a redshift occurs. For when a photon moves outwards against a gravitational field, it moves from a higher fabric density [φ2] to a lower [φ1], where the relativistic prediction suggest the photon frequency will reduce as it moves outwards, hence it will be redshifted. Equation [2.2] appears to contradict this view, as it predicts that the frequency will increase as viewed in the co-moving frame, but highlighting that this equation only applies to matter particules. For a photon, the lower fabric density [φ1] causes an increase in the velocity, as per [2.4], hence a stretch of the wavelength and a reduction in frequency. Thus, the Cordus model also predicts that the photon will display gravitational redshift, though attributes this to the change in fabric density rather than the gravitational field. As such, the Cordus model makes the falsifiable prediction that the gravitational redshift will depend not only on the gravitational potential, but also on the background fabric density. For situations with higher background [φ] the extent of the redshift will be reduced.

Many aspects of this description appear confusing. While there is a difference on the Cordus models for matter and photon particules in terms of velocity, i.e. [v] versus [c], the basic mechanism of field line emissions from the reactive ends appears generally equivalent. Equally, in the context of general relativity, the causal reason for an increase in the fabric density [φ2] would be attributed to the gravitational mass [M], which would affect both matter and photon particules alike, i.e. both would be subject to increasing time dilation as a function of decreasing radial position [r] with respect to mass [M]. If so, then both would be emitting a reduced frequency, which would correspond to an increased wavelength, i.e. a redshift, when approaching mass [M].

But, how do variable velocities [v] and [c] affect the Cordus model?

The standard model assumes [c] to be a universal constant, while [2.4] of the Cordus model suggests that it is a variable of fabric density [φ2], which the extended form of [2.4] suggests falls from [c1] to [c2] as a function of decreasing radial position within the gravitation well of mass [M]. Based on the earlier revised geometric model involving a particle [B] moving with velocity [v], while emitting a frequency with a propagating velocity [c], we might attempt a modification of [1.1], which is now present in [3.1]

[3.1]          

Note: The argument for the changed geometry was made on the basis that [RP] was the comoving equivalent of [OQ] in a stationary frame, when [vB=0], such that [RP] might be considered in terms of length contraction in the moving frame with respect to the stationary frame. However, the implication of changing fabric densities [φ1] and [φ2] now needs to be taken into consideration as suggested by [3.2].

[3.2]          

Again, it is highlighted that the Cordus model assumes that the ratio [φ12] affects both the frequency emissions propagating away with velocity [c] and the velocity [v] of the particule [B] equally. The causal explanation of this effect is not understood. However, it appears that [3.2] is anchored to [2.3] and [2.4], where [c1] and [vB] are assumed to be the velocities measured by a stationary observer surrounded by a fabric density [φ1], where the normal velocity of light is [c=c1]. On this basis, the final form in [3.2] has simply adopted the unprimed notation [c,v]. Finally, as before, we can convert the form of [3.2] to reflect a revised form of the Lorentz [γ] factor, as presented as [3.4] in the paper, but now extended to include the form for gravitation.

[3.4]          

So, while we have reached the same form as defined by the Cordus model, some of its assumptions have been challenged along the way. It might be highlighted that the form of [3.4] is still required to collapse to a Newton approximation in the absence of any relativistic factors, which the addition of [φ12=1] would maintain. However, what is absence from [3.4] is any causal explanation how the fabric density changes as a function of velocity [v] or in the proximity of the gravitational mass [M], although we might still consider the net effect of this revised formulation.  We know that normal relativistic effects defined by either the ratio [v/c] or [Rs/c] causes [γ] to increase from unity as [v] approaches [c] or [r] approaches [Rs]. However, the description of the fabric density ratio [φ12] suggests that this component of γ(φ) will decrease as the fabric density increases from [φ1] to [φ2] as implied in the case of an increasing gravitational field, although it is unclear which factor is most significant.

Note: The standard relativistic model is anchored in the idea of time dilation in the moving frame relative to a stationary frame. Normally time is denoted as [dτ] in the moving or gravitational frame and [dt] in the stationary frame of flat spacetime. We can express the ratio of [dτ/dt] in the form of the [γ] variable defined in [3.4], but where the ratio of [φ12] is removed.

[A.4]             

As suggested by [A.4], [dτ] is normally always less than or equal to [dt], i.e. [dτ] is time dilated with respect to [dt]. Therefore, we might reasonably assume that frequency sourced within the moving frame would also conform to this relationship, i.e. [fτ] is less or equal to [ft]. If so, we might consider a modified form of [2.2], but then add the implications in [A.4]:

[A.5]             

It is highlighted that the form of [A.5] in the note above is similar, but different in the subscript notation, where frequency [f2] corresponds to a particule in a higher fabric density [φ2] as measured by an observer positioned in a lower fabric density [φ1], such that the frequency is subject to both the ratio [φ12] and the Lorentz [γ] factor associated with time dilation. However, while we can quantify the scaling of the Lorentz time dilation to either velocity ratio [v/c] or [Rs/r], it is unclear how the ratio [φ12] scales under the same conditions or explained by the following statement:

The fabric density is determined by the spatial distribution of matter in the accessible universe around the location under examination. Thus, the fabric density is proposed to be the deeper causal mechanism that subsumes gravitational time dilation.

With some of the previous issues now noted, we will continue with the idea that the Lorentz transforms can be scaled to macroscopic objects, e.g. a billiard ball. Of course, such an object must consist of billions of particules that may emit different frequencies, but which it is assumed all scale by the same factor, i.e. γ(φ). However, recognising that these macroscopic objects also have a multitude of different interactions across all scales, i.e. sub-atomic and atomic, molecular and chemical bonding, field and particle kinematics, we might question how the revised Lorentz γ(φ) factor scales under all these causal mechanisms. However, having outline some of the primary assumptions of the Cordus model in relation to the Lorentz transforms, the next discussion will consider some of the issues associated with the Doppler effect.

Note: Before leaving the discussion of the Lorentz transforms, it might be highlighted that the Cordus model appears to say little about length contraction, which is another key assumption of the development of the Lorentz transforms used to  explain the null results of the Michelson-Morley experiment; although the link discusses this issue in terms of a wave model.