A Lorentz variant theory that passes fundamental tests of special relativity and makes diverging, testable but as of yet untested predictions

Background Tests of special relativity have been conducted over the past century with increasing accuracy and none have showed violations of Lorentz invariance. In this paper we will examine whether these tests are together sufficient to rule out theories that violate observational symmetry. Methods A variant theory is outlined where relativistic effects such as length contraction and time dilation are purely local consequences of the relative velocity between a system and its medium. The outlined theory is tested against the fundamental tests of special relativity. Results It is found that although this alteration does not align with the principle of relativity, it quantitatively aligns with the experimental results of the fundamental tests of special relativity and their modern variations, and makes diverging, testable but as of yet untested predictions concerning Doppler shift and time dilation. Conclusions These results warrant a closer theoretical inspection of the outlined theory, and could provide a direction to test for new physics. A modified Ives-Stilwell experiment is proposed to test between this model and special relativity.


Introduction
Although all our physical theories are ultimately based on assumptions, some of our theories have been so predictive and well-tested that it's easy to mistake their underlying assumptions for proven laws of nature.Perhaps the foremost example of this is the principle of relativity, stating that the laws of physics are invariant in all inertial frames of reference.
Of course the principle is fundamental to both the special 1 and the general theory of relativity developed by Einstein in the early 20th century, as well as to Lorentz' ether theory, 2,3 but even before that it had been an integral part of physics for almost three hundred years, being a cornerstone of Newtonian mechanics.The principle was formulated first by Galileo in 1632.In his 'Dialogue Concerning the Two Chief World Systems' 4 he argues one cannot measure absolute motion.Using a thought experiment concerning a ship that sails in uniform motion on a perfectly smooth sea, he showed that an observer below deck would observe all to be the same as when the ship had not been moving at all.Examining Galileo's thought experiment we can say it is no coincidence that it took place within the ship as opposed to on deck.If the ship had been in motion relative to the medium of air it was sailing through, physical experiments conducted on deck would be affected by that relative motion.Of course one could still not ascertain if the ship was absolutely in motion or the air was, but there would be no observational symmetry between the reference frame at rest relative to the air and one in motion.Observational symmetry between two reference frames in relative motion seems to necessitate at least one of two conditions to be true: Either the medium surrounding the reference frames needs to be locally co-moving with the reference frames, or the reference frames need to be within a vacuum.
In this paper we will assume that there is no such thing as vacuum, but that every single part of is space if 'filled' with matter, so that every body may be considered to be surrounded by a material medium.If all space contains matter, all motion through space naturally influences observation.As such, the principle of relativity would not be a general law at all, only a special case that arises when special local conditions are met.
In essence the inquiry underlying this paper is simple.Could the relativistic effects we have directly or indirectly measured be caused not by the relative motion between a body and an observer, but by the relative motion between a body and its material medium?
The basic assumptions The principal assumption in this paper is that all of space is completely 'filled' with matter, meaning that there is not a single 'point' of emptiness in the universe, and that which we now perceive as empty space can be treated as a medium with extremely low mass density.Secondly we assume, ad hoc, that as the local velocity between a reference frame and this 'vacuum medium' increases, quantities such as time, length and energy in that frame will differ in magnitude from the same quantities observed in a frame at rest relative to the medium.Specifically, for the purpose of this investigation we will assume the time (t ALT ) given by a clock with a velocity of v μ relative to the medium be retarded compared to the time (t 0 ) given by a clock at rest with the medium by a factor of: Where c μ is some natural physical limit associated with the medium, which we will set to be so close to c, the speed of light, that upon approximation c μ ¼ c.This relation is of course reminiscent of the Lorentz factor (γ) within the context of special relativity, 1 where v is the relative velocity between two reference frames and c the invariant speed of light in vacuum.

REVISED Amendments from Version 1
Upon the reviewer's advice a more formal definition of the principle of relativity is provided in the introduction, and assumptions regarding the nature and assumed dynamics of the 'vacuum medium' have been made more explicit.The section on superluminal jets has been altered to make it more clear that there is experimental evidence for global superluminal velocities, possibly explained by a theory which supposes there are only local limits to velocity.Reference to relevant research showing that time dilation could be caused by the relative velocity between a (signal) clock and its medium has been added.
Any further responses from the reviewers can be found at the end of the article In special relativity relativistic effects could be called kinematic in nature, arising from the relative velocity between two inertial reference frames.Following this alternative reinterpretation of the Lorentz factor, we should interpret relativistic effects as mechanical in nature, arising from the local relative velocity between a body and its medium.If we take for example time dilation, we could say that physically a clock, simply, is an oscillation of a material system.So when we increase the velocity of a clock relative to the material medium surrounding it, the clock's oscillation is expected to be retarded.The higher the velocity of a system relative to the medium it is travelling through, the more resistance or inertia it encounters, the more the motions of a clock carried with it will be retarded.At very high velocities the effects on the system will be substantial, even if we consider the medium it's traveling through to have such a low mass density that we would call it a 'vacuum'.
In this paper it will be investigated whether we can assume the above and still align with fundamental tests of special relativity.

Aligning with experiments
Michelson -Morley Following special relativity the null result of the Michelson -Morley experiment 5 can easily be explained.In a comoving frame the apparatus can be considered at rest, thus the beam travel times are the same.
The experiment's null result demands the dynamics of the 'vacuum medium' surrounding Earth to remain constant to a high degree regardless of the motion of Earth relative to the solar system or another arbitrary body in the universe.As of now it is as much an assumption to say the medium should be at rest relative to the sun, Sagittarius A* or some other arbitrary body as it is to assume it is at rest with the Earth, in a flow more or less synchronous with Earth's orbit.
Considering the Michelson Morley apparatus, assuming for now the Earth and the medium to be at rest or their relative dynamics to be constant regardless of Earth's orbit, then regardless of the orientation of the arms, the conditions in both arms are uniform.Therefore, the travel time of the split light remains the same and the light remains in phase.

Kennedy -Thorndike
Kennedy's and Thorndike's experiment 6 is much the same as Michelson's and Morley's.A similar kind of interferometer was used with the difference being that one arm was shorter than the other, and measurements were made throughout the year, to measure the possible effects of the (orbital) velocity of the apparatus relative to a luminiferous ether.This experiment returned a null result as well.It can be explained for by special relativity in much the same manner as Michelson -Morley type experiments.
Again, we also find a simple explanation for experimental results within the framework of our material universe.The lack of fringe shift is easily explained for by considering that the conditions in the apparatus are uniform regardless of the orientation or orbital velocity the apparatus.
Modern variations of Kennedy -Thorndike experiments 7 have confirmed the original with much greater precision, but they may be explained for in exactly the same way.We could continue improving the accuracy of Kennedy -Thorndike experiments for years to come, getting the same results with increasing precision and yet they would not allow us to differentiate between the special relativity and our alternative theory.

Ives -Stilwell
Ives and Stilwell utilized hydrogen canal rays, beams of positive ions, to test if the frequency of light emitted by particles travelling with high velocities would follow classical or relativistic theory.
Their experiments 8,9 did not return a null result.Michelson -Morley and Kennedy -Thorndike experiments could be explained for by any theory that supposes that the motion of Earth through the solar system does not physically influence the measurements made in our laboratories or any theory that supposes it does, but in a manner that prevents us from detecting it.Ives' and Stillwell's experimental results are much more solid evidence for special relativity (or Lorentz' ether theory) because they align with relativistic Doppler shift as opposed to classical Doppler shift. 8As such, they are direct, positive evidence for relativistic time dilation.
Doppler shift, a change in the frequency of waves experienced by the receiver and the source of a signal that are moving in relation to each other can classically be described as such: Where f r is the observed frequency of the receiver, v w is the speed of propagation of waves in the medium, v r the speed of the receiver relative to the medium, v s is the speed of the source relative to the medium and f s is the frequency observed by the source.The convention used here is that v is negative when source and receiver are approaching each other.
Longitudinal Doppler shift in the context of special relativity can be described 10 as such: Where β is the relative velocity between source and receiver in terms of c.
It shows the expression for observed Doppler shift in the context of our theory would have to be different from special relativity.For one, relativistic Doppler shift doesn't allow for the source and receiver to approach each other faster than c, while our reinterpretation only limits local velocities.Secondly the reference frames of the source and the receiver are not interchangeable like they are in special relativitythey are uniquely dependent on local principle.More specifically, the time dilation a source or receiver experiences is a local phenomenon that relates to their velocity relative to their medium.

A thought experiment concerning alternative Doppler shift
Let us imagine a source and a receiver approaching each other through a medium.Both of them have some velocity relative to the medium they are moving through.Both the source and the receiver carry with them a Cesium atomic clock.
They have agreed to record the times at which they send and receive signals and compare these when they meet.To visualize this we could say the source will send a signal, or we could say the waves it emits will crest each 9.192.631.770periods of its Cesium atomic clock. 11As receiver and source approach or recede from each other the frequency of the signal the receiver measures blue-or redshifts because the distance and time between emission and measurement decrease or increase, following classical Doppler shift.But that is not all: as the source moves through the medium the Cesium atoms' periods will be retarded compared to when it would be 'at rest' relative to the medium.For each second that seems to pass in the frame of the source 1= ffiffiffiffiffiffiffiffiffiffiffiffi 1 À β 2 s q : seconds would seem to pass in a frame locally at rest, where β s is the velocity of the source relative to the medium in terms of c.Thus the source's local time, which dictates the frequency by which it sends and records its signals would be observed to be retarded by a factor of , and the receiver would observe the frequency by which the source sends out wave crests to decrease by a factor of Furthermore, the receiver has its own local time dilation dependent on its own velocity relative to the medium.The higher the velocity, the more its own Cesium clock is retarded, and the higher it would observe the frequency by which the signals reach him (by a factor of 1= ).If receiver and source would meet and compare their logs of signals sent and received, it follows that their results would align with the equation: Where: • f r ¼ the frequency observed by the receiver or the velocity of the wave relative to the medium in terms of c • β r ¼ v c or the velocity of the receiver relative to the medium in terms of c • β s ¼ v c or the velocity of the source relative to the medium in terms of c

Conditional equivalence Lorentz variant alternative and special relativistic Doppler shift
Equation 4 converges to Equation 2 when considering non-relativistic velocities and it can be shown that it is exactly equivalent to Equation 3 if at least one of the source or the receiver can be considered 'at rest' relative to the medium: Let us consider a source in motion (β s > 0) relative to its medium and a receiver in and at rest relative to the same medium (β r ¼ 0).We assume here that β w ! 1 so that: Where β is both the local velocity of the source relative to its medium and the global relative velocity between source and receiver.When examining the ratio between the predicted Doppler shifted frequencies of special relativity and the Lorentz variant alternative we find: And if we consider a receiver in motion and a source at rest, so that β r > 0 and β s ¼ 0 and again assuming that β w ! 1, we can see that again Equation 3 and Equation 4 are equal: Where β is both the local velocity of the receiver relative to its medium and the global relative velocity between source and receiver.

Conditional equivalent Doppler shift in arbitrary direction of motion
Equation 2-7 are valid for cases where the velocity of source and receiver can be considered to be aligned with the line between source and receiver, or one could say parallel or antiparallel to the direction of motion of the signal being sent.

Special relativity
Within the context of special relativity the observed Doppler shift when considering relative motion in an arbitrary direction can be described as 1 : Where θ s is the angle of the line between source -receiver with respect to the velocity of the receiver, as seen from a system of co-ordinates which is at rest relatively to the source.
If we instead describe the relation as seen from a system of co-ordinates which at rest with the receiver the equation is: Where θ r is given by the angle between the direction of the line receiversource and the direction of the velocity of the source.

Lorentz variant alternative
Within the context of the Lorentz variant alternative theory discussed in this paper the more generalized form of Equation 4 is described as such: Where cos θ r and cos θ s are the cosines of the angle of the velocity of the receiver or source relative to the direction of the line sourcereceiver and receiversource respectively.
Let β s ¼ 0 and β r > 0 and β w ! 1.The alternative theory describes it as such: Where β: is both the local velocity of the receiver relative to its medium and the global relative velocity between source and receiver and γ a function of β.
Let β r ¼ 0 and β s > 0 and β w ! 1.The alternative theory describes it as such: Where β is both the local velocity of the source relative to its medium and the global relative velocity between source and receiver and γ a function of β.

Conditional equivalence
When examining the relation between the angles utilized in special relativity (SR) and the alternative (ALT) one can see that θ ALT r ¼ θ SR s and θ ALT s ¼ θ SR r and thus in the case of a source at rest: And in the case of a receiver at rest: Due to the relativistic aberration of light the relation between cos θ ALT s and cos θ ALT r and consequently between Equation 13 and 14 is given by the equation 1 : Interesting to note is that when we set the angle to π=2 we obtain the transverse Doppler effect: Where in the context of the alternative theory ffiffiffiffiffiffiffiffiffiffiffiffi 1 À β 2 p f s is a more apt description, because it is the time dilation that the source experiences that is indicative of the change in observed frequency.

General inequality special relativity and variant Doppler shift
We can remark that up until now there's a striking similarity between the predictions made by special relativity and the alternative.But this equivalence ceases to exist when considering (most) cases where both source and receiver can be considered in motion relative to their medium.The difference between the predictions of Equation 10 and those of Equation 8 or 9 becomes more and more pronounced as β, β r and β s approach 1.

Aligning with Ives -Stilwell tests
Ives -Stilwell type experiments have been conducted throughout the years with increasing accuracy [12][13][14] but as far as the author knows up until now have always involved either a receiver or a source that could be considered at rest relative to the 'vacuum' medium surrounding it, or at least at rest relative to the surface of the Earth where the dynamics of the medium are considered uniform regardless of Earth's orbit.These tests have thus not been able to differentiate between the alternative theory outlined in this paper and special relativity.

On superluminal velocities
One might argue that since c μ is a local limit as opposed to a global one like the speed of light in the context of special relativity, the theory would allow us to observe global superluminal, faster-than-light, velocities, provided the local limit is maintained, and because we haven't the outlined theory is likely to be of little practical interest.
Yet we might be observing superluminal velocities on a regular basis, for example in so called relativistic jets of matter expulsed by active galactic nuclei.These jets of matter have apparent superluminal speeds of up to almost 10 times the speed of light, 15 with the innermost parts of the jets attaining the highest speeds while the outermost parts appear slower and are themselves propagating through and interacting with a constant pressure cocoon, either a very hot gas or a magnetized sheath. 16In essence these jets are layered cylinders, with the highest velocities achieved in the inner layers.Precisely the conditions in which you would expect to possibly observe superluminal velocities, following our theory.
As of now the most commonly used explanation for these seemingly superluminal observations is that due to relativistic beaming (Doppler effect) we're looking at something like an optical illusion.We assume that the jets move with relativistic subluminal velocities, but that the angles of the jets are very close to our line of sight, so that we observe the motion as superluminal.What's problematic is that assuming that jet angles relative to us are randomly distributed, only very few AGN ($ 1%) would have the small (< 8°Þ viewing angles required to explain for our observations of extremely high apparent superluminal motion.Instead we observe about 33% of the radio-loud AGNs with high redshift to exhibit such highly superluminal motions. 17rthermore almost all jets display significant changes in their position angles over a 16 year interval, 18 and often display bends with misalignments of 90°or more, up to $ 180°: 19 Any change in alignment should be cause for significant changes in brightness, but these changes are often not observed. 20The commonly used explanation is that even though we see large misalignments in the jets, actual bendings may be small, and because the jet's angle is very close our line of sight, even small intrinsic bending may appear as a large misalignment. 20The problem with this is that if misalignment angles were randomly distributed, then it is statistically highly unlikely to observe the distribution of angles we seem to observe, and secondly even very small bendings in jets that are close to our line-of-sight would affect brightness in a significant manner. 20us it could be argued there is at least some practical application for a theory that allows for global superluminal velocities.

Testing new physics
To test between the theory outlined in this paper and the special theory of relativity we would need to conduct experiments where both the source and the receiver can be considered to be in motion relative to their supposed medium with some velocity.The experiment would have to be conducted in such a way that we can assume the clocks of the source and receiver to be sufficiently exposed to the medium they're travelling through.
Curiously, although the predictions of the alternative theory diverge with those of special relativity, the convention currently used to ascertain the accuracy of Ives -Stilwell type experiments cannot be used to differentiate between the two.Modern Ives -Stilwell tests 13,14 try to experimentally confirm the validity of special relativity's prediction that: Where f a and f p are the frequencies of the lasers propagating antiparallel and parallel to the ion beam and f 1 and f 2 are the rest-frame transition frequencies.
The most accurate Ives-Stilwell experiment that has been conducted up until now, by Botermann et al in 2014, 14 found experiment to align with special relativity's prediction with an accuracy of α j j≤2, 0 Â 10 À8 , where α Setting f 1 f 2 to unity, we can show that in the case that if at least one of β r ¼ 0 or β s ¼ 0 is true, our outlined theory predicts ε β ð Þ ¼ 0 just as special relativity, assuming β w !c: And: As is to be expected considering For the more general case where β r > 0 and β s > 0 it is necessary to employ Equation 4.
Considering that: It shows that even in the general case the alternative predicts Interestingly, Ives-Stillwell type experiments do provide us with an upper limit of the value of c μ within the context of the alternative theory.Equation 20 assumes that β w ! 1 and therefore β w ¼ 1.Following the theory light would never physically be able to reach a velocity equal to c. Since β w 6 ¼ 1, an accurate enough Ives -Stillwell experiment would in theory be able to distinguish between special relativity and the alternative.Looking at the accuracy achieved in 2014, 14 jαj≤ AE 2 Â 10 À8 , and assuming that the velocity of light in our 'vacuum' chambers is v light ¼ 299:792:458 m=s then it follows that v light < c μ ≤ AE 299:792:461 m=s.

Speculation regarding gravity
For the sake of simplicity we have assumed the 'vacuum medium' to be more or less at rest with the surface of the Earth, but perhaps it would be more natural to assume that it has a motion associated with gravity radially inward towards Earth, and that this inflow is largely unaffected by Earth's motion relative to the rest frame of the solar system or Milky Way.Following the assumption that the drag of the 'vacuum medium' is causal to the dilation of physical clocks, it would follow that the time dilation associated with gravity should indicate a flow of the medium relative to a frame at rest with a gravitational well.The time dilation predicted by General Relativity is given by: If the relative motion between a clock and the medium is causal to this dilation then following Equation 1 it should mean that the relative velocity between a clock held at the surface of a gravitational well is given by: Which is also the escape speed at given r from the centre of a gravitational mass M.So the velocity field surrounding a mass M in spherical coordinates, with the mass at the centre of coordinates and the flow field spherically symmetric, would be: Deriving the field of acceleration from a flow field involves calculating the spatial and temporal derivatives of the velocity field: Assuming steady flow the velocity field does not change with time, we only need to consider the convective acceleration term: Or, using spherical coordinates: We can formulate the rate of change of velocity with respect to position as: Thus: And thus the field of acceleration described by the medium is: Interestingly, a body completely at rest with the medium accelerating towards a gravitational mass in this manner would locally experience all to be the same as if it were in the bow of Galileo's ship.

Conclusions
This paper shows that by assuming that the universe contains absolutely no free space we can derive an ad hoc reinterpretation and slight alteration of the mathematical framework of the theory of special relativity, that is incompatible with both the postulates of special relativity and with the Galilean principle of relativity, yet can quantitatively account for fundamental tests of the special theory of relativity.
The outlined model's predictions diverge with special relativity's regarding Doppler shift in most situations where both the source and the receiver would be considered in motion relative to their medium, but would exactly align with special relativity's predictions otherwise.
Of course, the basis of the argument needs to be more thoroughly examined before we follow it to its conclusions.A more thorough investigation is needed to ascertain if there are indeed no tests that have experimentally ruled out the outlined theory, and perhaps a more thorough theoretical examination of the outlined theory is in order.Already research has been done showing that time dilation could be caused by the relative velocity between a (signal) clock and its medium, 21 and it might be possible to provide a more foundational basis for the theory investigated in this paper by deriving the 'local Lorentz factor' from fundamental principles like conservation of energy.Also, a theoretical inquiry into a fluid model for gravitation, which treats gravitational masses as sinks into which the 'vacuum fluid', or indeed space, flows, with the amount per unit time being proportional to their mass or energy, could yield interesting results, whatever their relation to our universe.
In any case, as of now the idea outlined in this paper seems to show there could be a Lorentz variant theory that is not ruled out by the fundamental tests of special relativity that we have conducted up until now.We could however conduct modified Ives-Stilwell experiments to differentiate between them and to possibly probe for new physics.
The manuscript attempts to study the potential of an aether-like fluid present in special relativity and more broadly general relativity.The manuscript involves a series of textbook exercises involving three inertial frames and is focused on the influence on the Doppler effect.This additional degree of freedom seems to be poorly motivated both theoretically and experimentally, besides not solving any outstanding problem present in special relativity experiments today.The work needs more theoretical or to be supported by outcomes from data analysis or experiments results.
For these reasons, I do not recommend this manuscript for indexing in its present form.Below, I detail the precise issue to be reconsidered by the authors.

Section 1 -Introduction
The opening covers some historical aspects of the principal of relativity including how this assumption permeates through Galileo's work, as well as Newton's and then in Einstein's special and general theories of relativity.However, this part of the manuscript then takes a turn and makes unfounded statements such as "no such thing as vacuum" rather than suggesting that these challenges to traditional physics are probes, they are directly stated, making vague whether these are questions to explore or statements to be verified.Moreover, the motivational part has no references to the literature rendering this a rather isolated work.On this point, the terse content in this section makes it hard to understand what framework the author is adopting, as an example, a distinction is made between inertial frames of reference and the background material (air) through which these measurements are being made.My strong suggestion would be to flesh out the motivation for this work, and write a clear research hypothesis that covers the main results of the work.
Section 2 -The basic assumptions This short section lays the groundwork for the theoretical development that then takes place in later sections.The core idea is to adopt the aether theory within special relativity where an additional degree of freedom is introduced in the form of the aether dynamics.The motivation or need for this remains an open question.Similarly, a direct measurement also would seem to remain elusive given that the vanilla version of special relativity remains in strong agreement with observational measurements.
Section 3 -Aligning with experiments In this section, three classic tests (and some of their variants) are discussed in support of special relativity.More modern tests are not discussed, which is unexpected since their increased precision may provide evidence for or against the proposal in this work.The section then transitions into a discussion of the Doppler effect where a textbook calculation of three frames under this effect is calculated.The next part, similarly, performs a basic limiting exercise to show that as the medium speed decreases, that the equations match.Another standard exercise is then discussed wherein the Doppler effect takes place at an arbitrary angle.This is then generalized for the three frame setting in the next subsection.On this, is this derived from first principles, or simply a substitution of the equations already shown?The section eventually closes with a discussion on conjecture of possible experimental results, why not perform data analysis to assess the strength of any possible such aether-like medium?
Section 4 -Testing new physics The hypothetical experiment setting under which this idea could be tested in outlined here.The author finds a best-case scenario in which there would be a marginal deviation from the speed of light but this lacks appropriate uncertainties so it is difficult to assess the statistical significance of the suggestion.The bigger question would be, don't special relativity experiments already test objects moving relative to a background aether, provided the aether structure that does have such small-scale evolution profiles?
Section 5 -Speculation regarding gravity The gravitational part of the manuscript is contained in this section.The author takes the Earth to be stationary with respect to this aether-like medium from the start which seems to negate the underlying proposal of this work.The author then convolutes the Schwarzschild time dilation formula with that in special relativity in Eq. ( 1).Ultimately, this is an expansion exercise where the leading order acceleration of this possible medium turns out to observe the regular Newtonian acceleration equation, as expected.

Section 6 -Conclusions
The Conclusion closes with a brief summary of the main observations of the work but is light on quantitative results.This is mainly a discussion of possible future theoretical, data analysis and experimental work.

Sincerely, Referee
Is the work clearly and accurately presented and does it cite the current literature?

Santanu Raut
Mathabhanga College, Cooch Behar, West Bengal, India This article derives some interesting results assuming that the universe contains absolutely no free space.Altering the mathematical framework of the theory of special relativity with the Galilean principle of relativity, some tests of special relativity have been conducted over the past century with increasing accuracy and none have showed violations of Lorentz invariance.In the present article, the author examines whether these tests are together sufficient to rule out theories that violate observational symmetry.The author used a variant theory through which relativistic effects such as length contraction and time dilation are purely local consequences.The results are satisfactory for indexing the article.

Roman Szostek
Rzeszów University of Technology, Rzeszów, Poland The article deals with a very interesting topic, i.e. the mathematical study of a theory that is an alternative to the Special Theory of Relativity and the problem of testing such a theory.
The work is interesting.In my opinion, the article should be indexed, but after taking into account the following additions and corrections: 1.I suggest that the author provide an explicit definition of the principle of relativity.In the 'Introduction' the author discusses this topic, but does not explicitly define the principle of relativity.
1.I have included a more explicit definition of the principle of relativity in the introduction.
2. You are completely right in that the assumptions regarding the dynamics between the medium and the apparatus or lack thereof should be more thoroughly explained.I have provided a more thorough explanation of the physical picture of the 'vacuum medium' that is assumed in my paper in the introduction, and explained why it is as natural to assume the medium is at rest with the apparatus as to assume it isn't in the Michelson Morley section.
For the sake of completeness, I also included a section near the end of the paper which considers a scenario where the apparatus and the medium are not at rest, but their relative motion uniform regardless of Earth's orbit by considering gravity.
3. Perhaps as you suspected: A mistake on my part, I meant to say as β approaches 1, or v approaches c.Thanks for catching this mistake.
4. I have tried to explain more thoroughly what I meant by super-c velocities and have included some studies in the field to highlight the potential need for a theory that allows for global velocities that exceed the speed of light.I'm not sure if the transformations you've provided, their merit notwithstanding, could be extended without additional considerations to these superluminal jets, whose dynamics might not be well suited to be treated purely kinematically, although it might be an exciting topic for future work.For the sake of simplicity I would just want to state here that if we're relieved of SR's second postulate, we could in theory observe global velocities greater than the speed of light, but still we would expect there to be physical, local limits.Therefore these layered cylinders are places where we could expect to possibly observe these high velocities.Although of course, as you posit, even from the perspective of SR one could argue that c should not necessarily be treated as an universal, global limit.
5. Thank you for introducing me to your work!I have included a reference to your work on a physical explanation for time dilation, which I think is a very fundamental approach that aligns with the investigation in my paper, and should indeed be referenced.I have made a preliminary inquiry into formulating time dilation by treating a clock as an underdamped harmonic oscillator, whose damped frequency of oscillation is given by ω_1=ω_0sqrt(1-ζ^2) and the damping ratio ζ∝ v/c.When the relative velocity between the medium and the clock reaches c the oscillator becomes critically damped, and no longer functions as a clock, much like the light clock considered in your paper.Perhaps sometime we could discuss further.

Thank you again for your time and consideration, DMF Bischoff van Heemskerck
Competing Interests: No competing interests were disclosed.
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Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Yes If applicable, is the statistical analysis and its interpretation appropriate? Not applicable Are all the source data underlying the results available to ensure full reproducibility? No source data required Are the conclusions drawn adequately supported by the results? Yes
Competing Interests: No competing interests were disclosed.Reviewer Expertise: Nonlinear dynamics.I confirm that I

have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.
https://doi.org/10.5256/f1000research.162306.r246481© 2024 Szostek R.This is an open access peer review report distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Yes If applicable, is the statistical analysis and its interpretation appropriate? Yes Are all the source data underlying the results available to ensure full reproducibility? Yes Are the conclusions drawn adequately supported by the results? Yes
The author took my comments into account in a satisfactory manner.IsCompeting Interests: No competing interests were disclosed.I confirm that I

have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. Version 1
https://doi.org/10.5256/f1000research.141795.r200395© 2023 Szostek R.This is an open access peer review report distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.