Please read the paper, may be helpful for you.

A new approach to the correction of

Galilean transformation

@ Robert J. Low : Thank you.

@ Sheng Liu : Of what I understand of the logic of the document, the mathematical formulation of the Hubble law must come from experimental data and the model parameters will be adjusted to reflect this reality. The proposed kinematics model does not attempt to predict the mathematical description of the observed expansion.

 In following article Hubble parameter simultaneously replaces time in special relativity and Newton's inverse square law, and also particle frequency in quantum mechanics. See eqns. 11 to 15, and eq. 36.

Article Periodic quantum gravity and cosmology

Zeeman defines the causality group to be the group of all transformations of R^4 which are bijective, but otherwise arbitrary and which preserve causal order. He then shows that this group in fact consists only of linear transformations, namely translations, rotations, dilations and special Lorentz transformations. So it is not clear how your non-linear transformations fit into the picture. They must, in some essential way, not be defined everywhere, or they must transform events which are causally ordered into such that are not, or viceversa.

Reference: E. C. Zeeman: ``Causality Implies the Lorentz Group’’, J. Math. Phys. 5, 490 (1964);

I think the point about the hubble expansion, is a noninertial accelarating system, at least over cosmological distances and times. Therefore special relativity is not relevant, except over limited space and time.

I can mathematically prove that the Special Theory of Relativity is false and does not represent physical and experimental reality.  So no, one cannot  formulate Hubble's law and the expansion of the universe within the context of special relativity and preserve physical, experimental reality. The proof is on my website www.jmkingsleyiii.info   

@ Juan Weisz :

Indeed, because special relativity (the original formulation without the Minkowski formalism which in my opinion introduces external mathematical elements to physical reality) only uses three-dimensional spaces with a uniform translational motion some with regard to the others, the fact of adding others three-dimensional spaces to the set (I would not say they are non-inertials as you do) means we are no longer under this theory. I use voluntarily a misuse of language.

The Big Bang observational reference frame R0 is the unique three-dimensional space whose cartesian spatial coordinates take all the values of R^3 and whose cartesian temporal parameters take all the values of R. Any other three-dimensional space has an origin in time and a finite volume every moment , and can be considered as an expansion of R0.

@ F. Leyvraz :

Events causally ordered keep this relation under these transformations (along any world line, two cartesian temporal parameters t and t' are always such as dt'/dt will never be zero) but actually, these transformations are not defined everywhere (on R^4).

I did not know the result of Zeeman and I was simply obsessed with the need to describe the fact that immobility or movement of a body (the stationary or non-stationary wave of probability) has to have a mathematical sense with respect to every conceivable human experimenter and not only with respect to an arbitrary Gaussian coordinates of a continuum of four dimensions.

I hoped at first to obtain solutions defined everywhere and even if I limit myself to the solutions having a certain level of regularity (differentiability), I notice that mathematically a cartesian temporal parameter (which is simply a formulation of Einstein's synchronization convention) can take values on all the mathematical real line only in a unique three-dimensional Euclidian space that I called the Big Bang observational reference frame. In all others three-dimensional Euclidian space a cartesian temporal parameter will necessarily take values in a real open interval which is unbounded and is different from R, and spatial coordinates are defined only in an open ball whose radius increases linearly with the cartesian time.

Also, although the expression of equations to be solved is completely symmetrical (except the imprecisions on the domains of functions), it appears an asymmetry in solutions with the discovery of the Big Bang observational reference frame R0 which has this property: with respect to any other three-dimensional space R, every point M (fixed spatial position) of R0 moves in a straight line with a constant speed, this straight line always passes by a unique world line O(R) which is constantly at rest both in R and R0, and this constant speed of estrangement of M with respect to O(R) gives the formula of the expansion of the universe: d/dt D(M,O(R)) = (1/t) D(M,O(R))

Mathematical difference with general relativity.

a) In general relativity we admit by definition that an affine parametrization (proper time) of a world line describing a trajectory of material body must mathematically result from a metric tensor on a manifold by integrating ds^2. Thus, if a clock is placed in the center of a disk in a laboratory (ie terrestrial), the relation between the proper time of the clock and the natural time of the laboratory (cosmic time) will not depend on whether the disk is put or is not put in rotation around the center.

b) In the model I propose, according to the presentation I gave by introducing this discussion, it is not a requirement that two cartesian temporal parameters of two separate three-dimensional spaces coincide along a world line which is constantly motionless in both spaces (when such a world line exists).

The answer is yes it is possible to give a form of Hubble’s Law valid in special relativity. But it is necessary to redefine both redshift and velocity. The present definition of redshift is not additive and consequently Earth-centred and it must be redefined logarithmically. The velocity must also be changed to relativistic velocity (rapidity). The relativistic law then takes exactly the same form as the known Hubble Law. I did this in a 1992 paper  ‘Logarithmic redshift and a relativistic Hubble Law’ which ia available under my name on RG. 

Robert Low is very correct in referring to the Milne model because the relativistic Hubble Law fits well here.

I would like to add a further comment.  The new measure of redshift is just log (1+z) where z is redshift as presently defined. It reduces to z for small redshifts. It can also be used for the Hubble law in General Relativity. 

Dear Rommel, I think it is more feasible to work with general relativity, which work with curved spaces generated by mass. The expansion of the universe depends on many factors, including dark mass and dark energy, the scale factor, etc. Therefore you must link Hubble with general relativity on curved spaces.

Actually the original Hubble's law v = (H_0)  d  is only valid for nearby galaxies with redshift z less than about 0.1. Moreover, galaxies do not really move with speed v.  For instance, the observed redshifts in the case of extragalactic sources like quasars and radio-galaxies are unlikely to be Doppler shifts. This is obvious from the fact that a large fraction of these sources have redshifts $1 \lesssim z \lesssim 7 $. Doppler shift origin for such large values would imply galactic size objects undergoing relativistic motion with very high Lorentz factor $\gamma \equiv ( 1 - v^2/c^2)^{-1/2}$, in which case, not only would they be stripped off their gaseous contents as they plough through the intergalactic medium with speed $v \approx c$, but would also have much greater observed mass $\gamma M_{gal}$, where $M_{gal} $ is the galactic mass in the rest frame.

Since, neither of these have any observational support, a much simpler explanation of such large redshifts is that of cosmological stretching of wavelengths due to expansion of the universe, which ensues from GR solutions predicting expanding (or contracting) universe for homogeneous and isotropic matter distribution. 

The short answer is No-an expanding Universe doesn't describe a Minkowski spacetime, except at any single point;  the two metrics aren't related by a diffeomorphism. And Hubble's law requires more than one point to make sense. The correct way is to use general relativity, precisely for this reason.  All the so-called paradoxes can be traced to the fact that general relativity isn't used-its symmetries aren't taken into account. In particular, the freedom to choose coordinates at any point implies that only non-local quantities can be given meaning-and these can be defined only on the boundary, or at infinity. The statement that any particular coordinate system is of physical significance is incorrect. This doesn't mean one can't make local measurements-only that they don't provide any physical information by themselves. So any apparent disagreement between two local observers doesn't mean anything-what matters is what the definition at infinity of their measurements implies for invariant quantities.

In my last post I wrote:

>

I should clarify:

>

In fact this is not the basic condition of the causal relationship which is always satisfied in these transformations:

If the instantaneous speed along a world line is always smaller or equal to that of light in one of the three-dimensional spaces, then this world line has this property in all other three-dimensional spaces.

I don't understand what is going on here. The question was 'can we formulate the Hubble Law in Special Relativity'. I answered and said 'yes, I have done it - my paper is on RG'. Silence, no comment at all but then someone said 'I think we should use General Relativity instead' i.e. we should answer a different question and ignore an answer to the one under discussion' How is it possible to proceed in this way? Is this science or some kind of game?

John Frederick Barrett,

I had read your article. I will not comment the section 7 which introduces the Robertson-Walker metric because I do not understand the physics of the ds^2 of general relativity. Besides, this dicussion is on Special Relativity therefore physics must be studied in a family of Galilean reference frames which are in uniform translational motion with each other.

In order to satisfy some formalation of the principle of Huygens, you suggest that equation (29) Z = ln(1 + z) = C D must be the exact theoretical formalation in a Galilean reference frame of the empirical law of Hubble z = C D .

You notice that this formula can be derived by assuming that the empirical form of Hubble is locally valid at each spatial position occupied by the light as it travels into the Galilean reference frame according to Equation 33. So it seems that you offer that in a Galilean frame of reference, the *energy* ( or wavelength ) of light should vary on its path a bit as it is the case for the intensity of a spherical wave. I think this looks like a tired light theory without any explanation on the physical mechanism. I'm not a physicist, but I read this on Wikipedia :

Regarding the model I propose :

Based on the theorem of Alexandrov - Zeeman, Andrew Vogt establishes in its article "A derivation of the Poincare group" that any mapping that is defined on all the four-real numbers to itself, and that turns any path light beam in a similar trajectory, is necessarily an element of the Poincare group extended by dilatation. This suggests that by keeping the use of Euclidean geometries to determine the spatial distances, the unique physics that can be extracted from the empirical principle of the constancy of the speed of light in vacuum is the theory of special relativity which is limited to the comparison of observations of the elements of a family of three-dimensional spaces in uniform translational motion with each other.

However, in reality, an extrapolation of the experimental Hubble law on leak motions of distant celestial objects suggests that the observable universe had a beginning in time. Accordingly, it would be well advised to seek bijective transformations which are not defined on all the four-real numbers but between parts of this set. We then obtain the transformations that I propose.

In classical kinematic, a relative motion between two three-dimensional Euclidean spaces is described by a translation vector with three components plus a rotation vector with three components. THERE IS NO spin vector. In the new kinematic, a relative motion between two three-dimensional spaces is described by a translation vector with three components plus a rotation vector with three components, PLUS a spin vector with three components.

In fact I solve the same equations as Zeeman but with a different physical interpretation (no Minkowski geometry) and with imprecisions on domains of functions.

I don't understand you at all. You are using a lot of sophisticated ideas in a completely irrelevant way whereas I just use standard Special Relativity, simple mathematics and a little common sense. The way you talk leads me to wonder whether you are even familiar with Einstein's Doppler shift formula since you refer to my use of it as a 'tired light' theory. The ds2 theory which you say you do not understand is just an addition to show that a similar idea can apply to general Relativity; it is not the main argument. But I must give up - it is hopeless to try to make myself understood or to make progress.

On so called "Hubble law"

There is no Hubble Law. Hubble measured redshift  z and found it to be proportional to distance d, z=pd. He naively assumed  that shift in spectra can be ONLY caused by linear Doppler shift z=kv. From genuine measurements and false assumption he concluded pd=kv, or v=Hd (, H=p/k)  what is ,by all idiots, since then called a Hubble law and a proof of expansion. See our works on that site or elsewhere.

You speak too wildly. To do him justice it should be said that he had considerable doubts in assuming the redshift was Doppler shift but he could not think of another explanation. In my paper I show a relativistic version follows directly from using Einstein's Doppler shift formula. For this you will regard me as an idiot except that I do have in mind a possible alternative explanation.

Regarding Hubble that is correct. His genuine observations was manipulated by people from the Einstein camp to fit  Rdot over R term. So inflation started from "politics" and  not from a science. What causes shifts - is negative curvature , due to position space and/or due to velocity space.  Einstein in my view has nothing to do with the story of redsfift beyond the  fact  he diverted all cosmological science into dead end street

I do not think about you the way you wrote. I think that you are one good track and have an open mind. I received your paper. Thanks.

John Frederick Barrett,

I want to respond to this:

>

By considering that in a Galilean reference frame the trajectory of a beam of light is always rectilinear between the source and the receiver, and that its speed is always equal to the relativist constant, the formula whom you propose Z = C D effectively solves the problem of  "faster than light" when the distance between the source and the receiver is arbitrarily large but this formula has the same shape in all the Galilean reference frames under the following conditions:

+ You propose a tired light theory without explaining the mechanism which tires the light. And your "Hubble parameter" does not depend on time (what is against the observations).

++ You do not propose a tired light theory and as Hubble (J. G. von Brzeski) you assume that the frequency shift comes only from a Doppler effect at the time of emission of the light signal. Then you place the Earth at the center of the universe and you believe that the earth is constantly motionless in a Galilean reference frame according to the following explanation:

(1) Your formula (24) is not the general formula of the Doppler shift (in standard Special Relativity) when the motion of the source and/or receiver are accelerated in the Galilean reference frame where we chose to make the observations.

(2) Your description is exact only if the receiver is on earth because if two separate receivers are aligned with the source in the Galilean reference frame where the observation is made, there will be conflict between the equation (26) which links Z which the speed of the source in the Galilean reference frame where the observation is made and equations (29) and (36) which connect Z with the travel time between source and receiver (There will be a travel time between the two receptors).

In all cases, your Hubble parameter does not vary in time.

Rommel Nana Dutchou

Sorry I really cannot answer you as I cannot understand what you are saying.. First you say I propose a tired light theory then immediately after you say the opposite. Then you say I place the Earth at the centre of the universe which I do not. Then you. talk about accelerated frames which special relativity never considers. Finally you say the Hubble parameter does not vary with time which it does not in special relativity but does in general relativity which I consider at the end of he paper showing time dependence... .

John Frederick Barrett, we agree. 

Rommel Nana Dutchou: good, best wishes

Dear Rommel Nana,

So called "Hubble Law" is a fake. There is no such thing as the "Hubble Law" and nobody  should use this terminology. Why and how Hubble did a crucial mistake and all astrophysicists after him is fully explained in " Illusions of space expansion...." paper on this site.

J. G. von Brzeski,

You state that an observer has a three-dimensional space whose geometry is Lobachevskian and you state that it is the mathematical formulas of this geometry that explain the fact that the geometry of the space on the local scales appears Euclidean, that explain the cosmological redshift and its linear approximation that was highlighted by the Hubble analysis.

On page 3 you write that "physics of Lobachevskian velocity space L^3_V, in a representation of a Poincare ball, is known as Special Relativity".

Let me ask you the following question in order to better understand your presentation and the notion of Lobachevskian velocity space:

If in an observer's three-dimensional and Lobaschevkian space we measure the length of a curve segment by adding the local measurements made by experimenters who are constantly at rest and aligned on the curve, and if for this observer The time is indicated by clocks which are constantly at rest and which are initialized step by step according to the Einstein convention, then how is defined a state of relative motion between two observers of the universe? How one describes a state of rotational motion ?

Dear Mr. Dutchou:

The geometry of a physical space is determined by the sensitivity of your instruments only. Experiments done within small domains in space and in small ranges of relative velocities shows no departure from the laws of Euclidean geometry. So in all your problems , in this case , you can use Euclidean mathematics to compute whatever you want to compute.

On cosmological scales and in case of relative velocities comparable to c your instruments clearly show departure from formulas of Euclidean geometry and you have to use non Euclidean geometry -Lobachevskian geometry in all your experimental data reduction problems .

Length of the curve  is defined ( by integration)  in both geometries and it can be found in any standard book on  geometry,  Euclidean and non Euclidean .

On a state of rotational motion see our new paper which will appear in a week on this site " Existence of an Absolute Motion and an Absolute Reference"

J. G. von Brzeski,

I think I have established that the cosmological redshift comes from the existence of a light medium which is an absolute reference in this sense.

However, all the mathematics of the Lorentz transformation in the context of special relativity remains valid (relativity of the notion of simultaneity, reciprocal contractions of lengths, reciprocal dilations of durations, ...).

Please read the comment I wrote on the following page:

https://www.researchgate.net/publication/296485231_On_the_Ehrenfest_Paradox_and_the_Expansion_of_the_Universe

Research On the Ehrenfest Paradox and the Expansion of the Universe

There is no expansion off the Universe. 

In classical physics, we can consider a boat moving uniformly with respect to the wharf. If a stationary observer on the boat chooses to describe the trajectory of a body using a coordinate system that he qualifies as Cartesian, and if another observer who is motionless on the platform also chooses to describe the trajectory of the same body using also a system of coordinates which he qualifies as Cartesian, then the formulas of transformation between the descriptions of these two observers will mathematically have the form to which we are accustomed.

This transformation between the descriptions of these observers could have an unusual mathematical form (without the physics of the problem changing) if at least one of them chooses to use rather a system of coordinates which he qualifies as spherical.

Thus, in order to determine the physics that is predicted by a transformation formula between two coordinate systems, it is necessary to specify the meaning (the concrete construction) that each observer gives to the system he uses.

What characterizes the physical theory that Einstein associated with Lorentz's mathematical formulas is the fact that a luminous signal emitted in all directions by a single source will represent a topological limit for possible future motions of a material body which is within such a signal.

This topological characterization is necessarily observed by all imaginable observers as soon as one of them can state it. And that is what identifies the physics that Einstein associated with observers whose motions are (he says) inertial.

In this physics called the theory of special relativity, if the observers in inertial motions each choose to use the Einstein-Poincaré procedure to date different events, and if in addition each one chooses to use the coordinates that he qualifies as Cartesian for space, then we must get mathematically that each uses a single numerical value to describe the movement of a light signal in any direction in a vacuum (the ratio between two numerical values used by two distinct observers will highlighting the fact that measurement standards for lengths and durations can be arbitrary), and we can see that the transformations between these coordinate systems are similar to those attributed to Lorentz.

The physics predicted by Einstein would not change if one of the two observers in inertial motions does not use the Einstein-Poincaré procedure to date different events, or does not use Cartesian coordinates to describe three-dimensional space. The mathematical expression of the transformation between chosen exotic coordinate systems will not be usual, but physics does not change because an electron will still not be able to catch a light signal emitted by itself in a vacuum.

If all the empirical laws (which are summed up by the maxwell equations; these are the first two Newton's laws of motion associated with the Lorentz force and the laws of Coulomb, Ampere, Biot and Savart, Faraday) were strictly accurate only in relation to a particular family of observers constantly at rest relative to each other, then the historical ether would exist and one might consider the possibility that nature can validate the transformation laws known in classical physics (or other speculations) when one wants to study physics for observers in motions with respect to this ether. The historical ether becomes completely useless (possesses no particular meaning) as soon as it is stated that each inertial observer can confidently formulate Maxwell's theory by considering himself as constantly motionless.

But if a possible medium of propagation of light signals in a vacuum was not likely to shelter observers constantly at rest, then the empirical laws that lead to Maxwell could be valid for all observers imaginable (on the planet earth, on the planet Mars, elsewhere...) and this ether would not be the historical ether sought by the ancients.

In fact, if the historical ether existed (the unique inertial reference frame where observers who are constantly at rest with suitably initialized identical clocks can notice the accuracy of all the empirical laws that ground Maxwell's theory), and if one supposed that the laws of transformation stated in classical kinematics are valid, then even if one can already affirm that the equations of Maxwell will be exact only in the ether's frame of reference because the speed the electromagnetic wave will not be isotropic in other inertial reference frame, we can add that the Coulomb law could no longer be valid in a reference frame that is in uniform translational motion with respect to the ether. Indeed, if an electric charge is animated by a uniform translational motion with respect to the ether, and if in its reference frame of rest it is necessary to use Coulomb's law to calculate its action on a test particle, then the acceleration of the test particle in this mobile reference frame will not depend on its own motion, and the formulas of the classical kinematics will imply that the acceleration of this test particle in the ether reference frame does not depend on its own motion and this is a contradiction because, since the source is moving relative to the ether, a magnetic field is produced and the Lorentz force must depend on the proper motion of the test particle.

Using the same analysis, we can see that the transformation of Lorentz is not enough to give up the existence of the historical ether. We must also consider the particular modification of Newton's second law (established in the theory of special elativity) in order to obtain that the Coulomb law (or Gauss's theorem for a continuously static electric charge distribution) can rigorously be formulated in any inertial reference frame.

Absolute Space of Isaac Newton and Samuel Clarke versus Relative Space of Gottfried Leibniz

Dear Sydney,

Whoever has the possibility to perform a certain number of elementary calculations, for example in geometry, does not need to have students in order to be interested in a relevant description of some phenomena that are accessible or that are of common knowledge. It is a need that can be felt like hunger and thirst. Moreover, other is the excitation of the experimenter, other is that of the theorist.

Physical science wants to establish relationships that explain (or predict) the interactions that occur between different material bodies in the environment of an alleged experimenter (observer). In order to be able to use empirical observations as a predictive tool, the same causes are considered to produce the same effects everywhere and at any time, and one would like to be able to identify, during an experiment observed by an experimenter, the responsibility which must be attribute to the different intrinsic natures of the bodies in presence, and the responsibility which must be attributed to the initial disposition of the bodies, in other words the distances and orientations that the experimenter observes between the bodies, and the responsibility which must be attributed to the states of motion that the experimenter observes for each body.

All bodies in the universe should normally be considered in the description of each experiment but since a causal principle has been assumed that the same causes must produce the same effects, locally repeated experiments are expected to be influenced in the same way by bodies which retain practically the same state during these experiences (same distances and directions, same states of motions, same intrinsic constitutions).

If the states of motion of the most distant bodies are not identical during different experiments conducted locally, it is not surprising that their influences are not the same so that this can explain the existence of inertial forces and call into question the need for the definition of a privileged reference system with respect to which there would be movements that must be qualified as non-inertial.

Dr. Clarke assumes the existence of a privileged reference system with respect to which the notions of positions must be defined for the description of the interactions between the bodies, and he states that if all the bodies of the universe are moved so to keep the same relations between them in the kinematic quantities defined within this privileged reference system, then one could at least state that their positions have changed. Obviously, this reasoning is only relevant if one assumes that the notions of positions and motions must be defined a priori in relation to a unique reference space, and such a hypothesis is superfluous in the description of any physics experiment, as Mr. Leibniz points out. The superior reasons mentioned by Mr. Leibniz are not necessarily of a theological nature and may simply be the empirical laws of electrodynamics.

Regardless of whether an experimenter is on the planet Earth or on the planet Mars, or elsewhere, it is desirable and it is assumed that in each of these situations a continuously static distribution of electrical charge always generates a field of electrostatic nature (attraction or repulsion of the test particles), and in addition it is assumed that mobile charges generate an electromagnetic field (characterized by the appearance of a Lorentz force). Thus a magnetized needle which is disposed immobile near a lead wire will be disturbed when passing an electric current in the wire. Thus two current-carrying wires aligned one near the other will attract or repel each other when introducing electric currents of the same direction or opposite direction.

The question that arises for me is how to translate these findings into a mathematical model and the answer I propose is a new formulation of the Mach principle, and a generalization of the expression of the electromagnetic force of Lorentz, in other words the the appearance of fields of non-electromagnetic nature when the motion of the electrical source is not rectilinear and uniform with respect to the experimenter. My document containing these equations was written in 2012: https://www.researchgate.net/post/Has_this_interpretation_of_the_Machs_principle_already_been_explored

In the theory of general relativity, there are four-dimensional tangent spaces at each event, a metric tensor field for assigning lengths to certain world lines, a mathematical notion of covariant derivative to define the variations of tangent vectors along world lines. There is no tool to characterize the fact that a body can be constantly at rest or in motion from the point of view of a designated experimenter, so that comparisons can be made between the characterizations of two separate experimenters (whatever their proximity): https://www.researchgate.net/post/Error_in_the_mathematical_formulation_of_Einsteins_equivalence_principle

The need to be able to state in a mathematically coherent way that two different experimenters (regardless of their proximity) must each have a reference system attached to him and within which one can define the kinematic quantities (constant immobility or motions of the bodies) seems to me to be a more fundamental preoccupation than the particular considerations which justified the introduction of the quantum mechanics formalism: if the results of the measurements made during a physics experiment are necessarily of a probabilistic nature when we consider bodies at the atomic scale, this cannot call into question the fact that an experimenter must be in a particular state of motion to carry out any measurement.

It is remarkable to note that the rigor of relativistic calculations can highlight new notions in the field of application of classical kinematic. Thus, we know that the expression of the electromagnetic force of Lorentz does not require relativistic experiments in order to be appreciated. And when we consider the transformation of the coordinates of classical kinematics in association with Newton's second law, this Lorentz force remains a postulate of experimental physics that cannot be reconciled with the principle of Galilean relativity (It should then be admitted that the known laws of electrodynamics can be strictly exact only in a privileged reference system that can be called ether). What is extraordinary is that this Lorentz force becomes a simple theoretical consequence of the association of the relativistic Lorentz transformation of coordinates and the appropriate rewrite of Newton's second law in a model which integrates a priori the principle of Galilean relativity. This is how I think that my relativistic calculations highlight the non-relativistic notion of spin of elementary particles, which it is usual to describe by operators introduced in this objective in the formalism that is postulated in quantum mechanics.

Dear Sydney Ernest Grimm, thank you for this information.

I would like to add that the error in Dr. Clarke's reasoning stems from the fact that, as is always the case in experimental physics, if he had assumed that a set of spatial positions (which is a mathematical space at within which we can state that bodies are either moving or constantly at rest) has a priori existence only in relation to at least one material object that is chosen as a physical reference, not necessarily as the origin of a mathematical coordinates system, then would have understood that in such a space one cannot mentally move all the bodies of the universe while keeping between them the same relations in the kinematic quantities because the material object which is the reference cannot be displaced with respect to itself.

I approve of the point of view of Mr. Leibniz who also wrote (Third letter to Samuel Clarke):

"I have said more than once, that I hold space to be something purely relative, as time; an order of coexistences, as time is an order of successions."

Reviewers' comments:

January 08, 2019

Reviewer #1: Very surprisingly to me the author of this work mixes different topics, among all: special and general relativity with quantum theory, etc. all together. This mixing is very useful as the readers are non-experts to introduce dummies to different topics. In the opposite case, it appears clear that nothing special has been reported. In fact, deeply looking at the manuscript, it is soon evident that the main topic of the work is wrong. Also the results, which make use of the Ehrenfest paradox, are meaningless. In particular:

1) there's no explanation on which metric the author lies on. Which is the topology of the universe? Which is the cosmological principle which is based on observations? in which ways the main hypotheses of modern cosmology are used?

2) the author ignores a century of literature and all the consequences developed during it. The work just takes a paradox and claims to use it in a confusing way.

3) The author mixes quantum mechanics with special relativity, without using second quantization. The basic strategy is to follow the mixing between special relativity and first quantization. Here completely ignored.

Furthermore:

Eq. 14 is trivial and the main results are unessential. Also: the paper is often hard to be read. Summing up, the paper cannot be accepted for publication.

................................

If the equations presented, which are already the subject of the E.C. Zeeman's paper:

Causality implies the lorentz group, Journal of Mathematical Physics, Volume 5, Issue 4, p.490-493 (1964)

were to have no solution other than those which are explained in my document, I would have at least put Evidence of new mathematics in the field of Euclidean geometry (possible transformations between Euclidean spaces). The logic of my work would then be similar to that presented in the invention of Jean-Marc Levy-Leblond's Carroll group:

Une  nouvelle  limite  non-relativiste du  groupe  de  Poincaré, Annales de l'Institut Henri Poincaré, section A, Vol.  III,  no 1, (1965), p.  1-12

What do you advise me to do so that my results can be cited as possible mathematics of a universe that is not ours?

Steen Markvorsen :

Speaking about physicists and the necessity of communication across disciplines, let me ask you about your discovery that 'causality implies the Lorentz group'.

Sir Christopher Zeeman :

...The homeomorphisms of this fine topology turned out to be the elements of the Lorentz group, so surprisingly the homeomorphisms really are constrained to be linear. You can build the whole of special relativity just on the notion of that fine topology.

When I gave a seminar to the physicists at IHES about this I proved that a homeomorphisms in this topology could eventually just be described in terms of a map that preserved the causal relation: a point 'a' is less than a point 'b' if 'b' is in the future time cone of 'a'. In other words an event at 'a' can cause an event at 'b'.

I then said that from there on it is an easy lemma to obtain the Lorentz group. At that point the physicists suddenly sat bolt up-right and said: "We don't know anything about topology, but that result you just mentioned - that preserving that causal relationship implies the Lorentz group - is enormously interesting. How did you prove that ?"

So my little lemma turned out to be important. I published a paper about it, which proved to be seminal in some sense because many people took up the idea and proved other results with it. So it is nice to have physicists around to tell you what you prove is interesting.

European Mathematical Society, newsletter No. 30, December 1998

Max Planck, The Principle of Relativity and the Fundamental Equations of Mechanics (1906):

"However that may be: a physical idea of that simplicity and universality, as contained in the relativity principle, deserves more than to be examined in a single way, and if it is incorrect it deserves to be led ad absurdum; and that can done in no better way than by exploring the consequences to which it leads. So perhaps at least from this point of view, the following investigation can provide some benefit."

"Remark 3. The condition for f to be a causal automorphism is a global condition, but is equivalent (by an elementary compactness argument using the transitivity of