## Abstract

In this chapter, cosmological models and the processes accompanying the propagation of the cosmic rays on cosmological scales are considered based on particle dynamics, electrodynamics and general relativity (GR) developed from the basic concepts of the ‘relativity with a preferred frame’. The ‘relativity with a preferred frame’, designed to reconcile the relativity principle with the existence of the cosmological preferred frame, incorporates the preferred frame at the fundamental level of special relativity (SR) while retaining the fundamental space-time symmetry which, in the standard SR, manifests itself as Lorentz invariance. The cosmological models based on the modified GR of the ‘relativity with a preferred frame’ allow us to explain the SNIa observational data without introducing the dark energy and also fit other observational data, in particular, the BAO data. Applying the theory to the photo pion-production and pair-production processes, accompanying the propagation of the Ultra-High Energy Cosmic Rays (UHECR) and gamma rays through the universal diffuse background radiation, shows that the modified particle dynamics, electrodynamics and GR lead to measurable signatures in the observed cosmic rays spectra which can provide an interpretation of some puzzling features found in the observational data. Other possible observational consequences of the theory, such as the birefringence of light propagating in vacuo and dispersion, are discussed.

### Keywords

- general relativity
- FRW models
- late-time cosmic acceleration
- dark energy
- UHECR
- gamma rays
- photo pion-production
- pair-production

## 1. Introduction

Lorentz symmetry is arguably the most fundamental symmetry of physics, at least in its modern conception. Physical laws are Lorentz-covariant among inertial frames; namely, the form of a physical law is invariant under the Lorentz group of space-time transformations. Therefore, the Lorentz symmetry sets a fundamental constraint for physical theories. Nevertheless, modifications of special relativity (SR) and possible violations of Lorentz invariance have recently obtained increased attention. Although, the success of general relativity (GR) to describe all observed gravitational phenomena proves the fundamental importance of Lorentz invariance in our current understanding of gravitation, some of the modern theories (unification theories, extensions of the standard model and so on) suggest a violation of special relativity. The aim of most of the Lorentz violating theories is to modify a Lorentz invariant theory by introducing small phenomenological Lorentz-violating terms into the basic relations of the theory (Lagrangian density, dispersion relation and so on) and predict what can be expected from it. Reviews of the most popular approaches [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26] to parameterizing Lorentz violating physics in the context of their relation to the ‘relativity with a preferred frame’ can be found in [27, 28]. Some of those studies are discussed in the following sections about the results obtained in the present paper.

The theory termed ‘relativity with a preferred frame’ developed in [27, 28, 29] represents a very special type of a Lorentz violating theory that is conceptually different from others found in the literature. It is not even a preferred frame that makes a difference—all violations of Lorentz invariance, made by distorting Lorentz-invariant relations of the theory, imply the existence of a preferred frame for the formulation of the physical laws, the one in which all the calculations need to be carried out, since breaking relativistic invariance also invalidates the transformations that allow us to change reference frame. The first major difference of the present analysis from the above-mentioned studies is that the Lorentz violation * is not introduced* into the theory but it is a result of using freedom in formulation one of two basic principles of special relativity, the principle of universality of the speed of light. In other terms, Lorentz’s violation is ingrained into the framework of the theory at some fundamental level. The second major difference is that the relativistic invariance, in the sense that the form of a physical law is invariant under the space-time transformations between inertial frames, is not violated—it is a

*.*Lorentz violation without violation of relativistic invariance

To outline the framework of the theory named ‘relativity with a preferred frame’ one has to start from the definition of the preferred frame. In the ‘relativity with a preferred frame’, the preferred frame is defined as the only frame where propagation of light is isotropic, while it is anisotropic in all other frames moving relative to the preferred one (it is a common definition in the studies investigating the fundamentals of special relativity and its potential breaking).^{1} Discussing the anisotropy of propagation of light one has to distinguish between the * two-way* speed of light, i.e. the average speed from source to observer and back, and the

*speed which is a speed of light in one direction—*one-way

*from source to observer*either

*back. In the ‘relativity with a preferred frame’, it is the*or

*speed of light that is assumed to be anisotropic in all the frames except the preferred frame, while the*one-way

*speed of light is isotropic and equal to*two-way

^{2}The analysis is based on the invariance of the equation of (anisotropic) light propagation for the space-time transformations between inertial frames and the group structure of the transformations plays a central role in the analysis. Although, the existence of the preferred frame seems to be in contradiction both with the basic principles of special relativity and with the group property of the transformations, in the framework of the ‘relativity with a preferred frame’, those principles are retained. The crucial element, which allows retaining the relativistic invariance and the group property of the space-time transformations, is that the anisotropy parameter

The above-described generalization of special relativity cannot be validated by experiments measuring the speed of light since only the two-way speed of light, the same in all the frames, can be measured. For creating a physical theory, predictions of which can be compared with observational data, it is needed to identify the preferred frame of the present analysis, which is defined by the property of isotropy of the one-way speed of light, with a frame possessing the property that velocity of any other frame relative to it can be measured using some physical phenomena. In the present analysis, that preferred frame is a comoving frame of cosmology or the CMB frame (note that identifying the preferred frame with the CMB frame is a common feature of practically all Lorentz-violating theories). It is the only frame possessing the property, that motion of any other frame relative to it is distinguishable, and, in addition, this frame, like the preferred frame of the present analysis, is defined based on the isotropy property. As a result of specifying the preferred frame, all the relations of the ‘relativity with a preferred frame’, as well as of its extensions, contain only one universal constant

Identifying the preferred frame with the cosmological comoving frame implies that the theory should be applied to phenomena on cosmological scales. Studying different phenomena requires extensions of the modified SR kinematics to different areas of physics. The purpose of this chapter is to present a unified view of the extensions and their applications based on the concept of the modified space-time symmetry. This includes extension to general relativity (Section 4.1) and constructing cosmological models based on the modified general relativity (Section 4.2); extension to the dynamics of the free particles (Section 3.1) and its application to the processes accompanying the Ultra High Energy Cosmic Rays (UHECR) and the gamma-rays propagation (Sections 5.1 and 5.2); extension to electromagnetic field (Section 3.2) and studying electromagnetic waves based on the modified electrodynamics (Section 3.3) with application to the gamma-rays propagation (Section 5.3).

## 2. Special relativity kinematics

Kinematics of the ‘relativity with a preferred frame’ will be only outlined in this section, for a detailed presentation see [27, 28, 29].

The transformations between two arbitrary inertial reference frames

where

Eq. (1) incorporates both the anisotropy of the * one-way* speed of light as equation (2) shows and the universality of the

*speed of light in the sense that it is equal to*two-way

^{3}The transformations involve both the space and time coordinates

where

where, based on the symmetry arguments, it is assumed that the transformations of the variables

Proceeding by the usual Lie group technique (see [27, 28, 29] for details) one can define the form of the transformations in

where

with

Furthermore, introducing the new variables

converts the invariant combination (7) into the Minkowski interval

while the transformations take the form of rotations in the

The expression (7) for the modified interval and the transformations (9) contain the function

where

which allows to calculate the factor

In the subsequent analysis, those general relations are specified using an approximation for

With this approximation, the group generator

and, correspondingly, the factors

Thus, after the specification, all the equations contain only one undefined parameter, a universal constant

## 3. Extensions to other areas of physics

### 3.1 Free particle dynamics

In this section, the free particle dynamics of the ‘relativity with a preferred frame’ developed in [28] is presented in a shortened form. The modified dynamics is developed based on the existence of the invariant combination

where the integral is along the world line between two given world points and

where

and

are components of the velocity vector. Then the Lagrange function is defined by

which is used to obtain expressions for the momentum

and

Proceeding with the four-dimensional formulation, we will use the variables

we define the contrainvariant four-velocity vector as

where the superscript

where

and the following relations hold

where a common rule of summation over repeated indexes is assumed.

Next, recalling that the momentum four-vector is defined by

and using the principle of the least action [37] we find (see [28] for details) that

while the contravariant components of the four-momentum vector are

Then from the identity (32) we get

Recalling that

with allowance for (26) and (33), we have

which, upon using (34) and (30), yields the relations (24) and (25) for the three-momentum and energy. Solving equations (38) for the components of the four-momentum vector we get

Then using (39) in (36) yields a dispersion relation for a free particle which can be represented in the form

where the speeds of light

while for massless particles moving in the negative

### 3.2 Electromagnetic field equations

The invariant action integral for a charged material particle in the electromagnetic field is made up of two parts: the action for the free particle defined by (19) and a term describing the interaction of the particle with the field. The invariance is provided by using the combinations that are invariant in the Minkowskian variables (26) so that the action integral takes the form [37]

where the coordinates

Upon representing the four-potential as

where

Here and in what follows, ‘tilde’ indicates that variables and operations are in Minkowskian space-time variables (26). Note that, while scalars and components of three-dimensional vectors in the Minkowskian formulation appear with ‘tilde’, four-dimensional Minkowskian variables are not supplied with ‘tilde’. It does not lead to any confusion since the four-dimensional notation does not applicable to the formulation in physical variables.

In the electrodynamics of the standard special relativity (which, in our case, is electrodynamics in Minkowskian variables), the electric and magnetic field intensities are defined based on equations of motion of a charged particle obtained from the Lagrange equations.

where, in the Lagrange function

where

The same line of arguments is used to derive equations describing the electromagnetic field in physical variables

where

As the result, the Lagrangian function

where

yields

Thus, upon using the modified potentials, equations of motion in physical variables have the same form as in the standard relativity and the physical electric and magnetic field intensities are expressed through the modified potentials by the relations

of the same form (49) as in the standard relativity.

It is evident that the first pair of the Maxwell equations in physical variables, which is derived from Eq. (55), have the same form as in the standard relativity

To obtain the second pair of Maxwell equations in physical variables let us calculate the components of the electromagnetic field tensor

Expressing

while

Note that the terms with

The electromagnetic field equations are obtained with the aid of the principle of least action [37] in the form

(only fields in a vacuum, that are relevant to the subject of this paper, are considered). Substituting (59) into (60) and transforming the equations to physical space-time variables, upon combining equations with different ‘

An important feature of Eq (61) is their linearity in

Note the existence of an alternative way of the derivation of the modified Maxwell Eqs. (56) and (61). Based on Eqs. (49), (51), and (55), the electric and magnetic field intensities

The same relations are seen in the expressions (58) for the components of the electromagnetic field tensor. It is readily verified that substituting the relations (62) into the Maxwell equations of the standard relativity

as

where

### 3.3 Electromagnetic waves

Like the electromagnetic wave equation of the standard relativity electrodynamics, the equation describing electromagnetic waves in the electrodynamics of the relativity with a preferred frame can be derived straight from the modified Maxwell equations (reproduced below for convenience)

Eliminating

where

Alternatively, the wave Eq. (68) can be derived from (60) expressed in terms of the potentials using (57) while imposing the Lorentz gauge condition

Converting the derivatives in the resulting equation

into derivatives in physical space-time variables yields equations of the form (68) with

Much of the propagation behavior of the electromagnetic wave is encoded in its dispersion relation, which provides spectral information for the modes. To find the dispersion relation the ansatz in the form of monochromatic plane waves is used, as follows

where

The dispersion relation (72) can be also represented in the form

where

Like as in the standard relativity case, the two roots (74) are obtained from each other by changing the sign of

More insight about the wave motion implied by Eq. (68) can be gained from the modified Maxwell Eqs. (66) and (67). Eq. (66), which are unaffected by the modifications, reduce with the ansatz (71) to

The first of these equations shows that the magnetic field remains transverse to

Eq. (76) implies the existence of two modes.

The first one corresponds to the electric field with

The second mode corresponds to the case

Thus, the second mode represents electromagnetic wave, in which the magnetic field

It is worthwhile to note a distinguishing feature of the above analysis as compared with other studies of electromagnetic waves in the presence of the Lorentz violation. Typically, different modes arising due to the Lorentz violation correspond to different roots of the modified dispersion relation (see, e.g., [6, 38, 39, 40]). The present analysis provides an unusual example when two different modes correspond to the same root of the dispersion relation (for the waves propagating to the observer. it is the second root of (74)). The existence of two modes is revealed only when one studies the corresponding solutions of the modified Maxwell equations. It is worth also noting that the present analysis is performed solely in terms of field intensities

## 4. Cosmology

### 4.1 General relativity

The basic principle of general relativity, the Equivalence Principle (see, e.g. [41]), which asserts that at each point of spacetime it is possible to choose a ‘locally inertial’ coordinate system where objects obey Newton’s first law, is valid independently of the law of propagation of light assumed. In other terms, it can be applied when the processes in the locally inertial frame are governed by the laws of ‘relativity with a preferred frame’. Based on that there exists the invariant combination (7), which by the change of variables (9) is converted into the Minkowski interval, one can state that the general relativity equations in arbitrary coordinates

where

The ‘true’ time and space intervals can be determined using a procedure similar to that described in [37]. Applying that procedure (see [27] for details) yields the following relations for the ‘true’ proper time interval

where

### 4.2 Cosmological models

Modern cosmological models assume that, at each point of the universe, the ‘typical’ (freely falling) observer can define the (preferred) Lorentzian frame in which the universe appears isotropic. The metric derived based on isotropy and homogeneity (the * Robertson-Walker metric*) has the form [41, 42]

where a comoving reference system, moving at each point of space along with the matter located at that point, is used. This implies that the coordinates

Introducing, instead of

and replacing the time

converts (82) into the form

The information about the scale factor

where

where

The relation expressing the

where the deceleration parameter

In the relativity with a preferred frame, solving the modified GR equations for a matter-dominated model lead to the luminosity distance-redshift relation of the form, which allows fitting the results of observations with supernovae so that the acceleration problem can be naturally resolved—there is no acceleration and so no need in introducing the dark energy. Below, the calculations leading to the modified luminosity distance-redshift relation are outlined (for more details see [27]).

In the relativity with a preferred frame, the expression for

which coincides with a common form of the relation for

where

and the parameters

where

The Friedmann Eq. (91) allows us to calculate the radial coordinate

where

In the standard cosmology, Eq. (88) (with

so that (96) becomes a closed-form relation for

Here and in what follows, quantities with a superscript “

In the framework of the present analysis, expressing ^{4} With that assumption and the presumption, that in the cosmology based on the relativity with a preferred frame there is no need in introducing dark energy,

The dependence

Substituting (101) and (100) into (88) reduces the problem to a transcendental equation for

Representing the solution of (102) as a series in

Then the relation

To compare the results produced by the model with those, obtained from an analysis of type Ia supernova (SNIa) observations, one needs some fitting formulas for the dependence

The Baryon Acoustic Oscillations (BAO) data are commonly considered as confirming the accelerated expansion and imposing constraints on the dark energy parameters. Applying the cosmological models based on the ‘relativity with a preferred frame’ to the interpretation of the BAO data provides an alternative view on the role of the BAO observations in cosmology. Comparing the predictions of the present model with the recently released galaxy clustering data set of the Baryon Oscillation Spectroscopic Survey (BOSS), part of the Sloan Digital Sky Survey III (SDSS III), shows that the BAO data can be well fit to the present cosmological model. The BAO data include two independent sets of data: the BAO scales in transverse and line-of-sight directions which can be interpreted to yield the comoving angular diameter distance

## 5. Propagation of cosmic rays

### 5.1 Attenuation of the UHECR due to the pion photoproduction process

In this section, the application of the theory to the description of the effects due to the interactions of the Ultra-High Energy Cosmic Rays (UHECR) with universal diffuse background radiation in the course of the propagation of cosmic rays from their sources to Earth over long distances (see, e.g., review articles [45, 46, 47]) is considered. The interactions of the UHECR with the CMB photons are characterized by a well-defined energy threshold for the energy suppression due to pion photoproduction by UHECR protons—the Greisen-Zatsepin-Kuzmin (GZK) cutoff [48, 49]. The fluxes of cosmic ray protons with energies above this threshold would be strongly attenuated over distances of a few tens of Mpc so that the cosmic ray protons from the sources at a larger distance, even if they were accelerated to energies higher than the threshold, would not be able to survive the propagation. The energy position of the GZK cutoff can be predicted based on special relativity as a theoretical upper limit (‘GZK limit’) on the energy of UHECR set by pion photoproduction in the interactions of cosmic ray particles with the microwave background radiation. Calculating the GZK limit based on the particle dynamics of the special relativity with a preferred frame developed in Section 3.1 (see [28] for details) yields

where

It is seen that the expression (105) for the threshold energy of the proton differs from the common one by the factor

This effect may contribute to the interpretation of the data on the mass composition of UHECR which is a key observable in the context of the physics of UHECR as it fixes few fundamental characteristics of the sources. The mass composition of UHECR became a matter of active debate after that the Pierre Auger Collaboration (Auger) reported on its recent observations [50, 51]. The observations of Auger, far the largest experiment set-up devoted to the detection of UHECR, have shown that the UHECR mass composition is dominated by protons only at energies around and below

The complexity of the scenario and the severe constraints on the model parameters, required in the case of a composition with heavy nuclei, are not present in the case if the UHECR mass composition is dominated by protons. In the latter case, the scenario is much simpler, only protons are accelerated with very high maximum injection energies. The view that the UHECR are mostly protons is, theoretically, a natural possibility. Proton is the most abundant element in the universe and several different astrophysical objects, at present and past cosmological epochs could provide efficient acceleration even if it requires very high luminosities and maximum acceleration energies. The models of interaction of UHECR with the astrophysical background are also much simpler if the UHECR are mostly protons. In this case, the only relevant astrophysical background is the CMB [53, 54]. This fact makes the propagation of UHE protons free from the uncertainties related to the background, being the CMB exactly known as a pure black body spectrum that evolves with red-shift through its temperature.

The results of the present study allow reconciling (at least, partially) the view, that, the primary UHECR flux at the sources is dominated by protons accelerated with very high maximum injection energies, with the observational evidence that the fraction of protons in the UHECR is decreasing towards higher energies. The apparent contradiction can be resolved by taking into account the effect, predicted by the present analysis, that the number of sources, which may contribute to the observed flux of protons at a given energy, is progressively decreasing with the energy increases. This effect is a consequence of the threshold condition (105) which implies that, among protons produced by a source at some

### 5.2 Attenuation due to the pair-production process

Gamma rays (

which has the effect of a significant energy attenuation in the flux of high-energy gamma rays. Such interaction takes place for gamma rays with energies (

The following assumptions should be made if we intend to calculate the threshold value of the energy of the gamma-rays photons:

It is needed to take the lowest energy the high-energy photon can have to react with the background photon to yield the two particles which correspond to the situation when they both are produced at rest in their center of mass frame after the collision.

To maximize the energy available from the collision, the initial momenta of the two particles in the lab frame should be pointing in opposite directions.

Let us equate the square of the total

The right-hand side of (107) is calculated, as follows

where Eq. (39) are to be substituted into (108), with the three-momentum and energy defined by equations (24), (25) and (21) in which it is set

Note that, although

The left-hand side of Eq. (107) is to be expressed in terms of the high-energy and background photons energies using the relations between the particle’s momentum and energies obtained from the dispersion relation (40). The high-energy photons move to the observer, in the direction opposite to the direction the velocity of the lab frame relative to the observer (relative to the preferred frame) which is chosen to be a positive direction of the

where

Substituting (110) for

Then using Eqs. (112) and (109) in (107) and solving the resulting equation for

or the expression for the threshold energy of the background photon (minimum energy to produce

The factor

In a cosmological context, where

where

Attenuation of gamma rays with the energy

where

Then the integral over distance

The threshold energy of background photons

The preferred frame effects may influence the optical depth also via the cosmological part of the expression (119). In the Robertson-Walker metric (82) (or (85)), the distance element

In the cosmology of the relativity with a preferred frame,

where the quantity

In the concordance model relation (120), the value

### 5.3 Astrophysical tests for vacuum dispersion and vacuum birefringence

In the literature on Lorentz violation, as major features of the behavior of electromagnetic waves in vacuum in the presence of Lorentz violation, vacuum dispersion and vacuum birefringence are considered. Astrophysical tests for vacuum dispersion of light from astrophysical sources seek differences in the velocity of light at different wavelengths due to Lorentz violation which should result in observed arrival-time differences. For differences in the arrival times of different wavelengths to be interpreted as caused by differences in the light velocities, explosive or pulsed sources of radiation that produce light over a wide range of wavelengths in a short period, such as gamma-ray bursts, pulsars, or blazars, are to be used. All those are point sources, which have the disadvantage (to impose constraints on Lorentz violation) that a single line of sight is involved, which provides sensitivity to only a restricted portion of space for free coefficients of the Lorentz violating models.

The same is valid for the present theory leading to the dispersion relation (72). In the case of the waves propagating along the

which corresponds to the waves propagating in the opposite directions. For a wave propagating to the observer from a cosmological source, with the

It does not depend on

Another test, that is commonly used for setting constraints on the parameters of the Lorentz-violating theories in electrodynamics, is the vacuum birefringence test. In birefringent scenarios, the two eigenmodes propagate at slightly different velocities. This implies that the superposition of the modes is altered as light propagates in free space. Since the two modes differ in polarization, the change in superposition causes a change in the net polarization of the radiation. However, it does not apply to the present theory leading to the dispersion relation (72). The two roots of the dispersion relation correspond to the waves propagating in different directions. Thus, no two eigenmodes are propagating in the same direction and so there is no possibility for vacuum birefringence. Thus, neither tests for vacuum dispersion nor tests for vacuum birefringence can impose restrictions, additional to those imposed by cosmological data, on the values of the only parameter of the theory

The vacuum birefringence and vacuum dispersion are widely discussed in the literature as astrophysical tests of Lorentz violation in the pure photon sector of the standard-model extension (e.g., [6, 38, 59, 60, 61, 62]). Therefore it is of interest, in that context, to compare the Lorentz-violating terms, appearing in the Lagrangian due to the preferred frame effects in the present study, with those introduced as a formal SME extension. Extracted from the SME, the Lorentz-violating electrodynamics can be written in terms of the usual field strength

In what follows, we calculate the Lagrangian of the electrodynamics with a preferred frame and compare the Lorentz violating terms in that Lagrangian with those in (125). Calculating

It is seen that the form (126) is in a sense more general than (125) because of the Lorentz violating multiplier

which fits the form (125) with the coefficients

while other

## 6. Discussion

The ‘relativity with a preferred frame’ incorporates the existence of the cosmological preferred frame into the framework of the theory while preserving fundamental principles of the SR: the principle of relativity and the principle of universality of the light propagation. The relativistic invariance is preserved in the sense, that the physical laws are covariant (their form does not change) under the group of transformations between inertial frames, and the relativistic symmetry is preserved (although modified) in the sense that there exists a combination, a counterpart of the interval of the standard relativity theory, which is invariant under the transformations. The existence of the modified symmetry provides an extension of the theory to general relativity such that the general covariance is also preserved. Thus, the ‘relativity with a preferred frame’ is a relativity theory, both in the special relativity and in the general relativity parts. Except for identifying the preferred frame with a comoving frame of cosmology, the theory does not include any assumptions. No approximations are involved besides approximating the universal function

The problem of defining allowed values of

In the applications of the theory to the BAO data, the conceptual and quantitative aspects go together. The BAO observations provide two different sets of data: BAO scales in transverse and line-of-sight directions. Measurements of the angular distribution of galaxies yield the quantity

Next, it might be expected that some constraints on allowed values of

Applying the modified particle dynamics to the pair-production process, which is responsible for attenuation of the gamma-rays flux, does not provide quantitative constraints on the values of the parameter

In general, the fact, that applying the theory containing only one universal parameter to several different phenomena does not lead to any contradictions, proves a consistency of its basic principles. The presence of only one parameter in the theory is a consequence of the fact that, as distinct from the popular Lorentz-violating theories, where Lorenz violation is introduced phenomenologically by adding Lorentz-violating terms to the Lorentz invariant relations, the ‘relativity with a preferred frame’ starts from the physically reasonable modification of the basic postulates of the SR. The generalized relativistic invariance, and so the Lorentz invariance violation, are ingrained in the theory at the most fundamental level being imbedded into the metric. It is also worth to emphasize that the conceptual basis of the theory has been developed without having in mind possible applications. It is aimed at designing the framework which would allow to incorporate the preferred frame into special relativity while retaining the relativity principle and the fundamental space-time symmetry. Nevertheless, the theory provides explanations of some observational data, that were regarded as puzzling after their discovery (like the SNIa luminosity distance-redshift relation indicating the acceleration of the universe and the absence of high energy protons in the UHECR flux). As the result, the concepts (among which dark energy is the most striking one), introduced to explain those puzzling features, become redundant. All the above justifies treating the ‘relativity with a preferred frame’ as an alternative to some currently accepted theories.

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## Notes

- It is worth noting that, although the anisotropy of speed of light is one of the central features of the present analysis, this theory stands apart from the ample literature on the conventionality of simultaneity and clock synchronization. A discussion of those issues in the context of the ‘relativity with a preferred frame’ can be found in [29, 30].
- In the modern versions of the experiments designed to test special relativity and the so-named ‘test theories’ (e.g., [31, 32], see a discussion in [27, 29, 30]), the tests are meant to detect the anisotropy of the two-way speed of light.
- Although the form (1) seems to be attributed to the one-dimensional formulation, in the three-dimensional case, the equation has the same form if the anisotropy vector k is directed along the x-axis [30]. In the present analysis, the x-axis defines also the line of relative motion of the two frames but it does not lead to any ambiguity. The assumption, that the anisotropy vector k is along the direction of relative motion of the frames S′ and S, is justified by that one of the frames in a set of frames with different values of k is a preferred frame. Since the anisotropy is attributed to the motion with respect to the preferred frame, it is expected that the axis of anisotropy is either in the direction of motion or opposite to it.
- In the present model, this assumption is not obligatory. It is worthwhile to note that, despite what is frequently claimed, a flatness of the universe is not stated in modern cosmology. Given the fact, that there is no direct measurement procedure of the curvature of space independent of the cosmological model assumed, the flatness of the space is the result valid only within the framework of the ΛCDM model.