Special relativity with a preferred frame and the relativity principle: cosmological implications

The modern view, that there exists a preferred frame of reference related to the cosmic microwave background (CMB), is in apparent contradiction with the principles of special relativity. The purpose of the present study is to develop a counterpart of the special relativity theory that is consistent with the existence of a preferred frame but, like the standard relativity theory, is based on the relativity principle and universality of the (\textit{two-way}) speed of light. In the framework developed, a degree of anisotropy of the one-way velocity acquires meaning of a characteristic of the really existing anisotropy caused by motion of an inertial frame relative to the preferred frame. The anisotropic special relativity kinematics is developed based on the first principles: (1) Space-time transformations between inertial frames leave the equation of anisotropic light propagation invariant and (2) A set of the transformations possesses a group structure. The Lie group theory apparatus is applied to define groups of space-time transformations between inertial frames. Applying the consequences of the transformations to the problem of calculating the CMB temperature distribution yields an equation in which the angular dependence coincides with that obtained on the basis of the standard relativity theory but the mean temperature is corrected by the terms second order in the observer velocity. From conceptual point of view, it eliminates the inconsistency of the usual approach when formulas of the standard special relativity are applied to define effects caused by motion with respect to the preferred frame.


Abstract
The modern view, that there exists a preferred frame of reference related to the cosmic microwave background (CMB), is in apparent contradiction with the principles of special relativity. The purpose of the present study is to develop a counterpart of the special relativity theory that is consistent with the existence of a preferred frame but, like the standard relativity theory, is based on the relativity principle and universality of the (two-way) speed of light. The synthesis of those seemingly incompatible concepts is possible at the expense of the freedom in assigning the one-way speeds of light that exists in special relativity. In the framework developed, a degree of anisotropy of the one-way velocity acquires meaning of a characteristic of the really existing anisotropy caused by motion of an inertial frame relative to the preferred frame. The anisotropic special relativity kinematics is developed based on the first principles: (1) Space-time transformations between inertial frames leave the equation of anisotropic light propagation invariant and (2) A set of the transformations possesses a group structure. The Lie group theory apparatus is applied to define groups of space-time transformations between inertial frames. The correspondence principle, that the coordinate transformations should turn into the Galilean transformations in the limit of small velocities, and the argument, that the anisotropy parameter k in a particular inertial frame is determined by its velocity relative to the preferred frame, are used to specify the transformations.
The parameter of anisotropy k becomes a variable which takes part in the transformations so that the preferred frame naturally arises as the frame with k = 0. The transformations between inertial frames obtained as the result of the analysis do not leave the interval between two events invariant but modify it by a conformal factor. Applying the consequences of the transformations to the problem of calculating the CMB temperature distribution yields an equation in which the angular dependence coincides with that obtained on the basis of the standard relativity theory but the mean temperature is corrected by the terms second order in the observer velocity. From conceptual point of view, it eliminates the inconsistency of the usual approach when formulas of the standard special relativity are applied to define effects caused by motion with respect to the preferred frame.

I. INTRODUCTION
Special relativity underpins nearly all of present day physics. Lorentz invariance is one of the cornerstones of general relativity and other theories of fundamental physics. However, the discovery of the cosmic microwave background (CMB) radiation has shown that cosmologically a preferred system of reference does exist which is in apparent contradiction with the principles of the special relativity theory. Nevertheless, the formulas of special relativity are commonly used in cosmological context when there is a need to relate physical effects in the frames moving with respect to each other. Applying the Doppler effect and the light aberration equations based on the Lorentz transformations for calculating the CMB temperature anisotropies due to our galaxy's peculiar motion with respect to the CMB provides an example of such an approach.
The view, that there exists a preferred frame of reference, seems to unambiguously lead to the abolishment of the basic principles of the special relativity theory: the principle of relativity and the principle of universality of the speed of light. The modern versions of experimental tests of special relativity and the "test theories" of special relativity [1]- [4] presume that a preferred inertial reference frame ("rest" frame), identified with the CMB frame, is the only frame in which the two-way speed of light (the average speed from source to observer and back) is isotropic. Furthermore, it seems that accepting the existence of a preferred frame forces one to abandon the group structure for the set of spacetime transformations between inertial frames. In the test theories, transformations between 'moving' frames are not considered, only a form of the transformation between a preferred frame and a particular moving frame is postulated.
The purpose of the present study is to develop a counterpart of the special relativity kinematics, that is consistent with the existence of a preferred frame but, like the standard relativity theory, is based on the universality of the (two-way) speed of light and the relativity principle. The group structure of a set of transformations between inertial frames is also preserved in the theory developed. The reconciliation and synthesis of those concepts with the existence of a preferred frame is possible at the expense of the freedom in assigning the one-way speeds of light. The one-way speed of light is commonly considered as irreducibly conventional in view of the fact that it cannot be defined separately from the synchronization choice (see, e.g., [5]- [8]). Nevertheless, the analysis shows that, despite the inescapable entanglement between remote clock synchronization and one-way speed of light, a specific value of the one-way speed of light and corresponding synchronization are selected from others in some objective way. In the framework developed, the argument that the anisotropy of the one-way speed of light in a particular inertial frame is due to its motion relative to the preferred frame, being combined with the requirements of invariance of the equation of (anisotropic) light propagation and the group structure of a set of transformations between inertial frames, defines a specific value of the one-way speed of light for that frame. The parameter of anisotropy of the one-way speed of light k becomes a variable that takes part in the group transformations. The preferred frame, commonly defined by that the propagation of light in that frame is isotropic, is naturally present in the analysis as the frame with k = 0 and it does not violate the relativity principle since the transformations from/to that frame are not distinguished from other members of the group.
The space-time transformations between inertial frames derived as a result of the analysis differ from the Lorentz transformations. Since the theory is based on the special relativity principles, it means that the Lorentz invariance is violated without violation of the relativistic invariance. The theory equations contain one undefined universal constant q such that the case of q = 0 corresponds to the standard special relativity with isotropic one-way speed of light in all inertial frames. The measurable effects following from the theory equations can provide estimates for q and define deviations from the standard relativity that way. Applying the theory to the problem of calculating the CMB temperature distribution eliminates the inconsistency of the usual approach when formulas of the standard special relativity, which does not allow a preferred frame, are used to define effects caused by motion with respect to the preferred frame. The CMB temperature angular dependence predicted by the present theory coincides with that obtained on the basis of the standard relativity equations while the mean temperature is corrected by the terms second order in the observer velocity.
A. Anisotropy of the one-way speed of light in special relativity The issue of anisotropy of the one-way speed of light is traditionally placed into the context of conventionality of distant simultaneity and clock synchronization [5]- [8]. Simultaneity at distant space points of an inertial system is defined by a clock synchronization that makes use of light signals. If a light ray is emitted from the master clock and reflected off the remote clock one has a freedom to give the reflection time t at the remote clock any intermediate time in the interval between the emission and reception times t 0 and t R at the master clock where ǫ is Reichenbach's synchrony parameter [9]. The thesis that the value of the synchrony parameter ǫ may be freely chosen in 0 < ǫ < 1 is known as the conventionality of simultaneity. Reichenbach's "nonstandard" synchronization reduces to the "standard" Einstein synchronization when ǫ = 1/2. Any choice of ǫ = 1/2 corresponds to assigning different one-way speeds of light signals in each direction which must satisfy the condition that the average is equal to c. Speed of light in each direction is therefore If the described procedure is used for setting up throughout the frame of a set of clocks using signals from some master clock placed at the spatial origin, a difference in the standard and nonstandard clock synchronization may be reduced to a change of coordinates [5]- [8] where t (s) = (t 0 + t R )/2 is the time setting according to Einstein (standard) synchronization procedure.
The analysis can be extended to the three dimensional case. If a beam of light propagates (along straight lines) from a starting point and through the reflection over suitable mirrors covers a closed part the experimental fact is that the speed of light as measured over closed part is always c (Round-Trip Light Principle). In accordance with that experimental fact, if the speed of light is allowed to be anisotropic it must depend on the direction of propagation as [6], [7] where k ǫ is a constant vector and θ k is the angle between the direction of propagation n and k ǫ . Similar to the one-dimensional case, the law (4) may be considered as a result of the transformation from "standard" coordinatization of the four-dimensional space-time manifold, with k ǫ = 0, to the "nonstandard" one with k ǫ = 0: There were several studies exploring the kinematics of the special relativity theory when a definition of simultaneity other than that used by Einstein is adopted. In particular, the transformations, which are treated as replacing standard Lorentz transformations of special relativity in the case of the "nonstandard" synchronization (1) with ǫ = 1/2, have been repeatedly derived in the literature (see, e.g., [10] - [12]). Although somewhat different assumptions (in addition to the common round-trip light principle and the linearity assumption) are used in those studies, the transformations derived, in fact, are identicalthey either coincide or become coinciding after a parameter change. In what follows, those transformations will be called the "ǫ-Lorentz transformations", the name is due to [11], [12].
The ǫ-Lorentz transformations can be obtained from the standard Lorentz transformations by a change of coordinates (3). Thus, the ǫ-Lorentz transformations are in fact the Lorentz transformations of the standard special relativity represented using the "nonstandard" coordinatization of the four-dimensional space-time manifold. This might be expected in view of the fact that the kinematic arguments used in the derivations of the ǫ-Lorentz transformations in the aforementioned works are based on the assumption that, in the case of ǫ = 1/2, the relations of the special relativity theory in its standard formulation are valid.
It is commonly believed that, since the speed of light cannot be defined separately from the synchronization choice, the one-way speed of light is irreducibly conventional. Also, a possibility to introduce the ǫ-Lorentz transformations is considered as an illustration of conventionality of the one-way speed of light. Nevertheless, there are arguments showing that, despite the inescapable entanglement between remote clock synchronization and oneway speed of light, a specific value of the one-way speed of light and the corresponding synchronization can be distinguished from others. In particular, it can be shown that the ǫ-Lorentz transformations, usually considered as incorporating an anisotropy, are in fact not applicable to the situation when there is an anisotropy in a physical system and that, in the case of isotropy, the particular case of the transformations corresponding to the isotropic one-way speed of light and Einstein synchronization (standard Lorentz transformations) is privileged.
The first point is that the ǫ-Lorentz transformations do not satisfy the Correspondence Principle unless the standard (Einstein) synchrony is used. The correspondence principle was taken by Niels Bohr as the guiding principle to discoveries in the old quantum theory.
Since then the correspondence principle had germinated and was considered as a guideline for the selection of new theories in physical science. In the context of special relativity, the correspondence principle is traditionally mentioned as a statement that Einstein's theory of special relativity reduces to classical mechanics in the limit of small velocities in comparison to the speed of light. Nevertheless, the correspondence principle has not been properly used as a heuristic principle in developing the special relativity theory.
Being applied to the special relativity kinematics, the correspondence principle implies that the transformations between inertial frames should turn into the Galilean transformations in the limit of small velocities. Let us consider from this point of view the ǫ-Lorentz transformations. The ǫ-Lorentz transformations between two arbitrary inertial reference frames S and S ′ in the standard configuration, with the y-and z-axes of the two frames being parallel while the relative motion with velocity v is along the common x-axis, being written in terms of k ǫ (instead of ǫ as in [10] - [12]) take the forms Here the space and time coordinates in S and S ′ are denoted respectively as {X, Y, Z, T } and {x, y, z, t} and it is implied that clocks in the frames S and S ′ are synchronized according to the anisotropy degree k ǫ . In the limit of β → 0, the formula for transformation of the coordinate x (first equation of (6)) turns into which is not coinciding with the formula for transformation of the coordinate x of the Galilean transformation It should be noted that the relations t = T , y = Y and z = Z, which are commonly included into the system of equations called the Galilean transformations, are not required to be valid in the limit of small velocities. The fact that the first order (in v) terms do not appear in those relations does not obligatory imply that they should be absent in the first order approximations of the special relativity formulas. In particular, if an expansion of the Lorentz transformations with respect to β = v/c is made the first order term arises in the expansion of the time transformation (see, for example, discussion in [13], [14]). So only the relation (8), which does contain the first order term, provides a reliable basis for applying the correspondence principle.
The additional, as compared with (8), term appearing in the small velocity limit (7) of the ǫ-Lorentz transformations includes the synchronization parameter and light speed which are alien to the framework of the Galilean kinematics. Thus, the "ǫ-Lorentz transformations" (6) do not satisfy the correspondence principle unless k ǫ = 0 which means that applying the correspondence principle singles out the isotropic one-way speed of light and Einstein synchrony.
The next point is that the "ǫ-Lorentz transformations" are applicable only to the situation when there is no anisotropy in a physical system, for the reason that they leave the interval between two events invariant. Invariance of the interval is commonly considered as an of the transformations between inertial frames. Therefore the use of the interval invariance is usually preceded by a proof of its validity (see, e.g., [15], [16]) based on invariance of the equation of light propagation. However, those proofs are not valid if an anisotropy is present.
In such proofs, two reference frames S and S ′ in a standard configuration, with S ′ moving with respect to S with velocity v, are considered. First, it is stated that, under the assumption of linearity of coordinate transformations between the frames, the two equations ds 2 = 0 and dS 2 = 0, with ds 2 and dS 2 being the intervals between two events in the frames S ′ and S, can be valid only if ds 2 = λ(v)dS 2 where λ(v) is an arbitrary function. Next, the third frame S ′′ moving with velocity (−v) with respect to S ′ (being at rest to S) is introduced and the transformations are applied once more which results in Applying the same arguments to a transversal coordinate yields where κ(v) corresponds to the change of the transverse dimensions of the rod. It is concluded that, for reasons of symmetry, it should be independent of the direction of the velocity which, together with (11), leads to However, the symmetry arguments are not valid if an anisotropy in the physical system is present. As a physical phenomenon it influences all the processes so that any effects due to movement of frame S ′ relative to S in some direction are not equivalent to those due to movement of frame S ′′ relative to S in the opposite direction. Therefore should be valid. Thus, in the presence of the anisotropy, the interval should not be invariant. Moreover, λ(v) = 1 implies that strict invariance should be replaced by conformal invariance.
The "ǫ-Lorentz transformations" leave the interval between two events invariant (the derivations of the ǫ-Lorentz transformations in the aforementioned works [10] - [12] are also based on invariance of the interval or equivalent assumptions) and therefore they are applicable only to the case of no anisotropy. Although, even in that case, the anisotropic one-way speed of light satisfying equation (4) with k ǫ = 0 and the corresponding ǫ-Lorentz transformations are mathematically acceptable, it is conceptually inconsistent to apply the transformations with anisotropic speed of light to isotropic situation. Moreover, the transformations themselves are physically inconsistent since they do not satisfy the correspondence principle. Thus, the value of k ǫ = 0 is privileged in some objective way if no anisotropy is present in a physical system.
It follows from the above discussion that, in the case of an anisotropic system, there should also exist a privileged value of the light-speed anisotropy parameter k ǫ selected by the size of the anisotropy.

B. Conceptual framework
The special relativity kinematics applicable to an anisotropic system should be developed where k is a (constant) vector characteristic of the anisotropy. The change of notation, as compared with (4), from k ǫ to k is intended to indicate that k is a parameter value corresponding to the size of the really existing anisotropy while k ǫ defines the anisotropy in the one-way speeds of light due to the nonstandard synchrony equivalent to the coordinate change (5). The anisotropic equation of light propagation incorporating the law (14) has the form (see Appendix A) where (x, y, z) are coordinates and t is time. It is assumed that the x-axis is chosen to be along the anisotropy vector k. Note that although the form (15) is usually attributed to the one-dimensional formulation it can be shown that, in the three-dimensional case, the equation has the same form if the anisotropy vector k is directed along the x-axis (see Further, in the development of the anisotropic relativistic kinematics, a number of other physical requirements, associativity, reciprocity and so on are to be satisfied which all are covered by the condition that the transformations between the frames form a group. Thus, the group property should be taken as another first principle. The formulation based on the invariance and group property suggests using the Lie group theory apparatus for defining groups of space-time transformations between inertial frames. At this point, it should be clarified that there can exist two different cases: (1) The size of anisotropy does not depend on the observer motion and so is the same in all inertial frames; (2) The anisotropy is due to the observer motion with respect to a preferred frame and so the size of anisotropy varies from frame to frame.
In the first case, the group of transformations leaving equation (15) The transformations from {X, Y, Z, T } to {x, y, z, t} are sought such that equation (16) is converted into (17) under the transformations. Groups of transformations defined within this framework are considered in [17].
The second case is relevant to the purpose of the present study -developing the special relativity kinematics consistent with the existence of a preferred frame. Since the anisotropy parameter varies from frame to frame, the equations of light propagation in the frames S and S ′ are where k differs from K. Thus, the anisotropy parameter becomes a variable which takes part in the transformations and so groups of transformations in five variables {x, y, z, t, k} which convert (18) into (19) are sought. In such a framework, the preferred frame, commonly defined by that the propagation of light in that frame is isotropic, naturally arises as the frame in which k = 0. However, it does not violate the relativity principle since the transformations from/to that frame are not distinguished from other members of the group.
Nevertheless, the fact, that the anisotropy of the one-way speed of light in an arbitrary inertial frame is due to motion of that frame relative to the preferred frame, is a part of the paradigm which is used in the analysis, arises if invariance of the electrodynamic equations is studied [18], [19] (see reviews [20], [21] for further developments). Transformations which conformally modify Minkowski metric have been introduced in the context of the special relativity kinematics in the presence of space anisotropy in [22], [23] (two papers from the series) and [24] (see also [25]). As distinct from the present framework, in the works [22] - [24], the assumption that the form of the metric changes by a conformal factor is imposed. Therefore any values of the conformal factor are permissible and in particular, it may be equal to one which reduces the transformations to the Lorentz transformations or to the ǫ-Lorentz transformations if a nonstandard clock synchronization is accepted (see [17] for a more detailed discussion of the works [22] - [24]).
Although, in the present analysis, the conformal invariance of the metric is not imposed but arises as an intrinsic feature of special relativity based on invariance of the anisotropic equation of light propagation and the group property, the conformal factor includes an arbitrary element. Therefore, in order to complete the construction of the anisotropic relativistic kinematics, one needs to relate the conformal factor to the parameter k (or K in the case when this parameter varies from frame to frame) characterizing the anisotropy of the light speed. The correspondence principle, according to which the transformations between inertial frames should turn into the Galilean transformations in the limit of small velocities, is used for this purpose. (As a matter of fact, the isotropic case provides a guiding example for this: it is the correspondence principle that assigns a preferred status to k ǫ = 0 if the scale factor λ = 1.) It is evident that validity of this principle should not depend on whether propagation of light is assumed to be isotropic or anisotropic since, in the framework of the Galilean kinematics, there is no place for the issues of light speed and its anisotropy. Thus, the anisotropic special relativity kinematics is developed using the following three princi- Applying those principles for deriving transformations between inertial frames, in the case when the anisotropy parameter k does not vary from frame to frame [17], yields the transformations which include the scale (conformal) factor λ in such a way that the value λ = 1 is not allowed unless k = 0. Therefore, as distinct from the "ǫ-Lorentz transfor- In the case of the variable anisotropy parameter k, which is the subject of the present study, derivation of transformations between inertial frames, although being based on the same first principles as in the case of a constant k, differs conceptually and methodologically from that case. First, the fact, that now groups of transformations in five variables {x, y, z, t, k} are sought, changes both the derivation procedure and the resulting transformations. Further, since the law of variation of the anisotropy parameter from frame to frame is not completely defined by the determining equations, the scale (conformal) factor contained in the transformations includes an undefined function of the group parameter.
The conceptual argument, that the size of anisotropy of the one-way speed of light in an arbitrary inertial frame depends on its velocity relative to the preferred frame, allows to specify the transformations. As the result, the specified transformations include, instead of an arbitrary function, only one undefined universal parameter.
The only preferred frame one may think of is of course the cosmological frame in which the microwave background radiation is isotropic. Applying the consequences of the anisotropic relativity transformations developed in the present analysis to the problem of calculating the formula which differs from that obtained using equations of the standard relativity theory.
The angular dependence appears to be the same but the mean temperature is corrected and the corrections are of the order of (v/c) 2 wherev is the observer velocity with respect to the CMB. The formula for the Doppler frequency shift of the present theory can be also applied to the case when an observer in a frame moving with respect to the CMB (Earth) receives light from an object (galaxy) which is also moving with respect to that preferred frame.
The paper is organized, as follows. In Section 2, the method is outlined and the coordinate transformations between inertial frames incorporating anisotropy of the light propagation, with the anisotropy parameter varying from frame to frame, are derived. In Section 3, the transformations are specified using the argument that the anisotropy of the light propagation is due to the observer motion with respect to the preferred frame. The parameter of anisotropy k is allowed to vary from frame to frame which, in particular, implies that there exists a preferred frame in which the speed of light is isotropic.
Consider two arbitrary inertial reference frames S and S ′ in the standard configuration with the y-and z-axes of the two frames being parallel while the relative motion is along the common x-axis. The space and time coordinates in S and S ′ are denoted respectively as {X, Y, Z, T } and {x, y, z, t}. The velocity of the S ′ frame along the positive x direction in S, is denoted by v. It is assumed that the frame S ′ moves relative to S along the direction determined by the vector k from (14). This assumption is justified by that one of the frames in a set of frames with different values of k is a preferred frame, in which k = 0, so that the transformations must include, as a particular case, the transformation to that preferred frame. Since the anisotropy is attributed to the fact of motion with respect to the preferred frame it is expected that the axis of anisotropy is along the direction of motion (however, the direction of the anisotropy vector can be both coinciding and opposite to that of velocity).
Transformations between the frames are derived based on the following first principles: invariance of the equation of light propagation (underlined by the relativity principle), group property and the correspondence principle. Note that the group property is used not as in the traditional analysis which commonly proceeds along the lines initiated by [26] and [27] which are based on the linearity assumption and relativity arguments. The difference can be seen from the derivation of the standard Lorentz transformations in Appendix B.
Invariance of the equation of light propagation. The equations for light propagation in the frames S and S ′ are Correspondence principle. The correspondence principle requires that, in the limit of small velocities v ≪ c (small values of the group parameter a ≪ 1), the formula for transformation of the coordinate x turns into that of the Galilean transformation: Remark that the small v limit is not influenced by the presence of anisotropy of the light propagation. It is evident that there should be no traces of light anisotropy in that limit, the issues of the light speed and its anisotropy are alien to the framework of Galilean kinematics.
The group property and the requirement of invariance of the equation of light propagation suggest applying the infinitesimal Lie technique (see, e.g., [28], [29]). The infinitesimal transformations corresponding to (23) are introduced, as follows and equations (20) and (21)  The correspondence principle can be applied to specify partially the infinitesimal group generators. Equation (24) is used to calculate the group generator ξ(X, T ), as follows It can be set b = 1 without loss of generality since this constant can be eliminated by redefining the group parameter. Thus, the generator ξ is defined by Then substituting the infinitesimal transformations (25), with ξ defined by (27), into equation (21) with subsequent linearizing with respect to a and using equation (20) to eliminate dT 2 yields −Kc 2 τ X + 1 − K 2 (K + cτ T ) + χ (K) cK dX 2 +c c 2 τ X + cKτ T + 1 + K 2 − χ (K) c dXdT where subscripts denote differentiation with respect to the corresponding variable. In view of arbitrariness of the differentials dX, dY , dZ and, dT , the equality (28) can be valid only if the coefficients of all the monomials in (28) vanish which results in an overdetermined system of determining equations for the group generators.
The generators τ , η and ζ found from the determining equations yielded by (28) are where c 2 , c 3 and c 4 are arbitrary constants. The common kinematic restrictions that one event is the spacetime origin of both frames and that the x and X axes slide along another can be imposed to make the constants c 2 , c 3 and c 4 vanishing (space and time shifts are eliminated). In addition, it is required that the (x, z) and (X, Z) planes coincide at all times which results in ω = 0 and so excludes rotations in the plane (y, z).
The finite transformations are determined by solving the Lie equations which, after rescaling the group parameter asâ = a/c together withχ = χc and omitting hats afterwards, take the forms dk(a) da = χ (k (a)) ; k(0) = K, where R is defined by To complete the derivation of the transformations the group parameter a is to be related to the velocity v using the condition Substituting (38) into (34) and (35) yields where k is the value of k(a) calculated for a given by (38).
Solving equations (32) and using (38) in the result yields Calculating the interval with (39) and (40) yields Thus, in the case when the anisotropy exists, the interval invariance is replaced by conformal invariance with the conformal factor dependent on the relative velocity of the frames and the anisotropy degree. replacing (X, Y, Z, T ), β 1 = v 1 /c replacing β, a 1 replacing a, k replacing K and k 1 = k(a 1 ) replacing k. Here a 1 is given by equation (38) with K replaced by k and β replaced by β 1 .
It is readily checked that substituting the transformation formulas for (x, y, z, t) into the transformation formulas for (x 1 , y 1 , z 1 , t 1 ) yields again the formulas (39) and (40) but with a and β replaced by a 2 and β 2 , where β 2 = v 2 /c is the velocity of S ′′ with respect to S and a 2 = a 1 + a is the corresponding value of the group parameter. The three velocities v 1 , v 2 and v are related by Equation (43) could alternatively be obtained from the relation a 2 = a 1 + a, in accordance with the basic group property, using a properly specified equation (38).
Considering inverse transformations from the frame S ′ to S one has to take into account that, in the presence of the light speed anisotropy, the reciprocity principle is modified [11], [5]. The reasoning behind this is that all speeds are to be affected by the anisotropy of the light speed since the speeds are timed by their coincidences at master and remote clocks, and the latter are altered. Therefore the relative velocity v − of S to S ′ is not equal to the relative velocity v of S ′ to S. The modified reciprocity relation can be derived with the use of equation (5) which allows to relate the velocity measured by the clocks synchronized with anisotropic light speed to the velocity v s measured by the clocks synchronized with isotropic light speed -the latter is not dependent on the direction. Using equation (3), which for k ǫ directed along the x-axis is equivalent to (5), one obtains If this equation is used to calculate the velocity v of the frame S ′ as measured by an observer in the frame S, then k ǫ , dx/dt and dx/dt s are to be replaced by K, dX/dT = v and dX/dT s = v s respectively while, for calculating the velocity v − of the frame S as measured in S ′ , the quantities k ǫ , dx/dt and dx/dt s are to be replaced by k, −v − and −v s , as follows Eliminating the velocity v s from equations (45) yields So, according to the modified reciprocity principle, the group parameter value corresponding to the inverse transformation is calculated from (38) but with K replaced by k and β replaced by (−β − ), as follows which yields the expected result a − = −a. Thus, the relation between v − and v derived above using (3) (similar relations are commonly obtained using kinematic arguments [30]) may be considered as resulting from the group property of the transformations.
For deriving consequences of the transformations it is convenient to write the inverse transformations in terms of β (not β − ), as follows The formulas for the velocity transformation are readily obtained from (39) and (40), as where (U X , U Y , U Z ) and (u x , u y , u z ) are the velocity components in the frames S and S ′ respectively. Remark that the relation (43) derived above represents the properly specified first equation of (50).
The transformations (36) -(40) contain an indefinite function k(a). The scale factor R also depends on that function. The transformations are specified in the next section.

III. SPECIFYING THE TRANSFORMATIONS
In the derivation of the transformations in the previous section, the arguments, that there exists a preferred frame in which the light speed is isotropic and that the anisotropy of the one-way speed of light in a specific frame is due to its motion relative to the preferred frame, have not been used. In the framework of the derivation, nothing distinguishes the frame in which k = 0 from others and the transformations from/to that frame are members of a group of transformations that are equivalent to others. Thus, the theory developed above is a counterpart of the standard special relativity kinematics which incorporates an anisotropy of the light propagation, with the anisotropy parameter varying from frame to frame. Below the transformations between inertial frames derived in Section 2 are specified based on that anisotropy of the one-way speed of light in an inertial frame is caused by its motion with respect to the preferred frame.
First, this leads to the conclusion that the anisotropy parameter k s in an arbitrary frame s moving with respect to the preferred frame with velocityv s should be given by some (universal) function k s = F β s of that velocity. Equations (30) and (38) imply that k = k (a (β, K) , K) which being specified for the transformation from the preferred frame to the frame s by setting K = 0, k = k s , β =β s yields k s = F β s . (It could be expected, in general, that a size of the anisotropy depends on the velocity relative to the preferred frame but, in the present analysis, it is not a presumption but a part of the framework.) Next, consider three inertial reference framesS, S and S ′ . As in the preceding analy- Exponentiation of equation (52) yields Let us now choose the frameS to be a preferred frame. Then,k = 0 and for the frames S and S ′ we have Withβ s = f (k s ) being a function inverse to F β s , using in (54) the equalities inverse to those of (55) together withk = 0 yields If the function f (k s ) were known, the relation (56), that implicitly defines the anisotropy parameter k in the frame S ′ as a function of the anisotropy parameter K in the frame S and the relative velocity v of the frames, would provide a formula for the transformation of the anisotropy parameter k. This would allow to specify the transformations (39) and (40) by substituting that formula for k into the equation of transformation for t and calculating the scale factor R using that formula with β expressed as a function of a group parameter a from (38).
Although the function F β s is not known, a further specification can be made based on the argument that an expansion of the function F β s in a series with respect toβ s should not contain a quadratic term since it is expected that a direction of the anisotropy vector changes to the opposite if a direction of a motion with respect to a preferred frame is reversed: F β s = −F −β s . Thus, with accuracy up to the third order inβ s , the dependence of the anisotropy parameter on the velocity with respect to a preferred frame can be approximated by Introducing the last equation of (57) into (56) yields which is the expression to be substituted for k into (39). To calculate the scale factor in (39) and (40), β is expressed as a function of a group parameter a from (38), as follows β = sinh a K sinh a + cosh a (59) which, being substituted into (58), yields Then using (60) in (36), with (38) substituted for a in the result, yields Thus, after the specification, the transformations between inertial frames incorporating anisotropy of light propagation are defined by equations (39) and (40) with k given by (58) and the scale factor given by (61). It is readily checked that the specified transformations satisfy the correspondence principle. All the equations contain only one undefined parameter, a universal constant q.
It should be clarified that, although the specification relies on the approximate relation that defined by (57). Nevertheless, a straightforward check can be made that the specified transformation (60) obeys the group properties. Using the notation κ(a, k) = q (k cosh a + q sinh a) k sinh a + q cosh a and introducing, in addition to S and S ′ , the frame S 0 with the anisotropy parameter k 0 , one can check that κ (a, κ (a 0 , k 0 )) = κ (a + a 0 , k 0 ) Similarly it is readily verified that κ (−a, κ (a, k)) = k and κ(0, k) = k. Alternatively, one can calculate the group generator χ(k) as and solve the initial value problem to be assured that it, as expected, yields (60). Thus, as a matter of fact, what is specified using the approximate relation (57) is the form of the group generator χ(k) in the group of transformations defined on the basis of the first principles.

IV. CONSEQUENCES OF THE TRANSFORMATIONS
Length contraction. Consider a rod that is at rest along the x-axis in the frame S ′ with the coordinates of its ends being x 1 and x 2 . In order to obtain its length in the frame S one has to measure the coordinates of its front tip X 1 and of its end X 2 at the same time moment Using the transformations (39) we have So we obtain the length contraction relation in the form Note that, in the presence of the anisotropy, the terms "length contraction" and "time dilation" become conditional in a sense. In general, it could be, for example, length dilation rather than length-contraction but, as it is commonly accepted in the literature, the corresponding relations are referred to as the length-contraction and time-dilation relations.
Time dilation. Consider a clock C ′ placed at rest in S ′ at a point on the x-axis with the coordinate x = x 1 . When the clock records the times t = t 1 and t = t 2 the clock in S which the clock C ′ is passing by at those moments will record times T 1 and T 2 given by the transformations (48) where it should be evidently set x 2 = x 1 . Subtracting the two relations we obtain the time dilation relation If clock were at rest in the frame S the time dilation relation would be with β − defined by (46).
Aberration law. The light aberration law can be derived using the formulas (50) for the velocity transformation. The relation between directions of a light ray in the two inertial frames S and S ′ is obtained by setting U X = c cos Θ/(1 + K cos Θ) and u x = c cos θ/(1 + k cos θ) in the first equation of (50). Then solving for cos θ yields where θ and Θ are the angles between the direction of motion and that of the light propagation in the frames of a moving observer (the Earth) and of an immovable source (star or Doppler effect. Consider a source of electromagnetic radiation (light) in a reference frame S very far from the observer in the frame S ′ moving with velocity v with respect to S along the X-axis with Θ being the angle between the direction of the observer motion and that of the light propagation as measured in a frame of the source. Let two pulses of the radiation are emitted from the source with the time interval (δT ) e (period). Then the interval (δT ) r between the times of arrival of the two pulses to the observer, as measured by a clock in the frame of the source S, is where δL is a difference of the distances traveled by the two pulses, measured in the frame of the source S, and V is the speed of light in the frame S given by The interval (δt) r between the moments of receiving the two pulses by the observer in the frame S ′ , as measured by a clock at rest in S ′ , is related to (δT ) r by the time dilation relation (68), as follows Thus, the periods of the electromagnetic wave measured in the frames of the source and the receiver are related by so that the relation for the frequencies is where ν e is the emitted wave frequency and ν r is the wave frequency measured by the observer moving with respect to the source. (This formula could be derived in several other ways, for example, using the condition of invariance of the wave phase.) To complete the derivation of the formula for the Doppler shift, the relation (77) is to be transformed such that the angle θ between the wave vector and the direction of motion measured in the frame of the observer S ′ figured instead of Θ which is the corresponding angle measured in the frame of the source. Using the aberration formula (70), solved for cos Θ, as follows in the relation (77) yields Finally, introducing the angleθ = θ − π between the line of sight and the direction of the observer motion one obtains the relation for a shift of frequencies due to the Doppler effect in the form

V. COSMOLOGICAL IMPLICATIONS
According to the modern view, there exists a preferred frame of reference related to the cosmic microwave background (CMB), more precisely to the last scattering surface (LSS).
Let us apply the equations of the anisotropic special relativity developed above to describe effects caused by an observer motion (our galaxy's peculiar motion) with respect to the CMB. Using equations of the standard special relativity in that context is inconsistent. The standard relativity theory framework is in contradiction with existence of a preferred frame while the anisotropic special relativity developed in the present paper naturally combines the special relativity principles with the existence of a preferred frame. In order to apply equations of the anisotropic special relativity for describing the physical phenomena in a frame moving with respect to the LSS, let choose the frame S to be a preferred frame and the frame S ′ to be a frame of an observer moving with respect to the preferred frame. Then the coordinate transformations from the preferred frame S to the frame S ′ of the moving observer are obtained by setting K = 0 in equations (39), (40), (61) and (58) which yields The inverse R −1 of (61) for K = 0 takes the form Substituting (84) into (83) yields Thus, in terms of the angle Θ between the direction of the observer motion and that of the light propagation as measured in a frame of the source, the Doppler frequency shift is a pure dipole pattern as it is in the standard relativity. However, the amplitude of the shift includes an additional factor which depends on the value of the universal constant q.
Equation (80) incorporating the effect of light aberration and thus relating the frequency ν e of the light emitted at the LSS to the frequency ν r measured by an observer moving with respect to the LSS, with the use of (84) becomes whereθ is the angle between the line of sight and the direction of the observer motion as measured in the frame of the observer. In the context of the CMB anisotropy, one should switch from the frequencies to effective thermodynamic temperatures of the CMB blackbody radiation using the relation [31] T where T 0 is the effective temperature measured by the observer, that is at rest relative to the LSS and sees strictly isotropic blackbody radiation, and T (θ) is the effective temperature of the blackbody radiation for the moving observer looking in the fixed directionθ. Substituting (86) into (87) yields Thus, the angular distribution of the CMB effective temperature seen by an observer moving with respect to the CMB is not altered by the light speed anisotropy. However, the anisotropy influences the mean temperature which now does not coincide with the temperature T 0 measured by the observer, that is at rest relative to the LSS, but differs from it by the factor (1 − β 2 ) − q 2 . Developing equation (88) up to the second order in β yields which implies that, up to the order β 2 , the amplitudes of the dipole and quadrupole patterns remain the same, only the constant term is modified.
It is worth reminding that, even though the specified law (57) is linear in β, it does include the second order term which is identically zero. Thus, describing the anisotropy effects, which are of the order of β 2 , by equations (88) and (89) is legitimate.
Equations (80) and (61) can be used to derive the Doppler frequency shift in the case when an observer in the frame moving with respect to the CMB (Earth) receives light from an object (galaxy) which is also moving with respect to that preferred frame. The present formulation, which assumes that all motions are along the same axis, implies that the relative motion of the object and the observer is along the direction of the observer motion relative to the CMB. It is straightforward, within the framework developed in the present study, to extend the analysis to defining a group of transformations which is not restricted by that assumption. It is worthwhile to note that the results related to the CMB temperature distribution considered above are not influenced by that assumption since only two frames, moving and preferred frames, figure in the analysis.
To obtain a formula convenient for applications some alterations are needed. First, the frequency shift is to be expressed in terms of the velocity v g (or β g ) of the object relative to the observer which, in the presence of anisotropy, is not equal to the velocity v (or β) of the observer with respect to the object that figures in equations (80) and (61). To do this β is expressed through β g from equation (46) where β − stays for β g , as follows Next, it is needed to express the frequency shift in terms of the observer velocity relative to the CMBv but not of the object velocity relative to the CMB. Since the velocities with respect to the CMB are related to the anisotropy parameters by (57) it implies that the anisotropy parameter in the frame of the observer K is to be expressed through k. It is done using equation (58) with β replaced by (90) which yields The final formula for the frequency shift is obtained from equations (80) and (61) by substituting (90) for β and next substituting (91) for K and expressing k as k = qβ afterwards. In order not to complicate matters, the case when the object motion is along the line of sight, θ = 0 is considered and the resulting expression for the frequency shift is expanded up to the order of β 2 g andββ g which yields ν r = ν e 1 + 1 + qβ β g + 1 Thus, corrections to the Doppler shift due to the presence of the anisotropy (the terms multiplied by q in (92)) are of the second order in velocities.
A counterpart of the special relativity kinematics has been developed to remedy the situation when the principles of special relativity are in contradiction with a commonly accepted view that there exists a preferred universal rest frame, that of the cosmic background radiation. Analysis of the present paper shows that, despite the general consensus that the special relativity principles should be abolished if the existence of the preferred frame is accepted, a synthesis of those seemingly incompatible concepts is possible. The framework developed doesn't abolish the basic principles of special relativity but simply uses the freedom in applying those principles in order to incorporate a preferred frame into the theory.
A degree of anisotropy of the one-way velocity, which is commonly considered as irreducibly conventional, acquires meaning of a characteristic of the really existing anisotropy caused by motion of an inertial frame relative to the preferred frame. In that context, the fact, that there exists the inescapable entanglement between remote clock synchronization and one-way speed of light, does not imply conventionality of the one-way velocity but means that the synchronization procedure is to be made using the one-way velocity selected by the size of the really existing anisotropy (like as the Einstein synchronization using the isotropic one-way velocity is selected in the case of an isotropic system).
Incorporating the anisotropy of the one-way speed of light into the framework based on the relativity principle and the principle of constancy of the two-way speed of light yields equations differing from those of the standard relativity. The deviations depend on the value of an universal constant q where q = 0 corresponds to the standard relativity theory with the isotropic one-way speed of light and Einstein synchronization in all the frames. The measurable effects following from the theory equations can be used to validate the theory and provide estimates for q. From somewhat different perspective, it means that, even though direct measuring of the one-way speed of light is not possible, the anisotropy of the one-way speed of light may reveal itself in measurable effects.
Applying the theory to the problem of calculating the CMB temperature distribution is conceptually attractive since it removes the inconsistency of the usual approach when formulas of the standard special relativity, in which a preferred frame is not allowed, are applied to define effects caused by motion with respect to the preferred frame. Nevertheless, other measurable effects, in particular, the Doppler frequency shift measured by the Earth observer receiving light from an object that is also moving relative to the GMB frame, might be easier to identify. The object could also be a light emitter in laboratory experiments.
The constant q has a definite physical meaning of the coefficient in the formula (57) defining dependence of the parameter k of anisotropy of the one-way speed of light in a particular frame on the frame velocity with respect to a preferred frame. Nevertheless a direct measuring of that constant is not possible since no experiment is a "one-way experiment".
(We leave aside a discussion of the papers in which measuring of the one-way speed of light is reported, as well as of the papers refuting them.) The present theory provides the possibility of obtaining estimates for that fundamentally important constant. It is worthwhile to note that even though it were found that the constant q is very small, which would mean that applying the present theory yields results practically identical to those of the standard relativity, this would not reduce the importance of the present framework which reconciles the principles of special relativity with the existence of the privileged CMB frame. As a matter of fact, it would justify the application of the standard relativity in that situation.

Appendix A: Equation of light propagation
We will define the form of the equation of light propagation based on the law (14) for the light speed variation. If we use the spherical coordinate system x = r cos θ, y = r sin θ sin φ, z = r sin θ cos φ (A1) with the axis x directed along the anisotropy vector k, then the angle θ k in (14) coincides with the polar angle θ so that the law of the variation of speed of light in space becomes To derive the equation for light propagation corresponding to the law (A2) we start from g ik dx i dx k = 0; (A3) with i and k running from 0 to 3 (g 00 > 0) and x 0 = ct, x 1 = x, x 2 = y, x 3 = z. To define g ik such that (A3) corresponded to the law (A2) we will use the expression for the light velocity (see, e.g., [? ]): where Greek letters run from 1 to 3 as distinct from Latin letters that run from 0 to 3.. We will also use the relation γ µν n µ n ν = 1; γ µν = −g µν + γ µ γ ν (A5) Based on the symmetry of the problem we have g 20 = g 30 = 0, g 22 = g 33 = −1 ⇒ γ 2 = γ 3 = 0, γ 22 = γ 33 = 1 (A6) Then it follows from (A4), (A2) and (A5) that g 00 = 1, −g 10 n 1 = k cos θ; (−g 11 + g 2 10 )(n 1 ) 2 + (n 2 ) 2 + (n 3 ) 2 = 1 (A7) With (n 1 = cos θ, n 2 = sin θ sin φ, n 3 = sin θ cos φ) we obtain g 10 = −k and g 11 = k 2 − 1 so that the equation for light propagation becomes It is worth remarking that the linearity assumption is not imposed.
Having the infinitesimal group generators defined by (B6) the finite group transformations can be found via solving the Lie equations with proper boundary conditions. As in Section 2, the common kinematic restrictions can be imposed to make the constants c 1 , c 2 , c 3 and ω vanishing which eliminates space and time shifts and excludes rotations in the plane (y, z). The initial data problems (B7) and (B8) are readily solved to give x = X cosh a − cT sinh a, ct = cT cosh a − X sinh a; y = Y, z = Z (B9) The group parameter a is related to the velocity v using the condition Substitution of (B11) into (B9) results in the Lorentz transformations