+ q sinh a ) k sinh a + q cosh a (48)

and introducing, in addition to S and S', the frame ${S}_{0}$ with the anisotropy parameter ${k}_{0}$ , one can check that

$\kappa \left(a,\kappa \left({a}_{0},{k}_{0}\right)\right)=\kappa \left(a+{a}_{0},{k}_{0}\right)$ (49)

Similarly it is readily verified that $\kappa \left(-a,\kappa \left(a,k\right)\right)=k$ and $\kappa \left(0,k\right)=k$. Alternatively, one can calculate the group generator $\chi \left(k\right)$ as

$\chi \left(k\right)={\frac{\partial \kappa \left(a,k\right)}{\partial a}|}_{a=0}=q-\frac{{k}^{2}}{q}$ (50)

and solve the initial value problem

$\frac{\text{d}k\left(a\right)}{\text{d}a}=q-\frac{k{\left(a\right)}^{2}}{q},\text{ }k\left(0\right)=K$ (51)

to be assured that it, as expected, yields (46). Thus, as a matter of fact, what is specified using the approximate relation (43) is the form of the group generator $\chi \left(k\right)$ in the group of transformations defined on the basis of the first principles.

The relation (42) allows defining a form of the group generator $\kappa \left(k\right)$ for arbitrary $F\left(\stackrel{¯}{\beta }\right)$ , not restricted by the approximate relation (43). Representing (42) in the form

$f\left(k\left(K;a\right)\right)=\frac{f\left(K\right)+\beta \left(a\right)\left(1-Kf\left(K\right)\right)}{1+\beta \left(a\right)\left(-K+f\left(K\right)\right)}$ (52)

substituting (45) for $\beta \left(a\right)$ and differentiating the result with respect to a, with $\partial k\left(K;a\right)/\partial a$ separated, yields

$\frac{\partial k\left(K;a\right)}{\partial a}=\frac{1-f{\left(K\right)}^{2}}{{\left(\mathrm{cosh}a+f\left(K\right)\mathrm{sinh}a\right)}^{2}{f}^{\prime }\left(k\left(a\right)\right)}$ (53)

Then the relation (52), with $\beta$ substituted from (45), is used again to express $f\left(K\right)$ through $f\left(k\right)$ and a. Substituting that expression into (53) yields

$\frac{\text{d}k\left(a\right)}{\text{d}a}=\frac{1-{f}^{2}\left(k\left(a\right)\right)}{{f}^{\prime }\left(k\left(a\right)\right)}$ (54)

Equation (54) is the Lie equation defining (with the initial condition $k\left(0\right)=K$) the group transformation $k\left(K;a\right)$ which implies that the expression on the right-hand side is the group generator

$\kappa \left(k\right)=\frac{1-{f}^{2}\left(k\right)}{{f}^{\prime }\left(k\right)}$ (55)

6. Time Dilation, Aberration Law and Doppler Effect

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 (33) where it should be evidently set ${x}_{2}={x}_{1}$. Subtracting the two relations we obtain the time dilation relation

$\Delta T=\frac{{R}^{-1}}{\sqrt{{\left(1-K\beta \right)}^{2}-{\beta }^{2}}}\Delta t$ (56)

If clock were at rest in the frame S the time dilation relation would be

$\Delta t=\frac{R\left(1-K\beta -k\beta \right)}{\sqrt{{\left(1-K\beta \right)}^{2}-{\beta }^{2}}}\Delta T=\frac{R}{\sqrt{{\left(1-k{\beta }_{-}\right)}^{2}-{\beta }_{-}^{2}}}\Delta T$ (57)

with ${\beta }_{-}$ defined by (32).

Aberration law. The light aberration law can be derived using the formulas (35) for the velocity transformation. The relation between directions of a light ray in the two inertial frames S and S' is obtained by setting ${U}_{X}=c\mathrm{cos}\Theta /\left(1+K\mathrm{cos}\Theta \right)$ and ${u}_{x}=c\mathrm{cos}\theta /\left(1+k\mathrm{cos}\theta \right)$ in the first equation of (35). Then solving for $cos\theta$ yields

$\mathrm{cos}\theta =\frac{\mathrm{cos}\Theta -\beta \left(1+K\mathrm{cos}\Theta \right)}{1-\beta \left(\mathrm{cos}\Theta +K\right)}$ (58)

where $\theta$ and $\Theta$ are the angles between the direction of motion and that of the light propagation in the frames of a moving observer and of an immovable source respectively. (Equation (58) could be obtained in several other ways, for example, straight from the transformations (27) and (28) by rewriting them in spherical coordinates and then specifying to radial light rays.) Introducing $\stackrel{˜}{\theta }=\theta -\pi$ and $\stackrel{˜}{\Theta }=\Theta -\pi$ as the angles between the direction of motion and the line of sight one gets the aberration law

$\mathrm{cos}\stackrel{˜}{\theta }=\frac{\mathrm{cos}\stackrel{˜}{\Theta }+\beta \left(1-K\mathrm{cos}\stackrel{˜}{\Theta }\right)}{1+\beta \left(\mathrm{cos}\stackrel{˜}{\Theta }-K\right)}$ (59)

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 $\Theta$ 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 ${\left(\delta T\right)}_{e}$ (period). Then the interval ${\left(\delta T\right)}_{r}$ 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

${\left(\delta T\right)}_{r}={\left(\delta T\right)}_{e}+\frac{\delta L}{V}$ (60)

where $\delta L$ 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

$\delta L=v{\left(\delta T\right)}_{r}\mathrm{cos}\Theta ,\text{ }V=\frac{c}{1+K\mathrm{cos}\Theta }$ (61)

Substituting (61) into (60) yields

${\left(\delta T\right)}_{e}={\left(\delta T\right)}_{r}\left(1-\beta \mathrm{cos}\Theta \left(1+K\mathrm{cos}\Theta \right)\right)$ (62)

The interval ${\left(\delta t\right)}_{r}$ 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 ${\left(\delta T\right)}_{r}$ by the time dilation relation (56), as follows

${\left(\delta T\right)}_{r}=\frac{{R}^{-1}}{\sqrt{{\left(1-K\beta \right)}^{2}-{\beta }^{2}}}{\left(\delta t\right)}_{r}$ (63)

Thus, the periods of the electromagnetic wave measured in the frames of the source and the receiver are related by

${\left(\delta T\right)}_{e}=\frac{{R}^{-1}\left(1-\beta \mathrm{cos}\Theta \left(1+K\mathrm{cos}\Theta \right)\right)}{\sqrt{{\left(1-K\beta \right)}^{2}-{\beta }^{2}}}{\left(\delta t\right)}_{r}$ (64)

so that the relation for the frequencies is

${\nu }_{r}={\nu }_{e}\frac{{R}^{-1}\left(1-\beta \mathrm{cos}\Theta \left(1+K\mathrm{cos}\Theta \right)\right)}{\sqrt{{\left(1-K\beta \right)}^{2}-{\beta }^{2}}}$ (65)

where ${\nu }_{e}$ is the emitted wave frequency and ${\nu }_{r}$ 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 (65) is to be transformed such that the angle $\theta$ between the wave vector and the direction of motion measured in the frame of the observer S' figured instead of $\Theta$ which is the corresponding angle measured in the frame of the source. Using the aberration formula (58), solved for $cos\Theta$ , as follows

$\mathrm{cos}\Theta =\frac{\mathrm{cos}\theta +\beta \left(1-K\mathrm{cos}\theta \right)}{1+\beta \left(\mathrm{cos}\theta -K\right)}$ (66)

in the relation (65) yields

${\nu }_{r}={\nu }_{e}\frac{{R}^{-1}\left(1+\beta \mathrm{cos}\theta \left(1-K\mathrm{cos}\theta \right)\right)\sqrt{{\left(1-K\beta \right)}^{2}-{\beta }^{2}}}{{\left(1-K\beta +\beta \mathrm{cos}\theta \right)}^{2}}$ (67)

Finally, introducing the angle $\stackrel{˜}{\theta }=\theta -\pi$ 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

${\nu }_{r}={\nu }_{e}\frac{{R}^{-1}\left(1-\beta \mathrm{cos}\stackrel{˜}{\theta }\left(1+K\mathrm{cos}\stackrel{˜}{\theta }\right)\right)\sqrt{{\left(1-K\beta \right)}^{2}-{\beta }^{2}}}{{\left(1-K\beta -\beta \mathrm{cos}\stackrel{˜}{\theta }\right)}^{2}}$ (68)

7. The CMB Effective Temperature

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 frame. It is more consistent than using equations of the standard special relativity in that context―the standard relativity framework is in contradiction with existence of a preferred frame while the anisotropic special relativity naturally combines a preferred frame concept with the special relativity principles. 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 (27), (28), (47) and (44) which yields

$x=\left(X-cT\beta \right){\left(1-{\beta }^{2}\right)}^{\frac{q-1}{2}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}ct=\left(cT\left(1-q{\beta }^{2}\right)-X\beta \left(1-q\right)\right){\left(1-{\beta }^{2}\right)}^{\frac{q-1}{2}}$

$y=Y{\left(1-{\beta }^{2}\right)}^{\frac{q}{2}},\text{ }z=Z{\left(1-{\beta }^{2}\right)}^{\frac{q}{2}}$ (69)

where q is a universal constant. Equation of aberration of light (59) with K = 0 converts into the common aberration law of the standard theory

$\mathrm{cos}\stackrel{˜}{\theta }=\frac{\mathrm{cos}\stackrel{˜}{\Theta }+\beta }{1+\beta \mathrm{cos}\stackrel{˜}{\Theta }}$ (70)

while Equation (65), describing the Doppler frequency shift for the light emitted at the last scattering surface (LSS) and received by a moving observer, differs from its counterpart of the standard relativity by the factor ${R}^{-1}$ , as follows

${\nu }_{r}={\nu }_{e}\frac{{R}^{-1}\left(1-\beta \mathrm{cos}\Theta \right)}{\sqrt{1-{\beta }^{2}}}$ (71)

The inverse ${R}^{-1}$ of (47) for K = 0 takes the form

${R}^{-1}={\left(1-{\beta }^{2}\right)}^{-\frac{q}{2}}$ (72)

Substituting (72) into (71) yields

${\nu }_{r}={\nu }_{e}{\left(1-{\beta }^{2}\right)}^{-\frac{1}{2}-\frac{q}{2}}\left(1-\beta \mathrm{cos}\Theta \right)$ (73)

Thus, in terms of the angle $\Theta$ 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 (68) incorporating the effect of light aberration and thus relating the frequency ${\nu }_{e}$ of the light emitted at the LSS to the frequency ${\nu }_{r}$ measured by a moving observer, with the use of (72) becomes

${\nu }_{r}={\nu }_{e}\frac{{\left(1-{\beta }^{2}\right)}^{\frac{1}{2}-\frac{q}{2}}}{1-\beta \mathrm{cos}\stackrel{˜}{\theta }}$ (74)

where $\stackrel{˜}{\theta }$ 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 

$\frac{T\left(\stackrel{˜}{\theta }\right)}{{\nu }_{r}}=\frac{{T}_{0}}{{\nu }_{e}}$ (75)

where ${T}_{0}$ is the effective temperature measured by the observer which sees strictly isotropic blackbody radiation, and $T\left(\stackrel{˜}{\theta }\right)$ is the effective temperature of the blackbody radiation for the moving observer looking in the fixed direction $\stackrel{˜}{\theta }$. Substituting (74) into (75) yields

$T\left(\stackrel{˜}{\theta }\right)=\frac{M}{1-\beta \mathrm{cos}\stackrel{˜}{\theta }}\text{\hspace{0.17em}};\text{ }M={T}_{0}{\left(1-{\beta }^{2}\right)}^{\frac{1}{2}-\frac{q}{2}}$ (76)

Thus, the angular distribution of the CMB effective temperature seen by an observer moving with respect to the CMB frame is not altered by the light speed anisotropy. However, the anisotropy influences the mean temperature which differs from the value yielded by applying the standard relativity by the factor ${\left(1-{\beta }^{2}\right)}^{-\frac{q}{2}}$ (it may be also considered as a correction to the temperature ${T}_{0}$). Dependence of the amplitude factor M (normalized by ${T}_{0}$) on $\beta$ for different values of the parameter q is shown in Figure 1. It is seen that, for negative values of q, the amplitude factor decreases with $\beta$ , like as it does in the standard SR (q = 0), but the dependence becomes steeper. For positive values of q, the factor M may both decrease and increase with $\beta$ and it does not depend on $\beta$ for a specific value q = 1. Note, however, that q is expected to be negative both from intuitive considerations and on the basis of some arguments considering of which is beyond the scope of the current study.

Developing Equation (76) up to the second order in $\beta$ yields

$T\left(\stackrel{˜}{\theta }\right)={T}_{0}\left(1+q\frac{{\beta }^{2}}{2}+\beta \mathrm{cos}\stackrel{˜}{\theta }+\frac{{\beta }^{2}}{2}\mathrm{cos}2\stackrel{˜}{\theta }\right)$ (77)

which implies that, up to the order ${\beta }^{2}$ , 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 (43) is linear in $\beta$ , it does include the second order term which is identically zero. Thus, describing the anisotropy effects, which are of the order of ${\beta }^{2}$ , by Equations (76) and (77) is legitimate.

Figure 1. Dependence of the amplitude factor M (normalized by T0) on the observer velocity β for different values of the parameter q.

8. Discussion

Analysis of the present paper, incorporating the existence of a preferred frame of reference into the special relativity framework, does not abolish the basic principles of special relativity but simply uses the freedom in applying those principles. 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 (if the synchronization is made using light signals), does not imply conventionality of the one-way velocity but means that, in the synchronization procedure, the one-way speed determined by the size of the anisotropy is used. The analysis 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 in all the frames. The measurable effects following from the theory equations can be used to provide estimates for q and validate the theory.

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. 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.

It is worthwhile, at the end of the discussion, to return to the much debated issues of conventionality of simultaneity and relativity of simultaneity in special relativity and discuss the approach and results of the present paper in the light of the debates. First of all, an important difference between motivations (and, correspondingly, conceptual frameworks) of the analyses devoted to those issues and the approach of the present study should be clarified and emphasized again.

The concept of anisotropy of light propagation is always discussed in the literature in relation with the concept of remote clock synchronization. Considering different synchronization procedures, as the rule, is aimed at obtaining the transformations possessing some specific properties. For example, in the work by Tangherlini  , a special method of synchronizing two clocks in an inertial frame is proposed in order to achieve a universal synchronization, such that spatially separated clocks remain synchronous between themselves thus establishing the common time of the moving system. In  , it is achieved by using clocks synchronized with absolute signals, that is, signals travelling with infinite or arbitrarily large velocity. Using these signals, one arrives at the view of an absolute rest frame (or ether frame), in which the velocity of light is the same in all directions, but for observers in motion relative to this frame velocity of light is not the same in all directions. Another method of synchronization of spatially separated clocks, which leads to the same transformations that Tangherlini obtained in  , is the so-called “external synchronization” (see, e.g.,   ). The external synchronization is based on the assumption that there is a preferred (“rest”) inertial frame in which the one-way speed of light in vacuum is c in all directions. The clocks from the rest system, S, are synchronized using Einsteins procedure with light signals. Then, in any moving inertial frame S', the common time can be established using these already synchronized clocks of the rest inertial frame. It can be done simply by adjusting clocks of moving inertial frame to zero during those moments of time when they meet in space a clock at rest that shows zero as well. Applying any of two synchronization methods described above, together with the postulate of constancy of the two-way speed of light, yields the transformations

${x}^{\prime }=\gamma \left(x-vt\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{t}^{\prime }=\frac{t}{\gamma };\text{\hspace{0.17em}}\text{\hspace{0.17em}}\gamma =\frac{1}{\sqrt{1-{\beta }^{2}}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\beta =\frac{v}{c}$ (78)

where $\left(x,t\right)$ and $\left({x}^{\prime },{t}^{\prime }\right)$ are space and time coordinates of a certain event in the rest frame S and in a moving frame S' respectively and v is a velocity of the frame S' relative to S. Thus, using the synchronization method, that is different from synchronization by light signals, yields the transformations (78) which exhibit absolute simultaneity. They also exhibit non-invariant one-way speed of light so that, in that approach, the anisotropy of the velocity of light in a moving inertial frame is a feature that emerges due to synchronization procedure designed to keep simultaneity unchanged between all inertial frames of reference.

The principal difference of the present analysis from those in the literature on the synchronization problem is that, in the present analysis, the one-way speed of light in an inertial frame is a primary issue and its anisotropy is governed entirely by a physical lawc (36) (or its approximate version (43)). If a preferred frame is identified then the law (36) defines unequivocally the anisotropy size. Note that there is no ambiguity in determining the velocity $\stackrel{¯}{\beta }$ since it is measured in a preferred frame where the one-way speed of light is c in all directions. At the same time, the relativity principle is not violated since transformations of the parameter of anisotropy k from one inertial frame to another possess a group property and, in this respect, transformations from/to the preferred frame with k = 0 are not distinguished from other members of the group of transformations. Specifying the function $F\left(\stackrel{¯}{\beta }\right)$ (or the inverse function $f\left(k\right)$) is equivalent to specifying the group generator for the variable k according to (55). In such a framework, synchronization is a concomitant issue if the remote clocks are set using light signals. In particular, since the transformations (27) are derived based on invariance of the equation of anisotropic light propagation, they correspond to the synchronization procedure using light signals with the one-way velocities defined by the relation (6) but, provided that the velocity of the frame relative to the preferred frame $\stackrel{¯}{\beta }$ is known, in the relation (6), k is a definite value determined by the law (36). That value cannot be altered by changing the synchronization method.

The same is valid if another method of synchronization, as, for example, the above discussed “external synchronization”, is used. Changing the synchronization method results in a change of the form of transformations for the time and space variables which is equivalent to a change of coordinates. The Lorentz transformations

${{x}^{\prime }}_{L}=\gamma \left(x-vt\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{{t}^{\prime }}_{L}=\gamma \left(t-\frac{vx}{{c}^{2}}\right)$ (79)

can be obtained from the Tangherlini transformations (78) by the change of coordinates 

${{t}^{\prime }}_{L}={t}^{\prime }-\frac{v{x}^{\prime }}{{c}^{2}}$ (80)

where ${t}^{\prime }$ and ${x}^{\prime }$ are defined by (78). Substituting (78) in (80), one gets the Lorentz transformations. The same can be done for the transformations (27) obtained in the present paper. In the case of the transformations from a preferred frame with the anisotropy parameter K = 0 to an arbitrary frame with the anisotropy parameter k, the transformations (27) take the form

$x=R\gamma \left(X-cT\beta \right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}ct=R\gamma \left(cT\left(1-k\beta \right)-X\left(\beta -k\right)\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\gamma =\frac{1}{\sqrt{1-{\beta }^{2}}}$ (81)

where R is the scale factor defined by (24) (for the sake of clearness, we do not use the law (43) in these calculations). The transformations that exhibit absolute simultaneity, a counterpart of the Tangherlini transformations, are

${x}^{\left(T\right)}=R\gamma \left(X-cT\beta \right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}c{t}^{\left(T\right)}=R\frac{cT}{\gamma }$ (82)

and the change of variables converting (82) into (81) is

$ct=c{t}^{\left(T\right)}-\left(\beta -k\right){x}^{\left(T\right)}$ (83)

It is readily verified that substituting (82) in (83) yields (81). Thus, any event that can be described by the transformations (81) can be described as well by the transformations with absolute simultaneity (82). Descriptions using clocks set as in (81) and clocks set as in (82) are equivalent in a sense that they are describing one and the same reality, which is independent of the coordinates chosen.

It is also worth remarking that alterations, as compared with the standard relativity, in the formulas describing physical effects caused by motion with respect to a preferred frame depend only on the scale factor R as, for example, a correction to the distribution (76) of the CMB effective temperature seen by an observer moving with respect to the CMB frame. In (76), R is given by the expression

$R={\left(1-{\beta }^{2}\right)}^{\frac{q}{2}}$ (84)

defining R as a function of the velocity of a moving frame measured in a preferred frame. The expression (84) has been obtained from (47) evaluated for K = 0 and so it corresponds to the approximate law (43) but it is possible to represent R defined by the general expression (24) as a function of $\stackrel{¯}{\beta }$ for arbitrary $F\left(\stackrel{¯}{\beta }\right)$. It is evident that the form $R\left(\stackrel{¯}{\beta }\right)$ of the scale factor does not depend on the synchronization (or on the space-time coordinates) chosen.

To conclude the discussion, the present analysis, which combines the basic principles of special relativity with the existence of a preferred frame, stands apart from the ample literature devoted to the conventionality of simultaneity, relativity of simultaneity and synchronization issues. In the present analysis, anisotropy of the one way speed of light in an inertial frame is governed by a physical law which is not influenced by changing the synchronization procedure. Synchronization emerges as a complementary issue needed for defining transformations of the space-time coordinates but physical effects are not changed by the way the clocks have been set.

Acknowledgements

Conflicts of Interest

The authors declare no conflicts of interest.

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