Abstract
Mansouri and Sexl (MS) presented a general framework for coordinate transformations between inertial frames, presupposing a preferred reference frame the spacetime of which is isotropic. The relative velocity between inertial frames in the standard synchronization is shown to be determined by the first row of the transformation matrix based on the MS framework. Utilizing this fact, we investigate the relativistic velocity addition. To effectively deal with it, we employ a diagram of velocity that consists of nodes and arrows. Nodes, which are connected to each other by arrows with relative velocities, represent inertial frames. The velocity composition law of special relativity has been known to be inconsistent with the reciprocity principle of velocity, through the investigation of a simple case where the inertial frames of interest are connected via a single node. When they are connected through more than one node, many inconsistencies including the violation of the reciprocity principle are found, as the successive coordinate transformation is not reduced to a Lorentz transformation. These inconsistencies can be cured by introducing a reference node such that the velocity composition is made in conjunction with it. The reference node corresponds to the preferred frame. The relativistic velocity composition law has no inconsistencies under the uniqueness of the isotropic frame.
1 Introduction
Special relativity (SR) was formulated based on the principle of relativity and the constancy of the speed of light [1], which lead to the Lorentz transformation (LT). Usually the LT has been utilized for a direct coordinate transformation from one inertial frame
The relativistic velocity composition has been investigated, as in the case of the Mocanu paradox, under a single intermediate frame, but such an investigation would not provide sufficient information on the consistency in the velocity composition law of SR. Considering the case where
Though all inertial frames are equivalent according to the relativity principle in SR, sometimes a preferred (or privileged) frame that can provide absolute simultaneity has been attracted and investigated [11,16,17,18,19,20,21], mainly in terms of clock synchronization. Presupposing a preferred reference frame the spacetime of which is isotropic, Mansouri and Sexl (MS) presented a general framework for the transformation between the preferred frame and an arbitrary inertial frame to investigate the role of convention in various definitions of clock synchronization and simultaneity [21]. The formulation is general in that it has been derived from fundamental kinematics and the isotropy of the preferred frame, and can be applied to various synchronizations.
It is shown based on the MS general framework that when the standard synchronization is adopted, the relative velocity of
The inconsistency associated with the reciprocity of velocity results from the noncommutativity of the velocity composition. When
Throughout the article, the following notations are used. We represent vectors with lowercase boldface letters and matrices with uppercase boldface letters. Superscript T stands for transpose. Matrix
where
2 General framework for inertial transformation
The matrices that make the transformation of coordinates between inertial frames include the information on relative velocities. Hence we can discover the velocities from them. This section addresses this matter in general transformations, employing the MS general framework. In the general framework, a preferred reference frame
where
with
The transformation coefficients
The transformation between arbitrary inertial frames
where
Using Eqs. (3), (4), and (6),
where
The transformation between
where
It should be noted that the direction of
PT, which is the time by the clock of an observer, is well known to be independent of the synchronization of clocks. We use a subscript “
The differential coordinate vector of an observer
The matrix
Comparing Eqs. (12b) and (13b), we see that the differential PT of
From Eq. (12a)
The relative velocity can be expressed as
3 Relativistic velocity composition
As mentioned in Section 1, the predictions of SR have been in agreement with numerous experimental results. The transformation coefficients in Eq. (2) are employed in accordance with SR and are given by:
where
It is identical with the LT matrix when the relative velocity is
The transformation matrix from
Then, the first row of A is written as:
It should be noted that in the standard synchronization, the relative velocity can be found from the first row of the transformation matrix. The
and
3.1 In SR
When making the analysis of velocity composition, it is convenient to employ a diagram of velocity, as in Figures 1–5, where inertial frames are designated by nodes so that node
where
and the reverse velocity
The
To resolve the nonequality, a Thomas rotation [13,14,15] is employed in such a way that
where the spatial rotation matrix
where
Besides the inconsistencies mentioned above, numerous inconsistencies can be found when the connection between the two inertial frames of interest is made via more than one node. As illustrated in Figure 2, nodes
Then, the relative velocities between nodes
In Eq. (27), the velocity addition is performed from right to left, rather than from left to right. For example,
The composition operation is not associative and has the following properties for arbitrary velocities
where
with
The relative velocities Eqs. (27a) and (27b) each should remain the same for every
In SR,
The composition operation is not associative and thus generally
Now let us turn our eyes to reverse velocities. The reverse velocity composition law Eq. (27b) itself is inconsistent since
The nonequality in (32a) results from the nonequality of
3.2 From inconsistency to consistency
We can make the velocity composition consistent by putting some constraints on it. A reference frame
Let us illustrate the consistency of it, selecting the isotropic frame
If
Then,
The velocity addition operation is not associative and thus the
In PFT, in which the isotropic frame is unique, the transformation between
The equality of Eq. (38) results from Eq. (37).
Now, it is time to examine the validity of the constrained composition law. From
When nodes
The multiple composition proceeds from left to right. Substituting Eq. (39) in Eq. (36) yields:
The velocities
Each relative velocity from Eqs. (40a) and (40b), which are dependent only on
3.3 Discussion
If SR is consistent, its velocity composition should also satisfy both Eqs. (40a) and (40b) because the LT is employed for coordinate transformations between
and from Eqs. (14), (11), and (18a),
Note that Eq. (41b) is valid even if
The nonequality Eq. (23) in SR, which results in Eq. (25), shows that the principle of relativity is not satisfied. It causes inconsistencies and paradoxes. The origin of the relativity principle is generally recognized to be attributed to Galileo [18]. In the Galilean transformation (GaT), if the velocity of
It is obvious that
where
Given
where
Though the composition law is not associative, the investigation under the assumption of the associativity would be helpful for deepening the understanding of its inconsistency. Suppose that the relativistic velocity addition is associative, but not commutative. Then, the parentheses in Eq. (27a) can be eliminated, and the terms of
SR is based on the principle of relativity and the isotropy of the speed of light. The inconsistency in the velocity composition law of SR results from the nonequality (23), which indicates that the principle of relativity is infeasible under the light speed constancy. The anisotropy of the speed of light has been observed in the experiments of the Sagnac effect. Recently, under the uniqueness of the isotropic frame, the generalized Sagnac effect [23,25,26,27], which involves linear motion as well as circular motion, has been comprehensively analyzed based on the MS general framework [23] and by using the transformation under constant light speed (TCL) [26,27]. The analysis results show that the speed of light is anisotropic in inertial frames as well as in rotating ones. TCL provides a coordinate transformation between the rotating and the inertial frames, holding the constancy of the twoway speed of light also in the rotating one. Though the rotating system is in motion with acceleration, it can be regarded as locally inertial. Accordingly, a transformation for inertial frames can be derived from TCL through the limit operation of circular motion to linear motion [26,27]. The derived inertial transformation, the predictions of which are in agreement with the results of the experiments for the validation of SR, indicates the anisotropy of the speed of light as well. Moreover, even if the LT is employed for the transformation of inertial frames, the actual speed of light, with respect to proper time, is shown to be anisotropic in inertial frames [26]. TCL and the inertial transformation, which are consistent with each other, are consistent with PFT.
4 Conclusion
It has been known that the velocity composition of SR is not consistent with the reciprocity principle, in a situation where two nodes
The problems can be cured by introducing a reference node

Funding information: The authors state no funding involved.

Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

Conflict of interest: The authors state no conflict of interest.
Appendix
If a coordinate transformation matrix
C
is known in the standard synchronization, the relative velocity can be obtained from its first row, as shown in Eq. (19). We denote the relative velocity from
C
by
where
B
is an identity matrix or an arbitrary matrix associated with coordinate transformation. As the relative velocity is determined by the first row of the transformation matrix, the equalities in Eq. (A1) mean that the first rows of the total matrices in square brackets on one side are equal to the respective ones on the other side. It is straightforward to see that the first rows of
AT
_{
L
}(
u
) and
T
_{
L
}(
u
) are identical and so are those of
AT
_{
L
}(
u
)
B
and
T
_{
L
}(
u
)
B
. Similarly the equality (A1b) is shown since the first rows of
T
_{
L
}(
u
)
A
and
The successive transformation
where A is calculated as Eq. (29). Using Eqs. (A1a) and (A2), we have
where
where
A
is given by Eq. (29) with
Using (A2), we have
where
Eqs. (A4a) and (A4b) lead to Eq. (28b).
Eq. (27a) can be derived by repeatedly applying Eq. (A1a). It is clear that
Continuing such a computation leads to Eq. (27a). Similarly Eq. (27b) can be obtained. The inverse of
It is straightforward to see that
References
[1] Einstein A. On the electrodynamics of moving bodies. Ann Phys Leipzig. 1905;322(10):891–921.10.1016/B9780080069951.500144Search in Google Scholar
[2] Gwinner G. Experimental tests of time dilation in special relativity. Mod Phys Lett A. 2005;20(11):791–805.10.1142/S0217732305017202Search in Google Scholar
[3] MacArthur DW. Special relativity: Understanding experimental tests and formulations. Phys Rev D. 1986;33(1):1–5.10.1103/PhysRevA.33.1Search in Google Scholar PubMed
[4] Will CM. Clock synchronization and isotropy of the oneway speed of light. Phys Rev D. 1992;45(2):403–11.10.1103/PhysRevD.45.403Search in Google Scholar
[5] Kretzschmar M. Doppler spectroscopy on relativistic particle beams in the light of a test theory of special relativity. Z Phys A. 1992;342(4):463–9.10.1007/BF01294957Search in Google Scholar
[6] Mocanu CI. Some difficulties within the framework of relativistic electrodynamics. Arch Elektrotech. 1986;69(2):97–110.10.1007/BF01574845Search in Google Scholar
[7] Koczan GM. Relative binary, ternary and pseudobinary 4D velocities in the special relativity: Part I. Covariant Lorentz transformation. arXiv:2108.10725 [Preprint]; 2021. https://arxiv.org/abs/2108.10725.Search in Google Scholar
[8] Koczan GM. Relativistic relative velocity and relativistic acceleration. Acta Physica Polonica A. 2021;139(4):401–6.10.12693/APhysPolA.139.401Search in Google Scholar
[9] Oziewicz Z. Ternary relative velocity; astonishing conflict of the Lorentz group with relativity. arXiv:1104.0682 [Preprint]; 2011. https://arxiv.org/abs/1104.0682.Search in Google Scholar
[10] Oziewicz Z. How do you add relative velocities? Group theoretical methods in physics. Vol. 185. Conference Series. CRC Press; 2004. p. 439–44. Available from https://www.researchgate.net/publication/253628364_How_do_you_add_relative_velocities/link/56fc9e8108aef6d10d91c11b/download.Search in Google Scholar
[11] Oziewicz Z. Æther needs relativitity group: Lorentz group if and only if Æther. Academia; 2006. https://www.academia.edu/12338869/%C3%86ther_Needs_Relativity_Group_Lorentz_Group_if_and_only_if_%C3%86ther.Search in Google Scholar
[12] FernándezGuasti M. Alternative realization for the composition of relativistic velocities. Proc SPIE. 2011;8121:812108.10.1117/12.894342Search in Google Scholar
[13] Ungar AA. The relativistic velocity composition paradox and the Thomas rotation. Found Phys. 1989;19(11):1385–96.10.1007/BF00732759Search in Google Scholar
[14] Ungar AA. The Thomas rotation and the parameterization of the Lorentz group transformation group. Found Phys Lett. 1988;1(1):57–89.10.1007/BF00661317Search in Google Scholar
[15] Kholmetskii AL, Yarman T. ThomasWigner rotation and Thomas precession in covariant ether theories: novel approach to experimental verification of special relativity. Can J Phys. 2015;93(5):1232–40.10.1139/cjp20140340Search in Google Scholar
[16] Tangherlini FR. The velocity of light in uniformly moving frames. [Ph.D. thesis]. USA: Stanford University; 1958.Search in Google Scholar
[17] Selleri F. Noninvariant oneway velocity of light. Found Phys. 1996;26(7):641–64.10.1007/BF02058237Search in Google Scholar
[18] de Abreu R, Guerra V. The principle of relativity and the indeterminacy of special relativity. Eur J Phys. 2008;29(1):33–52.10.1088/01430807/29/1/004Search in Google Scholar
[19] Szostek K, Szostek R. The derivation of the general form of kinematics with the universal reference system. Results in Phys. 2018;8:429–37.10.1016/j.rinp.2017.12.053Search in Google Scholar
[20] Szostek R. Derivation of all linear transformations that meet the results of MichelsonMorley’s experiment and discussion of the relativity basics. Moscow Univ Phys Bull. 2020;75(6):684–704.10.3103/S0027134920060181Search in Google Scholar
[21] Mansouri R, Sexl RU. A test theory of special relativity: I. Simultaneity and clock synchronization. Gen Relativ Gravit. 1977;8(7):497–513.10.1007/BF00762634Search in Google Scholar
[22] Choi YH. Uniqueness of the isotropic frame and usefulness of the Lorentz transformation. J Kor Phys Soc. 2018;72(10):1110–20.10.3938/jkps.72.1110Search in Google Scholar
[23] Choi YH. Theoretical analysis of generalized Sagnac effect in the standard synchronization. Can J Phys. 2017;95(8):761–6.10.1139/cjp20160953Search in Google Scholar
[24] Berzi V, Gorini V. Reciprocity principle and the Lorentz transformations. J Math Phys. 1969;10(8):1518–24.10.1063/1.1665000Search in Google Scholar
[25] Wang R, Zheng Y, Yao A. Generalized Sagnac effect. Phys Rev Lett. 2004;93(14):143901.1–143901.3.10.1103/PhysRevLett.93.143901Search in Google Scholar PubMed
[26] Choi YH. Consistent coordinate transformation for relativistic circular motion and speeds of light. J Kor Phys Soc. 2019;75(3):176–87.10.3938/jkps.75.176Search in Google Scholar
[27] Choi YH. Coordinate transformation between rotating and inertial systems under the constant twoway speed of light. Eur Phys J Plus. 2016;131(9):296.10.1140/epjp/i201616296xSearch in Google Scholar
© 2022 YangHo Choi, published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.