Mansouri and Sexl (MS) presented a general framework for coordinate transformations between inertial frames, presupposing a preferred reference frame the space-time 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.
Special relativity (SR) was formulated based on the principle of relativity and the constancy of the speed of light , which lead to the Lorentz transformation (LT). Usually the LT has been utilized for a direct coordinate transformation from one inertial frame into another when the relative velocity of with respect to is known. In such a situation, the results of the experiments to test the validity of SR have been known to be in agreement with its predictions [2,3,4,5]. In accordance with the relativity principle, the reverse velocity, the velocity of relative to , should be of equal magnitude and in the direction opposite to the forward one. When considering an intermediate frame between and , however, the Mocanu paradox is raised so that the reciprocity relation is not satisfied . The velocity composition of SR is neither commutative nor associative, which causes inconsistencies in the composition. Some alternative approaches have been presented that aim at overcoming these inconsistencies [7,8,9,10,11,12]. The Mocanu paradox has been explained mostly by resorting to the Thomas rotation [13,14,15]. It seems to be generally accepted that the paradox is resolved by the Thomas rotation.
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 and are connected via more than one inertial frame, this article deals with the problem of the relativistic velocity composition. To this end, we introduce a diagram of velocity where inertial frames are represented as nodes that are connected by arrows indicating relative velocities. The diagram is useful to see the relationships between the relative velocities of the nodes.
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 space-time 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 . 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 with respect to can be found from the first row of the transformation matrix from the latter to the former. Exploiting this fact together with the velocity diagram allows us to readily approach the problems of the relativistic velocity composition, extending them to the case of multiple connecting nodes. Associativity-related properties in the velocity composition may have been known to be difficult to prove because of seemingly high mathematical complexity involved. It is stated in ref.  (p. 72) that “A direct proof … is lengthy and, hence, requires the use of computer algebra.” The properties are easily derived using the relationship between the relative velocity and the first row of the transformation matrix.
The inconsistency associated with the reciprocity of velocity results from the non-commutativity of the velocity composition. When and are connected through more than one node, many inconsistencies in the composition law of SR are raised in addition to the violation of the velocity reciprocity since the composition operation is neither commutative nor associative. As a result, the velocity of with respect to , depending on the connecting nodes, has not only multiple directions but also multiple magnitudes. Therefore the proper time (PT) in is not uniquely determined though it should have the same value from frame to frame regardless of synchronization schemes. To cure these inconsistencies and contradictions, a reference node is introduced so that the composition law is applied in conjunction with it. The reference node corresponds to the unique isotropic frame. In the preferred frame theory (PFT), which refers to a theory based on the uniqueness of the isotropic frame , there are no inconsistencies and no contradictions.
Throughout the article, the following notations are used. We represent vectors with lower-case boldface letters and matrices with upper-case boldface letters. Superscript T stands for transpose. Matrix is an identity matrix of appropriate size. For a real vector , denotes its magnitude. The vector with hat represents its unit vector, i.e., . For convenience we introduce a partitioned matrix to be used in place of transformation matrices:
where , , and denote the -entry, mth row, and nth column, respectively, and is a single quantity. The preferred reference frame is denoted as where the speed of light is a constant regardless of the propagation direction. We use imaginary time , where designates time.
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 is presumed whose space-time is isotropic. An inertial frame is in uniform linear motion at a velocity with respect to and its normalized velocity is . Representing time as an imaginary number, the space-time coordinate vector of is expressed as , where is a spatial vector. Similarly the coordinate vector of is expressed as . The coordinate vector of is transformed into and the resulting coordinate vector is written as:
where is a transformation matrix, which is given in the MS framework as [21,22,23]:
The transformation coefficients , , and can depend on . When the standard synchronization is employed, they are given as Eq. (16) below. As approaches zero, tends to . Then, is reduced to an identity matrix so that , , and all become equal to one. The vector is a synchronization vector, which is determined by a clock synchronization in . Indeed, the spatial vector of is irrelevant to . One can confirm it from Eq. (2). The last three rows of are independent of , which leads to the irrelevance of to .
The transformation between arbitrary inertial frames and is written from Eq. (2) as:
Using Eqs. (3), (4), and (6), is expressed as [22,23]:
The transformation between and is dependent on both and . Given and , the velocity of relative to is given by [5,22,23]:
It should be noted that the direction of is independent of and whereas its magnitude is dependent on .
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 “ ” at PT, say , to distinguish it from the time adjusted through a synchronization procedure. For simplicity is referred to as PT, though is different from the clock time of since . Let . The quantity , which represents the time dilation factor, is given by [22,23]:
The differential coordinate vector of an observer who is at rest in can be represented as . Substituting the differential vector of in Eq. (5) with subscripts and interchanged, we have:
The matrix is the inverse of A . Since the normalized velocity of is in , the differential vector can be generally expressed as:
Comparing Eqs. (12b) and (13b), we see that the differential PT of is related to by
From Eq. (12a) , which is written using Eqs. (13b) and (14) as
The relative velocity can be expressed as . The first column of the matrix that transforms coordinates from to contains the information on the velocity of relative to so that the relative velocity can be discovered from it. This fact holds regardless of the transformation coefficients and the synchronization vectors.
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 . In case the coefficients of Eq. (16) are used, the two-way speed of light in becomes irrespective of the synchronization gauge, . We adopt the standard synchronization, which leads to . Then, Eqs. (4a) and (4b) are written as and . When employing the coefficients of (16) in the standard synchronization, the is expressed as:
It is identical with the LT matrix when the relative velocity is . One can confirm from Eq. (17) that tends to an identity matrix as goes to zero.
The transformation matrix from to is given as Eq. (6) with the , , of Eq. (17). As , the inverse of also becomes equal to its transpose. Recalling that the transpose of A is the same as and using the relationship of and Eq. (15), we have [22,23]:
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 is calculated from Eqs. (9), (16), and (18) as [22,23]:
and can be expressed as in terms of and , which confirms that . Eq. (20) can also be expressed as Eq. (36) below by using the velocity composition of SR. If and are collinear, the reverse velocity is identical to . In the event that the absolute velocities are non-collinear, however it is not, though the magnitudes of and are the same.
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 , for example, represents and denotes the velocity of relative to . Consider a case where and are connected via an inertial frame , as shown in Figure 1. In SR, the coordinate transformation from to via is given by:
where with representing the LT matrix for velocity . The denotes a successive LT with relative velocities and . Generally, the coordinate vector in calculated by the right side of Eq. (21) will be different from of Eq. (5) with the coefficients of (16) and thus, a different notation is used in Eq. (21). The forward velocities and are initially given, and then the reverse velocities are dependently determined by the reciprocity principle in SR  so that they, having the same magnitude as the respective forward ones, are in the opposite directions like and . Reflecting the reciprocity principle, Figure 1 has been drawn just for the analysis of relative velocity in SR, though the correct reverse velocities under the standard synchronization may be different from them. Hence question marks are put beside the velocities given by the reciprocity principle. It is well known that according to the velocity composition law of SR, the velocity of relative to is expressed as:
and the reverse velocity is given by
The should be equal to in accordance with the reciprocity principle, but unless and are collinear it is not, which represents the Mocanu paradox. According to the principle of relativity, the frame is nothing more than one of an infinite number of isotropic frames. Nonetheless, the is not identical to the of Eq. (20) either. These inconsistencies result from the fact that is not the same as :
To resolve the non-equality, a Thomas rotation [13,14,15] is employed in such a way that
where the spatial rotation matrix is determined such that the equality is fulfilled. However, the rotation, which is a function of and , is not uniquely given since it depends on the intermediate node . Suppose that is the correct relative velocity. Following the path for SR  that Einstein walked to discover the transformation of coordinates from to will lead to the LT , not . As the successive transformation depends on the connecting node, in general:
where . Spatial rotations may explain the non-equality (23), but cannot resolve the non-equality (25), which is irrelevant to them.
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 and are connected through nodes. In the case of , Figure 2 is reduced to Figure 1 for forward velocities. Given the forward velocities, say and , the reverse velocities with respect to them are given as and by the reciprocity principle. The forward velocities are assumed to be non-collinear. The successive transformation matrix from node to is written as:
Then, the relative velocities between nodes and are calculated as
In Eq. (27), the velocity addition is performed from right to left, rather than from left to right. For example, is computed as . Refer Appendix for the derivation of Eq. (27).
The composition operation is not associative and has the following properties for arbitrary velocities , [13,14]:
where , which has a form of Eq. (1), is a block diagonal matrix and is given by
with and . If and are collinear, the velocity composition is equal to and is reduced to an identity matrix. The associativity-related properties can be readily derived, as shown in Appendix.
The relative velocities Eqs. (27a) and (27b) each should remain the same for every and arbitrary connecting nodes in order that SR becomes consistent. At first glance, the composition law (27a) for the forward velocities appears to be consistently applied because is the same for every . Clearly is equal to since . However, there exist many inconsistencies. Let us first consider simple configurations of connection with two added nodes as shown in Figure 3. One can see two connection configurations for : and . The relative velocities for the connections are:
In SR, and . Substituting these in Eqs. (30a) and (30b), we have :
The composition operation is not associative and thus generally . In the case of , one can confirm that the is the same as , which is identical to Eq. (31a). If the order of connection is changed so that node is first connected to node and node is connected to node , then the is given as , which is equal to Eq. (31b) but different from the previous one. As more nodes are added, the inconsistency comes to be more deepened since the relative velocity is dependent on more nodes. Given intermediate nodes as in Figure 2, there are permutations taking account of the order of connection. If SR is consistent in the composition law, should be constant for all the permutations. The composition operation should be commutative and associative in order that the relative velocity remains the same for the permutations. However, it is neither commutative nor associative and the velocity, depending on the arrangement of intermediate nodes, varies with the permutation.
Now let us turn our eyes to reverse velocities. The reverse velocity composition law Eq. (27b) itself is inconsistent since . For N = 1 and 2, the reverse velocities can be written as:
The non-equality in (32a) results from the non-equality of , which indicates that the reciprocity principle is not satisfied in SR. The is not identical to the .
3.2 From inconsistency to consistency
We can make the velocity composition consistent by putting some constraints on it. A reference frame , as shown in Figure 4, is introduced such that the composition law is always applied in conjunction with . The reverse velocity that is the opposite of the corresponding forward one according to the reciprocity principle is allowed only between and other nodes, but not between two nodes neither of which is . For example, is the velocity of node relative to and then the velocity in the opposite direction from to is , but since neither of the nodes and is . The validity of the constrained velocity composition law will be discussed later.
Let us illustrate the consistency of it, selecting the isotropic frame as . Then, and are equal to and . Suppose the absolute velocity is available. Given and , the velocity is written as 
If , , and are known, is obtained as Eq. (33) with and , and then as (33) with and . The velocities can also be discovered by using the constrained composition law. Referring to Figure 4, we compute and as:
Then, is obtained as:
The velocity addition operation is not associative and thus the -related terms in Eq. (36) with the substitution of Eq. (35) are not cancelled. Consequently becomes a function of the absolute velocity and the relative velocities and . It is straightforward to see that Eqs. (36) and (34) are equal to Eqs. (20) and (33) with and , respectively. The reverse velocity is given by , which is not the same as . Even if is calculated through node other than node , the resultant is identical to Eq. (36) since
In PFT, in which the isotropic frame is unique, the transformation between and is independent of the intermediate frame, so are the relative velocities between nodes and . The relationship Eq. (37) results in the consistency of the constrained composition law. As is independent of intermediate nodes, we have
The equality of Eq. (38) results from Eq. (37).
Now, it is time to examine the validity of the constrained composition law. From , where , the coordinate vector is written as . The velocity composition law of SR is derived from the successive LTs. If A has a form of LT, can be obtained by the composition law, but A is different from . Nonetheless we can discover by it, as explained in the following. The first rows of A and are equal, so are the first rows of and . When the standard synchronization is employed, the relative velocity is determined only by the first row of the transformation matrix, as shown in Eq. (19). Hence, the velocity of is given by Eq. (34). The transformation from to is , where , and similarly is calculated as Eq. (35). The constrained composition law can be successively applied.
When nodes and are connected through nodes as in Figure 5, we can calculate and from and the relative velocities by successively applying the composition law. The velocity is given by
The multiple composition proceeds from left to right. Substituting Eq. (39) in Eq. (36) yields:
The velocities , , , , and are needed to obtain the reverse velocity and can be found from , , , and . If they are available, then the reverse velocity is calculated as:
Each relative velocity from Eqs. (40a) and (40b), which are dependent only on and , remains the same for every and arbitrary intermediate nodes in virtue of the equality (37). The constrained velocity composition law is consistent.
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 inertial frames. However, the of Eq. (27a), which is not a function of , is other than the of Eq. (40a) depending on . The former is dependent on connecting nodes and so it is not uniquely determined, varying with them. As a result, PT in SR is not uniquely determined either. The PT of an observer at rest in is absolute in that it has the same value from frame to frame irrespective of the synchronization of clocks. Since the velocity of in by the composition law is dependent on the connecting nodes, so is its magnitude, which leads the PT to depend on them as well so that it is not uniquely determined. The absoluteness of PT holds under the uniqueness of the isotropic frame. From Eqs. (2) and (17) with , the differential PT of is given by :
and from Eqs. (14), (11), and (18a),
Note that Eq. (41b) is valid even if and are interchanged despite . Eq. (5) is a different representation of the same expressed as Eq. (2) with and Eq. (41b) is a different representation of the same expressed as Eq. (41a). Under the uniqueness of the isotropic frame, PT is absolute, having the same value as Eq. (41a) from frame to frame regardless of the intermediate frames.
The non-equality 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 . In the Galilean transformation (GaT), if the velocity of relative to is β ji , the velocity of relative to is since where is the GaT matrix when the relative velocity is , namely:
It is obvious that
where . In Figure 4, given and , the relative velocities between nodes and are clearly given by:
Given , , , and as in Figure 5, they are written as:
where , , and . Both Eqs. (45a) and (45b) are satisfied for every and arbitrary node . The consistency in GaT results from the equality (43). In PFT, each relative velocity given by Eqs. (40a) and (40b) remains the same for every and all the permutations of the connecting nodes. The consistency in PFT results from the equality (37).
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 in Eq. (40a) are canceled so that the resulting becomes equal to the . If the velocity addition was associative, the relative velocity could be obtained by successively applying it without the absolute velocity, but Eq. (27a) does not remain the same with respect to the permutations because it is not commutative. As for the reverse velocity, Eq. (27b) itself is still inconsistent on account of , as can be seen from Eq. (32), even though the associativity is assumed. The inconsistency results from the non-commutativity of the composition. The associativity is not sufficient to hold the principle of relativity. In order to fulfil it so that each velocity in Eqs. (27a) and (27b) is the same for every and all the permutations, the velocity addition should be associative and commutative, as in GaT.
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 non-equality (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  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 two-way 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 . TCL and the inertial transformation, which are consistent with each other, are consistent with PFT.
It has been known that the velocity composition of SR is not consistent with the reciprocity principle, in a situation where two nodes and of interest are connected through a single intermediate node , as illustrated in Figure 1. When they are connected via multiple nodes, besides the violation of the reciprocity principle, a large number of inconsistencies have been found as the velocity by the composition law is dependent on the connecting nodes. Though the velocity of relative to should remain the same regardless of them, it does not so because of the non-equality Eq. (23) or Eq. (25). As the number of connecting nodes increases, the inconsistency is more deepened since the velocity comes to depend on more nodes. When the inertial frames and are connected via nodes as in Figure 2, the forward and reverse velocities are expressed as Eqs. (27a) and (27b), respectively. If SR is consistent, both velocities should be independent of the connecting nodes such that they do not vary with respect to the permutations of the nodes. However neither of them remains the same for the permutations since the velocity addition is not commutative or associative. Moreover, Eq. (27b) itself is inconsistent because . As a result of these inconsistencies, PT also becomes dependent on the connecting nodes so that it is not uniquely determined.
The problems can be cured by introducing a reference node such that the velocity composition is carried out in conjunction with it. In fact, the reference node is the unique isotropic frame. In the PFT, the relative velocity between the inertial frames, which depends only on their absolute velocities, is independent of the connecting nodes and remains the same for the permutation. According to the postulates of SR, the frame is nothing more than one of an infinite number of isotropic frames. Hence, Eqs. (45a) and (45b) with ‘ ’ replaced by ‘ ’ when performed from right to left should be satisfied for every and arbitrary connecting nodes, which is mathematically infeasible. If the isotropic frame is unique, the velocity composition law is consistent and PT is absolute, not depending on connecting nodes. As far as kinematics is concerned, the equivalence of inertial frames conforms to the GaT, which is, however, incompatible with time dilation. The isotropic frame is unique . Nature itself reveals the uniqueness of the isotropic frame.
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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 . According to the designation, . A partitioned matrix A is a block diagonal matrix with , , and . The following fundamental properties are important for the derivation of Eq. (28):
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 are identical.
The successive transformation can be written as:
where A is calculated as Eq. (29). Using Eqs. (A1a) and (A2), we have
where and A is given by Eq. (29) with and . It follows from (A1) and (A2) that
where A is given by Eq. (29) with and . Eqs. (A3a) and (A3b) lead to Eq. (28a).
Using (A2), we have
where and A is given by Eq. (29) with and . Since A is a block diagonal matrix, A −1 is also a block diagonal matrix and its (2,2)-entry is . It is easy to see by using Eq. (29) that , which results in . From Eqs. (A1) and (A2)
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 is written as
It is straightforward to see that is given as Eq. (27b).
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