Multiple velocity composition in the standard synchronization

Abstract

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.

Relativistic analysis of the Michelson-Gale experimental result

“…Equation ( 19 ) indicates that can be obtained if the first row of is known so that is found, even though the rest is unknown. The first row of is given by where is the normalized velocity of relative to 5 , 15 . Then is calculated as, …”

“…Disguising the inertial frame as isotropic via the standard synchronization and then using instead of , nonetheless, we can exactly obtain some physical quantities such as PTs, Doppler shifts, spatial lengths, and speeds with respect to PT 5 . It is because the first rows of and are identical 5 , 15 . One can readily see in the analysis of Subsection " Based on the MS framework " that even if is used in place of the same time difference as ( 22 ) is obtained.…”

“…Then the differential coordinate vector of S is transformed into as where is the Lorentz transformation matrix, with, and an identity matrix. Since , the transformation from one inertial frame to another is expressed as 5 , 15 . where, …”

Explanation of What Time in Kinematics Is and Dispelling Myths Allegedly Stemming from the Special Theory of Relativity

“…There are also published papers showing the paradoxes of the Special Theory of Relativity concerning rotating frames of reference [23]. Article [24] investigates the subject of relativistic velocity addition. There are many papers on relativistic mechanics with significant theoretical results.…”

“…Let the angle measured in the resting system of the body as α has a value α in the ether system. Acting in a symmetrical way, as in the case of Equations ( 20)- (24), it is possible to derive the relation to contraction at any angle α in the form of…”