Abstract
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.
Highlights
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
We introduce a diagram of velocity where inertial frames are represented as nodes that are connected by arrows indicating relative velocities
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 Sj with respect to Si can be found from the first row of the transformation matrix from the latter to the former
Summary
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 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. It is shown based on the MS general framework that when the standard synchronization is adopted, the relative velocity of Sj with respect to Si 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.
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