On the Concept of Velocity and Its Use in Special Relativity

The concept of velocity is central to both classical kinematics and special relativity (SR), yet its definition and application differ fundamentally between these frameworks. This article demonstrates that SR's treatment of light propagation contains an internal inconsistency arising from an imprecise and inconsistent use of the velocity concept. Velocity is defined as displacement per unit time within a specified reference frame. A fundamental consistency requirement follows: operations such as addition or comparison of velocities require that all quantities involved be defined within the same reference frame. Using examples from everyday experience, we establish a clear distinction between the physical distance traveled by a signal and the spatial coordinate assigned to an event in a given reference frame. These two quantities are in general not equal. We show that SR's identification of the spatial coordinate $x' = ct'$ with the physical distance traveled by light violates a consistent application of the velocity concept: while the physical distance traveled by light during time $t'$ is $s(t') = ct'$, the spatial coordinate of the wavefront in $F'$ is $x'(t') = (c - v)t'$, which differs from $ct'$ whenever $v \neq 0$. Furthermore, SR performs operations on velocities -- specifically $c$ and $v$ -- without ensuring that both quantities are defined within the same reference frame, in violation of the consistency requirement fundamental to any well-defined velocity concept.