On the Foundation of the Relativistic Dynamics with the Tachyon

TL;DR

This paper establishes a classical relativistic dynamics framework for tachyons, revealing a crucial sign-factor affecting mass transformation and addressing ambiguities in light-speed objects, thereby filling a theoretical gap in faster-than-light particle physics.

Contribution

It introduces a novel sign-factor in tachyon dynamics, clarifies mass-energy-momentum relations, and resolves description ambiguities for light-speed objects within a consistent relativistic framework.

Findings

The sign-factor is crucial for tachyon mass transformation.

Energy and momentum transform as in subluminal objects.

Zero-point energy constraints are derived for consistency.

Abstract

The theoretical foundation of the object moving faster than light in vacuum ({\it tachyon}) is still missing or incomplete. Here we present the classical foundation of the relativistic dynamics including the tachyon. An anomalous sign-factor extracted from the transformation of ${ \sqrt{1-u^{2}/c^{2} } }$ under the Lorentz transformation, which has been always missed in the usual formulation of the tachyon, has a crucial role in the dynamics of the tachyon. Due to this factor the mass of the tachyon transforms in the unusual way although the energy and momentum, which are defined as the conserved quantities in all uniformly moving systems, transform in the usual way as in the case of the object moving slower than light ({\it bradyon}). We show that this result can be also obtained from the least action approach. On the other hand, we show that the ambiguities for the description of the dynamics for the object moving with the velocity of light ({\it luxon}) can be consistently removed only by introducing a new dynamical variable. Furthermore, by using the fundamental definition of the momentum and energy we show that the zero-point energy for any kind of the objects, {\it i.e.}, the tachyon, bradyon, and luxon, which has been known as the undetermined constant, should satisfy some constraints for consistency, and we note that this is essentially another novel relativistic effect. Finally, we remark about the several unsolved problems.