### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 005, 21 pages      arXiv:1309.3713      https://doi.org/10.3842/SIGMA.2014.005

### Why Do the Relativistic Masses and Momenta of Faster-than-Light Particles Decrease as their Speeds Increase?

Judit X. Madarász a, Mike Stannett b and Gergely Székely a
a) Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, P.O. Box 127, Budapest 1364, Hungary
b) University of Sheffield, Department of Computer Science, 211 Portobello, Sheffield S1 4DP, United Kingdom

Received September 17, 2013, in final form January 07, 2014; Published online January 11, 2014

Abstract
It has recently been shown within a formal axiomatic framework using a definition of four-momentum based on the Stückelberg-Feynman-Sudarshan-Recami ''switching principle'' that Einstein's relativistic dynamics is logically consistent with the existence of interacting faster-than-light inertial particles. Our results here show, using only basic natural assumptions on dynamics, that this definition is the only possible way to get a consistent theory of such particles moving within the geometry of Minkowskian spacetime. We present a strictly formal proof from a streamlined axiom system that given any slow or fast inertial particle, all inertial observers agree on the value of $\mathsf{m}\cdot \sqrt{|1-v^2|}$, where $\mathsf{m}$ is the particle's relativistic mass and $v$ its speed. This confirms formally the widely held belief that the relativistic mass and momentum of a positive-mass faster-than-light particle must decrease as its speed increases.

Key words: special relativity; dynamics; faster-than-light particles; superluminal motion; tachyons; axiomatic method; first-order logic.

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