Alternative relativistic mechanics and Cherenkov radiation

In this article we explore implications of the Alternative Relativistic Mechanics (ARM) of a material particle, proposed by us [\textit{Advanced Studies in Theoretical Physics} \textbf{19, No.4}, 173 - 182 (2025)]. This ARM is a framework that introduces Lorentz invariance violation through the presence of a new fundamental assumption: variable rest mass. The rest mass of a particle $m_0$ is not a constant as in Einstein's special relativity, but a known function of the Lorentz factor $\gamma$, $m_0(\gamma)=M_0(1+\alpha \ln{\gamma})$, where the free parameter $\alpha=\text{const.} \geq 0$ quantifies the modification of Einstein's special reality, $M_0=m_0(1)=\text{const.}>0$. Our research focuses on the impact of the ARM on the Cherenkov angle (the characteristic angle of Cherenkov radiation) taking into consideration the deviations from Einstein's special relativity, which we derived in the ARM. These ARM's deviations lead to different predictions for the Cherenkov angle compared to the framework of classical electrodynamics and the framework of Einstein's special relativity. We derive the expression for the Cherenkov angle within the ARM framework. This allows us to use Cherenkov detectors (like the LHCb experiment at CERN) to test the ARM and to set limits on its free parameter $\alpha$. Since in the ARM the parameter $\alpha$ governs the non-linear increase of mass-energy and mass-momentum at high $\gamma$, to protect the ARM from being immediately debunked by the ultra-precise electron g-2 and muon g-2 data, we abandon the idea of $\alpha$ as a global constant. We propose that the parameter $\alpha$ scales with the square of the particle's initial (at $v=0$) rest mass: $\alpha_i=kM_{0,i}^2$, where $k$ is a new fundamental constant of the "variable mass field" and $M_{0,i}$ is the initial rest mass of the i~particle (lepton).