Mass Energy and Space-Time: On the Mass Increase of Moving Matter and Negative Energy Occurrence

The equation describing the relationship between mass and energy has long been a subject of debate among physicists. One perspective holds that the mass of an object changes with velocity, referred to as relativistic mass, while another maintains that mass remains constant regardless of velocity, referred to as invariant mass. However, many experiments have shown that the concept of relativistic mass is required to explain experimental results. Moreover, when analyzed using Newtonian or classical mechanics under the assumption that mass depends on velocity, the result coincides with Einstein's mass-energy relation. This makes it difficult to deny the validity of relativistic mass. A question arises, however: if the increase of mass with velocity is indeed real, how can an object's velocity continue to increase when all the work or energy applied to it seemingly contributes solely to increasing its mass? To resolve this while remaining consistent with the law of conservation of energy, the authors propose a theory derived from classical mechanics, under the assumption that mass varies with velocity, together with the consideration that space-time itself imposes constraint on motion. This space-time constraint introduces a resistive constraint force against motion. The result shows that the mass of an object increasing with velocity as relativistic mass given $m_v=\frac{m_o}{\sqrt{1-{\left(v/c\right)}^2}}$ is possible and that this increase in mass is accompanied by the occurrence of negative energy, with a value of $-\frac{1}{2}{m}_v{v}^2$ .