Defining of three-dimensional acceleration and inertial mass leading to the simple form F=MA of relativistic motion equation

Abstract

Newton second law of dynamics is a law of motion but also a useful definition of force (F=M A) or inertial mass (M =F/A), assuming a definition of acceleration and parallelism of force and acceleration. In the special theory of relativity, out of these three only the description of force (F=d p/dt) does not raise doubts. The greatest problems are posed by mass, which may be invariant rest mass or relativistic mass or even directional mass like longitudinal mass. This results from breaking the assumption of parallelism of force and standard acceleration. It turns out that these issues disappear if the relativistic acceleration A is defined as a relativistic velocity subtraction formula. This basic fact is obscured by some subtlety related to the calculation of the relativistic differential of velocity. It is based on the direction of force rather than on transformation to a resting system. The reference to a non-resting system generates a (seemingly) different velocity subtraction formula. Thus, the relativistic three-dimensional acceleration is neither rest acceleration, nor four-acceleration, nor standard acceleration. As a consequence, inertial mass in any direction of the force has the same value as relativistic mass. In other words, the concepts of transverse mass and longitudinal mass, which depend on velocity, have been unified. In this work a full relativistic equation is derived for the motion of a body with variable mass whose form confirmed the previously introduced definitions. In addition, these definitions are in line with the general version of the principle of mass and energy equivalence. The work presents a detailed review and discussion of different approaches to the subject in relation to original historical and contemporary texts. On this basis, a proposal is made for consistent definition of relativistic quantities associated with velocity change.