Resolution of two fundamental issues in the dynamics of relativity and exposure of a real version of the emperor's new clothes
In this paper, we aim to resolve two fundamental issues in the dynamics of relativity: (i) Under what condition, the time-column space integrals of a Lorentz four-tensor constitute a Lorentz four-vector, and (ii) under what condition, the time-element space integral of a Lorentz four-vector is a Lorentz scalar; namely two "conservation laws", which are mispresented in traditional textbooks, and widely used in fundamental research, such as relativistic analysis of the momentum of light in a medium, and the proofs of the positive mass theorem in general relativity. To resolve issue (i), we have developed a generalized Lorentz \emph{four-vector} theorem based on the principles of classical mathematical analysis, with a simplified analytic example given to illustrate how to transform a space integral from one inertial frame to another, and a strict mathematical derivation provided to confirm the effect of Lorentz contraction. We use this four-vector theorem to verify M{\o}ller's theorem, and surprisingly find that M{\o}ller's theorem is fundamentally wrong. We provide a corrected version of M{\o}ller's theorem. We also use this four-vector theorem to analyze a plane light wave in a moving uniform medium, and find that the momentum and energy of Minkowski quasi-photon constitute a Lorentz four-vector and Planck constant is a Lorentz invariant. To resolve issue (ii), we have developed a generalized Lorentz \emph{scalar} theorem. We use this theorem to verify the "invariant conservation law" in relativistic electrodynamics, and unexpectedly find that it is also fundamentally wrong. Thus the two "conservation laws" in traditional textbooks, which have magically attracted several generations of most outstanding scientists, turned out to be imaginary, just like the emperor's new clothes; creating a scientific myth in the modern theoretical and mathematical physics: Believing is seeing.