The notion of the center of masses

Even such a simple notion as "the center of masses of a system" becomes ambiguous in SRT in considering the mutual motion of system's components. So, in [33] the "paradox of a center of masses" is considered: in the reference frame of a rocket two identical cannon balls are fired off simultaneously inside a tube, and the ends of a tube are tightly closed immediately by plugs $A$ and $B$ (Fig. 4.1).

Figure 4.1: The center of mass of a tube with cannon balls.

In the classical physics no contradictions arise in this case: the center of masses in any frame of reference will always coincide with the center of the tube. It can be determined by various methods, namely: by weighing and direct calculation (the mass and distances are invariant in the classics), as a center of zero momentum, as a center of a baryon number (the number of nucleons in nuclei), as a center of gravitational attraction. The notion of the center of baryon number was declared "non-productive" in [33], since the world line of this center occurs to be irrelevant to the SRT laws (that is, it simply contradicts them!). The gravitation is organically not included into SRT, so that one should transfer to GRT, but the book [33] declares the coincidence of the center of gravitational attraction with the middle of a tube in the laboratory coordinate system (but in this case "the center of zero momentum" is studied). However, immediately after the first collision with a plug (non-simultaneous in the laboratory system) it becomes necessary to refuse from the universality of SRT and to recall about a specific compensation mechanism (for "saving" SRT) - on the acoustic waves in a tube and on the energy (mass) transfer by them. These waves, coming from tube's ends, then suppress each other. But in such a case one should have to postulate various velocities of acoustic waves in various systems for two opposite directions. And if we will change the material of a tube and the geometrical characteristics of the experiment? And if the tube is absent at all and only the plugs of very great mass are present, and the sensitivity of local gravitation measurements will allow for determining the motion of cannon balls? And what should be done with the compensation mechanism in the cases listed above?

If in the given problem we shall determine the mass from the momentum transfer on plugs $A$ and $B$ or on barriers parallel to them (the "longitudinal" mass), then we obtain some single world line of the center of masses. If, however, the mass will be determined from the pressure on the tube bottom (from the gravitation; from the electrical force for charged cannon balls or from the magnetic force for cannon balls-magnets, etc.), then for this ("transversal") mass the other world lines will exist. Generally speaking, in SRT all these world lines will be different. Some of them have to be postulated as senseless (non-productive for SRT), in some cases it would be necessary to transfer to particular mechanisms "explaining" the contradiction, and in other cases the change of objective characteristics should be postulated. For example, let the plug to be retained on a massive tube with the force slightly greater, than that required for a plug to be torn-off by a cannon ball (with "relativistic" mass) in rocket's frame of reference. Then in the laboratory frame of reference one of cannon balls (with a greater "relativistic" mass in this case) will beat the plug out. So, is the observer behind this plug alive or dead? Or, again, for "saving" SRT it is necessary to postulate that the plug-retaining limit in SRT is not an objective characteristic (but depends on the frame of reference)? And if at tube's ends there will be the "traps" at the bottom, in order that in rocket's frame of reference the ("transversal relativistic") mass be slightly insufficient for a cannon ball to be fallen down there. Then, again, in the laboratory frame of reference one of cannon balls (with a greater "relativistic" mass) will fall down. So, shall we postulate again the change of the threshold strength for "saving" SRT? Note that it would be necessary to postulate different threshold characteristics: both the longitudinal and transversal (generally, tensor) ones. Whether the SRT price is not too great - the price of postulating a loss of the majority of objective characteristics? Whether the number of problems, questions and contradictions is not too great in SRT "at the empty place" - where in the classical physics everything would be elementary simple? And, you see, SRT can not refuse from the concept of the center of masses, since the Einsteinian derivation of the $E=m_0c^2$ equivalence for the "rest mass" is based on this particular concept.