The author [35] set the goal to show that the most general relation (for parallel velocities!), compatible with the principle of relativity, is the law for velocity addition

How a special case can be the most general thing: is it possible in reality to guarantee a strict parallelism of velocities? Obviously not! For two velocities u and v with given modules, the case of their parallelism is a set of measure zero. And for noncollinear vectors, the result of relativistic addition already depends on the order of its application (on the order of addition of velocities)!

The value of is not an “invariant velocity”, but a boundary velocity: the addition of two quantities less than this speed gives a value that is also smaller than this speed, but the addition of two quantities larger than this speed also gives a value less than this speed! Only if at least one of the quantities is exactly equal to this boundary speed, the result of “addition” will again be this speed. As you can see, additional groundless postulates have been introduced about the impossibility of achieving velocities, more than and about the existence of a strange boundary speed from which it is impossible to “jump off” and on which it is impossible to “jump”.

Mermin [35] states about the method of reducing the function of two variables to a function of one variable. But this is not always possible in mathematics; therefore, some additional hypotheses and limitations will be artificially introduced, and even through thought experiments! The author of [35] presupposes the fulfillment of the relativity principle, that is, that we are dealing with isolated systems (identical systems without interaction, that is already a limitation of Nature), but it also seeks an open relationship between relative velocities. In expression (2.3) from [35] , he specially introduced other variables, so that the meaning of the previous expression (2.2) became invisible: pay attention to the subscripts! In expression (2.2), the subscripts are clearly joined together: that corresponds to the physical meaning of the velocity addition. If the author wanted to write an expression for through a change of sign, then it was necessary to write. Thus, instead of (2.3), the equality must be written, and no symmetry (2.6) with respect to the arguments follows from any “general considerations”. Moreover, our viewpoint is confirmed by the fact that the general relativistic law for velocity addition depends on the order of the velocities for noncollinear vectors (it is noncommutative!). Therefore, a particular case of parallel velocities does not need to be symmetric (commutative) either.

We make the following remark. It is necessary to accurately subdivide the measurable velocities (related to the measuring device located in some system) and the calculated velocities (not related to the system in which the measuring instrument is located). Obviously, in our case, the speed is the calculated speed, because just for this reason some function f is introduced, but the variables of this function-the velocities and―are the measurable velocities. But then the measuring device can only be in system B. Therefore, the addition of a new point D in [35] only leads to the fact that new calculated velocities were simply introduced in expression (2.7), but the measuring instrument cannot measure them in the system B. In doing this, in the first of the expressions (2.8), the measurable and calculated velocities were mutually replaced, that changes the physical meaning of the seeked computational function. The possibility of interchanges the measurable and calculated quantities in (2.9) is an additional physical hypothesis. We cannot assume in advance that when the measurable and calculated quantities are replaced, the form of the unknown function remains one and the same. For classical physics (linear dependence), the calculated velocity does not really depend on the motion of the observation system, but in relativistic physics for non-collinear vectors this is no longer so.

Note that in mathematics there is no such general property that any function of two variables can be expressed as a function of one variable, even if it is “continuous and differentiable”. And the plausible phrases about “parametric dependence”, “fixation of a variable” and the replacement of the partial derivative in (2.10) by the total derivative (2.14) are intended to hide the obvious deception. Each can elementarily find examples when this does not work. Thus, (2.17) does not hold in the general case, to which the “proof” of Mermin allegedly claims. Since we have seen earlier that the symmetry (2.6) does not hold in the SRT, then the equality (2.18) does not work all the more so. Then the expression (2.19) and the search for the function h lose meaning. Also the value could be equal to infinity if the derivative at zero turns out to be zero.

Further, instead of (3.1), we must write other self-consistent expressions:

Expression (3.5) is correct, since it uses only classical relativity. It is obvious that (3.6) no longer corresponds to the previous definitions. But even if you forget about everything, said above, including the absence of meaning in the search for h, then the simplest solution (3.9) will be. Note, firstly, that in any case, it can only be about determining the calculated speeds (so the best choice is the simplest option). But measurable speeds without our games with mathematics are determined from experiments. Secondly, take an attention that Mermin from the expression (3.9) tries to justify a certain unified constant for all cases of life. Note that the tortoise and the hare meet in any case: if one or both from them stand, or move with arbitrary speeds. Choosing u = 0, we again get the simplest choice in the particular case. But the most important thing is that the integration of this fictitious function does not give any law for velocity addition due to non-commutativity.

If we allow the possibility of exotic (relativistic) transformations from the belief in the relativity principle, that is, assuming a possible dependence of a row of quantities on the relative velocity, then an additional hypothesis is the assumption of the dependence of these quantities on the modulus of the relative velocity. Then we cannot even be sure of the equality of the quantities measured when going back and forth. For example, then you can doubt that in the system of reference of the train. Further, again it is not necessary to confuse the measurable and calculated quantities: instead of (4.1) it is necessary (for consistency with the function f) to check the value. The author’s ar-

guments relate to the train’s motion system, that is, (4.3) and, instead of (4.6), we can only write.

Then the author postulates (this is again an additional hypothesis) that this relation will be preserved in the v-system also. We will not correct all the intermediate formulas of the analyzed article, but directly write the final expression

and the limit:

But again, from here no special functions should follow.

Further, the author notes that for a negative value of K, the law for velocity addition (5.2) can lead to the result, if and . But for some reason the author ignores another strangeness with a positive value of K. The boundary velocity breaks up the phenomena into three strange “Worlds”: I), II) c, III). In this case, , , , but with the addition of velocities, each of which is greater than c, the particles “fall” into I “World”: (the same result will be for).

There is no evidence of the invariance of the light speed in a vacuum. The speed of wave propagation does not depend on the velocity of the source for any waves and at any speed of their propagation (the set among them occurs). This is just a property of wave motion, including in classical physics. The velocity determines the local velocity of wave propagation inside the measuring device. And determination of the value of c on the eclipses of the Io―the satellite of Jupiter, speaks of the dependence of the light speed on the speed of the receiver. In any case, there is no other evidence.

Mermin proposes to determine the value of K from expression (5.3), forgetting that in the system B only two velocities are measurable: and. In essence, the expression (5.3) is a definition for the not measurable value of velocity in the system B. But one expression cannot simultaneously determine two unknown quantities: and K. The author suggests to “ask” about the value in the system A. Relativity turns strange! For some reason, we cannot believe the observer in system A that he knows about the lengths and times that we do not measure in his system (this is unprofitable from the relativistic viewpoint). We ostensibly must calculate these lengths and times by the artificial relativistic rules. But we must blindly believe in the observer’s testimony from the system A about velocities. In general, “here we read, here we do not read” ... and as it is sang in the song “And the rest, beautiful marquise All right, all right” (the SRT must be defended at any cost?)!

Generally speaking, the synchronization method using an infinitely remote source on the median perpendicular to the motion line [21] unambiguously leads to all classical quantities (spatial, temporal and motion characteristics).

We also give brief remarks to the “justification” of the relativistic law for velocity addition in [36] . The requirement that the inverse transformation to a linear transformation and the product of transformations preserve the corresponding structure (constitute a group) are additional requirements (and for noncollinear motions are not satisfied in the SRT). When Terletskii claims about the homogeneity of space, but at the same time tries to artificially introduce some strange transformations, it would be worthwhile first to answer the question of what to expect from parallel transformations for such a fictitious “physics” (how to avoid paradoxes). In his expression (7.6) the constant can depend on other coordinates:. The form of the transformation (7.7) itself is a hypothesis: if we talk about generalization, then there may be cross-dependencies of coordinates.

Further, the replacement only changes the orientation of the triple of basis vectors. Consequently, to have changed the nothing in the transformation formulas (as the author wishes), it is necessary to swap (this is immediately noticeable for a non-spherical object). The coincidence of the form of direct and inverse transformations remains in question. Large problems with “group properties” arise in the transition to non-collinear vectors, so that all these mathematical exercises look artificial.

Finally, the dependence of mass on speed is fantasized: not the mass grows at a speed, but the effective force decreases as the speed of the body approaches the transfer rate of the interactions (to the momentum transfer rate)! In classical physics, there is also a similar decrease in effective force.

Thus, the paper [36] also cannot be considered rigorous in terms of justifying relativistic invariance and the law for velocity addition.

7. Conclusions

The principle of relativity and a mathematical skew, with an exaggeration of the role of invariance in physical research, were discussed in the paper. It was analyzed in detail a “bloated soap bubble” with the invariance of the Maxwell equations. Further, the Michelson-Morley experiment was analyzed in terms of various theoretical expectations. This experiment cannot distinguish between Galilean and Lorentz invariance. The fundamental problems associated with the speed of light and with the law for velocity addition were discussed. Some pseudo- proofs of the necessity of the invariant velocity existence are critically examined.

Thus, there is no strong theoretical evidence for the necessity of the invariant velocity existence, as well as experimental confirmation of this statement, including for light.

Conflicts of Interest

The authors declare no conflicts of interest.

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