On the Notion of the Measure of Inertia in the Special Relativity Theory

Transcription

1 Applied Physis Researh Vol. 4, No. ; 1 On the Notion of the Measure of Inertia in the Speial Relativity Theory Sergey A. Vasiliev 1 1 Sientifi Researh Institute of Exploration Geophysis VNIIGeofizika (retired), Mosow 1714, Russia Correspondene: Sergey A. Vasiliev, 38 Nazliu Str., Palio Faliro, Athens 17564, Greee. Tel: disput@gmail.om, disput@mail.ru Reeived: November 11, 11 Aepted: Deember 8, 11 Online Published: May 1, 1 doi:1.5539/apr.v4np136 URL: Abstrat The onept of the relativisti mass and its equivalene to the energy was reently negated within the framework of the speial relativity theory (SRT). As a onsequene, the relativisti mass notion was exluded from majority modern textbooks and books on SRT. The analysis of this negation is arried out in this paper. By the definition, the mass is the measure of inertia. Therefore everywhere, where inertia exists, the measure of this inertia, that is the mass, should exist. The inertia exists at relativisti veloities. Hene, the relativisti mass is obliged to be presented in SRT. The founders of SRT were right in their formulations from the very beginning and there is no need to revise their physial approahes onerning the relativisti measure of inertia. At the orret approah to the problem, the relativisti mass is returned to SRT. The relativisti mass is the measure of inertia, but it is not a salar in SRT. It is the omponent of a 4-vetor here. Sine the relativisti mass is the omponent of the 4-vetor, the fundamental equivalene of the measure of inertia and the energy is valid at all veloities (less or equal to the light veloity). The above mentioned negation is not harmless for siene beause it loses the road to some basi researhes and generates the onfusion in the students' brains. Keywords: relativisti mass, measure of inertia, speial relativity theory, 4-vetor, mass-energy equivalene, Okun', notion 1. The Aim of the Paper In the speial relativity theory (SRT) the Einstein's mass-energy relationship E m (1) began to be subjeted to the negation afterwards. In it m is the relativisti mass of a physial body, E is its energy, is the light veloity. Most sequentially and fully the Conept of the opponents is expressed in the papers (Okun', 1989, ). Aording to this Conept (Okun', 1989, ): 1) The relation (1) is not real and has no physial sense beause the relativisti mass has no physial sense (I shall speak simply "mass" in follow-up); ) Only the relation E m () between a rest-mass m and rest energy E has the physial sense; 3) The relativisti mass does not play the role of the measure of inertia and of the gravitational mass; 4) From all masses, only the rest-mass is the invariant of the Lorentz transformation. The rest-mass of the system of not interating, freely flying partiles depends on the angles between the momenta of the partiles. In partiular (Okun', ), the paradox of the photon system takes plae: the rest-mass of a pair of photons, eah having energy E, is E/ if they move in opposite diretions and vanishes if they propagate in the same diretion. This is diffiult to omprehend for an inexperiened reader who has never before dealt with the theory of relativity, but this is an established fat!. 136

2 Applied Physis Researh Vol. 4, No. ; 1 Within the framework of this Conept 1-4, the item 4 is methodologially assoiated with the items 1-3. Aording to (Okun', ), the formula (1) was frequently applied to a massless photon as well, muddling the students' brains to utter onfusion (on the one hand, the photon is massless; on the other hand, it has a mass). The Conept 1-4 was widely spread up to the exeptions of a relativisti mass from the manuals of physis in Russia (under L.B. Okun's insisting (Okun', 1989, )). Aording to the review (Wong & Yap, 5), the same has taken plae in a number of other ountries. The majority of the modern books on SRT adhere to this Conept 1-4, but the alternate onepts on the energy-mass relationship are also existing (Wong & Yap, 5). Meanwhile, the Conept 1-4 is not so firm, as it seems. Therefore it is neessary to state the ritial estimation of the Conept 1-4. The purpose of the present paper is to show, that all items 1-4 of the Conept are annulated, if at the dedution of these items to not infringe the requirement of a orretness that should promote elimination of the above muddle and the onfusion from the students' brains. Everywhere further, for brevity, we shall term as small veloities, the veloities, muh less than the light veloity, and as relativisti veloities - all other veloities, less or equal.. The Priniple of Existene of the Mass and the Requirement of Corretness Almost all depends on how to understand the term "mass". By the definition, the mass is a measure of inertia. Just so the mass is onsidered in the present paper. Therefore we shall formulate the priniple of existene of a mass: everywhere, where a inertia exists, the measure of this inertia, that is the mass, should exists. At relativisti veloities the inertia exists. Hene, the relativisti mass should exist. For understanding of the inauray of the Conept 1-4, the following has the deisive importane: at the transition from the lassial to the relativisti mehanis, it is impossible to require a priori: 1) that the measure of inertia - the mass - is a salar; ) that the vetor of fore F and the vetor of aeleration a, aused by this fore, are parallel; 3) that the mass was determined by the ratio of the module of the fore F to the module of the aeleration a. The given requirements are not imposed a priori also in the lassial mehanis, but follow there from a set of experiments, generalized by the Newton's seond law of motion. But the lassial mehanis knows nothing that really happens at the relativisti veloities, i.e. outside of the appliability of the lassial mehanis. Hene, it is impossible to transfer a priori the above onditions from the lassial mehanis to the relativisti mehanis. At the relativisti veloities in order to deide orretly the problems on the salariy or nonsalariy of masses, on the parallelism or no parallelism of the vetors F and a, on the sense of the ration of the modules of the fore and the aeleration, it is neessary to be based or on the diret measurings of the physial quantities (in partiular, F and a) at relativisti veloities, or on the equations of SRT, following of other diret experiments (for example, on the fixing of the onstany of the light veloity), but the problems to not solve in any way a priori. Many other things are impossible to transfer a priori from the lassial to the relativisti mehanis, for example, the Newton's seond law. In the lassial mehanis, the Newton's seond law and the masses salariy are leanly the experimental fats, besides not having any explanation. So why the Newton's seond law is not transferred a priori from the lassial to the relativisti mehanis, but it is modified by known manner, and the masses salariy is transferred a priori sometimes? This does not have a reasonable explanation. The above prohibitions for the a priori transferenes we shall term as the requirements of orretness. From the told learly: as inertia exists at the relativisti veloities the relativisti mass is obliged to exist. And what will be a relativisti mass - a salar, a vetor, a tensor - depends on transformations of the Newton's laws and some other relations at an exit out the limits of appliability of the lassial mehanis. May be, the masses nonsalariy is not blended in some theories. But, if the masses nonsalariy will follow from the experiments or from the equations basing on experiments, the author is predisposed to give the priority to the Nature. In fat only so the physis an ome nearer to the truth. At the substantiation of the Conept 1-4, all the above requirements of orretness are infringed. At the attempts of the definition of the relativisti mass, a priori the mass is supposed with a salar (Okun', 1989, ) (NOTE 1), and, for the definition of the mass in the relativisti mehanis, the Newton's seond law from the lassial mehanis was used (Okun', 1989, ), that is simply the module of fore is divided by the module of the aeleration (hene, it is a priori was onsidered, that the mass has sense only at the parallelism of the vetors F and a). As a result of the infringements of the requirements of a orretness, the dedutions about senselessness of the relativisti mass, about injustie of the fundamental law E = m and about validity of all items 1-4 have been obtained. Supporters of the Conept 1-4, obviously, have ome in the ontradition with the priniple of the mass existene. Moreover, the paradox of the photon system has arisen as an artefat, i. e. as the result of the not entirely orret building-up of the paradox (this is in detail the lower). Quite the ontrary: the more paradoxes of the physial sense arise at the negation of the relativisti mass existene. 137

3 Applied Physis Researh Vol. 4, No. ; 1 3. The Paradoxes of the Conept 1-4 The onept 1-4 lead to the following inonsistent no adequaies of the physial sense. In aordane with the Conept 1-4, the progressing eletromagneti waves and photons have no any mass. But for refletion of the photon it is required to transmit the momentum to the photon. Signifies, photons have the inertia. Then the following arises: the inertia is, and the measure of the inertia is not, the gravitational ation on the photon is, and the gravitational mass of the photon is not. And for any partile with a nonzero rest-mass m there is the same inadequay: the partile has the inertia of the motion, but, aording to the onept 1-4, has no the measure of this inertia during the motion. Or, as is known (Landau & Lifshitz, 1967), the mass of the physial body onsisting of the partiles, is inremented at the inrease of the veloities of the partiles, i. e. the kineti energy of the partiles ontributes to the measure of the inertia of the physial body, but, aording to the Conept 1-4, for one reason the kineti energy does not ontribute to the measure of the inertia of the partiles. The list of the no adequaies, arising from the Conept 1-4, is possible to prolong. Einstein was right from the very outset. In present paper the relativisti definition of the measure of inertia is advaned. In result, the paradox of the system of photons is eliminated; the strangenesses of the physial sense and the infringements of the priniple of the existene of a mass disappear. At the relativisti definition of the notion of the inertia measure, the relativisti mass m play role of the inertia measure and of the measure of the gravitational mass at any veloities (smaller or equal ). Energies of the motion of the partiles ontributes in their masses. The eletromagneti waves and photons have the relativisti mass, the gravitational field ats them aording to their gravitational mass, that is equivalent their inertia measure. 4. On the Existene of the Relativisti Mass The purpose of a relativisti mehanis is generalization of the rules of the lassial mehanis on the ase of the relativisti veloities. If sequentially to be guided by this purpose, and by virtue of the priniple of the existene of a mass, in SRT it is neessary to generalize the lassial notion of the inertia measure on the ase of the relativisti veloities. Therefore the relativisti mass appears an essential part of SRT. Hene, it is neessary to searh for the relativisti measure of inertia instead of to be foused to examinations of extremely that, the ration of the modules of the fore and the aeleration is whether or not the relativisti mass. The searhing of the relativisti measure of inertia with the help of division of the module of the fore on the module of the aeleration has not lead to suess (Okun', 1989, ). Hene, it is neessary to searh for the inertia measure in SRT by some other way. At that it is not required to devise new formulas of SRT. It is enough to use the known equations of SRT, resulting of the diret experiments and the relativity priniple. The inertia is appeared in two interrelations: 1) in the interrelation of the fore F and the aeleration a; ) in the interrelation of the momentum p and the veloity v (NOTE ). The kernel of the problem onsist not in that, the result of the division of the module of the fore on the module of the aeleration is whether or not the inertia measure in SRT, that is explored in works (Okun', 1989, ). The kernel of the problem onsist in the searhing of suh the inertia measure whih unambiguously determinates the aeleration a at the given fore F, or the momentum p - at the given veloity v. To find the onrete form of the relativisti inertia measure - the relativisti mass m - it is enough to define it by the rule (1) and to show, that the defined by suh a way relativisti mass is really the inertia measure. Let's be onvined, that defined by the given a way the relativisti mass m of a partile, at first, is the measure of its inertia in SRT, seondly, is equivalent to its gravitational mass in SRT, thirdly, follows to the equality entered for the relativisti mass by the founders SRT, where m m при m, (3) 1 1, β v, v β, (4) v - the vetor of the veloity of the partile, v v. Really, the equations of the relativisti mehanis for the free partile has form [3] 4 E p m, (5) 138

4 Applied Physis Researh Vol. 4, No. ; 1 E p v. (6) Here p - the vetor of the momentum of the partile. These equations are true and at the zero rest-mass m. Inertia of the partile is exhibited, in partiular, by the transmission to the partile of momentum. At that the role of the inertia measure of the partile is played by the multiplier that is situated in expression (6) before veloity v (NOTE 3). The multiplier in expression (6) is magnitude E /. Hene, the definition (1) really expresses the inertia measure in the relativisti mehanis. The substitution (6) in (5) gives a relation E 4 (1 ) m, (7) and by virtue of the definition (1) we obtain the formula (3). Besides ontrary to the Conept 1-4, by virtue of the formula (1), the interrelationship of the relativisti measure of inertia and the energy ( E m ) appears valid both at small, and at relativisti veloities as it was required to show onerning on the inertial mass. The energy is the omponent of the energy momentum 4-vetor. Hene, and the relativisti mass is not a salar, but is the omponent of the 4-vetor that orresponds to the above requirements of a orretness (NOTE 4). Stritly speaking, and at the small veloities, the relativisti mass m remains the vetor omponent, but varies so feebly, that is pratially indistinguishable from the salar m. Using the reasons from the book (Landau and Lifshitz, 1967), we ome to the onlusion about validity of the made developments and for a omposite physial body (Landau and Lifshitz, 1967): Let's underline, that though we speak here about a "partile", but its "simpliity" is not used anywhere. Therefore the obtained formulas are equally appliable and to any omposite physial body onsisting of many partiles, and under a mass m it is neessary to understand the omplete rest-mass of the physial body, and as the veloity - the veloity of its motion as the whole. - the end of the itation. 5. On the Infringements of the Requirements of a Corretness In work (Okun', ) it is a priori posited: mass is a relativisti salar while energy is a 4-vetor omponent. This is the first infringement of the requirements of orretness. Further, in view of formulas (3), (4), the Newton's seond law of the lassial mehanis F m dv a, (8) dt is transformed in SRT to the form (Landau & Lifshitz, 1967; Okun', 1989) or, that is the same, dp d( mv) F dt dt 3 m a m ( β, a) β, (9) F ( F, ββ ) m a, (1) where the parentheses with the omma in the middle mean a salar produt of vetors. Aording to paper (Okun', 1989): Despite the unusual appearane of Eq. (9) from the point of view of Newtonian mehanis (we should say, rather, preisely beause of this unusual appearane), this equation orretly desribes the motion of relativisti partiles. From the beginning of the entury it was frequently submitted to experimental verifiation for different onfigurations of the eletri and magneti fields. This equation is the foundation of the engineering alulations for relativisti aelerators. Thus, if F v, then F ma, but if F// v, then 3 m F a. Thus, if one attempts to define an inertial mass as the ration of the fore to the aeleration, then in the theory of relativity this quantity depends on the diretion of the fore relative to the veloity, and therefore annot be unambiguously defined. - the end of the itation. Evidently, within the framework of the Conept 1-4, for the definition of the relativisti inertia measure it is used a priori the no relativisti Newton's law (8) - the mass a priori is determined by the division of the fore magnitude on the aeleration magnitude. It is the seond, the most essential, infringement of the requirements of orretness. From the above replaement of the relativisti Newton's law (9) by its no relativisti analog (8) at the attempts of the definition of the notion of the relativisti inertia measure, follows the negation of the relativisti inertia measure, the ontradition with the priniple of existene of a mass and the priniple inaurate onlusions desribed in the items 1 4 of the first setion (please, see lower about a gravitational mass). From the above 139

5 Applied Physis Researh Vol. 4, No. ; 1 itations of the papers (Okun', 1989, ) it is well visible, that the searhing of the relativisti inertia measure in the frame of the Conept 1-4 is realized a priori so, as if at the exit out of the limits of the lassial mehanis validity there is no expansions of physial notions (as if outside of the indiated limits, as before, only the ratio of the modules of the fore and aeleration should unambiguously to determinate the mass). 6. On the Relativisti Mass as the Inertia Measure in the Relativisti Generalization of the Newton's seond Law Taking into aount the definition (1), the equation (9) permit its rewriting in the shape similar to the lassial Newton's seond law, where F m 14 W( β; a ), (11) W( β; a) a ( β, a) β. (1) Thus, though the Newton's seond law varies at the passage to relativisti veloities, but its general struture is remained: the left-hand parts of the equations (8), (11) is the result of division of the fore on the magnitude m or m, and the right members of these equations does not depend neither from F, nor from m or m. In the lassial mehanis the ratio F/m unambiguously determinates the aeleration a of the partile through the lassial Newton's laws (8), stritly speaking, only at the zero veloity of the partile. In the relativisti mehanis the ratio F/m unambiguously determinates the aeleration a at the given both zero, and any nonzero veloity v of the partile (smaller с) through the relativisti Newton's laws (11). Therefore determinate by equality (1) the relativisti mass m is the inertia measure inluding and from the point of view of the interdependene of the fore F and the aeleration a indued by this fore. Laws of the Nature are dilated at the transition from the lassial to the relativisti mehanis. Therefore it is quite natural that, in ontrast to the lassial mehanis, in SRT the aeleration depends not only on the ratio of the fore and the inertia measure, but else depends on the veloity of the partile, due to the ontribution of the term γ (β,α )β in equality (1). Thus the rule of the parallelism of the fore and aeleration is violated the aeleration has the omponent parallel to the fore, but besides aquires the omponent parallel to the veloity v of the partile that it is easy to see taking into aount the ontribution of the item ( F, ββ ) to the formula (1). It happens as though the "leeway" of the aeleration in the diretion of the veloity. The physial sense of the last is known and lear. The last is natural sine Lorentz transformations do not hange the spatial oordinates perpendiular to the veloity, but hange only one spatial oordinate, namely - parallel to the veloity. In this sense there is the seleted diretion (the diretion of the veloity) to whih the aeleration has "leeway". 7. On the Mass - energy Equivalene As it mentioned above, the formula E m of the interdependene of a mass and an energy, is valid in SRT at any veloities (smaller or equal ). Aording to paper (Okun', 1989): the formula E=m (I again set =1) is ugly, sine it represents an extremely unfortunate designation of the energy E by a further letter and term, these being, moreover, the letter and term assoiated in physis with another important onept. The only justifiation of the formula is the historial justifiation - at the beginning of the entury it helped the reators of the theory of relativity to reate this theory. From the historial point of view, this formula and everything assoiated with it an be regarded as the remnants of the saffolding used in the onstrution of the beautiful edifie of modern siene. But to judge from the literature, it is today regarded as almost the prinipal portal of this edifie. Probably, the formula m = E/ an seem ugly, but only as long as it is merely the denotation. To tell the truth, at that remains not lear why, aording to paper (Okun', 1989), the relation m = E / is not ugly. In fat, with the magnitude m, as before, is assoiated in physis with another important onept, than the notion of the energy E. However, starting with the moment when it is shown, that the magnitude m is the inertia measure and the 4-vetor omponent, the formula m = E/ is not ugly, and expresses the fundamental physial law: the energy (whih, in the final analysis, is the ability to make work) unambiguously determinates the inertia measure, and vie versa. Starting with this moment, the above literature is right: the formula m = E/ really represents the prinipal portal of SRT. When it will be understood, why the physially dissimilar features - the energy E and the inertia measure m - appear one-to-one oupled, the new horizon of the understanding of a physial reality will open. Seemingly, this law means, in essene, that there is a ertain ommon physial substane whih generates both the property of matter to have the energy and the property of mater to have the inertia. If it is so, to rejet the indiated law means to rejet a fundamental diretion of physial examinations and to exlude from examinations the essential physial properties. Then the Conept 1-4 is not harmless for the siene.

6 Applied Physis Researh Vol. 4, No. ; 1 8. On the Item 4 of the Conept 1-4 It is easy to see, that, at the orret definition (1), the relativisti mass of the system of the free partiles (perhaps and photons) always is equal to the sum of the relativisti masses of the partiles, irrespetively of the diretions of the partiles momenta and the paradox of the photons system is automatially exluded. It follows immediately from the definition (1) if to take into aount, that the energy of the system of the free partiles is equal to the sum of the energies of these partiles (NOTE 5). Aording to item 4 of the Conept 1-4, from all masses, only the rest-mass is an invariant of Lorentz transformation. Let v is the zero veloity of some partile (v = ), and v 1 is anyone, but the fixed veloity of this partile, v 1 <. The relativisti mass of the partile m depends on its veloity v by the rule (3). In other words, the magnitude m is funtion of the veloity v, m = m (v). In any inertial referene frame S, on the definition, the rest-mass m is the relativisti mass m (v ) at the fixed, zero veloity v partile, m m (v ). In any inertial referene frame S, we shall define the mass m 1 at anyone, but the fixed veloity v 1 by the rule m 1 m (v 1 ). If in eah system S foredly to stop the partile then in all the systems S the mass of the partile will be idential and equal to a rest-mass of a partile m m (v ). Only in suh respet, the rest-mass m is invariant of Lorentz transformation. But, evidently, if in eah system S foredly to impel the partile to move with anyone, but the same veloity v 1 then in all systems S the mass of the partile will be again idential, and it is equal to a mass m 1, m 1 m (v 1 ). Hene, any mass m 1 at anyone, but the fixed veloity v 1 is the same invariant of Lorentz transformation, as well as the rest-mass m taken at the fixed zero veloity. Hene, the masses m and m 1 are equivalent in the speial relativity theory. (But they are the onditional invariants of Lorentz transformation, i.e. the invariane of the mass is observed under the ondition of "obtrusion" to the partile of the same fixed veloity in different inertial referene frames. If to the partile to impose nothing and simply to observe it from the point of view of different inertial referene frames S the mass is not a salar, but is the 4-vetor omponent in relation to Lorentz transformation.) 9. On the Equivalene of an Inertial and Gravitational Masses Let there is a motion of the partile of the mass m (for example an eletron or a photon) under the influene of a onstant gravitational field of the heavy, not shifted body of a mass M S, i.e. M S m. Then, at the passage from the lassial to relativisti mehanis, the lassial Newton s gravitation law F mgm S 3 r r (13) it is transformed to the form (Okun', 1989): 141 GM S m[(1 ) r ( r, ββ ) ] F, (14) 3 r where m E/ - the relativisti inertial mass of the partile with an energy E ; F - the fore of gravitational ation on the partile; r - the radius-vetor joining the partile and the soure of the field, diretional from the soure; G - the gravitational onstant; v - the veloity of the partile. Aording to work (Okun', 1989): Whereas in Newtonian theory the fore of the gravitational interation is determined by the masses of the interating bodies, in the relativisti ase the situation is muh more ompliated. It is ase to see that for a slow eletron, with << 1, the expression in the square braket redues to r, and, bearing in mind that E / m, we return to Newton s nonrelativisti formula. However, for 1 or 1 we enounter a fundamentally new phenomenon, namely, the quantity that plays the role of the gravitational mass of the relativisti partile depends not only on this energy but also on the mutual diretion of the vetors r and v. If v// r, then the gravitational mass is E /, but is v r, it is ( E / )(1 ),and for a photon E /. We use the quotation marks to emphasize that for a relativisti body the onept of gravitational mass of a photon this quantity is half that for one traveling horizontally. - the end of the itation. In the quoted Position again as earlier onerning an inertial mass, for definition of the relativisti gravitational mass of the partile the nonrelativisti law of gravitation (13) is used and by that the requirements of the orretness are violated. However, the role of a relativisti gravitational mass onsist not at all to define uniquely the gravitational fore through the nonrelativisti law of gravitation, but in to define uniquely the gravitational fore through the relativisti law of gravitation. The relativisti formula (14) is redued to the lassial formula (13) only, when. So, the lassial relation (13) orretly defines the gravitational fore F, stritly speaking, only at the zero veloity of the partile. Let the

7 Applied Physis Researh Vol. 4, No. ; 1 radius-vetor r is anyone, but the fixed. Then in the lassial mehanis the inertial mass m unambiguously and orretly defines the gravitational fore F, stritly speaking, only at the zero veloity of the partile through the lassial Newton s law of gravity (13). Therefore the inertial mass m is equivalent to the gravitational mass, stritly speaking, only at the zero veloity of the partile. In the relativisti mehanis the inertial relativisti mass m unambiguously and orretly defines the gravitational fore F at the given both zero, and any nonzero veloity v of the partile ( v is smaller с) through the relativisti variant of the Newton s law of gravity (14). Therefore the given by equation (1) relativisti inertial mass m is the gravitational mass at any veloities v of the partile (smaller с). The law of gravity is varied at the passage from the lassial to relativisti mehanis. Therefore it is quite natural, that, as against to the lassial mehanis, in SRT the gravitational fore F depends not only on the gravitational mass of the partile, but else depends on the veloity of the partile. Due to adding of the term β r (r,β)β in the equation (14), the gravitational fore obtains the omponent, parallel to the veloity of the partile, that is again happens the "leeway", but now not of the aeleration, and of the gravitational fore. Aording to experiene of the development of physis, similar modifiations of physial laws of the lassial physis, at an exit out of the limits of its appliability, are quite natural. There is no sense to try "to pen" the adding β r (r, β) β into the gravitational mass. 1. Resume Aording to the priniple of existene of the mass, the relativisti measure of inertia should be presented in the speial relativity theory (SRT). But the mass is not obliged to remain a salar at the passage from the lassial to relativisti mehanis. We have demonstrated that the relativisti measure of inertia, i.e. the relativisti mass, exists in SRT and it is not a salar quantity but it is the omponent of the 4-vetor. The relativisti mass plays the role of the measure of inertia in the interrelation between the fore and aeleration aused by this fore as well as in the interrelation between the momentum and the veloity. At that any mass m 1 at anyone, but the fixed veloity v 1, is the same invariant of Lorentz transformation as well as the rest-mass m taken at the fixed zero veloity (please, look above setion 8). The use of the relativisti mass in SRT is sequential fulfillment of the purpose of SRT and is inalienable of SRT. Sine the relativisti mass is the omponent of the 4-vetor, the fundamental interrelation E m between the measure of inertia and the energy is orret at any veloity (less or equal с). Probably, it means, that there is a ertain ommon physial substane whih generates both the property of matter to have energy, and the property of matter to have inertia. It is possible to show, that the relativisti inertial mass plays in SRT the role of the relativisti gravitational mass, at least, at onstant gravitational field. The opposite onlusions of the supporters of the Conept 1-4 (setion 1), are obtained as result of the violation of the requirements of a orretness. At the orret analysis: all items 1-4 of the Conept 1-4 are vanished; the relativisti mass of the system of not interating, freely flying partiles is equal always to the sum of the relativisti masses of partiles and does not depend on the angles between the momenta of the partiles; the paradox of the photons system does not arise. The onept 1-4 is not harmless for a siene sine this Conept loses the road to the investigations of the physial reasons of the mass-energy E = m equivalene, to the investigations of the reasons of the inertia variations at the variations of the physial body motions and to study of the ontiguous problems. It is stated sometimes, that it is useless to introdue the relativisti mass m beause by the definition (1) the relativisti mass m is simply other measure of the energy. However, at first, it is not the strong argument, at least, as long as it is not told: it is useless to introdue the rest-mass m too sine by virtue of the fundamental relation (), m is other measure of the energy too, but now of the rest energy. Seondly, the kernel of the problem is not in that to introdue or to not introdue the designation m. The kernel of the problem is in that the relativisti measure of inertia is neessary in SRT, and in that the energy appears to be the measure of this inertia. Unfortunately, the majority of the modern books on SRT adhere to the Conept 1-4 (Wong and Kueh, 5). Therefore the inexat and distorted representation on the speial relativity theory an arise among the students, if eduational proess to not supplement with similar explanations. The author is grateful to professor D. D. Rabounski and professor I. V. Veselovskiy for ontansive disussions. Notes Note 1. It ould be understood somehow still if the mass was a measure not of inertia but some measure of the amount of the material invariant relatively Lorentz transforms. 14

8 Applied Physis Researh Vol. 4, No. ; 1 Note. The more inertia of the physial body - the variation of its veloity should be less at the delivery of the predetermined momentum to the physial body. Note 3. In fat, the more this multiplier, the more the inertia of the partile, that is, the less the partile veloity variation at the transmission to the partile of the fixed momentum. Note 4. In priniple, the researher ould define the relativisti measure of inertia ambiguously. For example, any funtion f (m) from a mass m ould serve as a measure of inertia if funtion f (m) approah to m when the veloity v approah to zero and if between funtion f (m) and the mass m there is an one-to-one orrespondene. The same ambiguity takes plae and in the lassial mehanis. But, hardly, it is onvenient to use other measure of inertia rather than magnitude m. Note 5. Applying here the formula (1) to the analysis of the system of the free partiles, we obtain the result formally. In order that the formal result has aquired the physial sense, it is neessary from the beginning to have defined the notion of the motion of the system of the independent partiles, as the single whole. (Differently it is not lear, in relation to what aeleration the inertia of the independent partiles system is disussed when speak on the mass of the independent partiles system.) For this purpose it is possible to enter the speial system of oordinates relative to whih the physial properties of the free partiles (their masses, veloities, momenta, energies) remain onstants - "frozen" in this system of oordinates - but whih an aquire aeleration. Suh "frozen" population of the oordinates and the partiles, appropriate to the hosen set of the partiles, the author terms as the appropriate system, or briefly as a-system. Then aeleration of the system of the free partiles (perhaps and photons) aquire the sense, as the aeleration of their a-system. The summarized relativisti mass of the partiles appears the rest-mass of their a-system. The aeleration of the a-system of a photon gives rise to hanging not of the veloity of the photon, but to hanging of its energy. Appliations of the а-systems and the disrepanies of the onstrution of the apparent paradox of the photons system (Okun', ) are in details onsidered in the work (Vasiliev, 9). Referenes Landau, L. D., & Lifshitz, E. M. (1967). The Theory of Fields. Mosow, Nauka, 458. Okun', L. B. (1989). The onept of mass. Usp. Fiz. Nauk, 158, (in Russian). (Sov. Phys. Usp., 3, , 1989, in English) Okun', L. B. (). Reply to the letter ''What is mass?'' by R I Khrapko. Usp. Fiz. Nauk, 17, (in Russian). (Sov. Phys. Usp., 43, ,, in English) Vasiliev, S. A. (9). On the role of the relativisti mass in the speial relativity theory. The Earth Planet System, Proeedings of XVII-th Sientifi Seminar, 15 years to the interdisiplinary sientifi seminar, Mosow State University, Mosow, The monography. Mosow, LENAND, p (in Russian). Retrieved from or Wong, Chee Leong, & Yap, Kueh Chin. (5), Coneptual Development of Einstein s Mass-Energy Relationship. New Horizons in Eduation, 51, May, Retrieved from 143