Derivation of Non-Einsteinian Relativistic Equations from Momentum Conservation Law

Transcription

1 Asian Journal of Applied Siene and Engineering, Volue, No 1/13 ISSN X(p); (e) Derivation of Non-Einsteinian Relativisti Equations fro Moentu Conservation Law M.O.G. Talukder Varendra University, Rajshahi 64, BANGLADESH Abstrat In this paper, we present a relativisti expression for the oentu onservation law using a foralis derived fro the Lorentz-Einstein law for the addition of veloities. It deonstrates that a ass-less partile an ove only at the speed of light and it has oentu like a photon. Hene, onsideration of relativisti oentu onservation of a photon-gun syste reveals that the ass of the gun and the tie period, aplitude and the speed of photon vary with the speed of the gun. That is the ass, tie, length and veloity are relative quantities. However, the orresponding expressions are different fro those in Einstein s theory of speial relativity. Hene, we all those as Non-Einsteinian relativisti expressions. Moreover, eah relative quantity has been found to have two values one in the longitudinal and the other in the transverse diretions, suh that the produt of the two reains invariant. Nevertheless, the longitudinal and transverse values have been denoted by the suffixes NL and respetively. It should be pointed here that, in the previous papers, the orresponding expressions in Einsteinian relativisti ase have been denoted by the suffixes EL and ET respetively. Keywords: Moentu onservation, Lorentz Einstein law, Non-Einsteinian relativity, photon, relative ass, tie, length and veloity. IRODUCTION As a way of illustrating the lassial oentu onservation law, let us take the exaple of a gun. Suppose, the ass of the gun is and that of a bullet is '. When the bullet is fired, it oves with a veloity V in the forward diretion. As a onsequene, the gun oves in the bakward diretion with a veloity v. Then, aording to Newton's third law of otion or the oentu onservation law 1 (1) V v 'V is the oentu of the bullet and v is that of the gun. The relativisti ultipliation of a veloity u by any nuber N is given by -6 N 1 u 1 1 u N u () N 1 u 1 1 u the sybol indiates the relativisti ultipliation aording to Lorentz - Einstein (L E) law 7-9 and is the speed of light. Copyright CC-BY-NC, Asian Business Consortiu AJASE Page 69

2 Talukder: Derivation of Non-Einsteinian Relativisti Equations fro Moentu Conservation Law (69-79) On the other hand, aording to Einstein s speial theory of relativity 1, the relative ass () of a partile of rest ass () oving at a veloity (v) is given by (3) v 1 is the speed of light. In this paper, we present a relativisti expression for the oentu onservation law aking use of Eq. () in Eq. (1). It has been deonstrated that lassially a ass-less partile will have an infinite speed, whereas, relativistially it an ove only at the speed of light. Its overall behavior has been haraterized as that of a photon. Hene, we have studied the oentu onservation of a photon eitted fro a gun alled photon-gun. In order to have better understanding of the relativisti effets, Non-Einsteinian relativisti ase has been introdued to desribe the variation of relative quantities like ass, tie, length and the speed of photon with veloity of the gun. For eah relative quantity, we have also onsidered two values - longitudinal and transverse. Further, Einstein proposed a partile or photon odel, whih is also known as the quantu odel, of light. In that odel, he viewed light as onsisting of streas of partiles, alled photons, rather than of wave. The energy ontent of eah photon is equal to the produt of Plank s onstant and the frequeny of light 11. That is E h (4) E = energy of photon h = Plank s onstant ν = frequeny of light It should be pointed here that the rest ass of a photon is zero. However, it has oentu whih an be obtained fro the relation p h (5) and the equivalent ass ( ) of a photon is given by h (6) The energy of a photon an be expressed in ters of its oentu as E p (7) p is the oentu of the photon and is the speed of light. In the study, presented here, we have used the above entioned photon odel of light to desribe the kineatis of a photon-gun syste eitting a photon. As a onsequene, we have Page 7 Copyright CC-BY-NC, Asian Business Consortiu AJASE

3 Asian Journal of Applied Siene and Engineering, Volue, No 1/13 ISSN X(p); (e) found the expressions for relative ass, tie, length and veloity in the Non- Einsteinian relativisti ase. RELATIVISTIC DESCRIPTION OF MOMEUM CONSERVATION LAW Fro the oentu onservation law as given by Eq. (1), we an write v V (8) However, Eq. (8) does not liit the value of V within that of the speed of light as required by the priniple of relativity. For exaple, if we put (ass-less partile) in the above equation, V beoes infinite (i.e. V ). It should be pointed here that in lassial ehanis, the speed of light is onsidered to be infinite. But we an ensure V using the foralis given by Eq. () in Eq. (8) as deonstrated below. Let us express the above Equation as V v (9) Multiplying both sides by 1/, we get 1 1 V v Aording to the properties of the foralis desribed in a previous paper, the above equation takes the for v (1) V (11) Then, sine /' is a nuber, the above equation an be written, following Eq. (), as 1 v 1 v 1 v 1 v 1 V (1) 1 The above equation shows that the value of V annot exeed. Further, it an be shown that in the lassial liit, v <<, it redues to the lassial oentu onservation law given by Eq. (1) as follows: Equation (1) an also be written as Copyright CC-BY-NC, Asian Business Consortiu AJASE Page 71

4 Talukder: Derivation of Non-Einsteinian Relativisti Equations fro Moentu Conservation Law (69-79) V 1 v/ 1 v/ 1 v/ 1 v/ (13) If v<<, then V v v 11 (14) whih is the lassial oentu onservation law represented by Eq. (8). Hene, we an onlude that Eq. (1) represents the relativisti expression for the oentu onservation law. The equation also indiates that V if /'. One possibility is when ' (ass-less partile). It eans that a ass less partile oves at the speed of light. Further, fro Eq. (1), V = for v =, whereas, V = for any non zero value of v (i.e. v ) and if (i.e. ). These results indiate the following: / ' A ass less partile does not ove unless soe oentu is iparted to it. Therefore, aording to the oentu onservation law, its oentu ust be equal to the reoil oentu of the soure. However, its oentu or hange of oentu annot be aounted for by its ass () and veloity (). These are the harateristis possessed by a photon aording to Einstein s odel of light IMPLICATIONS OF THE RESULTS (I) RELATIVE MASS In the above disussion, the ass of the bullet was onsidered to be negligible opared to that of the gun. If the ass of the bullet is taken into onsideration, the oentu onservation law beoes V (15) v is the total ass of the gun and bullet at rest and is the ass of the h bullet. Now, if a photon having energy is eitted fro the gun, sine the rest ass of a photon is zero, we an write fro the energy onservation law h 1 (16) v The ter on the RHS of the above equation represents the kineti energy of the gun. Further, sine a photon has oentu, the oentu onservation law given by Eq. (15), an be written using Eqs. (5) and (6) as h h v (17) Page 7 Copyright CC-BY-NC, Asian Business Consortiu AJASE

5 Asian Journal of Applied Siene and Engineering, Volue, No 1/13 ISSN X(p); (e) or h h v v The seond ter on the RHS is (v/) ties the oentu of the photon. Using Eq. (16) in Eq. (17), we get h v (18) (19) () The above equation indiates that the ass of the gun dereases with veloity. Hene, we an infer that the ass is relative. Now, in the Einsteinian relativisti ase, all the relative quantities were found to be dereasing with veloity in the longitudinal diretion. Hene, in the above ase, the variation of ass with veloity an be regarded as the relative ass in the longitudinal diretion. However, it is different fro the orresponding expression in Einsteinian relativity. Hene, we all this ase as Non-Einsteinian relativity. So, let us denote the relative ass, in this ase, by NL. Where, in the suffix, N stands for Non-Einsteinian and L for longitudinal. Heneforth, we will use the sae suffix for all relative quantities in the longitudinal diretion and the suffix for transverse relative values for the. Therefore, the RHS of the above equation an be expressed as NL 1 (1) Now, ultiplying both sides of the above equation by 1 NL v and rearranging ters, we get () whih is the energy onservation law. This eans that the total energy reains onstant at the rest ass energy. Further, dividing both sides of the above equation by 1, or 1 v 1 1 (3) Copyright CC-BY-NC, Asian Business Consortiu AJASE Page 73

6 Talukder: Derivation of Non-Einsteinian Relativisti Equations fro Moentu Conservation Law (69-79) 1 v (4) 1 (5) is the transverse ass in the Non-Einsteinian relativisti ase. It is lear that inreases with inreasing veloity and hene it is oparable with relative ass given by Eq. (3) in Einsteinian relativity. However, the seond ter on the RHS of Eq. (4) represents the relativisti kineti energy. Therefore, fro Eqs. (1) and (5), we get NL (6) The above equation indiates that the produt of the longitudinal and transverse asses is equal to the square of the rest ass. It eans that the rest ass is an invariant quantity and is equal to the geoetri ean of the longitudinal and transverse asses. (II) RELATIVE MOMEUM AND ENERGY OF THE GUN Aording to Einstein s ass-energy equivalene priniple, the rest ass energy (E) is given by: E (7) E (8) Using Eq. (8) in Eq. (6), we obtain or E E NL (9) NLE E (3) ENL NL E 1 (31) is the relative energy in the longitudinal diretion and E E 1 (3) Page 74 Copyright CC-BY-NC, Asian Business Consortiu AJASE

7 Asian Journal of Applied Siene and Engineering, Volue, No 1/13 ISSN X(p); (e) is the relative energy in the transverse diretion. Hene, fro Eq. (3), we an onlude that the produt of the relative energies in the longitudinal and transverse diretions is equal to the square of the rest ass energy. It eans that the rest ass energy is an invariant quantity and is equal to the geoetri ean of the longitudinal and transverse energies. Equation (9) an also be written as E NL (33) p p (34) NL p p E (35) is the oentu related to the rest ass energy or Copton oentu, pnl NL p 1 (36) is the relative oentu in the longitudinal diretion and p p 1 (37) is the relative oentu in the transverse diretion. Hene, fro Eq. (34), we an onlude that the produt of the oentus in the longitudinal and transverse diretions is equal to the square of the Copton oentu. It eans that the Copton oentu is an invariant quantity and is equal to the geoetri ean of the longitudinal and transverse oentus. (III) RELATIVE VELOCITY OF THE PHOTON Equation (33) an also be expressed as v (38) NL v vnlv (39) v 1 (4) NL is the relative veloity in the longitudinal diretion and v 1 (41) Copyright CC-BY-NC, Asian Business Consortiu AJASE Page 75

8 Talukder: Derivation of Non-Einsteinian Relativisti Equations fro Moentu Conservation Law (69-79) is the relative veloity in the transverse diretion. Hene, fro Eq. (39), we an onlude that the produt of the relative veloities in the longitudinal and transverse diretions is equal to the square of the veloity of the photon or the speed of light. It eans that the speed of light is an invariant quantity and is equal to the geoetri ean of the longitudinal and transverse veloities. (IV) RELATIVE TIME Putting = λ/t, (λ is the wavelength and t is the tie period of the photon) in Eq. (39), we obtain or 1 t t t NL t t 1 (4) t (43) t t (44) NL t It should be pointed here that the tie period (t), generally, indiates an interval of tie whih is the notion of tie in odern physis. In other words, the tie period an orretly represent the disrete nature of tie whih is onsistent with the onept of relativity. Therefore, in Eq. (44), t NL t 1 (45) represents the relative tie in the longitudinal diretion whih is ontrated and t t 1 represents the relative tie in the transverse diretion whih is dilated. Thus, we an onlude that tie is ontrated in the longitudinal diretion but dilated in the transverse diretion. Moreover, fro Eq. (44), we an onlude that t is an invariant quantity and is equal to the geoetri ean of the longitudinal and transverse ties. (V) RELATIVE LENGTH Sine, λ is an interval between two spae points, it an be onsidered as a orret representation of length (l) in relativisti ase. Hene, Eq. (4) an be written as (46) Page 76 Copyright CC-BY-NC, Asian Business Consortiu AJASE

9 Asian Journal of Applied Siene and Engineering, Volue, No 1/13 ISSN X(p); (e) or l l 1 (47) t t1 l t l l l NL t t (48) l l (49) NL l l NL l 1 (5) represents the relative length in the longitudinal diretion whih is ontrated and l 1 l (51) represents the relative length in the transverse diretion whih is dilated. Thus, we an onlude that length is ontrated in the longitudinal diretion but dilated in the transverse diretion. Moreover, fro Eq. (49), we an onlude that l is an invariant quantity and is equal to the geoetri ean of the longitudinal and transverse lengths. Fro the above disussions we an onlude that, as a general rule, any relative quantity will have longitudinal and transverse values suh that their produt will reain invariant at the square of its value at rest. Sybolially, it an be written as X NL X X (5) X is any relative quantity and X is its value when v. This is valid for Eq. (39) as well beause it is lear fro Eqs. (4) and (41) that v NL = v = when v. Further, it is obvious fro the above results that all relative quantities derease in the longitudinal diretion but inrease in the transverse diretion with inreasing veloity. We suarize the results obtained in the following table. The table ontains a oplete set of self onsistent equations for the relative ass, tie, length, veloity, oentu and energy in the ase of Non-Einsteinian view of relativity. For eah of the relative quantities, both the longitudinal and transverse values and the value of their produt are given. Copyright CC-BY-NC, Asian Business Consortiu AJASE Page 77

10 Talukder: Derivation of Non-Einsteinian Relativisti Equations fro Moentu Conservation Law (69-79) Table 1: Relative quantities in Non-Einsteinian view of relativity Relative Quantity Mass Tie t Transverse Longitudinal Produt 1 t 1 1 NL 1 t NL t t NL t t NL Length Veloity Moentu Energy l l l 1 1 v v 1 1 p p p1 1 E E E 1 1 The relativisti Kineti Energy (K.E.), fro Eq. (4), is CONCLUSIONS 1 K. E. l NL l l l NL NL v v NL p NL p NL p p E NL E E E NL v (53) Through using the relativisti ultipliation rule for the veloities we have found: (a) Relativisti expression for the oentu onservation law. (b) A ass less partile an ove only at the speed of light and has oentu. () Conservation of oentu for a photon-gun syste reveals (as ipliations): (i) A oplete set of self onsistent equations of relative quantities in non- Einsteinian relativisti ase. In this ase there, is no restrition on any partile s speed being equal to or greater than that of light. Page 78 Copyright CC-BY-NC, Asian Business Consortiu AJASE

11 Asian Journal of Applied Siene and Engineering, Volue, No 1/13 ISSN X(p); (e) (ii) The relative ass inreases in the transverse diretion but dereases in the longitudinal diretion. However, the rest ass is an invariant quantity. (iii) The relative length is ontrated in the longitudinal diretion but dilated in the transverse diretion. However, the length at rest is an invariant quantity. (iv) The relative tie is dilated in the transverse diretion but ontrated in the longitudinal diretion. However, the tie at rest is an invariant quantity. (v) The relative veloity dereases in the longitudinal diretion but inreases in the transverse diretion. However, the produt of the longitudinal and transverse veloities is equal to the square of the speed of light. It eans the speed of light is an invariant quantity and is equal to the geoetri ean of the two relative veloities. REFERENCES 1. I. Newton, in The Prinipia (1687).. M. Ahad, Phys. Essays., 44 (9). 3. M. Ahad, J. of S. Researh 1, 7 (9). DOI: 1.339/jsr.v1i M. Ahad and M.O.G. Talukder, Phys. Essays, 4, 593 (11). 5. M. Ahad and M. O. G. Talukder, e-print arxiv: M. Ahad and M.O.G. Talukder, Sent for publiation in Phys. Essays (11). 7. H.A Lorentz, KNAW, Proeedings, Asterda, 6, (194). 8. A. Einstein, Annalen der Physik, 17, 891 (195). 9. L. de Broglie, Ann. Phys. (Paris) 3, (195). Reprinted in Ann. Found. Louis de Broglie 17 (199) p.. 1. A. Einstein, Annalen der Physik, 17, 13 (195). 11. A. Einstein, Annalen der Physik,, 67 (196). Copyright CC-BY-NC, Asian Business Consortiu AJASE Page 79