Derivation of Einstein s Equation, E = mc 2, from the Classical Force Laws

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1 Apeiron, Vol. 14, No. 4, Otober Derivation of Einstein s Equation, E = m, from the Classial Fore Laws N. Hamdan, A.K. Hariri Department of Physis, University of Aleppo, Syria nhamdan59@hotmail.om, haririak@yahoo.om J. López-Bonilla SEPI-ESIME-Zaateno, Instituto Politénio Naional, Edif. Z-4, 3er. Piso, Col. Lindavista CP Méio DF joseluis.lopezbonilla@gmail.om In several reent papers we showed that hoosing new sets of postulates, inluding lassial (pre-einstein) physis laws, within the main body of Einstein s speial relativity theory (SRT) and applying the relativity priniple, enables us to anel the Lorentz transformation from the main body of SRT. In the present paper, and by following the same approah, we derive Einstein s equation E = m from lassial physial laws suh as the Lorentz fore law and Newton s seond law. Einstein s equation is obtained without the usual approahes of thought eperiment, onservation laws, onsidering 7 C. Roy Keys In.

2 Apeiron, Vol. 14, No. 4, Otober ollisions and also without the usual postulates of speial relativity. In this paper we also identify a fundamental oneptual flaw that has persisted for the past 1 years. The flaw is interpreting the formula E = m as the equivalene between inertial mass and any type of energy and in all ontets. It is shown in several reent papers that this is inorret, that this is a misinterpretation. What Einstein onsidered to be a entral onsequene of speial relativity is in fat derivable from (pre- Einstein) lassial onsiderations. E = m beomes seondary, not fundamental, and whilst no doubt useful in ertain irumstanes, need not be valid in all generality. Keywords: lassial fore laws, speial relativity theory, the Einstein s equation, E = m. Introdution The SRT [1] has removed the barrier between matter and energy, but it has reated a new barrier that annot be transended. This barrier separates what is known as non-relativisti from relativisti physis. The physial laws for non-relativisti physis annot transend this barrier and hene they form lassial physis. The physial laws adequate for relativisti physis an, however, over the nonrelativisti physis domain through the known approimation of the Lorentz transformation (LT): The LT beoming a Galilean transformation where appropriate. In lassial physis, we know that a partile m moving with veloity v has a momentum p= mv and a kineti energy 1 T = m v, 7 C. Roy Keys In.

3 Apeiron, Vol. 14, No. 4, Otober however, in relativisti physis the momentum and relativisti mass of a partile are mv p= = m v, (1) v m m v = =γ 7 C. Roy Keys In. m, () whereas its kineti energy is T = m m, (3) where as we see m represents the rest mass. This m aounts for the inertia of the partile at the moment when its aeleration starts from a state of rest. Relativists also introdue the onept of rest energy E = m, and of relativisti energy E T E m = + =. (4) In his paper, Einstein [] derived equation (4) through a thought eperiment. Many further papers have been devoted to the derivation of equation (4) using the popular approahes of onservation laws, onsideration of ollisions and the postulates of SRT [3-8]. In a previous paper [9] we suggest another way to aount for the kinematial effets in relativisti eletrodynamis. This method does not use the LT for a harged partile: Instead it involves inserting the Lorentz fore within the main body of SRT and applying the priniple of relativity (that the laws of physis have the same formulation relative to any inertial system) rather than the priniple of speial relativity (that the laws of physis are invariant under the LT). We

4 Apeiron, Vol. 14, No. 4, Otober have also presented in another paper [1] our laim that not only relativisti eletromagnetism, but also relativisti mehanis an easily be derived using this approah. As in the previous papers [9,1] we present a derivation of equation (4) starting from lassial laws, without alling upon the usual approahes. Further, we larify that the derivation of E = m an be studied without using the LT or its kinematial effets, i.e. the relation E = m does not need SRT, ontrary to Einstein s assertion [11]. Derivation of Einstein s equation E = m from the Lorentz fore Einstein was the first to derive mass-energy equivalene from the priniples of SRT in his artile titled "Does the Inertia of a Body Depend Upon Its Energy Content?" []. Sine this derivation was published, it has been the subjet of ontinuing ontroversy. For instane, the relativisti mass, Eq. (), is applied in the ase of an inertial frame o-moving with the partile, so it is a onsequene of the time dilation effet between the two frames i.e. a o-moving referene frame for a partile moving with a uniform veloity where the referene frame is supposed to be at rest at every moment during the motion of the partile. Therefore, the relativisti mass in this ase is a purely kinematial effet. Hene, aording to Einstein the relativisti mass is not a physial effet but rather the result of the effet of relative motion on observation. Einstein then derives his equation mathematially and under speial onditions, whih requires the above speulation. Therefore Eqs. () and (4) an appear mysterious and are put in doubt [1,13]. We will show in this paper a purely dynamial derivation of 7 C. Roy Keys In.

5 Apeiron, Vol. 14, No. 4, Otober E = m and the mass inrease relation, based solely on the Lorentz fore law and Newton s seond law (NSL). Let us onsider two inertial systems S and S with a relative veloity u // o between them. Einstein in his SRT assumes the equivalene of all inertial referene frames, but makes use of different assumptions, inluding: time dilation- length ontration and the relativity of simultaneity. These effets are epressed mathematially by the Lorentz transformation. Our paper makes a distintion between the Lorentz transformation and inertial frames, even though within the ontet of speial relativity they are the same thing. Our definition of inertial referene frame, on the other hand, is in fat the usual one (before Einstein): those referene frames in whih Newton's first and seond laws of motion are valid. Our approah in this paper is to onsider a single partile that moves along a given diretion with ommon o( o ) aes. As demonstrated in [9], ontrary to what is often laimed in SRT, the relativisti epressions an then be derived starting from the relativity priniple and the lassial Lorentz's law, i.e. F= q( E+ v B ). (5) Now assuming that a harged partile q moving with veloity v in the frame S, subjet to an eletri field E and a magneti flu density B, then the Cartesian omponents of (5) in frame S are F = q(e + vybz vzb y ) (6a) Fy = q(ey + vb z vb z ) (6b) Fz = q(ez + vby vyb ) (6) Applying the relativity priniple to Eqs. (6), we have 7 C. Roy Keys In.

6 Apeiron, Vol. 14, No. 4, Otober 7 44 F = q(e + v B v B ) (7a) y z z y F = q(e + vb vb ) (7b) y y z z F = q(e + v B v B ) (7) z z y y So following a similar approah to that used in [9], we an obtain the relativisti transformation equation for veloity: v u vy v = (a), v uv = vz y (b), v uv = z () (8) u γ γ v where the salar fator γ was fied by applying the relativity priniple to Eqs. (8): u 1 γ (1 ) = 1 or γ = 1 u (9) The onventional way to derive the relativisti transformation equations of the omponents of the relativisti veloity are obtained from SRT. However, we have already eplained that there is an alternative path that does not use the Lorentz transformation in the derivation of the 3-vetor relativisti veloity transformations appertaining to a harged partile. With Eqs. (8) this enables us to onstrut the following relativisti identities: uv 1 1 = (a), v u v 7 C. Roy Keys In.

7 Apeiron, Vol. 14, No. 4, Otober v v u = (b) (1) v u v v y vy v z vz = ( ), = (d) v v v v In spite of the objetions by L.B. Okun [14] who states that the proper definition of relativisti momentum is p= γ mv, the traditional definition of momentum, i.e. p= mv, ombined with Eqs. (8) gives all the relativisti epressions and relativisti transformation relations onerning momentum, energy and the relativisti mass as follows: When viewed from S the harged partile q that we have mentioned above has momentum p= mv whose omponents are: p = mv ( a), py = mvy ( b), pz = mvz ( ) (11) Viewed from S the momentum is p = m v having the omponents: p = mv ( a), p y = mv y ( b), p z = mv z ( ) (1) in aordane with the relativity priniple. Combining (8a), (11a) and (1a) we obtain: p p um =, m uv m (1 ) whih means that uv p = k( p um) ( a), m = mk(1 ) ( b) (13) 7 C. Roy Keys In.

8 Apeiron, Vol. 14, No. 4, Otober 7 44 where k represents an unknown onstant and this salar fator k an be fied by applying the relativity priniple to Eqs. (8a), (11a) and (1a) to get: p p + um = m uv m (1 + ) Then uv p = k( p + um ) ( a), m= mk (1 + ) ( b) (14) Multiplying (13b) and (14b) side by side we dedue: uv uv 1 = k (1 )(1 + ) and employing (8a) in the last equation, leads to u 1 k (1 ) = 1 or k = γ = () (14) u The onstant k in Eq.(14) is equal to the salar fator γ in Eq.(9). For simpliity, take the speial ase that the harged partile is at rest in frame S, thus: v =, v = u These results, when substituted into (13b) leads to m = γ m (15a) Observers of frame S measure the rest mass m, observers from S measure the mass m given by (15a). We an now assume that the harged partile is at rest in frame S, so: 7 C. Roy Keys In.

9 Apeiron, Vol. 14, No. 4, Otober v =, v = u These results, when substituted into (14b) lead to m= γ m (15b) Observers of frame S measure its rest mass m, observers from S measuring mass m given by (15b). We obtain the transformation equation for the oy( o y ) omponent of the momentum if we ombine (11b) and (1b) with (8b). Thus we dedue: p = mv = mv = p (13) y y y y In a similar way, we get p z = mv z = mvz = pz (13d) Now, using (9) in (13a,b) and (15), leads to the transformation equations uv m = γ m(1 ), p = γ ( p um) (16a,b) p y = py, p z = pz (16,d) and the epression for the relativisti mass in both frames: m m m =, m = (17a,b) v v If we start with the relativisti identities (1) multiplying both sides with m, the result is equations (16) and (17) (see [9]). To obtain the same relativisti ombination between the momentum and energy of the same harged partile q, we start from (14), and write this equation as: 7 C. Roy Keys In.

10 Apeiron, Vol. 14, No. 4, Otober u γ γ = 1, 4 Multiplying both its sides with m it beomes: 4 4 γ m γ mu = m (18) We reognize that the term p = γ m u = m v, u = v represents the square of the momentum in S. Moreover, the root of the first term presented is 1 v E = m (1 ) = γ m (19) = m Eq.(19) is the relativisti energy E, telling us that the hange of the mass of a partile is aompanied by a hange in its energy and vie versa [15]. With the new notation, (18) beomes: 4 E = p + mo () Eq. () represents the ombination of the momentum and the energy of the same partile. Multiplying both sides of (16a,b) with we obtain the transformation equation for energy and momentum: u E = γ ( E up ), p = γ p E (1) The onventional way to derive the relativisti transformation equations of energy-momentum is to onsider a ollision between two partiles from two inertial referene frames and imposing energymomentum onservation. However, the purpose of our paper is to 7 C. Roy Keys In.

11 Apeiron, Vol. 14, No. 4, Otober show that the relativisti dynamis, and espeially Einstein s equation E = m, ould be approahed without the popular approahes. The kineti energy T should equate the differene between relativisti energy E and rest energy E, i.e. T = m m 1 = m ( 1) () v Derivation of Einstein s equation E = m from Newton s seond law (NSL). With the disovery of the SRT, energy was found to be one omponent of an energy-momentum 4-vetor. Eah of the four omponents (one of energy and three of momentum) of this vetor is separately onserved in any given inertial referene frame. The vetor length is also onserved and is the rest mass. The relativisti energy of a single massive partile ontains a term related to its rest mass in addition to its kineti energy of motion. In the limit of zero kineti energy or equivalently in the rest frame of the massive partile the energy is related to its rest mass via the famous equation E = m. However, many tetbooks on SRT often devote great effort to disussing the proess of elasti ollision between two partiles to derive E = m and the relativisti mass m= γ m. Our goal on the other hand is to establish Einstein s formula E = m based solely on the NSL without using the LT or the ideas of Einstein. As we demonstrated in [16,17], the Lorentz fore and the relativity priniple, are more natural for desribing the physis of relativisti eletrodynamis. 7 C. Roy Keys In.

12 Apeiron, Vol. 14, No. 4, Otober F= q( E+ v B ) (a), de = Fv (b) (3) dt Moreover, we an now go further to get all of SRT's relations from lassial mehanis: d F = p (a), de = Fv (b) (4) dt dt Let us onsider two inertial systems S and S with a relative veloity u // o between them and onsider from S a partile whih has the momentum p= mv. The Cartesian omponents of (4) in frame S are: dp dp y dpz = F ( a ), = Fy ( b ), = Fz ( ) (5) dt dt dt and de Fv Fv y y Fv z z dt = + + (d) Applying the relativity priniple to (5), we have: dp dp y dp z = F ( a ), = F y ( b ), = F z ( ) (6) dt dt dt and de = Fv + Fv y y + Fv z z (d) dt By following the same approah to that used in [1] we an obtain the relativisti transformation equation for momentum, energy and veloity: u p = γ p E (a), p y = py (b), p z = pz (), 7 C. Roy Keys In.

13 Apeiron, Vol. 14, No. 4, Otober E = γ ( E up )(d) (7) v u vy v = (a), v uv = y (b), uv γ We may write Eqs. (8) as: vz v = z (b), (8) uv γ uv (1 ) 1 = v u v and we an put (8b) into (7b) to get (9) uv m = γ m(1 ) (3) Multiplying (9) with m, and omparing it with (3), we dedue: m m m = (a), m = (b) (31) v v From (5d) the total energy is given by: de = Fvdt = d ( mv) v = vdm+ mvdv and from (31a) we have: mvdv dm = i.. e m v dv = 1 dm v v (3) (33) 7 C. Roy Keys In.

14 Apeiron, Vol. 14, No. 4, Otober Substituting (33) in (3), we get: de = dm and by integration, from v 1 to v, we dedue that: E = m (34) 1 In the partiular ase when v 1 = and v = v then E should equal the kineti energy T, i.e. T = m = m m 1 So the quantities m and m are the total energy E and E in frames S and S, respetively. It is simple to prove, that Eqs. (7) and (31) lead to 4 E ( p + p + p ) = E ( p + p + p ) = m or y z y z E = P + m = m 4 4, E = P + m = m 4 4 We show that Einstein s formula E = m an be reahed without using onservation laws and avoiding onsideration of ollisions or Einstein s thought eperiment. The derivations are based on the NSL and the priniple of relativity and on its diret onsequene, the addition law of relativisti veloities. Einstein's original derivation ould have been made learer using methods shown here. Einstein s E = m is derived here using different postulates to those of Einstein and in two different ways. As a result Einstein s equation is shown not to be a ore result of Einstein s speial relativity but somewhat seondary to the wider ontet of lassial physis. Its fundamental nature is therefore hallenged. 7 C. Roy Keys In.

15 Apeiron, Vol. 14, No. 4, Otober The Generalized Mass-Energy Equation E = Am Einstein's 195 derivation of E = m has been ritiized for being irular and for the fat that Einstein s original derivation is not at all lear [18]. It is often said that (inertial) energy E and the so-alled relativisti mass m are the same thing, presumably due to the relation E = m. However, this argument is demonstrably wrong. Suh erroneous onlusions may have ome about due to the lak of appliation of speial relativity to partiles. When an objet is a partile then the relation E = m does not hold: it was demonstrated in Refs. [19,,1] that Einstein's relation E = m led to serious errors and inepliable results. As was pointed out in Refs. [17,1] to remove the ontraditions with Einstein's relation E = m, we onsidered that Eqs. (19) and (34) ould be written as: E = m = mv + m v (35) Speifially, it was shown in paper [17,1] that the new total energy, = mv, (36) Ev is an intrinsi energy of a partile. Perhaps this will allow us to reonstrut ompatibility between the framework of De Broglie wave theory and SRT without the usual ontraditions. Combining Eq.(36) and Eqs.(19) and (34) gives, v Ev = m = Am, (37) whih an be used suessfully for a moving partile. We see now that A = 1 for a wave travelling with the speed v=, and in this ase 7 C. Roy Keys In.

16 Apeiron, Vol. 14, No. 4, Otober 7 45 Eqs. (35), (36) and (37) are idential to Eqs. (19) and (34); otherwise, the value of A an be equal, less than, or more than unity, depending on the situation. For this reason some authors argue that Eqs. (19) and (34) are inorret [18,19]. A. Sharma [18] has onsidered the total energy as Δ Etotal = AΔm rather than E = m. So that in Eqs. (36) and (37) A beomes a onversion oeffiient that an be determined by eperiment, and the value of A an be less, more, or equal to unity. E. Bakhoum [19], on the other hand, has taken the approah of defining the total energy aording to Eq. (36). A. Sharma has provided eperimental evidene that the value of the onversion fator between mass m and energy E is not always. The integration of speial relativity theory with quantum mehanis has yielded many paradoes that remained unsolved until reent years; like the zitterbewegung problem as well as the fat that the spin predition of the Dira equation ould only be identified with non-relativisti approimations (Pauli and Foldy-Wouthysen). The most prominent attempt to eliminate suh problems was by E. G. Bakhoum and requires a modifiation in the mass-energy equivalene priniple. Bakhoum introdued a new total relativisti energy formula E = m instead of Einstein's E = m. This paper arries Bakhoum's work a step further as we have derived Einstein s equation E = m without using speial relativity theory, instead starting from the lassial physis laws like the Lorentz fore law and Newton s seond law. Further, the energy formula of a partile E = m allows reoniliation between the de Broglie wave theory and the framework of the relativisti physis without the usual ontraditions [17-1]. Einstein has derived the relation E = m under speial onditions, i.e.; he derived the equation only for when a body moves with uniform veloity in the relativisti domain. This is the main reason 7 C. Roy Keys In.

17 Apeiron, Vol. 14, No. 4, Otober that authors have derived the relation E = m by new methods. Now onurrently with deriving the mass-energy equation by new methods, eisting eperimental data [18] has been analysed and shows that mass does not in fat have equivalene to energy for a moving partile, i.e.; the value of the onversion fator between mass m and energy E is not always. Therefore Eqs. (19) and (34) are speial ases of Eq.(37). Conlusion No one appears to have ever asked the question what is needed in the most general sense to derive Einstein s formula E = m? Approahing this question ourselves, we started from the lassial laws to rebuild the theory of speial relativity (SRT) [9,1]. This enables us to derive Einstein s formula E = m from the lassial laws without any of the well-known approahes and to show that Einstein s formula E = m an be derived in many different ways, even without the usual methods of thought eperiment, onservation laws, onsidering ollisions or the postulates of speial relativity. Currently the formula E = m is often interpreted as the equivalene between inertial mass and any type of energy. It is shown in papers [17-1] that this is inorret, that this is a misinterpretation. What Einstein onsidered to be entral to speial relativity is in fat derivable from more lassial onsiderations, rather than as a entral onsequene only of speial relativity. E = m beomes seondary, not fundamental, and whilst no doubt useful in ertain irumstanes may not even be valid in all generality. The root of this problem is due to an erroneous interpretation of speial relativity where the formula E = m is taken to be true outside of the ontet of relativisti kinematis. 7 C. Roy Keys In.

18 Apeiron, Vol. 14, No. 4, Otober 7 45 Aknowledgments It is our pleasure to thank the referees for their support and inspiring remarks onerning the paper. Where their notes ompleted the paper is gratefully aknowledged. Referenes 1- Einstein, A On the eletrodynamis of moving bodies. Ann. Phys. 17:891 - Einstein, A Does the inertia of a body depend on its energy ontent?, Ann. Phys. 18: Stek, D. J An elementary development of mass-energy equivalene. Am.J.Phys. 51: Peters, P. C An alternate derivation of relativisti momentum. Am. J. Phys. 54: Fegenbaum, M., Mermin, N E = m. Am. J. Phys. 56: Rohrlih, F An elementary derivation of E = m. Am. J. Phys. 58: Plakhotnik, T. 6. Epliit derivation of the relativisti mass-energy relation for internal kineti and potential energies of a omposite system. Eur. J. Phys. 7: 13 17, Sebastiano, S., Massimo P. 5. Deriving relativisti momentum and energy Eur.J.Phys. 6: Hamdan, N. 3. Abandoning the Ideas of Length Contration and time dilation, Galilean Eletrodynamis. 14: Hamdan, N. 5. Newton s Seond Law is a Relativisti law without Einstein s relativity. Galilean Eletrodynamis. 16: Einstein, A An Elementary Derivation of the Equivalene of Mass and Energy. Bulletin of the Amerian mathematial soiety. 41: Whitney, C. 5. Does Mass Really Inrease?. Galilean Eletrodynamis. 16, SI No. 3, Adler, C.G Does mass really depend on veloity, dad? Am. J.Phys. 55: Okun, L.B Note on the Meaning and Terminology of Speial Relativity. Eur. J. Phys. 15: C. Roy Keys In.

19 Apeiron, Vol. 14, No. 4, Otober Thomas, E.G. 5. What s so speial about E = m. Eur. J. Phys. 6: Hamdan, N. 5. Derivation of the Relativisti Doppler Effet from the Lorentz Fore. Apeiron. 1: Hamdan, N. 6. On the Interpretation of the Doppler Effet in the Speial Relativity (SRT). Galilean Eletrodynamis. 17: Sharma, A. 4. The Origin of the Generalized Mass-Energy Equation E = Am and Its Appliations in General Physis and Cosmology. Physis Essays. 17: Bakhoum, E.. Fundamental Disagreement of Wave Mehanis with Relativity. Physis Essays 15: 87-1 : physis/661 - Hamdan, N. 7. Derivation of de Broglie's Relations from Newton s Seond Law. Galilean Eletrodynamis. 18: Hamdan, N. 5. A dynami approah to De Broglie s theory. Apeiron 1: C. Roy Keys In.