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The Generalized Conversion Factor in Einstein's Mass-Energy Equation

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The Generalized Conversion Factor in Einstein's Mass-Energy Equation Volume 3 PROGRESS IN PHYSICS July, 2008 The Generalized Conversion Factor in Einstein’s Mass-Energy Equation Ajay Sharma Fundamental Physics Society, His Mercy Enclave Post Box 107 GPO Shimla 171001, India ajay.sharmaa@rediffmail.com Einstein’s September 1905 paper is origin of light energy-mass inter conversion equa- tion (L = mc2) and Einstein speculated E = mc2 from it by simply replacing L by E. From its critical analysis it follows that L = mc2 is only true under special or ideal conditions. Under general cases the result is L / mc2 (E / mc2). Conse- quently an alternate equation E = Ac2M has been suggested, which implies that energy emitted on annihilation of mass can be equal, less and more than predicted by E = mc2. The total kinetic energy of fission fragments of U235 or Pu239 is found experimentally 20–60 MeV less than Q-value predicted by mc2. The mass of parti- cle Ds (2317) discovered at SLAC, is more than current estimates. In many reactions including chemical reactions E = mc2 is not confirmed yet, but regarded as true. It implies the conversion factor than c2 is possible. These phenomena can be explained with help of generalized mass-energy equation E = Ac2M. 1 Introduction bomb explosions, volcanic reactions etc. Whatever may be the case E = Ac2M is capable of explaining the phe- Mass energy inter-conversion processes are the oldest in na- nomena. Thus conversion factor other than c2 is possible, in ture and constitute the basis of various phenomena. Before Einstein’s September 1905 derivation and confirmed experi- Einstein’s work, Newton [1] stated that “Gross bodies and mentally also. light are convertible into one another. ”. Einstein derived light energy-mass inter-conversion equation for Newton’s 2 Einstein’s light energy — mass equation L = mc2 L = mc2 perception as . Before Einstein scientists such as and its hidden aspects S. Tolver Preston [2] Olinto De Pretto [3], Fritz Hasenohrl [4, 5] Frederick Soddi [6] contributed to the topic. Einstein [11] perceived that let there be a luminous body at 2 Einstein’s derivation of L = mc (from which Einstein rest in co-ordinate system (x; y; z). The system (, , ) is in 2 speculated E = mc ), is true under special conditions uniform parallel translation w.r.t. system (x, y, z); and origin (where selective values of variables are taken). Under gen- of system (, , ) moves along x-axis with relative velocity eral conditions (when all possible values of parameters are v. Let a system of plane light waves have energy ` relative 2 2 taken) equations like L = 0:0011mc , L = 0:999988mc to system (x, y, z), the ray direction makes angle with x- 2 etc. are obtained i.e. L / mc . Thus conversion factor axis of the system (, , ). The quantity of light measured in 2 other than c is possible in Einstein’s derivation. Further system (, , ) has the energy [11, 12]. the generalized mass–energy equation E = Ac2M, is de- E = mc2 1 v cos rived, and is special case of the former depend- ` = ` q c (1) ing upon value of A (depends upon the characteristics condi- 1 v2 tions of the process). Thus apart from theoretical limitations, c2 E = mc2 has experimental limitations e.g. sometimes ex- Einstein has given Eq. (1) in his paper known as Special The- perimental results differ from it and in many cases it is not ory of Relativity [12] and called Eq. (1) as Doppler principle confirmed. Under such cases E = Ac2M is widely use- 235 for any velocities whatever. ful and applicable. The fission fragments result from U E H x y z 239 Let 0 and 0 are energies in coordinate system ( , , ) and Pu have total kinetic energy 20–60 MeV less than and system (, , ) before emission of light energy, further Q-value (200 MeV) of reaction predicted by E = mc2 E1 and H1 are the energies of body in the both systems after [7, 8, 9]. Palano [10] has confirmed that mass of particle it emits light energy. Thus Einstein wrote various equations Ds (2317) has been found more than current estimates based as Energy of body in system (x, y, z) upon E = mc2. Also E = mc2 does not give consis- tent results in explaining the binding energy, as it violates the E0 = E1 + 0:5L + 0:5L = E1 + L; (2) universal equality of masses of nucleons. All these facts can be explained by E = Ac2M with Energy of body in system (, , ) 2 value of A less or more than one. E = mc is not con- h v v i H = H + 0:5 L 1 cos + 1 + cos (3) firmed in many processes such chemical reactions, atom 0 1 c c 76 Ajay Sharma. The Generalized Conversion Factor in Einstein’s Mass-Energy Equation July, 2008 PROGRESS IN PHYSICS Volume 3 where = 1=[1 v2=c2]1=2; 3 The conversion factor between mass-energy other than c2 is also supported by Einstein’s derivation H0 = H1 + L; (4) under general conditions or (H0 E0) (H1 E1) = L ( 1) : (5) As already mentioned Einstein’s September 1905 derivation of L = mc2 is true under special or ideal conditions (se- Einstein calculated, kinetic energy of body before emis- lected values of parameters is taken) only, this aspect is stud- 2 sion of light energy, K0(mbv =2) and kinetic energy of body ied critically with details by the author [23–36] discussing 2 after emission of light energy, K(mav =2) as those aspects which have not been raised earlier. Thus the 0 1 value of conversion factor other than c2 is also supported 1 from Einstein’s derivation under general conditions (all pos- @ q A K0 K = L 1 (6) sible values of variables). The law or phenomena of inter- 1 v2 c2 conversion of mass and energy holds good in all cases for all bodies and energies under all conditions. Einstein considered the velocity in classical region thus ap- 2 In the derivation of L = mc there are FOUR vari- plying binomial theorem, ables e.g. 2 4 v 3v (a) Number of waves emitted, K0 K = L 1 + + + 2c2 8c4 l (7) (b) magnitude of light energy, 15v6 105v8 + + + ::: 1 : (c) Angle at which light energy is emitted and 48c6 384c8 (d) Uniform velocity, v. Further Einstein quoted [16] “Neglecting magnitudes of Einstein has taken special values of parameters and in fourth and higher orders, we may place” general for complete analysis the derivation can be repeated v2 with all possible values of parameters i.e. under general con- K K = L (8) 0 2c2 ditions taking in account the momentum conservation (which is discussed in next sub-section). v2 v2 v2 M M = L (9) (A) The body can emit large number of light waves but Ein- b a 2 2 2 2c stein has taken only TWO light waves emitted by lumi- or 2 2 nous body. L = (Mb Ma) c = mc ; (10) (B) The light waves emitted may have different magnitudes or Mass of body after emission (Ma) = Mass of body before but Einstein has taken EQUAL magnitudes 2 emission (Mb L=c ). (C) Body may emit large number of light waves of different Now replacing L (light energy) by E (total energy or ev- magnitudes of energy making DIFFERENT ANGLES ery energy) Einstein wrote (other than 0 and 180) assumed by Einstein. 2 2 v c E = (Mb Ma) c = mc (11) (D) Einstein has taken velocity in classical region ( ) has not at all used velocity in relativistic region. If ve- or Mass of body after emission (Ma) = Mass of body before locity is regarded as in relativistic region (v is compa- 2 emission (Mb E=c ). rable with c), then equation for relativistic variation of Thus Einstein derived that conversion factor between mass with velocity i.e. mass and light energy is precisely equal to c2, this aspect Mrest is elaborated by Fadner [13]. But Einstein’s this derivation Mrel = q (12) v2 has been critically discussed by many such a Planck [14], 1 c2 Stark [15], Ives [16], Stanchel [17], Okun [18] and N. Ham- dan [19] etc. At the same time in some references [20, 21] it is taken in account. It must be noted that before Ein- is expressed that Einstein has taken hints to derive equation stein’s work this equation was given by Lorentz [37, E = mc2 and from existing literature without acknowledg- 38] and firstly confirmed by Kaufman [39] and after- ing the work of preceding scientists. Max Born [22] has ex- wards more convincingly by Bucherer [40]. Einstein pressed that Einstein should have given references of existing on June 19, 1948 wrote a letter to Lincoln Barnett [41] literature. and advocated abandoning relativistic mass and sug- Thus Einstein’s work on the topic has been critically an- gested that is better to use the expression for the mo- alyzed by scientists since beginning, in views of its scientific mentum and energy of a body in motion, instead of rel- and procedural aspects. ativistic mass. Ajay Sharma. The Generalized Conversion Factor in Einstein’s Mass-Energy Equation 77 Volume 3 PROGRESS IN PHYSICS July, 2008 It is strange suggestion as Einstein has used relativistic in mass is (0:000038077L=cv + L=c2) thus there is no def- mass in his work including in the expression of rela- inite value of decrease in mass in Einstein’s derivation. In tivistic kinetic energy [12] from which rest mass energy this case decrease in mass is more than as predicted by Ein- is derived. stein, hence again the conversion factor other than c2 is con- L / mc2 (E) In addition Einstein has assumed that body remains at firmed i.e.
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