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 = Ac2
M 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 = Ac2
M.

1 Introduction bomb explosions, volcanic reactions etc. Whatever may be the case
E = Ac2
M 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 = 0:0011
mc , L = 0:999988
mc 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 = Ac2
M, 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 = Ac2
M 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 = Ac2
M 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. . Like this there are many examples rest before and after emission of light energy. But the of this type. body may be at rest i.e. v = 0, velocity may be in clas- (b) The central equation in Einstein’s derivation is Eq. (1) sical region and velocity may be in relativistic region and binomial theorem is equally applicable to it at any (v  c), the law of inter-conversion of mass and energy stage i.e. in the beginning or end. Einstein applied bi- holds good under all conditions. nomial theorem in the end and obtained L =
mc2 , In electron-positron annihilation, the material particles but the same equation is not obtained if binomial theo- are in motion before and after annihilation. In mate- rem is applied in the beginning. The binomial theorem rialization of energy, a gamma ray photon is converted is simply a mathematical tool and its application at any to electron positron pair, which move in opposite direc- stage should not affect results i.e. make or mar equation tions to conserve momentum. In nuclear fission and fu- L =
mc2. sion particles remain in motion in the process of mass The reason is that typical nature of derivation and Eq. (1) energy inter conversion. The thermal neutron which is different from other relativistic equations. The energy is causes fission has velocity 2185 m/s. scalar quantity and independent of direction but Eq. (1) is di- rectional in nature due to angle . In contrast if binomial 4 L /
mc2 is mathematically consistent in Einstein’s theorem is applied to Relativistic Kinetic Energy in the be- derivation, under general conditions ginning or at the end then result is same i.e. classical form 2 of kinetic energy (mrestv =2). So there is inconsistency in Under general conditions (all possible values of variables) the applications in this case. value of conversion factor other than c2 can be easily justified Applying binomial theorem to Eq. (1) and repeating the mathematically in Einstein’s derivation [23–36]. This aspect calculations as Einstein did, altogether different results are is not touched by the preceding authors [13–21]. obtained,   (a) In Einstein’s derivation if one wave is regarded as to  v  v2 3v4   ` = ` 1 cos  1 + + + ::: : (16) form angle 0.5 rather than 0 then c 2c2 8c4

H0 = H1 + 0:5 L  Here v=c  1, hence v2=c2 and higher terms can be ne- h v   v i (13)  1 cos 0:5 + 1 cos 180 ; glected. Thus c c  v  ` = ` 1 cos  c or  v  H = H + L 1 + 0:000018038 ; or 0 1 c (H E ) (H E ) = 0 ; or 0 0 1 1 v v2 or K0 K = 0:000019038lL + L ; c 2c2 Kb Ka = 0 ;

L L or
m (Mb Ma) = 0:000038077 + ; (14) 1 1 cv c2 2 2 Mb v Ma v = 0 ; or 2 2
mc2 L = = 0:000876
mc2; (15) or M = 1141 Mass of body before emission ( b) (17) = Mass of body after emission (Ma):
L /
mc2: Thus light energy is being emitted, but under this condi- Further, Ma (mass after emission of light energy) = Mb (mass tion Einstein’s this derivation does not provide any relation- before emission of light energy): 0:000038077L=cv = L=c2 ship (equality or proportionality) between mass annihilated in (14). and energy created. Similar is the situation if velocity v = 0. According to Einstein if body emits two light waves of en- Hence Einstein’s derivation gives decrease in mass of body ergy 0:5L each in opposite directions then decrease in mass equal to L=c2 only under certain conditions. Thus in this case is given by Eq. (10) i.e.
m = L=c2 and in this case decrease derivation is not valid.

78 Ajay Sharma. The Generalized Conversion Factor in Einstein’s Mass-Energy Equation July, 2008 PROGRESS IN PHYSICS Volume 3

Sr. No Values of various parameters WhetherL =
mc2 or L /
mc2 1 0:5L, 0:5L,  = 0 ,  = 180 L =
mc2 2 0:5L, 0:5L,  = 0:5 ,  = 180 L =
mc2=901 or L /
mc2 3 0:5001L, 0:49999L,  = 0,  = 180 L = 0:9999988
mc2 or L /
mc2 4 0:5L, 0:5L,  = 0 ,  = 180 but v = 0 No relation between L and
m 5 0:5L, 0:5L,  = 0,  = 180 No relation between L and
m but Binomial Theorem is applied in beginning. 6 For other energies than light Equations not considered

Table 1: Einstein’s Sep 1905 derivation gives L =
mc2 under certain conditions and L /
mc2 under general conditions

(c) Let the body emits two light waves of slightly different (viii) energy emitted in cosmological and astrophysical phe- energies i.e. 0:5001L and 0:4999L in opposite direc- nomena, (ix) energy emitted volcanic reactions, (x) energies tions and other parameters remain the same as assumed co-existing in various forms etc., etc. by Einstein. In this case Now Eq. (1) i.e.
 v  v   1 c cos  H0 = H1 + 0:4999 L 1 cos 0 + ` = ` q c   (18) v2 v 1 2 + 0:5001 L 1 cos 180 : c c is put forth for light energy by Einstein in June 1905 paper Now proceeding in the same way as Einstein did (` is light energy in moving frame), it is not meant for other v v2 possible energies as quoted above. K K = 0:0002L + L (19) Einstein never justified Eq. (1) for all the energies cited 0 c 2c2 or above. The parameters used in Einstein’s equation are de- fined for light energy only, not for all the energies. Thus by
m = Mass of body before emission(Mb) this derivation L =
mc2 is derived under special conditions Mass of body after emission(Ma); for light energy only and replacing L by E in Eq. (10) is not L L = 0:0004 + (20) justified. cv c2 There are evidences that Einstein worked hurriedly in or L L other case also e.g. in theory of static universe the introduc- M = 0:004 + M tion of cosmological constant proved to be incorrect and Ein- a cv c2 b stein accepted the mistake later as quoted by Gamow [42]. or 2
mc2 The various cases when
E /
mc is justified are shown
L = c : in Table 1. 0:0004 v + 1 The velocity v is in classical region, say 10 m/s, 5 Conservation of momentum in general cases   L =
mc2 0:000083 ; (21) The momentum is conserved irrespective of the fact that body remains at rest or recoils or tends to recoil after emission of
L /
mc2: light energy [43]. The law of conservation of momentum Thus,
E /
mc2. Hence conversion factor other than can be used to calculate the velocity of recoil in this case c2 follows from Einstein’s derivation under general condi- also. Let the body of mass 10 kg emits two waves of en- _ tions. ergy in visible region of wavelength 5000A it corresponds to 19 energy 7:951210 J. This energy is emitted in two waves (d) Energy emitted in various reactions. In his September 19
L =
mc2 i.e., as obvious, 0:5001L (3:9763951210 J) and 0:4999L 1905 paper Einstein derived Eq. (10) i.e. 19 3:9748048810 and then replaced L (light energy) by E (total energy) ( J). Applying the conservation of mo- mentum [43] the recoil velocity, recoil momentum and recoil and speculated 32 31
E =
mc2: (11) kinetic energy comes out to be 5:310 m/s, 5:310 62 kgm/s and 1:40410 J respectively. This recoil velocity In Eq. (11) E stands for all possible energies of the uni- (Vr) will change the uniform velocity v as Vr + v, but it will verse e.g.: (i) sound energy, (ii) heat energy, (iii) chemical not make any difference to final result of change in mass as in energy, (iv) nuclear energy, (v) magnetic energy, (vi) electri- Eq. (21), due to negligible value of Vr [27]. Hence in the law cal energy, (vii) energy emitted in form of invisible radiations, of conservation of momentum is obeyed in this case also.

Ajay Sharma. The Generalized Conversion Factor in Einstein’s Mass-Energy Equation 79 Volume 3 PROGRESS IN PHYSICS July, 2008

6 Experimental feasibility with conversion factors other less than predicted by E =
mc2 i.e. E /
mc2 is feasible, than c2 it is an open possibility unless ruled out.

(a) Dirac [44] was one of the first physicists to suggest that, Reactions in nuclear physics in connection with his theory of large numbers, fundamental (ii) Less efficiency: The efficiency of the nuclear weapons as dimensional constants may vary in time during the expansion well as nuclear reactors is far less than the theoretical value of the universe. The idea of variation of the speed of light predicted by E =
mc2. Robert Serber (member of first is suggested in various cosmological models [45, 46] and has American team entered Hiroshima and Nagasaki in Septem- been the subject of attention by physicists in investigations of ber 1945 to assess loses), has indicated [51] that the effi- extra dimensions, strings and branes [47]. Webb [48] has re- ciency of “Little Boy” weapon (U235, 49 kg) that was used ported variations in fine structure constant over cosmological against Hiroshima was about 2% only. It is assumed that time scales and hence variations in c. This suggestion implies all the atoms don’t undergo fission, thus material is wasted. 2
E /
mc . But no such waste material is specifically measured quantita- (b) Einstein has derived L =
mc2 (conversion factor be- tively. Thus the waste material (nuclear reactor or weapon) tween mass and energy is precisely equal to c2) under the ex- must be measured and corresponding energy be calculated, tremely special or ideal conditions , which are even difficult and it must quantitatively explain that why efficiency is less. to attain practically. The work of scientists before Einstein It may require the measurements of all types of energies (may also justifies
E /
mc2. co-exist in various forms) in the processes and experimental This discussion does not confront with existing experi- errors. Until such experiments are specifically conducted and mental situation but addresses those theoretical and exper- E =
mc2 is confirmed,
E /
mc2 is equally feasible. imental issues for which
E =
mc2 is not analyzed yet. (iii) Less energy: In laboratory it is confirmed [7, 52, 53] The mass energy inter-conversion equation, with conversion that using thermal neutrons the total kinetic energy (TKE) of factor equal to c2 i.e.
E =
mc2 has been confirmed in fission fragments that result from of U235 and Pu239 is 20– nuclear physics and is also basis of nuclear physics. Even el- 60 MeV less than Q-value (200 MeV) of reaction predicted ementary units of atomic mass (1 amu) or and energy (eV) by
E =
mc2. This observation is nearly four decades are based upon it. Thus it will remain standard in measure- old. Bakhoum [7] has explained it on the basis of equation ments as seven days in a week; its validity in this regard is not H = mv2 (energy emitted is less than E =
mc2). Hence doubted at all. here E /
mc2 is justified. The aim is to discuss experimentally those phenomena in (iv) More mass: Palano [10] has confirmed that mass of par- which
E =
mc2 is not applied yet. The mass energy con- ticle Ds (2317) has been found more than current estimates version processes are weird in nature and all have not been based upon
E =
mc2. Thus in this case E /
mc2 is at all studied in view of
E =
mc2. The conversion fac- justified. tor other than c2 is discussed for such elusive cases, not for (v) Binding energy and mass defect in deuteron: There are those it is already confirmed. Hence there is no confrontation two inherent observations [23, 28, 29] about nucleus: firstly, with the established experimental situation at all, but aim is masses of nucleons are fundamental constants, i.e. they are to open a mathematical front (
E /
mc2) for numerous the same universally (inside and outside the nucleus in all experimentally unstudied phenomena in nature. This devel- cases); and secondly nuclei possess Binding Energy opment can be discussed as below. (BE =
mc2) owing to a mass defect. To explain these ob- servations, in the case of the deuteron (BE = 2:2244 MeV), 7 Most abundant chemical reactions the mass defect of nucleons must be 0.002388 amu or about 0.11854% of the mass of nucleons, i.e., nucleons must be (i) Unconfirmed chemical reactions. When Einstein de- lighter in the nucleus. This is not experimentally justified, rived E =
mc2, chemical reactions were the most abundant as masses of nucleons are universal constants. Thus observa- 2 sources of energy in nature. Till date E =
mc2 is not con- tions and predictions based upon
E =
mc are not justi- 2 firmed in the chemical reaction and the reason cited for this is fied, hence
E /
mc is equally feasible. that equipments are not enough sensitive [49, 50]. Consider burning of 1kg straw or paper or petrol in controlled way i.e. 8 Mathematical form of extended equation in such a way that masses, ashes, gases and energy produced can be estimated. Even if 0.001 kg or 1gm of matter is annihi- Until E =
mc2 is not precisely confirmed experimentally 13 lated then energy equal to 910 J (can drive a truck of mass in ALL CASES , it is equally feasible to assume that the en- 7 2 2 1000 kg to distance of 910 km) will be produced. Until ergy emitted may be less than E =
mc (or E /
mc ). the equation is not confirmed in such reactions, then scien- It does not have any effect on those cases where E =
mc2 tifically E =
mc2 may not be regarded as precisely true in is confirmed, it simply scientifically stresses confirmation of such cases. It is equally possible that energy emitted may be E =
mc2 in all cases. Also when reactants are in bulk

80 Ajay Sharma. The Generalized Conversion Factor in Einstein’s Mass-Energy Equation July, 2008 PROGRESS IN PHYSICS Volume 3

amount and various types of energies are simultaneously constant, just like universal gravitational constant G and k in emitted and energies may co-exist. Thus both the possibilities Newton’s Second Law of Motion. The reason is that mass en- are equally probable until one is not specifically ruled out. In ergy inter-conversion are the bizarre processes in nature and view of weirdness in reactions emitting energy in universe, not completely studied. some theoretical inconsistencies in the derivation and non- Now consider the case that when mass is converted into availability of data, one can explore the second possibility energy. Let in some conversion process mass decreases from even as a postulate. All the equations in science are regarded Mi(initial mass) to Mf (final mass), correspondingly energy as confirmed when specifically justified in all experiments increases from Ei (initial energy) to Ef (final energy). The time and again. The reactions involving inter-conversion of Eq. (22) gives infinitesimally small amount of energy dE cre- mass and energy are utmost diverse, weird and new phenom- ated on annihilation of mass dm. To get the net effect the ena are being added regularly, thus E = mc2 needs to be con- Eq. (22) can be integrated similarly Einstein has obtained the firmed in all cases. Thus in general, in view of above propor- relativistic form of kinetic energy in June 1905 paper [18] tionality it may be taken in account as Z Z dE = Ac2 dm ; dE / c2dm :

The above proportionality dE / c2dm can be changed Initial limit of mass = Mi, Initial limit of Energy = Ei , into equation by introducing a constant of proportionality. Final limit of mass = Mf , Final limit of Energy = Ef . The inception of proportionality constant is consistent with centuries old perception of constant of proportionality in Initially when mass of body is Mi, then Ei is the ini- physics since days of Aristotle and Newton. In second law of tial energy of the system. When mass (initial mass, Mi) motion (F = kma) the value of constant of proportionality, k is converted into energy by any process under suitable cir- is always unity (like universal constant) i.e. F = ma. When cumstances the final mass of system reduces to Mf . Conse- more and more complex phenomena were studied or values of quently, the energy of system increases to Ef the final en- constants of proportionality were determined then it showed ergy. Thus Mf and Ef are the quantities after the conversion. dependence on the inherent characteristics of the phenomena. Hence, Eq. (22) becomes In case constant of proportionality varies from one situation 2 to other then it is known as co-efficient of proportionality e.g. Ef Ei = Ac (Mf Mi) (23) co-efficient of thermal conductivity or viscosity etc. Thus re- or 2 moving the proportionality between dE and c2dm, we get
E = Ac
m (24) 2 dE = Ac2dm ; (22) Energy evolved = Ac (decrease in mass): (25) If the characteristic conditions of the process permit then A where is (a co-efficient) used to remove that sign of propor- whole mass is converted into energy i.e. after the reaction no tionality; it depends upon inherent characteristics of the pro- mass remains (Mf = 0) cesses in which conversion of mass to energy takes place and 2 it is dimensionless. It has nature precisely like Hubble’s con-
E = Ac Mi (26) stant (50 and 80 kilometers per second-Megaparsec, Mpc) or 3 6 In this case energy evolved is negative implies that energy coefficient of viscosity (1:0510 poise to 19:210 poise) is created at the cost of annihilation of mass and the process or co-efficient of thermal conductivity (0.02 Wm1K1 to is exo-energic nature (energy is emitted which may be in any 400 Wm1K1) etc. Thus, in fact Hubble’s constant may be form). Energy is scalar quantity having magnitude only, thus regarded Hubble’s variable constant or Hubble’s coefficient, no direction is associated with it. as it varies from one heavenly body to other. If “A” is equal Thus the generalized mass-energy equivalence may be to one, then we will get dE = dmc2 i.e. same as Einstein’s stated as equation. In Eq. (22) “A” is regarded as conversion factor as it de- “The mass can be converted into energy or vice-versa scribes feasibility and extent of conversion of mass into en- under some characteristic conditions of the process, 2 ergy. For example out of bulk mass, the mass annihilated to but conversion factor may or may not always be c 16 2 2 2 energy is maximum in matter-antimatter annihilation, appar- (910 m /s ) or c .” ently least in chemical reactions, undetermined in volcanic A reactions and cosmological reactions. It (the co-efficient ) 9 Applications of generalized mass energy inter conver- depends upon the characteristic conditions of a particular pro- sion equation
E = A c2
m cess. It may be constant for a particular process and varies for the other depending upon involved parameters or experi- (i) It is already mentioned in section (3) that if 0.001 kg or 13 mental situation. Thus “A” cannot be regarded as universal 1gm of matter is annihilated then energy equal to 910 J

Ajay Sharma. The Generalized Conversion Factor in Einstein’s Mass-Energy Equation 81 Volume 3 PROGRESS IN PHYSICS July, 2008

7 (can drive a truck of mass 1000 kg to distance of 910 km) Thus mass defect of deuteron must be infinitesimally will be produced. Such or similar predictions are not experi- small, only then masses of nucleons are same inside nucleus mentally confirmed and energy emitted can be found less than and outside nucleus. Also binding energy must be 2.2244 predictions. MeV as experimentally observed. Both these experimentally 13 2 Let the energy observed is 4:510 J corresponding to confirmed facts can be explained with help of
E =Ac
m. mass annihilated 0.001 kg , then value of A from
E = Let in this case the mass defect is negligibly small i.e. 2 13 40 = Ac
m will be 0.5 i.e. 2:38810 amu or 3:965310 kg. Then value of A 13
E 4:510 J (coefficient of proportionality or mass energy inter conver- A = = = 0:5: (27) 1010 c2
m 91016 sion coefficient) is i.e. for annihilation of infinitesimally small mass exceptionally large amount of energy is liberated. Thus in this case mass energy inter-conversion equation Thus 13 becomes
E 3:563410
E = 0:5c2
m : (28) A = = = 1010 ; (33) c2
m 91016 3:96531040 (ii) Let the TKE of fission fragments of U235 and Pu239 is 10 2 175 MeV (as experimentally it is observed less), instead of
E = 10 c
m : (34)
E = expected 200 MeV. It can be explained with help of (v) Webb [48] has reported results for time variability of the = Ac2
m A with value of is equal to 0.875 i.e. fine structure constant or Sommerfeld fine structure constant
E 175 ) A =
m = = 0:875 : (29) ( using absorption systems in the spectra of distant quasars. c2 200 The variation in magnitude of alpha has been observed as 235 239 Thus energy of fission fragments of U and Pu is
( then now) 5 = = 0:910 : (35) given by now
E = 0:875c2
m : (30) According to CODATA currently accepted value of alpha 3 Thus value of A less than one is justified experimentally ( now) is 7:29735210 . Hence from Eq. (35), in this case. = 0:007296 : (36) (iii) The anomalous observation of excess mass of then 2 Ds(2317) can be understood with help of
E = Ac
m, as Now corresponding to the reduced value of ( then = mass of the observed particle is found more [10] than pre- = 0.007296) the the speed of light can be determined from E =
mc2 A dictions of . In this case value of will be less equation e2 than one. For understanding consider energy equal to 106 J is c = (37) then 2 "h converted into mass, then corresponding mass must be then 8 11 2:99410 1:1110 kg. We are considering the case that mass is as m/s (where all terms have usual meanings). Cur- 11  8 found more than this. Let the mass be 1:1210 kg. The rently accepted value of the speed of light is 2:99729 10 m/s. value of A this case is 0.992, as calculated from
E = To explain the energy emitted with this value of the speed 2 = Ac2
m i.e. of light is the value A (
E = Ac
M) 6 10 c2 A = = 0:992 : (31) A = = 1:001 : (38) 10:8105 2 cthen Thus in this mass energy inter conversion equation be- Thus in this case mass energy inter conversion equation comes becomes 2
E = 0:992c2
m or
m = 1:008
E: (32)
E = 1:001c
m : (39)
E /
mc2 Thus corresponding to small energy more mass is emitted. Hence has both experimental and theo- (vi)
E = Ac2
m is useful in explaining the binding energy retical support, with emergence of new experimental data its 13 (2.2244MeV or 3:5590410 J), mass defect (0.002388 significance will increase. 3 amu or 2:38810 amu) and universal equality of mass of nucleons (mn = 1.008664 amu, mp = 1.006082 amu). Ob- Acknowledgements viously neutron and protons contribute equally towards the mass defect (0.001194 amu), then mass of neutron inside nu- The author is highly indebted to Stephen J. Crothers, cleus must be 1.00747 amu (mass outside nucleus i.e. in Free Dr. E. Bakhoum and Dr. N. Hamdan for critical discussions State is 1.008664 amu). Similarly corresponding mass of pro- of the topic and providing related materials. Also the au- ton in the nucleus must be 1.006082 amu (mass of proton thor is highly indebted to Dr. T. Ramasami, Secretary DST outside nucleus 1.007274 amu). But decrease in mass of nu- for arranging discussion on the topic with Physics Advisory cleons inside nucleus is not justified, as masses of nucleons Committee and providing financial assistance. are universally same [23, 28, 29]. Submitted on March 10, 2008 / Accepted on May 19, 2008

82 Ajay Sharma. The Generalized Conversion Factor in Einstein’s Mass-Energy Equation July, 2008 PROGRESS IN PHYSICS Volume 3

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