2Einstein‟s derivation of E=，mc also predicts ，E ?

2，mc

A SHARMA

ABSTRACT Web has reported variations in fine structure constant (hence in speed of light) and have

2direct effect on E=，mc. Depending upon recent WMAP data, Erickcek has suggested that time could exist before big bang and universe may have created from empty space. These aspect is also investigated in view of Einstein‟s Sep. 1905 paper which is origin of light energy-mass inter conversion equation

22 and interesting results are obtained. Now it follows that s only true under special L！，mcL！，mc

222LEconditions (from, is speculated) i.e. in general cases ? ， ( ? L！，mcE！，mcmc

22 ) also follows. Thus an alternate equation has been suggested, which implies ，mc，E！Ac，m

that energy emitted on annihilation of mass (or vice versa) can be equal, less and more than predicted by

22. However is yet unconfirmed in the most abundant chemical reactions. The total E！，mcE！，mc

235239kinetic energy of fission fragments of U or Pu is found experimentally 20-60 MeV less than Q-value

22predicted by ，mc, it is explainable with with value of A less than one. Energy emitted by ，E！Ac，m

45Gamma Ray Bursts (most energetic event after Big Bang) in duration 0.1-100s, is 10J , can be explained

2with help of . The mass of particle Ds (2317) discovered at SLAC, have mass lower than ，E！Ac，m

2current estimates; it can be explained with value of A more than one. , is capable of ，E！Ac，m

55 explaining the formation of primitive atom (10kg ) in state of singularity from dwindling amount of energy -50-122(10J or less) i.e. expceptionally-2 small and A is 1.11?10 J or less. This perception implies

gravitational energy is another form of mass (like other energies) and has been originated simultaneously with mass.

222EKey Words : mass, energy Interconversion, , ? , E！，mc，mc，E！Ac，m

1.INTRODUCTION

The word „energy‟ derives from energeia which was coined by Aristotle for first time [1].

2Leibniz [1646-1716] put forth idea of vis viva (from the latin living force) as (m as mv

2mass and velocity of body) and stated that it is conserved [2-3]. In 1807, Thomas v

Young [1773 -1829] was first to use term „energy‟ instead of vis viva [4-5]. Gustave

Coriolis [1792-1843] was first to define work as product of force and distance; in 1829 he

Fundamental Physics Society. His Mercy Enclave Post Box 107 GPO Shimla 171001 India

ajay.sharmaa@rediffmail.com

2Mvrestdescribed kinetic energy as [4-5]. Further mass is quantity of matter contained 2

in the body, the real understanding of mass started when Newton defined second law of motion in the Principia. [6]. Newton also stated inter conversion of light energy to mass [7], thus initiated important debate on this issue.

Mass energy inter-conversion processes are the oldest in nature and constitute the basis of various phenomena. Further the energies have various forms (e.g. sound energy, heat energy, chemical energy, energy emitted volcanic reactions nuclear energy, magnetic energy, electrical energy, energy emitted in form of invisible radiations, energy emitted in cosmological and astrophysical phenomena energies co-existing in various forms etc.) which are converted into mass. At different times various scientists have studied this significant topic in different ways and study is continuous process.

Aristotle [384-322 BC] believed that all matter on earth consisted of four pure substances or elements, which were earth, air, fire, and water [1]. Here fire may be regarded as energy.

Antoine Lavoisier (1743-1794) French Chemist was the first to formulate a law of conservation of matter in chemical reactions i.e. matter can neither be created nor be destroyed but can be transformed from one to other form [8].

Newton [7] has quoted in his book „Opticks‟ in 1704 that "Gross bodies and light are

convertible into one another...", It implies that energy is other form of mass. Neither Lavoisier nor Newton gave any mathematical equation relating to mass and energy, hence the deduction is qualitative only.

S. Tolver Preston [9] proposed that a vast amount of energy can be produced from matter in his book (now rare book) Physics of the Ether in 1875. Preston determined

that one grain (one grain = 0.0647989 grams) could lift a 100,000-ton object up to a

2Eheight of 1.9 miles. This deduction yields ? . ，mc

Jules Henri Poincaré [10,11] in 1900 applied the calculations in a recoil process and

E?：creached at the conclusion in the form, mv = From the viewpoint of dimensional (，2c)?

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E?：analysis, takes on the role of a „mass‟ associated with radiation, which yields (，2c)?

2. E！mc

22Olinto De Pretto [12] speculated ., implying that when v = c, then . E！mcE！mv

2( Leibniz‟s vis viva) becomes , in 1903-04. E！mc

Fritz Hasenohrl [13, 14] in 1904, concluded “to the mechanical mass of our system

8E must be added an apparent mass which is given by, m=where E is the energy of 23c

the radiation.” In a later paper he further improved result that mass exchanged is, m=

4E2E Thus in this case also ? ，mc23c

2Ebenezer Cunningham [15] has further improved Hasenohrl„s equation as . E！，mc

Frederick Soddi [16] and M. Henri Becquerel both have predicted that in radioactive emissions the mass of body decreases i.e. energy of radiations is at the cost of mass. Thus higher the decrease in mass more would be energy of radiation and no conversion

E，mfactor was given, this inference is like above one. It also corresponds to ?

2Eor ? . ，mc

Einstein [17] derived under certain conditions the conversion factor for mass and light

2energy is precisely equal to . This perception of light energy –mass inter conversion c

2was given by Newton. Then Einstein generalized conversion factor for „every energy‟ c

in speculative way.

Einstein [17] perceived that let there be a luminous body at rest in co-ordinate system (x, y, z). The system (ξ, η, ζ ) is in uniform parallel translation w.r.t. system (x, y, z); and origin of system (ξ, η, ζ ) moves along x-axis with relative velocity v. Let a system of

plane light waves have energy ℓ relative to system (x, y, z), the ray direction makes angle φ with x-axis of the system (ξ, η, ζ ). The quantity of light measured in system [ξ, η, ζ] has the energy [17-18].

v??~1，cos，?c??？;;！ (1) 2v1，2c

Einstein has given eq.(1) in his paper known as Special Theory of Relativity [18] and called eq.(1) as Doppler principle for any velocities whatever.

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Let and Hare energies in coordinate system (x, y, z) and system (ξ, η, ζ ) before E00

emission of light energy, further and are the energies of body in the both EH11

systems after it emits light energy. Thus Einstein wrote various equations as Energy of

body in system ( x,y,z )

= + 0.5L +0.5L = + L (2) EEE 011

Energy of body in system (ξ, η, ζ )

vvcos~cos~H = H + 0.5 β L{1 – + ( 1+ } (3) o1cc

1where ：！2v1，2c

H = + β L H01

(4)

LHOr, ( – E) – ( – ) = [β –1] (5) HE 0011

2MvbKEinstein calculated, kinetic energy of body before emission of light energy, ( ) 02

2MvaKand kinetic energy of body after emission of light energy, () as 2

1KLK – = {– 1} (6) 02v1，2c

Einstein considered the velocity in classical region thus applying binomial theorem,

24?：vv(，KLK – = (7) 1？？3？.....................，1024(，cc28)?

Further Einstein quoted [18]

Neglecting magnitudes of fourth and higher orders, we may place.

2vKLK – = (8) 022c

222MvMvvbaL – = (9) 2222c22Lor = (M– M) = (10) ，mccb a

2MMor Mass of body after emission () = Mass of body before emission () – L /c. Now ba

replacing L ( light energy) by E (total energy or every energy) Einstein wrote:

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22or E = (M– M)c = (11) ，mc a b

2or Mass of body after emission () = Mass of body before emission (M) – E /c Mba

Thus Einstein derived that conversion factor between mass and light energy is precisely

2equal to c , this aspect is elaborated by Fadner [19]. But Einstein‟s this derivation has

been critically discussed by many such a Planck [20], Stark [21 ], Ives [22], Stanchel [23], Okun [24] and N. Hamdan [25] etc. At the same time in some references [26-27]

2it is expressed that Einstein has taken hints to derive equation and from E！，mc

existing literature without acknowledging the work of preceding scientists in some of his papers [29-30]. Max Born [28] has expressed that Einstein should have given references of existing literature like other scientists as Planck [20, 29-30] mentioned Einstein‟s work.

Max Planck [29-30] in 1907 made an in-depth investigation of the energy "confined" within a body, but he did not use Einstein‟s approach at all. Planck derived an

Eexpression m-M= , for heat energy and mass and interpreted that ”The inertia mass 2c

of body is altered by absorption or emission of heat energy. The increments of mass of body are equal to heat energy divided by square of speed of light.”

Planck acknowledged Einstein‟s previous derivation but did not agree with correctness

of Einstein‟s derivation.

Ezzat Bakhoum [31-32] has proposed that a total energy equation that satisfies the Compton-de Broglie wave mechanics as well as theory of special relativity is

2 (12) H！mv

where H is total energy of particle, m is its mass and v is velocity. Bakhoum has put forth

22that (or widely regarded as ) cannot do so simultaneously. The H！mvE！mc

perception is applied to explain various phenomena. Thus again the conversion factor

2between mass and energy is other than c is suggested. Hence,

2 ，E ? ，mc

The similar ideas have also expressed by Hamdan [25] and Olinto De Pretto [12] has

22also suggested: from E！mcE！mv

22.0 THE CONVERSION FACTOR BETWEEN MASS-ENERGY OTHER THAN C IS

ALSO SUPPORTED BY SPECIAL CONDITIONS USED IN EINSTEIN‟S DERIVATION.

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2As already mentioned Einstein‟s Sep. 1905 derivation of is true under L！，mc

special conditions only, this aspect is studied critically with details by the author [33-45] discussing those aspects which have not been raised earlier. It automatically implies that

2cconversion factor between mass and energy is under those conditions only. Thus the

2cconversion factor other than is possible, under those conditions which are not taken in account in Einstein‟s Sep. 1905 derivation. Thus the value of conversion factor other

2cthan is also supported from Einstein‟s derivation. Einstein‟s derivation [17-18] is only

for light energy mass inter-conversion, as Einstein has considered only light energy in

2the derivation. In the derivation of there are FOUR variables e.g. L！，mc

(a) Number of waves emitted,

(b) l magnitude of light energy,

(c) Angle ~ at which light energy is emitted and

(d) Uniform velocity, v

Einstein has taken special values of parameters and in general for complete analysis the derivation can be repeated with all possible values of parameters taking in account the momentum conservation (which is discussed in next sub-section).

(A) The body can emit large number of light waves but Einstein has taken only TWO light waves emitted by luminous body.

(B) The light waves emitted may have different magnitudes but Einstein has taken EQUAL magnitudes.

(C) Body may emit large number of light waves of different magnitudes of energy making DIFFERENT ANGLES (other than 0º and 180º ) assumed by Einstein.

(D) Einstein has taken velocity in classical region (v<

MrestM = rel2v1，2c

is taken in account. It must be noted that before Einstein‟s work this equation was given by Lorentz [46-47] and firstly confirmed by Kaufman [48] and afterwards more convincingly by Bucherer [49]. Einstein on June 19, 1948 wrote a letter to Lincoln Barnett [50] and advocated abandoning relativistic mass and suggested that is better to

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use the expression for the momentum and energy of a body in motion, instead of relativistic mass.

It is strange suggestion as Einstein has used relativistic mass in his work including in the expression of relativistic kinetic energy [18] from which rest mass energy is derived [51-53].

(E) In addition Einstein has assumed that body remains at rest before and after emission of light energy. But the body may be at rest i.e. v = 0, velocity may be in classical region and velocity may be in relativistic region (v~c), the law of inter-conversion of mass and energy holds good under all conditions.

In electron-positron annihilation, the material particles are in motion before and after annihilation. In materialization of energy, a gamma ray photon is converted to electron positron pair, which moves in opposite directions to conserve momentum. Whereas in annihilation of mass, electron-positron pair combine to form a gamma ray. In nuclear fission and fusion particles remain in motion in the process of mass energy inter conversion. The thermal neutron which causes fission has velocity 2185m/s.

The law or phenomena of inter-conversion of mass and energy holds good in all cases for all bodies and energies under all conditions. But Einstein has taken special

2values of parameters under special conditions to derive equation i.e. to L！，mc

2 cconfirm that conversion factor between mass and energy is .

2L2.1 ? IS MATHEMATICALLY CONSISTENT IN EINSTEIN‟S DERIVATION, mc

UNDER GENERAL CONDITIONS.

In Einstein‟s derivation conversion factor between mass and energy is precisely equal to

2c and the derivation utilizes only special values of the involved variables. It itself

2cimplies that conversion factor is precisely equal under special conditions otherwise

2cits value will vary. The value of conversion factor other than can be easily justified

mathematically in Einstein‟s derivation [33-45]. This aspect is not touched by the

preceding authors [19-32].

(a) In Einstein‟s derivation if one wave is regarded as to form angle 0.5º rather than 0º

then

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vvH = + 0.5 β L{( 1 – cos0.5º) + ( 1– cos 180º) } (13) Ho1cc

vor H = + β L [ 1+0.000019038] H 01c

2vv Kor K – = 0.000019038L +L 02c2c

LLMΔm (–M) = 0.000038077 + (14) ba2cvc

If v is 10m/s then

2，mc2L = = 0.000876 (15) ，mc1141

2，L ? ，mc

Further,

LLMM ( mass after emission of light energy) = – 0.000038077 – + (14) ba2cvc

According to Einstein if body emits two light waves of energy 0.5L each in opposite

Ldirections then decrease in mass is given by eq.(10) i.e. Δm = and in this case 2c

LLdecrease in mass is (0.000038077 + ) thus there is no definite value of 2cvc

decrease in mass in Einstein‟s derivation. In this case decrease in mass is more than as

2predicted by Einstein, hence again the conversion factor other than c is confirmed i.e.

2L ? ， Like this there are many examples of this type. mc

(b) The central equation in Einstein‟s derivation is eq.(1) and binomial theorem is equally applicable to it at any stage i.e. in the beginning or end. Einstein applied binomial

2theorem in the end and obtained, but the same equation is not obtained if L！，mc

binomial theorem is applied in the beginning. The reason is that typical nature of derivation and eq.(1) is different from other relativistic equations. The energy is scalar quantity and independent of direction but eq.(1) directional in nature due to angle ~. In

contrast if binomial theorem is applied to Relativistic Kinetic Energy in the beginning or

2Mvrestat the end then result is same i.e. classical form of kinetic energy . So there is 2

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inconsistency in applications. Applying binomial theorem to eq. (1) and repeating the calculations as Einstein did, altogether different results are obtained.

24vv3vℓ* = ℓ (1 – cos~ ) (1+ + +…………….) (16) 24c2c8c

22Here v/c<<1, hence v/c and higher terms can be neglected. Thus

vℓ* = ℓ (1 – cos~ ) c

HOr, ( – E) – ( – ) = 0 HE 0011

Or K – K= 0 aa

22MvMvba– = 0 22

Or,

MMass of body before emission () b

= Mass of body after emission (M) (17) a

Thus light energy is being emitted, but under this condition Einstein‟s this derivation does not provide any relationship (equality or proportionality) between mass annihilated and energy created. Similar is the situation if velocity v = 0. Hence Einstein‟s derivation

Lgives decrease in mass of body equal to only under certain conditions. Thus in all 2c

conditions this derivation is not valid.

(c) Let the body emits two light waves of slightly different energies i.e. 0.5001L and 0.4999L in opposite directions and other parameters remain the same as assumed by Einstein. In this case

vv o oH = H + 0.4999 β L [1 – cos 0] +0.5001 βL[1– cos 180] (18) 01cc

Now proceeding in the same way as Einstein did

2vvKK – = 0.0002L +L (19) 02c2c

MMOr, Δm = Mass of body before emission ()–Mass of body after emission () ba

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LL = 0.0004 + (20) 2cvc

LLMor M = 0.0004 – + ba2cvc

2，mcor L = 0.0004c？1v

2L = [0.000083] (21) ，mc

2 ，L ? ，mc

2 Thus eq.(21) means that emitted light energy is proportional to . It must be ，mc

noted that neither Newton nor Lavoisier has given any conversion factor between mass

and energy. Similar is the approach of Frederick Soddi [16] and M. Henri Becquerel. S.

Tolver Preston [9] and Fritz Hasenohrl [13, 14] has given the mass energy inter

2conversion equation in form of ，E ? . ，mc

(d) Energy emitted in various reactions. In Sep. 1905 paper Einstein derived eq.(10 )

2i.e. ，L = and then replaced L by E (total energy) and speculated ，mc

2，E = (11) ，mc

In eq.(11) E stands for all possible energies of the universe e.g

(i) sound energy, (ii) heat energy, (iii) chemical energy,

(iv) nuclear energy, (v) magnetic energy, (vi) electrical energy,

(vii) energy emitted in form of invisible radiations,

(viii) energy emitted in cosmological and astrophysical phenomena

(xi) energy emitted volcanic reactions

(x) energies co-existing in various forms etc. etc. Now eq.(1) i.e.

v??~1，cos，?c??？;; ！ 2v1，2c

is put forth for light energy by Einstein in June 1905 paper ( as L is light energy , L is

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