Derivation of E= mc 2 Revisited

Transcription

1 Apeiron, Vol. 18, No. 3, July Deriation of E=mc Reisited Ajay Sharma Fundamental Physics Society His ercy Enclae Post Box 107 GPO Shimla HP India Einstein s September 1905 paper in which L=mc (light energy-mass equation) is deried, is not completely studied; and is only alid under special conditions of inoled parameters. The origin of E=mc from L=mc is completely speculatie in nature. The factor c has been arbitrarily brought into the picture by Einstein. To obtain L =mc Einstein retained the term /c (compared to unity) without giing numerical alues to. If the alue of is considered in a typical classical region, 1cm/s say ( /c = is negligible) then the result is b = a. Thus the conersion factor c is arbitrarily brought into the picture as both the results i.e. L=mc and b = a are equally probable. Einstein deried L=mc under special conditions and speculated from it that E=mc without mathematical deriation. These are the limitations of the deriation. The same deriation also gies L mc or L = A mc, where A is a coefficient of proportionality. There are numerous alues of coefficients of proportionality in the existing physics. This is a mathematical critical analysis, hae no implications on the experimentally established status of L=mc (E=mc ). 011 C. Roy Keys Inc.

2 Apeiron, Vol. 18, No. 3, July Description and critical analysis of Einstein s Thought Experiment In Einstein s deriation the basic equation is 1 cos c (1) 1 c where is light energy emitted by a body in (x,y,z) frame, * is the light energy measured in (,, ) system, and is the elocity with which the frame or (,, ) system is moing. This is the equation for the Doppler principle for light for any elocity whatsoeer []. Einstein s perception [1]: Let a system of plane waes of light, referred to the (x, y, z) system of coordinates, possesses the energy l. Let the direction of the ray (the wae-normal) make an angle with the x-axis of the system [1]. Energy is a scalar quantity, haing magnitude only, but according to eq. (1) it also depends upon angle. Introduce a new system of co-ordinates (,, ) moing in uniform parallel translation with respect to the (x, y, z) system, and haing its origin of coordinates in motion along the x-axis with the elocity. Thus is the relatie elocity between system (x, y, z) and system (,, ). The body which emits light energy is considered stationary in the system (x,y,z) and also remains stationary after emission of light energy in the system (x,y,z). Let E 0 and H 0 be the energies in the (x, y, z) coordinate system and the (,, ) coordinate system respectiely, before emission of light energy. Furthermore, E 1 and H 1 are the energies of the body in both systems after it emits light energy. E i and H i include all the 011 C. Roy Keys Inc.

3 Apeiron, Vol. 18, No. 3, July energies possessed by the body in the two systems respectiely. The arious meanings of the E i s and H i s are shown in Table I. Table I. Energies emitted before and after emission by a body in Einstein s September 1905 deriation. Sr No System (x,y,z) at rest System(,, ) moing with elocity 1 Before Emission E 0 Before Emission H 0 After Emission E 1 After Emission H 1 Then Einstein assumed that the body emits two light waes, each of energy 0.5L, in the (x,y,z) system where the energy is E 0. Thus, Energy before Emission = Energy after emission +0.5L + 0.5L (1) E 0 = E 1 + Energy of the body in system (,, ) is where or 0.5L + 0.5L = E 1 + L () H 0 H1 0.5 L 1 cos 1 cos (3) c c 1 (4) 1 c H 0 H 1 L (5) 011 C. Roy Keys Inc.

4 Einstein maintained that Apeiron, Vol. 18, No. 3, July E H E L 1 H (6) b H E K C C a H1 E1 K1 C C Einstein defined C as an additie constant which depends on the choice of the arbitrary additie constants of the energies H and E. The arbitrary additie constant C is regarded as equal in both cases. Kinetic energy of the body before emission of light energy, K 0, b and the kinetic energy of the body after emission of light energy, K 1, a gies 011 C. Roy Keys Inc. 1 K 0 K1 L 1 (7) 1 c Einstein considered the elocity in the classical region, applying the binomial theorem thus, 4 K 0 K1 L (8) 4 c 8c Further Einstein [1] neglected magnitudes of the fourth ( 4 /c 4 ) and higher ( 6 /c 6, 8 /c 8.) orders so that,

5 or or Apeiron, Vol. 18, No. 3, July K 0 K1 L (9) c b a L (10) c L = ( ) b a c = mc (11) ass of body after emission ( a ) = ass of body before emission L b. (1) c Then Einstein generalized the result for eery energy and asserted that the mass of a body is a measure of the energy content (eery energy that is included in a collection). Fadner [11] has mentioned that in the paper Einstein neither wrote E= mc nor E in the paper. It is concluded that Einstein s statement means E = mc. It can be obtained by replacing L (light energy) by E (energy-content or eery energy). Einstein wrote, or E c mc (13) b a E ass of body after emission ( a ) = ass of body before emission b (14) c When energy is emitted the mass decreases. Thus Einstein did not differentiate between Light Energy and other energies in the deriation. 011 C. Roy Keys Inc.

6 Apeiron, Vol. 18, No. 3, July Typical comments regarding the classical region of elocity (not gien by Einstein). Einstein s deriation also offers the most mysterious situation in physics, explained below with the help of this equation, rest motion (15) 1 c Let the elocity be in classical region e.g. 10m/s (36 km/hr, i.e. ordinary speed of a motor ehicle), then there is no increase in the mass of the object when it moes with this elocity. The speed of an aeroplane is oer 400km/hr, and no increase in mass is obsered. 4 3 rest c 8c motion (16) (i) If the elocity of the (x,y,z) system is = 0, then motion = rest. (ii) If = 1cm/s (0.036 km/hr) then motion = rest [ ] (17) motion = rest + rest rest Here een the term rest is regarded as negligible compared to rest, and rest is further negligible; thus motion rest (18) Thus the term can be neglected only when both masses are equal. motion = rest [1+ /c +3 4 /8c 4 + ] = rest [ ] motion = rest + rest rest (19) 011 C. Roy Keys Inc.

7 Apeiron, Vol. 18, No. 3, July The mass of Earth remains the same i.e kg always. Thus here also the term /c ( ) is neglected compared to unity; or it implies that the term rest has to be neglected compared to rest. It implies increase in mass of the Earth of about %. If the term /c ( ) is neglected then (iii) Similarly the orbital elocity of the Earth is 30km/s or 3,0000m/s i.e. /c =10 4 thus motion [mass of earth in motion] = rest [mass of earth at rest] (0) The ari Table II: Terms neglected in calculations and their effects. Sr. No. elocity rel = rest [ 1+ /c +3 4 /8c 4 +. ] Neglected term Result 1 0 rel = rest none rel = rest Earth s orbital elocity 30km/s or m/s rel = rest [ ] rel = rest 3 =1cm/s or 0.036km/s K b K a = L [ ] or b = a b = a ass before emission = ass after Emission 011 C. Roy Keys Inc.

8 Apeiron, Vol. 18, No. 3, July Appearance of c in L= mc is apparently arbitrary. 4 K 0 K1 L (8) 4 c 8c Now consider the same case when elocity is 1cm/s or 0.036km/hr. Under this conditions eq. (8) becomes b a 4 L (1) (i) Einstein has neglected the term 3 4 /8c 4 retained the term as /c, and obtained equation L = mc. (ii) If the elocity is ery small then /c can be neglected compared to unity. If the elocity is 1cm/s (classical region), then /c is Depending upon the orbital elocity of the Earth (30km/s or 3,0000m/s i.e. /c =10 4 ) the term /c ( ) can be neglected compared to unity; only then the equation i.e motion [mass of earth in motion] = rest [mass of earth at rest] is justified. In the typical classical region ( =1cm/s), /c = is neglected compared to unity ( as is neglected ) then b (mass before emission) = a ( mass after emission) (1) Thus both L= mc and b = a are equally probable but hae an entirely different nature. This discussion also alidates the necessity of categorisation of sub ranges of elocity in the classical region or up to which magnitude of the term to be neglected comparatiely. 011 C. Roy Keys Inc.

9 Apeiron, Vol. 18, No. 3, July Einstein took only super special alues of ariables and its effects. The following arguments can be gien that Einstein s deriation is true under special conditions [11-37]. 1. Einstein [1] has imposed a condition on state of the body: Let there be a stationary body in the system (x, y, z), and let its energy referred to the system (x, y, z) be E 0. Let the energy of the body relatie to the system (,, ) moing as aboe with the elocity, be H 0. The body also remains stationary in system (x, y, z) after emission of energy. Einstein also assumed that the body remains stationary before and after emission of light energy, which is a super special condition. But practically this condition (light emitting body is stationary) is not obeyed in many other cases. (i) Nuclear fission is caused by the thermal neutrons which hae elocity,185m/s. The uranium atom also moes as it is split up into barium and krypton, and emits energy. (ii) When a gamma ray photon of energy at least 1.0eV, moes near the field of a nucleus it is split up into an electron and positron pair []. The gamma ray photon is in motion and similar is the state of the electron and positron pair. (iii) Similarly particle and antiparticle moe towards each other for annihilation. The particle and antiparticle collide then annihilation takes place. In nuclear fusion the atoms are set in motion. Fission is only caused by thermal neutrons (0.05eV or haing elocity,185m/s). Thus there are characteristic or inherent conditions on the process in inter-conersion of mass and energy. These phenomena were not discoered in Einstein s time. 011 C. Roy Keys Inc.

10 Apeiron, Vol. 18, No. 3, July (i) When a piece of paper burns it also sets in motion, and energy is emitted arious forms. () When deuterium and tritium fuse, but only after these are set in motion under conditions of high temperature. In the nuclear fusion of deuterium tritium the energy of emitted neutrons is 14.1eV (moing at 5,000km/s); their mass must increase by about 15.36%. It may increase the mass considerably. The elocity of the reactants is not necessarily uniform and gradually they oercome the force of electrostatic repulsion. Chemical reactions were discoered in Einstein s time. Einstein neer discussed this phenomenon in his works. Thus deriation under the condition that body remains stationary in the emission process, is not conceptually useful or applicable in other cases. Other conditions on Einstein s deriation. Einstein s September 1905 deriation [1] of L = mc is true under super special conditions or handpicked conditions only. It is justified below. In the deriation of L = mc there are FOUR ariables, i.e. (a) Number of waes emitted, (b) magnitude of light energy l, (c) Angle at which light energy is emitted, and (d) Uniform elocity (relatie elocity),. The fast neutrons are slowed down and are called thermal neutrons; thus their elocities are not necessarily uniform and can be ariable while they cause fission of other nuclei. Nature of According to Einstein: is the relatie elocity between system (x, y, z) and system (,, ). If system (x,y,z) is at rest and system (,, ) moes with elocity, then is relatie elocity. If the system 011 C. Roy Keys Inc.

11 Apeiron, Vol. 18, No. 3, July (x,y,z) and system (,, ) both moe with same elocity then relatie elocity is zero. Further Einstein strictly took the alue of elocity as uniform. The law of inter-conersion of mass and energy holds good if (i) Velocity is in the classical region (ii) Velocity is in the relatiistic region (iii) Velocity is zero i.e. if both systems moe with same elocity or system (,, ) is at rest. (i) Velocity is ariable or uniform. These ariables hae numerous alues. The law of inter-conersion of mass and energy holds good under all conditions, but Einstein has considered just one i.e. elocity in the classical region. It does not hold good under relatiistic conditions. Such a significant deriation must be independent of elocity..1 Genuine cases neglected in Einstein s deriation Einstein took super special or handpicked alues of parameters in the deriation. Thus for complete analysis the deriation can be repeated with all possible alues of parameters. In all cases the law of conseration of momentum is obeyed (which is discussed in next subsection). (i) The body can emit a large number of light waes but Einstein has taken only TWO light waes emitted by a luminous body. Why were one or n light energy waes neglected? (ii) The energy of two emitted light waes may hae different magnitudes but Einstein has taken two light waes of EQUAL magnitudes (0.5L each). Why were other magnitudes ( L and L) neglected by Einstein? 011 C. Roy Keys Inc.

12 Apeiron, Vol. 18, No. 3, July (iii) A body may emit a large number of light waes of different magnitudes of energy making different angles (other than 0º and 180º as assumed by Einstein). Why were other angles (such as 0º and º, º and 180º etc.) neglected by Einstein? Thus a body needs to be specially fabricated; other forms of energy such as inisible energy are not taken in account. Furthermore Einstein has considered only light energy, not other forms of energy which can be emitted by body. (i) Einstein has taken elocity in the classical region (<<c and applied the binomial theorem at the end) has not at all used elocity in the relatiistic region. If elocity is regarded as in the relatiistic region ( is comparable with c), then the equation for relatiistic ariation of mass with elocity i.e. rest motion (15) 1 c is taken in account. It must be noted that before Einstein s work this equation was gien by Lorentz [3-4] and firstly confirmed by Kaufman [5] and afterwards more conincingly by Bucherer [6]. Einstein, on June 19, 1948 wrote a letter to Lincoln Barnett [7] and adocated abandoning relatiistic mass and suggested that it is better to use the expression for the momentum and energy of a body in motion, instead of relatiistic mass. It is a strange suggestion as Einstein has used relatiistic mass in his work, including it in the expression of relatiistic kinetic energy [8] from which rest mass energy is deried [9-10]. So Einstein s equation for inter-conersion of mass to energy highly depends upon elocity theoretically whereas practically the mass energy inter-conersion phenomena are applicable in all cases. 011 C. Roy Keys Inc.

13 Apeiron, Vol. 18, No. 3, July () Einstein has considered body emits light energy only, but simultaneously a body may also emit heat energy which is not taken into account in Einstein s deriation. A burning body emits heat, sound, light energies and energy in the form of inisible radiations simultaneously, along with isible radiations. For a proper description of heat energy-mass inter-conersion we need an equation equialent to eq.(1). Similar is the case for other energies. In a nuclear explosion energies exist in many forms e.g. light energy, sound energy, heat energy, energy in form of arious inisible radiations. Einstein has only considered light energy and other energies are neglected to derie L = mc. Furthermore Einstein has considered that a body emits energy in the isible region. But energy can also be emitted in the inisible region and Einstein did not mention at all heat and sound energies (emitted along with the light energy). Thus energies other than light energy are also emitted but not taken in account. So energies are not taken in account completely. The arious alues of parameters neglected and taken into account in the deriation are shown in Table III. 011 C. Roy Keys Inc.

14 Apeiron, Vol. 18, No. 3, July Table III The alues of arious parameters considered by Einstein and neglected by Einstein in the deriation of Light Energy ass equation L = mc. Sr No Parameters Einstein considered Einstein neglected (No reason was gien by Einstein why parameters are neglected). 1 No. of light waes Two Light Waes One, three, four or n waes Energy of light wae Equal 0.5L and 0.5L each Energies of the order of L and L are also possible. There are numerous such possibilities, which need to be probed. Bodies can emit more than two waes. The inisible waes of energy are not taken in account. 3 Angle 0 and 180 The angles can be 0 and or and 180 are also possible. There are numerous such possibilities which need to be probed. 4 Velocity Classical region The elocity may tend to zero. The elocity can also be zero i.e. = 0. The elocity can be in the relatiistic region; ~c and mass increases. 5 Velocity Velocity is uniform in the classical region The law of inter-conersion of mass to energy also holds good, when elocity is ariable. Deductions: Einstein has taken only super-special alues of parameters, and neglected many realistic alues. 011 C. Roy Keys Inc.

15 Apeiron, Vol. 18, No. 3, July Effects of general alues of ariables If a body recoils when waes of different light energy are emitted: eq.(1) is the main equation in Einstein s deriation. (i) If a light emitting body is at rest then the relatie elocity of the measuring system (,, ) is only. (ii) If a light emitting body recoils away from the body with elocity V R, then relatie elocity will be + V R. (iii) If a light emitting body drifts towards the measuring system (,, ) with elocity V R then relatie elocity will be V R. But in this case the recoil elocity V R is of the order of m/s and with this elocity a body can trael a distance of m in 10 years ( which is undetectable; hence by definition the body is at rest). Thus in this case the body is regarded as at rest. The body may moe towards or away from the obserer, after emitting light energy; then the elocity becomes (± V R ) or say ( + V R ). Thus VR 1 cos c (3) [ VR] 1 c Now this equation can be applied to study ariations in mass when light energy is emitted as in case of eq. (1). 011 C. Roy Keys Inc.

16 Apeiron, Vol. 18, No. 3, July Conseration of momentum in general cases helps in calculations of recoil elocity. The law of conseration of energy and momentum are two significant laws. (i) In the deriation of L = mc, the law of conseration of energy is taken into account as in eqs. () and (3). Similar is the status in the deriation under general conditions. (ii) Furthermore, the law of conseration of momentum holds good in such cases and can be used to calculate the recoil elocity, V R. The alue of the recoil elocity in this case turns out to be much less, i.e m/s; thus the body can be regarded as at rest. This case is similar to the recoil of the gun when a bullet is fired, but here two waes are emitted. The momentum is consered irrespectie of the fact that the body remains at rest or recoils after emission of light energy [38]. In case of Einstein s deriation momentum is confirmed in special and general cases. (i) When a light emitting body remains at rest after emission. In this case the recoil elocity is zero i.e. V R =0; it is calculated by applying the law conseration of momentum. (ii) When a light emitting body recoils due to emission of two waes in different directions. In this case the elocity of recoil is non-zero and is calculated by applying the law of conseration of momentum. Einstein did not consider that case. Calculations of recoil elocity in system (x,y,z) The recoil elocity V R can be calculated by applying the law of conseration of momentum. In this case V R will be non-zero. Now in 011 C. Roy Keys Inc.

17 Apeiron, Vol. 18, No. 3, July arious calculations eq.(3) will be used as eq. (1) is used in Einstein s deriation. The law of conseration of momentum can be used to calculate the elocity of recoil in this case too. Let the body of mass 1 kg emit two waes in the isible region of waelength 5000ºA; it corresponds to energy hc/ or J. Let this energy be diided in two waes. Let body emits light energy (towards the obserer, = 0º) L E 1 = J (4) and momentum p 1 = E 1 /c = kg m/s (5) Secondly, the body emits a light wae of energy (away from the obserer, = 180º) L i.e. E = J (6) omentum p = E /c = kg m/s. (7) Let us assume that when the body emits light waes of energy in the system (x,y,z) and recoils (if it actually does) with elocity V R (say). The body will recoil opposite in direction to that of the wae haing energy E 1 (more energetic wae). athematically, Initial momentum of waes + initial momentum of luminous body = (8) Final momentum of waes + final momentum of body due to recoil= p 1 +p V b (9) One wae haing momentum p moes towards the direction in which the body recoils and the other wae moes in the opposite direction. Here the energy of wae 1 is E 1 (E 1 = J, p 1 = E 1 /c = kg m/s), which is more than the energy of wae E ( E = J, p = E /c = kg m/s). The wae 1 is emitted towards the obserer s system (,, ). 011 C. Roy Keys Inc.

18 Apeiron, Vol. 18, No. 3, July Thus the body recoils in the direction opposite to which wae 1 is emitted. It is like a bullet fired from a gun; then the system recoils in the backward direction. Hence here momentum p 1 (greater in magnitude) haing a backward direction to the measuring system is taken as negatie. The direction of V R is also regarded as the same as that of p 1.The recoil elocity is calculated as m/s; hence relatie elocity of the system (,, ) becomes (+V r ). Then according to law of conseration of momentum we get 0 p p V 1 b r (30) V R = ( p 1 +p ) / b = ( ) 10 7 = = m/s (31) The elocity m/s means the body remains at rest. A body is said to at rest if it does not change its position w.r.t. its surroundings. The elocity is too small to be detected. It is analogous to the obseration that a car cannot moe when head and rear lights are switched on. Thus conseration of momentum requires that a body should moe with elocity m/s away from the obserer. So the law of conseration of energy is obeyed. The distances traelled by the body in 100 or 10 years can be calculated as, S(100 years) = m/s = m (3) S(10 years) = m which is undetectable by any means and hence the body can be regarded as at rest. Thus the body will tend to moe with elocity m/s (away from the obserer) which is immeasurably small or undetectable by any means; hence the body remains at rest by the definition of rest. 011 C. Roy Keys Inc.

19 Apeiron, Vol. 18, No. 3, July This recoil elocity (V R ) i.e m/s is negligible compared to the elocity of the measuring system i.e. +V R = m/s = : Size of nucleus = m (33) S(100 years) = size of nucleus (34) S(10 years) = size of nucleus (35) Thus the body moes a distance of m which is immeasurable. Hence the body can be regarded as at rest, as in the case of Einstein s deriation when two waes of energy are emitted. Een if this elocity is taken into account for the sake of completeness then the results are the same as in the preious case. Een greater numerical alues are neglected in calculations, e.g. in the relatiistic ariation of mass rest motion (15) 1 c motion 4 3 rest c 8c rest Here elocity is regarded as 1m/s (3.6km/hr) in the classical region and een the term is regarded as negligible, thus motion = rest. Hence recoil elocity m/s is also negligible. Thus equations for recoil momentum and recoil kinetic energy KE recoil will be P recoil = kgm/s (36) 011 C. Roy Keys Inc. KE recoil = kgm /s (37)

20 Apeiron, Vol. 18, No. 3, July Due to this uniform relatie elocity, the system (,, ) will not change within measurable limits, howeer the effect of V R can be considered for completeness. If a body does not change its position so it can be regarded at rest by definition, as m/s =, x m = x, in 10 years. Thus the law of conseration of momentum helps us in calculations of recoil elocity, which changes the magnitude of in eq.(1). Hence the equation of relatiistic ariation of light energy i.e. Doppler principle for light for any elocities whatsoeer, becomes VR 1 cos c (3) [ VR] 1 c Now if the body recoils then eq.(3) has to be used instead of eq.(1) in the calculation of energy. Howeer the magnitude of eq. (3) and eq.(1) is the practically the same as +V R = m/s =. 4.0 L mc or L=Amc is equally feasible In Einstein s deriation [1] all types of energies of a body in E and H are taken into account. Hence the equation for eery energy source or energy content is speculated as E = mc. While a body recoils a small amount of heat energy, sound energy etc. may also be produced and energy is dissipated against friction, depending upon elocity of recoil. In Einstein s case recoil elocity is zero. Thus when a body recoils then energies after emission are denoted by E 1 and H 1 or different notations may be gien to them. These energies (E i s and H i s) include all types of energies and Einstein has applied the law of conseration of energy. Einstein has considered a body emitting two light waes of energy 0.5L each just in opposite directions. Let in this 011 C. Roy Keys Inc.

21 Apeiron, Vol. 18, No. 3, July case the luminous body emit two light waes of energy L and L in system (x,y,z) emitted in opposite directions. As discussed in eq. (31), the body remains at rest in this case also. Then the amount of light energy measured in both systems are related as (equialent to case of Einstein) H 0 where E 0 E 1 L () H L 1 cos L 1 cos180 (38) c c 1 (39) 1 c The eqs.(37) and (38) correspond to the law of conseration of energy, like eqs.() and (3). H 0 H1 L (40) c H 0 E0 H1 E1 l L (41) c H (4) c E H E L K 0 K1 L c 011 C. Roy Keys Inc.

22 Apeiron, Vol. 18, No. 3, July L c c L c c K 0 K1 L (43) c c b a L (44) c c b c a L c b a L (45) c c L c c mc L 1 (46) mc L (47) c / C. Roy Keys Inc. 8 (48) mc mc L (49)

23 Apeiron, Vol. 18, No. 3, July L mc or 011 C. Roy Keys Inc. L Amc Een if the recoil elocity V R is included i.e. eq. (3) is taken into account, the results remain the same. In the deriation Einstein used eq. (1) for relatiistic ariation of light energy, which was speculated in the preious paper [8]. But this equation is only meant for light energy, not at all for other energies; hence any deduction from it must be applicable for light energy only. The equation is not meant for (i) 1 cos c (1) 1 c Sound energy. In the Doppler effect change in frequency of sound is estimated, not ariation in mass. The speed of sound is 33m/s. Eq. (1) is not associated with any other energy. (ii) Heat energy. There is no equation like eq. (1) which relates ariation of heat energy. Similar is the case for other types of energies. (iii) Chemical energy. (i) Nuclear energy. () magnetic energy. (i) Electrical energy. (ii) Energy emitted in form of inisible radiations. (iii)attractie binding energy of nucleus. (ix) Energy emitted in cosmological and astrophysical phenomena. (x) Energy emitted in olcanic reactions.

24 Apeiron, Vol. 18, No. 3, July (xi) Energies co-existing in arious forms etc. Then why are results based upon eq. (1) applied to the aboe energies (i xi)? The reason is that all energies hae a different type of nature, and the energies are not confirmed to obey the same equation i.e. eq.(1). Einstein initially deried a light energy mass inter-conersion equation L =mc, then speculated the eery energy mass inter conersion equation E =mc from L =mc. Eq. (1) is only meant for light energy, not for other energies. Hence speculatie transition to E =mc from L =mc is absolutely without any mathematical basis. 4. If the measuring system is at rest (=0) and a body emits two light waes as in Einstein s deriation then the deriation is not applicable; can also be zero if system (x,y,z) and system (,, ) moe with the same elocity. Howeer in this case experimentally when light energy is emitted mass decreases. It is a serious limitation of Einstein s deriation. When the measuring system (,, ) is at rest = 0 then H 0 * l l 011 C. Roy Keys Inc. (50) L L H1 (51) E 0 E 1 L () E H E 0 H (5) As a body is at rest and the measuring system (,, ) is also at rest, then (H o E o ) or (H 1 E 1 ) cannot be interpreted as kinetic energy. Hence further deriation is not applicable.

25 Apeiron, Vol. 18, No. 3, July Thus a critical analysis of Einstein s deriation leads to many interesting facts. Einstein deried L =mc under certain conditions. Acknowledgements The Author is highly indebted to Dr. T. Ramasami, Secretary DST for arranging discussion on the topic with the Physics Adisory Committee, and Stephen J. Crothers for assistance at arious stages of this work. References [1] Einstein, A Ann. der Phys. 18 (1905) [] Arthur Beiser, Concepts of odern Physics, 4 th edition (cgraw-hill International Edition, New York, 1987), pp. 5, 7, 40, 48 (1987). [3] Lorentz H A, Proc. Roy.Soc. Amst (1899) [4] Lorentz H A, Proc. Roy.Soc. Amst (1904) [5] Kaufmann, W. Phys. Z 4, 55 (190) and Gott. Nacchr. ath. Phys. 4, 90 (1903) [6] Bucherer, A.H. Phys. Z, 9,755 (1908); Ann. Phys. 8, 513 (1909). [7] Einstein A 1948 Letter to Lincoln Barnett, quoted in `The concept of mass' by Le Okum Physics Today (June 1989) [8] A. Einstein, Annalen der Physik 17, (1905). [9] Robert Resnick, Introduction to Special Theory of Relatiity (John Wiley and Sons, New York, 1968), p. 10. (1968) [10] A. Pais, Subtle Is the Lord (Oxford Uniersity Press, Oxford, 1983), p. 149 (1983). [11] Fadner, W.L. Am. J. Phys. 56, 144 (1988). [1] Sharma, A. Proceedings of Physical Science of 98 th Indian Science Congress, Chennai, p.194 (011) [13] Sharma, A Proceedings of the 4 th international conference of IBIC, Kolkata, pp. 11 (010). [14] Sharma, A Proceedings of the Natural Philosophy Alliance, Vol.6 No. pp (011). 011 C. Roy Keys Inc.

26 Apeiron, Vol. 18, No. 3, July [15] Sharma, A International Journal of Nuclear Science and Technology Vol. 3 No 4 pp (007) [16] Sharma, A. Galilean Electrodynamics, Vol. 18, No. 5, pp (007). [17] Sharma, A. Kurt Godel Society Collegium Logicum Volume IX pp (007) [18] Sharma, A Progress in Physics, Vol. 3 pp (008) [19] Sharma, A. Concepts of Physics, Vol. III, No. 4, pp (006), [0] Sharma, A. Abstract Book: 38th European Group of Atomic Systems (Euro physics Conference), Isachia (Naples), Italy, p.53 (006). [1] Sharma, A. Abstract Book: A Century After Einstein Physics 005, (Organiser Institute of Physics, Bristol) Uniersity of Warwick, UK, April. (005) [] Sharma, A. Abstract Book 19th International Conference on the Applications of Accelerators in Research and Industry, Fort Worth, Texas, USA, 0 5 August. 005 [3] Sharma, A. Abstract Book, Oral presentation (lecture) on 1 st September 005, in The 5 th British Graity eeting, Oxford, England. [4] Sharma, A. Physics Essays, Vol. 17, pp.195 (004) [5] Sharma, A. Proc. Int. Conf. on Number, Time, Relatiity, United Physical Society of Russian Federation, oscow, p.81 (004) [6] Sharma, A Acta Ciencia Indica Vol. XXIV P No. 4 pp (1998). [7] Sharma, A to be published in Galilean Electrodynamics, assachusetts, USA [8] Sharma, A submitted for publication. [9] Sharma, A Acta Ciencia Indica Vol. XXVI P. No. 1 pp (1998). [30] Sharma, A International Conference on World Year of Physics, Uniersity of Rajasthan, Jaipur pp.14 (005) [31] Sharma, A. 3 rd international Young Scientists Conference, Kyi National Uniersity, Scientific works pp. 19 ( 00). [3] Sharma, A Journal of Theoretics Vol.6-5, pp [33] Sharma, A Beyond Einstein (monograph) accepted for publication to be published. [34] Sharma, A Einstein s ass Energy Equation, Lambert Academic Publishers (009),Saarbrucken, Germany [35] Sharma, A. Journal of Vectorial Relatiity. JRV (4) 1-8 (009) 011 C. Roy Keys Inc.

27 Apeiron, Vol. 18, No. 3, July [36] Sharma, A. Concepts of Physics, Vol. V, No. 3, pp (006), [37] Sharma A., Proceedings of International Conference on Computational ethods in Sciences and Engineering 003 World Scientific Co. USA, (003). [38] R. Resnick and D. Halliday, Physics Part I, Forty Second Reprints (Wiley Eastern Limited, New Delhi, 1987), pp., 45, 81 (1987). 011 C. Roy Keys Inc.