The Equivalence of Mass and Energy: Blackbody Radiation in Uniform Translational Motion

Transcription

1 The Africn Review of Physics (2015) 10: The Equivlence of Mss nd Energy: Blckbody Rdition in Uniform Trnsltionl Motion Rndy Wyne * Lbortory of Nturl Philosophy, Section of Plnt Biology, School of Integrtivee Plnt Science, Cornell University, Ithc, New York 1485, USA Uniform trnsltionl motion cusess hollow cylinder filled with blckbody rdition to gin n pprent inertil mss. Bsed on n electrodynmic nlysis, Hsenöhrl determined tht the reltionship between the velocity-dependent pprent inertil mss () nd the energy () of rdition ws given by. The reltionship between mss nd energy derived dynmiclly by Hsenöhrl differs from derived kinemticlly by Einstein. Here I use the reltionship between the energy nd liner momentum ( ) of photon ( ) nd the dynmic effects produced in Eucliden spce nd Newtonin time by the Doppler effect expnded to second order to show tht Einstein s eqution for the mss-energy reltion is the correct one for rdition-filled cylinder in uniform motion. The increse in liner momentum cused by the velocity-induced Doppler effect expnded to the second order results in n increse in the internl energy density of the rdition. The velocity-dependent increse in the internl energy results from both n increse in temperture nd n increse of entropy consistent with Plnck s blckbody rdition lw. The velocity-induced increse in the internl energy density is equivlent to velocity-dependent increse in the pprent inertil mss of the rdition within the cylinder. 1. Introduction As result of kinemticl nlysis, Einstein [1,2,3] derived the following eqution to describe the reltionship between the energy () nd inerti () of light: (1) By contrst, Hsenöhrl [4,5,6] took dynmic pproch to determine the reltionship between the energy nd inerti of light, by considering the work done by the nisotropic rdition pressure [7] inside uniformly trnslting cylinder (Fig. 1). Hsenöhl s eqution describing the reltionship between the energy nd inerti of blckbody rdition in uniformly moving cylinder differs from Eqn. (1) by n lgebric prefctor: The gol of this pper is to show tht Eqn. (1) is the correct eqution for describing the reltionship between the energy nd inerti of rdition for uniformly trnslting blckbody rdition-filled cylinder lthough the ssumption of reltive time used by Einstein to derive Eqn. (1) is unnecessry. * row1@cornell.edu (2) Fig.1: Hsenöhrl s gednkenexperiment. A cylinder with mtte blck nd dibtic wlls nd constnt nd invrint volume contining blckbody rdition chrcterized by bsolute temperture T moves through vcuum t bsolute zero. According to the Third Lw of Thermodynmics, bsolute zero is unttinble, so in ny rel experiment, the cylinder will hve to move through vcuum with finite temperture, which itself will result in n optomechnicl friction due to the Doppler effect expnded to the second order nd the need for dditionl pplied potentil energy to move the cylinder t given uniform velocity for given time. The orienttion of reltive to the x xis is shown, the orienttion of is perpendiculr to the x xis nd prllel to the end wlls. The length () of the cylinder is equl to twice the rdius () of n end wll so tht 2 2. With this geometry, one-hlf of the photons intercting with wll interct with n end wll. The time it tkes photon to propgte from end wll to end wll is.

2 The Africn Review of Physics (2015) 10: Results nd Discussion The derivtion I give here is bsed on the postultes of Eucliden spce, Newtonin time, nd the Doppler effect expnded to second order [8]. I ssume tht the wlls of the cylinder re dibtic, tht the volume is constnt nd invrint, tht the thermodynmic system is closed, nd tht energy nd liner momentum re conserved. I discuss sitution where the wlls re mtte blck nd re perfect bsorbers nd emitters. I consider the possibility tht the portion of the potentil energy pplied to the moving cylinder tht is not trnsformed into kinetic energy is trnsformed t the moving mtte-blck wlls into thermodynmic potentil energy such s the internl energy (). According to the work-energy theorem, the pplied potentil energy ( ) needed to move n object qusi-stticlly from stte of rest to nother stte with higher uniform velocity () for time intervl () is given by the following formul: $!"# (3) $ & Where,!"# is the verge force pplied over time intervl. If one were to move blckbody rditioncontining cylinder t constnt velocity through vcuum t bsolute zero where there is no externl mteril - nor rdition-friction, the rdition inside the cylinder would still dd velocity-dependent pprent inertil mss to the cylinder. This is becuse the rdition inside the cylinder tht strikes the bck wll would be blue-shifted while the blckbody rdition inside the cylinder tht strikes the front wll would be red-shifted (Fig. 1). The Doppler-induced frequency shift expnded to second order of ech photon tht mkes up the blckbody rdition due to reltive motion is given by the following eqution [8]: ' ( ' ) * -./01, 2 3, 4-./ (4) Where, ' ) is the frequency of the photon t the pek of Plnck s blckbody rdition curve tht describes the blckbody rdition in the cvity t rest (7 8), ' ( is the frequency of the photon t the pek of Plnck s blckbody rdition curve tht describes the blckbody rdition in the cvity when the cylinder is trveling t reltive velocity 7, nd θ is the ngle subtending the bsorbed or emitted rdition nd the velocity vector 7:. The Doppleriztion of the rdition within the cylinder will lso result in chnge in the energy of ech photon: h' ( h' ) * -./01, 2 3, 4-./ (5) When the cylinder is t rest, the energy of photon ( ) ) with pek frequency (' ) ) striking norml to either end wll is given by: ) = h' ) (6) However, when the cylinder is moving t uniform velocity over given time, the energy of photon t the pek of the blckbody rdition curve tht is propgting norml to the end wlls will decrese s result of being bsorbed nd emitted by the front end wll (θ 05 nd increse s result of being bsorbed nd emitted by the bck end wll (θ 5. As result of the second order effects, the energy increse t the bck wll will be greter thn the energy decrese t the front wll. Consequently, s result of the uniform trnsltionl movement, the energy (h' ( ) of the photons bsorbed nd emitted by both wlls will increse over time. As result of the two bsorptions nd the two emissions tht tke plce t the front nd bck wlls during, the velocitydependent energy of the photon t the pek of the blck body rdition curve is given by: 4=)>?5( 2h' ) *, - 2 2h' ) * h' ) * 2 6 h' ) * B (7) Since the photons in the blckbody rdition cn strike the end wlls t ny ngle () from 0 to ± D nd ny ngle () from 0 to ±D, the energy trnsfer depends on the cosines of the ngles of incidence nd the ngles of emission reltive to the norml: H ( 4=)>?5( IH cos J 4KLMNOP5- H IH cos (8)

3 The Africn Review of Physics (2015) 10: Thus the verge energy trnsfer per photon t the pek of the blckbody rdition curve t ny ngle is one-fourth of the energy trnsfer of photon t the pek of the blckbody rdition curve tht strikes the end wll perpendiculr to the surfce. Consequently, when the cylinder is in uniform motion, the verge photon with frequency t the pek of the blckbody rdition curve will hve its energy incresed in time by: ( h' ( h' ) * 2 6 (9) As result of the Doppler shift expnded to second order, fter given time intervl, the energy of the photons intercting with the mtte blck end wlls of the cylinder moving t constnt velocity will contribute to blckbody distribution of rdition chrcterized by pek with greter frequency nd thus higher temperture. The firstorder Doppler effect lone cnnot ccount for the nonliner reltionship between pplied energy nd kinetic energy becuse the first-order Doppler shifts t the end wll t the front of the cylinder nd the end wll t the bck of the cylinder cncel ech other nd the verge frequency of the photons in the cvity does not chnge [9]. However, the expnsion of the Doppler effect to second order gives the symmetry necessry to convert motion into internl energy. As result of the Doppler shift expnded to second order, s 7, the energy of the verge photon with frequency t the pek of the blckbody rdition curve pproches infinity. Since this energy increses t the expense of the conversion of the pplied potentil energy into the kinetic energy of the cylinder, the incresed energy of the rdition cn be considered to be the equivlent of velocity-dependent increse in the pprent inertil mss of the rdition. Since the pprent inertil mss of the blckbody rdition increses s result of the uniform movement of the cylinder through the vcuum, the pplied force (!"# ) needed to mintin constnt velocity () cnnot be constnt but must increse over the time intervl consistent with Eqn. (3). Thus the pprent increse in the inertil mss is mnifesttion of the fct tht the pplied potentil energy is trnsformed not only into the kinetic energy of the rdition-filled cylinder but lso into the internl energy of the blckbody rdition or photon gs itself. A complete conversion of the pplied potentil energy into the kinetic energy of the cylinder would only hppen when the inside of the cylinder is t bsolute zero nd there would not be ny photons within the cvity [10,11]. As bsolute zero is unttinble, this is limiting condition wht would not hppen in nture. The temperture (R) of the photon gs within cylinder moving t constnt velocity is relted to the pek frequency of the rdition in the cylinder t rest nd on the velocity of the cylinder. It is obtined by tking Eqn. (9) into considertion in order to crete the reltivistic version of Wien s displcement lw given below: ( h' ( h' ) * YR (10) Where, Y is Boltzmnn s constnt. I ssume tht while the pek frequency nd pek energy chnge with velocity, the spectrl distribution, which is described by Plnck s blckbody rdition lw, remins invrint. The increse in the temperture (R) of the blckbody rdition tht is relted to the increse in the energy ( ( ) of the photons t the pek of the blckbody rdition curve resulting from the Doppler shift is obtined by differentiting Wien s displcement lw: ( Y R (11) Plnck s blckbody rdition lw reltes the spectrl distribution of energy to the internl energy. The internl energy () of the blckbody rdition is relted to the temperture of the blckbody rdition by the Stefn-Boltzmnn lw: ZD[ \ ] ^_`` br (12) The increse in the internl energy of the blckbody rdition tht is due to the velocityinduced increse in the temperture of the blckbody rdition is given by differentiting Eqn. (12): Where, ZD [ \ ] ^_`` d D[ \ ] ^_`` br R (13) is the product of the Stefn- Boltzmnn (c) constnt nd one-fourth the speed of light (d e = J m -3 K -4 ). Thus prt of the pplied potentil energy used in moving the

4 The Africn Review of Physics (2015) 10: cylinder through vcuum cuses n increse in the internl energy of the blckbody rdition in the cylinder consistent with the conservtion of energy. Thus the mount of potentil energy needed to move the cylinder t given constnt velocity is greter thn the potentil energy predicted to move the cylinder if the internl energy of the rdition did not chnge with movement, nd the rdition did not cquire n pprent inertil mss. In order to formlly relte the increse in the potentil energy needed to mintin constnt velocity to the pprent inertil mss, we hve to consider the liner momentum of the photons tht mke up the blckbody rdition. The liner momentum ( ) of photon is given by [12]: J _f (14) When the cylinder is t rest, ccording to Eqn. (8), the liner momentum trnsferred by the verge photon with pek frequency (' ) ) moving in ny direction when they re bsorbed or emitted by the end wll is given by: ) = _f L (15) When the cylinder is in uniform motion, the verge photon with the pek frequency (7 ( ) trveling in ny direction tht is bsorbed nd emitted by the front end wll nd then bsorbed nd emitted by the bck end wll, will hve its liner momentum incresed by: ( 2 _f L *, _fl 2 * B _f L * 6 _fl * 6 (16) 2 2 After given durtion of time, the liner momentum of the photons trveling in ny direction in the cylinder moving t uniform velocity would increse. By definition, the liner momentum of the photon with the new pek frequency is equl to the product of its mss ( ( ) nd velocity: ( _f g * 2 6 ( 7 (17) Since the velocity of the photons in the cvity is constnt nd equl to, the totl differentil of the liner momentum is given by: ( ( (18) As result of the Doppler effect expnded to second order, the rdition produces n increse in liner momentum in given durtion of time, on the cylinder moving t constnt velocity. The increse in liner momentum results in n increse in the pprent inertil mss () of the photon with pek frequency moving t velocity c in ny direction within the cylinder ccording to the following formul: ( _ ' ) * 2 6 _ ' ) ( (19) The blckbody rdition within the cylinder in uniform trnsltionl motion provides liner momentum tht increses with velocity nd time. Consequently, the pplied force needed to mintin constnt velocity during tht durtion of time cnnot remin constnt but must lso increse to compenste for the incresed liner momentum. Interestingly, while he ws working on the quntum nture of rdition, Einstein [12,13] relized tht the momentum of rdition would exert rdition friction on moving body, but ws too engged in the Generl Theory of Reltivity to re-interpret the electrodynmics of moving bodies in terms of dynmics. The pprent increse in the inertil mss of photon is obtined by dividing ll terms in Eqn. (19) by : - _ ' ) * After substitution of ( with J - we get: J - _ ' ) * After rerrngement, we get: ( h' ) * 2 16 ( (20) ccording to [12], 2 16 ( (21) 2 16 ( (22)

5 The Africn Review of Physics (2015) 10: which is reminiscent of the eqution given by Einstein [1]. The increse in the pprent inertil msses of the photons ( ( ) t the pek of the spectrl distribution given by Plnck s rdition lw is relted to the totl increse in the pprent inertil mss () of the blckbody rdition in the cylinder. The time needed for the photons within the cylinder to go from n initil stte whose blckbody rdition is described by pek t one frequency to finl stte whose blckbody rdition is described by pek t higher frequency my depend on the detils of the force nd the dimensions of the cylinder. To simplify mtters, I ssume tht the durtion of time needed for the blckbody rdition to rech ech stte is constnt nd the force is not constnt. Eqn. (22) describes how chnge in the uniform velocity of rdition-filled cylinder is trnsformed into chnge in the energy nd inerti of the rdition inside the cylinder. The chnge in the energy nd inerti of rdition results from the Doppler effect expnded to second order nd the reltionship between energy nd inerti depends on the defined reltionship between the energy nd momentum of rdition ( (. Below I will use Plnck s blckbody rdition lw, lw tht I contend holds in ny inertil frme, to relte the spectrl distribution of rdition t ny velocity to the internl energy of the rdition. In cylinder contining photon gs t constnt volume, the pplied potentil energy needed to move the cylinder t constnt velocity from stte 1 to stte 2 is distributed between the increse in the kinetic energy of the cylinder itself nd the energy equivlent of the pprent inertil msses of ll the photons in the photon gs. Bsed on Eqn. (22), I consider the increse in the energy equivlent of the pprent inertil msses to be equl to the increse in the internl energy () of the photon gs: (23) Since the internl energy of photon gs is relted thermodynmiclly to both therml energy nd pressure-volume energy [14,15], we cn try to chrcterize further the increse in the energy of rdition tht occurs when closed thermodynmic system is moved t constnt velocity. In closed thermodynmic system with constnt volume moving t constnt velocity, the velocity-dependent increse in internl energy () of the photon gs cn be considered thermodynmiclly s velocity-dependent increse in therml energy (i) nd pressure-volume energy (b): i b (24) However, b vnishes since the volume is constnt by definition. The complete differentil of the therml energy (i) is given by the following eqution: i Rj jr (25) Where, R is the bsolute temperture nd j is the entropy of rdition in the cvity with constnt volume (b). Note tht the increse in therml energy in the cylinder with dibtic wlls does not result from het trnsfer from therml reservoir, s it does in trditionl thermodynmic systems [16], but from the dynmic effects of the velocityinduced Doppler effect expnded to second order. The velocity-induced Doppler effect cn result in n increse in temperture nd/or n increse of entropy. The increse in the therml energy of the photon gs tht hppens in the bsence of het flow my help in the kinemtic understnding of the contentious nd contrdictory field of reltivistic thermodynmics [17]. If the wlls of the cvity re mtte blck, consistent with Plnck s blckbody rdition lw, the photon number is not conserved nd the increse in internl energy due to the uniform trnsltionl motion for given durtion will result in n increse in the temperture (due to n increse in the frequency of the photons t the pek of the blckbody rdition curve) nd n increse in entropy (due to n increse in the number of photons). Neither of these quntities lone re considered here to be invrints. The bility of the blckbody rdition to tke up therml energy is given by the specific het cpcity t constnt volume (k l ) tht is obtined by differentiting the internl energy t constnt volume with respect to temperture: k l m n o p l D[ \ ] ^_`` VR (26) The spectrl number density (r) of photons, ech with given frequency, in the cvity t given temperture (R), is obtined by dividing the internl energy (), given by Plnck s blckbody rdition lw, by h'. Plnck s [18] blckbody rdition lw gives the spectrl energy density:

6 The Africn Review of Physics (2015) 10: s4',r5' n l 4',R5' ZDf ` _f u- vw, ' (27) The spectrl number density or the number of photons with given frequency is given by: r4',r5' x4f,o5 _f ' ZD_`f u- _``vw, ' (28) The totl number density ( y ) of photons within l the cylinder vries with the velocity of the cylinder in the sme wy tht the totl number density vries with temperture. The totl number density of photons comprising blckbody distribution of rdition chrcterized by given temperture t constnt volume is obtined by integrting Eqn. (28): y l z ZD\`o` r4',r5' { _`` ZD\`o` m u vw p u} z vw _`` { u- vw, z _`f { u- \`o`vw, ' (29) The integrl in Eqn. (29) cn be solved by letting ~ _( \o nd ~ _ \o ': y z r4',r5' l { z {, ZD\`o` z 45 _`` {, (30) Since , the totl number density of photons in cvity of temperture R is: y l 60.42m\o _ p 60.42m \ _ p R ƒ R (31) Differentiting Eqn. (31), we get the chnge in the totl number density of photons with chnge of temperture t constnt volume: l m\ _ p R R (32) The totl entropy of photons within the cylinder vries with the velocity of the cylinder in the sme wy tht the totl entropy vries with temperture. The increse of entropy of the photon gs with temperture R is obtined by dividing the internl energy differentil given in Eqn. (13) by the temperture: j n o D[ \ ] ^_`` br R (33) Since j vnishes t bsolute zero where R lso vnishes, the bsolute entropy j [19] of the photon gs t ny temperture R is obtined by integrting Eqn. (33): j D[ \ ] ^_`` VR (34) The entropy density of photons within the cylinder vries with the velocity of the cylinder in the sme wy tht the entropy density vries with temperture. The vlue of the entropy obtined from Eqn. (34) cn be used to find the entropy density of the blckbody rdition in the cylinder chrcterized by temperture R: D[ \ m \o l ^ _ p (35) By tking the rtio of Eqn. (35) nd Eqn. (31), the entropy per photon in the blckbody rdition is found to be constnt independent of velocity nd temperture, nd is given by: l D[ \ m \o y yl ^ _ p {.m vw p` D[\ ^ u2 {. 3.60Y (36) How the entropy of photon reltes to the degrees of freedom of photon is mystery. If entropy is defined by the degrees of freedom, with ech degree of freedom contributing to the entropy, then the entropy of photon is close to tht of ditomic gs molecule with trnsltionl, rottionl nd vibrtionl degrees of freedom [20]. For ditomic gs molecule, the three trnsltionl degrees of freedom contribute Y, the two rottionl degrees of freedom contribute Y nd the two vibrtionl degrees of freedom contribute Y for totl of 3.5Y. If photon were point prticle with three degrees of freedom for trnsltion nd one degree of freedom for polriztion, the entropy would be close to 2.5Y. Quntum mechniclly speking [21], entropy is relted to the number of microsttes (ˆ), which for the photon would be 36.6, since j Y lnˆ. The entropy of photon given in Eqn. (35) is more consistent with the model of n extended nd dynmic photon consisting of trnsltionl nd rottionl oscilltors [22-25], nd less consistent with the model of kinemtic photon being mthemticl point with integer spin. The verge internl energy of the photons ( ) composing the photon gs in the cylinder is relted

7 The Africn Review of Physics (2015) 10: to the velocity of the cylinder in the sme wy tht the verge internl energy of the photons is relted to the temperture. The verge internl energy of the photons in the blckbody rdition is given by the rtio of the internl energy density given in Eqn. (12) nd the totl number of photons per unit volume given in Eqn. (31): nl ZD[ yl ^ {. YR 2.70 YR (37) Consistent with the shpe of the curve tht describes blckbody rdition, the vlue for the verge internl energy of the photon is slightly lower thn the vlue of the internl energy of the photon t the pek of the blckbody rdition curve given by the Wien displcement lw: 4\ YR (38) When the wlls of the cvity re mtte blck, both the energy of the photon t the pek of the curve tht chrcterizes the blckbody rdition nd the number of photons t the pek of the curve increses s result of the velocity-induced Doppler effect (Fig. 2). This is best chrcterized by the velocity-dependent temperture of the photon gs inside the cylinder. The velocityinduced temperture increse hs greter effect on photon number thn on the energy of the photon t the pek of the blckbody rdition curve since the energy of the photon t the pek of the blckbody rdition curve is directly proportionl to temperture while the number of photons in the blckbody rdition is proportionl to the third power of temperture. The velocity-induced Doppler effect expnded to the second order results in n increse in the internl energy density, the temperture, the entropy density, nd the pprent inertil mss (Tble 1). When the wlls of the cvity re not blck, but re perfectly speculr reflecting, uniform trnsltionl motion will cuse n increse in the energy of the photon t the pek of the curve tht chrcterizes the rdition but there will be neither chnge in photon number ( 0) nor entropy (j 0). Consequently, the rdition in the cvity will no longer be distributed ccording to Plnck s blckbody rdition lw nd the nlysis given here will not hold. Nevertheless, whether the wlls of the cvity re mtte blck or perfectly speculr reflecting, the velocity-induced chnge in the pek of rdition would be irreversible nd the potentil energy irretrievble becuse moving the blckbody rdition-filled cylinder t uniform velocity in the opposite direction would not cuse decrese in internl energy but further increse. Indeed I hve shown tht the interction of Doppler-shifted photons with moving bodies is the fundmentl nd inevitble cuse of irreversibility [10]. Spectrl Energy Density, J m -3 3E E-18 2E E-18 1E-18 5E E00 Frequency, s E14 Fig.2: The velocity-induced chnge in the distribution of blckbody rdition in cylinder with mtte blck nd dibtic wlls nd constnt nd invrint volume trnslting uniformly t vrious speeds (0, 0.5c, nd 0.9c). Note tht the spectrl energy density (in J/m 3 ) t given velocity multiplied by the squre of the speed of light gives the spectrl mss density (in kg/m 3 ) t given velocity. Likewise the integrl of the spectrl energy density (in J/m 3 ) t given velocity multiplied by the squre of the speed of light gives the totl mss density (in kg/m 3 ) of the enclosed rdition t given velocity. As result of the increse in the internl energy () of the photon gs in the moving cylinder described nd explined by the velocity-induced Doppler effect expnded to second order, the blckbody rdition within the cvity cn be considered to hve velocity-dependent pprent inertil mss () tht provides resistnce to movement t constnt velocity (Tble 1). For the cylinder to move t constnt velocity, n dditionl mount of potentil energy must be pplied to the moving cylinder in order to compenste for the pprent inertil mss of the blckbody rdition within. In light of wht we hve deduced bout the velocity-induced increse in internl energy tht will tke plce s result of the Doppler effect expnded to second order, we cn write the reltionship between the pplied potentil energy nd the chnge in kinetic energy of the rdition filled cylinder like so: 7 (39) Where, is the differentil in the internl energy density tht results from the differentil velocity 0.5c.9c

8 The Africn Review of Physics (2015) 10: chnge 7. Given tht Eqn. (23) sttes tht, 7 (40) Tble 1: The velocity-induced increse in the frequency ('), energy (h') nd momentum ( _f ) of photon t the pek of the curve chrcterizing blckbody rdition; the velocity-induced increse in the temperture (R), internl energy density ( n ), photon number density l (y) nd entropy density l ( ) of the blckbody rdition; nd l the velocity-induced increse in the pprent mss ( n ) of blckbody rdition initilly t rest t 300 K for cylinder with mtte blck nd dibtic wlls nd with constnt nd invrint volume. The vlues re given for n initil stte of zero velocity nd finl stte with constnt velocity of 0.9c. 7 ' s -1 h' J h' kg m s -1 R K b J m -3 b m -3 j b J K -1 m -3 kg c This mens tht the verge force clculted from the work-energy theorem given in Eqn. (3) to be sufficient to ccelerte the cvity with constnt mss t rest to given velocity will ctully be n underestimte of the required force. The dditionl force necessry is result of the velocity-dependent increse in pprent inerti () of the photon gs within the cylinder tht must lso be considered when clculting the reltionship between the verge pplied force (!"# ) nd ccelertion ():!"# 4 5 (41) The optomechnicl counterforce ( Ž " ) resulting from the Doppler effect expnded to second order ws lredy found to describe nd explin the nonliner reltionship between pplied force nd the ccelertion of prticles with chrge nd/or mgnetic moment [11,26]:!"# Ž " (42) nd here I hve shown tht the Doppler effect cn describe nd explin the nonliner reltionship between pplied potentil energy nd constnt velocity for n extended object: $ $ Ž "!"# $ & $ & (43) The discrepncy between Hsenöhrl s nd Einstein s equtions for the reltionship between the energy nd inerti of rdition hs been difficult to reconcile [27-35]. Here I hve obtined Einstein s reltionship for the energy equivlent of mss bsed on Hsenöhrl s gednkenexperiment by tking into considertion the reltionship between the energy nd liner momentum of photon nd the dynmic processes tht result from the Doppler effect expnded to the second order in Eucliden spce nd Newtonin time. References [1] A. Einstein, Does the inerti of body depend upon its energy content? In: The Collected Ppers of Albert Einstein. Volume 2. Doc. 24. Trnslted by Ann Beck. Princeton University Press, Princeton, NJ pp (1905). [2] A. Einstein, On the inerti of energy required by the reltivity principle. In: The Collected Ppers of Albert Einstein. Volume 2. Doc. 45. Trnslted by Ann Beck. Princeton University Press, Princeton, NJ pp (1907). [3] A. Einstein, On the reltivity principle nd the conclusions drwn from it. In: The Collected Ppers of Albert Einstein. Volume 2. Doc. 47. Trnslted by Ann Beck. Princeton University Press, Princeton, NJ pp (1907).

9 The Africn Review of Physics (2015) 10: [4] F. Hsenöhrl, F. Sitzungsberichte der mthemtisch-nturwissenschftliche Klsse der kiserlichen Akdemie der Wissenschften, Wien 113 II, 1039 (1904). [5] F. Hsenöhrl, Annlen der Physik 320, 344 (1904). [6] F. Hsenöhrl, Annlen der Physik 321, 589 (1905). [7] J. H. Poynting, Philosophicl Trnsctions of the Royl Society of London A 202, 525 (1904). [8] R. Wyne, The Africn Physicl Review 4, 43 (2010); v/rticle/view/369/175 [9] L. Pge, Proceedings of the Ntionl Acdemy of Science USA 4, 47 (1918). [10] R. Wyne, The Africn Review of Physics 7, 115 (2012); v/rticle/view/539/233 [11] R. Wyne, Act Physic Polonic B. 41, 2297 (2010). [12] A. Einstein, On the quntum theory of rdition. In: The Collected Ppers of Albert Einstein. Volume 6. Doc. 38. Trnslted by Alfred Engel. Princeton University Press, Princeton, NJ pp (1916). [13] A. Einstein, On the development of our views concerning the nture nd constitution of rdition. In: The Collected Ppers of Albert Einstein. Volume 2. Doc. 60. Trnslted by Ann Beck. Princeton University Press, Princeton, NJ pp (1909). [14] H. S. Leff, Americn Journl of Physics 70, 792 (2002). [15] R. E. Kelly, Americn Journl of Physics 49, 714 (1981). [16] B.-Z. Ginzburg nd R. Wyne, Turkish Journl of Physics 36, 155 (2012). [17] H. Cllen nd G. Horwitz, Americn Journl of Physics 39, 938 (1971). [18] M. Plnck, The Theory of Het Rdition (Blkiston s Son & Co., Phildelphi, 1914). [19] D. V. Schroeder, An Introduction to Therml Physics (Addison Wesley Longmn, Sn Frncisco, 2000). [20] R. Clusius, Philosophicl Mgzine 4 th Series 14, 108 (1857). [21] H. A. Bent, The Second Lw. An Introduction to Clssicl nd Sttisticl Thermodynmics, second edition (Oxford University Press, New York. 1965). [22] R. Wyne, Light nd Video Microscopy (Elsevier/Acdemic Press, Amsterdm, 2009). [23] R. Wyne, Turkish Journl of Physics 36, 165 (2012). [24] R. Wyne, The Africn Review of Physics 9, 183 (2012). [25] R. Wyne, Nture of Light from the Perspective of Biologist: Wht is Photon? In: Hndbook of Photosynthesis, Third Edition, M. Pessrkli, ed. CRC Press, Tylor nd Frncis Group, London. [26] R. Wyne, The Africn Review of Physics 8, 283 (2013); v/rticle/view/761/320 [27] P. Lenrd, Gret Men of Science. A History of Scientific Progress (G. Bell nd Sons, London, 1933) p.371. [28] M. von Lue, Inerti nd Energy. In: Albert Einstein: Philosopher-Scientist. P. A. Schilpp, ed. Tudor Publishing Co., New York. pp (1949). [29] E. Whittker, A History of the Theories of Aether nd Electricity. Volume 2. The Modern Theories. Thoms Nelson nd Sons, London. pp (1953). [30] W. Puli, Theory of Reltivity, Dover, New York. pp (1981). [31] M. Jmmer, Concepts of Mss in Contemporry Physics nd Philosophy (Princeton University Press, Princeton, NJ, 2000) p.72. [32] A. Einstein, The principle of conservtion of motion of the center of grvity nd the inerti of energy. In: The Collected Ppers of Albert Einstein. Volume 2. Doc. 35. Trnslted by Ann Beck. Princeton University Press, Princeton, NJ pp (1906). [33] S. Boughn. nd T. Rothmn, Hsenöhrl nd the equivlence of mss nd energy (2011); [34] S. Boughn, Europen Physicl Journl H. 38, 261 (2013); rt253a fepjh252fe pdf?uth66= _ fdc b eee9&ext=.pdf [35] A. Einstein, Reltivity. The Specil nd the Generl Theory (Bonnz Books, New York, 1961) p.44. Received: 16 September, 2014 Accepted: 4 Mrch, 2015