Mass & Energy. Q. ter Spill. P. Chesmond. English corrected by. 's Gravesande Institute of Physics Education

Transcription

1 Mass & Energy by Q. ter Spll Englsh corrected by P. Chesmond 's Graesande Insttute of Physcs Educaton Jan an Houtkade 26a, 2311 PD Leden Netherlands tel+fax+answ.m

2 Mass and Energy, Contents. page 1 CONTENTS 1, Mass and Energy, Introducton... page 3 2, E = m.c 2 msleadngly explaned... page 5 3, E = m.c 2 and Poyntng s theorem... page 17 4, Gratatonal mass... page 21 5, Rest mass... page 23 6, Wthn the same nertal system Newtonan and relatstc dynamcs are dentcal...page 27 7, Conseraton, Inarance and Addtty... page 35 8, Problems I hae been unable to sole... page 43 9, Relatstc mass s the mass... page 47 Recommended lterature... page 51 Appendx I, Conseraton of momentum and the centre of mass... page 53 Appendx II, An example of rest mass sum beng confused wth rest mass...page 65 Appendx III, Rest mass sum s not a relatstc narant... page 69

3 page 2 Mass and Energy.

4 Mass and Energy, Introducton. page 3 1 Mass and Energy INTRODUCTION 2 "Mass dsappears and energy appears" s found as an explanaton of E = m c n secondary schoolbooks. Ths s an ncorrect statement. The released energy has mass of tself, exactly the mass whch seemngly dsappeared. The formula E has been explaned wrongly up untl now. 2 = m c In September 1905 the frst publcaton on the relaton between mass and energy 2 appeared n the form n whch t has become generally known: E = m c. The author was Ensten. Een before 1905 some scentsts started to suspect that energy, at least n some of ts forms, should hae mass. They calculated that the electrcal feld of a charged body should hae nerta and therefore mass. Ths mples a relatonshp between mass and energy, because an electrcal feld s thought to be the resdence of energy. These calculatons fnally reealed some nconsstences n the theory of electromagnetsm but dd not lead to the correct relatonshp between mass and energy, although they were not too far off the mark. At ths tme Lorentz too came to realze that energy possesses nerta. He had concluded that n certan nteractons between charged partcles Newton's thrd law could no longer be ald, unless mass s attrbuted to the electrcal felds. Ensten was the frst to ge the proper energy-mass relaton. From a thoughtexperment, an object that emts two lght waes of equal frequency and energy n two 2 opposte drectons, he dered E = m c. Nowadays ths formula has become one of the best known n physcs and can be found n eery secondary schoolbook on physcs. 2 Yet E = m c s ncorrectly explaned. One takes m to be the rest mass and one subsequently arres at confusng and nconsstent statements, whch wll ge the 2 dscernng user of E = m c whch ths formula plays a role. less rather than more nsght nto the processes n

5 page 4 Mass and Energy, Introducton. Actually the pont s that one attrbutes to rest mass a leadng role n these processes, whereas relatstc mass s seen as.... well, as what n fact? As a mathematcal trck perhaps. As an abstract noton, whch at best ads the mathematcs somewhat wthout representng a physcal realty. A strkng example of ths concept s to be found n an artcle n Physcs Today (The concept of mass, june 1989, page 31), wrtten by Le Okun, head of the laboratory of elementary-partcle theory at the Insttute of Theoretcal and Expermental Physcs, n Moscow, Russa. In ths artcle 2 Okun states, the m n E = m c represents the rest mass and certanly not the relatstc mass. Ths pont of ew s most unfortunate, to such a degree that I would call t wrong. It s precsely the relatstc mass, whch plays the leadng role n 2 relatstc processes and whch we hae to substtute n E = m c. An argument often heard n faour of the mportance of rest mass s ts narance when Lorentz transformatons are appled. Opposed to ths s the fact that the sum of relatstc masses n a system s constant (relate to an nertal system) as long as no work s done on the system by the outsde world. It should be noted that n that case the mass of the felds (we shall later see that felds possess mass also, namely ther energy dded by c 2 ) should also be added to these relatstc masses. I wll therefore from now on defne these feldmasses as relatstc masses. Thus t can be stated: f the total relatstc mass of a system s consered, so s ts total energy. Ths s agan a powerful argument n faour of the relatstc mass (and aganst Okun's statement). "Rest mass s mportant, when we want to ascertan the dentty of an elementary partcle" s often heard as an argument. Howeer, the relatstc mass at a gen speed can sere ths purpose just as well. Ths amounts to the same thng as usng rest mass, because ths s nothng other than the relatstc mass at speed zero. Restmass s a specal case of relatstc mass and the latter s the only successor to the classcal mass, that s to say the mass whch obeys Newton's three laws, and whch can be used as a measure of the quantty of matter and whch s consered for a closed system. The am of the argument that now follows s to make plausble that: 1 e statements as "mass can be conerted nto energy" are not true n ther generalty. 2 e 2 n E = m c the m represents relatstc mass and not the rest mass. 3 e the relatstc mass obeys Newton's three laws, so the Newtonan and relatstc dynamcs appear to be one and the same, as soon as we admt felds to hae relatstc mass too. *

6 Mass and Energy, 2, E = mc 2 msleadngly explaned n secondary educaton. page 5 2 E = mc 2 msleadngly explaned n secondary educaton From now on I wll ndcate rest mass by m rest and relatstc mass by m rel. Only n 2 the formula E = m c for the tme beng no ndex has been gen to m, because one dsagrees about whch m we are dealng wth here. The ntenton n ths paper s, amongst other thngs, to make clear that we are, n fact, dealng wth m rel. 2 A clear example of the msleadng explanaton of E = m c whch s gen n Dutch secondary schoolbooks, goes as follows: '... when mass dsappears energy appears. If on the other hand energy dsappears, mass s created.' Other books suggest a smlar thng, although mostly not so explctly stated. The problem wth ths statement s that the author nowhere says whether he means rest mass or relatstc mass. But n both cases the statement s ncorrect, at least n certan stuatons. For the followng argument t s mportant to dstngush clearly between the arous knds of mass known n the theory of relatty. Three knds of mass are defned: Defnton 1: Restmass of a body s the mass measured n an nertal system mong along wth the centre of mass of the system. Defnton 2: Inarant mass of a system s the mass of a system consstng of relate to each other mong parts, measured n an nertal system mong along wth the centre of mass of the system. Defnton 3: Relatstc mass relate to an nertal system s restmass, or narant 2 2 mass, tmes γ M. Heren γ M stands for 1/ 1 / c, beng the elocty of the centre of mass wth respect to that nertal system. M M

7 page 6 Mass and Energy, 2, E = mc 2 msleadngly explaned n secondary educaton. When measurng rest mass eloctes should be used of magntudes approachng zero n order to aod relatstc effects. In the case of a system of free partcles whch we can regard as pont masses, we can calculate the narant mass of ths system from the rest masses of the materal partcles and the photons; m = Σ m. γ + Σ E. c 2, γ = 1/ 1 2 / c 2 ( 1) nar rest, photon, j wth n whch and m rest, are respectely the elocty and the rest mass of the -st materal partcle, and E photon, j the energy of the j-st photon, all measured wth respect to the centre of mass of the system. Defntons 1, 2 and 3 demand a defnton of the poston of the centre of mass. It wll be gen here only for a system of free partcles, whch can be consdered as pont masses: Defnton 4: r M = 2 Σm γ r + ΣE c r rest, photon, j j Σm γ + Σ E c rest, photon, j 2 where and are the postons of the -st materal partcle and the j-st photon r respectely. rj If on the system no forces from the outsde world are exerted two conseraton laws are ald wth respect to a arbtrary nertal system: 1 e the narant mass s consered. 2 e the elocty of the centre of mass does not change. The last law can be deduced from the law of conseraton of momentum. Defnton 1 s famlar to eeryone who knows somethng about the theory of relatty. Normally the restmass (of a body consdered as a pont mass) s smply measured by weghng. The relatstc mass of a body consdered as a pont mass, can be defned by defnton 3, that s to say as the restmass tmes γ. These defntons are n agreement wth Mach s defnton of classcal mass. Mach defned classcal mass n the followng way. The unknown mass and a known testmass, both at rest, are placed sde by sde wth a compressed sprng of neglgble mass between them (of course, the sprng can be replaced by any force nteracton between the two masses, for nstance by the coulomb force). Forces from the outsde world are absent. Now the sprng s released and both masses moe away from each other n opposte drectons. After an arbtrary tme has elapsed, ther dstances from ther common pont of departure are measured smultaneously. When we defne the rato of ther masses as beng equal to the rato of ther dstances, the unknown mass s defned as well. Ths procedure can be carred oer lterally to the theory of relatty n order to defne the

8 Mass and Energy, 2, E = mc 2 msleadngly explaned n secondary educaton. page 7 rest mass of a pont shaped body. A necessary extra precauton s to choose the force of the nteracton to be so small, that eloctes stay n the non-relatstc regon. For the relatstc pont mass, Machs defnton can also be used. Now a grazng collson between the unknown relatstc pont mass (whch s mong at full speed) and the standard mass (whch s at rest) must be arranged. Afterwards, the dstances of the masses from the carrer lne of the ntal elocty must be measured. In ths case only for the test mass care has to be taken, that t does not acqure a relatstc elocty. For the rest, the procedure s the same as that descrbed for the rest mass. For the narant mass and the relatstc mass n the case of a system of free partcles, Mach s method can also be used by applyng t to each partcle nddually and then calculatng the poston of the centre of mass before and after all the collsons wth the formula of defnton 4 (see page 6). Of course n practce, Mach s method s too dffcult. Howeer, t s mportant to pont out that the defntons of rest mass, narant mass, relatstc pont mass and relatstc non-pont mass are n agreement wth Mach s defnton. In practce, the restmass of a pont shaped body s measured by weghng. The relatstc mass of the same object at speed s calculated by multplyng ts restmass wth γ. The narant mass of a system of free pont masses can be found n practce by weghng each pont mass at rest and then calculatng the narant mass by formula (1). The narant mass of a coherent system such as a sold object or an on stll hang some electrons, can also be found by weghng t at rest (and ths tme there s of course no need for formula (1)). The essence of the narant mass s reflected by defnton 3. The relatstc mass of the system when the centre of mass has elocty M can be calculated by magnng the whole of the narant mass as beng concentrated n the centre of mass. So the relatstc mass s found by multplyng the narant mass by γ M. The correctness of ths statement can be proen for a system of free pont masses out of the lnearty of the Lorentz transformaton. Then, for the narant mass of ths system, we fnd the expresson on the rght hand sde of formula (1) on the preous page. Formula (1) can also be used as a defnton of narant mass. Note that n formula (1) the relatstc mass agan plays an mportant role; we fnd the narant mass by addng the relatstc masses, as measured relate to the centre of mass of the system. If photons are also noled, we hae to add for each photon a term E photon /c 2, or h.f/c 2, wth f measured relate to the centre of mass of the system. Because of ths I wll n future call E foton /c 2 the relatstc mass of the photon, so we can say: "The narant mass s the sum of the relatstc masses measured wth respect to the centre of mass." Defnton 4 and formula (1) together seem to ge a crcular defnton; for the poston of the centre of mass you need the relatstc masses and for the relatstc masses you need the elocty, that means the tme derate of the poston, of the centre of mass. Howeer, the relatstc masses n defnton 2 are measured relate to the centre of mass, whereas the relatstc masses n defnton 4 are measured relate to an arbtrary nertal system. Therefore there s no queston of a crcular defnton. Notce that relatstc mass n the defnton of the relatstc centre of mass plays exactly the same role as classcal mass n the defnton of the classcal centre of mass.

9 page 8 Mass and Energy, 2, E = mc 2 msleadngly explaned n secondary educaton. An mportant objecton to defnton 4 seems to be, that especally for photons, the Hesenberg uncertanty relaton s so mportant that postons of photons are far from beng defned exactly. To follow through ths thought would lead to quantum mechancs, to the Ensten Podolsky Rosen "paradox", to the theorem of Bell and to the experment of Aspect. Ths s not my ntenton. Suffce t to menton what Aspect has demonstrated for a system of one atom and two photons. When detectng one of the partcles, so at the moment of collapse of ts waefuncton, the waefunctons of the other partcles collapse smultaneously (for all obserers) n such a way that all conseraton laws reman ald. Ths means therefore, that expermental determnaton of the centre of mass s meanngful, that s to say t s n agreement wth the theory. It s clear that defnton 1 only for the theoretcal case of a pont mass does not rase questons. In all other cases there s the problem that one neer can say wth certanty that n each nfntely small scale there are no mong parts n a body. Neertheless one often speaks of "the rest mass of an atom", whereas n an atom the electrons are certanly not at rest wth respect to the nucleus. One should speak here of the narant mass of an atom. The atom s seen here as a pont mass. In fact there s nothng wrong wth that, as long as t s justfed by our measurng apparatus, as long as an atom looks lke a pont to our measurng system. In that case rest mass and narant mass are the same. When a system s noled wth dmensons sgnfcantly greater than our measurng accuracy, there are two possbltes: frst, the body has a centre of mass wth a fxed poston wth respect to fxed and recognzable ponts on the body. Take for example a sold body (although a sold body also conssts of parts mong relate to each other, merely because of the thermal moement of the molecules, talkng of fxed ponts on the body neertheless mples no contradcton, because we then look at a much larger scale than the molecular one, so the thermal moement aerages zero. Besdes, we assume that stresses n the body are not so great that the body s notceably deformed). In ths case the centre of mass can be determned by calculaton f the densty functon s known. The narant mass can be determned n the same way as the rest mass, the coered dstance of the unknown mass after the collson beng that of ts centre of mass. Secondly, the body s less coherent, the centre of mass hang no fxed poston wth respect to fxed, recognzable ponts of the body. For nstance, a number of nucle flyng away from each other after a nuclear reacton. Then the narant mass could be determned wth the method gen on page 7, lne 9 to lne 12, appled on all pont masses n the system. Agan, n practce almost mpossble to perform, but theoretcally not mpossble. Compare a sngle pont mass and a system-consstng-of-seeral-pont-massesmong- relate-to-each-other, the separate pont mass and the centre of mass of the system beng at rest. Suppose we take the sngle pont mass as well as the centre of mass of the system to obtan a elocty. Ths elocty s obtaned not by exertng a force on them, but on us, the obserers. After our relate acceleraton has proded the relate elocty, the force s brought back to zero and we can consder ourseles agan as nertal obserers. Then m rel of the pont mass s a factor γ ( ) tmes ts rest mass, whereas Σm rel pont masses, appears to be also a factor γ of the system, so the sum of the relatstc masses of ts tmes ts narant mass. Therefore t s

10 Mass and Energy, 2, E = mc 2 msleadngly explaned n secondary educaton. page 9 meanngful to call Σm rel the (total) relatstc mass of the system. So n magnaton we can replace een such a system of pont masses, no matter how bg ther relate freedom of moement s, by one pont mass as far as the relatstc mass ncrease s concerned, proded we magne ths pont mass to be localzed n the centre of mass and take as the total relatstc mass the sum of the relatstc masses of ts pont masses. Suppose we accelerate the two systems and not ourseles. Are the dynamcs analogous n both cases, s Ftotal = d( m ) / dt ald for both sngle pont mass and the centre of mass of the system of pont masses? In general the answer s no. See the appendx I, page 53 to 63. The complex system wll generally see ts narant mass changed. Howeer, f the complex system s a sold and the nternal stresses are not to great, I assume that the answer s yes. When an onsed atom or molecule s accelerated, to all lkelhood ts centre of mass obeys ths equaton as long as no nternal degrees of freedom are excted. If we look at t ths way the rest mass (of a pont mass) s nothng else other than a specal case of the narant mass and the relatstc mass of a pont mass s nothng else other than a specal case of the relatstc mass of a system of parts mong relate to each other. If we are only nterested n the behaour of the centres of masses and f the nternal energy of the body s not changed, not one sngle essental dfference can be ponted out between rest mass and narant mass on the one hand, and the relatstc mass of a pont mass and relatstc mass of a composed system on the other hand. As far as the dynamcs of centres of masses s concerned we can use "rest mass" and "narant mass" as synonyms. Let us now look at an atomc nucleus at rest, whch s about to undergo fsson. Let us call the pont where the nucleus sts P. Let us assume that t splts nto two fsson products (lghter nucle) and that no other partcles such as photons are produced (the fact that photons are produced, s not an essental objecton to our reasonng: Takng photons nto account doesn't change our argument, t merely complcates t). For the sake of smplcty, we shall assume that both fsson products hae the same mass. After the fsson, the fsson products fly away at hgh speed n opposte drectons: both hae knetc energy. Ther masses can be measured after they hae come to rest. Now, the sum of ther rest masses appears to be less than the rest mass of the orgnal nucleus. "The dfference s conerted nto energy", students learn n secondary school. I consder ths a msleadng statement. Look agan at the stuaton mmedately after the fsson, when the fsson products are at full speed, so before they hae been slowed down (such as by collson wth surroundng molecules). The fsson products hae knetc energy. Accordng to the secondary school books, ths energy would hae been created by the "dsappearance" of mass suggestng that the fsson products together hae less mass than the orgnal nucleus. Ths s ndeed true f one looks at the rest masses of the fsson products separately, but s not the case f one consders ther relatstc masses. When the relatstc masses are added, one obtans the mass of the orgnal nucleus. Ths follows from the fact that the narant mass does not change, because no work s done on the system by the outsde world. It can be seen n a dfferent way also, as the followng example wll llustrate. In fgure (a), see below,

11 page 10 Mass and Energy, 2, E = mc 2 msleadngly explaned n secondary educaton. the orgnal nucleus s denoted by A. An addtonal partcle (such as an atom), remanng at rest and undergong no changes s denoted by D. For smplcty's sake, I assume A and D to hae equal rest masses. Now consder fgure (b), the stuaton mmedately after nucleus A has fssoned nto the two fragments B and C. By symmetry, the center of mass of B and C taken together has to reman at pont P. Because there s a mass assocated wth each center of mass pont, we can ask here, whch mass? It must be the mass of A. Ths can be seen by consderng n fgure (a) the center of mass of A and D. Ths must le at the geometrcal centre between A and D, snce they hae the same rest mass. Now, f n fgure (b), the (narant) mass assocated wth the centre of mass of B and C would dffer from the mass of A, then the centre of mass of B, C, and D would hae moed from the mdpont between P and D. Ths cannot be the case, accordng to the prncple we stated on page 6, lne 16 (statement 2 e ) and 15 from bottom, snce no forces external fg. (a) fg. (b) the system of A, B, C, and D hae been noled. Clearly, the mass assocated wth the center of mass of B and C cannot be the sum of ther rest masses: ths s less than the rest mass of A, because the dfference s "conerted" nto the knetc energes of B and C. If, on the other hand, we consder the relatstc masses of B and C, we see that ther sum s equal to the rest mass of A. Ths has been erfed both expermentally and theoretcally. In smple terms, the mass assocated wth the knetc energy has to be taken nto account. We hae just seen that rest mass s not an absolute noton. The system of B, C, and D has a centre of mass at rest, so t s meanngful to call ths a system at rest. Yet, t s the relatstc mass of the nddual partcles (of B, C and D), not ther rest mass, that we must add up to obtan the (rest) mass of the system. The knetc energy of the moton of the nddual partcles relate to the centre of mass of the system contrbutes to the systems rest mass. Now consder the followng excerpts from a course on the theory of relatty for students of secondary school by a professor of physcs at Leden Unersty (1993, prof. Nenhus):... The enormous quanttes of energy whch can be released n

12 Mass and Energy, 2, E = mc 2 msleadngly explaned n secondary educaton. page 11 And: nuclear reactons orgnate from a dsappearance of a part of the mass. Mass s conerted nto energy (ref. 9, page 52). In the old physcs, conseraton of mass was consdered a fundamental property of matter. Whateer happened n collsons, durng combuston or explosons, the total mass could not change. Ths expressed the mpershableness of matter. Matter could change n nature, t could conert from solds nto gases, and ce ersa, but the amount of matter, as expressed n the total mass, always remaned the same. Ths dea now has to be abandoned. Matter can come nto exstence and can persh. Not matter s consered, but energy. Therefore the concept of energy acqures a more fundamental mportance than that of matter. Summary: Mass s energy n the rest frame. Energy s consered, mass s not. Mass can come nto exstence and can persh. At the bottom of the page n small letters: Note: If desred, we can defne the mass of an arbtrary physcal system as the energy dded by c². If we do so, then the mass s, of course, agan a consered quantty. The consequences, howeer, are far reachng. Frstly, the mass of a partcle s no longer ndependent of ts moton. [...] An object becomes heaer as t moes faster, and the mass becomes arbtrarly large as the speed of lght s approached. Furthermore, we must attrbute mass also to lght, or to an electrcal or magnetc feld, because they represent energy. The frst two quotatons of Prof. Nenhus apparently refer to the rest mass, but whch rest mass? For we hae just seen that rest mass s not an absolute noton. If reference s made to rest mass attrbuted to the centre of mass of a quantty of radoacte gas under low pressure, whch s completely solated (both thermally and otherwse) from the outsde world, and whch warms tself up by ts own radoactty, then the statement that ths heat energy orgnates "from a dsappearance of a part of the mass" s ncorrect. The (narant) mass, whch ths generated heat possesses, s equal to the decrease n the rest mass of the consttuent atoms (I assume a monatomc gas). Because the heat together wth ts mass remans n the system and contrbutes n ths way to the rest mass of the system, the total rest mass does not change. A smlar example can also be gen for a sold body, although thngs then become a bt more complcated; a pece of uranum warms tself up by ts own radoactty, whle t s n a contaner of thermally solatng materal and whle radaton partcles are preented from escapng (by a thck layer of lead for nstance). In ths case too there s a decrease of rest mass f we compare the rest mass of the uranumnucleus wth the sum of the rest masses of the nucle of ts fsson products (the energes of the electrons can be neglected). Now agan there s a release of (nternal) energy, but ths tme that energy conssts not only of knetc, but also of potental energy, namely that of the ntermolecular forces. Both knds of energy contrbute to the restmass of

13 page 12 Mass and Energy, 2, E = mc 2 msleadngly explaned n secondary educaton. the pece of uranum-wth-contaner. The knetc energy of each atom contrbutes m { γ ( ) 1} rest to the restmass, n whch s the elocty of the atom wth respect to the centre of mass. The potental energy of each atom contrbutes E pot /c 2 to the restmass. When all these (relatstc) masses are added up, we fnd exactly the mass whch we see dsappear when we look at those separate restmasses of uranumnucle and fssonproducts. In fact we only thought to see restmass dsappearng. It ddn't dsappear n realty. We moed n thought subsequenty along wth eery nucleus apart, because we looked at ts restmass and then added all those restmasses. Then t looked lke mass hang dsappeared. Of course, we (Lorentz)transformed away the knetc energy of the thermal moement and therefore ts mass also. To make thngs worse, we ddn't take nto account that potental energes also contrbute to the restmass. Checkng a conseraton law s lke book-keepng whle the foregong s lke fraud. If we wanted to do a thng lke that n classcal mechancs loud protests would arse, as the followng example clarfes: " In contrast to what has been sad for a long tme, energy s not a consered quantty. Launch a sled on an ar-cushon track wth a sprng. There s no frcton anywhere. The potental energy of the sprng has dsappeared completely and has not been conerted nto other forms of energy. Ths becomes clear when we start mong along wth the sled; ths then has no knetc energy." The edent decet of ths reasonng s agan n the use of a (gallean) transformaton. We hae transformed away the knetc energy by startng to moe along as an obserer. Nobody wll take ths reasonng serously. The reasonng just mentoned, n whch the same mstake s made, remarkably enough s neer contradcted. We should be well aware of the fact that conseraton laws of energy and mpulse are ald only as long as we do not step oer to another nertal frame. E = m.c 2 s about such a conseraton law, and n fact tells us that the law of conseraton of energy and of mass are one and the same. When workng wth ths law we should not change the nertal system. The example of the pece of uranum s so gratfyng because t apparently noles the quantty whch we call restmass. In fact I hae been reproached seeral tmes for confusng rest- and narant mass. When dealng wth the example of the rado-acte gas I was told: "But you are lookng at the narant mass, you wrongly call ths the restmass." As remarked earler, t s the other way round. Eerybody calls the mass of such a pece of uranum the restmass, whle n fact we should call t the narant mass. One oerlooks the thermal moement. As already sad on page 8, drectly under the whte lne, n practce rest- and narant mass are one and the same. Suppose the two quotatons of prof. Nenhus of page 10 and 11 refer to the separate restmasses of the gas atoms n our example. Then the aboe mentoned mstake n book-keepng s made. But apart from that the queston arses, why the restmass of a gas s seen as the sum of the restmasses of the atoms separately. Why not look nsde the atom? Ths holds the knetc (and also the potental) energy of the electrons, whch are ndeed not at rest wth respect to the nucleus. Looked at n ths way, we should take the restmass of the electrons, add t to the restmass of the nucleus and consder the knetc and potental energy of the electrons separately; those energes do not contrbute to the restmasses of the elementary partcles. But do protons and neutrons n the nucleus not hae knetc and potental energy also? The

14 Mass and Energy, 2, E = mc 2 msleadngly explaned n secondary educaton. page 13 same can be sad perhaps for neutrons and protons on ther own. There are ndcatons that quarks also hae knetc energy wthn the baryon. Those who argue that t s logcal to compare atoms (or atomc nucle) before the reacton wth atoms (or atomc nucle) after the reacton, that t s not obous or een wrong to compare atoms before the reacton wth elementary partcles after the reacton, forget that ths s always done at the fsson of an uranum nucleus; one nucleus (that of uranum) splts nto two nucle plus two or three neutrons. The "mass conerted nto energy" s calculated by comparng the restmasses of the fsson product nucle and the neutrons (elementary partcles) wth the restmass of the uranum nucleus. Whoeer argues that you hae to see as one entty those partcles whch are bound to each other and hae to compare ther restmasses wth one another, s mssng the pont also; the whole fsson process could be happenng n a contaner wth thck lead walls, out of whch nether neutrons nor nucle could escape. In ths way all partcles are bound partcles. Do you hae to see an alpha partcle n the uranum nucleus as bound or not? After all t can escape. The noton "bound" s too ague to justfy statements such as "you hae to compare only nucle wth nucle". The noton of restmass therefore wthout further specfcatons s not unambguous ether. For t s mpossble to state wether, on an nfntely small scale, a body s not composed of parts mong relate to each other. Therefore the ew dscussed aboe doesn't seem to me to be a frutful one. It seems to me more justfed to state that eerythng we called energy earler appears to hae mass and contrbutes to the total mass of the system, n whch that energy s lodged, to the restmass (=narant mass) f the centre of mass of the system s at rest, to the relatstc mass f the centre of mass s n moton. Moreoer, what s called restmass here n fact s no restmass, but somethng what I call wth a neologsm the restmass sum. Further on I wll demonstrate, that the restmass sum s not a relatstc narant. The thrd part of the quotaton leans towards our ew, yet the author seems unwllng to make the whole step. He foresees far reachng consequences f mass s attrbuted to lght and electromagnetc felds. From my pont of ew t s the other way round: the consequences would be far reachng f we ddn't do that. If the gas mentoned aboe starts glowng because of ts radoactty, lght wll be permanently exchanged between ts atoms. The mass of ths lght must contrbute to the rest mass of the gas. Must, because otherwse we could present the same argument about a dsplacng centre of mass as n fgures (a) and (b). Another example: an atom ntally at rest that emts a photon. The atom experences a recol. The centre of mass of the atom and photon must reman mmoble. If we fg. (c) fg. (d)

15 page 14 Mass and Energy, 2, E = mc 2 msleadngly explaned n secondary educaton. don t attrbute mass to the photon, the centre of mass would trael along wth the atom. Because the system does not experence forces from the outsde world, ths s mpossble. Thus, the photon must hae mass, see fgures (c) and (d). Mass also has to be attrbuted to a statc electrc feld. In ths case we can safely speak of rest mass, contrary to the case of separate photons. To see ths, let us replace the gas by a charged capactor. An electrcal feld exsts between the plates. If we dscharge the capactor through a long conductng wre (consdered to form part of our solated system), heat s generated. Ths thermal energy has mass whch contrbutes to the total rest mass of the system. The rest mass has to reman the same, so an equal amount of mass must hae dsappeared as appeared n the form of heat. The dsappearance of mass can only be attrbuted to the dsappearance of the electrc feld. Thus, one can safely state that a statc electrc feld has a rest mass. Returnng to the statement quoted n the frst lne of 1, page 3, Mass dsappears and energy appears : f we assume that the relatstc mass s meant, then the statement s ncorrect n the example of the splttng nucleus, snce the total relatstc masses of B and C are the same as that of A. If rest mass s ntended, then the statement s wrong n the example usng the capactor (more correctly: here the statement Energy dsappears and mass appears s ncorrect). Another quotaton, from a Dutch Secondary school book, the book "Scoop", ref. 16 on page 52, states: "Accordng to that law, mass s one of the forms n whch energy can exst." Ths suggests the followng: energy can exst n many forms such as knetc energy (flyng bullet), chemcal energy (gunpowder), potental energy (a stretched sprng) and as mass. It seems qute clear to me that rest mass s meant here. Then ths quotaton suggests, wrongly, that the frst three forms of energy don't represent mass. Wrongly, because a compressed sprng has more rest mass than the same sprng n a released state (after bratons are damped out and nternal frcton energy has escaped n the form of heat). A quantty of gunpowder has a bgger rest mass than ts combuston products (after the energy of the exploson has been dsspated). Een modern measurng technques cannot reeal ths small mass dfference, but t does exst. The dfference n mass s the mass of the energy contaned n the stretched sprng or the gunpowder. When we shoot off a bullet wth the sprng or the gunpowder, ths energy, together wth ts mass, leaes the sprng or gunpowder and adds to the bullets relatstc mass (whch before the shot s equal to ts restmass). The bullet's mass s thereby ncreased to the relatstc mass correspondng to the speed acqured by the bullet. If we follow "Scoop", and consder the rest mass when the bullet s shot off by a sprng, we are saddled wth an unattracte descrpton: the sprng energy has mass. As soon as t has passed oer to the bullet t suddenly has no mass anymore. The argument from "Scoop" can also be found n "Systematsche Natuurkunde", another Dutch Secondary schoolbook whch states that E = mc 2 should be replaced by E = m. c 2. It would be somethng lke E = E, or n other words, the ncrease n knetc energy equals the decrease of potental energy and ce ersa. So, E = m. c 2 kn would mean: the decrease n mass equals the ncrease of other forms pot

16 Mass and Energy, 2, E = mc 2 msleadngly explaned n secondary educaton. page 15 of energy and ce ersa. Probably Ensten would hae wrtten t ths way f ths was n hs mnd. To me t seems more lkely that Ensten meant: energy and matter are dentcal, up to now we hae made an unjustfable dstncton between the two. To make that clear, t s not suffcent to know that all energy has mass. Ths only means that energy and matter hae a property n common, namely they hae mass. We must also show that all matter represents energy and ce ersa. I hae ntroduced the term matter here. Ths s necessary, snce pre-relatstc physcs (pror to the frst publcaton of E = mc 2 on 27 September, 1905) consdered the terms matter and mass as nterchangeable. We hae just seen that energy has mass too, so the dfference between matter and energy temporarly requres redefnton. Let us defne matter as eerythng made of atoms and ther consttuents (electrons, protons, and neutrons). The defnton of energy s: the ablty to do work. Now, electrons, protons and neutrons all hae ther so called ant-partcles. If, for example, a proton colldes wth an ant-proton, both partcles are destroyed and photons are created. Een n prerelatstc physcs, t was already clear that photons (lght) could be conerted nto all the other forms of energy known to us, and ce ersa. Also, photons can be conerted to partcle ant-partcle pars. Indeed, matter and energy can be freely conerted from one to the other (proded we consder ant-matter to be ncluded n the defnton of matter). The defnton of matter just gen for the sake of argument s too lmted. The only meanngful defnton of matter s: eerythng that has mass. But now t has become mpossble to pont out any essental dfference between matter and energy: they are dentcal. Mass can be taken as a measure of both. Why do practcally all physcsts manly thnk n terms of rest mass? Ths s probably because the mportance of rest mass s emphassed so much by unersty lecturers. Some of them een suggest that relatstc mass s not a useful concept. Ths emphass comes manly from the fact that the rest mass s narant under Lorentz transformatons. In other words: the rest mass plays an mportant and elegant role n the mathematcs of relatty theory. But ths role s n no way dsturbed by my ew of relatstc mass. My concluson s as follows: t would be better to say that mass and energy are dentcal, that E = mc² must be seen as an dentty, not as a reacton equaton. Consequently, we can use the same unt for mass and energy. By force of habt we stll express them usng dfferent unts, namely klograms for mass, and joules for energy, but we must be able to conert these unts nto each other. Ths can be done wth E = mc². Compare ths wth the tme calores were stll n use. It also came from a tme when a dstncton was wrongly made between two forms of energy: heat (measured n calores) and energy (measured n joules). Untl qute recently, the formula 1 calore = 4.18 joule was common n school physcs books. One could just as well hae wrtten: E = 4.18(J/cal)*Q, wth E for energy (n joules) and Q for heat (n calores). The 4.18 plays the same role as the c² n the formula E = mc². The law of conseraton of mass, whose aldty s no longer recognsed n the course on relatty theory by prof. Nenhus (page 10 and 11), can now be restored. We can say: The law of conseraton of mass, as formulated amongst others by

17 page 16 Mass and Energy, 2, E = mc 2 msleadngly explaned n secondary educaton. Laoser, mpled that when matter s locked n a essel and none of t can escape, then ts total mass wll reman the same, rrespecte of ts physcal (for nstance meltng or eaporatng) and/or chemcal (for nstance burnng) changes. Laoser dd not thnk t necessary to preent energy from passng through the walls of the essel as well. He dd not know energy possesses mass too. Now we know ths, the old law of conseraton of mass can retan ts aldty, proded the contaner s closed to flows of mass n all ts forms (ncludng the ones we were used to call energy untl recently). *

18 Mass and Energy, 3, E = mc 2 and Poyntngs theorem. page 17 3 E = mc 2 and Poyntngs theorem Poyntngs theorem assgns the EM-feld as the resort of the EM-energy, expresses the energy densty n the electrcal feldstrength E and the magnetc nducton B and ges an equaton of contnuty, that s to say t suggests that the decrease per second of the EM-energy n a certan olume s equal to the energy flux that passes the boundary of that olume n an outward drecton plus the EM-energy that, through the work done by the Lorentz forces, s conerted nto other forms of energy: U t ( f ) d V = d V ( S d A) V wth V 1 U = ε E + c B ( ) A and Poyntngs theorem (2) S = c E B 2 ε 0 ( ) Where f s the Lorentz force densty on an nfntesmal part of an electrcally charged system, s the elocty of that part, U s the energy densty of the EM-feld S and s the so called Poyntng ector. Ths ector s the energy stream densty of the EM-feld. So the theorem states, work done by the EM-feld on a charge n a certan olume V (left hand sde of (2)) equals the decrease of EM-energy that s located n the olume plus the EM-energy possbly enterng the olume by ts boundary. Ths equaton s a relatstc coarant, so we can take t oer unmodfed n the theory of relatty. Its left hand sde s the work done on an electrcally charged system; note that n the theory of relatty too work s equal to force ntegrated to dsplacement. Work equals the energy suppled to a system, so work dded by c 2 s the mass suppled to a system. Whch mass? Edently the relatstc mass, as wll become clear n the followng. Only f we take for m and E from E = mc 2 the relatstc mass respectely the total energy, eerythng matches. If we dde equaton (2) by c 2 we get the statement: the relatstc mass ncrease of a charged system beng accelerated by an EM-feld s equal to the mass decrease of the EM-feld n ts mmedate cnty plus the mass of the EM-feld that possbly streams from elsewhere towards that cnty. Ths because f U s the EM-energy densty, then U/c 2 s the EM-mass densty. If S s the EM-energy stream densty, then S /c 2 s the mass stream densty. So: V U c 2 f V c 2 2 t ( ) d = dv c ( S da) V That the left hand sde of (3) s ndeed equal to the relatstc mass ncrease, can be proed for the case of a pontcharge as follows: frst state the left hand sde s equal A (3)

19 page 18 Mass and Energy, 3, E = mc 2 and Poyntngs theorem. to the ncrease of m rel. Then ths ncrease can be proen to be equal to m.γ m, whch means the statement to be correct; rest c F o rest dmrel dmrel dmrel = = c dt dt dt ( ) ( o ) d dm 2 2 rel 1 2 mrel = ( c ) dt dt 2 c 2 2 ( ) 2 d 1 dm = dt m dt rel rel ntegratng from a moment at whch = 0 (and therefore m rel = m rest ) tll a moment at whch = end (and so m rel = m rel ( end ) ) ges: [ ] [ ] 1 t 2 2 end tend 2 log( c ) = log mrel from where: ( ) 0 0 m = m rel end rest 1 1 c 2 2 so m rel = m.γ rest We obtan the correct formula for relatstc mass (so also the correct one for relatstc mass ncrease) of a pont mass wthout nternal degrees of freedom, therefore n ths case the left hand sde s ndeed equal to the mass ncrease. It s reasonable to suppose that ths s also true for more complex systems, such as systems wth nternal degrees of freedom. These can enlarge not only ther relatstc, but also ther narant mass, for example by absorpton of EM-radaton (ther mass ncrease s no longer, as for pont masses, a functon of ther elocty alone). As Formula (3) makes completely clear, the relatstc mass ncrease (for pont masses as well as for narant masses) caused by EM-forces s not merely an abstract mathematcal descrpton, t s caused by an ncomng flow of mass stored n the EM-feld. From formula (3) we can see that t follows the law of conseraton of mass (relatstc mass) for electrodynamcs f we ntegrate t oer tme and choose surface A n such a way, that the crcutal ntegral oer t s zero, that s to say f we take a closed system 1. Let us assume that for each type of force (so not only for the electromagnetc force) an equaton lke (3) can be wrtten, then t can be sad that for any system, each relatstc and/or narant and/or feld mass ncrease always goes hand n hand wth an equal decrease of relatstc and/or narant and/or feld mass n the outsde 1 Equaton (3) s also useful n another case. The B-feld of, for nstance, an electromagnet at rest and the E-feld of an electrcal charge at rest n the cnty ge a poyntng ector feld, whch ndcates that energy s crculatng n closed orbts n ths feld. Now n such an EM-feld an amount of angular momentum appears to be present (see Feynmans paradox on ths, Feynmans Lectures on physcs, part II, 17-4 ). Ths becomes clear when we swtch off the current through the electromagnet. Then the charge wll be brought nto moton as a result of the electromagnetc nducton and wll acqure a certan amount of angular momentum wth respect to the electromagnet. Ths angular momentum appears at frst sght to come out of nothng. Looked at t ths way we hae a paradox. Howeer, as s clear from equaton (3), such an EM-feld has an angular momentum, because t tells us that a quantty of mass s rotatng n closed orbts. Now the paradox s soled, because t s edent that the angular momentum obtaned by the system was stored n the EM-feld before.

20 Mass and Energy, 3, E = mc 2 and Poyntngs theorem. page 19 world. Ths supposton, namely that somethng lke (3) s true for all types of forces s completely justfed because t s nothng other than the relatstc prncple. Ths states not only that electrodynamc forces, but also all other types of forces, can be brought nto relatstc coarant form, an ndspensable postulate n the theory of relatty. It now becomes clear that statements lke 'ascrbng relatstc mass to a body ges a false pcture' and 'rest mass s the only mass you must work wth, ths we hae to call the mass' are completely wrong. The only dsadantage of speakng of the relatstc mass of a body s that you hae to say at the same tme n whch nertal system the mass s measured. On the other hand we hae a bg adantage here; as long as we don't swtch to another nertal frame and no work s done on the system by forces from the outsde world, the total relatstc mass does not change (the mass of the felds I also call relatstc mass, see page 7, lne 13 from bottom and page 13, text between the two blank lnes). Thnkng n rest masses, we get on the contrary a confusng mage of rest masses beng conerted nto knetc energy and ce ersa, rest masses beng "conerted" nto potental energes or ce ersa (n whch case one mostly forgets potental energy has rest mass too, thnk of the example wth the capactor), of a total rest mass that s dfferent each tme someone decdes to splt up the system n a dfferent way n supposed "pontmasses" (bllardballs as pont masses, or molecules, or elementary partcles, or fundamental partcles). A much more logcal approach s to call the relatstc mass of a system the mass and to see rest mass as a specal case of the mass, namely the mass at M = 0, wth m as centre of mass of the system. Note how much clearer the frst phrase of the preous paragraph then becomes:.... then t can be sad that for any system, each mass ncrease always goes hand n hand wth an equal decrease of mass n the outsde world. *

21 page 20 Mass and Energy, 3, E = mc 2 and Poyntngs theorem.

22 Mass and Energy, 4, Gratatonal mass. page 21 4 Gratatonal mass Another argument n faour of the relatstc mass s, that n one and the same nertal frame the sum of the relatstc masses of a system s equal to ts gratatonal mass. He who adds up the rest masses of, for example, the separate atoms of the system, defntely doesn't fnd the gratatonal mass. Ths can be shown as follows. We place a mnature nuclear reactor on one of the scales of a balance and brng t n equlbrum. 'The reactor s capable of conertng rest mass nto thermal energy. So rest mass dsappears, a part of the rest mass of, for example, uranum nucle.' The reactor has walls mpermeable to eerythng, ncludng heat, so all energy and matter stay nsde the reactor. Now f gratatonal mass should dsappear, the balance would dp. There s no law of nature forbddng the 'conerson of ths thermal energy a a fuson reacton back agan nto a rest mass ncrease (of atomc nucle).' Suppose the reactor does so. Then the balance would regan ts equlbrum. From ths moement, energy could be extracted, whereas n the reactor no net change of rest mass has taken place. Ths s n contradcton of energymass conseraton. The gratatonal mass therefore s not equal to the sum of the rest masses of the nucle, but s equal to the sum of the relatstc masses. For only then does the paradox dsappear, because the thermal energy has exactly the mass (relatstc mass), whch the nucle hae lost n ther fsson, so the balance mantans ts equlbrum. An obous crtcsm of ths reasonng s that the second law of thermodynamcs blocks the depcted process. Howeer, the second law forbds nothng. At most t states that such a cyclc process s hghly unlkely. *

23 page 22 Mass and Energy, 4, Gratatonal mass.

24 Mass and Energy, 5, Restmass. page 23 5 Rest mass Rest mass (or narant mass, I wll use them as synonyms) s usually seen as the most meanngful noton of mass, the relatstc mass s usually seen as of mnor mportance. To account for ths, t s ponted out that c tmes the rest mass s equal to the length of a four-ector, namely the energy-mpulse ector. The energy-mpulse ector s ( E c, p r ). The length of ths four-ector s, by defnton, (E 2 /c 2 - p 2 ) 1/2. Ths s another way of sayng that rest mass s narant for Lorentz transformatons, because f we swtch to an nertal system wth a dfferent elocty the length of a four-ector remans unaltered. I don't thnk ths makes rest mass nto somethng more than the relatstc mass at speed zero. Whoeer says: "The rest mass s narant" thnks thereafter "And relatstc mass s not so". But n ths way one compares mass at a gen speed (namely zero) to a mass at arable speed and therefore one compares ncompatble objects. To me "rest mass s narant for Lorentz transformatons" would only become somethng specal, f relatstc mass at a gen speed would not be so, f the relatstc mass at a speed of, for example, four ffths of c would not be relatstcally narant. But of course t s, because γ then s fe thrds, so m rel s then always fe thrds of the rest mass. So m rel (=0,8c) s also a Lorentz narant. The same can be sad for each speed between zero and c. The obous thng to say would then be: "The length of the energy-mpulse ector s m rest c" and not: ".... s m rel (0,8c)c3/5", but that s merely a choce for smplcty, not a real dstncton. I thnk the most mportant role of the rest mass les n somethng dfferent; f the rest mass of a body changes, then ts nternal energy changes. If a body has no nternal degrees of freedom, ts rest mass wll always be the same. A free electron cannot absorb a photon. If on paper we let an electron absorb a photon and calculate ts fnal elocty out of energy conseraton (=relatstc mass conseraton), mpulse s not consered. So n ths example we calculate from m. γ = m + E. c rest rest photon 2. Notce we ourseles put n the rest mass of the electron as unaltered. Ths becomes clear by the notaton, we ge rest mass n both sdes the same symbol. If the rest mass of an electron could ncrease, absorbton would be possble. Ths can easly be calculated, but can be seen wthout calculaton also. Transform to the

25 page 24 Mass and Energy, 5, Restmass. centre of mass system of photon + electron. In ths the electron s at rest after the absorbton, so the photon energy must hae been "conerted" to an ncrease of rest mass of the electron. So far ths process has neer been obsered and therefore we suppose for the tme beng that the electron has no nternal degrees of freedom. An atom can absorb a photon. It does hae nternal degrees of freedom, so t can ncrease ts nternal energy and therefore ts rest mass. Rest mass plays an mportant role n defnng knetc energy, amongst other thngs, as we know t n collsons. To llustrate ths we frst look at the classcal, so the non relatstc, knetc energy. In classcal mechancs the defnton n formula form of the classcal knetc energy for a pont mass s: E kn = 0,5m 2. If we look at a system composed of seeral pont masses, thngs get more complcated. What do we mean by the knetc energy? What do we mean wth: "An nelastc collson s a collson by whch the total knetc energy dmnshes"? We mean that the macroscopc knetc energy dmnshes, so the knetc energy of the centres of masses, not the macroscopc plus the thermal knetc energy (we could arrange a collson durng whch a chemcal reacton n the bodes causes a temperature rse, but we would neer add here the noled thermal knetc energy to the total knetc energy of the colldng bodes). If we speak wthout further specfcaton of the knetc energy of a body, we practcally always mean 0,5m M M 2, n whch m M s the mass of the body and M s the elocty of the centre of mass. A careful defnton of the (centre of mass-) knetc energy of a body composed of pont masses s: It s that part of the knetc energy of the pont masses, that can be transformed away by steppng oer to the centre of mass reference frame. The transformaton beng of course Gallean, we are stll dealng here wth non relatstc mechancs. In ths way t can be proen that the maxmum part of the knetc energy (knetc energy as the sum of the knetc energes of the separate atoms) s transformed away. Wth all other transformatons a smaller part of the knetc energy dsappears. Note that the knetc energy of a pont mass can be defned n a smlar way as aboe: It s the form of energy that becomes zero, f we step oer to the CMS of the pont mass. The relatstc knetc energy of a pont mass can be defned n exactly the same way as the classcal knetc energy. So for the relatstc knetc energy we hae: E = m ( γ 1) c 2. Ths formula follows from the erbal defnton, from kn rest E = mc 2 (proded we read for m the relatstc mass) and from the fact that a Lorentz transformaton leaes the rest mass unaltered. The relatstc knetc energy (of the centre of mass) of a system of pont masses can agan be defned n exactly the same way as n the precedng paragraph for the classcal knetc energy: E = m ( γ 1) c 2. For the n γ we must fll n the kn rest elocty of the centre of mass. I use m rest to mean narant mass. As already mentoned on page 9, last phrase before the blank lne, these two masses are the same n practce. Now agan the total knetc energy s mnmal n the centre of mass system. So you can say that rest mass (=narant mass) s the mnmum alue of the relatstc mass,