On the Electrodynamics of Moving Bodies by A. Einstein 1905 June 30

Transcription

1 On he lerodnamis of Moing Bodies b insein 95 June 3 I is nown ha Mawell s elerodnamis as usuall undersood a he presen ime when applied o moing bodies leads o asmmeries whih do no appear o be inheren in he phenomena Tae for eample he reiproal elerodnami aion of a magne and a onduor The obserable phenomenon here depends onl on he relaie moion of he onduor and he magne whereas he usomar iew draws a sharp disinion beween he wo ases in whih eiher he one or he oher of hese bodies is in moion For if he magne is in moion and he onduor a res here arises in he neighborhood of he magne an eleri field wih a erain definie energ produing a urren a he plaes where pars of he onduor are siuaed Bu if he magne is saionar and he onduor in moion no eleri field arises in he neighborhood of he magne In he onduor howeer we find an eleromoie fore o whih in iself here is no orresponding energ bu whih gies rise assuming equali of relaie moion in he wo ases disussed o eleri urrens of he same pah and inensi as hose produed b he eleri fores in he former ase amples of his sor ogeher wih he unsuessful aemps o disoer an moion of he earh relaiel o he ligh medium sugges ha he phenomena of elerodnamis as well as of mehanis possess no properies orresponding o he idea of absolue res The sugges raher ha as has alread been shown o he firs order of small quaniies he same laws of elerodnamis and opis will be alid for all frames of referene for whih he equaions of mehanis hold good We will raise his onjeure (he purpor of whih will hereafer be alled he Priniple of Relaii ) o he saus of a posulae and also inrodue anoher posulae whih is onl apparenl irreonilable wih he former namel ha ligh is alwas propagaed in emp spae wih a definie eloi whih is independen of he sae of moion of he emiing bod These wo posulaes suffie for he aainmen of a simple and onsisen heor of he elerodnamis of moing bodies based on Mawell s heor for saionar bodies The inroduion of a luminiferous eher will proe o be superfluous inasmuh as he iew here o be deeloped will no require an absoluel saionar spae proided wih speial properies nor assign a eloi-eor o a poin of he emp spae in whih eleromagnei proesses ae plae The heor o be deeloped is based lie all elerodnamis on he inemais of he rigid bod sine he asserions of an suh heor hae o do wih he relaionships beween rigid bodies (ssems of oordinaes) los and eleromagnei proesses Insuffiien onsideraion of his irumsane lies a he roo of he diffiulies whih he elerodnamis of moing bodies a presen enouners I Kinemaial Par Definiion of Simulanei Le us ae a ssem of oordinaes in whih he equaions of Newonian mehanis hold good In order o render our presenaion more preise and o disinguish his ssem of oordinaes erball from ohers whih will be inrodued hereafer we all i he saionar ssem If a maerial poin is a res relaiel o his ssem of oordinaes is posiion an be defined relaiel hereo b he emplomen of rigid sandards of measuremen and he mehods of ulidean geomer and an be epressed in Caresian oordinaes If we wish o desribe he moion of a maerial poin we gie he alues of is oordinaes as funions of he ime Now we mus bear arefull in mind ha a mahemaial desripion of his ind has no phsial meaning unless we are quie lear as o wha we undersand b ime We hae o ae ino aoun ha all our judgmens in whih ime plas a par are alwas judgmens of simulaneous eens If for insane I Translaed 998 Marh wih LaTeXTML 98p release b Nios Draos (nios@blleedsaus) CBLU Uniersi of Leeds; reformaed 8 Februar 6 n error (in ranslaion?) in he seond of he nd se of equaions in seion 6 was orreed Furher hanges in he noaion o bring i more up o dae hae been made (eg he use of 3-eor noaion) More deails are proided following he end of his paper

2 sa Tha rain arries here a 7 o lo I mean somehing lie his: The poining of he small hand of m wah o 7 and he arrial of he rain are simulaneous eens 3 I migh appear possible o oerome all he diffiulies aending he definiion of ime b subsiuing he posiion of he small hand of m wah for ime nd in fa suh a definiion is saisfaor when we are onerned wih defining a ime elusiel for he plae where he wah is loaed; bu i is no longer saisfaor when we hae o onne in ime series of eens ourring a differen plaes or wha omes o he same hing o ealuae he imes of eens ourring a plaes remoe from he wah We migh of ourse onen ourseles wih ime alues deermined b an obserer saioned ogeher wih he wah a he origin of he oordinaes and oordinaing he orresponding posiions of he hands wih ligh signals gien ou b eer een o be imed and reahing him hrough emp spae Bu his oordinaion has he disadanage ha i is no independen of he sandpoin of he obserer wih he wah or lo as we now from eperiene We arrie a a muh more praial deerminaion along he following line of hough If a he poin of spae here is a lo an obserer a an deermine he ime alues of eens in he immediae proimi of b finding he posiions of he hands whih are simulaneous wih hese eens If here is a he poin B of spae anoher lo in all respes resembling he one a i is possible for an obserer a B o deermine he ime alues of eens in he immediae neighborhood of B Bu i is no possible wihou furher assumpion o ompare in respe of ime an een a wih an een a B We hae so far defined onl an ime and a B ime We hae no defined a ommon ime for and B for he laer anno be defined a all unless we esablish b definiion ha he ime required b ligh o rael from o B equals he ime i requires o rael from B o Le a ra of ligh sar a he ime from owards B le i a he B ime B be refleed a B in he direion of and arrie again a a he ime In aordane wih definiion he wo los snhronie if B We assume ha his definiion of snhronism is free from onradiions and possible for an number of poins; and ha he following relaions are uniersall alid: If he lo a B snhronies wih he lo a he lo a snhronies wih he lo a B If he lo a snhronies wih he lo a B and also wih he lo a C he los a B and C also snhronie wih eah oher Thus wih he help of erain imaginar phsial eperimens we hae seled wha is o be undersood b snhronous saionar los loaed a differen plaes and hae eidenl obained a definiion of simulaneous or snhronous and of ime The ime of an een is ha whih is gien simulaneousl wih he een b a saionar lo loaed a he plae of he een his lo being snhronous and indeed snhronous for all ime deerminaions wih a speified saionar lo In agreemen wih eperiene we furher assume he quani B o be a uniersal onsan he eloi of ligh in emp spae I is essenial o hae ime defined b means of saionar los in he saionar ssem and he ime now defined being appropriae o he saionar ssem we all i he ime of he saionar ssem On he Relaii of Lenghs and Times The following refleions are based on he priniple of relaii and on he priniple of he onsan of he eloi of ligh These wo priniples we define as follows: The laws b whih he saes of phsial ssems undergo hange are no affeed wheher hese B

3 hanges of sae be referred o he one or he oher of wo ssems of oordinaes in uniform ranslaional moion n ra of ligh moes in he saionar ssem of oordinaes wih he deermined eloi wheher he ra be emied b a saionar or b a moing bod ene ligh pah eloi ime ineral where ime ineral is o be aen in he sense of he definiion in Le here be gien a saionar rigid rod; and le is lengh be l as measured b a measuring-rod whih is also saionar We now imagine he ais of he rod ling along he ais of of he saionar ssem of oordinaes and ha a uniform moion of parallel ranslaion wih eloi along he ais of in he direion of inreasing is hen impared o he rod We now inquire as o he lengh of he moing rod and imagine is lengh o be aserained b he following wo operaions: ( a ) The obserer moes ogeher wih he gien measuring-rod and he rod o be measured and measures he lengh of he rod direl b superposing he measuring-rod in jus he same wa as if ( b ) all hree were a res B means of saionar los se up in he saionar ssem and snhroniing in aordane wih he obserer aserains a wha poins of he saionar ssem he wo ends of he rod o be measured are loaed a a definie ime The disane beween hese wo poins measured b he measuring-rod alread emploed whih in his ase is a res is also a lengh whih ma be designaed he lengh of he rod In aordane wih he priniple of relaii he lengh o be disoered b he operaion ( a ) we will all i he lengh of he rod in he moing ssem mus be equal o he lengh l of he saionar rod The lengh o be disoered b he operaion ( b ) we will all he lengh of he (moing) rod in he saionar ssem This we shall deermine on he basis of our wo priniples and we shall find ha i differs from l Curren inemais ail assumes ha he lenghs deermined b hese wo operaions are preisel equal or in oher words ha a moing rigid bod a he epoh ma in geomerial respes be perfel represened b he same bod a res in a definie posiion We imagine furher ha a he wo ends and B of he rod los are plaed whih snhronie wih he los of he saionar ssem ha is o sa ha heir indiaions orrespond a an insan o he ime of he saionar ssem a he plaes where he happen o be These los are herefore snhronous in he saionar ssem We imagine furher ha wih eah lo here is a moing obserer and ha hese obserers appl o boh los he rierion esablished in for he snhroniaion of wo los Le a ra of ligh depar from a he ime 4 le i be refleed a B a he ime B and reah again a he ime Taing ino onsideraion he priniple of he onsan of he eloi of ligh we find ha rb rb B and B where r B denoes he lengh of he moing rod measured in he saionar ssem Obserers moing wih he moing rod would hus find ha he wo los were no snhronous while obserers in he saionar ssem would delare he los o be snhronous So we see ha we anno aah an absolue signifiaion o he onep of simulanei bu ha wo eens whih iewed from a ssem of oordinaes are simulaneous an no longer be looed upon as simulaneous eens when enisaged from a ssem whih is in moion relaiel o ha ssem

4 3 Theor of he Transformaion of Coordinaes and Times from a Saionar Ssem o noher Ssem in Uniform Moion of Translaion Relaiel o he Former Le us in saionar spae ae wo ssems of oordinaes ie wo ssems eah of hree rigid maerial lines perpendiular o one anoher and issuing from a poin Le he aes of X of he wo ssems oinide and heir aes of Y and Z respeiel be parallel Le eah ssem be proided wih a rigid measuring-rod and a number of los and le he wo measuring-rods and liewise all he los of he wo ssems be in all respes alie Now o he origin of one of he wo ssems ( K ) le a onsan eloi be impared in he direion of he inreasing of he oher saionar ssem ( K ) and le his eloi be ommuniaed o he aes of he oordinaes he relean measuring-rod and he los To an ime of he saionar ssem K here hen will orrespond a definie posiion of he aes of he moing ssem and from reasons of smmer we are eniled o assume ha he moion of K ma be suh ha he aes of he moing ssem are a he ime (his alwas denoes a ime of he saionar ssem) parallel o he aes of he saionar ssem We now imagine spae o be measured from he saionar ssem K b means of he saionar measuring-rod and also from he moing ssem K b means of he measuring-rod moing wih i; and ha we hus obain he oordinaes r ( ) and r ( ) respeiel Furher le he ime of he saionar ssem be deermined for all poins hereof a whih here are los b means of ligh signals in he manner indiaed in ; similarl le he ime of he moing ssem be deermined for all poins of he moing ssem a whih here are los a res relaiel o ha ssem b appling he mehod gien in of ligh signals beween he poins a whih he laer los are loaed To an ssem of alues whih ompleel defines he plae and ime of an een in he saionar ssem here belongs a ssem of alues deermining ha een relaiel o he ssem K and our as is now o find he ssem of equaions onneing hese quaniies In he firs plae i is lear ha he equaions mus be linear on aoun of he properies of homogenei whih we aribue o spae and ime If we plae i is lear ha a poin a res in he ssem K mus hae a ssem of alues independen of ime We firs define τ( ) as a funion of and To do his we hae o epress in equaions ha is nohing else han he summar of he daa of los a res in ssem K whih hae been snhronied aording o he rule gien in From he origin of ssem K le a ra be emied a he ime along he X -ais o and a he ime be refleed hene o he origin of he oordinaes arriing here a he ime ; we hen mus hae or b insering he argumens of he funion τ and appling he priniple of he onsan ( ) of he eloi of ligh in he saionar ssem: τ τ τ ene if be hosen infiniesimall small ( ) τ τ τ The original had τ (when wrien in our noaion here) This should be τ is purel onenional and immaerial (as indiaed in he paper) Oher snhroniaion onenions an be used and will all lead o differen oordinaiaions of he same geomer and he same phsis The one in frequen use in asronom for insane is he rearded ime The use of he ( )

5 or τ τ! I is o be noed ha insead of he origin of he oordinaes we migh hae hosen an oher poin for he poin of origin of he ra and he equaion jus obained is herefore alid for all alues of n analogous onsideraion applied o he aes of Y and Z i being borne in mind ha ligh is alwas propagaed along hese aes when iewed from he saionar ssem wih he eloi τ τ Sine τ is a linear funion i follows from hese equaions ha where a is a funion ( ) gies us a a presen unnown and where for brei i is assumed ha a he origin of K when Wih he help of his resul we easil deermine he quaniies ( ) r b epressing in equaions ha ligh (as required b he priniple of he onsan of he eloi of ligh in ombinaion wih he priniple of relaii) is also propagaed wih eloi when measured in he moing ssem For a ra of ligh emied a he ime in he direion of he inreasing or a Bu he ra moes relaiel o he iniial poin of K when measured in he saionar ssem wih he eloi so ha If we inser his alue of in he equaion for we obain a In an analogous manner we find b onsidering ras moing along he wo oher aes ha a when Thus a and a Subsiuing for is alue we obain! ieτ is linear in The alulus-based argumen is no neessar Boh he differeniabili of he funionτ and is lineari follow b a purel geomeri argumen; as firs shown b Robb in 95 The onl assumpion required is ha he dimension of spae be a leas One wa his migh be shown is o embed he geomer aboe ino a projeie geomer as an affine subspae and hen show ha he onl ransformaions onforming o he desripion aboe are projeie ransformaions In urn he onl suh ransformaions ha leae he affine subspae fied are affine ransformaions Finall he onl affine ransformaions ha are linear ransformaions leae he origin fied ( ) ( ) ( ) ( )

6 where ( ) ( ) ( ) ( ) ( ) and is an as e unnown funion of If no assumpion whaeer be made as o he iniial posiion of he moing ssem and as o he ero poin of an addiie onsan is o be plaed on he righ side of eah of hese equaions We now hae o proe ha an ra of ligh measured in he moing ssem is propagaed wih he eloi if as we hae assumed his is he ase in he saionar ssem; for we hae no as e furnished he proof ha he priniple of he onsan of he eloi of ligh is ompaible wih he priniple of relaii he ime when he origin of he oordinaes is ommon o he wo ssems le a spherial wae be emied herefrom and be propagaed wih he eloi in ssem K If r ( ) be a poin jus aained b his wae hen Transforming his equaion wih he aid of our equaions of ransformaion we obain afer a simple alulaion The wae under onsideraion is herefore no less a spherial wae wih eloi of propagaion when iewed in he moing ssem This shows ha our wo fundamenal priniples are ompaible 5 In he equaions of ransformaion whih hae been deeloped here eners an unnown funion of whih we will now deermine For his purpose we inrodue a hird ssem of oordinaes K whih relaiel o he ssem K is in a sae of parallel ranslaional moion parallel o he ais of X i suh ha he origin of oordinaes of ssem K moes wih eloi on he ais of X he ime le all hree origins oinide and when le he ime of he ssem K be ero We all he oordinaes measured in he ssem K r and b a wofold appliaion of our equaions of ransformaion we obain ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) The wording here needs orreion: he lous of is no a ime bu a plane a an insan namel he -plane a he insan i oinides wih he -plane i In insein s original paper he smbols ( Z ) defining hem In he 93 nglish ranslaion ( X Y Z ) Ξ for he oordinaes of he moing ssem K were inrodued wihou epliil were used reaing an ambigui beween X oordinaes in he fied ssem K and he parallel ais in moing ssem K ere and in subsequen referenes we use X when referring o he ais of ssem K along whih he ssem is ranslaing wih respe o K In addiion he referene o ssem K laer in his senene was inorrel gien as K in he 93 nglish ranslaion

7 Sine he relaions beween r and r do no onain he ime he ssems K and K are a res wih respe o one anoher and i is lear ha he ransformaion from K o K mus be he ideni ransformaion Thus ( ) ( ) We now inquire ino he signifiaion of ( ) We gie our aenion o ha par of he ais of Y of ssem K whih lies beween r and r ( l) This par of he ais of Y is a rod moing perpendiularl o is ais wih eloi relaiel o ssem K Is ends possess in K he oordinaes and l r ( ) r ( ) The lengh of he rod measured in K is herefore l ( ) ; and his gies us he meaning of he funion ( ) From reasons of smmer i is now eiden ha he lengh of a gien rod moing perpendiularl o is ais measured in he saionar ssem mus depend onl on he eloi and no on he direion and he sense of he moion The lengh of he moing rod measured in he saionar ssem does no hange herefore if and l l or are inerhanged ene follows ha ( ) ( ) ( ) ( ) I follows from his relaion and he one preiousl found ha ( ) so ha he ransformaion equaions whih hae been found beome ( ) where 4 Phsial Meaning of he quaions Obained in Respe o Moing Rigid Bodies and Moing Clos We enisage a rigid sphere 6 of radius R a res relaiel o he moing ssem K and wih is enre a he origin of oordinaes of K The equaion of he surfae of his sphere moing relaiel o he ssem K wih eloi is R The equaion of his surfae epressed in a he ime is R rigid bod whih measured in a sae of res has he form of a sphere herefore has in a sae of moion iewed from he saionar ssem he form of an ellipsoid of reoluion wih he aes R R R Thus whereas he Y and Z dimensions of he sphere (and herefore of eer rigid bod of no maer wha form) do no appear modified b he moion he X dimension appears shorened in he raio : ie he greaer he alue of he greaer he shorening For all moing objes iewed from he saionar ssem shriel up ino plane figures ii For eloiies greaer han ha of ligh ii In he original 93 nglish ediion his phrase was erroneousl ranslaed as plain figures I hae used he orre plane figures in his doumen

8 our deliberaions beome meaningless; we shall howeer find in wha follows ha he eloi of ligh in our heor plas he par phsiall of an infiniel grea eloi I is lear ha he same resuls hold good of bodies a res in he saionar ssem iewed from a ssem in uniform moion Furher we imagine one of he los whih are qualified o mar he ime when a res relaiel o he saionar ssem and he ime τ when a res relaiel o he moing ssem o be loaed a he origin of he oordinaes of K and so adjused ha i mars he ime Wha is he rae of his lo when iewed from he saionar ssem? Beween he quaniies and whih refer o he posiion of he lo we hae eidenl and Therefore whene i follows ha he ime mared b he lo (iewed in he saionar ssem) is slow b seonds per seond or negleing magniudes of fourh and higher order b From his here ensues he following peuliar onsequene If a he poins and B of K here are saionar los whih iewed in he saionar ssem are snhronous; and if he lo a is moed wih he eloi along he line B o B hen on is arrial a B he wo los no longer snhronie bu he lo moed from o B lags behind he oher whih has remained a B b magniudes of fourh and higher order) being he ime oupied in he journe from o B (up o I is a one apparen ha his resul sill holds good if he lo moes from o B in an polgonal line and also when he poins and B oinide If we assume ha he resul proed for a polgonal line is also alid for a oninuousl ured line we arrie a his resul: If one of wo snhronous los a is moed in a losed ure wih onsan eloi unil i reurns o he journe lasing seonds hen b he lo whih has remained a res he raeled lo on is arrial a will be seond slow Thene we onlude ha a balanelo 7 a he equaor mus go more slowl b a er small amoun han a preisel similar lo siuaed a one of he poles under oherwise idenial ondiions 5 The Composiion of Veloiies In he ssem K moing along he ais of X of he ssem K wih eloi le a poin moe in aordane wih he equaions w w where w and w denoe onsans Required: he moion of he poin relaiel o he ssem K If wih he help of he equaions of ransformaion deeloped in 3 we inrodue he quaniies ino he equaions of moion of he poin we obain w w w w

9 Thus he law of he parallelogram of eloiies is alid aording o our heor onl o a firs approimaion We se d d w V w w w a an iii d d w a is hen o be looed upon as he angle beween he eloiies and w fer a simple alulaion we obain i ( w w os a) w sin a V w os a I is worh of remar ha and w ener ino he epression for he resulan eloi in a smmerial manner If w also has he direion of he ais of X we ge w V w I follows from his equaion ha from a omposiion of wo eloiies whih are less han here alwas resuls a eloi less han For if we se κ w λ κ and λ being posiie and less han hen κ λ V < κ λ κλ I follows furher ha he eloi of ligh anno be alered b omposiion wih a eloi less han ha of ligh For his ase we obain w V w We migh also hae obained he formula for V for he ase when and w hae he same direion b ompounding wo ransformaions in aordane wih 3 If in addiion o he ssems K and K figuring in 3 we inrodue sill anoher ssem of oordinaes K moing parallel o K is iniial poin moing on he ais of Ξ wih he eloi w we obain equaions beween he quaniies and he orresponding quaniies of K whih differ from he equaions found in 3 onl in ha he plae of is aen b he quani w ; w from whih we see ha suh parallel ransformaions neessaril form a group We hae now dedued he requisie laws of he heor of inemais orresponding o our wo priniples and we proeed o show heir appliaion o elerodnamis II lerodnamial Par 6 Transformaion of he Mawell-er quaions for mp Spae On he Naure of he leromoie Fores Ourring in a Magnei Field During Moion Le he Mawell-er equaions for emp spae hold good for he saionar ssem K so ha we hae iii This equaion was inorrel gien in insein s original paper and he 93 nglish ranslaion as a an w w i The eponen of in he denominaor of he sine erm of his equaion was erroneousl gien as in he 93 ediion of his paper I has been orreed o uni here X in he 93 nglish ranslaion

10 where ( ) denoes he eor of he eleri fore and ( ) ha of he magnei fore If we appl o hese equaions he ransformaion deeloped in 3 b referring he eleromagnei proesses o he ssem of oordinaes here inrodued moing wih he eloi we obain he equaions or where Now he priniple of relaii requires ha if he Mawell-er equaions for emp spae hold good in ssem K he also hold good in ssem K ; ha is o sa ha he eors of he eleri and he magnei fore ( ) and ( ) of he moing ssem K whih are defined b heir ponderomoie effes on eleri or magnei masses respeiel saisf he following equaions: idenl he wo ssems of equaions found for ssem K mus epress eal he same hing sine boh ssems of equaions are equialen o he Mawell-er equaions for ssem K Sine furher he equaions of he wo ssems agree wih he eepion of he smbols for he eors i follows ha he funions ourring in he ssems of equaions a orresponding plaes mus agree wih he eepion of a faor ( ) ψ! whih is ommon for all funions of he one ssem of equaions and is independen of ( ) r and bu depends upon Thus we hae he relaions Furher ommens on he noaion ma be found in he ranslaion noes following he end of he paper! This is wrong: he need onl agree up o a ompleion ransformaion: ( ) ( ) θ ψ and ( ) ( ) θ ψ The preeding lause defining and was herefore added in he 8 reision so ha his orreion ould be poined ou here

11 where ( ) ψ ( ) ψ If we now form he reiproal of his ssem of equaions firsl b soling he equaions jus obained and seondl b appling he equaions o he inerse ransformaion (from K o K ) whih is haraeried b he eloi i follows when we onsider ha he wo ssems of equaions hus obained mus be ψ ψ Furher from reasons of smmer 8 and herefore idenial ha ( ) ( ) and our equaions assume he form ( ) ψ! s o he inerpreaion of hese equaions we mae he following remars: Le a poin harge of elerii hae he magniude one when measured in he saionar ssem K ie le i when a res in he saionar ssem eer a fore of one dne upon an equal quani of elerii a a disane of one m B he priniple of relaii his eleri harge is also of he magniude one when measured in he moing ssem If his quani of elerii is a res relaiel o he saionar ssem hen b definiion he eor is equal o he fore aing upon i If he quani of elerii is a res relaiel o he moing ssem (a leas a he relean insan) hen he fore aing upon i measured in he moing ssem is equal o he eor Consequenl he firs hree equaions aboe allow hemseles o be lohed in words in he wo following was: If a uni eleri poin harge is in moion in an eleromagnei field here as upon i in addiion o he eleri fore an eleromoie fore whih if we negle he erms muliplied b he seond and higher powers of is equal o he eor-produ of he eloi of he harge and he magnei fore diided b he eloi of ligh (Old manner of epression) If a uni eleri poin harge is in moion in an eleromagnei field he fore aing upon i is equal o he eleri fore whih is presen a he loali of he harge and whih we aserain b ransformaion of he field o a ssem of oordinaes a res relaiel o he elerial harge (New manner of epression) The analog holds wih magneomoie fores We see ha eleromoie fore plas in he deeloped heor merel he par of an auiliar onep whih owes is inroduion o he irumsane ha eleri and magnei fores do no eis independenl of he sae of moion of he ssem of oordinaes Furhermore i is lear ha he asmmer menioned in he inroduion as arising when we onsider he urrens produed b he relaie moion of a magne and a onduor now disappears Moreoer quesions as o he sea of elerodnami eleromoie fores (unipolar mahines) now hae no poin 7 Theor of Doppler s Priniple and of berraion In he ssem K er far from he origin of oordinaes le here be a soure of elerodnami waes whih in a par of spae onaining he origin of oordinaes ma be represened o a suffiien degree of approimaion b he equaions sin Φ sin Φ where r Φ ω! oweer o show ha he faor θ ( ) alluded o in he preious foonoe is also one needs o resor o he argumen in seion 9

12 ere ( ) and ( ) are he eors defining he ampliude of he waerain and ( ) he direion-osines of he wae-normals We wish o now he onsiuion of hese waes when he are eamined b an obserer a res in he moing ssem K ppling he equaions of ransformaion found in 6 for eleri and magnei fores and hose found in 3 for he oordinaes and he ime we obain direl Φ Φ Φ ω r sin sin where ( ) ( ) ( ) ω ω From he equaion for ω i follows ha if an obserer is moing wih eloi relaiel o an infiniel disan soure of ligh of frequen ν in suh a wa ha he onneing line soure-obserer maes he angle wih he eloi of he obserer referred o a ssem of oordinaes whih is a res relaiel o he soure of ligh he frequen ν of he ligh pereied b he obserer is gien b he equaion os ν ν This is Doppler s priniple for an eloiies whaeer When he equaion assumes he perspiuous form ν ν We see ha in onras wih he usomar iew when ν If we all he angle beween he wae-normal (direion of he ra) in he moing ssem and he onneing line soure-obserer he equaion for i assumes he form os os os This equaion epresses he law of aberraion in is mos general form If π he equaion beomes simpl os We sill hae o find he ampliude of he waes as i appears in he moing ssem If we all he ampliude of he eleri or magnei fore or respeiel aordingl as i is measured in he saionar ssem or in he moing ssem we obain ( ) os whih equaion if simplifies ino i rroneousl gien as l in he 93 nglish ranslaion propagaing an error despie a hange in smbols from he original 95 paper

13 I follows from hese resuls ha o an obserer approahing a soure of ligh wih he eloi his soure of ligh mus appear of infinie inensi 8 Transformaion of he nerg of Ligh Ras Theor of he Pressure of Radiaion ered on Perfe Refleors Sine 8π equals he energ of ligh per uni of olume we hae o regard 8π b he priniple of relaii as he energ of ligh in he moing ssem Thus would be he raio of he measured in moion o he measured a res energ of a gien ligh omple if he olume of a ligh omple were he same wheher measured in K or in K Bu his is no he ase If ( l m n) are he direionosines of he wae-normals of he ligh in he saionar ssem no energ passes hrough he surfae elemens of a spherial surfae moing wih he eloi of ligh: ( l) ( m) ( n) R r We ma herefore sa ha his surfae permanenl enloses he same ligh omple We inquire as o he quani of energ enlosed b his surfae iewed in ssem K ha is as o he energ of he ligh omple relaiel o he ssem K The spherial surfae iewed in he moing ssem is an ellipsoidal surfae he equaion for whih a he ime is l m n R If S is he olume of he sphere and S ha of his ellipsoid hen b a simple alulaion S S os Thus if we all he ligh energ enlosed b his surfae when i is measured in he saionar ssem and when measured in he moing ssem we obain S os S and his formula when simplifies ino I is remarable ha he energ and he frequen of a ligh omple ar wih he sae of moion of he obserer in aordane wih he same law Now le he oordinae plane be a perfel refleing surfae a whih he plane waes onsidered in 7 are refleed We see for he pressure of ligh eered on he refleing surfae and for he direion frequen and inensi of he ligh afer refleion Le he inidenal ligh be defined b he quaniies os ν (referred o ssem K ) Viewed from K he orresponding quaniies are os os os os ν ν os For he refleed ligh referring he proess o ssem K we obain os os ν ν

14 Finall b ransforming ba o he saionar ssem K we obain for he refleed ligh ( ) os os os os os os os os os ν ν ν The energ (measured in he saionar ssem) whih is iniden upon uni area of he mirror in uni ime is eidenl π 8 os The energ leaing he uni of surfae of he mirror in he uni of ime is ( ) π 8 os The differene of hese wo epressions is b he priniple of energ he wor done b he pressure of ligh in he uni of ime If we se down his wor as equal o he produ P where P is he pressure of ligh we obain ( ) os 8 π P In agreemen wih eperimen and wih oher heories we obain o a firs approimaion π P os 8 ll problems in he opis of moing bodies an be soled b he mehod here emploed Wha is essenial is ha he eleri and magnei fore of he ligh whih is influened b a moing bod be ransformed ino a ssem of oordinaes a res relaiel o he bod B his means all problems in he opis of moing bodies will be redued o a series of problems in he opis of saionar bodies 9 Transformaion of he Mawell-er quaions when Coneion-Currens are Taen ino oun We sar from he equaions ρ u ρ u ρ u ρ u where ρ denoes π 4 imes he densi of elerii and ( ) u u u u he eloi-eor of he harge If we imagine he eleri harges o be inariabl oupled o small rigid bodies (ions elerons) hese equaions are he eleromagnei basis of he Lorenian elerodnamis and opis of moing bodies Le hese equaions be alid in he ssem K and ransform hem wih he assisane of he equaions of ransformaion gien in 3 and 6 o he ssem K We hen obain he equaions

15 u ρ where u u u u u ( u ) ( u ) and u ρ ρ Sine as follows from he heorem of addiion of eloiies 5 he eor u is nohing else han he eloi of he eleri harge measured in he ssem K we hae he proof ha on he basis of our inemaial priniples he elerodnami foundaion of Loren s heor of he elerodnamis of moing bodies is in agreemen wih he priniple of relaii In addiion I ma briefl remar ha he following imporan law ma easil be dedued from he deeloped equaions: If an eleriall harged bod is in moion anwhere in spae wihou alering is harge when regarded from a ssem of oordinaes moing wih he bod is harge also remains when regarded from he saionar ssem K onsan Dnamis of he Slowl eleraed leron Le here be in moion in an eleromagnei field an eleriall harged parile (in he sequel alled an eleron ) for he law of moion of whih we assume as follows: If he eleron is a res a a gien epoh he moion of he eleron ensues in he ne insan of ime aording o he equaions d r m e d d d d m e m e m e d d d where r ( ) denoe he oordinaes of he eleron and m he mass of he eleron as long as is moion is slow Now seondl le he eloi of he eleron a a gien epoh be We see he law of moion of he eleron in he immediael ensuing insans of ime Wihou affeing he general haraer of our onsideraions we ma and will assume ha he eleron a he momen when we gie i our aenion is a he origin of he oordinaes and moes wih he eloi along he ais of X of he ssem K I is hen lear ha a he gien momen ( ) he eleron is a res relaiel o a ssem of oordinaes whih is in parallel moion wih eloi along he ais of X From he aboe assumpion in ombinaion wih he priniple of relaii i is lear ha in he immediael ensuing ime (for small alues of ) he eleron iewed from he ssem moes in aordane wih he equaions d r m e d refer o he ssem K If furher we deide in whih he smbols ( ) r and ( ) ha when and r hen τ and r he ransformaion equaions of 3 and 6 hold good so ha we hae r ( ( ) )

16 Wih he help of hese equaions we ransform he aboe equaions of moion from ssem K o ssem K and obain d r e 3 d m Taing he ordinar poin of iew we now inquire as o he longiudinal and he ranserse mass of he moing eleron We wrie he equaions () in he form 3 d d d m m m e e d d d e e e e are he omponens of he ponderomoie fore aing upon and remar firsl ha ( ) he eleron and are so indeed as iewed in a ssem moing a he momen wih he eleron wih he same eloi as he eleron (This fore migh be measured for eample b a spring balane a res in he las-menioned ssem) Now if we all his fore simpl he fore aing upon he eleron 9 and mainain he equaion mass aeleraion fore and if we also deide ha he aeleraions are o be measured in he saionar ssem K we derie from he aboe equaions m m Longiudinal Mass Transerse Mass 3 Wih a differen definiion of fore and aeleraion we should naurall obain oher alues for he masses This shows us ha in omparing differen heories of he moion of he eleron we mus proeed er auiousl We remar ha hese resuls as o he mass are also alid for ponderable maerial poins beause a ponderable maerial poin an be made ino an eleron (in our sense of he word) b he addiion of an eleri harge no maer how small We will now deermine he inei energ of he eleron If an eleron moes from res a he origin of oordinaes of he ssem K along he ais of X under he aion of an elerosai fore i is lear ha he energ wihdrawn from he elerosai field has he alue e dr e d s he eleron is o be slowl aeleraed and onsequenl ma no gie off an energ in he form of radiaion he energ wihdrawn from he elerosai field mus be pu down as equal o he energ of moion W of he eleron Bearing in mind ha during he whole proess of moion whih we are onsidering he firs of he equaions () applies we herefore obain 3 W e d m d m Thus when W beomes infinie Veloiies greaer han ha of ligh hae as in our preious resuls no possibili of eisene This epression for he inei energ mus also b irue of he argumen saed aboe appl o ponderable masses as well We will now enumerae he properies of he moion of he eleron whih resul from he ssem of equaions () and are aessible o eperimen From he seond equaion of he ssem () i follows ha an eleri fore and a magnei fore hae an equall srong defleie aion on an eleron moing wih he eloi when Thus we see ha i is possible b our heor o deermine he eloi of he eleron from he raio of he magnei power of defleion o he eleri power of defleion for an eloi b m e

17 appling he law m e This relaionship ma be esed eperimenall sine he eloi of he eleron an be direl measured eg b means of rapidl osillaing eleri and magnei fields From he deduion for he inei energ of he eleron i follows ha beween he poenial differene P raersed and he aquired eloi of he eleron here mus be he relaionship m P dr e 3 We alulae he radius of uraure of he pah of he eleron when a magnei fore is presen (as he onl defleie fore) aing perpendiularl o he eloi of he eleron From he seond of he equaions () we obain or d d R e m m R e These hree relaionships are a omplee epression for he laws aording o whih b he heor here adaned he eleron mus moe In onlusion I wish o sa ha in woring a he problem here deal wih I hae had he loal assisane of m friend and olleague M Besso and ha I am indebed o him for seeral aluable suggesions Noes on Translaion and Formaing This ediion of insein s On he lerodnamis of Moing Bodies reformaed in Februar 8 is deried from an on-line ersion prepared b John Waler in Noember 999 ha was based on he nglish ranslaion of his original 95 German-language paper (published as Zur lerodnami beweger Körper in nnalen der Phsi 7:89 95) whih appeared in he boo The Priniple of Relaii (hp://wwwamaonom/ee/obidos/sin/486685/fourmilabwwwfour) published in 93 b Mehuen and Compan Ld of London Mos of he papers in ha olleion are nglish ranslaions b W Perre and GB Jeffer from he German Das Relaiasprinip 4 h ed published b in 9 b Tübner ll of hese soures are now in he publi domain; his doumen deried from hem remains in he publi domain and ma be reprodued in an manner or medium wihou permission resriion aribuion or ompensaion Numbered foonoes are as he appeared in he 93 ediion and are lised here as endnoes dior s noes lised as foonoes agged in Roman numerals Oher foonoes speiall mared were added wih he 8 reformaing The 93 nglish ranslaion updaed he noaion in he 95 original o ha prealen in he 9 s (eg using for ligh speed insead of insein s V ) Furher hanges hae been made in he 8 ersion o updae he noaion o ha prealen hroughou mos of he h enur pas ha poin In pariular onersion o 3-eor noaion made been made hroughou In addiion oher hanges hae been made o improe he onsisen hroughou eg ssem of he original is denoed here K and is oordinaes ξ η ζ τ as is β is denoed here as as is he presen-da onenion; while he onenion hese das is o use β for original b ( l m n) are denoed here b ( ) insead The omponens of he wae eor denoed in he

18 The eleri fore ( ) and magnei fore ( ) ( X Y Z ) and ( L M N ) were denoed respeiel b in he original The hange was made neessar in par anhow o remoe he ambigui in he 95 original: X Y Z were also being used o denoe oordinae aes For he harge e is used in plae of ε The underling heor (er s) ail endorsed b insein in he 95 paper does no sril onform o Mawell s elerodnamis in ha i erapolaes beond he heor b posing in effe he i i onersions B and D ; ha is B i and D i for i These are equialen o he Loren relaions B µ and D ε ha haraerie he Mawell-Loren heor Bu boh are deparures from he Mawell To arrie a he more generalied heor ourrenes of and ma be inerpreed respeiel as B and D ndnoes The preeding memoir b Loren was no a his ime nown o he auhor ie o he firs approimaion 3 We shall no here disuss he ineaiude whih lurs in he onep of simulanei of wo eens a approimael he same plae whih an onl be remoed b an absraion 4 Time here denoes ime of he saionar ssem and also posiion of hands of he moing lo siuaed a he plae under disussion 5 The equaions of he Loren ransformaion ma be more simpl dedued direl from he ondiion ha in irue of hose equaions he relaion shall hae as is onsequene he seond relaion 6 Tha is a bod possessing spherial form when eamined a res 7 No a pendulum-lo whih is phsiall a ssem o whih he arh belongs This ase had o be eluded 8 If for eample ( ) and hen from reasons of smmer i is lear ha when hanges sign wihou hanging is numerial alue mus also hange sign wihou hanging is numerial alue 9 The definiion of fore here gien is no adanageous as was firs shown b M Plan I is more o he poin o define fore in suh a wa ha he laws of momenum and energ assume he simples form