,! ,pr9ciple ( ,relativity ,,! ,,pr9ciple ,,( ,,relativity ,,orig9al ,,pap]s ,,by ,a4 ,,e9/e9 ,,& ,h4 ,,m9k[ski ,,translat$ ,,into ,,engli% ,,by ,m4 ,n4 ,,saha ,,& ,s4 ,n4 ,,bose ,,lectur]s ,,on ,,physics ,,& ,,appli$ ,,ma!matics ,univ]sity ,college ( ,sci5ce1 ,calcutta ,univ]sity ,,) ,a ,,hi/orical ,,introduction ,,by ,p4 ,c4 ,,mahalanobis ,,pr(essor ,,( ,,physics1 ,,presid5cy ,,college1 ,,calcu4 ,,publi%$ ,,by ,,! ,,univ]sity ,,( ,,calcutta #aibj ,sole ,ag5ts ,r4 ,,cambray ,,co4 ,,pr9t$ ,,by ,,atul*&ra ,,bhatta*>yya1 ,,at ,,! ,,calcutta ,,univ]sity ,,press1 ,,s5ate ,,h\se1 ,,calcutta ,,table ,,( ,,cont5ts #a4 ,hi/orical ,introduction i-xxiii ,by ,mr4 ,p4 ,c4 ,mahalanobis4 #b4 ,on ! ,electrodynamics ( ,mov+ ,bodies #a-cd ,e9/e9's fir/ pap] on ! re/rict$ ,!ory ( ,relativity1 orig9ally publi%$ 9 ! ,annal5 d] ,physik 9 #aije4 ,translat$ f ! orig9al ,g]man by ,dr4 ,meghnad ,saha4 #c4 ,albre*t ,e9/e9 #ce-ci ,a %ort biographical note by ,dr4 ,meghnad ,saha4 #d4 ,pr9ciple ( ,relativity #a-eb ,h4 ,m9k[ski's orig9al pap] on ! re/rict$ ,pr9ciple ( ,relativity fir/ publi%$ 9 #aiji4 ,translat$ f ! orig9al ,g]man by ,dr4 ,meghnad ,saha4 #e4 ,app5dix to ! above by ,h4 ,m9k[ski #ec-hh ,translat$ by ,dr4 ,meghnad ,saha4 #f4 ,! ,g5]alis$ ,pr9ciple ( ,relativity #hi-afc ,a4 ,e9/e9's second pap] on ! ,g5]alis$ ,pr9ciple fir/ publi%$ 9 #aiaf4 ,translat$ f ! orig9al ,g]man by ,mr4 ,saty5drana? ,bose4 #g4 ,notes #afe-ahe ,transcrib]'s ,note3 ,! pla9 text v]sion ( ?is ebook includes complex ma!matical =mulas4 ,"s are simple 9-l9e expressions l k #a - #a_/b4 ,!y may include special notations su* z xy = x to ! p[] ( y1 xy = x ) a subscript ( y1 a = an 'a' ) a b> across ! top1 4a = an 'a' ) a dot ov] x1 44a = an 'a' ) two dots ov] x4 ,o!rs are m complex 8,,ascii ,art0 l ?is3 l l #blc #bl t ------ ------ -------- --- c - u c u c - u c ,"s are s complex t !y mu/ be r5d]$ 9 ! ,te,x ma!matical notation1 enclos$ betwe5 d\ble doll> signs1 l ?is3 beta 7#a - frac u#b;c#b7-;frac#ab ,,hi/orical ,,introduction ,"l ,kelv9 writ+ 9 #ahic1 9 8 preface to ! ,engli% edition ( ,h]tz's ,rese>*es on ,electric ,waves1 says 8_m work]s & _m ?9k]s h help$ to build up ! n9ete5? c5tury s*ool ( pl5um1 "o e!r = light1 heat1 electricity1 magnetism2 & ! ,g]man & ,engli% volumes conta9+ ,h]tz's electrical pap]s1 giv5 to ! _w 9 ! la/ decade ( ! c5tury1 w be a p]man5t monum5t ( ! spl5did consummation n[ realis$40 ,t5 ye>s lat]1 9 #aije1 we f9d ,e9/e9 decl>+ t 8! e!r w be prov$ to be sup]flu\s40 ,at fir/ sight ! revolution 9 sci5tific ?\ght br\ght ab\t 9 ! c\rse ( a s+le decade appe>s to be almo/ too viol5t4 ,a m c>eful ev5 ?\gh a rapid review ( ! subject w1 h[ev]1 %[ h[ ! ,!ory ( ,relativity gradually became a hi/orical necessity4 ,t[>ds ! beg9n+ ( ! n9ete5? c5tury1 ! lum9if]\s e!r came into prom95ce z a result ( ! brilliant successes ( ! wave !ory 9 ! h&s ( ,"y & ,fresnel4 ,9 its /ation>y aspect ! ela/ic solid e!r 0 ! outcome ( ! se>* = a m$ium 9 :i* ! light waves may 8undulate40 ,?is /ation>y e!r1 z %[n by ,"y1 also af=d$ a satisfactory explanation ( a/ronomical ab]ration4 ,b its v success gave rise to a ho/ ( new que/ions all be>+ on ! c5tral problem ( relative motion ( e!r & matt]4 ,arago's prism exp]im5t4--,! refractive index ( a glass prism dep5ds on ! incid5t velocity ( light outside ! prism & its velocity inside ! prism aft] refraction4 ,on ,fresnel's fix$ e!r hypo!sis1 ! incid5t light waves are situat$ 9 ! /ation>y e!r outside ! prism & move ) velocity c ) respect to ! e!r4 ,if ! prism moves ) a velocity u ) respect to ?is fix$ e!r1 !n ! incid5t velocity ( light ) respect to ! prism %\ld be c u4 ,?us ! refractive index ( ! glass prism %\ld dep5d on u ! absolute velocity ( ! prism1 i4e41 its velocity ) respect to ! fix$ e!r4 ,arago p]=m$ ! exp]im5t 9 #ahai1 b fail$ to detect ! expect$ *ange4 ,airy-,boscovit* wat]-telescope exp]im5t4--,boscovit* _h /ill e>li] 9 #agff1 rais$ ! v important "q ( ! dep5d5ce ( ab]ration on ! refractive index ( ! m$ium fill+ ! telescope4 ,ab]ration dep5ds on ! diff]5ce 9 ! velocity ( light outside ! telescope & its velocity inside ! telescope4 ,if ! latt] velocity *anges ow+ to a *ange 9 ! m$ium fill+ ! telescope1 ab]ration itself %\ld *ange1 t is1 ab]ration %\ld dep5d on ! nature ( ! m$ium4 ,airy1 9 #ahga fill$ up a telescope ) wat]--b fail$ to detect any *ange 9 ! ab]ration4 ,?us we get bo? 9 ! case ( ,arago prism exp]im5t & ,airy-,boscovit* wat]-telescope exp]im5t1 ! v />tl+ result t optical effects 9 a mov+ m$ium seem to be q indep5d5t ( ! velocity ( ! m$ium ) respect to ,fresnel's /ation>y e!r4 ,fresnel's convection coeffici5t k #a - #a_/b4--,possibly "s =m ( comp5sation is tak+ place4 ,work+ on ?is hypo!sis1 ,fresnel (f]$ 8 fam\s e!r convection !ory4 ,accord+ to ,fresnel1 ! pres5ce ( matt] implies a def9ite cond5sation ( e!r )9 ! region occupi$ by matt]4 ,?is 8cond5s$0 or excess portion ( e!r is suppos$ to be c>ri$ away ) its own piece ( mov+ matt]4 ,x %\ld be obs]v$ t only ! 8excess0 portion is c>ri$ away1 :ile ! re/ rema9s z /agnant z ev]4 ,a complete convection ( ! 8excess0 e!r ) ! full velocity u is optically equival5t to a p>tial convection ( ! total e!r 1 ) only a fraction ( ! velocity k4 u4 ,fresnel %[$ t if ?is convection coeffici5t k is #a - #a_/b 7 be+ ! refractive index ( ! prism71 !n ! velocity ( light aft] refraction )9 ! mov+ prism w\ld be alt]$ to j su* ext5t z w\ld make ! refractive index ( ! mov+ prism q indep5d5t ( its 8absolute0 velocity u4 ,! non-dep5d5ce ( ab]ration on ! 8absolute0 velocity u1 is also v easily expla9$ ) ! help ( ?is ,fresnelian convection-coeffici5t k4 ,/okes' visc\s e!r4--,x %\ld be rememb]$1 h[ev]1 t ,fresnel's /ation>y e!r is absolutely fix$ & is n at all di/urb$ by ! motion ( matt] ?r\gh x4 ,9 ?is respect ,fresnelian e!r _c be said to behave 9 any respectable physical fa%ion1 & ?is l$ ,/okes1 9 #ahde-df1 to con/ruct a m mat]ial type ( m$ium4 ,/okes assum$ t visc\s motion ensues ne> ! surface ( sep>ation ( e!r & mov+ matt]1 :ile at suffici5tly di/ant regions ! e!r rema9s :olly undi/urb$4 ,he %[$ h[ su* a visc\s e!r w\ld expla9 ab]ration if all motion 9 x 7 diff]5tially irrotational4 ,b 9 ord] to expla9 ! null ,arago effect1 ,/okes 0 compell$ to assume ! convection hypo!sis ( ,fresnel ) an id5tical num]ical value = k1 namely #a - #a_/b4 ,?us ! pre/ige ( ! ,fresnelian convection-coeffici5t 0 enhanc$1 if any?+1 by ! !oretical inve/igations ( ,/okes4 ,fizeau's exp]im5t4--,soon aft]1 9 #ahea1 x receiv$ direct exp]im5tal confirmation 9 a brilliant piece ( "w by ,fizeau4 ,if a divid$ beam ( light is re-unit$ aft] pass+ ?r\gh two adjac5t cyl9d]s fill$ ) wat]1 ord9>y int]f]5ce fr+es w be produc$4 ,if ! wat] 9 "o ( ! cyl9d]s is n[ made to fl[1 ! 8cond5s$0 e!r )9 ! fl[+ wat] w\ld be convect$ & w\ld produce a %ift 9 ! int]f]5ce fr+es4 ,! %ift actually obs]v$ agre$ v well ) a value ( k #a - #a_/b4 ,! ,fresnelian convection-coeffici5t n[ became firmly e/abli%$ z a consequ5ce ( a direct positive effect4 ,on ! o!r h&1 ! negative evid5ces 9 fav\r ( ! convection-coeffici5t _h also multipli$4 ,masc>t1 ,hoek1 ,maxwell & o!rs s\ght = def9ite *anges 9 diff]5t optical effects induc$ by ! motion ( ! e>? relative to ! /ation>y e!r4 ,b all su* attempts fail$ to reveal ! slighte/ trace ( any optical di/urbance due to ! 8absolute0 velocity ( ! e>?1 ?us prov+ conclusively t all ! diff]5t optical effects %>$ 9 ! g5]al comp5sation aris+ out ( ! ,fresnelian convection ( ! excess e!r4 ,x mu/ be c>efully not$ t ! ,fresnelian convection-coeffici5t implicitly assumes ! exi/5ce ( a fix$ e!r 7,fresnel7 or at lea/ a :olly /agnant m$ium at suffici5tly di/ant regions 7,/okes71 ) ref]5ce to :i* alone a convection velocity c h any significance4 ,?us ! convection-coeffici5t imply+ "s type ( a /ation>y or visc\s1 yet nev]!less 8absolute0 e!r1 succe$$ 9 expla9+ satisfactorily all kn[n optical facts d[n to #ahhj4 ,mi*elson-,morley ,exp]im5t4--,9 #ahha1 ,mi*elson & ,morley p]=m$ _! classical exp]im5ts :i* und]m9$ ! :ole /ructure ( ! old e!r !ory & ?us s]v$ to introduce ! new !ory ( relativity4 ,! fundam5tal idea und]ly+ ?is exp]im5t is q simple4 ,9 all old exp]im5ts ! velocity ( light situat$ 9 free e!r 0 comp>$ ) ! velocity ( waves actually situat$ 9 a piece ( mov+ matt] & presumably c>ri$ away by x4 ,! comp5satory effect ( ! ,fresnelian convection ( e!r af=d$ a satisfactory explanation ( all negative results4 ,9 ! ,mi*elson-,morley exp]im5t ! arrangem5t is q diff]5t4 ,if !re is a def9ite gap 9 a rigid body1 light waves situat$ 9 free e!r w take a def9ite "t 9 cross+ ! gap4 ,if ! rigid plat=m c>ry+ ! gap is set 9 motion ) respect to ! e!r 9 ! direction ( light propagation1 light waves 7:i* are ev5 n[ situat$ 9 free e!r7 %\ld presumably take a long] "t to cross ! gap4 ,we _c d bett] ?an quote ,edd+ton's description ( ?is fam\s exp]im5t4 8,! pr9ciple ( ! exp]im5t may be illu/rat$ by consid]+ a swimm] 9 a riv]4 ,x is easily realiz$ t x takes long] to swim to a po9t #ej y>ds up-/ream & back ?an to a po9t #ej y>ds across-/ream & back4 ,if ! e>? is mov+ ?r\gh ! e!r !re is a riv] ( e!r fl[+ ?r\gh ! laboratory1 & a wave ( light may be comp>$ to a swimm] travell+ ) con/ant velocity relative to ! curr5t4 ,if1 !n1 we divide a beam ( light into two p>ts1 & s5d "o-half swimm+ up ! /ream = a c]ta9 di/ance & !n 7by a mirror7 back to ! />t+ po9t1 & s5d ! o!r half an equal di/ance across /ream & back1 ! across-/ream beam %\ld arrive back fir/4 ----u ,o ,a----------44444444 x ,b ,let ! e!r be fl[+ relative to ! app>atus ) velocity u 9 ! direction ,ox1 & let ,,oa1 ,,ob1 be ! two arms ( ! app>atus ( equal l5g? l1 ,,oa be+ plac$ up-/ream4 ,let c be ! velocity ( light4 ,! "t = ! d\ble j\rney along ,,oa & back is l l #blc #bl t ------ ------ -------- --- c - u c u c - u c :]e beta 7#a - frac u#b;c#b7-;frac #ab a factor great] ?an unity4 ,= ! transv]se j\rney ! light mu/ h a compon5t velocity n up-/ream 7relative to ! e!r7 9 ord] to avoid be+ c>ri$ bel[ ,,ob3 & s9ce its total velocity is c1 its compon5t across-/ream mu/ be 7c - u71 ! "t = ! d\ble j\rney ,,ob is accord+ly t#b frac #b;asqrt c#b - u#b frac #b;ac beta s t t t4 ,b :5 ! exp]im5t 0 tri$1 x 0 f\nd t bo? p>ts ( ! beam took ! same "t1 z te/$ by ! int]f]5ce b&s produc$40 ,aft] a mo/ c>eful s]ies ( obs]vations1 ,mi*elson & ,morley fail$ to detect ! slighte/ trace ( any effect due to e>?'s motion ?r\gh e!r4 ,! ,mi*elson-,morley exp]im5t seems to %[ t !re is no relative motion ( e!r & matt]4 ,fresnel's /agnant e!r requires a relative velocity (--u4 ,?us ,mi*elson & ,morley !mselves ?\ght at fir/ t _! exp]im5t confirm$ ,/okes' visc\s e!r1 9 :i* no relative motion c ensue on acc\nt ( ! abs5ce ( slipp+ ( e!r at ! surface ( sep>ation4 ,b ev5 on ,/okes' !ory ?is visc\s fl[ ( e!r w\ld fall (f at a v rapid rate z we rec$e f ! surface ( sep>ation4 ,mi*elson & ,morley repeat$ _! exp]im5t at diff]5t heights f ! surface ( ! e>?1 b inv>iably obta9$ ! same negative results1 ?us fail+ to confirm ,/okes' !ory ( visc\s fl[4 ,lodge's exp]im5t4--,fur!r1 9 #ahic1 ,lodge p]=m$ 8 rotat+ sph]e exp]im5t :i* %[$ complete abs5ce ( any visc\s fl[ ( e!r due to mov+ masses ( matt]4 ,a divid$ beam ( light1 aft] repeat$ reflections )9 a v n>r[ gap betwe5 two massive hemisph]es1 0 all[$ to re-unite & ?us produce int]f]5ce b&s4 ,:5 ! two hemisph]es are set rotat+1 x is conceivable t ! e!r 9 ! gap w\ld be di/urb$ due to visc\s fl[1 & any su* fl[ w\ld be imm$iately detect$ by a di/urbance ( ! int]f]5ce b&s4 ,b actual obs]vation fail$ to detect ! slighte/ di/urbance ( ! e!r 9 ! gap1 due to ! motion ( ! hemisph]es4 ,lodge's exp]im5t ?us seems to %[ a complete abs5ce ( any visc\s fl[ ( e!r4 ,ap>t f ^! exp]im5tal discrepancies1 grave !oretical objections 7 urg$ aga9/ a visc\s e!r4 ,/okes himself _h %[n t 8 e!r mu/ be incompressible & all motion 9 x diff]5tially irrotational1 at ! same "t !re %\ld be absolutely no slipp+ at ! surface ( sep>ation4 ,n[ all ^! conditions _c be simultane\sly satisfi$ = any conceivable mat]ial m$ium )\t c]ta9 v special & arbitr>y assumptions4 ,?us ,/okes' e!r fail$ to satisfy ! v motive :i* _h l$ ,/okes to =mulate x1 namely1 ! desirability ( con/ruct+ a 8physical0 m$ium4 ,planck (f]$ modifi$ =ms ( ,/okes' !ory :i* seem$ capable ( be+ reconcil$ ) ! ,mi*elson-,morley exp]im5t1 b requir$ v special assumptions4 ,! v complexity & ! v arbitr>9ess ( ^! assumptions prev5t$ ,planck's e!r f atta9+ any degree ( practical importance 9 ! fur!r developm5t ( ! subject4 ,! sole crit]ion ( ! value ( any sci5tific !ory mu/ ultimately be its capacity = (f]+ a simple1 unifi$1 coh]5t & fruitful description ( obs]v$ facts4 ,9 proportion z a !ory becomes complex x loses 9 usefulness--a !ory :i* is oblig$ to requisition a :ole array ( arbitr>y assumptions 9 ord] to expla9 special facts is practically worse ?an useless1 z x s]ves to disjo91 r ?an to unite1 ! sev]al gr\ps ( facts4 ,! optical exp]im5ts ( ! la/ qu>t] ( ! n9ete5? c5tury %[$ ! impossibility ( con/ruct+ a simple e!r !ory1 :i* w\ld be am5able to analytic treatm5t & w\ld at ! same "t /imulate fur!r progress4 ,x %\ld be obs]v$ t x c\ld sc>cely be %[n t no logically consi/5t e!r !ory 0 possible2 inde$ 9 #aiaj1 ,h4 ,a4 ,wilson (f]$ a consi/5t e!r !ory :i* 0 at lea/ q neutral ) respect to all available optical data4 ,b ,wilson's e!r is almo/ :olly negative--its only virtue be+ t x does n directly contradict obs]v$ facts4 ,nei!r any direct confirmation nor a direct refutation is possible & x does n ?r[ any light on ! v>i\s optical ph5om5a4 ,a !ory l ?is be+ practically useless /&s self-condemn$4 ,we mu/ n[ consid] ! problem ( relative motion ( e!r & matt] f ! po9t ( view ( electrical !ory4 ,f #ahfj ! id5tity ( light z an electromagnetic vector became gradually e/abli%$ z a result ( ! brilliant 8displacem5t curr5t0 hypo!sis ( ,cl]k ,maxwell & 8 fur!r analytical inve/igations4 ,! ela/ic solid e!r became gradually trans=m$ into ! electromagnetic "o4 ,maxwell succe$$ 9 giv+ a fairly satisfactory acc\nt ( all ord9>y optical ph5om5a & little room 0 left = any s]i\s d\bts z reg>ds ! g5]al validity ( ,maxwell's !ory4 ,h]tz's rese>*es on electric waves1 fir/ c>ri$ out 9 #ahhf1 succe$$ 9 furni%+ a /rong exp]im5tal confirmation ( ,maxwell's !ory4 ,electric waves behav$ g5]ally l light waves ( v l>ge wave l5g?4 ,! or?odox ,maxwellian view locat$ ! dielectric pol>isation 9 ! electromagnetic e!r :i* 0 m]ely a trans=mation ( ,fresnel's /agnant e!r4 ,! magnetic pol>isation 0 look$ ^u z :olly second>y 9 orig91 be+ due to ! relative motion ( ! dielectric tubes ( pol>isation4 ,on ?is view ! ,fresnelian convection coeffici5t comes out to be 1 z %[n by ,j4 ,j4 ,?omson 9 #ahhj1 in/ead ( #a - 7#a_/7 z requir$ by optical exp]im5ts4 ,?is obvi\sly implies a complete failure to acc\nt = all ^? optical exp]im5ts :i* dep5d = _! satisfactory explanation on ! assumption ( a value = ! convection coeffici5t equal to #a - 7#a_/74 ,! modifications propos$ indep5d5tly by ,h]tz & ,heaviside f>e no bett]4#a ,!y po/ulat$ ! actual m$ium to be ! seat ( all electric pol>isation & fur!r emphasis$ ! reciprocal relation subsi/+ betwe5 electricity & magnetism1 ?us mak+ ! field equations m symmetrical4 ,on ?is view ! :ole ( ! pol>is$ e!r is c>ri$ away by ! mov+ m$ium1 & consequ5tly1 ! convection coeffici5t naturally becomes unity 9 ?is !ory1 a value q z discrepant z t obta9$ on ! orig9al ,maxwellian assumption4 ,?us nei!r ,maxwell's orig9al !ory nor its subsequ5t modifications z develop$ by ,h]tz & ,heaviside succe$$ 9 obta9+ a value = ,fresnelian coeffici5t equal to #a - 7#a_/b71 & consequ5tly /ood totally condemn$ f ! optical po9t ( view4 ,c]ta9 direct electromagnetic exp]im5ts involv+ relative motion ( pol>is$ dielectrics 7 no less conclusive aga9/ ! g5]alis$ !ory ( ,h]tz & ,heaviside4 ,accord+ to ,h]tz a mov+ dielectric w\ld c>ry away ! :ole ( its electric displacem5t ) x4 ,h5ce ! electromagnetic effect ne> ! mov+ dielectric w\ld be proportional to ! total electric displacem5t1 t is to ,k1 ! specific inductive capacity ( ! dielectric4 ,9 #aija1 ,blondlot work+ ) a /ream ( mov+ gas c\ld n detect any su* effect4 ,h4 ,a4 ,wilson repeat$ ! exp]im5t 9 an improv$ =m 9 #aijc & work+ ) ebonite f\nd t ! obs]v$ effect 0 proportional to ,k - #a in/ead ( to ,k4 ,= gases ,k is ne>ly equal to #a & h5ce practically no effect w be obs]v$ 9 _! case4 ,?is gives a satisfactory explanation ( ,blondlot's negative results4 ,r[l& _h %[n 9 #ahgf t ! magnetic =ce due to a rotat+ cond5s] 7! dielectric rema9+ /ation>y7 0 proportional to ,k1 ! sp4 ind4 cap4 ,on ! o!r h&1 ,rntg5 f\nd 9 #ahhh ! magnetic effect due to a rotat+ dielectric 7! cond5s] rema9+ /ation>y7 to be proportional to ,k - #a1 & n to ,k4 ,f9ally ,ei*5wald 9 #aijc f\nd t :5 bo? cond5s] & dielectric are rotat$ toge!r1 ! effect obs]v$ 0 q indep5d5t ( ,k1 a result q consi/5t ) ! two previ\s exp]im5ts4 ,! ,r[l& effect proportional to ,k1 toge!r ) ! opposite ,rntg5 effect proportional to #a - ,k1 makes ! ,ei*5wald effect indep5d5t ( ,k4 ,all ^! exp]im5ts toge!r ) ^? ( ,blondlot & ,wilson made x cle> t ! electromagnetic effect due to a mov+ dielectric 0 proportional to ,k - #a1 & n to ,k z requir$ by ,h]tz's !ory4 ,?us ! above gr\p ( exp]im5ts ) mov+ dielectrics directly contradict$ ! ,h]tz-,heaviside !ory4 ,! int]nal discrepancies inh]5t 9 ! classic e!r !ory _h n[ become too prom95t4 ,x 0 cle> t ! e!r concept _h f9ally outgr[n its usefulness4 ,! obs]v$ facts _h become too contradictory & too het]og5e\s to be r$uc$ to an organis$ :ole ) ! help ( ! e!r concept alone4 ,radical dep>tures f ! classical !ory _h become absolutely necess>y4 ,!re 7 sev]al out/&+ difficulties 9 connection ) anomal\s disp]sion1 selective reflection & selective absorption :i* c\ld n be satisfactory expla9$ 9 ! classic electromagnetic !ory4 ,x 0 evid5t t ! assumption ( "s k9d ( discret5ess 9 ! optical m$ium _h become inevitable4 ,su* an assumption naturally gave rise to an atomic !ory ( electricity1 namely1 ! mod]n electron !ory4 ,lor5tz _h po/ulat$ ! exi/5ce ( electrons s e>ly z #ahgh1 b x 0 n until "s ye>s lat] t ! electron !ory became firmly e/abli%$ on a satisfactory basis4 ,lor5tz assum$ t a mov+ dielectric m]ely c>ri$ away its own 8pol>isation d\blets10 :i* on 8 !ory gave rise to ! induc$ field proportional to ,k - #a4 ,! field ne> a mov+ dielectric is naturally proportional to ,k - #a & n to ,k4 ,lor5tz's !ory ?us gave a satisfactory explanation ( all ^? exp]im5ts ) mov+ dielectrics :i* requir$ effects proportional to ,k - #a4 ,lor5tz fur!r succe$$ 9 obta9+ a value = ! ,fresnelian convection coeffici5t equal to #a - #a_/b1 ! exact value requir$ by all optical exp]im5ts ( ! mov+ type4 ,we mu/ n[ g back to ,mi*elson & ,morley's exp]im5t4 ,we h se5 t bo? p>ts ( ! beam are situat$ 9 free e!r2 no mat]ial m$ium is involv$ 9 any portion ( ! pa?s actually trav]s$ by ! beam4 ,consequ5tly no comp5sation due to ,fresnelian convection ( e!r by mov+ m$ium is possible4 ,?us ,fresnelian convection comp5sation c h no possible application 9 ?is case4 ,yet "s m>vell\s comp5sation has evid5tly tak5 place :i* has completely mask$ ! 8absolute0 velocity ( ! e>?4 ,9 ,mi*elson & ,morley's exp]im5t1 ! di/ance travell$ by ! beam along ,,oa 7t is1 9 a direction p>allel to ! motion ( ! plat=m7 is #bl1 :ile ! di/ance travell$ by ! beam along ,,ob1 p]p5dicul> to ! direction ( motion ( ! plat=m1 is #bl4 ,yet ! mo/ c>eful exp]im5ts %[$1 z ,edd+ton says1 8t bo? p>ts ( ! beam took ! same "t z te/$ by ! int]f]5ce b&s produc$4 ,x w\ld seem t ,,oa & ,,ob c\ld n really h be5 ( ! same l5g?2 & if ,,ob 0 ( l5g? l1 ,,oa mu/ h be5 ( l5g? l_/4 ,! app>atus 0 n[ rotat$ ?r\gh #ij1 s t ,,ob became ! up-/ream4 ,! "t = ! two j\rneys 0 aga9 ! same1 s t #j;,b mu/ n[ be ! %ort] l5g?4 ,! pla9 mean+ ( ! exp]im5t is t bo? arms h a l5g? l :5 plac$ along ,oy 7p]p5dicul> to ! direction ( motion71 & automatically contract to a l5g? l_/1 :5 plac$ along ,ox 7p>allel to ! direction ( motion74 ,?is explanation 0 fir/ giv5 by ,fitz-,g]ald40 ,?is ,fitz-,g]ald contraction1 />tl+ en\gh 9 itself1 does n suffice4 ,assum+ ?is contraction to be a real "o1 ! di/ance travell$ ) respect to ! e!r is #bl & ! "t tak5 = ?is j\rney is #bl_/c4 ,b ! di/ance travell$ ) respect to ! plat=m is always #bl4 ,h5ce ! velocity ( light ) respect to ! plat=m is frac #blfrac #blbetac frac cbeta a v>iable quantity dep5d+ on ! 8absolute0 velocity ( ! plat=m4 ,b no trace ( su* an effect has ev] be5 f\nd4 ,! velocity ( light is always f\nd to be q indep5d5t ( ! velocity ( ! plat=m4 ,! pres5t difficulty _c be solv$ by any fur!r alt]ation 9 ! measure ( space4 ,! only rec\rse left op5 is to alt] ! measure ( "t z well1 t is1 to adopt ! concept ( 8local "t40 ,if a mov+ clock goes sl[] s t "o #8real' second becomes #a_/ second z measur$ 9 ! mov+ sy/em1 ! velocity ( light relative to ! plat=m w always rema9 c4 ,we mu/ adopt two v />tl+ hypo!ses1 namely1 ! ,fitz-,g]ald contraction & ! concept ( 8local "t10 9 ord] to give a satisfactory explanation ( ! ,mi*elson-,morley exp]im5t4 ,^! results 7 already rea*$ by ,lor5tz 9 ! c\rse ( fur!r developm5ts ( 8 electron !ory4 ,lor5tz us$ a special set ( trans=mation equations#b = "t :i* implicitly introduc$ ! concept ( local "t4 ,b he himself fail$ to atta* any special significance to x1 & look$ ^u x r z a m]e ma!matical artifice l imag9>y quantities 9 analysis or ! circle at inf9ity 9 projective geometry4 ,! orig9ality ( ,e9/e9 at ?is /age consi/s 9 8 successful physical int]pretation ( ^! results1 & view+ !m z ! coh]5t organis$ consequ5ces ( a s+le g5]al pr9ciple4 ,lor5tz e/abli%$ ! ,relativity ,!orem#c 7consi/+ m]ely ( a set ( trans=mation equations7 :ile ,e9/e9 g5]alis$ x into a ,univ]sal ,pr9ciple4 ,9 addition ,e9/e9 introduc$ fundam5tally new concepts ( space & "t1 :i* s]v$ to de/roy old feti%es & dem&$ a :olesale revision ( sci5tific concepts & ?us op5$ up new possibilities 9 ! syn!tic unification ( natural processes4 ,newton _h fram$ 8 laws ( motion 9 su* a way z to make !m q indep5d5t ( ! absolute velocity ( ! e>?4 ,uni=m relative motion ( e!r & matt] c\ld n be detect$ ) ! help ( dynamical laws4 ,accord+ to ,e9/e9 nei!r c\ld x be detect$ ) ! help ( optical or electromagnetic exp]im5ts4 ,?us ! ,e9/e9ian ,pr9ciple ( ,relativity ass]ts t all physical laws are indep5d5t ( ! #8absolute' velocity ( an obs]v]4 ,= diff]5t sy/ems1 ! =m ( all physical laws is cons]v$4 ,if we *ose ! velocity ( light#d to be ! fundam5tal unit ( measurem5t = all obs]v]s 7t is1 assume ! con/ancy ( ! velocity ( light 9 all sy/ems7 we c e/abli% a metric 8"o-"o0 correspond5ce betwe5 any two obs]v$ sy/ems1 su* correspond5ce dep5d+ only ! relative velocity ( ! two sy/ems4 ,e9/e9's ,relativity is ?us m]ely ! consi/5t logical application ( ! well kn[n physical pr9ciple t we c "k no?+ b relative motion4 ,9 ?is s5se x is a fur!r ext5sion ( ,newtonian ,relativity4 ,on ?is int]pretation1 ! ,lor5tz-,fitzg]ald contraction & 8local "t0 lose _! arbitr>y "*4 ,space & "t z measur$ by two diff]5t obs]v]s are naturally div]se1 & ! diff]5ce dep5ds only on _! relative motion4 ,bo? are equally valid2 !y are m]ely diff]5t descriptions ( ! same physical reality4 ,?is is ess5tially ! po9t ( view adopt$ by ,m9k[ski4 ,he consid]s "t itself to be "o ( ! co-ord9ate axes1 & 9 8 f\r-dim5sional _w1 t is 9 ! space-"t reality1 relative motion is r$uc$ to a rotation ( ! axes ( ref]5ce4 ,?us1 ! div]sity 9 ! measurem5t ( l5g?s & temporal rates is m]ely due to ! /atic diff]5ce 9 ! 8frame-"w0 ( ! diff]5t obs]v]s4 ,! above !ory ( ,relativity absorb$ practically ! :ole ( ! electromagnetic !ory bas$ on ! ,maxwell-,lor5tz sy/em ( field equations4 ,x comb9$ all ! advantages ( classic ,maxwellian !ory toge!r ) an electronic hypo!sis4 ,! ,lor5tz assumption ( pol>isation d\blets _h furni%$ a satisfactory explanation ( ! ,fresnelian convection ( e!r1 b 9 ! new !ory ?is is d$uc$ m]ely z a consequ5ce ( ! alt]$ concept ( relative velocity4 ,9 addition1 ! !ory ( ,relativity accept$ ! results ( ,mi*elson & ,morley's exp]im5ts z a def9ite pr9ciple1 namely1 ! pr9ciple ( ! con/ancy ( ! velocity ( light1 s t !re 0 no?+ left = explanation 9 ! ,mi*elson-,morley exp]im5t4 ,b ev5 m ?an all ?is1 x e/abli%$ a s+le g5]al pr9ciple :i* s]v$ to connect toge!r 9 a simple coh]5t & fruitful mann] ! kn[n facts ( ,physics4 ,! !ory ( ,relativity receiv$ direct exp]im5tal confirmation 9 sev]al directions4 ,repeat$ attempts 7 made to detect ! ,lor5tz-,fitzg]ald contraction4 ,any ord9>y physical contraction w usually h obs]vable physical results2 = example1 ! total electrical resi/ance ( a conductor w dim9i%4 ,tr\ton & ,noble1 ,tr\ton & ,rank9e1 ,rayleigh & ,brace1 & o!rs employ$ a v>iety ( diff]5t me?ods to detect ! ,lor5tz-,fitzg]ald contraction1 b inv>iably ) ! same negative results4 ,:e!r !re is an e!r or n1 uni=m velocity ) respect to x c nev] be detect$4 ,?is does n prove t !re is no su* ?+ z an e!r b c]ta9ly does r5d] ! e!r entirely sup]flu\s4 ,univ]sal comp5sation is due to a *ange 9 local units ( l5g? & "t1 or r1 be+ m]ely diff]5t descriptions ( ! same reality1 !re is no comp5sation at all4 ,!re 0 ano!r gr\p ( obs]v$ ph5om5a :i* c\ld sc>cely be fitt$ into a ,newtonian s*eme ( dynamics )\t do+ viol5ce to x4 ,! exp]im5tal "w ( ,kaufmann1 9 #aija1 made x abundantly cle> t ! 8mass0 ( an electron dep5d$ on its velocity4 ,s e>ly z #ahha1 ,j4 ,j4 ,?omson _h %[n t ! in]tia ( a *>g$ p>ticle increas$ ) its velocity4 ,abraham n[ d$uc$ a =mula = ! v>iation ( mass ) velocity1 on ! hypo!sis t an electron always rema9$ a rigid sph]e4 ,lor5tz proce$$ on ! assumption t ! electron %>$ 9 ! ,lor5tz-,fitzg]ald contraction & obta9$ a totally diff]5t =mula4 ,a v c>eful s]ies ( measurem5ts c>ri$ out indep5d5tly by ,b*]]1 ,wolz1 ,hupka & f9ally ,neumann 9 #aiac1 decid$ conclusively 9 fav\r ( ! ,lor5tz =mula4 ,?is 8contractile0 =mula foll[s imm$iately z a direct consequ5ce ( ! new ,!ory ( ,relativity1 )\t any assumption z reg>ds ! electrical orig9 ( in]tia4 ,?us ! complete agreem5t ( exp]im5tal facts ) ! pr$ictions ( ! new !ory mu/ be consid]$ z confirm+ x z a pr9ciple :i* goes ev5 beyond ! electron itself4 ,! greate/ triumph ( ?is new !ory consi/s1 inde$1 9 ! fact t a l>ge numb] ( results1 :i* _h =m]ly requir$ all k9ds ( special hypo!ses = _! explanation1 are n[ d$uc$ v simply z inevitable consequ5ces ( "o s+le g5]al pr9ciple4 ,we h n[ trac$ ! hi/ory ( ! developm5t ( ! re/rict$ or special !ory ( ,relativity1 :i* is ma9ly conc]n$ ) optical & electrical ph5om5a4 ,x 0 fir/ (f]$ by ,e9/e9 9 #aije4 ,t5 ye>s lat]1 ,e9/e9 =mulat$ 8 second !ory1 ! ,g5]alis$ ,pr9ciple ( ,relativity4 ,?is new !ory is ma9ly a !ory ( gravitation & has v little connection ) optics & electricity4 ,9 "o s5se1 ! second !ory is inde$ a fur!r g5]alisation ( ! re/rict$ pr9ciple1 b ! =m] does n really conta9 ! latt] z a special case4 ,e9/e9's fir/ !ory is re/rict$ 9 ! s5se t x only ref]s to uni=m rectil9e> motion & has no application to any k9d ( accel]at$ movem5ts4 ,e9/e9 9 8 second !ory ext5ds ! ,relativity ,pr9ciple to cases ( accel]at$ motion4 ,if ,relativity is to be univ]sally true1 !n ev5 accel]at$ motion mu/ be m]ely relative motion betwe5 matt] & matt]4 ,h5ce ! ,g5]alis$ ,pr9ciple ( ,relativity ass]ts t 8absolute0 motion _c be detect$ ev5 ) ! help ( gravitational laws4 ,all movem5ts mu/ be ref]r$ to def9ite sets ( co-ord9ate axes4 ,if !re is any *ange ( axes1 ! num]ical magnitude ( ! movem5ts w also *ange4 ,b accord+ to ,newtonian dynamics1 su* alt]ation 9 physical movem5ts c only be due to ! effect ( c]ta9 =ces 9 ! field4#e ,?us any *ange ( axes w introduce new 8geometrical0 =ces 9 ! field :i* are q indep5d5t ( ! nature ( ! body act$ on4 ,gravitational =ces also h ?is same rem>kable prop]ty1 & gravitation itself may be ( ess5tially ! same nature z ^! 8geometrical0 =ces introduc$ by a *ange ( axes4 ,?is leads to ,e9/e9's fam\s ,pr9ciple ( ,equival5ce4 ,a gravitational field ( =ce is /rictly equival5t to "o introduc$ by a trans=mation ( co-ord9ates & no possible exp]im5t c di/+ui% betwe5 ! two4 ,?us x may become possible to 8trans=m away0 gravitational effects1 at lea/ = suffici5tly small regions ( space1 by ref]r+ all movem5ts to a new set ( axes4 ,?is new 8framework0 may ( c\rse h all k9ds ( v complicat$ movem5ts :5 ref]r$ to ! old ,galilean or 8rectangul> unaccel]at$ sy/em ( co-ord9ates40 ,b !re is no reason :y we %\ld look ^u ! ,galilean sy/em z m fundam5tal ?an any o!r4 ,if x is f\nd simpl] to ref] all motion 9 a gravitational field to a special set ( co-ord9ates1 we may c]ta9ly look ^u ?is special 8framework0 7at lea/ = ! p>ticul> region conc]n$71 to be m fundam5tal & m natural4 ,we may1 /ill m simply1 id5tify ?is p>ticul> framework ) ! special local prop]ties ( space 9 t region4 ,t is1 we c look ^u ! effects ( a gravitational field z simply due to ! local prop]ties ( space & "t itself4 ,! v pres5ce ( matt] implies a modification ( ! *>act]i/ics ( space & "t 9 its neighb\rhood4 ,z ,edd+ton says 8matt] does n cause ! curvature ( space-"t4 ,x is ! curvature4 ,j z light does n cause electromagnetic oscillations2 x is ! oscillations40 ,we may look ^u ?is f a slightly diff]5t po9t ( view4 ,! ,g5]al ,pr9ciple ( ,relativity ass]ts t all motion is m]ely relative motion betwe5 matt] & matt]1 & z all movem5ts mu/ be ref]r$ to def9ite sets ( co-ord9ates1 ! gr\nd ( any possible framework mu/ ultimately be mat]ial 9 "*4 ,x is conv5i5t to take ! matt] actually pres5t 9 a field z ! fundam5tal gr\nd ( our framework4 ,if ?is is done1 ! special *>act]i/ics ( our framework w\ld naturally dep5d on ! actual di/ribution ( matt] 9 ! field4 ,b physical space & "t is completely def9$ by ! 8framework40 ,9 o!r words ! 8framework0 itself is space & "t4 ,h5ce we see h[ physical space & "t is actually def9$ by ! local di/ribution ( matt]4 ,!re are c]ta9 magnitudes :i* rema9 con/ant by any *ange ( axes4 ,9 ord9>y geometry di/ance betwe5 two po9ts is "o su* magnitude2 s t x y z is an inv>iant4 ,9 ! re/rict$ !ory ( light1 ! pr9ciple ( con/ancy ( light velocity dem&s t x y z - ct %\ld rema9 con/ant4 ,! sep>ation ds ( adjac5t ev5ts is def9$ by ds -dx - dy - dz cdt4 ,x is an ext5sion ( ! notion ( di/ance & ?is is ! new inv>iant4 ,n[ if x1 y1 z1 t are trans=m$ to any set ( new v>iables x1 x1 x1 x1 we %all get a quadratic expression = ds#b g#a2ax#ab #b;g#a2bx#ax#b 444 sum gi2jxixj :]e ! g's are functions ( x1 x1 x1 x dep5d+ on ! trans=mation4 ,! special prop]ties ( space & "t 9 any region are def9$ by ^! g's :i* are !mselves det]m9$ by ! actual di/ribution ( matt] 9 ! locality4 ,?us f ! ,newtonian po9t ( view1 ^! g's repres5t ! gravitational effect ( matt] :ile f ! ,relativity /&-po9t1 ^! m]ely def9e ! non-,newtonian 7& incid5tally non-,euclidean7 space 9 ! neighb\rhood ( matt]4 ,we h se5 t ,e9/e9's !ory requires local curvature ( space-"t 9 ! neighb\rhood ( matt]4 ,su* alt]$ *>act]i/ics ( space & "t give a satisfactory explanation ( an out/&+ discrepancy 9 ! obs]v$ advance ( p]ihelion ( ,m]cury4 ,! l>ge discordance is almo/ completely remov$ by ,e9/e9's !ory4 ,aga91 9 an int5se gravitational field1 a beam ( light w be affect$ by ! local curvature ( space1 s t to an obs]v] :o is ref]r+ all ph5om5a to a ,newtonian sy/em1 ! beam ( light w appe> to deviate f its pa? along an ,euclidean /raight l9e4 ,?is fam\s pr$iction ( ,e9/e9 ab\t ! deflection ( a beam ( light by ! sun's gravitational field 0 te/$ dur+ ! total sol> eclipse ( ,may1 #aiai4 ,! obs]v$ deflection is decisively 9 fav\r ( ! ,g5]alis$ ,!ory ( ,relativity4 ,x %\ld be not$ h[ev] t ! velocity ( light itself w\ld decrease 9 a gravitational field4 ,?is may appe> at fir/ sight to be a violation ( ! pr9ciple ( con/ancy ( light-velocity4 ,b :5 we rememb] t ! ,special ,!ory is explicitly re/rict$ to ! case ( unaccel]at$ motion1 ! difficulty vani%es4 ,9 ! abs5ce ( a gravitational field1 t is 9 any unaccel]at$ sy/em1 ! velocity ( light w always rema9 con/ant4 ,?us ! validity ( ! ,special ,!ory is completely pres]v$ )9 its own re/rict$ field4 ,e9/e9 has propos$ a ?ird crucial te/4 ,he has pr$ict$ a %ift ( spectral l9es t[>ds ! r$1 due to an int5se gravitational pot5tial4 ,exp]im5tal difficulties are v consid]able "h1 z ! %ift ( spectral l9es is a complex ph5om5on4 ,evid5ce is conflict+ & no?+ conclusive c yet be ass]t$4 ,e9/e9 ?\ght t a gravitational displacem5t ( ! ,fraunh(] l9es is a necess>y & fundam5tal condition = ! acceptance ( 8 !ory4 ,b ,edd+ton has po9t$ out t ev5 if ?is te/ fails1 ! logical conclusion w\ld seem to be t :ile ,e9/e9's law ( gravitation is true = matt] 9 bulk1 x is n true = su* small mat]ial sy/ems z atomic oscillator4 ,conclusion ,f ! conceptual /&-po9t !re are sev]al important consequ5ces ( ! ,g5]alis$ or ,gravitational ,!ory ( ,relativity4 ,physical space-"t is p]ceiv$ to be intimately connect$ ) ! actual local di/ribution ( matt]4 ,euclid-,newtonian space-"t is n ! actual space-"t ( ,physics1 simply because ! =m] completely neglects ! actual pres5ce ( matt]4 ,euclid-,newtonian cont9uum is m]ely an ab/raction1 :ile physical space-99"t is ! actual framework :i* has "s def9ite curvature due to ! pres5ce ( matt]4 ,gravitational ,!ory ( ,relativity ?us br+s out cle>ly ! fundam5tal di/9ction betwe5 actual physical space-"t 7:i* is non-isotropic & non-,euclid-,newtonian7 on "o h& & ! ab/ract ,euclid-,newtonian cont9uum 7:i* is homog5e\s1 isotropic & a purely intellectual con/ruction7 on ! o!r4 ,! measurem5ts ( ! rotation ( ! e>? reveals a fundam5tal framework :i* may be call$ ! 8in]tial framework40 ,?is con/itutes ! actual physical univ]se4 ,?is univ]se approa*es ,galilean space-"t at a great di/ance f matt]4 ,! prop]ties ( ?is physical univ]se may be ref]r$ to "s _w-di/ribution ( matt] or ! 8in]tial framework0 may be con/ruct$ by a suitable modification ( ! law ( gravitation itself4 ,9 ,e9/e9's !ory ! actual curvature ( ! 8in]tial framework0 is ref]r$ to va/ quantities ( undetect$ _w-matt]4 ,x has int]e/+ consequ5ces4 ,! dim5sions ( ,e9/e9ian univ]se w\ld dep5d on ! quantity ( matt] 9 x2 x w\ld vani% to a po9t 9 ! total abs5ce ( matt]4 ,!n aga9 curvature dep5ds on ! quantity ( matt]1 & h5ce 9 ! pres5ce ( a suffici5t quantity ( matt] space-"t may curve r\nd & close up4 ,e9/e9ian univ]se w !n r$uce to a f9ite sy/em )\t b\nd>ies1 l ! surface ( a sph]e4 ,9 ?is 8clos$ up0 sy/em1 light rays w come to a focus aft] travell+ r\nd ! univ]se & we %\ld see an 8anti-sun0 7correspond+ to ! back surface ( ! sun7 at a po9t 9 ! sky opposite to ! real sun4 ,?is anti-sun w\ld ( c\rse be equally l>ge & equally bright if !re is no absorption ( light 9 free space4 ,9 de ,sitt]'s !ory1 ! exi/5ce ( va/ quantities ( _w-matt] is n requir$4 ,b beyond a def9ite di/ance f an obs]v]1 "t itself /&s /ill1 s t to ! obs]v] no?+ c ev] 8happ50 !re4 ,all ^! !ories are /ill highly speculative 9 "*1 b !y h c]ta9ly ext5d$ ! scope ( !oretical physics to ! c5tral problem ( ! ultimate nature ( ! univ]se itself4 ,"o out/&+ peculi>ity /ill atta*es to ! concept ( electric =ce--x is n am5able to any process ( be+ 8trans=m$ away0 by a suitable *ange ( framework4 ,h4 ,weyl1 x seems1 has develop$ a geometrical !ory 79 hyp]-space7 9 :i* no fundam5tal di/9ction is made betwe5 gravitational & electrical =ces4 ,e9/e9's !ory connects up ! law ( gravitation ) ! laws ( motion1 & s]ves to e/abli% a v intimate relation%ip betwe5 matt] & physical space-99"t4 ,space1 "t & matt] 7or en]gy7 7 consid]$ to be ! ?ree ultimate elem5ts 9 ,physics4 ,! re/rict$ !ory fus$ space-"t into "o indissoluble :ole4 ,! g5]alis$ !ory has fur!r syn!sis$ space-"t & matt] into "o fundam5tal physical reality4 ,space1 "t & matt] tak5 sep>ately are m ab/ractions4 ,physical reality consi/s ( a syn!sis ( all ?ree4 ,p4 ,c4 ,,mahalanobis4 ,note ,a4 ,= example consid] a massive p>ticle re/+ on a circul> disc4 ,if we set ! disc rotat+1 a c5trifugal =ce appe>s 9 ! field4 ,on ! o!r h&1 if we trans=m to a set ( rotat+ axes1 we mu/ introduce a c5trifugal =ce 9 ord] to correct = ! *ange ( axes4 ,?is newly introduc$ c5trifugal =ce is usually look$ ^u z a ma!matical fiction--z 8geometrical0 r ?an physical4 ,! pres5ce ( su* a geometrical =ce is usually int]pret$ z be+ due to ! adoption ( a fictiti\s framework4 ,on ! o!r h& a gravitational =ce is consid]$ q real4 ,?us a fundam5tal di/9ction is made betwe5 geometrical & gravitational =ces4 ,9 ! ,g5]al ,!ory ( ,relativity1 ?is fundam5tal di/9ction is done away )4 ,! v possibility ( di/+ui%+ betwe5 geometrical & gravitational =ces is d5i$4 ,all axes ( ref]5ce may n[ be reg>d$ z equally valid4 ,9 ! ,re/rict$ ,!ory1 all 8unaccel]at$0 axes ( ref]5ce 7 recognis$ z equally valid1 s t physical laws 7 made indep5d5t ( uni=m absolute velocity4 ,9 ! ,g5]al ,!ory1 physical laws are made indep5d5t ( 8absolute0 motion ( any k9d4 ,footnote #a3 ,see ,note #a4 ,footnote #b3 ,see ,note #b4 ,footnote #c3 ,see ,note #d4 ,footnote #d3 ,see ,notes #i & #ab4 ,footnote #e3 ,note ,a4 ,on ,! ,electrodynamics ( ,mov+ ,bodies ,by ,a4 ,e9/e94 ,,introduction4 ,x is well kn[n t if we attempt to apply ,maxwell's electrodynamics1 z conceiv$ at ! pres5t "t1 to mov+ bodies1 we are l$ to asymmetry :i* does n agree ) obs]v$ ph5om5a4 ,let u ?9k ( ! mutual action betwe5 a magnet & a conductor4 ,! obs]v$ ph5om5a 9 ?is case dep5d only on ! relative motion ( ! conductor & ! magnet1 :ile accord+ to ! usual conception1 a di/9ction mu/ be made betwe5 ! cases :]e ! "o or ! o!r ( ! bodies is 9 motion4 ,if1 = example1 ! magnet moves & ! conductor is at re/1 !n an electric field ( c]ta9 en]gy-value is produc$ 9 ! neighb\rhood ( ! magnet1 :i* excites a curr5t 9 ^? p>ts ( ! field :]e a conductor exi/s4 ,b if ! magnet be at re/ & ! conductor be set 9 motion1 no electric field is produc$ 9 ! neighb\rhood ( ! magnet1 b an electromotive =ce :i* corresponds to no en]gy 9 itself is produc$ 9 ! conductor2 ?is causes an electric curr5t ( ! same magnitude & ! same c>e] z ! electric =ce1 x be+ ( c\rse assum$ t ! relative motion 9 bo? ( ^! cases is ! same4 #b4 ,examples ( a simil> k9d su* z ! unsuccessful attempt to sub/antiate ! motion ( ! e>? relative to ! 8,light-m$ium0 lead u to ! supposition t n only 9 me*anics1 b also 9 electrodynamics1 no prop]ties ( obs]v$ facts correspond to a concept ( absolute re/2 b t = all coord9ate sy/ems = :i* ! me*anical equations hold1 ! equival5t electrodynamical & optical equations hold also1 z has already be5 %[n = magnitudes ( ! fir/ ord]4 ,9 ! foll[+ we make ^! assumptions 7:i* we %all subsequ5tly call ! ,pr9ciple ( ,relativity7 & introduce ! fur!r assumption1--an assumption :i* is at ! fir/ sight q irreconcilable ) ! =m] "o--t light is propagat$ 9 vacant space1 ) a velocity c :i* is indep5d5t ( ! nature ( motion ( ! emitt+ body4 ,^! two assumptions are q suffici5t to give u a simple & consi/5t !ory ( electrodynamics ( mov+ bodies on ! basis ( ! ,maxwellian !ory = bodies at re/4 ,! introduction ( a 8,light!r0 w be prov$ to be sup]flu\s1 = accord+ to ! conceptions :i* w be develop$1 we %all introduce nei!r a space absolutely at re/1 & end[$ ) special prop]ties1 nor %all we associate a velocity-vector ) a po9t 9 :i* electro-magnetic processes take place4 #c4 ,l e o!r !ory 9 electrodynamics1 ! !ory is bas$ on ! k9ematics ( rigid bodies2 9 ! enunciation ( e !ory1 we h to d ) relations betwe5 rigid bodies 7co-ord9ate sy/em71 clocks1 & electromagnetic processes4 ,an insuffici5t consid]ation ( ^! circum/ances is ! cause ( difficulties ) :i* ! electrodynamics ( mov+ bodies h to fight at pres5t4 ,i4--,,k9ematical ,,portion4 #a4 ,def9ition ( ,syn*ronism4 ,let u h a co-ord9ate sy/em1 9 :i* ! ,newtonian equations hold4 ,= di/+ui%+ ?is sy/em f ano!r :i* w be introduc$ h]eaft]1 we %all always call x 8! /ation>y sy/em40 ,if a mat]ial po9t be at re/ 9 ?is sy/em1 !n its position 9 ?is sy/em c be f\nd out by a measur+ rod1 & c be express$ by ! me?ods ( ,euclidean ,geometry1 or 9 ,c>tesian co-ord9ates4 ,if we wi% to describe ! motion ( a mat]ial po9t1 ! values ( its coord9ates mu/ be express$ z functions ( "t4 ,x is always to be borne 9 m9d t su* a ma!matical def9ition has a physical s5se1 only :5 we h a cle> notion ( :at is meant by "t4 ,we h to take into consid]ation ! fact t ^? ( our conceptions1 9 :i* "t plays a "p1 are always conceptions ( syn*ronism4 ,= example1 we say t a tra9 arrives "h at #g o'clock2 ?is means t ! exact po9t+ ( ! little h& ( my wat* to #g1 & ! arrival ( ! tra9 are syn*ron\s ev5ts4 ,x may appe> t all difficulties connect$ ) ! def9ition ( "t c be remov$ :5 9 place ( "t1 we sub/itute ! position ( ! little h& ( my wat*4 ,su* a def9ition is 9 fact suffici5t1 :5 x is requir$ to def9e "t exclusively = ! place at :i* ! clock is /ation$4 ,b ! def9ition is n suffici5t :5 x is requir$ to connect by "t ev5ts tak+ place at diff]5t /ations1--or :at am\nts to ! same ?+1--to e/imate by means ( "t 7zeitli* w]t57 ! occurr5ce ( ev5ts1 :i* take place at /ations di/ant f ! clock4 ,n[ ) reg>d to ?is attempt2--! "t-e/imation ( ev5ts1 we c satisfy ourselves 9 ! foll[+ mann]4 ,suppose an obs]v]--:o is /ation$ at ! orig9 ( coord9ates ) ! clock--associates a ray ( light :i* comes to him ?r\gh space1 & gives te/imony to ! ev5t ( :i* ! "t is to be e/imat$1--) ! correspond+ position ( ! h&s ( ! clock4 ,b su* an association has ?is defect1--x dep5ds on ! position ( ! obs]v] provid$ ) ! clock1 z we "k by exp]i5ce4 ,we c atta9 to a m practicable result by ! foll[+ treatm5t4 ,if an obs]v] be /ation$ at ,a ) a clock1 he c e/imate ! "t ( ev5ts occurr+ 9 ! imm$iate neighb\rhood ( ,a1 by look+ = ! position ( ! h&s ( ! clock1 :i* are syn*ron\s ) ! ev5t4 ,if an obs]v] be /ation$ at ,b ) a clock1--we %\ld add t ! clock is ( ! same nature z ! "o at ,a1--he c e/imate ! "t ( ev5ts occurr+ ab\t ,b4 ,b )\t fur!r premises1 x is n possible to comp>e1 z f> z "t is conc]n$1 ! ev5ts at ,b ) ! ev5ts at ,a4 ,we h hi!rto an ,a-"t1 & a ,b-"t1 b no "t common to ,a & ,b4 ,?is la/ "t 7i4e41 common "t7 c be def9$1 if we e/abli% by def9ition t ! "t :i* light requires 9 travell+ f ,a to ,b is equival5t to ! "t :i* light requires 9 travell+ f ,b to ,a4 ,= example1 a ray ( light proce$s f ,a at ,a-"t t,a t[>ds ,b1 arrives & is reflect$ f ,b at ,b-"t t,b1 & returns to ,a at ,a-"t t,a4 ,accord+ to ! def9ition1 bo? clocks are syn*ron\s1 if t,b - t,a t,a - t,b4 ,we assume t ?is def9ition ( syn*ronism is possible )\t involv+ any inconsi/5cy1 = any numb] ( po9ts1 !re=e ! foll[+ relations hold3-- #a4 ,if ! clock at ,b be syn*ron\s ) ! clock at ,a1 !n ! clock at ,a is syn*ron\s ) ! clock at ,b4 #b4 ,if ! clock at ,a z well z ! clock at ,b are bo? syn*ron\s ) ! clock at ,c1 !n ! clocks at ,a & ,b are syn*ron\s4 ,?us ) ! help ( c]ta9 physical exp]i5ces1 we h e/abli%$ :at we und]/& :5 we speak ( clocks at re/ at diff]5t /ations1 & syn*ron\s ) "o ano!r2 & !reby we h arriv$ at a def9ition ( syn*ronism & "t4 ,9 accordance ) exp]i5ce we %all assume t ! magnitude frac #b ov]l9e,,abt',a - t,a c :]e c is a univ]sal con/ant4 ,we h def9$ "t ess5tially ) a clock at re/ 9 a /ation>y sy/em4 ,on acc\nt ( its adaptability to ! /ation>y sy/em1 we call ! "t def9$ 9 ?is way z 8"t ( ! /ation>y sy/em40 #b4 ,on ! ,relativity ( ,l5g? & ,"t4 ,! foll[+ reflections are bas$ on ! ,pr9ciple ( ,relativity & on ! ,pr9ciple ( ,con/ancy ( ! velocity ( light1 bo? ( :i* we def9e 9 ! foll[+ way3-- #a4 ,! laws accord+ to :i* ! nature ( physical sy/ems alt] are indep5d5t ( ! mann] 9 :i* ^! *anges are ref]r$ to two co-ord9ate sy/ems :i* h a uni=m translators motion relative to ea* o!r4 #b4 ,e ray ( light moves 9 ! 8/ation>y co-ord9ate sy/em0 ) ! same velocity c1 ! velocity be+ indep5d5t ( ! condition :e!r ?is ray ( light is emitt$ by a body at re/ or 9 motion4#f ,!re=e velocity ,pa? ( ,light_/,int]val ( "t1 :]e1 by #8int]val ( "t' we mean "t z def9$ 9 #a4 ,let u h a rigid rod at re/2 ?is has a l5g? l1 :5 measur$ by a measur+ rod at re/2 we suppose t ! axis ( ! rod is laid along ! ,x-axis ( ! sy/em at re/1 & !n a uni=m velocity v1 p>allel to ! axis ( ,x1 is imp>t$ to x4 ,let u n[ enquire ab\t ! l5g? ( ! mov+ rod2 ?is c be obta9$ by ei!r ( ^! op]ations4-- 7a7 ,! obs]v] provid$ ) ! measur+ rod moves along ) ! rod to be measur$1 & measures by direct sup]position ! l5g? ( ! rod3--j z if ! obs]v]1 ! measur+ rod1 & ! rod to be measur$ 7 at re/4 7b7 ,! obs]v] f9ds out1 by means ( clocks plac$ 9 a sy/em at re/ 7! clocks be+ syn*ron\s z def9$ 9 #a71 ! po9ts ( ?is sy/em :]e ! ends ( ! rod to be measur$ occur at a p>ticul> "t t4 ,! di/ance betwe5 ^! two po9ts1 measur$ by ! previ\sly us$ measur+ rod1 ?is "t x be+ at re/1 is a l5g?1 :i* we may call ! 8l5g? ( ! rod40 ,accord+ to ! ,pr9ciple ( ,relativity1 ! l5g? f\nd out by ! op]ation a71 :i* we may call 8! l5g? ( ! rod 9 ! mov+ sy/em0 is equal to ! l5g? l ( ! rod 9 ! /ation>y sy/em4 ,! l5g? :i* is f\nd out by ! second me?od1 may be call$ #8! l5g? ( ! mov+ rod measur$ f ! /ation>y sy/em4' ,?is l5g? is to be e/imat$ on ! basis ( our pr9ciple1 & we %all f9d x to be diff]5t f l4 ,9 ! g5]ally recognis$ k9ematics1 we sil5tly assume t ! l5g?s def9$ by ^! two op]ations are equal1 or 9 o!r words1 t at an epo* ( "t t1 a mov+ rigid body is geometrically replaceable by ! same body1 :i* c replace x 9 ! condition ( re/4 ,relativity ( ,"t4 ,let u suppose t ! two clocks syn*ron\s ) ! clocks 9 ! sy/em at re/ are br\ght to ! ends ,a1 & ,b ( a rod1 i4e41 ! "t ( ! clocks correspond to ! "t ( ! /ation>y sy/em at ! po9ts :]e !y happ5 to arrive2 ^! clocks are !re=e syn*ron\s 9 ! /ation>y sy/em4 ,we fur!r imag9e t !re are two obs]v]s at ! two wat*es1 & mov+ ) !m1 & t ^! obs]v]s apply ! crit]ion = syn*ronism to ! two clocks4 ,at ! "t t,a1 a ray ( light goes out f ,a1 is reflect$ f ,b at ! "t t,b1 & arrives back at ,a at "t t,a4 ,tak+ into consid]ation ! pr9ciple (1 con/ancy ( ! velocity ( light1 we h t,b - t,a r,,ab_/7c - v71 & t,a - t,b r,,ab_/7c v71 :]e r,,ab is ! l5g? ( ! mov+ rod1 measur$ 9 ! /ation>y sy/em4 ,!re=e ! obs]v]s /ation$ ) ! wat*es w n f9d ! clocks syn*ron\s1 ?\gh ! obs]v] 9 ! /ation>y sy/em mu/ decl>e ! clocks to be syn*ron\s4 ,we !re=e see t we c atta* no absolute significance to ! concept ( syn*ronism2 b two ev5ts :i* are syn*ron\s :5 view$ f "o sy/em1 w n be syn*ron\s :5 view$ f a sy/em mov+ relatively to ?is sy/em4 #c4 ,!ory ( ,co-ord9ate & ,"t-,trans=mation f a /ation>y sy/em to a sy/em :i* moves relatively to ?is ) uni=m velocity4 ,let !re be giv51 9 ! /ation>y sy/em two co-ord9ate sy/ems1 i4e41 two s]ies ( ?ree mutually p]p5dicul> l9es issu+ f a po9t4 ,let ! ,x-axes ( ea* co9cide ) "o ano!r1 & ! ,y & ,z-axes be p>allel4 ,let a rigid measur+ rod1 & a numb] ( clocks be giv5 to ea* ( ! sy/ems1 & let ! rods & clocks 9 ea* be exactly alike ea* o!r4 ,let ! initial po9t ( "o ( ! sy/ems 7k7 h a con/ant velocity 9 ! direction ( ! ,x-axis ( ! o!r :i* is /ation>y sy/em ,k1 ! motion be+ also communicat$ to ! rods & clocks 9 ! sy/em 7k74 ,any "t t ( ! /ation>y sy/em ,k corresponds to a def9ite position ( ! axes ( ! mov+ sy/em1 :i* are always p>allel to ! axes ( ! /ation>y sy/em4 ,by t1 we always mean ! "t 9 ! /ation>y sy/em4 ,we suppose t ! space is measur$ by ! /ation>y measur+ rod plac$ 9 ! /ation>y sy/em1 z well z by ! mov+ measur+ rod plac$ 9 ! mov+ sy/em1 & we ?us obta9 ! co-ord9ates 7x1 y1 z7 = ! /ation>y sy/em1 & 71 1 7 = ! mov+ sy/em4 ,let ! "t t be det]m9$ = ea* po9t ( ! /ation>y sy/em 7:i* are provid$ ) clocks7 by means ( ! clocks :i* are plac$ 9 ! /ation>y sy/em1 ) ! help ( light-signals z describ$ 9 #a4 ,let also ! "t ( ! mov+ sy/em be det]m9$ = ea* po9t ( ! mov+ sy/em 79 :i* !re are clocks :i* are at re/ relative to ! mov+ sy/em71 by means ( ! me?od ( light signals betwe5 ^! po9ts 79 :i* !re are clocks7 9 ! mann] describ$ 9 #a4 ,to e value ( 7x1 y1 z1 t7 :i* fully det]m9es ! position & "t ( an ev5t 9 ! /ation>y sy/em1 !re correspond a sy/em ( values 71 1 1 72 n[ ! problem is to f9d out ! sy/em ( equations connect+ ^! magnitudes4 ,prim>ily x is cle> t on acc\nt ( ! prop]ty ( homog5eity :i* we ascribe to "t & space1 ! equations mu/ be l9e>4 ,if we put x x - vt1 !n x is cle> t at a po9t relatively at re/ 9 ! sy/em k1 we h a sy/em ( values 7x y z7 :i* are indep5d5t ( "t4 ,n[ let u f9d out z a function ( 7x1 y1 z1 t74 ,= ?is purpose we h to express 9 equations ! fact t is n o!r ?an ! "t giv5 by ! clocks :i* are at re/ 9 ! sy/em k :i* mu/ be made syn*ron\s 9 ! mann] describ$ 9 #a4 ,let a ray ( light be s5t at "t f ! orig9 ( ! sy/em k along ! ,x-axis t[>ds x & let x be reflect$ f t place at "t t[>ds ! orig9 ( mov+ co-ord9ates & let x arrive !re at "t 2 !n we mu/ h 7 7 ,if we n[ introduce ! condition t is a function ( co-ord9ates1 & apply ! pr9ciple ( con/ancy ( ! velocity ( light 9 ! /ation>y sy/em1 we h frac #ab 7tau 7#j1j1j1t7 tau 7#j1j1j17t frac x'c-v frac x'cv777 tau 7x'1#j1j1 t frac x'c-v7 ,x is to be notic$ t in/ead ( ! orig9 ( co-ord9ates1 we c\ld select "s o!r po9t z ! exit po9t = rays ( light1 & !re=e ! above equation holds = all values ( 7x1 y1 z1 t174 ,a simil> conception1 be+ appli$ to ! y- & z-axis gives u1 :5 we take into consid]ation ! fact t light :5 view$ f ! /ation>y sy/em1 is always propagat$ along ^? axes ) ! velocity 7c - v71 we h ! que/ions ---- #j1 ---- #j4 y z ,f ^! equations x foll[s t is a l9e> function ( x & t4 ,f equations 7#a7 we obta9 vx a 7t - -------- 7 c - v :]e a is an unkn[n function ( v4 ,) ! help ( ^! results x is easy to obta9 ! magnitudes 71 1 7 if we express by means ( equations ! fact t light1 :5 measur$ 9 ! mov+ sy/em is always propagat$ ) ! con/ant velocity c 7z ! pr9ciple ( con/ancy ( light velocity 9 conjunction ) ! pr9ciple ( relativity requires74 ,= a "t #j1 if ! ray is s5t 9 ! direction ( increas+ 1 we h vx c1 i4e4 a c7t - ------------ 7 c - v ,n[ ! ray ( light moves relative to ! orig9 ( k ) a velocity c - v1 measur$ 9 ! /ation>y sy/em2 !re=e we h x ---------- t c - v ,sub/itut+ ^! values ( t 9 ! equation = 1 we obta9 c a ------------- x c - v ,9 an analog\s mann]1 we obta9 by consid]+ ! ray ( light :i* moves along ! y-axis1 vx c a c7t - ------------- 7 c - v :]e y ------------------ t1 x #j1 7c - v7 ,!re=e c a ------------------ y1 7c - v7 c a ----------------- z 4 7c - v7 ,if = x1 we sub/itute its value x - tv1 we obta9 v4c 7v74 7t - ----------- 1 c 7v74 7x - vt7 1 7v7 y 7v7 z 1 :]e beta frac #asqrt #a - frac v#b;c#b & 7v7 ac _/ 7c - v7 a _/ is a function ( v4 ,if we make no assumption ab\t ! initial position ( ! mov+ sy/em & ab\t ! null-po9t ( t1 !n an additive con/ant is to be add$ to ! "r h& side4 ,we h n[ to %[1 t e ray ( light moves 9 ! mov+ sy/em ) a velocity c 7:5 measur$ 9 ! mov+ sy/em71 9 case1 z we h actually assum$1 c is also ! velocity 9 ! /ation>y sy/em2 = we h n z yet adduc$ any pro( 9 support ( ! assumption t ! pr9ciple ( relativity is reconcilable ) ! pr9ciple ( con/ant light-velocity4 ,at a "t t #j let a sph]ical wave be s5t out f ! common orig9 ( ! two sy/ems ( co-ord9ates1 & let x spread ) a velocity c 9 ! sy/em ,k4 ,if 7x1 y1 z71 be a po9t rea*$ by ! wave1 we h x y z ct ) ! aid ( our trans=mation-equations1 let u trans=m ?is equation1 & we obta9 by a simple calculation1 c4 ,!re=e ! wave is propagat$ 9 ! mov+ sy/em ) ! same velocity c1 & z a sph]ical wave4#g ,!re=e we %[ t ! two pr9ciples are mutually reconcilable4 ,9 ! trans=mations we h got an undet]m9$ function 7v71 & we n[ proce$ to f9d x out4 ,let u introduce = ?is purpose a ?ird co-ord9ate sy/em k1 :i* is set 9 motion relative to ! sy/em k1 ! motion be+ p>allel to ! -axis4 ,let ! velocity ( ! orig9 be 7-v74 ,at ! "t t #j1 all ! initial co-ord9ate po9ts co9cide1 & = t x y z #j1 ! "t t ( ! sy/em k #j4 ,we %all say t 7x y z t7 are ! co-ord9ates measur$ 9 ! sy/em k1 !n by a two-fold application ( ! trans=mation-equations1 we obta9 v 7-v77-v7 ----- c 7v77-v7t1 x 7v77v77 v7 7v77-v7x1 etc4 ,s9ce ! relations betwe5 7x1 y1 z1 t71 & 7x1 y1 z1 t7 d n conta9 "t explicitly1 !re=e ,k & k are relatively at re/4 ,x appe>s t ! sy/ems ,k & k are id5tical4 7v77-v7 #a4 ,let u n[ turn our att5tion to ! "p ( ! -axis betwe5 7 #j1 #j1 #j71 & 7 #j1 #a1 #j74 ,let ?is piece ( ! y-axis be cov]$ ) a rod mov+ ) ! velocity v relative to ! sy/em ,k & p]p5dicul> to its axis2--! ends ( ! rod hav+ !re=e ! co-ord9ates x vt1 y l _/ 7v71 z #j x vt1 y #j1 z #j ,!re=e ! l5g? ( ! rod measur$ 9 ! sy/em ,k is l_/7v74 ,= ! sy/em mov+ ) velocity 7-v71 we h on gr\nds ( symmetry1 l l -------- --------- 7v7 7-v7 7v7 7-v71 7v7 #a4 #d4 ,! physical significance ( ! equations obta9$ conc]n+ mov+ rigid bodies & mov+ clocks4 ,let u consid] a rigid sph]e 7i4e41 "o hav+ a sph]ical figure :5 te/$ 9 ! /ation>y sy/em7 ( radius ,r :i* is at re/ relative to ! sy/em 7,k71 & ^: c5tre co9cides ) ! orig9 ( ,k !n ! equation ( ! surface ( ?is sph]e1 :i* is mov+ ) a velocity v relative to ,k1 is ,r4 ,at "t t #j1 ! equation is express$ by means ( 7x1 y1 z1 t17 z frac x#b7sqrt #a - frac v#b;c#b7b y#b z#b ,r#b4 ,a rigid body :i* has ! figure ( a sph]e :5 measur$ 9 ! mov+ sy/em1 has !re=e 9 ! mov+ condition--:5 consid]$ f ! /ation>y sy/em1 ! figure ( a rotational ellipsoid ) semi-axes ,r sqrt #a - frac v#b;c#b1 ,r1 ,r4 ,!re=e ! y & z dim5sions ( ! sph]e 7!re=e ( any figure also7 d n appe> to be modifi$ by ! motion1 b ! x dim5sion is %ort5$ 9 ! ratio #a 3 sqrt #a - frac v#b;c#b2 ! %ort5+ is ! l>g]1 ! l>g] is v4 ,= v c1 all mov+ bodies1 :5 consid]$ f a /ation>y sy/em %r9k into planes4 ,= a velocity l>g] ?an ! velocity ( light1 our propositions become mean+less2 9 our !ory c plays ! "p ( inf9ite velocity4 ,x is cle> t simil> results hold ab\t /ation>y bodies 9 a /ation>y sy/em :5 consid]$ f a uni=mly mov+ sy/em4 ,let u n[ consid] t a clock :i* is ly+ at re/ 9 ! /ation>y sy/em gives ! "t t1 & ly+ at re/ relative to ! mov+ sy/em is capable ( giv+ ! "t 2 suppose x to be plac$ at ! orig9 ( ! mov+ sy/em k1 & to be s arrang$ t x gives ! "t 4 ,h[ mu* does ! clock ga91 :5 view$ f ! /ation>y sy/em ,k8 ,we h1 tau frac #asqrt #a-;frac v#b;c#b 7t - frac vc#bx71 & x vt1 !re=e tau - t 7#a - sqrt #a - frac v#b;c#b7 t4 ,!re=e ! clock loses by an am\nt 7v_/c7 p] second ( motion1 to ! second ord] ( approximation4 ,f ?is1 ! foll[+ peculi> consequ5ce foll[s4 ,suppose at two po9ts ,a & ,b ( ! /ation>y sy/em two clocks are giv5 :i* are syn*ron\s 9 ! s5se expla9$ 9 #c :5 view$ f ! /ation>y sy/em4 ,suppose ! clock at ,a to be set 9 motion 9 ! l9e jo9+ x ) ,b1 !n aft] ! arrival ( ! clock at ,b1 !y w no long] be f\nd syn*ron\s1 b ! clock :i* 0 set 9 motion f ,a w lag beh9d ! clock :i* _h be5 all along at ,b by an am\nt t7v_/c71 :]e t is ! "t requir$ = ! j\rney4 ,we see =?) t ! result holds also :5 ! clock moves f ,a to ,b by a polygonal l9e1 & also :5 ,a & ,b co9cide4 ,if we assume t ! result obta9$ = a polygonal l9e holds also = a curv$ l9e1 we obta9 ! foll[+ law4 ,if at ,a1 !re be two syn*ron\s clocks1 & if we set 9 motion "o ( !m ) a con/ant velocity along a clos$ curve till x comes back to ,a1 ! j\rney be+ complet$ 9 t-seconds1 !n aft] arrival1 ! la/ m5tion$ clock w be beh9d ! /ation>y "o by t7v_/c7 seconds4 ,f ?is1 we conclude t a clock plac$ at ! equator mu/ be sl[] by a v small am\nt ?an a simil>ly con/ruct$ clock :i* is plac$ at ! pole1 all o!r conditions be+ id5tical4 #e4 ,addition-,!orem ( ,velocities4 ,let a po9t move 9 ! sy/em k 7:i* moves ) velocity v along ! x-axis ( ! sy/em ,k7 accord+ to ! equation xi wxi tau1 eta weta tau1 zeta #j1 :]e w & w are con/ants4 ,x is requir$ to f9d out ! motion ( ! po9t relative to ! sy/em ,k4 ,if we n[ introduce ! sy/em ( equations 9 #c 9 ! equation ( motion ( ! po9t1 we obta9 x 7frac wxi v#a;frac vwxic#b7 t 1 y frac 7#a-;frac v#b;c#b7;frac #ab wetat #a;frac vwxic#b 1 z #j 4 ,! law ( p>allelogram ( velocities hold up to ! fir/ ord] ( approximation4 ,we c put ,u#b 7frac p>tial xp>tial t7#b 7frac p>tial yp>tial t7#b 1 w#b wxi#b weta#b 1 & alpha tan-#a frac wwxi i4e41 is put equal to ! angle betwe5 ! velocities v1 & w4 ,!n we h-- ,u frac 7v#b w#b #b vw cos alpha7 - 7frac vw s9 alphac7#b;frac #ab #a frac vw cos alphac#b ,x %\ld be notic$ t v & w ent] into ! expression = velocity symmetrically4 ,if w has ! direction ( ! -axis ( ! mov+ sy/em1 ,u frac v w#a frac vwc#b ,f ?is equation1 we see t by comb9+ two velocities1 ea* ( :i* is small] ?an c1 we obta9 a velocity :i* is always small] ?an c4 ,if we put v c - 1 & w c - 1 :]e & are ea* small] ?an c1#h ,u c frac #b;c - *i - lambda#b;c - *i - lambda frac *i lambdac#b c ,x is also cle> t ! velocity ( light c _c be alt]$ by add+ to x a velocity small] ?an c4 ,= ?is case1 ,u frac c v#a frac cvc#b c ,we h obta9$ ! =mula = ,u = ! case :5 v & w h ! same direction2 x c also be obta9$ by comb9+ two trans=mations accord+ to section #c4 ,if 9 addition to ! sy/ems ,k1 & k1 we introduce ! sy/em k1 ( :i* ! initial po9t moves p>allel to ! -axis ) velocity w1 !n betwe5 ! magnitudes1 x1 y1 z1 t & ! correspond+ magnitudes ( k1 we obta9 a sy/em ( equations1 :i* diff] f ! equations 9 #c1 only 9 ! respect t 9 place ( v1 we %all h to write1 frac v w#a frac vwc#b ,we see t su* a p>allel trans=mation =ms a gr\p4 ,we h d$uc$ ! k9ematics correspond+ to our two fundam5tal pr9ciples = ! laws necess>y = u1 & we %all n[ pass ov] to _! application 9 electrodynamics4 ,,ii4--,,electrodynamical ,,"p4 #f4 ,trans=mation ( ,maxwell's equations = ,pure ,vacuum4 ,on ! nature ( ! ,electromotive ,=ce caus$ by motion 9 a magnetic field4 ,! ,maxwell-,h]tz equations = pure vacuum may hold = ! /ation>y sy/em ,k1 s t frac #a;c frac p>tialp>tial t ,x1 ,y1 ,z beg9vmatrix frac p>tialp>tial x frac p>tialp>tial y frac p>tialp>tial z ,l ,m ,n endvmatrix & frac #a;c frac p>tialp>tial t ,l1 ,m1 ,n beg9vmatrix frac p>tialp>tial x frac p>tialp>tial y frac p>tialp>tial z ,x ,y ,z endvmatrix 7#a7 :]e ,x1 ,y1 ,z are ! compon5ts ( ! electric =ce1 ,l1 ,m1 ,n are ! compon5ts ( ! magnetic =ce4 ,if we apply ! trans=mations 9 #c to ^! equations1 & if we ref] ! electromagnetic processes to ! co-ord9ate sy/em mov+ ) velocity v1 we obta91 frac #a;c frac p>tialp>tial tau ,x1 beta 7,y - frac vc ,n71 beta 7,z frac vc ,m7 beg9vmatrix frac p>tialp>tial xi frac p>tialp>tial eta frac p>tialp>tial zeta ,l beta7,m frac vc ,z7 beta7,n - frac vc ,y7 endvmatrix & frac #a;c frac p>tialp>tial tau ,l1 beta7,m frac vc ,z71 beta7,n - frac vc ,y7 - beg9vmatrix frac p>tialp>tial xi frac p>tialp>tial eta frac p>tialp>tial zeta ,x beta7,y - frac vc ,n7 beta7,z frac vc ,m7 endvmatrix 444 7#b7 :]e beta frac #asqrt #a - frac v#b;c#b ,! pr9ciple ( ,relativity requires t ! ,maxwell-,h]tzian equations = pure vacuum %all hold also = ! sy/em k1 if !y hold = ! sy/em ,k1 i4e41 = ! vectors ( ! electric & magnetic =ces act+ ^u electric & magnetic masses 9 ! mov+ sy/em k1 :i* are def9$ by _! pond]motive reaction1 ! same equations hold1 444 i4e4 444 frac #a;c frac p>tialp>tial tau 7,x'1 ,y'1 ,z'7 beg9vmatrix frac p>tialp>tial xi frac p>tialp>tial eta frac p>tialp>tial zeta ,l' ,m' ,n' endvmatrix 1 frac #a;c frac p>tialp>tial tau 7,l'1 ,m'1 ,n'7 - beg9vmatrix frac p>tialp>tial xi frac p>tialp>tial eta frac p>tialp>tial zeta ,x' ,y' ,z' endvmatrix 444 7#c7 ,cle>ly bo? ! sy/ems ( equations 7#b7 & 7#c7 develop$ = ! sy/em k %all express ! same ?+s1 = bo? ( ^! sy/ems are equival5t to ! ,maxwell-,h]tzian equations = ! sy/em ,k4 ,s9ce bo? ! sy/ems ( equations 7#b7 & 7#c7 agree up to ! symbols repres5t+ ! vectors1 x foll[s t ! functions occurr+ at correspond+ places w agree up to a c]ta9 factor 7v71 :i* dep5ds only on v1 & is indep5d5t ( 71 1 1 74 ,h5ce ! relations1 v v ,x1 ,y1 ,z 7v7 ,x1 7,y - ----- ,n71 7,z ------ ,m71 c c v v ,l1 ,m1 ,n 7v7 ,l1 7,m - ----- ,z71 7,n ----- ,y71 c c ,!n by reason+ simil> to t foll[$ 9 7#c71 x c be %[n t 7v7 #a4 v v ,x1 ,y1 ,z ,x1 7,y - ----- ,n71 7,z ------ ,m7 c c v v ,l1 ,m1 ,n ,l1 7,m - ------ ,z71 7,n ----- ,y71 c c ,= ! int]pretation ( ^! equations1 we make ! foll[+ rem>ks4 ,let u h a po9t-mass ( electricity :i* is ( magnitude unity 9 ! /ation>y sy/em ,k1 i4e41 x ex]ts a unit =ce ^u a simil> quantity plac$ at a di/ance ( #a cm4 ,if ?is quantity ( electricity be at re/ 9 ! /ation>y sy/em1 !n ! =ce act+ ^u x is equival5t to ! vector 7,x1 ,y1 ,z7 ( electric =ce4 ,b if ! quantity ( electricity be at re/ relative to ! mov+ sy/em 7at lea/ = ! mom5t consid]$71 !n ! =ce act+ ^u x1 & measur$ 9 ! mov+ sy/em is equival5t to ! vector 7,x1 ,y1 ,z74 ,! fir/ ?ree ( equations 7#a71 7#b71 7#c71 c be express$ 9 ! foll[+ way3-- #a4 ,if a po9t-mass ( electric unit pole moves 9 an electro-magnetic field1 !n besides ! electric =ce1 an electromotive =ce acts ^u x1 :i*1 neglect+ ! numb]s involv+ ! second & high] p[]s ( v_/c1 is equival5t to ! vector-product ( ! velocity vector1 & ! magnetic =ce divid$ by ! velocity ( light 7,old mode ( expression74 #b4 ,if a po9t-mass ( electric unit pole moves 9 an electro-magnetic field1 !n ! =ce act+ ^u x is equival5t to ! electric =ce exi/+ at ! position ( ! unit pole1 :i* we obta9 by ! trans=mation ( ! field to a co-ord9ate sy/em :i* is at re/ relative to ! electric unit pole ,new mode ( expression4 ,simil> !orems hold ) ref]5ce to ! magnetic =ce4 ,we see t 9 ! !ory develop$ ! electro-magnetic =ce plays ! "p ( an auxili>y concept1 :i* owes its introduction 9 !ory to ! circum/ance t ! electric & magnetic =ces possess no exi/5ce indep5d5t ( ! nature ( motion ( ! co-ord9ate sy/em4 ,x is fur!r cle> t ! asymmetry m5tion$ 9 ! introduction :i* occurs :5 we treat ( ! curr5t excit$ by ! relative motion ( a magnet & a conductor disappe>s4 ,also ! "q ab\t ! seat ( electromagnetic en]gy is se5 to be )\t any mean+4 #g4 ,!ory ( ,dppl]'s ,pr9ciple & ,ab]ration4 ,9 ! sy/em ,k1 at a great di/ance f ! orig9 ( co-ord9ates1 let !re be a s\rce ( electrodynamic waves1 :i* is repres5t$ ) suffici5t approximation 9 a "p ( space n conta9+ ! orig91 by ! equations3-- ,x ,x s9 ,y ,y s9 ,z ,z s9 ,l ,l s9 ,m ,m s9 ,n ,n s9 lx my nz 7t - ------------ 7 c ,"h 7,x1 ,y1 ,z7 & 7,l1 ,m1 ,n7 are ! vectors :i* det]m9e ! amplitudes ( ! tra9 ( waves1 7l1 m1 n7 are ! direction-cos9es ( ! wave-normal4 ,let u n[ ask ourselves ab\t ! composition ( ^! waves1 :5 !y are inve/igat$ by an obs]v] at re/ 9 a mov+ m$ium k3--,by apply+ ! equations ( trans=mation obta9$ 9 #f = ! electric & magnetic =ces1 & ! equations ( trans=mation obta9$ 9 #c = ! co-ord9ates1 & "t1 we obta9 imm$iately3-- ,x ,x s9 v ,y 7,y - --- ,n7 s9 c v ,z 7,z - --- ,m7 s9 c ,l ,l s9 v ,m 7,m - --- ,z7 s9 c v ,n 7,n - --- ,y7 s9 c l m n 7t - --------------- 7 c :]e omega' omega beta 7#a - frac lvc7 1 l' frac l - frac vc#a - frac lvc 1 m' frac mbeta 7#a - frac lvc7 1 n' frac nbeta 7#a - frac lvc7 ,f ! equation = x foll[s3--,if an obs]v] moves ) ! velocity v relative to an inf9itely di/ant s\rce ( light emitt+ waves ( frequ5cy 1 9 su* a mann] t ! l9e jo9+ ! s\rce ( light & ! obs]v] makes an angle ( ) ! velocity ( ! obs]v] ref]r$ to a sy/em ( co-ord9ates :i* is /ation>y ) reg>d to ! s\rce1 !n ! frequ5cy :i* is p]ceiv$ by ! obs]v] is repres5t$ by ! =mula nu' nu frac #a - cos ,phi frac vc sqrt #a - frac v#b;c#b ,?is is ,dppl]'s pr9ciple = any velocity4 ,if #j1 !n ! equation takes ! simple =m nu' nu 7frac #a - frac vc#a frac vc7frac #ab ,we see t--contr>y to ! usual conception-- 1 = v -c4 ,if angle betwe5 ! wave-normal 7direction ( ! ray7 9 ! mov+ sy/em1 & ! l9e ( motion ( ! obs]v]1 ! equation = l takes ! =m cos ,phi' frac cos ,phi - frac vc #a - frac vc cos ,phi ,?is equation expresses ! law ( obs]vation 9 its mo/ g5]al =m4 ,if _/#b1 ! equation takes ! simple =m v cos --- c ,we h /ill to inve/igate ! amplitude ( ! waves1 :i* occur 9 ^! equations4 ,if ,a & ,a be ! amplitudes 9 ! /ation>y & ! mov+ sy/ems 7ei!r electrical or magnetic71 we h ,a'#b ,a#b frac 7#a - frac vc cos ,phi7#b #a - frac v#b;c#b ,if #j1 ?is r$uces to ! simple =m ,a'#b ,a#b frac #a - frac vc #a frac vc ,f ^! equations1 x appe>s t = an obs]v]1 :i* moves ) ! velocity c t[>ds ! s\rce ( light1 ! s\rce %\ld appe> inf9itely int5se4 #h4 ,trans=mation ( ! ,en]gy ( ! ,rays ( ,light4 ,!ory ( ! ,radiation-pressure on a p]fect mirror4 ,s9ce ,a_/#h is equal to ! en]gy ( light p] unit volume1 we h to reg>d ,a_/#h z ! en]gy ( light 9 ! mov+ sy/em4 ,a_/,a w\ld !re=e d5ote ! ratio betwe5 ! en]gies ( a def9ite light-complex 8measur$ :5 mov+0 & 8measur$ :5 /ation>y10 ! volumes ( ! light-complex measur$ 9 ,k & k be+ equal4 ,yet ?is is n ! case4 ,if l1 m1 n are ! direction-cos9es ( ! wave-normal ( light 9 ! /ation>y sy/em1 !n no en]gy passes ?r\gh ! surface elem5ts ( ! sph]ical surface 7x - clt7 7y - cmt7 7z - cnt7 ,r1 :i* exp&s ) ! velocity ( light4 ,we c !re=e say1 t ?is surface always encloses ! same light-complex4 ,let u n[ consid] ! quantity ( en]gy1 :i* ?is surface encloses1 :5 reg>d$ f ! sy/em k1 i4e41 ! en]gy ( ! light-complex relative to ! sy/em k4 ,reg>d$ f ! mov+ sy/em1 ! sph]ical surface becomes an ellipsoidal surface1 hav+1 at ! "t #j1 ! equation3-- 7beta xi - l beta frac vc xi7#b 7eta - m beta frac vc xi7#b 7zeta - n beta frac vc xi7#b ,r#b ,if ,s volume ( ! sph]e1 ,s volume ( ?is ellipsoid1 !n a simple calculation %[s t3 frac ,s',s frac betasqrt#a - frac vc cos ,phi ,if ,e d5otes ! quantity ( light en]gy measur$ 9 ! /ation>y sy/em1 ,e ! quantity measur$ 9 ! mov+ sy/em1 :i* are enclos$ by ! surfaces m5tion$ above1 !n frac ,e',e frac frac ,a'#bhpi ,s'frac ,a#bhpi,s frac #a - frac vc cos ,phisqrt#a - frac v#b;c#b ,if #j1 we h ! simple =mula3-- frac ,e',e 7frac#a - fracvc#a fracvc7frac#ab ,x is to be notic$ t ! en]gy & ! frequ5cy ( a light-complex v>y accord+ to ! same law ) ! /ate ( motion ( ! obs]v]4 ,let !re be a p]fectly reflect+ mirror at ! co-ord9ate-plane #j1 f :i* ! plane-wave consid]$ 9 ! la/ p>agraph is reflect$4 ,let u n[ ask ourselves ab\t ! light-pressure ex]t$ on ! reflect+ surface & ! direction1 frequ5cy1 int5sity ( ! light aft] reflexion4 ,let ! incid5t light be def9$ by ! magnitudes ,a cos 1 v 7ref]r$ to ! sy/em ,k74 ,reg>d$ f k1 we h ! correspond+ magnitudes3 ,a' ,a frac#a - fracvc cos ,phisqrt#a - fracv#b;c#b cos ,phi' fraccos ,phi - fracvc#a - fracvc cos ,phi nu' nu frac#a - fracvc cos ,phisqrt#a - fracv#b;c#b ,= ! reflect$ light we obta91 :5 ! process is ref]r$ to ! sy/em k3-- ,a ,a1 cos -cos 1 ,by means ( a back-trans=mation to ! /ation>y sy/em1 we obta9 ,k1 = ! reflect$ light3-- ,a''' ,a'' frac#a fracvccos ,phi''sqrt#a - fracv#b;c#b ,a frac#a - #b;fracvc cos ,phi fracv#b;c#ba - fracv#b;c#b 1 cos ,phi''' fraccos ,phi'' fracvc#a fracvccos ,phi'' - frac7#a fracv#b;c#b7 cos ,phi - #b fracvc #a - #b fracvc cos ,phi frac v#b;c#b 1 nu''' nu'' frac#a fracvc cos ,phi''sqrt#a - fracv#b;c#b nu frac#a - #b fracvc cos ,phi fracv#b;c#b 7#a - fracvc7#b ,! am\nt or en]gy fall+ ^u ! unit surface ( ! mirror p] unit ( "t 7measur$ 9 ! /ation>y sy/em7 is ,a_/7#h 7c cos - v774 ,! am\nt ( en]gy go+ away f unit surface ( ! mirror p] unit ( "t is ,a_/7#h 7-c cos v774 ,! diff]5ce ( ^! two expressions is1 accord+ to ! ,en]gy pr9ciple1 ! am\nt ( "w ex]t$1 by ! pressure ( light p] unit ( "t4 ,if we put ?is equal to ,p4v1 :]e ,p pressure ( light1 we h ,p #b frac,a#bhpi frac7cos ,phi - fracvc7#b #a - 7fracvc7#b ,z a fir/ approximation1 we obta9 ,a ,p #b -- cos #h :i* is 9 accordance ) facts1 & ) o!r !ories4 ,all problems ( optics ( mov+ bodies c be solv$ aft] ! me?od us$ "h4 ,! ess5tial po9t is1 t ! electric & magnetic =ces ( light1 :i* are influ5c$ by a mov+ body1 %\ld be trans=m$ to a sy/em ( co-ord9ates :i* is /ation>y relative to ! body4 ,9 ?is way1 e problem ( ! optics ( mov+ bodies w\ld be r$uc$ to a s]ies ( problems ( ! optics ( /ation>y bodies4 #i4 ,trans=mation ( ! ,maxwell-,h]tz ,equations4 ,let u />t f ! equations3-- frac#a;c7rho ux fracp>tial ,xp>tial t7 fracp>tial ,np>tial y - fracp>tial ,mp>tial z frac#a;c7rho uy fracp>tial ,yp>tial t7 fracp>tial ,lp>tial z - fracp>tial ,np>tial x frac#a;c7rho uz fracp>tial ,zp>tial t7 fracp>tial ,mp>tial x - fracp>tial ,lp>tial y frac#a;c fracp>tial ,lp>tial t fracp>tial ,yp>tial z - fracp>tial ,zp>tial y frac#a;c fracp>tial ,mp>tial t fracp>tial ,zp>tial x - fracp>tial ,xp>tial z frac#a;c fracp>tial ,np>tial t fracp>tial ,xp>tial y - fracp>tial ,yp>tial x :]e rho fracp>tial ,xp>tial x fracp>tial ,yp>tial y fracp>tial ,zp>tial z d5otes #d times ! d5sity ( electricity1 & 7ux1 uy1 uz7 are ! velocity-compon5ts ( electricity4 ,if we n[ suppose t ! electrical-masses are b\nd un*angeably to small1 rigid bodies 7,ions1 electrons71 !n ^! equations =m ! electromagnetic basis ( ,lor5tz's electrodynamics & optics = mov+ bodies4 ,if ^! equations :i* hold 9 ! sy/em ,k1 are trans=m$ to ! sy/em k ) ! aid ( ! trans=mation-equations giv5 9 #c & #f1 !n we obta9 ! equations3-- frac#a;c 7rho' uxi fracp>tial ,x'p>tial tau7 fracp>tial ,n'p>tial eta - fracp>tial ,m'p>tial zeta 1 fracp>tial ,l'p>tial tau fracp>tial ,y'p>tial zeta - fracp>tial ,z'p>tial eta 1 frac#a;c 7rho' ueta fracp>tial ,y'p>tial tau7 fracp>tial ,l'p>tial zeta - fracp>tial ,n'p>tial xi 1 fracp>tial ,m'p>tial tau fracp>tial ,z'p>tial xi - fracp>tial ,x'p>tial zeta 1 frac#a;c 7rho' uzeta fracp>tial ,z'p>tial tau7 fracp>tial ,m'p>tial xi - fracp>tial ,l'p>tial eta 1 fracp>tial ,n'p>tial tau fracp>tial ,x'p>tial eta - fracp>tial ,y'p>tial xi 1 :]e fracux - v#a - fracuxvc uxi 1 fracuybeta7#a - fracvuxc#b7 ueta 1 rho' fracp>tial ,x'p>tial xi fracp>tial ,y'p>tial eta fracp>tial ,z'p>tial xi beta7#a - fracvuxc#b7 rho 1 fracuxbeta7#a - fracvuxc#b7 uzeta 1 ,s9ce ! vector 7u1 u1 u7 is no?+ b ! velocity ( ! electrical mass measur$ 9 ! sy/em k1 z c be easily se5 f ! addition-!orem ( velocities 9 #d--s x is h]eby %[n1 t by tak+ our k9ematical pr9ciple z ! basis1 ! electromagnetic basis ( ,lor5tz's !ory ( electrodynamics ( mov+ bodies correspond to ! relativity-po/ulate4 ,x c be briefly rem>k$ "h t ! foll[+ important law foll[s easily f ! equations develop$ 9 ! pres5t section3--if an electrically *>g$ body moves 9 any mann] 9 space1 & if its *>ge does n *ange !reby1 :5 reg>d$ f a sy/em mov+ along ) x1 !n ! *>ge rema9s con/ant ev5 :5 x is reg>d$ f ! /ation>y sy/em ,k4 #aj4 ,dynamics ( ! ,electron 7sl[ly accel]at$74 ,let u suppose t a po9t-%ap$ p>ticle1 hav+ ! electrical *>ge e 7to be call$ h5ce=? ! electron7 moves 9 ! electromagnetic field2 we assume ! foll[+ ab\t its law ( motion4 ,if ! electron be at re/ at any def9ite epo*1 !n 9 ! next 8p>ticle ( "t10 ! motion takes place accord+ to ! equations dx dy dz m ----- e,x1 m ----- e,y1 m ----- e,z dt dt dt ,:]e 7x1 y1 z7 are ! co-ord9ates ( ! electron1 & m is its mass4 ,let ! electron possess ! velocity v at a c]ta9 epo* ( "t4 ,let u n[ inve/igate ! laws accord+ to :i* ! electron w move 9 ! #8p>ticle ( "t' imm$iately foll[+ ?is epo*4 ,)\t influ5c+ ! g5]ality ( treatm5t1 we c & we w assume t1 at ! mom5t we are consid]+1 ! electron is at ! orig9 ( co-ord9ates1 & moves ) ! velocity v along ! ,x-axis ( ! sy/em4 ,x is cle> t at ?is mom5t 7t #j7 ! electron is at re/ relative to ! sy/em k1 :i* moves p>allel to ! ,x-axis ) ! con/ant velocity v4 ,f ! suppositions made above1 9 comb9ation ) ! pr9ciple ( relativity1 x is cle> t reg>d$ f ! sy/em k1 ! electron moves accord+ to ! equations d d d m ----- e,x1 m ----- e,y1 m ----- e,z 1 d d d 9 ! "t imm$iately foll[+ ! mom5t1 :]e ! symbols 71 1 1 1 ,x'1 ,y'1 ,z'7 ref] to ! sy/em k4 ,if we n[ fix1 t = t v y z #j1 #j1 !n ! equations ( trans=mation giv5 9 #c 7& #f7 hold1 & we h3 v 7t - ---- x71 7x - vt71 y1 z1 c v v ,x ,x1 ,y 7,y - --- ,n71 ,z 7,z --- ,m7 c c ,) ! aid ( ^! equations1 we c trans=m ! above equations ( motion f ! sy/em k to ! sy/em ,k1 & obta93-- 7,a7 fracd#b xdt#b fracem frac#a;beta ,x 1 fracd#b ydt#b fracem frac#a;beta 7,y - fracvc ,n7 1 fracd#b zdt#b fracem frac#a;beta 7,z fracvc ,m7 ,let u n[ consid]1 foll[+ ! usual me?od ( treatm5t1 ! longitud9al & transv]sal mass ( a mov+ electron4 ,we write ! equations 7,a7 9 ! =m dx m ----- e,x e,x dt dy v m ----- e 7,y - --- ,n7 e,y dt c dz v m ----- e 7,z - --- ,m7 e,z dt c & let u fir/ rem>k1 t e,x1 e,y1 e,z are ! compon5ts ( ! pond]omotive =ce act+ ^u ! electron1 & are consid]$ 9 a mov+ sy/em :i*1 at ?is mom5t1 moves ) a velocity :i* is equal to t ( ! electron4 ,?is =ce c1 = example1 be measur$ by means ( a spr+-balance :i* is at re/ 9 ?is la/ sy/em4 ,if we briefly call ?is =ce z 8! =ce act+ ^u ! electron10 & ma9ta9 ! equation3-- ,mass-numb] accel]ation-numb] =ce-numb]1 & if we fur!r fix t ! accel]ations are measur$ 9 ! /ation>y sy/em ,k1 !n f ! above equations1 we obta93-- ,longitud9al mass3 fracm7sqrt#a - fracv#b;c#b7;frac#cb ,transv]sal mass3 fracmsqrt#a - fracv#b;c#b ,naturally1 :5 o!r def9itions are giv5 ( ! =ce & ! accel]ation1 o!r numb]s are obta9$ = ! mass2 h5ce we see t we mu/ proce$ v c>efully 9 comp>+ ! diff]5t !ories ( ! motion ( ! electron4 ,we rem>k t ?is result ab\t ! mass hold also = pond]able mat]ial mass2 = 9 our s5se1 a pond]able mat]ial po9t may be made into an electron by ! addition ( an electrical *>ge :i* may be z small z possible4 ,let u n[ det]m9e ! k9etic en]gy ( ! electron4 ,if ! electron moves f ! orig9 ( co-ord9ates ( ! sy/em ,k ) ! initial velocity #j /eadily along ! ,x-axis "u ! action ( an electromotive =ce ,x1 !n x is cle> t ! en]gy drawn f ! electro/atic field has ! value e,xdx4 ,s9ce ! electron is only sl[ly accel]at$1 & 9 consequ5ce1 no en]gy is giv5 out 9 ! =m ( radiation1 !re=e ! en]gy drawn f ! electro-/atic field may be put equal to ! en]gy ,w ( motion4 ,consid]+ ! :ole process ( motion 9 que/ions1 ! fir/ ( equations ,a7 holds1 we obta93-- ,w int e,xdx int#jv mbeta#c vdv mc#b 7frac#asqrt#a - fracv#b;c#b - #a7 ,= v c1 ,w is inf9itely great4 ,z our =m] result %[s1 velocities exce$+ t ( light c h no possibility ( exi/5ce4 ,9 consequ5ce ( ! argum5ts m5tion$ above1 ?is expression = k9etic en]gy mu/ also hold = pond]able masses4 ,we c n[ enum]ate ! *>act]i/ics ( ! motion ( ! electrons available = exp]im5tal v]ification1 :i* foll[ f equations ,a74 #a4 ,f ! second ( equations ,a71 x foll[s t an electrical =ce ,y1 & a magnetic =ce ,n produce equal deflexions ( an electron mov+ ) ! velocity v1 :5 ,y ,nv_/c4 ,!re=e we see t accord+ to our !ory1 x is possible to obta9 ! velocity ( an electron f ! ratio ( ! magnetic deflexion ,am1 & ! electric deflexion ,ae1 by apply+ ! law3-- frac,am,ae fracvc ,?is relation c be te/$ by means ( exp]im5ts because ! velocity ( ! electron c be directly measur$ by means ( rapidly oscillat+ electric & magnetic fields4 #b4 ,f ! value :i* is d$uc$ = ! k9etic en]gy ( ! electron1 x foll[s t :5 ! electron falls ?r\gh a pot5tial diff]5ce ( ,p1 ! velocity v :i* is acquir$ is giv5 by ! foll[+ relation3-- ,p int ,xdx fracmec#b 7frac#asqrt#a - fracv#b;c#b - #a7 #c4 ,we calculate ! radius ( curvature ,r ( ! pa?1 :]e ! only deflect+ =ce is a magnetic =ce ,n act+ p]p5dicul> to ! velocity ( projection4 ,f ! second ( equations ,a7 we obta93 - fracd#bydt#b fracv#b,r fracem fracvc ,n sqrt#a - fracv#b;c#b or mvc ,r ---------- e,n ,^! ?ree relations are complete expressions = ! law ( motion ( ! electron accord+ to ! above !ory4 ,footnote #f3 ,vide ,note #i4 ,footnote #g3 ,vide ,note #i4 ,footnote #h3 ,vide ,note #ab4 ,,albre*t ,,e9/e9 ,a %ort biographical note4 ,! "n ( ,pr(4 ,albre*t ,e9/e9 has n[ spread f> beyond ! n>r[ pale ( sci5tific inve/igators ow+ to ! brilliant confirmation ( 8 pr$ict$ deflection ( light-rays by ! gravitational field ( ! sun dur+ ! total sol> eclipse ( ,may #bi1 #aiai4 ,b to ! s]i\s /ud5t ( sci5ce1 he has be5 kn[n f ! beg9n+ ( ! curr5t c5tury1 & _m d>k problems 9 physics has be5 illum9at$ ) ! lu/re ( 8 g5ius1 be=e1 ow+ to ! late/ s5sation j m5tion$1 he fla%es out be=e public imag9ation z a sci5tific /> ( ! fir/ magnitude4 ,e9/e9 is a ,swiss-,g]man ( ,jewi% extraction1 & began 8 sci5tific c>e] z a privat-doz5t 9 ! ,swiss ,univ]sity ( ,zri* ab\t ! ye> #aijb4 ,lat] on1 he migrat$ to ! ,g]man ,univ]sity ( ,prague 9 ,bohemia z auss]-ord5tli*e 7or associate7 ,pr(essor4 ,9 #aiad1 ?r\gh ! ex]tions ( ,pr(4 ,m4 ,planck ( ! ,b]l9 ,univ]sity1 he 0 appo9t$ a paid memb] ( ! ,royal 7n[ ,national7 ,prussian ,academy ( ,sci5ces1 on a sal>y ( #ah1jjj m>ks p] ye>4 ,9 ?is po/1 he has only to d & guide rese>* "w4 ,ano!r di/+ui%$ occupant ( ! same po/ 0 ,van't ,h(f1 ! em95t physical *emi/4 ,x is r difficult to give a detail$1 & consi/5t *ronological acc\nt ( 8 sci5tific activities1--!y are s v>iegat$1 & cov] su* a wide field4 ,! fir/ "w :i* ga9$ him di/9ction 0 an inve/igation on ,br[nian ,movem5t4 ,an admirable acc\nt w be f\nd 9 ,p]r9's book #8,! ,atoms4' ,/>t+ f ,boltzmann's !orem connect+ ! entropy1 & ! probability ( a /ate1 he d$uc$ a =mula on ! mean displacem5t ( small p>ticles 7colloidal7 susp5d$ 9 a liquid4 ,?is =mula gives u "o ( ! be/ me?ods = f9d+ out a v fundam5tal numb] 9 physics--namely--! numb] ( molecules 9 "o gm4 molecule ( gas 7,avogadro's numb]74 ,! =mula 0 %ortly aft]w>ds v]ifi$ by ,p]r91 ,pr(4 ( ,*emical ,physics 9 ! ,sorbonne1 ,p>is4 ,to ,e9/e9 is also due ! resuscitation ( ,planck's quantum !ory ( en]gy-emission4 ,?is !ory has n yet caught ! popul> imag9ation to ! same ext5t z ! new !ory ( ,"t1 & ,space1 b x is none ! less iconocla/ic 9 its scope z f> z classical concepts are conc]n$4 ,x 0 kn[n = a long "t t ! obs]v$ emission ( light f a heat$ black body did n correspond to ! =mula :i* c\ld be d$uc$ f ! old] classical !ories ( cont9u\s emission & propagation4 ,9 ! ye> #aijj1 ,pr(4 ,planck ( ! ,b]l9 ,univ]sity work$ out a =mula :i* 0 bas$ on ! bold assumption t en]gy 0 emitt$ & absorb$ by ! molecules 9 multiples ( ! quantity h1 :]e h is a con/ant 7:i* is univ]sal l ! con/ant ( gravitation71 & is ! frequ5cy ( ! light4 ,! conception 0 s radically diff]5t f all accept$ !ories t 9 spite ( ! great success ( ,planck's radiation =mula 9 expla9+ ! obs]v$ facts ( black-body radiation1 x did n meet ) mu* fav\r f ! physici/s4 ,9 fact1 "s "o rem>k$ jocul>ly t accord+ to ,planck1 en]gy flies out ( a radiator l a sw>m ( gnats4 ,b ,e9/e9 f\nd a support = ! new-born concept 9 ano!r direction4 ,x 0 kn[n t if gre5 or ultraviolet light 0 all[$ to fall on a plate ( "s alkali metal1 ! plate lo/ electrons4 ,! electrons 7 emitt$ ) all velocities1 b !re is g5]ally a maximum limit4 ,f ! inve/igations ( ,l5>d & ,lad5burg1 ! curi\s discov]y 0 made t ?is maximum velocity ( emission did n at all dep5d ^u ! int5sity ( light1 b ^u its wavel5g?4 ,! m violet 0 ! light1 ! great] 0 ! velocity ( emission4 ,to acc\nt = ?is fact1 ,e9/e9 made ! bold assumption t ! light is propagat$ 9 space z a unit pulse 7he calls x a ,light-cell71 & fall+ ^u an individual atom1 lib]ates electrons accord+ to ! en]gy equation #a h --- mv ,a1 #b :]e 7m1 v7 are ! mass & velocity ( ! electron4 ,a is a con/ant *>act]i/ic ( ! metal plate4 ,!re 0 little mat]ial = ! confirmation ( ?is law :5 x 0 fir/ propos$ 7#aije71 & elev5 ye>s elaps$ be=e ,pr(4 ,millikan e/abli%$1 by a set ( exp]im5ts sc>cely rivall$ = ! +5uity1 skill1 & c>e display$1 ! absolute tru? ( ! law4 ,z results ( ?is confirmation1 & o!r brilliant triumphs1 ! quantum law is n[ reg>d$ z a fundam5tal law ( ,en]getics4 ,9 rec5t ye>s1 ,x-rays h be5 add$ to ! doma9 ( light1 & 9 ?is direction also1 ,e9/e9's photo-electric =mula has prov$ to be "o ( ! mo/ fruitful conceptions 9 ,physics4 ,! quantum law 0 next ext5d$ by ,e9/e9 to ! problems ( decrease ( specific heat at l[ temp]ature1 & "h also 8 !ory 0 confirm$ 9 a brilliant mann]4 ,we pass ov] 8 o!r contributions to ! equation ( /ate1 to ! problems ( null-po9t en]gy1 & photo-*emical reactions4 ,! rec5t exp]im5tal works ( ,n]n/ & ,w>burg seem to indicate t ?r\gh ,e9/e9's g5ius1 we are probably = ! fir/ "t hav+ a satisfactory !ory ( photo-*emical action4 ,9 #aiae1 ,e9/e9 made an excursion into ,exp]im5tal ,physics1 & "h also1 9 8 *>act]i/ic way1 he tackl$ "o ( ! mo/ fundam5tal concepts ( ,physics4 ,x is well-kn[n t accord+ to ,amp]e1 ! magnetisation ( iron & iron-l bodies1 :5 plac$ )9 a coil c>ry+ an electric curr5t is due to ! excitation 9 ! metal ( small electrical circuits4 ,b ! conception ?\gh a v fruitful "o1 long rema9$ )\t a trace ( exp]im5tal pro(1 ?\gh aft] ! discov]y ( ! electron1 x 0 g5]ally believ$ t ^! molecul> curr5ts may be due to ! rotational motion ( free electrons )9 ! metal4 ,x is easily se5 t if 9 ! process ( magnetisation1 a numb] ( electrons be set into rotatory motion1 !n ^! w imp>t to ! metal itself a turn+ c\ple4 ,! exp]im5t is a r difficult "o1 & _m physici/s tri$ 9 va9 to obs]ve ! effect4 ,b 9 collaboration ) de ,haas1 ,e9/e9 plann$ & successfully c>ri$ out ?is exp]im5t1 & prov$ ! ess5tial correctness ( ,amp]e's views4 ,e9/e9's /udies on ,relativity 7 comm5c$ 9 ! ye> #aije1 & has be5 cont9u$ up to ! pres5t "t4 ,! fir/ pap] 9 ! pres5t collection =ms ,e9/e9's fir/ great contribution to ! ,pr9ciple ( ,special ,relativity4 ,we h rec\nt$ 9 ! introduction h[ out ( ! *aos & disord] into :i* ! electrodynamics & optics ( mov+ bodies _h fall5 previ\s to #ahie1 ,lor5tz1 ,e9/e9 & ,m9k[ski h succe$$ 9 build+ up a consi/5t1 & fruitful new !ory ( ,"t & ,space4 ,b ,e9/e9 0 n satisfi$ ) ! /udy ( ! special problem ( ,relativity = uni=m motion1 b tri$1 9 a s]ies ( pap]s beg9n+ f #aiaa1 to ext5d x to ! case ( non-uni=m motion4 ,! la/ pap] 9 ! pres5t collection is a translation ( a compreh5sive article :i* he contribut$ to ! ,annal5 d] ,physik 9 #aiaf on ?is subject1 & gives1 9 8 own words1 ! ,pr9ciples ( ,g5]aliz$ ,relativity4 ,! triumphs ( ?is !ory are n[ matt]s ( public k4 ,e9/e9 is n[ only #de1 & x is to be hop$ t sci5ce w cont9ue to be enri*$1 = a long "t to come1 ) fur!r a*ievem5ts ( 8 g5ius4 ,pr9ciple ( ,relativity ,,introduction4 ,at ! pres5t "t1 diff]5t op9ions are be+ held ab\t ! fundam5tal equations ( ,electro-dynamics = mov+ bodies4 ,! ,h]tzian#i =ms mu/ be giv5 up1 = x has appe>$ t !y are contr>y to _m exp]im5tal results4 ,9 #ahie ,h4 ,a4 ,lor5tz#aj publi%$ 8 !ory ( optical & electrical ph5om5a 9 mov+ bodies2 ?is !ory 0 bas$ ^u ! atomi/ic conception 7vor/ellung7 ( electricity1 & on acc\nt ( its great success appe>s to h ju/ifi$ ! bold hypo!ses1 by :i* x has be5 u%]$ into exi/5ce4 ,9 8 !ory1 ,lor5tz proce$s f c]ta9 equations1 :i* mu/ hold at e po9t ( 8!r02 !n by =m+ ! av]age values ov] 8,physically inf9itely small0 regions1 :i* h[ev] conta9 l>ge numb]s ( electrons1 ! equations = electro-magnetic processes 9 mov+ bodies c be successfully built up4 ,9 p>ticul>1 ,lor5tz's !ory gives a good acc\nt ( ! non-exi/5ce ( relative motion ( ! e>? & ! lum9if]\s 8!r02 x %[s t ?is fact is intimately connect$ ) ! cov>iance ( ! orig9al equation1 :5 c]ta9 simultane\s trans=mations ( ! space & "t co-ord9ates are effect$2 ^! trans=mations h !re=e obta9$ f ,h4 ,po9c>e#aa ! "n ( ,lor5tz-trans=mations4 ,! cov>iance ( ^! fundam5tal equations1 :5 subject$ to ! ,lor5tz-trans=mation is a purely ma!matical fact i4e4 n bas$ on any physical consid]ations2 ,i w call ?is ! ,!orem ( ,relativity2 ?is !orem re/s ess5tially on ! =m ( ! diff]5tial equations = ! propagation ( waves ) ! velocity ( light4 ,n[ )\t recogniz+ any hypo!sis ab\t ! connection betwe5 8!r0 & matt]1 we c expect ^! ma!matically evid5t !orems to h _! consequ5ces s f> ext5d$--t !reby ev5 ^? laws ( pond]able m$ia :i* are yet unkn[n may anyh[ possess ?is cov>iance :5 subject$ to a ,lor5tz-trans=mation2 by say+ ?is1 we d n inde$ express an op9ion1 b r a conviction1--& ?is conviction ,i may be p]mitt$ to call ! ,po/ulate ( ,relativity4 ,! position ( affairs "h is almo/ ! same z :5 ! ,pr9ciple ( ,cons]vation ( ,en]gy 0 po/ulat$ 9 cases1 :]e ! correspond+ =ms ( en]gy 7 unkn[n4 ,n[ if h]eaft]1 we succe$ 9 ma9ta9+ ?is cov>iance z a def9ite connection betwe5 pure & simple obs]vable ph5om5a 9 mov+ bodies1 ! def9ite connection may be /yl$ #8! ,pr9ciple ( ,relativity4' ,^! diff]5tiations seem to me to be necess>y = enabl+ u to *>act]ise ! pres5t day position ( ! electro-dynamics = mov+ bodies4 ,h4 ,a4 ,lor5tz#ab has f\nd out ! 8,relativity !orem0 & has creat$ ! ,relativity-po/ulate z a hypo!sis t electrons & matt] suff] contractions 9 consequ5ce ( _! motion accord+ to a c]ta9 law4 ,a4 ,e9/e9#ac has br\ght out ! po9t v cle>ly1 t ?is po/ulate is n an artificial hypo!sis b is r a new way ( compreh5d+ ! "t-concept :i* is =c$ ^u u by obs]vation ( natural ph5om5a4 ,! ,pr9ciple ( ,relativity has n yet be5 =mulat$ = electro-dynamics ( mov+ bodies 9 ! s5se *>act]iz$ by me4 ,9 ! pres5t essay1 :ile =mulat+ ?is pr9ciple1 ,i %all obta9 ! fundam5tal equations = mov+ bodies 9 a s5se :i* is uniquely det]m9$ by ?is pr9ciple4 ,b x w be %[n t none ( ! =ms hi!rto assum$ = ^! equations c exactly fit 9 ) ?is pr9ciple4#ad ,we w\ld at fir/ expect t ! fundam5tal equations :i* are assum$ by ,lor5tz = mov+ bodies w\ld correspond to ! ,relativity ,pr9ciple4 ,b x w be %[n t ?is is n ! case = ! g5]al equations :i* ,lor5tz has = any possible1 & also = magnetic bodies2 b ?is is approximately ! case 7if neglect ! squ>e ( ! velocity ( matt] 9 comp>ison to ! velocity ( light7 = ^? equations :i* ,lor5tz h]eaft] inf]s = non-magnetic bodies4 ,b ?is latt] accordance ) ! ,relativity ,pr9ciple is due to ! fact t ! condition ( non-magnetisation has be5 =mulat$ 9 a way n correspond+ to ! ,relativity ,pr9ciple2 !re=e ! accordance is due to ! =tuit\s comp5sation ( two contradictions to ! ,relativity-,po/ulate4 ,b mean:ile enunciation ( ! ,pr9ciple 9 a rigid mann] does n signify any contradiction to ! hypo!ses ( ,lor5tz's molecul> !ory1 b x %all become cle> t ! assumption ( ! contraction ( ! electron 9 ,lor5tz's !ory mu/ be introduc$ at an e>li] /age ?an ,lor5tz has actually done4 ,9 an app5dix1 ,i h gone into discussion ( ! position ( ,classical ,me*anics ) respect to ! ,relativity ,po/ulate4 ,any easily p]ceivable modification ( me*anics = satisfy+ ! requirem5ts ( ! ,relativity !ory w\ld h>dly af=d any noticeable diff]5ce 9 obs]vable processes2 b w\ld lead to v surpris+ consequ5ces4 ,by lay+ d[n ! ,relativity-,po/ulate f ! outset1 suffici5t means h be5 creat$ = d$uc+ h5ce=? ! complete s]ies ( ,laws ( ,me*anics f ! pr9ciple ( cons]vation ( ,en]gy alone 7! =m ( ! ,en]gy be+ giv5 9 explicit =ms74 ,,notations4 ,let a rectangul> sy/em 7x1 y1 z1 t17 ( ref]5ce be giv5 9 space & "t4 ,! unit ( "t %all be *os5 9 su* a mann] ) ref]5ce to ! unit ( l5g? t ! velocity ( light 9 space becomes unity4 ,al?\gh ,i w\ld pref] n to *ange ! notations us$ by ,lor5tz1 x appe>s important to me to use a diff]5t selection ( symbols1 = !reby c]ta9 homog5eity w appe> f ! v beg9n+4 ,i %all d5ote ! vector electric =ce by ,e1 ! magnetic induction by ,m1 ! electric induction by e & ! magnetic =ce by m1 s t 7,e1 ,m1 e1 m7 are us$ in/ead ( ,lor5tz's 7,e1 ,b1 ,d1 ,h7 respectively4 ,i %all fur!r make use ( complex magnitudes 9 a way :i* is n yet curr5t 9 physical inve/igations1 i4e41 in/ead ( op]at+ ) 7t71 ,i %all op]ate ) 7i t71 :]e i d5otes 7-#a74 ,if n[ in/ead ( 7x1 y1 z1 i t71 ,i use ! me?od ( writ+ ) indices1 c]ta9 ess5tial circum/ances w come into evid5ce2 on ?is w be bas$ a g5]al use ( ! suffixes 7#a1 #b1 #c1 #d74 ,! advantage ( ?is me?od w be1 z ,i expressly emphasize "h1 t we %all h to h&le symbols :i* h app>5tly a purely real appe>ance2 we c h[ev] at any mom5t pass to real equations if x is und]/ood t ( ! symbols ) indices1 su* ones z h ! suffix #d only once1 d5ote imag9>y quantities1 :ile ^? :i* h n at all ! suffix #d1 or h x twice d5ote real quantities4 ,an individual sy/em ( values ( 7x1 y1 z1 t7 i4 e41 ( 7x x x x7 %all be call$ a space-"t po9t4 ,fur!r let u d5ote ! velocity vector ( matt]1 ! dielectric con/ant1 ! magnetic p]meability1 ! conductivity ( matt]1 :ile d5otes ! d5sity ( electricity 9 space1 & x ! vector ( 8,electric ,curr5t0 :i* we %all "s across 9 #g & #h4 ,,"p ,i #b4 ,! ,limit+ ,case4 ,! ,fundam5tal ,equations = !r4 ,by us+ ! electron !ory1 ,lor5tz 9 8 above m5tion$ essay traces ! ,laws ( ,electro-dynamics ( ,pond]able ,bodies to /ill simpl] laws4 ,let u n[ adh]e to ^! simpl] laws1 :]eby we require t = ! limit+ case #a1 #a1 #j1 !y %\ld con/itute ! laws = pond]able bodies4 ,9 ?is ideal limit+ case #a1 #a1 #j1 ,e w be equal to e1 & ,m to m4 ,at e space "t po9t 7x1 y1 z1 t7 we %all h ! equations#ae 7i7 ,curl m - 7e_/t7 u 7ii7 div e 7iii7 ,curl e m_/t #j 7iv7 div m #j ,i %all n[ write 7x x x x7 = 7x1 y1 z1 t7 & 71 1 1 7 = 7rho ux1 rho uy1 rho uz1 irho7 i4e4 ! compon5ts ( ! convection curr5t u1 & ! electric d5sity multipli$ by -#a ,fur!r ,i %all write f#b #c1 f#c #a1 f#a #b1 f#a #d1 f#b #d1 f#c #d4 = mx1 my1 mz1 -iex1 -iey1 -iez4 i4e41 ! compon5ts ( m & 7-i4e47 along ! ?ree axes2 n[ if we take any two indices 7h4 k7 out ( ! s]ies #c1 #d71 fk h -fk h1 ,!re=e f -f1 f -f1 f -f f -f1 f -f1 f -f ,!n ! ?ree equations compris$ 9 7i71 & ! equation 7ii7 multipli$ by i becomes beg9vmatrix fracdelta f#a #b;delta x#b fracdelta f#a #c;delta x#c fracdelta f#a #d;delta x#d rho#a fracdelta f#b #a;delta x#a fracdelta f#b #c;delta x#c times fracdelta f#b #d;delta x#d rho#b fracdelta f#c #a;delta x#a times fracdelta f#c #b;delta x#b fracdelta f#c #d;delta x#d rho#c fracdelta f#d #a;delta x#a fracdelta f#d #b;delta x#b fracdelta f#d #c;delta x#c rho#d endvmatrix ,on ! o!r h&1 ! ?ree equations compris$ 9 7iii7 & ! 7iv7 equation multipli$ by 7i7 becomes beg9vmatrix fracdelta f#c #d;delta x#b fracdelta f#d #b;delta x#c fracdelta f#b #c;delta x#d fracdelta f#d #c;delta x#a fracdelta f#a #d;delta x#c fracdelta f#c #a;delta x#d #j fracdelta f#b #d;delta x#a fracdelta f#d #a;delta x#b fracdelta f#a #b;delta x#d #j fracdelta f#c #b;delta x#a fracdelta f#a #c;delta x#b fracdelta f#b #a;delta x#c - endvmatrix ,by means ( ?is me?od ( writ+ we at once notice ! p]fect symmetry ( ! #a/ z well z ! #bnd sy/em ( equations z reg>ds p]mutation ) ! indices1 7#a1 #b1 #c1 #d74 #c4 ,x is well-kn[n t by writ+ ! equations i7 to iv7 9 ! symbol ( vector calculus1 we at once set 9 evid5ce an inv>iance 7or r a 7cov>iance7 ( ! sy/em ( equations ,a7 z well z ( ,b71 :5 ! co-ord9ate sy/em is rotat$ ?r\gh a c]ta9 am\nt r\nd ! null-po9t4 ,= example1 if we take a rotation ( ! axes r\nd ! z-axis1 ?r\gh an am\nt 1 keep+ e1 m fix$ 9 space1 & introduce new v>iables x x x x in/ead ( x x x x :]e x x cos x s9 1 x -x s9 x cos 1 x x1 x x1 & introduce magnitudes 1 1 1 1 :]e cos s9 1 - s9 cos & f#a #b1 444 444 f#c #d1 :]e f f cos f s9 1 f - f s9 f cos 1 f f1 f f cos f s9 1 f - f s9 f cos 1 f f#c #d1 fk h - fk h 7h l k #a1 #b1 #c1 #d74 !n out ( ! equations 7,a7 w\ld foll[ a correspond+ sy/em ( da%$ equations 7,a7 compos$ ( ! newly introduc$ da%$ magnitudes4 ,s ^u ! gr\nd ( symmetry alone ( ! equations 7,a7 & 7,b7 conc]n+ ! suffixes 7#a1 #b1 #c1 #d71 ! !orem ( ,relativity1 :i* 0 f\nd out by ,lor5tz1 foll[s )\t any calculation at all4 ,i w d5ote by i a purely imag9>y magnitude1 & consid] ! sub/itution x x1 x x1 x x cos i x s9 i1 7#a7 x - x s9 i x cos i1 ,putt+ - i tan ipsi fracepsi - e-psiepsie-psi q 1 psi frac#ab log frac#a q#a - q 7#b7 ,we %all h cos i #a_/7a - q71 s9 i iq_/7#a - q7 :]e -#a q #a1 & 7#a - q7 is always to be tak5 ) ! positive sign4 ,let u n[ write x x1 x y1 x z1 x x 7#c7 !n ! sub/itution #a7 takes ! =m x x1 y y1 z 7z - qt7_/7#a - q71 t 7-qz t7_/7#a - q71 7#d7 ! coeffici5ts be+ ess5tially real4 ,if n[ 9 ! above-m5tion$ rotation r\nd ! ,z-axis1 we replace #a1 #b1 #c1 #d ?r\gh\t by #c1 #d1 #a1 #b1 & by i1 we at once p]ceive t simultane\sly1 new magnitudes 1 1 1 1 :]e 1 1 cos i s9 i1 - s9 i cos i71 & f#a #b 444 f#c #d1 :]e f#d #a f#d #a cos i f#a #c s9 i1 f#a #c - f#d #a s9 i f#a #c cos i1 f#c #d f#c #d1 f#c #b f#c #b cos i f#d #b s9 i1 f#d #b - f#c #b s9 i f#d #b cos i1 f#a #b f#a #b1 fk h - fk h1 mu/ be introduc$4 ,!n ! sy/ems ( equations 9 7,a7 & 7,b7 are trans=m$ into equations 7,a71 & 7,b71 ! new equations be+ obta9$ by simply da%+ ! old set4 ,all ^! equations c be writt5 9 purely real figures1 & we c !n =mulate ! la/ result z foll[s4 ,if ! real trans=mations #d7 are tak51 & x y z t be tak5 z a new frame ( ref]5ce1 !n we %all h 7#e7 7-quz #a7_/7a - q71 uz 7uz - q7_/7#a - q71 ux ux1 uy uy4 7#f7 ex 7ex - qmy7_/77#a - q771 mr 7qex my7_/77#a - q771 ez ez4 7#g7 mx 7mx - qey7_/77#a - q771 ey 7qmx ey7_/77#a - q771 mz mz4 ,!n we h = ^! newly introduc$ vectors u1 e1 m 7) compon5ts ux1 uy1 uz2 ex1 ey1 ez2 mx1 my1 mz71 & ! quantity a s]ies ( equations ,i71 ,,ii71 ,,iii71 ,,iv7 :i* are obta9$ f ,i71 ,,ii71 ,,iii71 ,,iv7 by simply da%+ ! symbols4 ,we rem>k "h t ex - qmy1 ey qmx are compon5ts ( ! vector e vm1 :]e v is a vector 9 ! direction ( ! positive ,z-axis1 & v q1 & vm is ! vector product ( v & m2 simil>ly -qex my1 mx qey are ! compon5ts ( ! vector m - ve4 ,! equations #f7 & #g71 z !y /& 9 pairs1 c be express$ z4 ex imx 7ex imx7 cos i 7ey imy7 s9 i1 ey imy - 7ex imx7 s9 i 7ey imy7 cos i1 ez imz ez imz4 ,if d5otes any o!r real angle1 we c =m ! foll[+ comb9ations3-- 7ex imx7 cos4 7ey imy7 s9 7ex imx7 cos4 7 i7 7ey imy7 s9 7 i71 7ex imx7 s9 7ey imy7 cos4 - 7ex imx7 s9 7 i7 7ey imy7 cos4 7 i74 #d4 ,special ,lor5tz ,trans=mation4 ,! rle :i* is play$ by ! ,z-axis 9 ! trans=mation 7#d7 c easily be transf]r$ to any o!r axis :5 ! sy/em ( axes are subject$ to a trans=mation ab\t ?is la/ axis4 ,s we came to a m g5]al law3-- ,let v be a vector ) ! compon5ts vx1 vy1 vz1 & let v q #a4 ,by we %all d5ote any vector :i* is p]p5dicul> to v1 & by rv1 r we %all d5ote compon5ts ( r 9 direction ( & v4 ,in/ead ( 7x1 y1 z1 t71 new magnetudes 7x y z t7 w be introduc$ 9 ! foll[+ way4 ,if = ! sake ( %ortness1 r is writt5 = ! vector ) ! compon5ts 7x1 y1 z7 9 ! fir/ sy/em ( ref]5ce1 r = ! same vector ) ! compon5ts 7x y z7 9 ! second sy/em ( ref]5ce1 !n = ! direction ( v1 we h 7#aj7 rv 7rv - qt7_/7#a - q7 & = ! p]p5dicul> direction 1 7#aa7 r r & fur!r 7#ab7 t 7-qrv t7_/7#a - q74 ,! notations 7r1 rv7 are to be und]/ood 9 ! s5se t ) ! directions v1 & e direction p]p5dicul> to v 9 ! sy/em 7x1 y1 z7 are always associat$ ! directions ) ! same direction cos9es 9 ! sy/em 7x y z74 ,a trans=mation :i* is accompli%$ by means ( 7#aj71 7#aa71 7#ab7 ) ! condition #j q #a w be call$ a special ,lor5tz-trans=mation4 ,we %all call v ! vector1 ! direction ( v ! axis1 & ! magnitude ( v ! mom5t ( ?is trans=mation4 ,if fur!r & ! vectors u1 e1 m1 9 ! sy/em 7x y z7 are s def9$ t1 7#ac7 7-quv #a7_/7a - q71 uv 7uv - q7_/7#a - q71 u uv1 fur!r 7#ad7 7e im7 77e im7 - iv1 7e im7'7_/7#a - q74 7#ae7 7e im7v 7e im7 - iu1 7e im7v4 ,!n x foll[s t ! equations ,i71 ,,ii71 ,,iii71 ,,iv7 are trans=m$ into ! correspond+ sy/em ) da%es4 ,! solution ( ! equations 7#aj71 7#aa71 7#ab7 leads to 7#af7 rv 7rv qt7_/7#a - q71 r r1 t 7qrv t7_/7#a - q71 ,n[ we %all make a v important obs]vation ab\t ! vectors u & u4 ,we c aga9 introduce ! indices #a1 #b1 #c1 #d1 s t we write 7x1 x1 x1 x7 in/ead ( 7x1 y1 z1 x7 & 1 1 1 in/ead ( 7ux1 uy1 uz1 i74 ,l ! rotation r\nd ! ,z-axis1 ! trans=mation 7#d71 & m g5]ally ! trans=mations 7#aj71 7#aa71 7#ab71 are also l9e> trans=mations ) ! det]m9ant #a1 s t 7#ag7 x x x x i4 e4 x y z - t1 is trans=m$ into x x x x i4 e4 x y z - t4 ,on ! basis ( ! equations 7#ac71 7#ad71 we %all h 7 7 7#a - ux1 -uy1 -uz7 7#a - u7 trans=m$ into 7#a - u7 or 9 o!r words1 7#ah7 7#a - u7 is an inv>iant 9 a ,lor5tz-trans=mation4 ,if we divide 71 1 1 7 by ?is magnitude1 we obta9 ! f\r values 71 1 1 7 7#a_/7a - u777ux1 uy1 uz1 i7 s t -#a4 ,x is app>5t t ^! f\r values are det]m9$ by ! vector u & inv]sely ! vector u ( magnitude #a foll[s f ! #d values 1 1 1 2 :]e 71 1 7 are real1 -i real & positive & condition 7#ai7 is fulfill$4 ,! mean+ ( 71 1 1 7 "h is1 t !y are ! ratios ( dx1 dx1 dx1 dx to 7#bj7 7-7dx dx dx dx77 dt7#a - u74 ,! diff]5tials d5ot+ ! displacem5ts ( matt] occupy+ ! spacetime po9t 7x1 x1 x1 x7 to ! adjac5t space-"t po9t4 ,aft] ! ,lor5tz-trans=mation is accompli%$ ! velocity ( matt] 9 ! new sy/em ( ref]5ce = ! same space-"t po9t 7x y z t7 is ! vector u ) ! ratios dx_/dt1 dy_/dt1 dz_/dt1 dl_/dt1 z compon5ts4 ,n[ x is q app>5t t ! sy/em ( values x 1 x 1 x 1 x is trans=m$ into ! values x 1 x 1 x 1 x 9 virtue ( ! ,lor5tz-trans=mation 7#aj71 7#aa71 7#ab74 ,! da%$ sy/em has got ! same mean+ = ! velocity u aft] ! trans=mation z ! fir/ sy/em ( values has got = u be=e trans=mation4 ,if 9 p>ticul> ! vector v ( ! special ,lor5tz-trans=mation be equal to ! velocity vector u ( matt] at ! space-"t po9t 7x1 x1 x1 x7 !n x foll[s out ( 7#aj71 7#aa71 7#ab7 t #j1 #j1 #j1 i ,"u ^! circum/ances !re=e1 ! correspond+ space-"t po9t has ! velocity v #j aft] ! trans=mation1 x is z if we trans=m to re/4 ,we may call ! inv>iant 7#a - u7 ! re/-d5sity ( ,electricity4#af #e4 ,space-"t ,vectors4 ,( ! #a/ & #bnd k9d4 ,if we take ! pr9cipal result ( ! ,lor5tz trans=mation toge!r ) ! fact t ! sy/em 7,a7 z well z ! sy/em 7,b7 is cov>iant ) respect to a rotation ( ! coord9ate-sy/em r\nd ! null po9t1 we obta9 ! g5]al relativity !orem4 ,9 ord] to make ! facts easily compreh5sible1 x may be m conv5i5t to def9e a s]ies ( expressions1 = ! purpose ( express+ ! ideas 9 a concise =m1 :ile on ! o!r h& ,i %all adh]e to ! practice ( us+ complex magnitudes1 9 ord] to r5d] c]ta9 symmetries q evid5t4 ,let u take a l9e> homog5e\s trans=mation1 beg9vmatrix x#a x#b x#c x#d endvmatrix beg9vmatrix a#a #a a#a #b a#a #c a#a #d a#b #a a#b #b a#b #c a#b #d a#c #a a#c #b a#c #c a#c #d a#d #a a#d #b a#d #c a#d #d endvmatrix beg9vmatrix x#a' x#b' x#c' x#d' endvmatrix ! ,det]m9ant ( ! matrix is #a1 all co-effici5ts )\t ! index #d occurr+ once are real1 :ile a1 a1 a1 are purely imag9>y1 b a is real & #j1 & x x x x trans=ms into x x x x4 ,! op]ation %all be call$ a g5]al ,lor5tz trans=mation4 7,?is notation1 :i* is due to ,dr4 ,c4 ,e4 ,cullis ( ! ,calcutta ,univ]sity1 has be5 us$ ?r\gh\t in/ead ( ,m9k[ski's notation1 x ax ax ax ax47 ,if we put x x1 x y1 x z1 x x1 !n imm$iately !re occurs a homog5e\s l9e> trans=mation ( 7x1 y1 z1 t7 to 7x1 y1 z1 t7 ) ess5tially real co-effici5ts1 :]eby ! aggregate -x - y - z t trans=ms into -x - y - z t1 & to e su* sy/em ( values x1 y1 z1 t ) a positive t1 = :i* ?is aggregate #j1 !re always corresponds a positive t'2 ?is la/ is q evid5t f ! cont9uity ( ! aggregate x1 y1 z1 t4 ,! la/ v]tical column ( co-effici5ts has to fulfil ! condition #bb7 a a a a #a4 ,if a a a #j1 !n a #a1 & ! ,lor5tz trans=mation r$uces to a simple rotation ( ! spatial co-ord9ate sy/em r\nd ! _w-po9t4 ,if a1 a1 a are n all z]o1 & if we put a 3 a 3 a 3 a vx 3 vy 3 vz 3 i q 7vx vy vz7 #a4 ,on ! o!r h&1 ) e set ( values ( a1 a1 a1 a :i* 9 ?is way fulfil ! condition #bb7 ) real values ( vx1 vy1 vz1 we c con/ruct ! special ,lor5tz trans=mation 7#af7 ) 7a1 a1 a1 a7 z ! la/ v]tical column1--& !n e ,lor5tz-trans=mation ) ! same la/ v]tical column 7a1 a1 a1 a7 c be suppos$ to be compos$ ( ! special ,lor5tz-trans=mation1 & a rotation ( ! spatial co-ord9ate sy/em r\nd ! null-po9t4 ,! totality ( all ,lor5tz-,trans=mations =ms a gr\p4 ,"u a space-"t vector ( ! #a/ k9d %all be und]/ood a sy/em ( f\r magnitudes 71 1 1 7 ) ! condition t 9 case ( a ,lor5tz-trans=mation x is to be replac$ by ! set 71 1 1 71 :]e ^! are ! values ( 7x1 x1 x1 x71 obta9$ by sub/itut+ 71 1 1 7 = 7x1 x1 x1 x7 9 ! expression 7#ba74 ,besides ! "t-space vector ( ! #a/ k9d 7x1 x1 x1 x7 we %all also make use ( ano!r space-"t vector ( ! fir/ k9d 7y1 y1 y1 y71 & let u =m ! l9e> comb9ation 7#bc7 f7xy - xy7 f7xy - xy7 f7xy - xy7 f7xy - xy7 f7xy - xy7 f7xy - xy7 ) six coeffici5ts f--f4 ,let u rem>k t 9 ! vectorial me?od ( writ+1 ?is c be con/ruct$ out ( ! f\r vectors4 x1 x1 x2 y1 y1 y2 f1 f1 f2 f1 f1 f & ! con/ants x & y1 at ! same "t x is symmetrical ) reg>d ! indices 7#a1 #b1 #c1 #d74 ,if we subject 7x1 x1 x1 x7 & 7y1 y1 y1 y7 simultane\sly to ! ,lor5tz trans=mation 7#ba71 ! comb9ation 7#bc7 is *ang$ to3 7#bd7 f7xy - xy7 f7xy - xy7 f 7xy - xy7 f7xy7 - xy7 f7xy - xy7 f7xy - xy71 :]e ! coeffici5ts f1 f1 f1 f1 f1 f1 dep5d solely on 7f f7 & ! coeffici5ts a 444 a4 ,we %all def9e a space-"t ,vector ( ! #bnd k9d z a sy/em ( six-magnitudes f1 f 444 f1 ) ! condition t :5 subject$ to a ,lor5tz trans=mation1 x is *ang$ to a new sy/em f 444 f1 444 :i* satisfies ! connection betwe5 7#bc7 & 7#bd74 ,i enunciate 9 ! foll[+ mann] ! g5]al !orem ( relativity correspond+ to ! equations 7,i7-7iv71--:i* are ! fundam5tal equations = !r4 ,if x1 y1 z1 x 7space co-ord9ates1 & "t x7 is subject$ to a ,lor5tz trans=mation1 & at ! same "t 7pux1 puy1 puz1 i7 7convection-curr5t1 & *>ge d5sity i7 is trans=m$ z a space "t vector ( ! #a/ k9d1 fur!r 7mx1 my1 mz1 -iex1 -iey1 -iez7 7magnetic =ce1 & electric induction 7-i7 is trans=m$ z a space "t vector ( ! #bnd k9d1 !n ! sy/em ( equations 7,i71 7,,ii71 & ! sy/em ( equations 7,,iii71 7,,iv7 trans=ms into ess5tially correspond+ relations betwe5 ! correspond+ magnitudes newly introduc$ into ! sy/em4 ,^! facts c be m concisely express$ 9 ^! words3 ! sy/em ( equations 7,i & ,,ii7 z well z ! sy/em ( equations 7,,iii7 7,,iv7 are cov>iant 9 all cases ( ,lor5tz-trans=mation1 :]e 7u1 i7 is to be trans=m$ z a space "t vector ( ! #a/ k9d1 7m - ie7 is to be treat$ z a vector ( ! #bnd k9d1 or m significantly1-- 7u1 i7 is a space "t vector ( ! #a/ k9d1 7m - ie7#ag is a space-"t vector ( ! #bnd k9d4 ,i %all add a few m rem>ks "h 9 ord] to elucidate ! conception ( space-"t vector ( ! #bnd k9d4 ,cle>ly1 ! foll[+ are inv>iants = su* a vector :5 subject$ to a gr\p ( ,lor5tz trans=mation4 7i7 m - e f f f f f f me i7ff ff ff74 ,a space-"t vector ( ! second k9d 7m - ie71 :]e 7m & e7 are real magnitudes1 may be call$ s+ul>1 :5 ! scal> squ>e 7m - ie7 #j1 ie m - e #j1 & at ! same "t 7m e7 #j1 ie ! vector m & e are equal & p]p5dicul> to ea* o!r2 :5 su* is ! case1 ^! two prop]ties rema9 cons]v$ = ! space-"t vector ( ! #bnd k9d 9 e ,lor5tz-trans=mation4 ,if ! space-"t vector ( ! #bnd k9d is n s+ul>1 we rotate ! spacial co-ord9ate sy/em 9 su* a mann] t ! vector-product me co9cides ) ! ,z-axis1 i4e4 mx #j1 ex #j4 ,!n 7mx1 -i ex7 7my1 -i ey7 #j4 ,!re=e 7ey i my7_/7ex i ex7 is diff]5t f i1 & we c !re=e def9e a complex argum5t 7 i7 9 su* a mann] t tan 7 i7 ey i my ------------------------- ex i mx ,if !n1 by ref]r+ back to equations 7#i71 we c>ry out ! trans=mation 7#a7 ?r\gh ! angle & a subsequ5t rotation r\nd ! ,z-axis ?r\gh ! angle 1 we p]=m a ,lor5tz-trans=mation at ! end ( :i* my #j1 ey #j1 & !re=e m & e %all bo? co9cide ) ! new ,z-axis4 ,!n by means ( ! inv>iants m - e1 7me7 ! f9al values ( ^! vectors1 :e!r !y are ( ! same or ( opposite directions1 or :e!r "o ( !m is equal to z]o1 w\ld be at once settl$4 #f4 ,concept ( ,"t4 ,by ! ,lor5tz trans=mation1 we are all[$ to effect c]ta9 *anges ( ! "t p>amet]4 ,9 consequ5ce ( ?is fact1 x is no long] p]missible to speak ( ! absolute simultaneity ( two ev5ts4 ,! ord9>y idea ( simultaneity r presupposes t six indep5d5t p>amet]s1 :i* are evid5tly requir$ = def9+ a sy/em ( space & "t axes1 are someh[ r$uc$ to ?ree4 ,s9ce we are accu/om$ to consid] t ^! limitations repres5t 9 a unique way ! actual facts v approximately1 we ma9ta9 t ! simultaneity ( two ev5ts exi/s ( !mselves4#ah ,9 fact1 ! foll[+ consid]ations w prove conclusive4 ,let a ref]5ce sy/em 7x1 y1 z1 t7 = space "t po9ts 7ev5ts7 be someh[ kn[n4 ,n[ if a space po9t ,a 7x1 y1 z7 ! "t t be comp>$ ) a space po9t ,p 7x1 y1 z7 at ! "t t1 & if ! diff]5ce ( "t t - t1 7let t t7 be less ?an ! l5g? ,a ,p i4e4 less ?an ! "t requir$ = ! propagation ( light f ,a to ,p1 & if q 7t - t7_/7,a ,p7 #a1 !n by a special ,lor5tz trans=mation1 9 :i* ,a ,p is tak5 z ! axis1 & :i* has ! mom5t q1 we c introduce a "t p>amet] t1 :i* 7see equation #aa1 #ab1 #d7 has got ! same value t #j = bo? space-"t po9ts 7,a1 t71 & 7,p1 t74 ,s ! two ev5ts c n[ be compreh5d$ to be simultane\s4 ,fur!r1 let u take at ! same "t t #j1 two diff]5t space-po9ts ,a1 ,b1 or ?ree space-po9ts 7,a1 ,b1 ,c7 :i* are n 9 ! same space-l9e1 & comp>e !re) a space po9t ,p1 :i* is outside ! l9e ,a ,b1 or ! plane ,a ,b ,c1 at ano!r "t t1 & let ! "t diff]5ce t - t 7t t7 be less ?an ! "t :i* light requires = propagation f ! l9e ,a ,b1 or ! plane 7,a ,b ,c7 to ,p4 ,let q be ! quoti5t ( 7t - t7 by ! second "t4 ,!n if a ,lor5tz trans=mation is tak5 9 :i* ! p]p5dicul> f ,p on ,a ,b1 or f ,p on ! plane ,a ,b ,c is ! axis1 & q is ! mom5t1 !n all ! ?ree 7or f\r7 ev5ts 7,a1 t71 7,b1 t71 7,c1 t7 & 7,p1 t7 are simultane\s4 ,if f\r space-po9ts1 :i* d n lie 9 "o plane1 are conceiv$ to be at ! same "t t1 !n x is no long] p]missible to make a *ange ( ! "t p>amet] by a ,lor5tz-trans=mation1 )\t at ! same "t de/roy+ ! "* ( ! simultaneity ( ^! f\r space po9ts4 ,to ! ma!matician1 accu/om$ on ! "o h& to ! me?ods ( treatm5t ( ! poly-dim5sional manifold1 & on ! o!r h& to ! conceptual figures ( ! s-call$ non-,euclidean ,geometry1 !re c be no difficulty 9 adopt+ ?is concept ( "t to ! application ( ! ,lor5tz-trans=mation4 ,! pap] ( ,e9/e9 :i* has be5 cit$ 9 ! ,introduction1 has succe$$ to "s ext5t 9 pres5t+ ! nature ( ! trans=mation f ! physical /&po9t4 ,,"p ,,ii4 ,,electro-,,magnetic ,,ph5om5a4 #g4 ,fundam5tal ,equations = bodies at re/4 ,aft] ^! prep>atory works1 :i* h be5 fir/ develop$ on acc\nt ( ! small am\nt ( ma!matics involv$ 9 ! limit+ case #a1 #a1 #j1 let u turn to ! electro-magnetic ph5om5a 9 matt]4 ,we look = ^? relations :i* make x possible = u--:5 prop] fundam5tal data are giv5--to obta9 ! foll[+ quantities at e place & "t1 & !re=e at e space-"t po9t z functions ( 7x1 y1 z1 t73--! vector ( ! electric =ce ,e1 ! magnetic induction ,m1 ! electrical induction e1 ! magnetic =ce m1 ! electrical space-d5sity 1 ! electric curr5t s 7^: relation h]eaft] to ! conduction curr5t is kn[n by ! mann] 9 :i* conductivity occurs 9 ! process71 & la/ly ! vector v1 ! velocity ( matt]4 ,! relations 9 "q c be divid$ into two classes4 ,fir/ly--^? equations1 :i*1--:5 v1 ! velocity ( matt] is giv5 z a function ( 7x1 y1 z1 t71--lead u to a k ( o!r magnitude z functions ( x1 y1 z1 t--,i %all call ?is fir/ class ( equations ! fundam5tal equations-- ,secondly1 ! expressions = ! pond]omotive =ce1 :i*1 by ! application ( ! ,laws ( ,me*anics1 gives u fur!r in=mation ab\t ! vector u z functions ( 7x1 y1 z1 t74 ,= ! case ( bodies at re/1 i4e4 :5 u 7x1 y1 z1 t7 #j ! !ories ( ,maxwell 7,heaviside1 ,h]tz7 & ,lor5tz lead to ! same fundam5tal equations4 ,!y are2-- 7#a7 ,! ,diff]5tial ,equations3--:i* conta9 no con/ant ref]r+ to matt]3-- 7i7 ,curl m - e_/t ,c1 7ii7 div e l4 7iii7 ,curl ,e ,m_/t #j1 7iv7 ,div ,m #j4 7#b7 ,fur!r relations1 :i* *>act]ise ! influ5ce ( exi/+ matt] = ! mo/ important case to :i* we limit ourselves i4e4 = isotopic bodies2--!y are compris$ 9 ! equations 7,v7 e ,e1 ,m m1 ,c ,e4 :]e dielectric con/ant1 magnetic p]meability1 ! conductivity ( matt]1 all giv5 z function ( x1 y1 z1 t2 s is "h ! conduction curr5t4 ,by employ+ a modifi$ =m ( writ+1 ,i %all n[ cause a lat5t symmetry 9 ^! equations to appe>4 ,i put1 z 9 ! previ\s "w1 x x1 x y1 x z1 x x1 & write s1 s1 s1 s = ,cx1 ,cy1 ,cz 7-#a74 ,fur!r f1 f1 f1 f1 f1 f = mx1 my1 mz1 -i7ex1 ey1 ez71 & ,f1 ,f1 ,f1 ,f1 ,f1 ,f = ,mx1 ,my1 ,mz1 -i7,ex1 ,ey1 ,ez7 la/ly we %all h ! relation fk h - fh k1 ,fk h -,fh k1 7! lett] f1 ,f %all d5ote ! field1 s ! 7i4e4 curr5t74 ,!n ! fundam5tal ,equations c be writt5 z 7,a7 f_/x f_/x f_/x s f_/x f_/x f_/x s f_/x f_/x f_/x s f_/x f_/x f_/x s & ! equations 7#c7 & 7#d71 are ,f_/x ,f_/x ,f_/x #j ,f_/x ,f_/x ,fx #j ,f_/x ,f_/x ,f_/x #j ,f_/x ,f_/x ,f_/x #j #h4 ,! ,fundam5tal ,equations4 ,we are n[ 9 a position to e/abli% 9 a unique way ! fundam5tal equations = bodies mov+ 9 any mann] by means ( ^! ?ree axioms exclusively4 ,! fir/ ,axion %all be1-- ,:5 a deta*$ region#ai ( matt] is at re/ at any mom5t1 !re=e ! vector u is z]o1 = a sy/em 7x1 y1 z1 t7--! neighb\rhood may be suppos$ to be 9 motion 9 any possible mann]1 !n = ! space-"t po9t x1 y1 z1 t1 ! same relations 7,a7 7,b7 7,v7 :i* hold 9 ! case :5 all matt] is at re/1 %all also hold betwe5 1 ! vectors ,c1 e1 m1 ,m1 ,e & _! diff]5tials ) respect to x1 y1 z1 t4 ,! second axiom %all be3-- ,e velocity ( matt] is #a1 small] ?an ! velocity ( propagation ( light4#bj ,! fundam5tal equations are ( su* a k9d t :5 7x1 y1 z1 x7 are subject$ to a ,lor5tz trans=mation & !reby 7m - ie7 & 7,m - i,e7 are trans=m$ into space-"t vectors ( ! second k9d1 7,c1 i7 z a space-"t vector ( ! #a/ k9d1 ! equations are trans=m$ into ess5tially id5tical =ms involv+ ! trans=m$ magnitudes4 ,%ortly ,i c signify ! ?ird axiom z3-- 7m1 -ie71 & 7,m1 -i,e7 are space-"t vectors ( ! second k9d1 7,c1 ip7 is a space-"t vector ( ! fir/ k9d4 ,?is axiom ,i call ! ,pr9ciple ( ,relativity4 ,9 fact ^! ?ree axioms lead u f ! previ\sly m5tion$ fundam5tal equations = bodies at re/ to ! equations = mov+ bodies 9 an unambigu\s way4 ,accord+ to ! second axiom1 ! magnitude ( ! velocity vector u is #a at any space-"t po9t4 ,9 consequ5ce1 we c always write1 in/ead ( ! vector u1 ! foll[+ set ( f\r alli$ quantities ux_/7#a - u71 uy_/7#a - u71 uz_/7#a - u71 i_/7#a - u7 ) ! relation 7#bg7 - ,f :at has be5 said at ! end ( #d1 x is cle> t 9 ! case ( a ,lor5tz-trans=mation1 ?is set behaves l a space-"t vector ( ! #a/ k9d4 ,let u n[ fix our att5tion on a c]ta9 po9t 7x1 y1 z7 ( matt] at a c]ta9 "t 7t74 ,if at ?is space-"t po9t u #j1 !n we h at once = ?is po9t ! equations 7,a71 7,b7 7,v7 ( #g4 ,if u #j1 !n !re exi/s accord+ to #af71 9 case u #a1 a special ,lor5tz-trans=mation1 ^: vector v is equal to ?is vector u 7x1 y1 z1 t71 & we pass on to a new sy/em ( ref]5ce 7x y z t7 9 accordance ) ?is trans=mation4 ,!re=e = ! space-"t po9t consid]$1 !re arises z 9 #d1 ! new values #bh7 #j1 #j1 #j1 i1 !re=e ! new velocity vector #j1 ! space-"t po9t is z if trans=m$ to re/4 ,n[ accord+ to ! ?ird axiom ! sy/em ( equations = ! trans=m$ po9t 7x y z t7 involves ! newly introduc$ magnitude 7u 1 ,c1 e1 m1 ,e1 ,m7 & _! diff]5tial quoti5ts ) respect to 7x1 y1 z1 t7 9 ! same mann] z ! orig9al equations = ! po9t 7x1 y1 z1 t74 ,b accord+ to ! fir/ axiom1 :5 u #j1 ^! equations mu/ be exactly equival5t to 7#a7 ! diff]5tial equations 7,a71 7,b71 :i* are obta9$ f ! equations 7,a71 7,b7 by simply da%+ ! symbols 9 7,a7 & 7,b74 7#b7 & ! equations 7,v7 e ,e1 ,m' m1 ,c ,e :]e 1 1 are ! dielectric con/ant1 magnetic p]meability1 & conductivity = ! sy/em 7x y z t7 i4e4 9 ! space-"t po9t 7x y1 z t7 ( matt]4 ,n[ let u return1 by means ( ! reciprocal ,lor5tz-trans=mation to ! orig9al v>iables 7x1 y1 z1 t71 & ! magnitudes 7u1 1 ,c1 e1 m1 ,e1 ,m7 & ! equations1 :i* we !n obta9 f ! la/ m5tion$1 w be ! fundam5tal equations s\ght by u = ! mov+ bodies4 ,n[ f #d1 & #f1 x is to be se5 t ! equations ,a71 z well z ! equations ,b7 are cov>iant = a ,lor5tz-trans=mation1 i4e4 ! equations1 :i* we obta9 backw>ds f ,a7 ,b71 mu/ be exactly ( ! same =m z ! equations ,a7 & ,b71 z we take !m = bodies at re/4 ,we h !re=e z ! fir/ result3-- ,! diff]5tial equations express+ ! fundam5tal equations ( electrodynamics = mov+ bodies1 :5 writt5 9 & ! vectors ,c1 e1 m1 ,e1 ,m1 are exactly ( ! same =m z ! equations = mov+ bodies4 ,! velocity ( matt] does n ent] 9 ^! equations4 ,9 ! vectorial way ( writ+1 we h ,i7 curl m - e_/t ,c1 ,,ii7 div e ,,iii7 curl ,e ,m_/t #j ,,iv7 div ,m #j ,! velocity ( matt] occurs only 9 ! auxili>y equations :i* *>act]ise ! influ5ce ( matt] on ! basis ( _! *>act]i/ic con/ants 1 1 4 ,let u n[ trans=m ^! auxili>y equations ,v7 into ! orig9al co-ord9ates 7x1 y1 z1 & t47 ,accord+ to =mula #ae7 9 #d1 ! compon5t ( e 9 ! direction ( ! vector u is ! same z t ( 7e u m71 ! compon5t ( m is ! same z t ( m - u e1 b = ! p]p5dicul> direction 1 ! compon5ts ( e1 m are ! same z ^? ( 7e u m7 & 7m - u e1 multipli$ by #a_/7a - u74 ,on ! o!r h& ,e & ,m %all /& to ,e u,m1 & ,m - u,e 9 ! same relation z e & m to e um1 & m - 7ue74 ,f ! relation e ,e1 ! foll[+ equations foll[ 7,c7 e um 7,e u,m71 & f ! relation ,m m1 we h 7,d7 ,m - u ,e 7m - ue71 ,= ! compon5ts 9 ! directions p]p5dicul> to u1 & to ea* o!r1 ! equations are to be multipli$ by 7#a - u74 ,!n ! foll[+ equations foll[ f ! trans=mation8 equations 7#ab71 7#aj71 7#aa7 9 #d1 :5 we replace q1 rv1 r1 t1 rv1 r1 t' by u1 ,cu1 ,c1 1 ,cu1 ,c1 7-u ,cu 7_/7#a - u71 ,c'u 7,cu - u7_/7#a - u71 ,c ,c1 ,e7 7,cu - u7_/7#a - u7 7,e u,m7u1 ,c 7,e u,m7u_/7#a - u74 ,9 consid]ation ( ! mann] 9 :i* ent]s into ^! relations1 x w be conv5i5t to call ! vector ,c - u ) ! compon5ts ,cu - u 9 ! direction ( u1 & ,c 9 ! directions p]p5dicul> to u ! 8,convection curr5t40 ,?is la/ vani%es = #j4 ,we rem>k t = #a1 #a ! equations e ,e1 m ,m imm$iately lead to ! equations e ,e1 m ,m by means ( a reciprocal ,lor5tz-trans=mation ) -u z vector2 & = #j1 ! equation ,c #j leads to ,c u2 t ! fundam5tal equations ( !r discuss$ 9 #b becomes 9 fact ! limitt+ case ( ! equations obta9$ "h ) #a1 #a1 #j4 #i4 ,! ,fundam5tal ,equations 9 ,lor5tz's ,!ory4 ,let u n[ see h[ f> ! fundam5tal equations assum$ by ,lor5tz correspond to ! ,relativity po/ulate1 z def9$ 9 #h4 ,9 ! article on ,electron-!ory 7,ency41 ,ma?41 ,wiss41 ,bd4 ,v4 #b1 ,art #ad7 ,lor5tz has giv5 ! fundam5tal equations = any possible1 ev5 magnetis$ bodies 7see !re page #bji1 ,eqn ,,xxx1 =mula 7#ad7 on page #gh ( ! same 7"p74 7,,iiia7 ,curl 7,h - u,e7 ,j d,d_/dt u div ,d - curl u,d4 7,i7 div ,d 7,,iv7 curl ,e - d,b_/dt1 ,div ,b #j 7,v7 ,!n = mov+ non-magnetis$ bodies1 ,lor5tz puts 7page #bbc1 #c7 #a1 ,b ,h1 & 9 addition to t takes acc\nt ( ! occurr5ce ( ! di-electric con/ant 1 & conductivity accord+ to equations 7q,,xxxiv1 p4 #cbg7 ,d - ,e 7 - #a7 ,e u,b 7q,,xxxiii1 p4 #bbc7 ,j 7,e u,b7 ,lor5tz's ,e1 ,d1 ,h are "h d5ot$ by ,e1 ,m1 e1 m :ile ,j d5otes ! conduction curr5t4 ,! ?ree la/ equations :i* h be5 j cit$ "h co9cide ) eqn 7,,ii71 7,,iii71 7,,iv71 ! fir/ equation w\ld be1 if ,j is id5tifi$ ) ,c1 u 7! curr5t be+ z]o = #j1 7#bi7 ,curl ,h - 7u1 ,e7 ,c d,d_/dt - curl u,d1 & ?us comes out to be 9 a diff]5t =m ?an 7#a7 "h4 ,!re=e = magnetis$ bodies1 ,lor5tz's equations d n correspond to ! ,relativity ,pr9ciple4 ,on ! o!r h&1 ! =m correspond+ to ! relativity pr9ciple1 = ! condition ( non-magnetisation is to be tak5 out ( 7,d7 9 #h1 ) #a1 n z ,b ,h1 z ,lor5tz takes1 b z 7#cj7 ,b - u,d ,h - u,d 7,m - u,e m - ue4 ,n[ by putt+ ,h ,b1 ! diff]5tial equation 7#bi7 is trans=m$ into ! same =m z eqn 7#a7 "h :5 m - ue ,m - u,e4 ,!re=e x s happ5s t by a comp5sation ( two contradictions to ! relativity pr9ciple1 ! diff]5tial equations ( ,lor5tz = mov+ non-magnetis$ bodies at la/ agree ) ! relativity po/ulate4 ,if we make use ( 7#cj7 = non-magnetic bodies1 & put accord+ly ,h ,b u1 7,d - ,e71 !n 9 consequ5ce ( 7,c7 9 #h1 7 - #a7 7,e u1 ,b7 ,d - ,e u4 u1 ,d - ,e1 i4e4 = ! direction ( u1 7 - #a7 7,e u,b7u 7,d - ,e7u & = a p]p5dicul> direction 1 7 - #a7 ,e 7u,b7u 7#a - u7 7,d - ,e7u i4e4 x co9cides ) ,lor5tz's assumption1 if we neglect u 9 comp>ison to #a4 ,also to ! same ord] ( approximation1 ,lor5tz's =m = ,j corresponds to ! conditions impos$ by ! relativity pr9ciple comp4 7,e7 #h--t ! compon5ts ( ,ju1 ,j are equal to ! compon5ts ( 7,e u ,b7 multipli$ by 7#a - u7 or #a _/ 7#a - u7 respectively4 #aj4 ,fundam5tal ,equations ( ,e4 ,cohn4 ,e4 ,cohn assumes ! foll[+ fundam5tal equations4 7#ca7 ,curl 7,m u ,e7 d,e_/dt u div4 ,e ,j - ,curl ,e - 7u4 ,m7 d,m_/dt u div4 ,m4 7#cb7 ,j ,e1 ,e - u ,m1 ,m 7m u ,e47 :]e ,e ,m are ! electric & magnetic field int5sities 7=ces71 ,e1 ,m are ! electric & magnetic pol>isation 7induction74 ,! equations also p]mit ! exi/5ce ( true magnetism2 if we d n take into acc\nt ?is consid]ation1 div4 ,m4 is to be put #j4 ,an objection to ?is sy/em ( equations1 is t accord+ to ^!1 = #a1 #a1 ! vectors =ce & induction d n co9cide4 ,if 9 ! equations1 we conceive ,e & ,m & n ,e - 7,u4 ,m71 & ,m ,u ,e z electric & magnetic =ces1 & ) a glance to ?is we sub/itute = ,e1 ,m1 ,e1 ,m1 div4 ,e1 ! symbols e1 ,m1 ,e ,u ,m1 m - u e1 1 !n ! diff]5tial equations trans=m to our equations1 & ! conditions 7#cb7 trans=m into ,j 7,e u ,m7 e u1 7m - u e7 7,e u ,m7 ,m - u1 7,e u ,m7 7m - u e7 !n 9 fact ! equations ( ,cohn become ! same z ^? requir$ by ! relativity pr9ciple1 if errors ( ! ord] u are neglect$ 9 comp>ison to #a4 ,x may be m5tion$ "h t ! equations ( ,h]tz become ! same z ^? ( ,cohn1 if ! auxili>y conditions are 7#cc7 ,e ,e1 ,m ,m1 ,j ,e4 #aa4 ,typical ,repres5tations ( ! ,fundam5tal ,equations4 ,9 ! /atem5t ( ! fundam5tal equations1 our lead+ idea _h be5 t !y %\ld reta9 a cov>iance ( =m1 :5 subject$ to a gr\p ( ,lor5tz-trans=mations4 ,n[ we h to deal ) pond]omotive reactions & en]gy 9 ! electro-magnetic field4 ,"h f ! v fir/ !re c be no d\bt t ! settlem5t ( ?is "q is 9 "s way connect$ ) ! simple/ =ms :i* c be giv5 to ! fundam5tal equations1 satisfy+ ! conditions ( cov>iance4 ,9 ord] to arrive at su* =ms1 ,i %all fir/ ( all put ! fundam5tal equations 9 a typical =m :i* br+s out cle>ly _! cov>iance 9 case ( a ,lor5tz-trans=mation4 ,"h ,i am us+ a me?od ( calculation1 :i* enables u to deal 9 a simple mann] ) ! space-"t vectors ( ! #a/1 & #bnd k9d1 & ( :i* ! rules1 z f> z requir$ are giv5 bel[4 ,a sy/em ( magnitudes ah k =m$ into ! matrix a44444444444444444444a#a q ap #a44444444444;ap q arrang$ 9 p horizontal r[s1 & q v]tical columns is call$ a p q s]ies-matrix1 & w be d5ot$ by ! lett] ,a4 ,if all ! quantities ah k are multipli$ by ,c1 ! result+ matrix w be d5ot$ by ,,ca4 ,if ! roles ( ! horizontal r[s & v]tical columns be int]*>g$1 we obta9 a q p s]ies matrix1 :i* w be kn[n z ! transpos$ matrix ( ,a1 & w be d5ot$ by 4 a 4444444444444444444444 ap #a a#a q 444444444444 ap q ,if we h a second p q s]ies matrix ,b1 ,b b 4444444444444444444444444 bq bp #a 4444444444444 bp q !n ,a ,b %all d5ote ! p q s]ies matrix ^: memb]s are ah k bh k4 #b ,if we h two matrices ,a a 444444444444444444444 a#a q ap #a 44444444444 ap q ,b b#a #a 44444444444444 b#a r bq #a 4444444444 bp r :]e ! numb] ( horizontal r[s ( ,b1 is equal to ! numb] ( v]tical columns ( ,a1 !n by ,,ab1 ! product ( ! matrices ,a & ,b1 w be d5ot$ ! matrix ,c c 4444444444444444444444 c#a r cp r 44444444444 cp p :]e ch k ah #a bk ah #b b#b h 444 ak s bs k 444 ak q bq h ^! elem5ts be+ =m$ by comb9ation ( ! horizontal r[s ( ,a ) ! v]tical columns ( ,b4 ,= su* a po9t1 ! associative law 7,,ab7,s ,a7,,bs7 holds1 :]e ,s is a ?ird matrix :i* has got z _m horizontal r[s z ,b 7or ,,ab7 has got v]tical columns4 ,= ! transpos$ matrix ( ,c ,,ba1 we h #c4 ,we %all h pr9cipally to deal ) matrices ) at mo/ f\r v]tical columns & = horizontal r[s4 ,z a unit matrix 79 equations !y w be kn[n = ! sake ( %ortness z ! matrix #a7 w be d5ot$ ! foll[+ matrix 7#d #d s]ies7 ) ! elem5ts4 7#cd7 e e e e #a #j #j #j e e e e #j #a #j #j e e e e #j #j #a #j e e e e #j #j #j #a ,= a #d #d s]ies-matrix1 ,det ,a %all d5ote ! det]m9ant =m$ ( ! #d #d elem5ts ( ! matrix4 ,if det ,a #j1 !n correspond+ to ,a !re is a reciprocal matrix1 :i* we may d5ote by ,a s t ,a,a #a4 ,a matrix f #j f f f f #j f f f f #j f f f f #j 9 :i* ! elem5ts fulfil ! relation fh k -fh k1 is call$ an alt]nat+ matrix4 ,^! relations say t ! transpos$ matrix -f4 ,!n by f99 w be ! dual1 alt]nat+ matrix 7#ce7 f99 #j f f f f #j f f f f #j f f f f #j ,!n 7#cf7 f99 f f f f f f f i4e4 ,we %all h a #d #d s]ies matrix 9 :i* all ! elem5ts except ^? on ! diagonal f left up to "r d[n are z]o1 & ! elem5ts 9 ?is diagonal agree ) ea* o!r1 & are ea* equal to ! above m5tion$ comb9ation 9 7#cf74 ,! det]m9ant ( f is !re=e ! squ>e ( ! comb9ation1 by ,detf we %all d5ote ! expression ,detf f f f f f f #d4 ,a l9e> trans=mation xh h#a x h#b x h#c x h#d x 7h #a1b1c1 :i* is accompli%$ by ! matrix ,a 1 1 1 1 1 1 1 1 1 1 1 1 w be d5ot$ z ! trans=mation ,a4 ,by ! trans=mation ,a1 ! expression x x x x is *ang$ into ! quadratic = m hk xh xk1 :]e hk #ak #ak #b;h #bk #c;h #ck #d;h #dk are ! memb]s ( a #d #d s]ies matrix :i* is ! product ( ,a1 ! transpos$ matrix ( ,a into ,a4 ,if by ! trans=mation1 ! expression is *ang$ to x x#b x#b x1 we mu/ h ,a #a4 ,a has to correspond to ! foll[+ relation1 if trans=mation 7#ch7 is to be a ,lor5tz-trans=mation4 ,= ! det]m9ant ( ,a7 x foll[s out ( 7#ci7 t 7,det ,a7 #a1 or ,det ,a #a4 ,f ! condition 7#ci7 we obta9 ,a 1 i4e4 ! reciprocal matrix ( ,a is equival5t to ! transpos$ matrix ( ,a4 ,= ,a z ,lor5tz trans=mation1 we h fur!r ,det ,a #a1 ! quantities involv+ ! index #d once 9 ! subscript are purely imag9>y1 ! o!r co-effici5ts are real1 & a #j4 #e4 ,a space "t vector ( ! fir/ k9d#ba :i* s repres5t$ by ! #a #d s]ies matrix1 7#da7 s s s s s is to be replac$ by s,a 9 case ( a ,lor5tz trans=mation ,a4 i4e4 s s s s s s s s s ,a2 ,a space-"t vector ( ! #bnd k9d#bb ) compon5ts f 444 f %all be repres5t$ by ! alt]nat+ matrix 7#db7 f #j f f f f #j f f f f #j f f f f #j & is to be replac$ by ,a f ,a 9 case ( a ,lor5tz trans=mation see ! rules 9 #e 7#bc7 7#bd74 ,!re=e ref]r+ to ! expression 7#cg71 we h ! id5tity ,det 7 f ,a7 ,det ,a4 ,det f4 ,!re=e ,det f becomes an inv>iant 9 ! case ( a ,lor5tz trans=mation see eq4 7#bf7 ,see4 #e4 ,look+ back to 7#cf71 we h = ! dual matrix 7f99,a7 7,af,a7 ,af99f,a ,det function4 ,a,a ,detf f :i* x is to be se5 t ! dual matrix f99 behaves exactly l ! prim>y matrix f1 & is !re=e a space "t vector ( ! ,,ii k9d2 f99 is !re=e kn[n z ! dual space-"t vector ( f ) compon5ts 7f1 f1 f171 7f1 f1 f74 #f4 ,if w & s are two space-"t rectors ( ! #a/ k9d !n by w 7z well z by s 7 w be und]/ood ! comb9ation 7#dc7 w s w s w s w s4 ,9 case ( a ,lor5tz trans=mation ,a1 s9ce 7w,a7 77 w s1 ?is expression is inv>iant4--,if w #j1 !n w & s are p]p5dicul> to ea* o!r4 ,two space-"t rectors ( ! fir/ k9d 7w1 s7 gives u a #b #d s]ies matrix w w w w s s s s ,!n x foll[s imm$iately t ! sy/em ( six magnitudes 7#dd7 w s - w s1 w s - w s1 w s - w s1 w s - w s1 w s - w s1 w s - w s1 behaves 9 case ( a ,lor5tz-trans=mation z a space-"t vector ( ! ,,ii k9d4 ,! vector ( ! second k9d ) ! compon5ts 7#dd7 are d5ot$ by w1 s4 ,we see easily t ,det w1 s #j4 ,! dual vector ( w1 s %all be writt5 z w1 s4 ,if is a space-"t vector ( ! #a/ k9d1 f ( ! second k9d1 w f signifies a #a #d s]ies matrix4 ,9 case ( a ,lor5tz-trans=mation ,a1 w is *ang$ into w w,a1 f into f ,a f ,a1--!re=e w f becomes 7w,a ,a f ,a7 w f ,a i4e4 w f is trans=m$ z a space-"t vector ( ! #a/ k9d4#bc ,we c v]ify1 :5 w is a space-"t vector ( ! #a/ k9d1 f ( ! #bnd k9d1 ! important id5tity 7#de7 w1 wf w1 wf9999 7w 7f4 ,! sum ( ! two space "t vectors ( ! second k9d on ! left side is to be und]/ood 9 ! s5se ( ! addition ( two alt]nat+ matrices4 ,= example1 = #j1 #j1 #j1 i1 f if1 if1 if1 #j 2 f99 if1 if1 if1 #j f #j1 #j1 #j1 f1 f1 f2 f9999 #j1 #j1 #j1 f1 f1 f4 ,! fact t 9 ?is special case1 ! relation is satisfi$1 suffices to e/abli% ! !orem 7#de7 g5]ally1 = ?is relation has a cov>iant "* 9 case ( a ,lor5tz trans=mation1 & is homog5e\s 9 71 1 1 74 ,aft] ^! prep>atory works let u engage ourselves ) ! equations 7,c17 7,d17 7,e7 by means :i* ! con/ants 1 w be introduc$4 ,in/ead ( ! space vector u1 ! velocity ( matt]1 we %all introduce ! space-"t vector ( ! fir/ k9d ) ! compon5ts4 ux_/7#a - u71 uy_/7#a - u71 uz_/7#a - u71 i_/7#a - u74 7#dj7 :]e -#a & -i #j4 ,by ,f & f %all be und]/ood ! space "t vectors ( ! second k9d ,m - i,e1 m - ie4 ,9 ,f1 we h a space "t vector ( ! fir/ k9d ) compon5ts ,f ,f ,f ,f ,f ,f ,f ,f ,f ,f ,f ,f ,! fir/ ?ree quantities 71 1 7 are ! compon5ts ( ! space-vector 7,e u1 ,m7_/7#a - u71 & fur!r 7 iu ,e_/7#a - u74 ,because ,f is an alt]nat+ matrix1 7#di7 #j4 i4e4 is p]p5dicul> to ! vector 2 we c also write ix y z 4 ,i %all call ! space-"t vector ( ! fir/ k9d z ! ,electric ,re/ ,=ce4#bd ,relations analog\s to ^? hold+ betwe5 -,f1 ,e1 ,m1 ,u1 hold among/ -f1 e1 m1 u1 & 9 p>ticul> -f is normal to 4 ,! relation 7,c7 c be writt5 z ,c f ,f4 ,! expression 7f7 gives f\r compon5ts1 b ! f\r? c be d]iv$ f ! fir/ ?ree4 ,let u n[ =m ! "t-space vector #a/ k9d1 - if991 ^: compon5ts are -i7 f f f7 -i7 f f f7 -i7 f f f7 -i7 f f f7 ,( ^!1 ! fir/ ?ree 1 1 1 are ! x1 y1 z compon5ts ( ! space-vector #ea7 7m - 7ue77_/7#a - u7 & fur!r 7#eb7 i7um7_/7#a - u74 ,among ^! !re is ! relation 7#ec7 #j :i* c also be writt5 z i 7ux uy uz 74 ,! vector is p]p5dicul> to 2 we c call x ! ,magnetic re/-=ce4 ,relations analog\s to ^! hold among ! quantities ,f991 ,m1 ,e1 u & ,relation 7,d7 c be replac$ by ! =mula ,d -,f99 f994 ,we c use ! relations 7,c7 & 7,d7 to calculate ,f & f f & we h ,f -1 ,f99 -i1 f -1 f99 -i4 & apply+ ! relation 7#de7 & 7#df71 we h ,f 4 i4 99 #ee7 f 4 i4 99 #ef7 i4e4 ,f 7 - 7 i - 1 etc4 f 7 - 7 i - 41 etc4 ,let u n[ consid] ! space-"t vector ( ! second k9d 1 ) ! compon5ts - 1 - 1 - - 1 - 1 - ,!n ! correspond+ space-"t vector ( ! fir/ k9d 1 vani%es id5tically ow+ to equations #i7 & #ec7 = 4 -77 77 ,let u n[ take ! vector ( ! #a/ k9d 7#eg7 i99 ) ! compon5ts -i 1 etc4 ,!n by apply+ rule 7#de71 we h 7#eh7 4 i99 i4e4 - i7 - 7 etc4 ,! vector fulfils ! relation 77 #j1 7:i* we c write z i7x y z7 & is also normal to 4 ,9 case #j1 we h #j1 #j1 #j1 & 1 1 4 ,i %all call 1 :i* is a space-"t vector #a/ k9d ! ,re/-,ray4 ,z = ! relation ,e71 :i* introduces ! conductivity we h -,s -7s s s s7 7- u ,cu 7_/7#a - u7 4 ,?is expression gives u ! re/-d5sity ( electricity 7see #h & #d74 ,!n #fa7 s 77 repres5ts a space-"t vector ( ! #a/ k9d1 :i* s9ce -#a1 is normal to 1 & :i* ,i may call ! re/-curr5t4 ,let u n[ conceive ( ! fir/ ?ree compon5t ( ?is vector z ! 7x-y-z7 co-ord9ates ( ! space-vector1 !n ! compon5t 9 ! direction ( u is ,cu - 7 u 7_/7#a - u7 7cu - u 7_/7#a - u7 ,ju_/7#a - u7 & ! compon5t 9 a p]p5dicul> direction is ,cu ,j4 ,?is space-vector is connect$ ) ! space-vector ,j ,c - u1 :i* we d5ot$ 9 #h z ! conduction-curr5t4 ,n[ by comp>+ ) -,f1 ! relation 7,e7 c be br\ght into ! =m ,e s 77 - ,f1 ,?is =mula conta9s f\r equations1 ( :i* ! f\r? foll[s f ! fir/ ?ree1 s9ce ?is is a space-"t vector :i* is p]p5dicul> to 4 ,la/ly1 we %all trans=m ! diff]5tial equations 7,a7 & 7,b7 into a typical =m4 #ab4 ,! ,diff]5tial ,op]ator ,lor4 ,a #d #d s]ies matrix #fb7 ,s ,s ,s ,s ,s ,skh ,s ,s ,s ,s ,s ,s ,s ,s ,s ,s ,s ,s ) ! condition t 9 case ( a ,lor5tz trans=mation x is to be replac$ by ,,sa1 may be call$ a space-"t matrix ( ! ,,ii k9d4 ,we h examples ( ?is 93-- #a7 ! alt]nat+ matrix f1 :i* corresponds to ! space-"t vector ( ! ,,ii k9d1-- #b7 ! product f,f ( two su* matrices1 = by a trans=mation ,a1 x is replac$ by 7,af,a,a,,fa7 ,af,,fa1 #c7 fur!r :5 71 1 1 7 & 71 1 1 7 are two space-"t vectors ( ! #a/ k9d1 ! #d #d matrix ) ! elem5t ,shk hk1 la/ly 9 a multiple ,l ( ! unit matrix ( #d #d s]ies 9 :i* all ! elem5ts 9 ! pr9cipal diagonal are equal to ,l1 & ! re/ are z]o4 ,we %all h to d con/antly ) functions ( ! space-"t po9t 7x1 y1 z1 x71 & we may ) advantage employ ! #a #d s]ies matrix1 =m$ ( diff]5tial symbols1-- _/x1 _/y1 _/z1 _/it1 or 7#fc7 _/x _/x _/x _/x ,= ?is matrix ,i %all use ! %ort5$ f 8lor40#be ,!n if ,s is1 z 9 7#fb71 a space-"t matrix ( ! ,,ii k9d1 by lor ,s w be und]/ood ! #a #d s]ies matrix ,k ,k ,k ,k :]e ,kk ,s#ak_/x ,s#bk_/x ,s#ck_/x ,s#d;h_/x4 ,:5 by a ,lor5tz trans=mation ,a1 a new ref]5ce sy/em 7x x x x7 is introduc$1 we c use ! op]ator lor _/x _/x _/x _/x ,!n ,s is trans=m$ to ,s ,s ,a ,shk 1 s by lor ',s is meant ! #a #d s]ies matrix1 ^: elem5t are ,k'k ,s#ak_/x ,s#bk_/x ,s#ck_/x ,s#dk_/x4 ,n[ = ! diff]5tiation ( any function ( 7x y z t7 we h ! rule _/xk _/x x_/xk _/x x_/xk _/x x_/xk _/x x_/xk _/x a#ak _/x a#bk _/x a#ck _/x a#dk4 s t1 we h symbolically lor lor ,a4 ,!re=e x foll[s t lor ,s lor 7,a ,a ,,sa7 7lor ,s7,a4 i4e41 lor ,s behaves l a space-"t vector ( ! fir/ k9d4 ,if ,l is a multiple ( ! unit matrix1 !n by lor ,l w be d5ot$ ! matrix ) ! elem5ts ,l_/x ,l_/x ,l_/x ,l_/x ,if s is a space-"t vector ( ! #a/ k9d1 !n lor s_/x s_/x s_/x s_/x4 ,9 case ( a ,lor5tz trans=mation ,a1 we h lor lor ,a4 s lor s4 i4e41 lor s is an inv>iant 9 a ,lor5tz-trans=mation4 ,9 all ^! op]ations ! op]ator lor plays ! "p ( a space-"t vector ( ! fir/ k9d4 ,if f repres5ts a space-"t vector ( ! second k9d1--lor f d5otes a space-"t vector ( ! fir/ k9d ) ! compon5ts f_/x f_/x f_/x1 f_/x f_/x f_/x1 f_/x f_/x f_/x1 f_/x f_/x f_/x ,s ! sy/em ( diff]5tial equations 7,a7 c be express$ 9 ! concise =m ,a lor f -s1 & ! sy/em 7,b7 c be express$ 9 ! =m ,b log ,f99 #j4 ,ref]r+ back to ! def9ition 7#fg7 = log 1 we f9d t ! comb9ations lor 77lor f771 & lor 77lor ,f9977 vani% id5tically1 :5 f & ,f99 are alt]nat+ matrices4 ,accord+ly x foll[s out ( ,a1 t 7#fh7 7s_/x7 7s_/x7 7s_/x7 7s_/x7 #j1 :ile ! relation 7#fi7 lor 7lor ,f997 #j1 signifies t ( ! f\r equations 9 ,b1 only ?ree repres5t indep5d5t conditions4 ,i %all n[ collect ! results4 ,let d5ote ! space-"t vector ( ! fir/ k9d 7u_/7#a - u71 i_/7#a - u77 7u velocity ( matt]71 ,f ! space-"t vector ( ! second k9d 7,m1-i,e7 7,m magnetic induction1 ,e ,electric =ce1 f ! space-"t vector ( ! second k9d 7m1-ie7 7m magnetic =ce1 e ,electric ,induction4 s ! space-"t vector ( ! fir/ k9d 7,c1 i7 7 electrical space-d5sity1 ,c - u conductivity curr5t1 dielectric con/ant1 magnetic p]meability1 conductivity1 !n ! fundam5tal equations = electromagnetic processes 9 mov+ bodies are#bf ,a lor f -s ,b log ,f99 #j ,c f ,f ,d ,f99 f99 ,e s 771 w - ,f4 -#a1 & ,f1 f1 ,f991 f991 s 7s7 :i* are space-"t vectors ( ! fir/ k9d are all normal to 1 & = ! sy/em ,b1 we h lor 7lor ,f997 #j4 ,be>+ 9 m9d ?is la/ relation1 we see t we h z _m indep5d5t equations at our disposal z are necess>y = det]m9+ ! motion ( matt] z well z ! vector u z a function ( x1 y1 z1 t1 :5 prop] fundam5tal data are giv54 #ac4 ,! ,product ( ! ,field-vectors f ,f4 ,f9ally let u enquire ab\t ! laws :i* lead to ! det]m9ation ( ! vector z a function ( 7x1 y1 z1 t47 ,9 ^! inve/igations1 ! expressions :i* are obta9$ by ! multiplication ( two alt]nat+ matrices f #j f f f f #j f f f f #j f f f f #j ,f #j ,f ,f ,f ,f #j ,f ,f ,f ,f #j ,f ,f ,f ,f #j are ( mu* importance4 ,let u write1 7#gj7 f,f ,s - ,l ,s ,s ,s ,s ,s - ,l ,s ,s ,s ,s ,s - ,l ,s ,s ,s ,s ,s - ,l ,!n 7#ga7 ,s ,s ,s ,s #j4 ,let ,l n[ d5ote ! symmetrical comb9ation ( ! indices #a1 #b1 #c1 #d1 giv5 by 7#gb7 ,l 7f ,f f,f f ,f f ,f f ,f f ,f7 ,!n we %all h 7#gc7 ,s 7f ,f f ,f f ,f - f ,f - f ,f f ,f7 ,s f ,f f ,f etc4444 ,9 ord] to express 9 a real =m1 we write 7#gd7 ,s ,s ,s ,s ,s ,s ,s ,s ,s ,s ,s ,s ,s ,s ,s ,s ,s ,xx ,yx ,zx -i,tx ,xy ,yy ,zy -i,ty ,xz ,yz ,zz -i,tz -i,xt -i,yt -i,zt ,tt ,n[ ,xx mx,mx - my,my - mz,mz ex,ex - ey,ey - ez,ez s 7#ge7 ,xy mx,my ey,ex1 ,yx my,mx ex,ey etc4 ,xt ey,mz - ez,my1 ,tx mx,ey - my,ez1 etc4 ,tt mx,mx my,my mz,mz ex,ex ey,ey ez,ez ,lt mx,mx my,my mz,mz - ex,ex - ey,ey - ez,ez ,^! quantities#bg are all real4 ,9 ! !ory = bodies at re/1 ! comb9ations 7,xx1 ,xy1 ,xz1 ,yz1 ,yy1 ,yz1 ,zx1 ,zy1 ,zz7 are kn[n z 8,maxwell's ,/resses10 ,tx1 ,ty1 ,tz are kn[n z ! ,poynt+'s ,vector1 ,tt z ! electromagnetic en]gy-d5sity1 & ,l z ! ,langrangian function4 ,on ! o!r h&1 by multiply+ ! alt]nat+ matrices ( f99 & ,f991 we obta9 7#gg7 ,f99f99 -,s - ,l1 -,s1 -,s4 -,s -,s1 -,s - ,l1 -,s1 -,s -,s -,s1 -,s - ,l1 -,s -,s -,s -,s -,s - ,l & h5ce1 we c put 7#gh7 f,f ,s - ,l1 ,f99f99 -,s - ,l1 :]e by ,l1 we mean ,l-times ! unit matrix1 i4e4 ! matrix ) elem5ts ,lehk 1 7ehh #a1 ehk #j1 h k h1 k #a1 #b1 #c1 #d74 ,s9ce "h ,,sl ,,ls1 we d$uce t1 ,f99f99f,f 7-,s - ,l77,s - ,l7 -,,ss ,l1 & f9d1 s9ce f99f ,detf1 ,f99,f ,det,f1 we arrive at ! int]e/+ conclusion 7#gi7 ,,ss ,l - ,detf ,det,f i4e4 ! product ( ! matrix ,s into itself c be express$ z ! multiple ( a unit matrix--a matrix 9 :i* all ! elem5ts except ^? 9 ! pr9cipal diagonal are z]o1 ! elem5ts 9 ! pr9cipal diagonal are all equal & h ! value giv5 on ! "r-h& side ( 7#gi74 ,!re=e ! g5]al relations 7#hj7 ,sh#a ,s#ak ,sh#b ,s#bk ,sh#c ,s#ck ,sh#d ,s#dk #j1 h1 k be+ unequal indices 9 ! s]ies #a1 #b1 #c1 #d1 & 7#ha7 ,sh#a ,s#a;h ,sh#b ,s#b;h ,sh#c ,s#c;h ,sh#d ,s#d;h ,l - ,detf ,det,f1 = h #a1 #b1 #c1 #d4 ,n[ if in/ead ( ,f1 & f 9 ! comb9ations 7#gb7 & 7#gc71 we introduce ! electrical re/-=ce 1 ! magnetic re/-=ce 1 & ! re/-ray 7#ee71 7#ef7 & 7#eg71 we c pass ov] to ! expressions1-- 7#hb7 ,l - 1 7#hc7 ,shk - ehk - ehk 7h k - 77 h k 7h k - h k7 - h k - h k 7h k #a1 #b1 #c1 #d74 ,"h we h 1 ehh #a1 ehk #j 7h k74 ,! "r side ( 7#hb7 z well z ,l is an inv>iant 9 a ,lor5tz trans=mation1 & ! #d #d elem5t on ! "r side ( 7#hc7 z well z ,sk h repres5t a space "t vector ( ! second k9d4 ,rememb]+ ?is fact1 x suffices1 = e/abli%+ ! !orems 7#hb7 & 7#hc7 g5]ally1 to prove x = ! special case #j1 #j1 #j1 i4 ,b = ?is case #j1 we imm$iately arrive at ! equations 7#hb7 & 7#hc7 by means 7#de71 7#ea71 7#fj7 on ! "o h&1 & e ,e1 ,m m on ! o!r h&4 ,! expression on ! "r-h& side ( 7#ha71 :i* equals 7m ,m - e,e7 7em7 7,,em71 is #j1 because 7em 1 7,,em7 2 n[ ref]r+ back to #gi71 we c d5ote ! positive squ>e root ( ?is expression z ,det#a_/d ,s4 ,s9ce -f1 & -,f1 we obta9 = 1 ! transpos$ matrix ( ,s1 ! foll[+ relations f 7#gh71 7#hd7 ,ff - ,l1 f99 ,f99 - - ,l1 ,!n is - ,s ,sh k - ,st k an alt]nat+ matrix1 & d5otes a space-"t vector ( ! second k9d4 ,f ! expressions 7#hc71 we obta91 7#he7 ,s - - 7 - #a7 1 1 f :i* we d$uce t see 7#eg71 7#eh74 7#hf7 7,s - 799 #j1 7#hg7 7,s - 7 7 - #a7 ,:5 ! matt] is at re/ at a space-"t po9t1 #j1 !n ! equation #hf7 d5otes ! exi/5ce ( ! foll[+ equations ,zy ,yz1 ,xz ,zx1 ,yx ,xy1 & f #hc71 ,tx 1 ,ty 1 ,tz ,xt 1 ,yt 1 ,zt ,n[ by means ( a rotation ( ! space co-ord9ate sy/em r\nd ! null-po9t1 we c make1 ,zy ,yz #j1 ,xz ,zx #j1 ,xx ,xy #j1 ,accord+ to #ga71 we h 7#hh7 ,xx ,yy ,zz ,tt #j1 & accord+ to #hc71 ,tt #j4 ,9 special cases1 :]e vani%es x foll[s f #ha7 t ,xx ,yy ,zz ,tt1 7,det#a_/d ,s71 & if ,t1 & "o ( ! ?ree magnitudes ,xx1 ,yy1 ,zz are ,det#a_/d ,s1 ! two o!rs -,det#a_/d ,s4 ,if does n vani% let #j1 !n we h 9 p>ticul> f #hj7 ,tz ,xt #j1 ,tz ,yt #j1 ,zz ,tz ,tz ,tt #j1 & if #j1 #j1 ,zz -,tt ,x foll[s f 7#ha71 7see also #hc7 t ,xx -,yy ,det#a_/d ,s1 & -,zz ,tt 7,det ,s 7 ,det#a_/d,s4 ,! space-"t vector ( ! fir/ k9d 7#hi7 ,k lor ,s1 is ( v great importance = :i* we n[ want to demon/rate a v important trans=mation ,accord+ to #gh71 ,s ,l f,f1 & x foll[s t lor ,s lor ,l lor f,f4 ,! symbol #8lor' d5otes a diff]5tial process :i* 9 lor f,f1 op]ates on ! "o h& ^u ! compon5ts ( f1 on ! o!r h& also ^u ! compon5ts ( ,f4 ,accord+ly lor f,f c be express$ z ! sum ( two p>ts4 ,! fir/ "p is ! product ( ! matrices 7lor f7 ,f1 lor f be+ reg>d$ z a #a #d s]ies matrix4 ,! second "p is t "p ( lor f,f1 9 :i* ! diff5tiations op]ate ^u ! compon5ts ( ,f alone4 ,f #gh7 we obta9 f,f -,f99f99 - #b,l2 h5ce ! second "p ( lor f,f -7lor ,f997f99 ! "p ( -#b lor ,l1 9 :i* ! diff]5tiations op]ate ^u ! compon5ts ( ,f alone4 ,we ?us obta9 lor ,s 7lor f7,f - 7lor ,f997f99 ,n1 :]e ,n is ! vector ) ! compon5ts ,nh 7f_/xh ,f f_/xh ,f f_/xh ,f f_/xh ,f f_/xh ,f f_/xh ,f - ,f_/xh f - ,f_/xh f - ,f_/xh f - ,f_/xh f - ,f_/xh f - ,f_/xh f71 7h #a1 #b1 #c1 #d7 ,by us+ ! fundam5tal relations ,a7 & ,b71 #ij7 is trans=m$ into ! fundam5tal relation 7#ia7 lor ,s -s,f ,n4 ,9 ! limitt+ case #a1 #a1 f ,f1 ,n vani%es id5tically4 ,n[ ^u ! basis ( ! equations 7#ee7 & 7#ef71 & ref]r+ back to ! expression 7#hb7 = ,l1 & f #eg7 we obta9 ! foll[+ expressions z compon5ts ( ,n1-- 7#ib7 ,nh - _/xh - _/xh 7 - #a77 _/xh _/xh _/xh _/xh7 = h #a1 #b1 #c1 #d4 ,n[ if we make use ( 7#ei71 & d5ote ! space-vector :i* has 1 1 z ! x1 y1 z compon5ts by ! symbol ,w1 !n ! ?ird compon5t ( #ib7 c be express$ 9 ! =m 7#ic7 7 - #a7_/7a - u7 7,w u_/xh71 ,! r\nd bracket d5ot+ ! scal> product ( ! vectors )9 x4 #ad4 ,! ,pond]omotive ,=ce4#bh ,let u n[ write out ! relation ,k lor ,s -s,f ,n 9 a m practical =m2 we h ! f\r equations 7#id7 ,k ,xx_/x ,xy_/y ,xy_/z - ,xt_/t ,ex sy,mz - sz,mx - _/x - _/x 7 - #a7_/7a - u7 7,wu_/x71 7#ie7 ,k ,yx_/x ,yy_/y ,yz_/z - ,yt_/t ,ey sz,mx - sx,my - _/y - _/y 7 - #a7_/7a - u7 7,wu_/y71 7#if7 ,k ,zx_/x ,zy_/y ,zz_/z - ,zt_/t ,e sx,my - sy,m - _/z - _/z 7 - #a7_/7a - u7 7,wu_/z71 7#ig7 7#a_/;i7,k ,ty_/x - ,ty_/y - ,tz_/z - ,tt_/t sx,ex sy,ey sz,ez - _/t - _/t 7 - #a7_/7a - u7 7,wu_/t74 ,x is my op9ion t :5 we calculate ! pond]omotive =ce :i* acts ^u a unit volume at ! space-"t po9t x1 y1 z1 t1 x has got1 x1 y1 z compon5ts z ! fir/ ?ree compon5ts ( ! space-"t vector ,k 7,k71 ,?is vector is p]p5dicul> to 2 ! law ( ,en]gy f9ds its expression 9 ! f\r? relation4 ,! e/abli%m5t ( ?is op9ion is res]v$ = a sep>ate tract4 ,9 ! limit+ case #a1 #a1 #j1 ! vector ,n #j1 ,s 1 ,k #j1 & we obta9 ! ord9>y equations 9 ! !ory ( electrons4 ,footnote #i3 ,vide ,note #a4 ,footnote #aj3 ,note #b4 ,footnote #aa3 ,vide ,note #c4 ,footnote #ab3 ,vide ,note #d4 ,footnote #ac3 ,note #e4 ,footnote #ad3 ,see notes on #h & #aj4 ,footnote #ae3 ,see note #i4 ,footnote #af3 ,see ,note4 ,footnote #ag3 ,vide ,note4 ,footnote #ah3 ,j z be+s :i* are conf9$ )9 a n>r[ region surr\nd+ a po9t on a sph]ical surface1 may fall into ! error t a sph]e is a geometric figure 9 :i* "o diamet] is p>ticul>ly di/+ui%$ f ! re/4 ,footnote #ai3 ,e9zelne /elle d] ,mat]ie4 ,footnote #bj3 ,vide ,note4 ,footnote #ba3 ,vide note #ac4 ,footnote #bb3 ,vide note #ad4 ,footnote #bc3 ,vide note #ae4 ,footnote #bd3 ,vide note #af4 ,footnote #be3 ,vide note #ag4 ,footnote #bf3 ,vide note #ai4 ,footnote #bg3 ,vide note #ah4 ,footnote #bh3 ,vide note #dj4 ,,app5dix ,me*anics & ! ,relativity-,po/ulate4 ,x w\ld be v unsatisfactory1 if ! new way ( look+ at ! "t-concept1 :i* p]mits a ,lor5tz trans=mation1 7 to be conf9$ to a s+le "p ( ,physics4 ,n[ _m au?ors say t classical me*anics /& 9 opposition to ! relativity po/ulate1 :i* is tak5 to be ! basis ( ! new ,electro-dynamics4 ,9 ord] to decide ?is let u fix our att5tion ^u a special ,lor5tz trans=mation repres5t$ by 7#aj71 7#aa71 7#ab71 ) a vector v 9 any direction & ( any magnitude q #a b diff]5t f z]o4 ,= a mom5t we %all n suppose any special relation to hold betwe5 ! unit ( l5g? & ! unit ( "t1 s t in/ead ( t1 t1 q1 we %all write ct1 ct1 & q_/c1 :]e c repres5ts a c]ta9 positive con/ant1 & q is c4 ,! above m5tion$ equations are trans=m$ into r r1 rv c7rv - qt7_/7c - q71 t 7qrv ct7_/c7c - q7 ,!y d5ote1 z we rememb]1 t r is ! space-vector 7x1 y1 z71 r is ! space-vector 7x y z7 ,if 9 ^! equations1 keep+ v con/ant we approa* ! limit c 1 !n we obta9 f ^! r r1 rv rv - qt1 t t4 ,! new equations w\ld n[ d5ote ! trans=mation ( a spatial co-ord9ate sy/em 7x1 y1 z7 to ano!r spatial co-ord9ate sy/em 7x y z7 ) p>allel axes1 ! null po9t ( ! second sy/em mov+ ) con/ant velocity 9 a /raight l9e1 :ile ! "t p>amet] rema9s un*ang$4 ,we c1 !re=e1 say t classical me*anics po/ulates a cov>iance ( ,physical laws = ! gr\p ( homog5e\s l9e> trans=mations ( ! expression -x - y - z c 7#a7 :5 c 4 ,n[ x is r confus+ to f9d t 9 "o bran* ( ,physics1 we %all f9d a cov>iance ( ! laws = ! trans=mation ( expression 7#a7 ) a f9ite value ( c1 9 ano!r "p = c 4 ,x is evid5t t accord+ to ,newtonian ,me*anics1 ?is cov>iance holds = c & n = c velocity ( light4 ,may we n !n reg>d ^? traditional cov>iances = c only z an approximation consi/5t ) exp]i5ce1 ! actual cov>iance ( natural laws hold+ = a c]ta9 f9ite value ( c4 ,i may "h po9t out t by if in/ead ( ! ,newtonian ,relativity-,po/ulate ) c 1 we assume a relativity-po/ulate ) a f9ite c1 !n ! axiomatic con/ruction ( ,me*anics appe>s to ga9 consid]ably 9 p]fection4 ,! ratio ( ! "t unit to ! l5g? unit is *os5 9 a mann] s z to make ! velocity ( light equival5t to unity4 ,:ile n[ ,i want to introduce geometrical figures 9 ! manifold ( ! v>iables 7x1 y1 z1 t71 x may be conv5i5t to leave 7y1 z7 out ( acc\nt1 & to treat x & t z any possible pair ( co-ord9ates 9 a plane1 ref]r$ to oblique axes4 ,a space "t null po9t #j 7x1 y1 z1 t #j1 #j1 #j1 #j7 w be kept fix$ 9 a ,lor5tz trans=mation4 ,! figure -x - y - z t #a1 t #j 444 7#b7 :i* repres5ts a hyp] boloidal %ell1 conta9s ! space-"t po9ts ,a 7x1 y1 z1 t #j1 #j1 #j1 #a71 & all po9ts ,a :i* aft] a ,lor5tz-trans=mation ent] into ! newly introduc$ sy/em ( ref]5ce z 7x1 y1 z1 t #j1 #j1 #j1 #a74 ,! direction ( a radius vector #j;,a drawn f #j to ! po9t ,a ( 7#b71 & ! directions ( ! tang5ts to 7#b7 at ,a are to be call$ normal to ea* o!r4 ,let u n[ foll[ a def9ite position ( matt] 9 its c\rse ?r\gh all "t t4 ,! totality ( ! space-"t po9ts 7x1 y1 z1 t7 :i* correspond to ! positions at diff]5t times t1 %all be call$ a space-"t l9e4 ,! task ( det]m9+ ! motion ( matt] is compris$ 9 ! foll[+ problem3--,x is requir$ to e/abli% = e space-"t po9t ! direction ( ! space-"t l9e pass+ ?r\gh x4 ,to trans=m a space-"t po9t ,p 7x1 y1 z1 t7 to re/ is equival5t to introduc+1 by means ( a ,lor5tz trans=mation1 a new sy/em ( ref]5ce 7x1 y1 z1 t71 9 :i* ! t axis has ! direction #j;,a1 #j;,a indicat+ ! direction ( ! space-"t l9e pass+ ?r\gh ,p4 ,! space t con/1 :i* is to be laid ?r\gh ,p1 is ! "o :i* is p]p5dicul> to ! space-"t l9e ?r\gh ,p4 ,to ! increm5t dt ( ! "t ( ,p corresponds ! increm5t d 7dt - dx - dy7 - dz dt7#a - u7 ( ! newly introduc$ "t p>amet] t4 ,! value ( ! integral d 7-7dx dx dx dx77 :5 calculat$ ^u ! space-"t l9e f a fix$ initial po9t ,p to ! v>iable po9t ,p1 7bo? be+ on ! space-"t l9e71 is kn[n z ! #8,prop]-"t' ( ! position ( matt] we are conc]n$ ) at ! space-"t po9t ,p4 7,x is a g5]alization ( ! idea ( ,positional-"t :i* 0 introduc$ by ,lor5tz = uni=m motion47 ,if we take a body ,r :i* has got ext5sion 9 space at "t t1 !n ! region compris+ all ! space-"t l9e pass+ ?r\gh ,r & t %all be call$ a space-"t filam5t4 ,if we h an analytical expression 7x y1 z1 t7 s t 7x1 y z t7 #j is int]sect$ by e space "t l9e ( ! filam5t at "o po9t1--:]eby -7_/x71 -7_/y71 -7_/z71 -7_/t7 #j1 _/t #j4 !n ! totality ( ! int]sect+ po9ts w be call$ a cross section ( ! filam5t4 ,at any po9t ,p ( su* across-section1 we c introduce by means ( a ,lor5tz trans=mation a sy/em ( ref]5ce 7x1 y1 z t71 s t accord+ to ?is _/x #j1 _/y #j1 _/z #j1 _/t #j4 ,! direction ( ! uniquely det]m9$ t--axis 9 "q "h is kn[n z ! upp] normal ( ! cross-section at ! po9t ,p & ! value ( d,j dx dy dz = ! surr\nd+ po9ts ( ,p on ! cross-section is kn[n z ! elem5t>y cont5ts 7,inhalts-elem5t7 ( ! cross-section4 ,9 ?is s5se ,r is to be reg>d$ z ! cross-section normal to ! t axis ( ! filam5t at ! po9t t t1 & ! volume ( ! body ,r is to be reg>d$ z ! cont5ts ( ! cross-section4 ,if we all[ ,r to conv]ge to a po9t1 we come to ! conception ( an inf9itely ?9 space-"t filam5t4 ,9 su* a case1 a space-"t l9e w be ?\ght ( z a pr9cipal l9e & by ! t]m #8,prop]-"t' ( ! filam5t w be und]/ood ! #8,prop]-"t' :i* is laid along ?is pr9cipal l9e2 "u ! t]m normal cross-section ( ! filam5t1 we %all und]/& ! cross-section ^u ! space :i* is normal to ! pr9cipal l9e ?r\gh ,p4 ,we %all n[ =mulate ! pr9ciple ( cons]vation ( mass4 ,to e space ,r at a "t t1 belongs a positive quantity--! mass at ,r at ! "t t4 ,if ,r conv]ges to a po9t 7x1 y1 z1 t71 !n ! quoti5t ( ?is mass1 & ! volume ( ,r approa*es a limit 7x1 y1 z1 t71 :i* is kn[n z ! mass-d5sity at ! space-"t po9t 7x1 y1 z1 t74 ,! pr9ciple ( cons]vation ( mass says--t = an inf9itely ?9 space-"t filam5t1 ! product d,j1 :]e mass-d5sity at ! po9t 7x1 y1 z1 t7 ( ! filam5t 7i4e41 ! pr9cipal l9e ( ! filam5t71 d,j cont5ts ( ! cross-section normal to ! t axis1 & pass+ ?r\gh 7x1 y1 z1 t71 is con/ant along ! :ole filam5t4 ,n[ ! cont5ts d,jn ( ! normal cross-section ( ! filam5t :i* is laid ?r\gh 7x1 y1 z1 t7 is 7#d7 d,jn 7#a_/7a - u77d,j -i d,j 7dt_/d7d,j4 & ! function _/-i 7#a - u77 7_/t4 7#e7 may be def9$ z ! re/-mass d5sity at ! position 7x y z t74 ,!n ! pr9ciple ( cons]vation ( mass c be =mulat$ 9 ?is mann]3-- ,= an inf9itely ?9 space-"t filam5t1 ! product ( ! re/-mass d5sity & ! cont5ts ( ! normal cross-section is con/ant along ! :ole filam5t4 ,9 any space-"t filam5t1 let u consid] two cross-sections ,q & ,q1 :i* h only ! po9ts on ! b\nd>y common to ea* o!r2 let ! space-"t l9es inside ! filam5t h a l>g] value ( t on ,q ?an on ,q4 ,! f9ite range enclos$ betwe5 ,q & ,q %all be call$ a space-"t si*el1#bi ,q is ! l[] b\nd>y1 & ,q is ! upp] b\nd>y ( ! si*el4 ,if we decompose a filam5t into elem5t>y space-"t filam5ts1 !n to an entrance-po9t ( an elem5t>y filam5t ?r\gh ! l[] b\nd>y ( ! si*el1 !re corresponds an exit po9t ( ! same by ! upp] b\nd>y1 :]eby = bo?1 ! product d,jn tak5 9 ! s5se ( 7#d7 & 7#e71 has got ! same value4 ,!re=e ! diff]5ce ( ! two integrals d,jn 7! fir/ be+ ext5d$ ov] ! upp]1 ! second ^u ! l[] b\nd>y7 vani%es4 ,accord+ to a well-kn[n !orem ( ,integral ,calculus ! diff]5ce is equival5t to lor dx dy dz dt1 ! integration be+ ext5d$ ov] ! :ole range ( ! si*el1 & 7comp4 7#fg71 #ab7 lor 7_/x7 7_/x7 7_/x7 7_/x74 ,if ! si*el r$uces to a po9t1 !n ! diff]5tial equation lor #j1 7#f7 :i* is ! condition ( cont9uity 7ux_/x7 7uy_/y7 7uz_/z7 7_/t7 #j4 ,fur!r let u =m ! integral ,n dx dy dz dt 7#g7 ext5d+ ov] ! :ole range ( ! space-"t si*el4 ,we %all decompose ! si*el into elem5t>y space-"t filam5ts1 & e "o ( ^! filam5ts 9 small elem5ts d ( its prop]-"t1 :i* are h[ev] l>ge comp>$ to ! l9e> dim5sions ( ! normal cross-section2 let u assume t ! mass ( su* a filam5t d,jn dm & write 1 l = ! #8,prop]-"t' ( ! upp] & l[] b\nd>y ( ! si*el4 ,!n ! integral 7#g7 c be d5ot$ by d,jn d 7-7 dm4 tak5 ov] all ! elem5ts ( ! si*el4 ,n[ let u conceive ( ! space-"t l9es inside a space-"t si*el z mat]ial curves compos$ ( mat]ial po9ts1 & let u suppose t !y are subject$ to a cont9ual *ange ( l5g? inside ! si*el 9 ! foll[+ mann]4 ,! entire curves are to be v>i$ 9 any possible mann] inside ! si*el1 :ile ! end po9ts on ! l[] & upp] b\nd>ies rema9 fix$1 & ! individual sub/antial po9ts ^u x are displac$ 9 su* a mann] t !y always move =w>d normal to ! curves4 ,! :ole process may be analytically repres5t$ by means ( a p>amet] 1 & to ! value #j1 %all correspond ! actual curves inside ! si*el4 ,su* a process may be call$ a virtual displacem5t 9 ! si*el4 ,let ! po9t 7x1 y1 z1 t7 9 ! si*el #j h ! values x x1 y y1 z z1 t t1 :5 ! p>amet] has ! value 2 ^! magnitudes are !n functions ( 7x1 y1 z1 t1 74 ,let u n[ conceive ( an inf9itely ?9 space-"t filam5t at ! po9t 7x y z t7 ) ! normal section ( cont5ts d,jn & if d,jn d,jn be ! cont5ts ( ! normal section at ! correspond+ position ( ! v>i$ filam5t1 !n accord+ to ! pr9ciple ( cons]vation ( mass--7 d be+ ! re/-mass-d5sity at ! v>i$ position71 7#h7 7 7 7d,jn d,jn7 d,jn dm4 ,9 consequ5ce ( ?is condition1 ! integral 7#g7 tak5 ov] ! :ole range ( ! si*el1 v>ies on acc\nt ( ! displacem5t z a def9ite function ,n ,n ( 1 & we may call ?is function ,n ,n z ! mass action ( ! virtual displacem5t4 ,if we n[ introduce ! me?od ( writ+ ) indices1 we %all h 7#i7 d7xh xh7 dxh k xh_/xk xh_/ d k #a1 #b1 #c1 #d h #a1 #b1 #c1 #d ,n[ on ! basis ( ! rem>ks already made1 x is cle> t ! value ( ,n ,n1 :5 ! value ( ! p>amet] is 1 w be3-- 7#aj7 ,n ,n 77d7 77_/d7dx dy dz dt1 ! integration ext5d+ ov] ! :ole si*el d7 7 :]e d7 7 d5otes ! magnitude1 :i* is d$uc$ f 7-7dx dx7 - 7dx dx7 - 7dx dx7 - 7dx dx77 by means ( 7#i7 & dx d1 dx d1 dx d1 dx d1 d #j !re=e3-- 7#aa7 7d7 77_/d 7 -7h 7xh_/xk7k77 k #a1 #b1 #c1 #d4 h #a1 #b1 #c1 #d4 ,we %all n[ subject ! value ( ! diff]5tial quoti5t 7#ab7 77d7,n ,n77_/d7 7 #j7 to a trans=mation4 ,s9ce ea* xh z a function ( 7x1 y1 z1 t7 vani%es = ! z]o-value ( ! p>amet] 1 s 9 g5]al dxk_/7xh #j1 = #j4 ,let u n[ put 7xh_/7 h 7h #a1 #b1 #c1 #d7 7#ac7 #j !n on ! basis ( 7#aj7 & 7#aa71 we h ! expression 7#ab73-- - h77h_/x7 7h_/x7 7h_/x7 7h_/x77 dx dy dz dt = ! sy/em 7x x x x7 on ! b\nd>y ( ! si*el1 7x x x x7 %all vani% = e value ( & !re=e 1 1 1 are nil4 ,!n by p>tial integration1 ! integral is trans=m$ into ! =m h7h_/x h_/x h_/x h_/x7 dx dy dz dt ! expression )9 ! bracket may be writt5 z h k_/xk kh_/xk4 ,! fir/ sum vani%es 9 consequ5ce ( ! cont9uity equation 7b74 ,! second may be writt5 z 7h_/x77dx_/d7 7h_/x77dx_/d7 7h_/x77dx_/d7 7h_/x77dx_/d7 dh_/d 7d_/d77dxh_/d7 :]eby 7d_/d7 is meant ! diff]5tial quoti5t 9 ! direction ( ! space-"t l9e at any position4 ,= ! diff]5tial quoti5t 7#ab71 we obta9 ! f9al expression 7#ad7 77_/7 7_/7 7_/7 7_/77 dx dy dz dt4 ,= a virtual displacem5t 9 ! si*el we h po/ulat$ ! condition t ! po9ts suppos$ to be sub/antial %all advance normally to ! curves giv+ _! actual motion1 :i* is #j2 ?is condition d5otes t ! h is to satisfy ! condition w w w w #j4 7#ae7 ,let u n[ turn our att5tion to ! ,maxwellian t5sions 9 ! electrodynamics ( /ation>y bodies1 & let u consid] ! results 9 #ab & #ac2 !n we f9d t ,hamilton's ,pr9ciple c be reconcil$ to ! relativity po/ulate = cont9u\sly ext5d$ ela/ic m$ia4 ,at e space-"t po9t 7z 9 #ac71 let a space "t matrix ( ! #bnd k9d be kn[n 7#af7 ,s ,s ,s ,s ,s ,xx ,yx ,zx -i,tx ,s ,s ,s ,s ,xy ,yy ,zy -i,ty ,s ,s ,s ,s ,xz ,yz ,zz -i,tz ,s ,s ,s ,s -i,xt -i,yt -i,zt ,tt :]e ,xn ,yx 44444,xz1 ,tt are real magnitudes4 ,= a virtual displacem5t 9 a space-"t si*el 7) ! previ\sly appli$ designation7 ! value ( ! integral 7#ag7 ,w ,w 7,sh k 77xk xk77_/xh dx dy dz dt ext5d$ ov] ! :ole range ( ! si*el1 may be call$ ! t5sional "w ( ! virtual displacem5t4 ,! sum :i* comes =? "h1 writt5 9 real magnitudes1 is ,xx ,yy ,zz ,tt ,xx 7x7_/x ,xy 7x7_/y 444 ,zz 7z7_/z - ,xt 7x_/t - 444 ,tx 7t7_/x 444 ,tt 7t7_/t we c n[ po/ulate ! foll[+ m9imum pr9ciple 9 me*anics4 ,if any space-"t ,si*el be b\nd$1 !n = ea* virtual displacem5t 9 ! ,si*el1 ! sum ( ! mass-works1 & t5sion works %all always be an extremum = t process ( ! space-"t l9e 9 ! ,si*el :i* actually occurs4 ,! mean+ is1 t = ea* virtual displacem5t1 7d7,n ,w7_/d7 #j #j 7#ah7 ,by apply+ ! me?ods ( ! ,calculus ( ,v>iations1 ! foll[+ f\r diff]5tial equations at once foll[ f ?is m9imal pr9ciple by means ( ! trans=mation 7#ad71 & ! condition 7#ae74 7#ai7 wh_/ ,kh wh 7h #a1 #b1 #c1 #d7 :5ce ,kh ,s#a h_/x ,s#b h_/x ,s#c h_/x ,s#d h_/x1 7#bj7 are compon5ts ( ! space-"t vector #a/ k9d ,k lor ,s1 & ,x is a factor1 :i* is to be det]m9$ f ! relation w - #a4 ,by multiply+ 7#ai7 by wh1 & summ+ ! f\r1 we obta9 ,x ,k1 & !re=e cle>ly ,k 7,k7w w be a space-"t vector ( ! #a/ k9d :i* is normal to w4 ,let u write out ! compon5ts ( ?is vector z ,x1 ,y1 ,z1 i,t ,!n we arrive at ! foll[+ equation = ! motion ( matt]1 7#ba7 d_/d 7dx_/d7 ,x1 d_/d 7dy_/d7 ,y1 d_/d 7dz_/d7 ,z1 d_/d 7dx_/d7 ,t1 & we h also 7dx_/d7 7dy_/d7 7dz_/d7 7dt_/d7 -#a1 & ,x dx_/d ,y dy_/d ,z dz_/d ,t dt_/d4 ,on ! basis ( ?is condition1 ! f\r? ( equations 7#ba7 is to be reg>d$ z a direct consequ5ce ( ! fir/ ?ree4 ,f 7#ba71 we c d$uce ! law = ! motion ( a mat]ial po9t1 i4e41 ! law = ! c>e] ( an inf9itely ?9 space-"t filam5t4 ,let x1 y1 z1 t1 d5ote a po9t on a pr9cipal l9e *os5 9 any mann] )9 ! filam5t4 ,we %all =m ! equations 7#ba7 = ! po9ts ( ! normal cross section ( ! filam5t ?r\gh x1 y1 z1 t1 & integrate !m1 multiply+ by ! elem5t>y cont5ts ( ! cross section ov] ! :ole space ( ! normal section4 ,if ! integrals ( ! "r side be ,rx ,ry ,rz ,rt & if m be ! con/ant mass ( ! filam5t1 we obta9 7#bb7 m d_/d dx_/d ,rx1 m d_/d dy_/d ,ry1 m d_/d dz_/d ,rz1 m d_/d dt_/d ,rt ,r is n[ a space-"t vector ( ! #a/ k9d ) ! compon5ts 7,rx ,ry ,rz ,rt7 :i* is normal to ! space-"t vector ( ! #a/ k9d w1--! velocity ( ! mat]ial po9t ) ! compon5ts dx_/d1 dy_/d1 dz_/d1 i dt_/d4 ,we may call ?is vector ,r ! mov+ =ce ( ! mat]ial po9t4 ,if in/ead ( integrat+ ov] ! normal section1 we integrate ! equations ov] t cross section ( ! filam5t :i* is normal to ! t axis1 & passes ?r\gh 7x1 y1 z1 t71 !n ,see 7#d7 ! equations 7#bb7 are obta9$1 b are n[ multipli$ by d_/dt2 9 p>ticul>1 ! la/ equation comes out 9 ! =m1 m d_/dt 7dt_/d7 wx ,rx d_/dt wy ,ry d_/dt wz ,rz d_/dt4 ,! "r side is to be look$ ^u z ! am\nt ( "w done p] unit ( "t at ! mat]ial po9t4 ,9 ?is equation1 we obta9 ! en]gy-law = ! motion ( ! mat]ial po9t & ! expression m 7dt_/d - #a7 m #a_/7a - w7 - #a m 7 w #c_/h w 7 may be call$ ! k9etic en]gy ( ! mat]ial po9t4 ,s9ce dt is always great] ?an d we may call ! quoti5t 7dt - d7_/d z ! 8,ga90 7vorgeh57 ( ! "t ov] ! prop]-"t ( ! mat]ial po9t & ! law c !n be ?us express$2--,! k9etic en]gy ( a mat]ial po9t is ! product ( its mass into ! ga9 ( ! "t ov] its prop]-"t4 ,! set ( f\r equations 7#bb7 aga9 %[s ! symmetry 9 7x1 y1 z1 t71 :i* is dem&$ by ! relativity po/ulate2 to ! f\r? equation h[ev]1 a high] physical significance is to be atta*$1 z we h already se5 9 ! analog\s case 9 electrodynamics4 ,on ! gr\nd ( ?is dem& = symmetry1 ! triplet consi/+ ( ! fir/ ?ree equations are to be con/ruct$ aft] ! model ( ! f\r?2 rememb]+ ?is circum/ance1 we are ju/ifi$ 9 say+1-- 8,if ! relativity-po/ulate be plac$ at ! head ( me*anics1 !n ! :ole set ( laws ( motion foll[s f ! law ( en]gy40 ,i _c refra9 f %[+ t no contradiction to ! assumption on ! relativity-po/ulate c be expect$ f ! ph5om5a ( gravitation4 ,if ,b997x991 y991 z991 t997 be a solid 7fe/]7 space-"t po9t1 !n ! region ( all ^? space-"t po9ts ,b 7x1 y1 z1 t71 = :i* 7#bc7 7x - x997 7y - y997 7z - z997 7t - t997 t - t99 #j may be call$ a 8,ray-figure0 7,/rahl-gebilde7 ( ! space "t po9t ,b994 ,a space-"t l9e tak5 9 any mann] c be cut by ?is figure only at "o p>ticul> po9t2 ?is easily foll[s f ! convexity ( ! figure on ! "o h&1 & on ! o!r h& f ! fact t all directions ( ! space-"t l9es are only directions f ,b99 t[>ds to ! concave side ( ! figure4 ,!n ,b99 may be call$ ! light-po9t ( ,b4 ,if 9 7#bc71 ! po9t 7x y z t7 be suppos$ to be fix$1 ! po9t 7x99 y99 z99 t997 be suppos$ to be v>iable1 !n ! relation 7#bc7 w\ld repres5t ! locus ( all ! space-"t po9ts ,b991 :i* are light-po9ts ( ,b4 ,let u conceive t a mat]ial po9t ,f ( mass m may1 ow+ to ! pres5ce ( ano!r mat]ial po9t ,f991 exp]i5ce a mov+ =ce accord+ to ! foll[+ law4 ,let u picture to ourselves ! space-"t filam5ts ( ,f & ,f99 along ) ! pr9cipal l9es ( ! filam5ts4 ,let ,,bc be an inf9itely small elem5t ( ! pr9cipal l9e ( ,f2 fur!r let ,b99 be ! light po9t ( ,b1 ,c99 be ! light po9t ( ,c on ! pr9cipal l9e ( ,f992 s t ,,oa is ! radius vector ( ! hyp]boloidal fundam5tal figure 7#bc7 p>allel to ,b99,c991 f9ally ,d99 is ! po9t ( int]section ( l9e ,b99,c99 ) ! space normal to itself & pass+ ?r\gh ,b4 ,! mov+ =ce ( ! mass-po9t ,f 9 ! space-"t po9t ,b is n[ ! space-"t vector ( ! fir/ k9d :i* is normal to ,,bc1 & :i* is compos$ ( ! vectors 7#bd7 mm997,,oa_/,b99,d997 ,,bd99 9 ! direction ( ,,bd991 & ano!r vector ( suitable value 9 direction ( ,b99,c994 ,n[ by 7,,oa_/,b99,d997 is to be und]/ood ! ratio ( ! two vectors 9 "q4 ,x is cle> t ?is proposition at once %[s ! cov>iant "* ) respect to a ,lor5tz-gr\p4 ,let u n[ ask h[ ! space-"t filam5t ( ,f behaves :5 ! mat]ial po9t ,f99 has a uni=m translatory motion1 i4e41 ! pr9cipal l9e ( ! filam5t ( ,f99 is a l9e4 ,let u take ! space "t null-po9t 9 ?is1 & by means ( a ,lor5tz-trans=mation1 we c take ?is axis z ! t-axis4 ,let x1 y1 z1 t1 d5ote ! po9t ,b1 let 99 d5ote ! prop] "t ( ,b991 reckon$ f ,o4 ,our proposition leads to ! equations 7#be7 dx_/d - m99x_/7t - 9971 dy_/d - m99y_/7t - 997 dz_/d -m99z_/7t - 9971 7#bf7 dt_/d -m99_/7t - 997 d7t - 997_/dt :]e 7#bg7 x y z 7t - 997 & 7#bh7 7dx_/d7 7dy_/d7 7dz_/d7 7dt_/d7 - #a4 ,9 consid]ation ( 7#bg71 ! ?ree equations 7#be7 are ( ! same =m z ! equations = ! motion ( a mat]ial po9t subject$ to attraction f a fix$ c5tre accord+ to ! ,newtonian ,law1 only t in/ead ( ! "t t1 ! prop] "t ( ! mat]ial po9t occurs4 ,! f\r? equation 7#bf7 gives !n ! connection betwe5 prop] "t & ! "t = ! mat]ial po9t4 ,n[ = diff]5t values ( 1 ! orbit ( ! space-po9t 7x y z7 is an ellipse ) ! semi-major axis a & ! ecc5tricity e4 ,let ,e d5ote ! ecc5tric anomaly1 ! increm5t ( ! prop] "t = a complete description ( ! orbit1 f9ally n #b1 s t f a prop]ly *os5 initial po9t 1 we h ! ,kepl]-equation 7#bi7 n ,e - e s9 ,e4 ,if we n[ *ange ! unit ( "t1 & d5ote ! velocity ( light by c1 !n f 7#bh71 we obta9 7#cj7 7dt_/d7 - #a 7m99_/ac7 7#a e cos ,e7_/7#a - e cos ,e7 ,n[ neglect+ c ) reg>d to #a1 x foll[s t ndt nd #a m99_/ac 7#a e cos ,e7_/7#a - e cos ,e7 f :i*1 by apply+ 7#bi71 7#ca7 nt con/ 7#a m99_/ac7 n m99_/ac ,s9 ,e4 ! factor m99_/ac is "h ! squ>e ( ! ratio ( a c]ta9 av]age velocity ( ,f 9 its orbit to ! velocity ( light4 ,if n[ m99 d5ote ! mass ( ! sun1 a ! semi major axis ( ! e>?'s orbit1 !n ?is factor am\nts to #aj4 ,! law ( mass attraction :i* has be5 j describ$ & :i* is =mulat$ 9 accordance ) ! relativity po/ulate w\ld signify t gravitation is propagat$ ) ! velocity ( light4 ,9 view ( ! fact t ! p]iodic t]ms 9 7#ca7 are v small1 x is n possible to decide out ( a/ronomical obs]vations betwe5 su* a law 7) ! modifi$ me*anics propos$ above7 & ! ,newtonian law ( attraction ) ,newtonian me*anics4 ,footnote #bi3 ,si*el--a ,g]man ^w mean+ a cresc5t or a scy!4 ,! orig9al t]m is reta9$ z !re is no suitable ,engli% equival5t4 ,,space ,,& ,,"t ,a ,lecture deliv]$ be=e ! ,natur=s*] ,v]sammlung 7,congress ( ,natural ,philosoph]s7 at ,cologne--7#ba/ ,septemb]1 #aijh74 ,g5tlem51 ,! conceptions ab\t "t & space1 :i* ,i hope to develop be=e y to-day1 has gr[n on exp]im5tal physical gr\nds4 ,h]e9 lies its /r5g?4 ,! t5d5cy is radical4 ,h5ce=?1 ! old conception ( space = itself1 & "t = itself %all r$uce to a m]e %ad[1 & "s sort ( union ( ! two w be f\nd consi/5t ) facts4 ,i ,n[ ,i want to %[ y h[ we c arrive at ! *ang$ concepts ab\t "t & space f me*anics1 z accept$ n[-a-days1 f purely ma!matical consid]ations4 ,! equations ( ,newtonian me*anics %[ a tw(old inv>iance1 7i7 _! =m rema9s unalt]$ :5 we subject ! fundam5tal space-coord9ate sy/em to any possible *ange ( position1 7ii7 :5 we *ange ! sy/em 9 its nature ( motion1 i4 e41 :5 we impress ^u x any uni=m motion ( translation1 ! null-po9t ( "t plays no "p4 ,we are accu/om$ to look ^u ! axioms ( geometry z settl$ once = all1 :ile we seldom h ! same am\nt ( conviction reg>d+ ! axioms ( me*anics1 & !re=e ! two inv>iants are seldom m5tion$ 9 ! same brea?4 ,ea* "o ( ^! d5otes a c]ta9 gr\p ( trans=mations = ! diff]5tial equations ( me*anics4 ,we look ^u ! exi/5ce ( ! fir/ gr\p z a fundam5tal *>act]i/ics ( space4 ,we always pref] to leave (f ! second gr\p to itself1 & ) a light he>t conclude t we c nev] decide f physical consid]ations :e!r ! space1 :i* is suppos$ to be at re/1 may n f9ally be 9 uni=m motion4 ,s ^! two gr\ps lead q sep>ate exi/5ces besides ea* o!r4 ,_! totally het]og5e\s "* may sc>e u away f ! attempt to comp\nd !m4 ,yet x is ! :ole comp\nd$ gr\p :i* z a :ole gives u occasion = ?\ght4 ,we wi% to picture to ourselves ! :ole relation graphically4 ,let 7x1 y1 z7 be ! rectangul> coord9ates ( space1 & t d5ote ! "t4 ,subjects ( our p]ception are always connect$ ) place & "t4 ,no "o has obs]v$ a place except at a p>ticul> "t1 or has obs]v$ a "t except at a p>ticul> place4 ,yet ,i respect ! dogma t "t & space h indep5d5t exi/5ces4 ,i w call a space-po9t plus a "t-po9t1 i4e41 a sy/em ( values x1 y1 z1 t1 z a _w-po9t4 ,! manifoldness ( all possible values ( x1 y1 z1 t1 w be ! _w4 ,i c draw f\r _w-axes ) ! *alk4 ,n[ any axis drawn consi/s ( quickly vibrat+ molecules1 & besides1 takes "p 9 all ! j\rneys ( ! e>? 2 & !re=e gives u occasion = reflection4 ,! great] ab/raction requir$ = ! f\r-axes does n cause ! ma!matician any tr\ble4 ,9 ord] n to all[ any yawn+ gap to exi/1 we %all suppose t at e place & "t1 some?+ p]ceptible exi/s4 ,9 ord] n to specify ei!r matt] or electricity1 we %all simply /yle ^! z sub/ances4 ,we direct our att5tion to ! _w-po9t x1 y1 z1 t1 & suppose t we are 9 a position to recognise ?is sub/antial po9t at any subsequ5t "t4 ,let dt be ! "t elem5t correspond+ to ! *anges ( space coord9ates ( ?is po9t dx1 dy1 dz4 ,!n we obta9 7z a picture1 s to speak1 ( ! p]5nial life-c>e] ( ! sub/antial po9t71--a curve 9 ! _w--! _w-l9e1 ! po9ts on :i* unambigu\sly correspond to ! p>amet] t f to -4 ,! :ole _w appe>s to be resolv$ 9 su* _w-l9es1 & ,i may j deviate f my po9t if ,i say t accord+ to my op9ion ! physical laws w\ld f9d _! fulle/ expression z mutual relations among ^! l9es4 ,by ?is conception ( "t & space1 ! 7x1 y1 z7 manifoldness t #j & its two sides t #j & t #j falls asund]4 ,if = ! sake ( simplicity1 we keep ! null-po9t ( "t & space fix$1 !n ! fir/ nam$ gr\p ( me*anics signifies t at t #j we c give ! x1 y1 & z-axes any possible rotation ab\t ! null-po9t correspond+ to ! homog5e\s l9e> trans=mation ( ! expression x y z4 ,! second gr\p d5otes t )\t *ang+ ! expression = ! me*anical laws1 we c sub/itute 7x - t1 y - t1 z - t = 7x1 y1 z7 :]e 71 1 7 are any con/ants4 ,accord+ to ?is we c give ! "t-axis any possible direction 9 ! upp] half ( ! _w t #j4 ,n[ :at h ! dem&s ( or?ogonality 9 space to d ) ?is p]fect fre$om ( ! "t-axis t[>ds ! upp] half8 ,to e/abli% ?is connection1 let u take a positive p>amet] c1 & let u consid] ! figure ct - x - y - z #a ,accord+ to ! analogy ( ! hyp]boloid ( two %eets1 ?is consi/s ( two %eets sep>at$ by t #j4 ,let u consid] ! %eet1 9 ! region ( t #j1 & let u n[ conceive ! trans=mation ( x1 y1 z1 t 9 ! new sy/em ( v>iables2 7x'1 y'1 z'1 t'7 by means ( :i* ! =m ( ! expression w rema9 unalt]$4 ,cle>ly ! rotation ( space r\nd ! null-po9t belongs to ?is gr\p ( trans=mations4 ,n[ we c h a full idea ( ! trans=mations :i* we picture to ourselves f a p>ticul> trans=mation 9 :i* 7y1 z7 rema9 unalt]$4 ,let u draw ! cross section ( ! upp] %eets ) ! plane ( ! x- & t-axes1 i4e41 ! upp] half ( ! hyp]bola ct - x #a1 ) its asymptotes 7vide fig4 #a74 ,!n let u draw ! radius rector ,,oa1 ! tang5t ,a ,b at ,a1 & let u complete ! p>allelogram ,,oa ,b ,c2 also produce ,b ,c to meet ! x-axis at ,d4 ,let u n[ take ,ox1 ,,oa z new axes ) ! unit measur+ rods ,,oc #a1 ,,oa 7#a_/;c7 2 !n ! hyp]bola is aga9 express$ 9 ! =m ct - x #a1 t #j & ! transition f 7x1 y1 z1 t7 to 7x y z t7 is "o ( ! transitions 9 "q4 ,let u add to ?is *>act]i/ic trans=mation any possible displacem5t ( ! space & "t null-po9ts2 !n we get a gr\p ( trans=mation dep5d+ only on c1 :i* we may d5ote by ,gc4 ,n[ let u increase c to inf9ity4 ,?us 7#a_/;c7 becomes z]o & x appe>s f ! figure t ! hyp]bola is gradually %runk into ! x-axis1 ! asymptotic angle becomes a /raight "o1 & e special trans=mation 9 ! limit *anges 9 su* a mann] t ! t-axis c h any possible direction upw>ds1 & x m & m approximates to x4 ,rememb]+ ?is po9t x is cle> t ! full gr\p belong+ to ,newtonian ,me*anics is simply ! gr\p ,gc1 ) ! value ( c 4 ,9 ?is /ate ( affairs1 & s9ce ,gc is ma!matically m intelligible ?an ,g1 a ma!matician may1 by a free play ( imag9ation1 hit ^u ! ?\ght t natural ph5om5a possess an inv>iance n only = ! gr\p ,g1 b 9 fact also = a gr\p ,gc1 :]e c is f9ite1 b yet exce$+ly l>ge comp>$ to ! usual measur+ units4 ,su* a preconception w\ld be an extraord9>y triumph = pure ma!matics4 ,at ! same "t ,i %all rem>k = :i* value ( c1 ?is inv>iance c be conclusively held to be true4 ,= c1 we %all sub/itute ! velocity ( light c 9 free space4 ,9 ord] to avoid speak+ ei!r ( space or ( vacuum1 we may take ?is quantity z ! ratio betwe5 ! electro/atic & electro-magnetic units ( electricity4 ,we c =m an idea ( ! inv>iant "* ( ! expression = natural laws = ! gr\p-trans=mation ,gc 9 ! foll[+ mann]4 ,out ( ! totality ( natural ph5om5a1 we c1 by successive high] approximations1 d$uce a coord9ate sy/em 7x1 y1 z1 t72 by means ( ?is coord9ate sy/em1 we c repres5t ! ph5om5a accord+ to def9ite laws4 ,?is sy/em ( ref]5ce is by no means uniquely det]m9$ by ! ph5om5a4 ,we c *ange ! sy/em ( ref]5ce 9 any possible mann] correspond+ to ! above-m5tion$ gr\p trans=mation ,gc1 b ! expressions = natural laws w n be *ang$ !reby4 ,= example1 correspond+ to ! above describ$ figure1 we c call t ! "t1 b !n necess>ily ! space connect$ ) x mu/ be express$ by ! manifoldness 7x y z74 ,! physical laws are n[ express$ by means ( x1 y1 z1 t1--& ! expressions are j ! same z 9 ! case ( x1 y1 z1 t4 ,accord+ to ?is1 we %all h 9 ! _w1 n "o space1 b _m spaces1--q analog\s to ! case t ! ?ree-dim5sional space consi/s ( an inf9ite numb] ( planes4 ,! ?ree-dim5sional geometry w be a *apt] ( f\r-dim5sional physics4 ,n[ y p]ceive1 :y ,i said 9 ! beg9n+ t "t & space %all r$uce to m]e %ad[s & we %all h a _w complete 9 itself4 ,,ii ,n[ ! "q may be ask$1--:at circum/ances lead u to ^! *ang$ views ab\t "t & space1 are !y n 9 contradiction ) obs]v$ ph5om5a1 d !y f9ally gu>antee u advantages = ! description ( natural ph5om5a8 ,be=e we ent] into ! discussion1 a v important po9t mu/ be notic$4 ,suppose we h individualis$ "t & space 9 any mann]2 !n a _w-l9e p>allel to ! t-axis w correspond to a /ation>y po9t2 a _w-l9e incl9$ to ! t-axis w correspond to a po9t mov+ uni=mly2 & a _w-curve w correspond to a po9t mov+ 9 any mann]4 ,let u n[ picture to our m9d ! _w-l9e pass+ ?r\gh any _w po9t x1 y1 z1 t2 n[ if we f9d ! _w-l9e p>allel to ! radius vector ,,oa ( ! hyp]boloidal %eet1 !n we c introduce ,,oa z a new "t-axis1 & !n accord+ to ! new conceptions ( "t & space ! sub/ance w appe> to be at re/ 9 ! _w po9t conc]n$4 ,we %all n[ introduce ?is fundam5tal axiom3-- ,! sub/ance exi/+ at any _w po9t c always be conceiv$ to be at re/1 if we e/abli% our "t & space suitably4 ,! axiom d5otes t 9 a _w-po9t1 ! expression cdt - dx - dy - dz %all always be positive or :at is equival5t to ! same ?+1 e velocity ,v %\ld be small] ?an c4 c %all !re=e be ! upp] limit = all sub/antial velocities & h]e9 lies a deep significance = ! quantity c4 ,at ! fir/ impression1 ! axiom seems to be r unsatisfactory4 ,x is to be rememb]$ t only a modifi$ me*anics w occur1 9 :i* ! squ>e root ( ?is diff]5tial comb9ation takes ! place ( "t1 s t cases 9 :i* ! velocity is great] ?an c w play no "p1 some?+ l imag9>y coord9ates 9 geometry4 ,! impulse & real cause ( inducem5t = ! assumption ( ! gr\p-trans=mation ,gc is ! fact t ! diff]5tial equation = ! propagation ( light 9 vacant space possesses ! gr\p-trans=mation ,gc4 ,on ! o!r h&1 ! idea ( rigid bodies has any s5se only 9 a sy/em me*anics ) ! gr\p ,ginf9ity4 ,n[ if we h an optics ) ,gc1 & on ! o!r h& if !re are rigid bodies1 x is easy to see t a t-direction c be def9$ by ! two hyp]boloidal %ells common to ! gr\ps ,g1 & ,gc1 :i* has got ! fur!r consequ5ce1 t by means ( suitable rigid in/rum5ts 9 ! laboratory1 we c p]ceive a *ange 9 natural ph5om5a1 9 case ( diff]5t ori5tations1 ) reg>d to ! direction ( progressive motion ( ! e>?4 ,b all ef=ts direct$ t[>ds ?is object1 & ev5 ! celebrat$ int]f]5ce-exp]im5t ( ,mi*elson h giv5 negative results4 ,9 ord] to supply an explanation = ?is result1 ,h4 ,a4 ,lor5tz =m$ a hypo!sis :i* practically am\nts to an inv>iance ( optics = ! gr\p ,gc4 ,accord+ to ,lor5tz e sub/ance %all suff] a contraction #a377a - v_/c77 9 l5g?1 9 ! direction ( its motion l_/l #a_/7a - v_/c7 l l7#a - v_/c74 ,?is hypo!sis s\nds r phanta/ical4 ,= ! contraction is n to be ?\ght ( z a consequ5ce ( ! resi/ance ( e!r1 b purely z a gift f ! skies1 z a sort ( condition always accompany+ a /ate ( motion4 ,i %all %[ 9 our figure t ,lor5tz's hypo!sis is fully equival5t to ! new conceptions ab\t "t & space4 ,!reby x may appe> m intelligible4 ,let u n[1 = ! sake ( simplicity1 neglect 7y1 z7 & fix our att5tion on a two dim5sional _w1 9 :i* let upright /rips p>allel to ! t-axis repres5t a /ate ( re/ & ano!r p>allel /rip incl9$ to ! t-axis repres5t a /ate ( uni=m motion = a body1 :i* has a con/ant spatial ext5sion 7see fig4 #a74 ,if ,,oa is p>allel to ! second /rip1 we c take t z ! t-axis & x z ! x-axis1 !n ! second body w appe> to be at re/1 & ! fir/ body 9 uni=m motion4 ,we %all n[ assume t ! fir/ body suppos$ to be at re/1 has ! l5g? l1 i4e41 ! cross section ,,pp ( ! fir/ /rip ^u ! x-axis l4 ,,oc1 :]e ,,oc is ! unit measur+ rod ^u ! x-axis--& ! second body also1 :5 suppos$ to be at re/1 has ! same l5g? l1 ?is means t1 ! cross section ,q,q ( ! second /rip has a cross-section l ,,oc1 :5 measur$ p>allel to ! x-axis4 ,9 ^! two bodies1 we h n[ images ( two ,lor5tz-electrons1 "o ( :i* is at re/ & ! o!r moves uni=mly4 ,n[ if we /ick to our orig9al coord9ates1 !n ! ext5sion ( ! second electron is giv5 by ! cross section ,,qq ( ! /rip belong+ to x measur$ p>allel to ! x-axis4 ,n[ x is cle> s9ce ,q,q l ,,oc1 t ,,qq l ,,od4 ,if 7dc_/dt7 v1 an easy calculation gives t ,,od ,,oc 7#a-7v_/c771 !re=e 7,,pp_/,,qq7 7#a_/7a-7v_/c77 ,?is is ! s5se ( ,lor5tz's hypo!sis ab\t ! contraction ( electrons 9 case ( motion4 ,on ! o!r h&1 if we conceive ! second electron to be at re/1 & !re=e adopt ! sy/em 7x1 t17 !n ! cross-section ,p,p ( ! /rip ( ! electron p>allel to ,,oc is to be reg>d$ z its l5g? & we %all f9d ! fir/ electron %ort5$ ) ref]5ce to ! second 9 ! same proportion1 = x is1 ,p,p_/,q,q ,,od_/,,oc ,,od_/,,oc ,,qq_/,,pp ,lor5tz call$ ! comb9ation t ( 7t & x7 z ! local "t 7,ortszeit7 ( ! uni=mly mov+ electron1 & us$ a physical con/ruction ( ?is idea = a bett] compreh5sion ( ! contraction-hypo!sis4 ,b to p]ceive cle>ly t ! "t ( an electron is z good z ! "t ( any o!r electron1 i4e4 t1 t are to be reg>d$ z equival5t1 has be5 ! s]vice ( ,a4 ,e9/e9 ,ann4 d4 ,phys4 #hia1 p4 #aije1 ,jahrb4 d4 ,radis4 444 #d-d-aa-aijg4 ,!re ! concept ( "t 0 %[n to be completely & unambigu\sly e/abli%$ by natural ph5om5a4 ,b ! concept ( space 0 n arriv$ at1 ei!r by ,e9/e9 or ,lor5tz1 probably because 9 ! case ( ! above-m5tion$ spatial trans=mations1 :]e ! 7x1 t7 plane co9cides ) ! x-t plane1 ! significance is possible t ! x-axis ( space "s-h[ rema9s cons]v$ 9 its position4 ,we c approa* ! idea ( space 9 a correspond+ mann]1 ?\gh "s may reg>d ! attempt z r fanta/ical4 ,accord+ to ^! ideas1 ! ^w 8,relativity-,po/ulate0 :i* has be5 co9$ = ! dem&s ( inv>iance 9 ! gr\p ,g1 seems to be r inexpressive = a true und]/&+ ( ! gr\p ,gc1 & = fur!r progress4 ,because ! s5se ( ! po/ulate is t ! f\r-dim5sional _w is giv5 9 space & "t by ph5om5a only1 b ! projection 9 "t & space c be h&l$ ) a c]ta9 fre$om1 & !re=e ,i w\ld r l to give to ?is ass]tion ! "n 8,! ,po/ulate ( ! ,absolute _w0 ,_w-,po/ulate4 ,,iii ,by ! _w-po/ulate a simil> treatm5t ( ! f\r det]m9+ quantities x1 y1 z1 t1 ( a _w-po9t is possible4 ,!reby ! =ms "u :i* ! physical laws come =?1 ga9 9 intelligibility1 z ,i %all pres5tly %[4 ,above all1 ! idea ( accel]ation becomes mu* m /rik+ & cle>4 ,i %all aga9 use ! geometrical me?od ( expression4 ,let u call any _w-po9t ,o z a 8,space-"t-null-po9t40 ,! cone ct - x - y - z ,o consi/s ( two p>ts ) ,o z apex1 "o "p hav+ t #j1 ! o!r hav+ t #j4 ,! fir/1 :i* we may call ! =e-cone consi/s ( all ^? po9ts :i* s5d light t[>ds ,o1 ! second1 :i* we may call ! aft-cone1 consi/s ( all ^? po9ts :i* receive _! light f ,o4 ,! region b\nd$ by ! =e-cone may be call$ ! =e-side ( ,o1 & ! region b\nd$ by ! aft-cone may be call$ ! aft-side ( ,o4 7,vide fig4 #b74 ,on ! aft-side ( ,o we h ! already consid]$ hyp]boloidal %ell ,f ct - x - y - z #a1 t #j4 ,! region inside ! two cones w be occupi$ by ! hyp]boloid ( "o %eet -,f x y z - ct k1 :]e k c h all possible positive values4 ,! hyp]bolas :i* lie ^u ?is figure ) ,o z c5tre1 are important = u4 ,= ! sake ( cle>ness ! individual bran*es ( ?is hyp]bola w be call$ ! 8,int]-hyp]bola ) c5tre ,o40 ,su* a hyp]bolic bran*1 :5 ?\ght ( z a _w-l9e1 w\ld repres5t a motion :i* = t - & t 1 asymptotically approa*es ! velocity ( light c4 ,if1 by way ( analogy to ! idea ( vectors 9 space1 we call any direct$ l5g? 9 ! manifoldness x1 y1 z1 t a vector1 !n we h to di/+ui% betwe5 a "t-vector direct$ f ,o t[>ds ! %eet ,f #a1 t #j & a space-vector direct$ f ,o t[>ds ! %eet -,f #a4 ,! "t-axis c be p>allel to any vector ( ! fir/ k9d4 ,any _w-po9t betwe5 ! =e & aft cones ( ,o1 may by means ( ! sy/em ( ref]5ce be reg>d$ ei!r z syn*ron\s ) ,o1 z well z lat] or e>li] ?an ,o4 ,e _w-po9t on ! =e-side ( ,o is necess>ily always e>li]1 e po9t on ! aft side ( ,o1 lat] ?an ,o4 ,! limit c corresponds to a complete fold+ up ( ! w$ge-%ap$ cross-section betwe5 ! =e & aft cones 9 ! manifoldness t #j4 ,9 ! figure drawn1 ?is cross-section has be5 int5tionally drawn ) a diff]5t bread?4 ,let u decompose a vector drawn f ,o t[>ds 7x1 y1 z1 t7 into its compon5ts4 ,if ! directions ( ! two vectors are respectively ! directions ( ! radius vector ,,or to "o ( ! surfaces ,f #a1 & ( a tang5t ,,rs at ! po9t ,r ( ! surface1 !n ! vectors %all be call$ normal to ea* o!r4 ,accord+ly ctt - xx - yy - zz #j1 :i* is ! condition t ! vectors ) ! compon5ts 7x1 y1 z1 t7 & 7x y z t7 are normal to ea* o!r4 ,= ! measurem5t ( vectors 9 diff]5t directions1 ! unit measur+ rod is to be fix$ 9 ! foll[+ mann]2--a space-l vector f #j to -,f ,i is always to h ! measure unity1 & a "t-l vector f ,o to ,f #a1 t #j is always to h ! measure #a_/;c4 ,let u n[ fix our att5tion ^u ! _w-l9e ( a sub/antive po9t runn+ ?r\gh ! _w-po9t 7x1 y1 z1 t72 !n z we foll[ ! progress ( ! l9e1 ! quantity d 7#a_/;c7 7cdt - dx - dy - dz71 corresponds to ! "t-l vector-elem5t 7dx1 dy1 dz1 dt74 ,! integral d1 tak5 ov] ! _w-l9e f any fix$ initial po9t ,p to any v>iable f9al po9t ,p1 may be call$ ! 8,prop]-"t0 ( ! sub/antial po9t at ,p ^u ! _w-l9e4 ,we may reg>d 7x1 y1 z1 t71 i4e41 ! compon5ts ( ! vector ,,op1 z functions ( ! 8prop]-"t0 2 let 74x1 4y1 4z1 4t7 d5ote ! fir/ diff]5tial-quoti5ts1 & 744x1 44y1 44z1 44t7 ! second diff]5tial quoti5ts ( 7x1 y1 z1 t7 ) reg>d to 1 !n ^! may respectively be call$ ! ,velocity-vector1 & ! ,accel]ation-vector ( ! sub/antial po9t at ,p4 ,n[ we h c 4t - 4x - 4y - 4z c c 4t44t - 4x44x - 4y44y - 4z44z #j i4e41 ! #8,velocity-vector' is ! "t-l vector ( unit measure 9 ! direction ( ! _w-l9e at ,p1 ! #8,accel]ation-vector' at ,p is normal to ! velocity-vector at ,p1 & is 9 any case1 a space-l vector4 ,n[ !re is1 z c be easily se51 a c]ta9 hyp]bola1 :i* has ?ree inf9itely contigu\s po9ts 9 common ) ! _w-l9e at ,p1 & ( :i* ! asymptotes are ! g5]ators ( a #8=e-cone' & an #8aft-cone4' ,?is hyp]bola may be call$ ! 8hyp]bola ( curvature0 at ,p 7vide fig4 #c74 ,if ,m be ! c5tre ( ?is hyp]bola1 !n we h to deal "h ) an #8,int]-hyp]bola' ) c5tre ,m4 ,let ,p measure ( ! vector ,,mp1 !n we easily p]ceive t ! accel]ation-vector at ,p is a vector ( magnitude c_/ 9 ! direction ( ,,mp4 ,if 44x1 44y1 44z1 44t are nil1 !n ! hyp]bola ( curvature at ,p r$uces to ! /raight l9e t\*+ ! _w-l9e at ,p1 & 4 ,,iv ,9 ord] to demon/rate t ! assumption ( ! gr\p ,gc = ! physical laws does n possibly lead to any contradiction1 x is unnecess>y to und]take a revision ( ! :ole ( physics on ! basis ( ! assumptions und]ly+ ?is gr\p4 ,! revision has already be5 successfully made 9 ! case ( 8,!rmodynamics & ,radiation10#cj = 8,electromagnetic ph5om5a01#ca & f9ally = 8,me*anics ) ! ma9t5ance ( ! idea ( mass40 ,= ?is la/ m5tion$ prov9ce ( physics1 ! "q may be ask$3 if !re is a =ce ) ! compon5ts ,x1 ,y1 ,z 79 ! direction ( ! space-axes7 at a _w-po9t 7x1 y1 z1 t71 :]e ! velocity-vector is 74x1 4y1 4z1 4t71 !n h[ are we to reg>d ?is =ce :5 ! sy/em ( ref]5ce is *ang$ 9 any possible mann]8 ,n[ x is kn[n t !re are c]ta9 well-te/$ !orems ab\t ! pond]omotive =ce 9 electromagnetic fields1 :]e ! gr\p ,gc is und\bt$ly p]missible4 ,^! !orems lead u to ! foll[+ simple rule2 if ! sy/em ( ref]5ce be *ang$ 9 any way1 !n ! suppos$ =ce is to be put z a =ce 9 ! new space-coord9ates 9 su* a mann]1 t ! correspond+ vector ) ! compon5ts 4t,x1 4t,y1 4t,z1 4t,t1 :]e ,t #a_/;c 74x_/4t ,x 4y_/4t ,y 4z_/4t ,z7 #a_/;c 7! rate ( :i* "w is done at ! _w-po9t71 rema9s unalt]$4 ,?is vector is always normal to ! velocity-vector at ,p4 ,su* a =ce-vector1 repres5t+ a =ce at ,p1 may be call$ a mov+ =ce-vector at ,p4 ,n[ ! _w-l9e pass+ ?r\gh ,p w be describ$ by a sub/antial po9t ) ! con/ant me*anical mass m4 ,let u call m-times ! velocity-vector at ,p z ! impulse-vector1 & m-times ! accel]ation-vector at ,p z ! =ce-vector ( motion1 at ,p4 ,accord+ to ^! def9itions1 ! foll[+ law tells u h[ ! motion ( a po9t-mass takes place "u any mov+ =ce-vector#cb3 ,! =ce-vector ( motion is equal to ! mov+ =ce-vector4 ,?is enunciation comprises f\r equations = ! compon5ts 9 ! f\r directions1 ( :i* ! f\r? c be d$uc$ f ! fir/ ?ree1 because bo? ( ! above-m5tion$ vectors are p]p5dicul> to ! velocity-vector4 ,f ! def9ition ( ,t1 we see t ! f\r? simply expresses ! 8,en]gy-law40 ,accord+ly c-times ! compon5t ( ! impulse-vector 9 ! direction ( ! t-axis is to be def9$ z ! k9etic-en]gy ( ! po9t-mass4 ,! expression = ?is is mc dt_/d mc _/7#a - v_/c7 i4e41 if we d$uct f ?is ! additive con/ant mc1 we obta9 ! expression mv ( ,newtonian-me*anics up to magnitudes ( ! ord] ( #a_/;c4 ,h5ce x appe>s t ! en]gy dep5ds ^u ! sy/em ( ref]5ce4 ,b s9ce ! t-axis c be laid 9 ! direction ( any "t-l axis1 !re=e ! en]gy-law comprises1 = any possible sy/em ( ref]5ce1 ! :ole sy/em ( equations ( motion4 ,?is fact reta9s its significance ev5 9 ! limit+ case c 1 = ! axiomatic con/ruction ( ,newtonian me*anics1 z has already be5 po9t$ out by ,t4 ,r4 ,s*tz4#cc ,f ! v beg9n+1 we c e/abli% ! ratio betwe5 ! units ( "t & space 9 su* a mann]1 t ! velocity ( light becomes unity4 ,if we n[ write -#a t l1 9 ! place ( l1 !n ! diff]5tial expression d -7dx dy dz dl71 becomes symmetrical 9 7x1 y1 r1 l72 ?is symmetry !n ent]s into ea* law1 :i* does n contradict ! _w-po/ulate4 ,we c clo! ! 8ess5tial nature ( ?is po/ulate 9 ! my/ical1 b ma!matically significant =mula #caj km -#a ,sec4 ,v ,! advantages aris+ f ! =mulation ( ! _w-po/ulate are illu/rat$ by no?+ s /rik+ly z by ! expressions :i* tell u ab\t ! reactions ex]t$ by a po9t-*>ge mov+ 9 any mann] accord+ to ! ,maxwell-,lor5tz !ory4 ,let u conceive ( ! _w-l9e ( su* an electron ) ! *>ge 7e71 & let u introduce ^u x ! 8,prop]-"t0 reckon$ f any possible initial po9t4 ,9 ord] to obta9 ! field caus$ by ! electron at any _w-po9t ,p let u con/ruct ! =e-cone belong+ to ,p 7vide fig4 #d74 ,cle>ly ?is cuts ! unlimit$ _w-l9e ( ! electron at a s+le po9t ,p1 because ^! directions are all "t-l vectors4 ,at ,p1 let u draw ! tang5t to ! _w-l9e1 & let u draw f ,p ! normal to ?is tang5t4 ,let r be ! measure ( ,p,q4 ,accord+ to ! def9ition ( a =e-cone1 r_/e is to be reckon$ z ! measure ( ,,pq4 ,n[ at ! _w-po9t ,p1 ! vector-pot5tial ( ! field excit$ by e is repres5t$ by ! vector 9 direction ,,pq1 hav+ ! magnitude e_/cr1 9 its ?ree space compon5ts along ! x-1 y-1 z-axes2 ! scal>-pot5tial is repres5t$ by ! compon5t along ! t-axis4 ,?is is ! elem5t>y law f\nd out by ,a4 ,li5>d1 & ,e4 ,wie*]t4#cd ,if ! field caus$ by ! electron be describ$ 9 ! above-m5tion$ way1 !n x w appe> t ! division ( ! field into electric & magnetic =ces is a relative "o1 & dep5ds ^u ! "t-axis assum$2 ! two =ces consid]$ toge!r be>s "s analogy to ! =ce-screw 9 me*anics2 ! analogy is1 h[ev]1 imp]fect4 ,i %all n[ describe ! pond]omotive =ce :i* is ex]t$ by "o mov+ electron ^u ano!r mov+ electron4 ,let u suppose t ! _w-l9e ( a second po9t-electron passes ?r\gh ! _w-po9t ,p4 ,let u det]m9e ,p1 ,q1 r z be=e1 con/ruct ! middle-po9t ,m ( ! hyp]bola ( curvature at ,p1 & f9ally ! normal ,,mn ^u a l9e ?r\gh ,p :i* is p>allel to ,,qp4 ,) ,p z ! initial po9t1 we %all e/abli% a sy/em ( ref]5ce 9 ! foll[+ way3 ! t-axis w be laid along ,,pq1 ! x-axis 9 ! direction ( ,,qp4 ,! y-axis 9 ! direction ( ,,mn1 !n ! z-axis is automatically det]m9$1 z x is normal to ! x-1 y-1 z-axes4 ,let 3x1 3y1 3z1 3t be ! accel]ation-vector at ,p1 4x1 4y 4z1 4t be ! velocity-vector at ,p4 ,!n ! =ce-vector ex]t$ by ! fir/ election e1 7mov+ 9 any possible mann]7 ^u ! second election e1 7likewise mov+ 9 any possible mann]7 at ,p is repres5t$ by -e e74t - 4x_/c7,f1 ,= ! compon5ts ,fx1 ,fy1 ,fz1 ,ft ( ! vector ,f ! foll[+ ?ree relations hold3-- c,ft - ,fx #a_/r1 ,fy 3y_/7cr71 ,fz #j1 & f\r?ly ?is vector ,f is normal to ! velocity-vector ,p1 & ?r\gh ?is circum/ance alone1 its dep5d5ce on ?is la/ velocity-vector arises4 ,if we comp>e ) ?is expression ! previ\s =mul#ce giv+ ! elem5t>y law ab\t ! pond]omotive action ( mov+ electric *>ges ^u ea* o!r1 !n we _c b admit1 t ! relations :i* occur "h reveal ! inn] ess5ce ( full simplicity fir/ 9 f\r dim5sions2 b 9 ?ree dim5sions1 !y h v complicat$ projections4 ,9 ! me*anics re=m$ accord+ to ! _w-po/ulate1 ! di%>monies :i* h di/urb$ ! relations betwe5 ,newtonian me*anics & mod]n electrodynamics automatically disappe>4 ,i %all n[ consid] ! position ( ! ,newtonian law ( attraction to ?is po/ulate4 ,i w assume t two po9t-masses m & m describe _! _w-l9es2 a mov+ =ce-vector is ex]cis$ by m ^u m1 & ! expression is j ! same z 9 ! case ( ! electron1 only we h to write mm in/ead -ee4 ,we %all consid] only ! special case 9 :i* ! accel]ation-vector ( m is always z]o3 !n t may be introduc$ 9 su* a mann] t m may be reg>d$ z fix$1 ! motion ( m is n[ subject$ to ! mov+-=ce vector ( m alone4 ,if we n[ modify ?is giv5 vector by writ+ -74#a_/7a-7v_/c77 in/ead ( 4t 74t #a up to magnitudes ( ! ord] 7#a4_/;c771 !n x appe>s t ,kepl]'s laws hold good = ! position 7x1 y1 z71 ( m at any "t1 only 9 place ( ! "t t1 we h to write ! prop] "t ( m4 ,on ! basis ( ?is simple rem>k1 x c be se5 t ! propos$ law ( attraction 9 comb9ation ) new me*anics is n less suit$ = ! explanation ( a/ronomical ph5om5a ?an ! ,newtonian law ( attraction 9 comb9ation ) ,newtonian me*anics4 ,also ! fundam5tal equations = electro-magnetic processes 9 mov+ bodies are 9 accordance ) ! _w-po/ulate4 ,i %all also %[ on a lat] occasion t ! d$uction ( ^! equations1 z taught by ,lor5tz1 are by no means to be giv5 up4 ,! fact t ! _w-po/ulate holds )\t exception is1 ,i believe1 ! true ess5ce ( an electromagnetic picture ( ! _w2 ! idea fir/ occurr$ to ,lor5tz1 its ess5ce 0 fir/ pick$ out by ,e9/e91 & is n[ gradually fully manife/4 ,9 c\rse ( "t1 ! ma!matical consequ5ces w be gradually d$uc$1 & en\gh sugge/ions w be =?com+ = ! exp]im5tal v]ification ( ! po/ulate2 9 ?is way ev5 ^?1 :o f9d x uncong5ial1 or ev5 pa9ful to give up ! old1 "t-hon\r$ concepts1 w be reconcil$ to ! new ideas ( "t & space1--9 ! prospect t !y w lead to pre-e/abli%$ h>mony betwe5 pure ma!matics & physics4 ,footnote #cj3 ,planck1 ,zur ,dynamik bewegt] sy/eme1 ,ann4 d4 physik1 ,bd4 #bf1 #aijh1 p4 #a4 ,footnote #ca3 ,h4 ,m9k[ski2 ! passage ref]s to pap] 7#b7 ( ! pres5t edition4 ,footnote #cb3 ,m9k[ski--,me*anics1 app5dix1 page #fe ( pap] 7#b74 ,planck--,v]h4 d4 ,d4 ,p4 ,g4 ,vol4 #d1 #aijf1 p4 #acf4 ,footnote #cc3 ,s*tz1 ,gtt4 ,na*r4 #ahig1 p4 #aaj4 ,footnote #cd3 ,li5>d1 ,l',eclairage lectrique ,t4 #af1 #ahif1 p4 #ec4 ,wie*]t1 ,ann4 d4 ,physik1 ,vol4 #d4 ,footnote #ce3 ,k4 ,s*w>zs*ild4 ,gtt-,na*r4 #aijc4 ,h4 ,a4 ,lor5tz1 ,enzyklopdie d] ,ma?4 ,wiss5s*aft5 ,v4 ,art #ad1 p4 #aii4 ,! ,f\ndation ( ! ,g5]alis$ ,!ory ( ,relativity ,by ,a4 ,e9/e94 ,f ,annal5 d] ,physik #d4di4aiaf4 ,! !ory :i* is sket*$ 9 ! foll[+ pages =ms ! mo/ wide-go+ g5]alization conceivable ( :at is at pres5t kn[n z 8! !ory ( ,relativity20 ?is latt] !ory ,i diff]5tiate f ! =m] 8,special ,relativity !ory10 & suppose x to be kn[n4 ,! g5]alization ( ! ,relativity !ory has be5 made mu* easi] ?r\gh ! =m giv5 to ! special ,relativity !ory by ,m9k[ski1 :i* ma!matician 0 ! fir/ to recognize cle>ly ! =mal equival5ce ( ! space l & "t-l co-ord9ates1 & :o made use ( x 9 ! build+ up ( ! !ory4 ,! ma!matical app>atus useful = ! g5]al relativity !ory1 lay already complete 9 ! 8,absolute ,diff]5tial ,calculus10 :i* 7 bas$ on ! rese>*es ( ,gauss1 ,riemann & ,*ri/(fel on ! non-,euclidean manifold1 & :i* h be5 %ap$ into a sy/em by ,ricci & ,levi-civita1 & already appli$ to ! problems ( !oretical physics4 ,i h 9 "p ,b ( ?is communication develop$ 9 ! simple/ & cle>e/ mann]1 all ! suppos$ ma!matical auxili>ies1 n kn[n to ,physici/s1 :i* w be useful = our purpose1 s t1 a /udy ( ! ma!matical lit]ature is n necess>y = an und]/&+ ( ?is pap]4 ,f9ally 9 ?is place ,i ?ank my fri5d ,grossmann1 by ^: help ,i 0 n only sp>$ ! /udy ( ! ma!matical lit]ature p]t95t to ?is subject1 b :o also aid$ me 9 ! rese>*es on ! field equations ( gravitation4 ,a ,pr9cipal consid]ations ab\t ! ,po/ulate ( ,relativity4 #a4 ,rem>ks on ! ,special ,relativity ,!ory4 ,! special relativity !ory re/s on ! foll[+ po/ulate :i* also holds valid = ! ,galileo-,newtonian me*anics4 ,if a co-ord9ate sy/em ,k be s *os5 t :5 ref]r$ to x1 ! physical laws hold 9 _! simple/ =ms ^! laws w\ld be also valid :5 ref]r$ to ano!r sy/em ( co-ord9ates ,k :i* is subject$ to an uni=m translational motion relative to ,k4 ,we call ?is po/ulate 8,! ,special ,relativity ,pr9ciple40 ,by ! ^w special1 x is signifi$ t ! pr9ciple is limit$ to ! case1 :5 ,k has uni=m translatory motion ) ref]5ce to ,k1 b ! equival5ce ( ,k & ,k does n ext5d to ! case ( non-uni=m motion ( ,k relative to ,k4 ,! ,special ,relativity ,!ory does n diff] f ! classical me*anics ?r\gh ! assumption ( ?is po/ulate1 b only ?r\gh ! po/ulate ( ! con/ancy ( light-velocity 9 vacuum :i*1 :5 comb9$ ) ! special relativity po/ulate1 gives 9 a well-kn[n way1 ! relativity ( syn*ronism z well z ! ,lor5z-trans=mation1 ) all ! relations betwe5 mov+ rigid bodies & clocks4 ,! modification :i* ! !ory ( space & "t has und]gone ?r\gh ! special relativity !ory1 is inde$ a pr(\nd "o1 b a weighti] po9t rema9s unt\*$4 ,accord+ to ! special relativity !ory1 ! !orems ( geometry are to be look$ ^u z ! laws ab\t any possible relative positions ( solid bodies at re/1 & m g5]ally ! !orems ( k9ematics1 z !orems :i* describe ! relation betwe5 measurable bodies & clocks4 ,consid] two mat]ial po9ts ( a solid body at re/2 !n accord+ to ^! conceptions !re corresponds to ^! po9ts a :olly def9ite ext5t ( l5g?1 indep5d5t ( k9d1 position1 ori5tation & "t ( ! body4 ,simil>ly let u consid] two positions ( ! po9t]s ( a clock :i* is at re/ ) ref]5ce to a co-ord9ate sy/em2 !n to ^! positions1 !re always corresponds1 a "t-int]val ( a def9ite l5g?1 indep5d5t ( "t & place4 ,x w\ld be soon %[n t ! g5]al relativity !ory c n hold fa/ to ?is simple physical significance ( space & "t4 #b4 ,ab\t ! reasons :i* expla9 ! ext5sion ( ! relativity-po/ulate4 ,to ! classical me*anics 7no less ?an7 to ! special relativity !ory1 is atta*$ an epi/eomological defect1 :i* 0 p]haps fir/ cleanly po9t$ out by ,e4 ,ma*4 ,we %all illu/rate x by ! foll[+ example2 ,let two fluid bodies ( equal k9d & magnitude swim freely 9 space at su* a great di/ance f "o ano!r 7& f all o!r masses7 t only t sort ( gravitational =ces are to be tak5 into acc\nt :i* ! p>ts ( any ( ^! bodies ex]t ^u ea* o!r4 ,! di/ance ( ! bodies f "o ano!r is inv>iable4 ,! relative motion ( ! diff]5t p>ts ( ea* body is n to occur4 ,b ea* mass is se5 to rotate by an obs]v] at re/ relative to ! o!r mass r\nd ! connect+ l9e ( ! masses ) a con/ant angul> velocity 7def9ite relative motion = bo? ! masses74 ,n[ let u ?9k t ! surfaces ( bo? ! bodies 7,s & ,s7 are measur$ ) ! help ( measur+ rods 7relatively at re/72 x is !n f\nd t ! surface ( ,s is a sph]e & ! surface ( ! o!r is an ellipsoid ( rotation4 ,we n[ ask1 :y is ?is diff]5ce betwe5 ! two bodies8 ,an answ] to ?is "q c only !n be reg>d$ z satisfactory f ! epi/eomological /&po9t :5 ! ?+ adduc$ z ! cause is an obs]vable fact ( exp]i5ce4 ,! law ( causality has ! s5se ( a def9ite /atem5t ab\t ! _w ( exp]i5ce only :5 obs]vable facts alone appe> z causes & effects4 ,! ,newtonian me*anics does n give to ?is "q any satisfactory answ]4 ,= example1 x says3--,! laws ( me*anics hold true = a space ,r relative to :i* ! body ,s is at re/1 n h[ev] = a space relative to :i* ,s is at re/4 ,! ,galiliean space1 :i* is "h introduc$ is h[ev] only a purely imag9>y cause1 n an obs]vable ?+4 ,x is ?us cle> t ! ,newtonian me*anics does n1 9 ! case treat$ "h1 actually fulfil ! requirem5ts ( causality1 b produces on ! m9d a fictiti\s complac5cy1 9 t x makes responsible a :olly imag9>y cause ,r = ! diff]5t behavi\rs ( ! bodies ,s & ,s :i* are actually obs]vable4 ,a satisfactory explanation to ! "q put =w>d above c only be ?us giv53--t ! physical sy/em compos$ ( ,s & ,s %[s = itself alone no conceivable cause to :i* ! diff]5t behavi\r ( ,s & ,s c be attribut$4 ,! cause mu/ ?us lie outside ! sy/em4 ,we are !re=e l$ to ! conception t ! g5]al laws ( motion :i* det]m9e specially ! =ms ( ,s & ,s mu/ be ( su* a k9d1 t ! me*anical behavi\r ( ,s & ,s mu/ be ess5tially condition$ by ! di/ant masses1 :i* we _h n br\ght into ! sy/em consid]$4 ,^! di/ant masses1 7& _! relative motion z reg>ds ! bodies "u consid]ation7 are !n to be look$ ^u z ! seat ( ! pr9cipal obs]vable causes = ! diff]5t behavi\rs ( ! bodies "u consid]ation4 ,!y take ! place ( ! imag9>y cause ,r4 ,among all ! conceivable spaces ,r & ,r mov+ 9 any mann] relative to "o ano!r1 !re is a priori1 no "o set :i* c be reg>d$ z af=d+ great] advantages1 aga9/ :i* ! objection :i* 0 already rais$ f ! /&po9t ( ! !ory ( k _c be aga9 reviv$4 ,! laws ( physics mu/ be s con/itut$ t !y %\ld rema9 valid = any sy/em ( co-ord9ates mov+ 9 any mann]4 ,we ?us arrive at an ext5sion ( ! relativity po/ulate4 ,besides ?is mom5t\s epi/eomological argum5t1 !re is also a well-kn[n physical fact :i* speaks 9 fav\r ( an ext5sion ( ! relativity !ory4 ,let !re be a ,galiliean co-ord9ate sy/em ,k relative to :i* 7at lea/ 9 ! f\r-dim5sional region consid]$7 a mass at a suffici5t di/ance f o!r masses move uni=mly 9 a l9e4 ,let ,k be a second co-ord9ate sy/em :i* has a uni=mly accel]at$ motion relative to ,k4 ,relative to ,k any mass at a suffici5tly great di/ance exp]i5ces an accel]at$ motion su* t its accel]ation & ! direction ( accel]ation is indep5d5t ( its mat]ial composition & its physical conditions4 ,c any obs]v]1 at re/ relative to ,k1 !n conclude t he is 9 an actually accel]at$ ref]5ce-sy/em8 ,?is is to be answ]$ 9 ! negative2 ! above-nam$ behavi\r ( ! freely mov+ masses relative to ,k c be expla9$ 9 z good a mann] 9 ! foll[+ way4 ,! ref]5ce-sy/em ,k has no accel]ation4 ,9 ! space-"t region consid]$ !re is a gravitation-field :i* g5]ates ! accel]at$ motion relative to ,k4 ,?is conception is feasible1 because to u ! exp]i5ce ( ! exi/5ce ( a field ( =ce 7namely ! gravitation field7 has %[n t x possesses ! rem>kable prop]ty ( imp>t+ ! same accel]ation to all bodies4 ,! me*anical behavi\r ( ! bodies relative to ,k is ! same z exp]i5ce w\ld expect ( !m ) ref]5ce to sy/ems :i* we assume f habit z /ation>y2 ?us x expla9s :y f ! physical /&-po9t x c be assum$ t ! sy/ems ,k & ,k c bo? ) ! same legitimacy be tak5 z at re/1 t is1 !y w be equival5t z sy/ems ( ref]5ce = a description ( physical ph5om5a4 ,f ^! discussions we see1 t ! work+ out ( ! g5]al relativity !ory mu/1 at ! same "t1 lead to a !ory ( gravitation2 = we c 8create0 a gravitational field by a simple v>iation ( ! co-ord9ate sy/em4 ,also we see imm$iately t ! pr9ciple ( ! con/ancy ( light-velocity mu/ be modifi$1 = we recognise easily t ! pa? ( a ray ( light ) ref]5ce to ,k mu/ be1 9 g5]al1 curv$1 :5 light travels ) a def9ite & con/ant velocity 9 a /raight l9e ) ref]5ce to ,k4 #c4 ,! "t-space cont9uum4 ,requirem5ts ( ! g5]al ,co-v>iance = ! equations express+ ! laws ( ,nature 9 g5]al4 ,9 ! classical me*anics z well z 9 ! special relativity !ory1 ! co-ord9ates ( "t & space h an imm$iate physical significance2 :5 we say t any arbitr>y po9t has x z its ,x co-ord9ate1 x signifies t ! projection ( ! po9t-ev5t on ! ,x-axis asc]ta9$ by means ( a solid rod accord+ to ! rules ( ,euclidean ,geometry is rea*$ :5 a def9ite measur+ rod1 ! unit rod1 c be c>ri$ x times f ! orig9 ( co-ord9ates along ! ,x axis4 ,a po9t hav+ x t z ! ,x co-ord9ate signifies t a unit clock :i* is adju/$ to be at re/ relative to ! sy/em ( co-ord9ates1 & co9cid+ 9 its spatial position ) ! po9t-ev5t & set accord+ to "s def9ite /&>d has gone ov] x t p]iods be=e ! occurr5ce ( ! po9t-ev5t4 ,?is conception ( "t & space is cont9ually pres5t 9 ! m9d ( ! physici/1 ?\gh (t5 9 an unconsci\s way1 z is cle>ly recognis$ f ! role :i* ?is conception has play$ 9 physical measurem5ts4 ,?is conception mu/ also appe> to ! read] to be ly+ at ! basis ( ! second consid]ation ( ! la/ p>agraph & imp>t+ a s5se to ^! conceptions4 ,b we wi% to %[ t we are to ab&on x & 9 g5]al to replace x by m g5]al conceptions 9 ord] to be able to "w out ?or\ghly ! po/ulate ( g5]al relativity1--! case ( special relativity appe>+ z a limit+ case :5 !re is no gravitation4 ,we introduce 9 a space1 :i* is free f ,gravitation-field1 a ,galiliean ,co-ord9ate ,sy/em ,k 7x1 y1 z1 t7 & also1 ano!r sy/em ,k 7x y z t7 rotat+ uni=mly relative to ,k4 ,! orig9 ( bo? ! sy/ems z well z _! z-axes might cont9ue to co9cide4 ,we w %[ t = a space-"t measurem5t 9 ! sy/em ,k1 ! above e/abli%$ rules = ! physical significance ( "t & space c n be ma9ta9$4 ,on gr\nds ( symmetry x is cle> t a circle r\nd ! orig9 9 ! ,,xy plane ( ,k1 c also be look$ ^u z a circle 9 ! plane 7,x1 ,y7 ( ,k4 ,let u n[ ?9k ( measur+ ! circumf]5ce & ! diamet] ( ^! circles1 ) a unit measur+ rod 7inf9itely small comp>$ ) ! radius7 & take ! quoti5t ( bo? ! results ( measurem5t4 ,if ?is exp]im5t be c>ri$ out ) a measur+ rod at re/ relatively to ! ,galiliean sy/em ,k we w\ld get 1 z ! quoti5t4 ,! result ( measurem5t ) a rod relatively at re/ z reg>ds ,k w\ld be a numb] :i* is great] ?an 4 ,?is c be se5 easily :5 we reg>d ! :ole measurem5t-process f ! sy/em ,k & rememb] t ! rod plac$ on ! p]iph]y suff]s a ,lor5z-contraction1 n h[ev] :5 ! rod is plac$ along ! radius4 ,euclidean ,geometry !re=e does n hold = ! sy/em ,k2 ! above fix$ conceptions ( co-ord9ates :i* assume ! validity ( ,euclidean ,geometry fail ) reg>d to ! sy/em ,k4 ,we _c simil>ly introduce 9 ,k a "t correspond+ to physical requirem5ts1 :i* w be %[n by all simil>ly prep>$ clocks at re/ relative to ! sy/em ,k4 ,9 ord] to see ?is we suppose t two simil>ly made clocks are arrang$ "o at ! c5tre & "o at ! p]iph]y ( ! circle1 & consid]$ f ! /ation>y sy/em ,k4 ,accord+ to ! well-kn[n results ( ! special relativity !ory x foll[s--7z view$ f ,k7--t ! clock plac$ at ! p]iph]y w g sl[] ?an ! second "o :i* is at re/4 ,! obs]v] at ! common orig9 ( co-ord9ates :o is able to see ! clock at ! p]iph]y by means ( light w see ! clock at ! p]iph]y go+ sl[] ?an ! clock beside him4 ,s9ce he _c all[ ! velocity ( light to dep5d explicitly ^u ! "t 9 ! way "u consid]ation he w int]pret 8 obs]vation by say+ t ! clock on ! p]iph]y actually goes sl[] ?an ! clock at ! orig94 ,he _c !re=e d o!rwise ?an def9e "t 9 su* a way t ! rate ( go+ ( a clock dep5ds on its position4 ,we !re=e arrive at ?is result4 ,9 ! g5]al relativity !ory "t & space magnitudes _c be s def9$ t ! diff]5ce 9 spatial co-ord9ates c be imm$iately measur$ by ! unit-measur+ rod1 & "t-l co-ord9ate diff]5ce ) ! aid ( a normal clock4 ,! means hi!rto at our disposal1 = plac+ our co-ord9ate sy/em 9 ! "t-space cont9uum1 9 a def9ite way1 !re=e completely fail & x appe>s t !re is no o!r way :i* w enable u to fit ! co-ord9ate sy/em to ! f\r-dim5sional _w 9 su* a way1 t by x we c expect to get a specially simple =mulation ( ! laws ( ,nature4 ,s t no?+ rema9s = u b to reg>d all conceivable co-ord9ate sy/ems z equally suitable = ! description ( natural ph5om5a4 ,?is am\nts to ! foll[+ law3-- ,t 9 g5]al1 ,laws ( ,nature are express$ by means ( equations :i* are valid = all co-ord9ate sy/ems1 t is1 :i* are cov>iant = all possible trans=mations4 ,x is cle> t a physics :i* satisfies ?is po/ulate w be unobjectionable f ! /&po9t ( ! g5]al relativity po/ulate4 ,because among all sub/itutions !re are1 9 e case1 conta9$ ^?1 :i* correspond to all relative motions ( ! co-ord9ate sy/em 79 ?ree dim5sions74 ,?is condition ( g5]al cov>iance :i* takes away ! la/ remnants ( physical objectivity f space & "t1 is a natural requirem5t1 z se5 f ! foll[+ consid]ations4 ,all our well-sub/antiat$ space-"t propositions am\nt to ! det]m9ation ( space-"t co9cid5ces4 ,if1 = example1 ! ev5t consi/$ 9 ! motion ( mat]ial po9ts1 !n1 = ?is la/ case1 no?+ else are really obs]vable except ! enc\nt]s betwe5 two or m ( ^! mat]ial po9ts4 ,! results ( our measurem5ts are no?+ else ?an well-prov$ !orems ab\t su* co9cid5ces ( mat]ial po9ts1 ( our measur+ rods ) o!r mat]ial po9ts1 co9cid5ces betwe5 ! h&s ( a clock1 dial-m>ks & po9t-ev5ts occurr+ at ! same position & at ! same "t4 ,! introduction ( a sy/em ( co-ord9ates s]ves no o!r purpose ?an an easy description ( totality ( su* co9cid5ces4 ,we fit to ! _w our space-"t v>iables 7x x x x7 su* t to any & e po9t-ev5t corresponds a sy/em ( values ( 7x x x x74 ,two co9cid5t po9t-ev5ts correspond to ! same value ( ! v>iables 7x x x x72 i4e41 ! co9cid5ce is *>act]is$ by ! equality ( ! co-ord9ates4 ,if we n[ introduce any f\r functions 7x x x x7 z co-ord9ates1 s t !re is an unique correspond5ce betwe5 !m1 ! equality ( all ! f\r co-ord9ates 9 ! new sy/em w /ill be ! expression ( ! space-"t co9cid5ce ( two mat]ial po9ts4 ,z ! purpose ( all physical laws is to all[ u to rememb] su* co9cid5ces1 !re is a priori no reason pres5t1 to pref] a c]ta9 co-ord9ate sy/em to ano!r2 i4e41 we get ! condition ( g5]al cov>iance4 #d4 ,relation ( f\r co-ord9ates to spatial & "t-l measurem5ts4 ,analytical expression = ! ,gravitation-field4 ,i am n try+ 9 ?is communication to d$uce ! g5]al ,relativity-!ory z ! simple/ logical sy/em possible1 ) a m9imum ( axioms4 ,b x is my *ief aim to develop ! !ory 9 su* a mann] t ! read] p]ceives ! psy*ological naturalness ( ! way propos$1 & ! fundam5tal assumptions appe> to be mo/ reasonable accord+ to ! light ( exp]i5ce4 ,9 ?is s5se1 we %all n[ introduce ! foll[+ supposition2 t = an inf9itely small f\r-dim5sional region1 ! relativity !ory is valid 9 ! special s5se :5 ! axes are suitably *os54 ,! nature ( accel]ation ( an inf9itely small 7positional7 co-ord9ate sy/em is h]eby to be s *os51 t ! gravitational field does n appe>2 ?is is possible = an inf9itely small region4 ,x1 ,x1 ,x are ! spatial co-ord9ates2 ,x is ! correspond+ "t-co-ord9ate measur$ by "s suitable measur+ clock4 ,^! co-ord9ates h1 ) a giv5 ori5tation ( ! sy/em1 an imm$iate physical significance 9 ! s5se ( ! special relativity !ory 7:5 we take a rigid rod z our unit ( measure74 ,! expression 7#a7 ds - d,x - d,x - d,x d,x _h !n1 accord+ to ! special relativity !ory1 a value :i* may be obta9$ by space-"t measurem5t1 & :i* is indep5d5t ( ! ori5tation ( ! local co-ord9ate sy/em4 ,let u take ds z ! magnitude ( ! l9e-elem5t belong+ to two inf9itely ne> po9ts 9 ! f\r-dim5sional region4 ,if ds belong+ to ! elem5t 7d,x1 d,x1 d,x1 d,x7 be positive we call x ) ,m9k[ski1 "t-l1 & 9 ! contr>y case space-l4 ,to ! l9e-elem5t consid]$1 i4e41 to bo? ! inf9itely ne> po9t-ev5ts belong also def9ite diff]5tials dx1 dx1 dx1 dx1 ( ! f\r-dim5sional co-ord9ates ( any *os5 sy/em ( ref]5ce4 ,if !re be also a local sy/em ( ! above k9d giv5 = ! case "u consid]ation1 d,x's w\ld !n be repres5t$ by def9ite l9e> homog5e\s expressions ( ! =m 7#b7 d,x adx ,if we sub/itute ! expression 9 7#a7 we get 7#c7 ds gdxdx :]e g w be functions ( x1 b w no long] dep5d ^u ! ori5tation & motion ( ! #8local' co-ord9ates2 = ds is a def9ite magnitude belong+ to two po9t-ev5ts inf9itely ne> 9 space & "t & c be got by measurem5ts ) rods & clocks4 ,! g's are "h to be s *os51 t g g2 ! summation is to be ext5d$ ov] all values ( & 1 s t ! sum is to be ext5d$1 ov] #d #d t]ms1 ( :i* #ab are equal 9 pairs4 ,f ! me?od adopt$ "h1 ! case ( ! usual relativity !ory comes out :5 ow+ to ! special behavi\r ( g 9 a f9ite region x is possible to *oose ! sy/em ( co-ord9ates 9 su* a way t g assumes con/ant values-- -#a1 #j1 #j1 #j 7#d7 #j -#a #j #j #j #j -#a #j #j #j #j #a ,we w\ld aft]w>ds see t ! *oice ( su* a sy/em ( co-ord9ates = a f9ite region is 9 g5]al n possible4 ,f ! consid]ations 9 #b & #c x is cle>1 t f ! physical /&-po9t ! quantities g are to be look$ ^u z magnitudes :i* describe ! gravitation-field ) ref]5ce to ! *os5 sy/em ( axes4 ,we assume fir/ly1 t 9 a c]ta9 f\r-dim5sional region consid]$1 ! special relativity !ory is true = "s p>ticul> *oice ( co-ord9ates4 ,! g's !n h ! values giv5 9 7#d74 ,a free mat]ial po9t moves ) ref]5ce to su* a sy/em uni=mly 9 a /raight-l9e4 ,if we n[ introduce1 by any sub/itution1 ! space-"t co-ord9ates x444x !n 9 ! new sy/em g's are no long] con/ants1 b functions ( space & "t4 ,at ! same "t1 ! motion ( a free po9t-mass 9 ! new co-ord9ates1 w appe> z curvil9e>1 & n uni=m1 9 :i* ! law ( motion1 w be indep5d5t ( ! nature ( ! mov+ mass-po9ts4 ,we c ?us signify ?is motion z "o "u ! influ5ce ( a gravitation field4 ,we see t ! appe>ance ( a gravitation-field is connect$ ) space-"t v>iability ( g's4 ,9 ! g5]al case1 we c n by any suitable *oice ( axes1 make special relativity !ory valid ?r\gh\t any f9ite region4 ,we ?us d$uce ! conception t g's describe ! gravitational field4 ,accord+ to ! g5]al relativity !ory1 gravitation ?us plays an exceptional rle z di/+ui%$ f ! o!rs1 specially ! electromagnetic =ces1 9 z mu* z ! #aj functions g repres5t+ gravitation1 def9e imm$iately ! metrical prop]ties ( ! f\r-dim5sional region4 ,b ,ma!matical ,auxili>ies = ,e/abli%+ ! ,g5]al ,cov>iant ,equations4 ,we h se5 be=e t ! g5]al relativity-po/ulate leads to ! condition t ! sy/em ( equations = ,physics1 mu/ be co-v>iants = any possible sub/itution ( co-ord9ates x1 444 x2 we h n[ to see h[ su* g5]al co-v>iant equations c be obta9$4 ,we %all n[ turn our att5tion to ^! purely ma!matical propositions4 ,x w be %[n t 9 ! solution1 ! inv>iant ds1 giv5 9 equation 7#c7 plays a fundam5tal rle1 :i* we1 foll[+ ,gauss's ,!ory ( ,surfaces1 /yle z ! l9e-elem5t4 ,! fundam5tal idea ( ! g5]al co-v>iant !ory is ?is3--,) ref]5ce to any co-ord9ate sy/em1 let c]ta9 ?+s 7t5sors7 be def9$ by a numb] ( functions ( co-ord9ates :i* are call$ ! compon5ts ( ! t5sor4 ,!re are n[ c]ta9 rules accord+ to :i* ! compon5ts c be calculat$ 9 a new sy/em ( co-ord9ates1 :5 ^! are kn[n = ! orig9al sy/em1 & :5 ! trans=mation connect+ ! two sy/ems is kn[n4 ,! ?+s h]efrom designat$ z 8,t5sors0 h fur!r ! prop]ty t ! trans=mation equation ( _! compon5ts are l9e> & homog5e\s2 s t all ! compon5ts 9 ! new sy/em vani% if !y are all z]o 9 ! orig9al sy/em4 ,?us a law ( ,nature c be =mulat$ by putt+ all ! compon5ts ( a t5sor equal to z]o s t x is a g5]al co-v>iant equation2 ?us :ile we seek ! laws ( =mation ( ! t5sors1 we also rea* ! means ( e/abli%+ g5]al co-v>iant laws4 #e4 ,contra-v>iant & co-v>iant ,f\r-vector4 ,contra-v>iant ,f\r-vector4 ,! l9e-elem5t is def9$ by ! f\r compon5ts dx1 ^: trans=mation law is express$ by ! equation 7#e7 dx'sigma sumnu fracp>tial x'sigmap>tial xnu dxnu ,! dx's are express$ z l9e> & homog5e\s function ( dx's2 we c look ^u ! diff]5tials ( ! co-ord9ates z ! compon5ts ( a t5sor1 :i* we designate specially z a contrav>iant ,f\r-vector4 ,ev]y?+ :i* is def9$ by ,f\r quantities ,a1 ) ref]5ce to a co-ord9ate sy/em1 & trans=ms accord+ to ! same law1 7#e;a7 ,asigma sumnu fracp>tial x'sigmap>tial xnu ,anu we may call a contra-v>iant ,f\r-vector4 ,f 7#e4 a71 x foll[s at once t ! sums 7,a ,b7 are also compon5ts ( a f\r-vector1 :5 ,a & ,b are s2 correspond+ relations hold also = all sy/ems aft]w>ds introduc$ z 8t5sors0 7,rule ( addition & subtraction ( ,t5sors74 ,co-v>iant ,f\r-vector4 ,we call f\r quantities ,a z ! compon5ts ( a cov>iant f\r-vector1 :5 = any *oice ( ! contra-v>iant f\r vector ,b 7#f7 ,a ,b ,inv>iant4 ,f ?is def9ition foll[s ! law ( trans=mation ( ! co-v>iant f\r-vectors4 ,if we sub/itute 9 ! "r h& side ( ! equation ,a ,b ,a ,b4 ! expressions sumsigma fracp>tial xnup>tial xsigma' ,bsigma' = ,b foll[+ f ! inv]sion ( ! equation 7#e;a7 we get sumsigma ,bsigma' sumnu fracp>tial xnup>tial xsigma' ,anu sumsigma ,bsigma' ,a'sigma ,z 9 ! above equation ,b are indep5d5t ( "o ano!r & p]fectly arbitr>y1 x foll[s t ! trans=mation law is3-- ,a'sigma sum fracp>tial xnup>tial xsigma' ,anu ,rem>ks on ! simplification ( ! mode ( writ+ ! expressions4 ,a glance at ! equations ( ?is p>agraph w %[ t ! indices :i* appe> twice )9 ! sign ( summation = example 9 7#e7 are ^? ov] :i* ! summation is to be made & t only ov] ! indices :i* appe> twice4 ,x is !re=e possible1 )\t loss ( cle>ness1 to leave (f ! summation sign2 s t we introduce ! rule3 :]ev] ! index 9 any t]m ( an expression appe>s twice1 x is to be summ$ ov] all ( !m except :5 x is n express$ly said to ! contr>y4 ,! diff]5ce betwe5 ! co-v>iant & ! contra-v>iant f\r-vector lies 9 ! trans=mation laws 7#g7 & 7#e74 ,bo? ! quantities are t5sors accord+ to ! above g5]al rem>ks2 9 x lies its significance4 ,9 accordance ) ,ricci & ,levi-civita1 ! contrav>iants & co-v>iants are designat$ by ! ov] & "u indices4 #f4 ,t5sors ( ! second & high] ranks4 ,contrav>iant t5sor3--,if we n[ calculate all ! #af products ,a ( ! compon5ts ,a ,b1 ( two contrav>iant f\r-vectors 7#h7 ,a ,a,b ,a1 w accord+ to 7#h7 & 7#e a7 satisfy ! foll[+ trans=mation law4 7#i7 ,asigma tau' fracp>tial x'sigmap>tial xmu fracp>tial x'taup>tial xnu ,amu nu ,we call a ?+ :i*1 ) ref]5ce to any ref]5ce sy/em is def9$ by #af quantities & fulfils ! trans=mation relation 7#i71 a contrav>iant t5sor ( ! second rank4 ,n e su* t5sor c be built f two f\r-vectors1 7accord+ to #h74 ,b x is easy to %[ t any #af quantities ,a1 c be repres5t$ z ! sum ( ,a,b ( prop]ly *os5 f\r pairs ( f\r-vectors4 ,f x1 we c prove 9 ! simple/ way all laws :i* hold true = ! t5sor ( ! second rank def9$ ?r\gh 7#i71 by prov+ x only = ! special t5sor ( ! type 7#h74 ,contrav>iant ,t5sor ( any rank3--,x is cle> t correspond+ to 7#h7 & 7#i71 we c def9e contrav>iant t5sors ( ! #crd & high] ranks1 ) #d1 etc4 compon5ts4 ,?us x is cle> f 7#h7 & 7#i7 t 9 ?is s5se1 we c look ^u contrav>iant f\r-vectors1 z contrav>iant t5sors ( ! fir/ rank4 ,co-v>iant t5sor4 ,if on ! o!r h&1 we take ! #af products ,a ( ! compon5ts ( two co-v>iant f\r-vectors ,a & ,b1 7#aj7 ,a ,a ,b4 = !m holds ! trans=mation law 7#aa7 ,asigma tau' fracp>tial x'mup>tial xsigma' fracp>tial x'nup>tial xtau' ,amu nu ,by means ( ^! trans=mation laws1 ! co-v>iant t5sor ( ! second rank is def9$4 ,all re-m>ks :i* we h already made conc]n+ ! contrav>iant t5sors1 hold also = co-v>iant t5sors4 ,rem>k3-- ,x is conv5i5t to treat ! scal> ,inv>iant ei!r z a contrav>iant or a co-v>iant t5sor ( z]o rank4 ,mix$ t5sor4 ,we c also def9e a t5sor ( ! second rank ( ! type 7#ab7 ,a ,a,b :i* is co-v>iant ) ref]5ce to & contrav>iant ) ref]5ce to 4 ,its trans=mation law is 7#ac7 ,atau'sigma fracp>tial xtau'p>tial xbeta fracp>tial alphap>tial xsigma' ,abetaalpha ,naturally !re are mix$ t5sors ) any numb] ( co-v>iant indices1 & ) any numb] ( contra-v>iant indices4 ,! co-v>iant & contra-v>iant t5sors c be look$ ^u z special cases ( mix$ t5sors4 ,symmetrical t5sors3-- ,a contrav>iant or a co-v>iant t5sor ( ! second or high] rank is call$ symmetrical :5 any two compon5ts obta9$ by ! mutual int]*ange ( two indices are equal4 ,! t5sor ,a or ,a is symmetrical1 :5 we h = any comb9ation ( indices 7#ad7 ,a ,a or 7#ad;a7 ,a ,a4 ,x mu/ be prov$ t a symmetry s def9$ is a prop]ty indep5d5t ( ! sy/em ( ref]5ce4 ,x foll[s 9 fact f 7#i7 rememb]+ 7#ad7 ,asigma tau' fracp>tial xsigma'p>tial xmu fracp>tial x'taup>tial xnu ,amu nu fracp>tial xsigma'p>tial xmu fracp>tial xtau'p>tial xnu ,anu mu ,atau sigma' ,antisymmetrical t5sor4 ,a contrav>iant or co-v>iant t5sor ( ! #bnd1 #crd or #d? rank is call$ antisymmetrical :5 ! two compon5ts got by mutually int]*ang+ any two indices are equal & opposite4 ,! t5sor or ,a or ,a is ?us antisymmetrical :5 we h 7#ae7 ,a -,a or 7#ae;a7 ,a -,a4 ,( ! #af compon5ts ,a1 ! f\r compon5ts ,a vani%1 ! re/ are equal & opposite 9 pairs2 s t !re are only #f num]ically diff]5t compon5ts pres5t 7,six-vector74 ,?us we also see t ! antisymmetrical t5sor ,a 7#crd rank7 has only #d compon5ts num]ically diff]5t1 & ! antisymmetrical t5sor ,a only "o4 ,symmetrical t5sors ( ranks high] ?an ! f\r?1 d n exi/ 9 a cont9uum ( #d dim5sions4 #g4 ,multiplication ( ,t5sors4 ,out] multiplication ( ,t5sors3--,we get f ! compon5ts ( a t5sor ( rank z1 & ano!r ( a rank z1 ! compon5ts ( a t5sor ( rank 7z z7 = :i* we multiply all ! compon5ts ( ! fir/ ) all ! compon5ts ( ! second 9 pairs4 ,= example1 we obta9 ! t5sor f ! t5sors ,a & ,b ( diff]5t k9ds3-- ,a,b1 ,a,b1 ,a,b4 ,! pro( ( ! t5sor "* ( 1 foll[s imm$iately f ! expressions 7#h71 7#aj7 or 7#ab71 or ! trans=mation equations 7#i71 7#aa71 7#ac72 equations 7#h71 7#aj7 & 7#ab7 are !mselves examples ( ! out] multiplication ( t5sors ( ! fir/ rank4 ,r$uction 9 rank ( a mix$ ,t5sor4 ,f e mix$ t5sor we c get a t5sor :i* is two ranks l[]1 :5 we put an index ( co-v>iant "* equal to an index ( ! contrav>iant "* & sum accord+ to ^! indices 7,r$uction74 ,we get = example1 out ( ! mix$ t5sor ( ! f\r? rank ,a1 ! mix$ t5sor ( ! second rank ,a ,a 7 ,a7 & f x aga9 by 8r$uction0 ! t5sor ( ! z]o rank ,a ,a ,a4 ,! pro( t ! result ( r$uction reta9s a truly t5sorial "*1 foll[s ei!r f ! repres5tation ( t5sor accord+ to ! g5]alisation ( 7#ab7 9 comb9ation ) 7#f7 or out ( ! g5]alisation ( 7#ac74 ,inn] & mix$ multiplication ( ,t5sors4 ,?is consi/s 9 ! comb9ation ( out] multiplication ) r$uction4 ,examples3--,f ! co-v>iant t5sor ( ! second rank ,a & ! contrav>iant t5sor ( ! fir/ rank ,b we get by out] multiplication ! mix$ t5sor ,d ,a ,b 4 ,?r\gh r$uction accord+ to indices & 7i4e41 putt+ 71 ! co-v>iant f\r vector ,d ,d ,a ,b is g5]at$4 ,^! we d5ote z ! inn] product ( ! t5sor ,a & ,b4 ,simil>ly we get f ! t5sors ,a & ,b ?r\gh out] multiplication & two-fold r$uction ! inn] product ,a ,b4 ,?r\gh out] multiplication & "o-fold r$uction we get out ( ,a & ,b1 ! mix$ t5sor ( ! second rank ,d ,a ,b4 ,we c fitly call ?is op]ation a mix$ "o2 = x is out] ) ref]5ce to ! indices & & inn] ) respect to ! indices & 4 ,we n[ prove a law1 :i* w be (t5 applicable = prov+ ! t5sor-"* ( c]ta9 quantities4 ,accord+ to ! above repres5tation1 ,a ,b is a scal>1 :5 ,a & ,b are t5sors4 ,we also rem>k t :5 ,a ,b is an inv>iant = e *oice ( ! t5sor ,b1 !n ,a has a t5sorial "*4 ,pro(3--,accord+ to ! above assumption1 = any sub/itution we h ,a ,b ,a ,b4 ,f ! inv]sion ( 7#i7 we h h[ev] ,bmu nu fracp>tial xmup>tial xsigma' fracp>tial xnup>tial tau' ,bsigma tau' ,sub/itution ( ?is = ,b 9 ! above equation gives 7,asigma tau' - fracp>tial xmup>tial xsigma' fracp>tial xnup>tial xtau'7 ,bsigma tau' #j ,?is c be true1 = any *oice ( ,b only :5 ! t]m )9 ! bracket vani%es4 ,f :i* by ref]r+ to 7#aa71 ! !orem at once foll[s4 ,?is law correspond+ly holds = t5sors ( any rank & "*4 ,! pro( is q simil>4 ,! law c also be put 9 ! foll[+ =m4 ,if ,b & ,c are any two vectors1 & if = e *oice ( !m ! inn] product ,a ,b ,c is a scal>1 !n ,a is a co-v>iant t5sor4 ,! la/ law holds ev5 :5 !re is ! m special =mulation1 t ) any arbitr>y *oice ( ! f\r-vector ,b alone ! scal> product ,a ,b ,b is a scal>1 9 :i* case we h ! additional condition t ,a satisfies ! symmetry condition4 ,accord+ to ! me?od giv5 above1 we prove ! t5sor "* ( 7,a ,a71 f :i* on acc\nt ( symmetry foll[s ! t5sor-"* ( ,a4 ,?is law c easily be g5]aliz$ 9 ! case ( co-v>iant & contrav>iant t5sors ( any rank4 ,f9ally1 f :at has be5 prov$1 we c d$uce ! foll[+ law :i* c be easily g5]aliz$ = any k9d ( t5sor3 ,if ! quantities ,a ,b =m a t5sor ( ! fir/ rank1 :5 ,b is any arbitr>ily *os5 f\r-vector1 !n ,a is a t5sor ( ! second rank4 ,if = example1 ,c is any f\r-vector1 !n ow+ to ! t5sor "* ( ,a ,b1 ! inn] product ,a ,c ,b is a scal>1 bo? ! f\r-vectors ,c & ,b be+ arbitr>ily *os54 ,h5ce ! proposition foll[s at once4 ,a few words ab\t ! ,fundam5tal ,t5sor g4 ,! co-v>iant fundam5tal t5sor--,9 ! inv>iant expression ( ! squ>e ( ! l9e> elem5t ds g dx dx dx plays ! rle ( any arbitr>ily *os5 contrav>iant vector1 s9ce fur!r g g1 x foll[s f ! consid]ations ( ! la/ p>agraph t g is a symmetrical co-v>iant t5sor ( ! second rank4 ,we call x ! 8fundam5tal t5sor40 ,aft]w>ds we %all d$uce "s prop]ties ( ?is t5sor1 :i* w also be true = any t5sor ( ! second rank4 ,b ! special rle ( ! fundam5tal t5sor 9 our ,!ory1 :i* has its physical basis on ! p>ticul>ly exceptional "* ( gravitation makes x cle> t ^? relations are to be develop$ :i* w be requir$ only 9 ! case ( ! fundam5tal t5sor4 ,! co-v>iant fundam5tal t5sor4 ,if we =m f ! det]m9ant s*eme g ! m9ors ( g & divide !m by ! det]m9ant g g we get c]ta9 quantities g g1 :i* z we %all prove g5]ates a contrav>iant t5sor4 ,accord+ to ! well-kn[n law ( ,det]m9ants 7#af7 g g :]e is #a1 or #j1 accord+ z or n4 ,in/ead ( ! above expression = ds1 we c also write g dx dx or accord+ to 7#af7 also 9 ! =m g g g dx dx ,n[ accord+ to ! rules ( multiplication1 ( ! =e-go+ p>agraph1 ! magnitudes d g dx =ms a co-v>iant f\r-vector1 & 9 fact 7on acc\nt ( ! arbitr>y *oice ( dx7 any arbitr>y f\r-vector4 ,if we introduce x 9 our expression1 we get ds g d d4 ,= any *oice ( ! vectors d d ?is is scal>1 & g1 accord+ to its def9ition is a symmetrical ?+ 9 & 1 s x foll[s f ! above results1 t g is a contrav>iant t5sor4 ,out ( 7#af7 x also foll[s t is a t5sor :i* we may call ! mix$ fundam5tal t5sor4 ,det]m9ant ( ! fundam5tal t5sor4 ,accord+ to ! law ( multiplication ( det]m9ants1 we h g g g g ,on ! o!r h& we h g g #a ,s t x foll[s 7#ag7 t g g #a4 ,inv>iant ( volume4 ,we see fir/ ! trans=mation law = ! det]m9ant g g 4 ,accord+ to 7#aa7 g' fracp>tial xmup>tial xsigma' fracp>tial xnup>tial xtau' gmu u ,f ?is by apply+ ! law ( multiplication twice1 we obta9 g' fracp>tial xmup>tial xsigma' fracp>tial xnup>tial xtau' gmu nu fracp>tial xmualphasigma' g or 7,a7 sqrtg' fracp>tial xmup>tial xsigma' sqrtg ,on ! o!r h& ! law ( trans=mation ( ! volume elem5t d dx dx dx dx is accord+ to ! wellkn[n law ( ,jacobi4 7,b7 dtau' fracdx'sigmadxmu dtau by multiplication ( ! two la/ equations 7,a7 & 7,b7 we get 7#ah7 g d g d4 ,in/ead ( g1 we %all aft]w>ds introduce 7-g7 :i* has a real value on acc\nt ( ! hyp]bolic "* ( ! "t-space cont9uum4 ,! inv>iant 7-g7d1 is equal 9 magnitude to ! f\r-dim5sional volume-elem5t measur$ ) solid rods & clocks1 9 accordance ) ! special relativity !ory4 ,rem>ks on ! "* ( ! space-"t cont9uum--,our assumption t 9 an inf9itely small region ! special relativity !ory holds1 leads u to conclude t ds c always1 accord+ to 7#a7 be express$ 9 real magnitudes d,x 444 d,xh4 ,if we call d ! 8natural0 volume elem5t d,x d,x d,x d,x we h ?us 7#ah;a7 d 7g7i4 ,%\ld 7-g7 vani% at any po9t ( ! f\r-dim5sional cont9uum x w\ld signify t to a f9ite co-ord9ate volume at ! place corresponds an inf9itely small 8natural volume40 ,?is c nev] be ! case2 s t g c nev] *ange its sign2 we w\ld1 accord+ to ! special relativity !ory assume t g has a f9ite negative value4 ,x is a hypo!sis ab\t ! physical nature ( ! cont9uum consid]$1 & also a pre-e/abli%$ rule = ! *oice ( co-ord9ates4 ,if h[ev] 7-g7 rema9s positive & f9ite1 x is cle> t ! *oice ( co-ord9ates c be s made t ?is quantity becomes equal to "o4 ,we w\ld aft]w>ds see t su* a limitation ( ! *oice ( co-ord9ates w\ld produce a significant simplification 9 expressions = laws ( nature4 ,9 place ( 7#ah7 x foll[s !n simply t d d f ?is x foll[s1 rememb]+ ! law ( ,jacobi1 7#ai7 fracp>tial x'sigmadxmu #a ,) ?is *oice ( co-ord9ates1 only sub/itutions ) det]m9ant #a are all[able4 ,x w\ld h[ev] be errone\s to ?9k t ?is /ep signifies a p>tial r5unciation ( ! g5]al relativity po/ulate4 ,we d n seek ^? laws ( nature :i* are co-v>iants ) reg>d to ! trans=mations hav+ ! det]m9ant #a1 b we ask3 :at are ! g5]al co-v>iant laws ( nature8 ,fir/ we get ! law1 & !n we simplify its expression by a special *oice ( ! sy/em ( ref]5ce4 ,build+ up ( new t5sors ) ! help ( ! fundam5tal t5sor4 ,?r\gh inn]1 out] & mix$ multiplications ( a t5sor ) ! fundam5tal t5sor1 t5sors ( o!r k9ds & ( o!r ranks c be =m$4 ,example3-- ,a g ,a ,a g ,a ,we w\ld po9t out specially ! foll[+ comb9ations3 ,a g g ,a ,a g g ,a 7complem5t to ! co-v>iant or contrav>iant t5sors7 & ,b g g ,a ,we c call ,b ! r$uc$ t5sor relat$ to ,a4 ,simil>ly ,b gg,a4 ,x is to be rem>k$ t g is no o!r ?an ! 8complem5t0 ( g = we h1-- ggg g g4 #i4 ,equation ( ! geodetic l9e 7or ( po9t-motion74 ,z ! 8l9e elem5t0 ds is a def9ite magnitude indep5d5t ( ! co-ord9ate sy/em1 we h also betwe5 two po9ts ,p & ,p ( a f\r dim5sional cont9uum a l9e = :i* ds is an extremum 7geodetic l9e71 i4e41 "o :i* has got a significance indep5d5t ( ! *oice ( co-ord9ates4 ,its equation is 7#bj7 ,p,p ds #j ,f ?is equation1 we c 9 a wellkn[n way d$uce #d total diff]5tial equations :i* def9e ! geodetic l9e2 ?is d$uction is giv5 "h = ! sake ( complet5ess4 ,let 1 be a function ( ! co-ord9ates x2 ?is def9es a s]ies ( surfaces :i* cut ! geodetic l9e s\ght-= z well z all neighb\r+ l9es f ,p to ,p4 ,we c suppose t all su* curves are giv5 :5 ! value ( its co-ord9ates x are giv5 9 t]ms ( 4 ,! sign corresponds to a passage f a po9t ( ! geodetic curve s\ght-= to a po9t ( ! contigu\s curve1 bo? ly+ on ! same surface 4 ,!n 7#bj7 c be replac$ by d #j 7#bj;a7 g7dx_/d77dx_/d7 ,b 7#a_/77;g_/x7 7dx_/d7 7dx_/d7 x g7dx_/d77dx_/d7 ,s we get by ! sub/itution ( 9 7#bj;a71 rememb]+ t 7dx_/d7 7d_/d77x7 aft] p>tial integration1 d k x #j 7#bj;b7 :]e k 7d_/d77g_/7 7dx_/d7 - 7#a_/7b777;g_/x 7dx_/d7 7dx_/d74 ,f :i* x foll[s1 s9ce ! *oice ( is p]fectly arbitr>y t k's %\ld vani%4 ,!n 7#bj;c7 k #j 7 #a1 #b1 #c1 #d7 are ! equations ( geodetic l9e2 s9ce along ! geodetic l9e consid]$ we h ds #j1 we c *oose ! p>amet] 1 z ! l5g? ( ! arc measur$ along ! geodetic l9e4 ,!n w #a1 & we w\ld get 9 place ( 7#bj;c7 gmunu fracp>tial#b xmup>tial s#b fracp>tial gmunup>tial xsigma fracp>tial xsigmap>tial s fracp>tial xmup>tial s - frac#ab fracp>tial gmusigmap>tial xnu fracp>tial xmup>tial s fracp>tial xsigmap>tial s #j ,or by m]ely *ang+ ! notation suitably1 7#bj;d7 galphasigma fracd#b xalphads#b beg9bmatrixmunusigmaendbmatrix fracdxmuds fracdxnuds #j :]e we h put1 foll[+ ,*ri/(fel1 7#ba7 beg9bmatrixmunusigmaendbmatrix frac#ab beg9bmatrixfracp>tial gmusigmap>tial xnu fracp>tial gnusigmap>tial xmu - fracp>tial gmunup>tial sigmaendbmatrix ,multiply f9ally 7#bj;d7 ) g 7out] multiplication ) ref]5ce to 1 & inn] ) respect to 7 we get at la/ ! f9al =m ( ! equation ( ! geodetic l9e-- fracd#b xtauds#b beg9,bmatrixmunutauend,bmatrix fracdxmuds fracdxnuds #j ,"h we h put1 foll[+ ,*ri/(fel1 beg9,bmatrixmunutauend,bmatrix gtaualpha beg9bmatrixmunualphaendbmatrix #aj4 ,=mation ( ,t5sors ?r\gh ,diff]5tiation4 ,rely+ on ! equation ( ! geodetic l9e1 we c n[ easily d$uce laws accord+ to :i* new t5sors c be =m$ f giv5 t5sors by diff]5tiation4 ,= ?is purpose1 we w\ld fir/ e/abli% ! g5]al co-v>iant diff]5tial equations4 ,we a*ieve ?is ?r\gh a repeat$ application ( ! foll[+ simple law4 ,if a c]ta9 curve be giv5 9 our cont9uum ^: po9ts are *>act]is$ by ! arc-di/ances s1 measur$ f a fix$ po9t on ! curve1 & if fur!r 1 be an inv>iant space function1 !n d_/ds is also an inv>iant4 ,! pro( foll[s f ! fact t d z well z ds1 are bo? inv>iants ,s9ce fracd phids fracp>tial phip>tial xmu fracp>tial xmup>tial s s t psi fracp>tial phip>tial xmu fracdxmuds is also an inv>iant = all curves :i* g out f a po9t 9 ! cont9uum1 i4e41 = any *oice ( ! vector dx4 ,f :i* foll[s imm$iately t ,a _/x is a co-v>iant f\r-vector 7gradi5t ( 74 ,accord+ to our law1 ! diff]5tial-quoti5t _/s tak5 along any curve is likewise an inv>iant4 ,sub/itut+ ! value ( 1 we get *i fracp>tial#b phip>tial xmu p>tial xnu fracdxmuds fracdxnuds fracp>tial phip>tial xmu fracd#b xmuds#b ,"h h[ev] we c n at once d$uce ! exi/5ce ( any t5sor4 ,if we h[ev] take t ! curves along :i* we are diff]5tiat+ are geodesics1 we get f x by replac+ dx_/ds accord+ to 7#bb7 *i beg9bmatrixfracp>tial#b phip>tial xmup>tial xnu - beg9,bmatrixmunutauend,bmatrix fracp>tial phip>tial xtau endbmatrix fracdxmuds fracdxnuds ,f ! int]*angeability ( ! diff]5tiation ) reg>d to & 1 & also accord+ to 7#bc7 & 7#ba7 we see t ! bracket beg9,bmatrixmunutauend,bmatrix is symmetrical ) respect to & 4 ,z we c draw a geodetic l9e 9 any direction f any po9t 9 ! cont9uum1 x_/ds is ?us a f\r-vector1 ) an arbitr>y ratio ( compon5ts1 s t x foll[s f ! results ( #g t 7#be7 ,amunu fracp>tial#b phip>tial xmu p>tial xnu - beg9,bmatrixmunutauend,bmatrix fracp>tial phip>tial xtau is a co-v>iant t5sor ( ! second rank4 ,we h ?us got ! result t out ( ! co-v>iant t5sor ( ! fir/ rank ,a _/x we c get by diff]5tiation a co-v>iant t5sor ( #bnd rank 7#bf7 ,amunu fracp>tial ,amup>tial xnu - beg9,bmatrixmunutauend,bmatrix ,atau ,we call ! t5sor ,a ! 8ext5sion0 ( ! t5sor ,a4 ,!n we c easily %[ t ?is comb9ation also leads to a t5sor1 :5 ! vector ,a is n repres5table z a gradi5t4 ,9 ord] to see ?is we fir/ rem>k t 7d_/x7 is a co-v>iant f\r-vector :5 & are scal>s4 ,?is is also ! case = a sum ( f\r su* t]ms 3-- ,smu psi7#a7 fracp>tial phi7#a7p>tial xmu 444 psi7#d7 fracp>tial phi7#d7p>tial xmu :5 7#a71 7#a7 444 7#d71 7#d7 are scal>s4 ,n[ x is h[ev] cle> t e co-v>iant f\r-vector is repres5table 9 ! =m ( ,s4 ,if = example1 ,a is a f\r-vector ^: compon5ts are any giv5 functions ( x1 we h1 7) ref]5ce to ! *os5 co-ord9ate sy/em7 only to put 7#a7 ,a 7#a7 x 7#b7 ,a 7#b7 x 7#c7 ,a 7#c7 x 7#d7 ,a 7#d7 x4 9 ord] to arrive at ! result t ,s is equal to ,a4 ,9 ord] to prove !n t ,a is a t5sor :5 on ! "r side ( 7#bf7 we sub/itute any co-v>iant f\r-vector = ,a we h only to %[ t ?is is true = ! f\r-vector ,s4 ,= ?is latt] case1 h[ev]1 a glance on ! "r h& side ( 7#bf7 w %[ t we h only to br+ =? ! pro( = ! case :5 ,a _/x4 ,n[ ! "r h& side ( 7#be7 multipli$ by is psi fracp>tial#b phip>tial xmu p>tial xnu - beg9,bmatrixmunutauend,bmatrix psi fracp>tial phip>tial xtau :i* has a t5sor "*4 ,simil>ly1 7_/x7 7_/x7 is also a t5sor 7out] product ( two f\r-vectors74 ,?r\gh addition foll[s ! t5sor "* ( fracp>tialp>tial xnu 7psi fracp>tial phip>tial xmu7 - beg9,bmatrixmunutauend,bmatrix 7psi fracp>tial phip>tial xtau7 ,?us we get ! desir$ pro( = ! f\r-vector1 _/x & h5ce = any f\r-vectors ,a z %[n above4 ,) ! help ( ! ext5sion ( ! f\r-vector1 we c easily def9e 8ext5sion0 ( a co-v>iant t5sor ( any rank4 ,?is is a g5]alisation ( ! ext5sion ( ! f\r-vector4 ,we conf9e ourselves to ! case ( ! ext5sion ( ! t5sors ( ! #bnd rank = :i* ! law ( =mation c be cle>ly se54 ,z already rem>k$ e co-v>iant t5sor ( ! #bnd rank c be repres5t$ z a sum ( ! t5sors ( ! type ,a ,b4 ,x w\ld !re=e be suffici5t to d$uce ! expression ( ext5sion1 = "o su* special t5sor4 ,accord+ to 7#bf7 we h ! expressions fracp>tial ,amup>tial xsigma - beg9,bmatrixsigmamutauend,bmatrix ,atau fracp>tial ,bnup>tial xsigma - beg9,bmatrixsigmamutauend,bmatrix ,btau are t5sors4 ,?r\gh out] multiplication ( ! fir/ ) ,b & ! #bnd ) ,a we get t5sors ( ! ?ird rank4 ,_! addition gives ! t5sor ( ! ?ird rank 7#bg7 ,amunusigma fracp>tial ,amunup>tial xsigma - beg9,bmatrixsigmamutauend,bmatrix ,ataunu - beg9,bmatrixsigmanutauend,bmatrix ,amutau :]e ,a is put ,a ,b4 ,! "r h& side ( 7#bg7 is l9e> & homog5e\s ) ref]5ce to ,a1 & its fir/ diff]5tial co-effici5t1 s t ?is law ( =mation leads to a t5sor n only 9 ! case ( a t5sor ( ! type ,a ,b b also 9 ! case ( a summation = all su* t5sors1 i4e41 9 ! case ( any co-v>iant t5sor ( ! second rank4 ,we call ,a ! ext5sion ( ! t5sor ,a4 ,x is cle> t 7#bf7 & 7#bd7 are only special cases ( 7#bg7 7ext5sion ( ! t5sors ( ! fir/ & z]o rank74 ,9 g5]al we c get all special laws ( =mation ( t5sors f 7#bg7 comb9$ ) t5sor multiplication4 ,"s special cases ( ,p>ticul> ,importance4 ,a few auxili>y lemmas conc]n+ ! fundam5tal t5sor4 ,we %all fir/ d$uce "s ( ! lemmas mu* us$ aft]w>ds4 ,accord+ to ! law ( diff]5tiation ( det]m9ants1 we h 7#bh7 dg g g dg -g gdg4 ,! la/ =m foll[s f ! fir/ :5 we rememb] t g g 1 & !re=e gg #d1 consequ5tly gdg g dg #j4 ,f 7#bh71 x foll[s t 7#bi7 frac#asqrt-g fracp>tial sqrt-gp>tial xsigma frac#ab fraclog 7-g7p>tial xsigma frac#ab gmunu fracp>tial gmunup>tial xsigma - frac#ab gmunu fracp>tial gmunup>tial xsigma ,aga91 s9ce g g 1 we h1 by diff]5tiation1 gmusigma dgnusigma -gnusigma dgmusigma or gmusigma fracp>tial gnusigmap>tial xlambda - gnusigma fracp>tial gmusigmap>tial xlambda ,by mix$ multiplication ) g & g respectively we obta9 7*ang+ ! mode ( writ+ ! indices74 7#ca7 dg -g g dg g_/x -g g dg & 7#cb7 dg -g g dg g_/x -g g g_/x4 ,! expression 7#ca7 all[s a trans=mation :i* we %all (t5 use2 accord+ to 7#ba7 7#cc7 fracp>tial galphabetap>tial xsigma beg9bmatrixalpha sigma beta endbmatrix beg9bmatrixbeta sigma alphaendbmatrix ,if we sub/itute ?is 9 ! second ( ! =mula 7#ca71 we get1 rememb]+ 7#bc71 7#cd7 fracp>tial gmunup>tial xsigma - 7 gmutau beg9,bmatrixtau sigma nuend,bmatrix gnutau beg9,bmatrixtau sigma muend,bmatrix 7 ,by sub/itut+ ! "r-h& side ( 7#cd7 9 7#bi71 we get 7#bi;a7 frac#asqrt-g fracp>tial sqrt-gp>tial xsigma beg9,bmatrixmu sigmamuend,bmatrix ,div]g5ce ( ! contrav>iant f\r-vector4 ,let u multiply 7#bf7 ) ! contrav>iant fundam5tal t5sor g 7inn] multiplication71 !n by a trans=mation ( ! fir/ memb]1 ! "r-h& side takes ! =m 7,a7 fracp>tialp>tial xnu 7gmunu ,amu7 - ,amu fracp>tial gmunup>tial xnu - frac#ab gtaualpha 7fracp>tial gmualphap>tial xnu fracp>tial g ualphap>tial xmu - fracp>tial gmunup>tial xalpha7 gmunu ,atau ,accord+ to 7#ca7 & 7#bi71 ! la/ memb] c take ! =m 7,b7 frac#ab fracp>tial gtaunup>tial xnu ,atau frac#ab fracp>tial gmutaup>tial xmu ,atau frac#asqrt-g fracp>tial sqrt-gp>tial xalpha gmualpha ,atau ,bo? ! fir/ memb]s ( ! expression 7,b71 & ! second memb] ( ! expression 7,a7 cancel ea* o!r1 s9ce ! nam+ ( ! summation-indices is immat]ial4 ,! la/ memb] ( 7,b7 c !n be unit$ ) fir/ ( 7,a74 ,if we put g ,a ,a1 :]e ,a z well z ,a are vectors :i* c be arbitr>ily *os51 we obta9 f9ally ,phi frac#asqrt-g fracp>tialp>tial xnu 7sqrt-g ,anu7 ,?is scal> is ! ,div]g5ce ( ! contrav>iant f\r-vector ,a4 ,rotation ( ! 7cov>iant7 f\r-vector4 ,! second memb] 9 7#bf7 is symmetrical 9 ! indices 1 & 4 ,h5ce ,a - ,a is an antisymmetrical t5sor built up 9 a v simple mann]4 ,we obta9 ,a ,a 7#cf7 ,b -------------- - ------------ x x ,antisymmetrical ,ext5sion ( a ,six-vector4 ,if we apply ! op]ation 7#bg7 on an antisymmetrical t5sor ( ! second rank ,a & =m all ! equations aris+ f ! cyclic int]*ange ( ! indices 1 1 1 & add all !m1 we obta9 a t5sor ( ! ?ird rank 7#cg7 ,b ,a ,a ,a ,a ,a ,a ------------ ------------- ------------ x x x f :i* x is easy to see t ! t5sor is antisymmetrical4 ,div]g5ce ( ! ,six-vector4 ,if 7#bg7 is multipli$ by g g 7mix$ multiplication71 !n a t5sor is obta9$4 ,! fir/ memb] ( ! "r h& side ( 7#bg7 c be writt5 9 ! =m fracp>tialp>tial xsigma 7gmualpha gnubeta ,amunu7 - gmualpha fracp>tial gnubetap>tial xsigma ,amunu - gnubeta fracp>tial gmualphap>tial xsigma ,amunu ,if we replace g g ,a by ,a1 g g ,a by ,a & replace 9 ! trans=m$ fir/ memb] g_/x & g_/x ) ! help ( 7#cd71 !n f ! "r-h& side ( 7#bg7 !re arises an expression ) sev5 t]ms1 ( :i* f\r cancel4 ,!re rema9s 7#ch7 ,aalphabetasigma fracp>tial ,aalphabetap>tial xsigma beg9,bmatrixsigma kappa alpha end,bmatrix ,akappabeta beg9,bmatrixsigma kappa betaend,bmatrix ,aalphakappa ,?is is ! expression = ! ext5sion ( a contrav>iant t5sor ( ! second rank2 ext5sions c also be =m$ = correspond+ contrav>iant t5sors ( high] & l[] ranks4 ,we rem>k t 9 ! same way1 we c also =m ! ext5sion ( a mix$ t5sor ,a 7#ci7 ,aalphamusigma fracp>tial ,aalphamup>tial xsigma - beg9,bmatrixsigma mu tauend,bmatrix ,aalphatau beg9,bmatrixsigma tau alphaend,bmatrix ,ataumu ,by ! r$uction ( 7#ch7 ) ref]5ce to ! indices & 7inn] multiplication ) 71 we get a contrav>iant f\r-vector ,aalpha fracp>tial ,aalphabetap>tial xbeta beg9,bmatrixbeta kappa betaend,bmatrix ,aalphakappa beg9,bmatrixbeta kappa alphaend,bmatrix ,akappabeta ,on ! acc\nt ( ! symmetry ( beg9,bmatrixbeta kappa alphaend,bmatrix ) ref]5ce to ! indices & 1 ! ?ird memb] ( ! "r h& side vani%es :5 ,a is an antisymmetrical t5sor1 :i* we assume "h2 ! second memb] c be trans=m$ accord+ to 7#bi;a72 we !re=e get 7#dj7 ,aalpha frac#asqrt-g fracp>tial7sqrt-g ,aalphabeta7p>tial xbeta ,?is is ! expression ( ! div]g5ce ( a contrav>iant six-vector4 ,div]g5ce ( ! mix$ t5sor ( ! second rank4 ,let u =m ! r$uction ( 7#ci7 ) ref]5ce to ! indices & 1 we obta9 rememb]+ 7#bi;a7 7#da7 sqrt-g ,amu fracp>tial7sqrt-g ,asigmamu7p>tial xsigma - beg9,bmatrixsigma mu tauend,bmatrix sqrt-g ,asigmatau ,if we introduce into ! la/ t]m ! contrav>iant t5sor ,a g ,a1 x takes ! =m - beg9bmatrixsigma mu rhoendbmatrix sqrt-g ,arhosigma ,if fur!r ,a or is symmetrical x is r$uc$ to - frac#ab sqrt-g fracp>tial grhosigmap>tial xmu ,arhosigma ,if in/ead ( ,a1 we introduce 9 a simil> way ! symmetrical co-v>iant t5sor ,a g g ,a1 !n ow+ to 7#ca7 ! la/ memb] c take ! =m frac#ab sqrt-g fracp>tial grhosigmap>tial xmu ,arhosigma ,9 ! symmetrical case treat$1 7#da7 c be replac$ by ei!r ( ! =ms 7#da;a7 sqrt-g ,amu fracp>tial 7sqrt-g ,asigmamu7p>tial xsigma - frac#ab fracp>tial grhosigmap>tial xmu sqrt-g ,arhosigma or 7#da;b7 sqrt-g ,amu fracp>tial 7sqrt-g ,asigmamu7p>tial xsigma frac#ab fracp>tial grhosigmap>tial xmu sqrt-g ,arhosigma :i* we %all h to make use ( aft]w>ds4 #ab4 ,! ,riemann-,*ri/(fel ,t5sor4 ,we n[ seek only ^? t5sors1 :i* c be obta9$ f ! fundam5tal t5sor g by diff]5tiation alone4 ,x is f\nd easily4 ,we put 9 7#bg7 in/ead ( any t5sor ,a ! fundam5tal t5sor g & get f x a new t5sor1 namely ! ext5sion ( ! fundam5tal t5sor4 ,we c easily conv9ce ourselves t ?is vani%es id5tically4 ,we prove x 9 ! foll[+ way2 we sub/itute 9 7#bg7 ,amunu fracp>tial ,amup>tial xnu - beg9,bmatrixmu nu rhoend,bmatrix ,arho i4e41 ! ext5sion ( a f\r-vector4 ,?us we get 7by slightly *ang+ ! indices7 ! t5sor ( ! ?ird rank ,amusigmatau fracp>tial#b ,amup>tial xsigma p>tial xtau - beg9,bmatrixmu sigma rhoend,bmatrix fracp>tial ,arhop>tial xtau - beg9,bmatrixmu tau rhoend,bmatrix fracp>tial ,arhop>tial xsigma - beg9,bmatrixsigma tau rhoend,bmatrix fracp>tial ,amup>tial xrho beg9bmatrix - fracp>tialp>tial xtau beg9,bmatrixmusigma rhoend,bmatrix beg9,bmatrixmutau alphaend,bmatrix beg9,bmatrixalphasigma rhoend,bmatrix beg9,bmatrixsigmatau alphaend,bmatrix beg9,bmatrixalphamu rhoend,bmatrix endbmatrix ,arho ,we use ^! expressions = ! =mation ( ! t5sor ,a - ,a4 ,!reby ! foll[+ t]ms 9 ,a cancel ! correspond+ t]ms 9 ,a2 ! fir/ memb]1 ! f\r? memb]1 z well z ! memb] correspond+ to ! la/ t]m )9 ! squ>e bracket4 ,^! are all symmetrical 9 1 & 4 ,! same is true = ! sum ( ! second & ?ird memb]s4 ,we ?us get 7#dc7 ,amusigmatau - ,amutausigma ,brhomusigmatau ,arho ,brhomusigmatau - fracp>tialp>tial xtau beg9,bmatrixmu sigma rhoend,bmatrix fracp>tialp>tial xsigma beg9,bmatrixmu tau rhoend,bmatrix - beg9,bmatrixmu sigma alphaend,bmatrix beg9,bmatrixalpha tau rhoend,bmatrix beg9,bmatrixmu tau alphaend,bmatrix beg9,bmatrixalpha sigma rhoend,bmatrix ,! ess5tial ?+ 9 ?is result is t on ! "r h& side ( 7#db7 we h only ,a1 b n its diff]5tial co-effici5ts4 ,f ! t5sor-"* ( ,a - ,a1 & f ! fact t ,a is an arbitr>y f\r vector1 x foll[s1 on acc\nt ( ! result ( #g1 t ,b is a t5sor 7,riemann-,*ri/(fel ,t5sor74 ,! ma!matical significance ( ?is t5sor is z foll[s2 :5 ! cont9uum is s %ap$1 t !re is a co-ord9ate sy/em = :i* g's are con/ants1 ,b all vani%4 ,if we *oose in/ead ( ! orig9al co-ord9ate sy/em any new "o1 s w\ld ! g's ref]r$ to ?is la/ sy/em be no long] con/ants4 ,! t5sor "* ( ,b %[s u1 h[ev]1 t ^! compon5ts vani% collectively also 9 any o!r *os5 sy/em ( ref]5ce4 ,! vani%+ ( ! ,riemann ,t5sor is ?us a necess>y condition t = "s *oice ( ! axis-sy/em g's c be tak5 z con/ants4 ,9 our problem x corresponds to ! case :5 by a suitable *oice ( ! co-ord9ate sy/em1 ! special relativity !ory holds ?r\gh\t any f9ite region4 ,by ! r$uction ( 7#dc7 ) ref]5ce to indices to & 1 we get ! cov>iant t5sor ( ! second rank 7#dd7 ,bmunu ,rmunu ,smunu ,rmunu - fracp>tialp>tial xalpha beg9,bmatrixmu nu alphaend,bmatrix beg9,bmatrixmu alpha betaend,bmatrix beg9,bmatrixnu beta alphaend,bmatrix ,smunu fracp>tial log sqrt-gp>tial xmu p>tial xnu - beg9,bmatrixmu nu alphaend,bmatrix fracp>tial log sqrt-gp>tial xalpha ,rem>ks ^u ! *oice ( co-ord9ates4--,x has already be5 rem>k$ 9 #h1 ) ref]5ce to ! equation 7#ah;a71 t ! co-ord9ates c ) advantage be s *os5 t 7-g7 #a4 ,a glance at ! equations got 9 ! la/ two p>agraphs %[s t1 ?r\gh su* a *oice1 ! law ( =mation ( ! t5sors suff]s a significant simplification4 ,x is specially true = ! t5sor ,b1 :i* plays a fundam5tal rle 9 ! !ory4 ,by ?is simplification1 ,s vani%es ( itself s t t5sor ,b r$uces to ,r4 ,i %all give 9 ! foll[+ pages all relations 9 ! simplifi$ =m1 ) ! above-nam$ specialisation ( ! co-ord9ates4 ,x is !n v easy to g back to ! g5]al cov>iant equations1 if x appe>s desirable 9 any special case4 ,c4 ,,! ,,!ory ,,( ,,! ,,gravitation-,,field #ac4 ,equation ( motion ( a mat]ial po9t 9 a gravitation-field4 ,expression = ! field-compon5ts ( gravitation4 ,a freely mov+ body n act$ on by ext]nal =ces moves1 accord+ to ! special relativity !ory1 along a /raight l9e & uni=mly4 ,?is also holds = ! g5]alis$ relativity !ory = any "p ( ! f\r-dim5sional region1 9 :i* ! co-ord9ates ,k#j c be1 & are1 s *os5 t g's h special con/ant values ( ! expression 7#d74 ,let u discuss ?is motion f ! /&-po9t ( any arbitr>y co-ord9ate-sy/em ,k2 x moves ) ref]5ce to ,k 7z expla9$ 9 #b7 9 a gravitational field4 ,! laws ( motion ) ref]5ce to ,k foll[ easily f ! foll[+ consid]ation4 ,) ref]5ce to ,k1 ! law ( motion is a f\r-dim5sional /raight l9e & ?us a geodesic4 ,z a geodetic-l9e is def9$ indep5d5tly ( ! sy/em ( co-ord9ates1 x w\ld also be ! law ( motion = ! motion ( ! mat]ial-po9t ) ref]5ce to ,k4 ,if we put 7#de7 ,gammataumunu - beg9,bmatrixmu nu tauend,bmatrix we get ! motion ( ! po9t ) ref]5ce to ,k1 giv5 by 7#df7 fracd#b xtauds#b ,gammataumunu fracdxmuds fracdxnuds ,we n[ make ! v simple assumption t ?is g5]al cov>iant sy/em ( equations def9es also ! motion ( ! po9t 9 ! gravitational field1 :5 !re exi/s no ref]5ce-sy/em ,k1 ) ref]5ce to :i* ! special relativity !ory holds ?r\gh\t a f9ite region4 ,! assumption seems to u to be all ! m legitimate1 z 7#df7 conta9s only ! fir/ diff]5tials ( g1 among :i* !re is no relation 9 ! special case :5 ,k exi/s4 ,if 's vani%1 ! po9t moves uni=mly & 9 a /raight l9e2 ^! magnitudes !re=e det]m9e ! deviation f uni=mity4 ,!y are ! compon5ts ( ! gravitational field4 #ad4 ,! ,field-equation ( ,gravitation 9 ! abs5ce ( matt]4 ,9 ! foll[+1 we diff]5tiate gravitation-field f matt] 9 ! s5se t ev]y?+ besides ! gravitation-field w be signifi$ z matt]2 !re=e ! t]m includes n only matt] 9 ! usual s5se1 b also ! electro-dynamic field4 ,our next problem is to seek ! field-equations ( gravitation 9 ! abs5ce ( matt]4 ,= ?is we apply ! same me?od z employ$ 9 ! =ego+ p>agraph = ! d$uction ( ! equations ( motion = mat]ial po9ts4 ,a special case 9 :i* ! field-equations s\ght-= are evid5tly satisfi$ is t ( ! special relativity !ory 9 :i* g's h c]ta9 con/ant values4 ,?is w\ld be ! case 9 a c]ta9 f9ite region ) ref]5ce to a def9ite co-ord9ate sy/em ,k4 ,) ref]5ce to ?is sy/em1 all ! compon5ts ,b ( ! ,riemann's ,t5sor equation #dc vani%4 ,^! vani% !n also 9 ! region consid]$1 ) ref]5ce to e o!r co-ord9ate sy/em4 ,! equations ( ! gravitation-field free f matt] mu/ ?us be 9 e case satisfi$ :5 all ,b vani%4 ,b ?is condition is cle>ly "o :i* goes too f>4 ,= x is cle> t ! gravitation-field g5]at$ by a mat]ial po9t 9 its own neighb\rhood c nev] be trans=m$ away by any *oice ( axes1 i4e41 x _c be trans=m$ to a case ( con/ant g's4 ,!re=e x is cle> t1 = a gravitational field free f matt]1 x is desirable t ! symmetrical t5sors ,b d$uc$ f ! t5sors ,b %\ld vani%4 ,we ?us get #aj equations = #aj quantities g :i* are fulfill$ 9 ! special case :5 ,b's all vani%4 ,rememb]+ 7#dd7 we see t 9 abs5ce ( matt] ! field-equations come out z foll[s2 7:5 ref]r$ to ! special co-ord9ate-sy/em *os547 7#dg7 fracp>tial ,gammaalphamunup>tial xalpha ,gammaalphamubeta ,gammabetamualpha #j sqrt-g #a ,gammaalphamunu - beg9,bmatrixmu nu alphaend,bmatrix ,x c also be %[n t ! *oice ( ^! equations is connect$ ) a m9imum ( arbitr>9ess4 ,= besides ,b1 !re is no t5sor ( ! second rank1 :i* c be built out ( g's & _! d]ivatives no high] ?an ! second1 & :i* is also l9e> 9 !m4 ,x w be %[n t ! equations aris+ 9 a purely ma!matical way out ( ! conditions ( ! g5]al relativity1 toge!r ) equations 7#df71 give u ! ,newtonian law ( attraction z a fir/ approximation1 & lead 9 ! second approximation to ! explanation ( ! p]ihelion-motion ( m]cury discov]$ by ,lev]ri] 7! residual effect :i* c\ld n be acc\nt$ = by ! consid]ation ( all sorts ( di/urb+ factors74 ,my view is t ^! are conv9c+ pro(s ( ! physical correctness ( my !ory4 #ae4 ,hamiltonian ,function = ! ,gravitation-field4 ,laws ( ,impulse & ,en]gy4 ,9 ord] to %[ t ! field equations correspond to ! laws ( impulse & en]gy1 x is mo/ conv5i5t to write x 9 ! foll[+ ,hamiltonian =m3-- 7#dg;a7 ,hd #j ,h g 7-g7 #a ,"h ! v>iations vani% at ! limits ( ! f9ite f\r-dim5sional integration-space consid]$4 ,x is fir/ necess>y to %[ t ! =m 7#dg;a7 is equival5t to equations 7#dg74 ,= ?is purpose1 let u consid] ,h z a function ( g & g 7 g_/x7 ,we h at fir/ ,h g #b;g - g #b 7g74 ,b delta7gmunu ,gammabetanualpha7 - frac#ab delta beg9bmatrixgmunu gbetalambdaendbmatrix 7fracp>tial gnulambdap>tial xalpha fracp>tial galphalambdap>tial xnu - fracp>tial galphanup>tial xlambda7 ,! t]ms aris+ out ( ! two la/ t]ms )9 ! r\nd bracket are ( diff]5t signs1 & *ange into "o ano!r by ! int]*ange ( ! indices & 4 ,!y cancel ea* o!r 9 ! expression = ,h1 :5 !y are multipli$ by 1 :i* is symmetrical ) respect to & 1 s t only ! fir/ memb] ( ! bracket rema9s = our consid]ation4 ,rememb]+ 7#ca71 we ?us h3-- ,h - g g ,!re=e 7#dh7 ,h_/g - ,h_/g ,if we n[ c>ry out ! v>iations 9 7#dg;a71 we obta9 ! sy/em ( equations 7#dg;b7 _/x 7 ,h_/g 7 - ,h_/g #j1 :i*1 ow+ to ! relations 7#dh71 co9cide ) 7#dg71 z 0 requir$ to be prov$4 ,if 7#dg;b7 is multipli$ by g1 s9ce g_/x g_/x & consequ5tly g _/x 7,h_/g7 _/x 7g ,h_/g7 - ,h_/g g_/x we obta9 ! equation _/x 7g ,h_/g7 - ,h_/x #j or t_/x #j 7#di7 -#bt g ,h_/g - ,h4 ,ow+ to ! relations 7#dh71 ! equations 7#dg7 & 7#cd71 7#ej7 t g - g 4 ,x is to be notic$ t t is n a t5sor1 s t ! equation 7#di7 holds only = sy/ems = :i* -g #a4 ,?is equation expresses ! laws ( cons]vation ( impulse & en]gy 9 a gravitation-field4 ,9 fact1 ! integration ( ?is equation ov] a ?ree-dim5sional volume ,v leads to ! f\r equations 7#di;a7 d_/dx t#d d,v 7t#a t t 7d,s :]e 1 1 are ! direction-cos9es ( ! inw>d-drawn normal to ! surface-elem5t d,s 9 ! ,euclidean ,s5se4 ,we recognise 9 ?is ! usual expression = ! laws ( cons]vation4 ,we d5ote ! magnitudes t z ! en]gy-compon5ts ( ! gravitation-field4 ,i w n[ put ! equation 7#dg7 9 a ?ird =m :i* w be v s]viceable = a quick realisation ( our object4 ,by multiply+ ! field-equations 7#dg7 ) g1 ^! are obta9$ 9 ! mix$ =ms4 ,if we rememb] t g _/x _/x 7g 7 - g_/x 1 :i* ow+ to 7#cd7 is equal to _/x 74g 7 - g - g 1 or slightly alt]+ ! notation1 equal to _/x 7g 7 - gmn m n - g 4 ,! ?ird memb] ( ?is expression cancels ) ! second memb] ( ! field-equations 7#dg74 ,9 place ( ! second t]m ( ?is expression1 we c1 on acc\nt ( ! relations 7#ej71 put 7t - t71 :]e t t ,!re=e 9 ! place ( ! equations 7#dg71 we obta9 7#ea7 _/x 7g 7 -7t - t7 7-g7 #a4 #af4 ,g5]al =mulation ( ! field-equation ( ,gravitation4 ,! field-equations e/abli%$ 9 ! prec$+ p>agraph = spaces free f matt] is to be comp>$ ) ! equation #j ( ! ,newtonian !ory4 ,we h n[ to f9d ! equations :i* w correspond to ,poisson's ,equation #d 7 signifies ! d5sity ( matt]74 ,! special relativity !ory has l$ to ! conception t ! in]tial mass 7,trge ,masse7 is no o!r ?an en]gy4 ,x c also be fully express$ ma!matically by a symmetrical t5sor ( ! second rank1 ! en]gy-t5sor4 ,we h !re=e to introduce 9 our g5]alis$ !ory en]gy-t5sor associat$ ) matt]1 :i* l ! en]gy compon5ts t ( ! gravitation-field 7equations #di1 & #ej7 h a mix$ "* b :i* h[ev] c be connect$ ) symmetrical cov>iant t5sors4 ,! equation 7#ea7 tea*es u h[ to introduce ! en]gy-t5sor 7correspond+ to ! d5sity ( ,poisson's equation7 9 ! field equations ( gravitation4 ,if we consid] a complete sy/em 7= example ! ,sol>-sy/em7 its total mass1 z also its total gravitat+ action1 w dep5d on ! total en]gy ( ! sy/em1 pond]able z well z gravitational4 ,?is c be express$1 by putt+ 9 7#ea71 9 place ( en]gy-compon5ts t ( gravitation-field alone ! sum ( ! en]gy-compon5ts ( matt] & gravitation1 i4e41 t ,t4 ,we ?us get in/ead ( 7#ea71 ! t5sor-equation 7#eb7 fracp>tialp>tial xalpha 7gsigmaeta ,gammalphamubeta7 - kappa 7tsigmamu ,tsigmamu7 - frac#ab deltasigmamu 7t ,t7 sqrt-g #a :]e ,t ,t 7,laue's ,scal>74 ,^! are ! g5]al field-equations ( gravitation 9 ! mix$ =m4 ,9 place ( 7#dg71 we get by work+ backw>ds ! sy/em 7#ec7 fracp>tial ,gammalphamu up>tial xalpha ,gammalphamubeta ,gammaetanualpha - kappa 7,tmunu - frac#ab gmunu ,t7 sqrt-g #a ,x mu/ be admitt$1 t ?is introduction ( ! en]gy-t5sor ( matt] _c be ju/ifi$ by means ( ! ,relativity-,po/ulate alone2 = we h 9 ! =ego+ analysis d$uc$ x f ! condition t ! en]gy ( ! gravitation-field %\ld ex]t gravitat+ action 9 ! same way z e o!r k9d ( en]gy4 ,! /ronge/ gr\nd = ! *oice ( ! above equation h[ev] lies 9 ?is1 t !y lead1 z _! consequ5ces1 to equations express+ ! cons]vation ( ! compon5ts ( total en]gy 7! impulses & ! en]gy7 :i* exactly correspond to ! equations 7#di7 & 7#di;a74 ,?is %all be %[n aft]w>ds4 #ag4 ,! laws ( cons]vation 9 ! g5]al case4 ,! equations 7#eb7 c be easily s trans=m$ t ! second memb] on ! "r-h& side vani%es4 ,we r$uce 7#eb7 ) ref]5ce to ! indices & & subtract ! equation s obta9$ aft] multiplication ) f 7#eb74 ,we obta91 7#eb;a7 _/x7g - g 7 -7t ,t7 we op]ate on x by _/x4 ,n[1 _/xx 7g7 - _/xx g g7g_/x g_/x - g_/x74 ,! fir/ & ! ?ird memb] ( ! r\nd bracket lead to expressions :i* cancel "o ano!r1 z c be easily se5 by int]*ang+ ! summation-indices 1 & 1 on ! "o h&1 & & 1 on ! o!r4 ,! second t]m c be trans=m$ accord+ to 7#ca74 ,s t we get1 7#ed7 _/xx 7g7 g_/xxx ,! second memb] ( ! expression on ! left-h& side ( 7#eb;a7 leads fir/ to - _/xx 7g7 or to #a_/d _/xx gg7 g_/x g_/x - g_/x74 ,! expression aris+ out ( ! la/ memb] )9 ! r\nd bracket vani%es accord+ to 7#bi7 on acc\nt ( ! *oice ( axes4 ,! two o!rs c be tak5 toge!r & give u on acc\nt ( 7#ca71 ! expression - g_/xxx ,s t rememb]+ 7#ed7 we h 7#ee7 _/xx 7g - g 7 #j4 id5tically4 ,f 7#ee7 & 7#eb;a7 x foll[s t 7#ef7 _/x 7t ,t7 #j ,f ! field equations ( gravitation1 x also foll[s t ! cons]vation-laws ( impulse & en]gy are satisfi$4 ,we see x mo/ simply foll[+ ! same reason+ :i* lead to equations 7#di;a72 only in/ead ( ! en]gy-compon5ts ( ! gravitational-field1 we are to introduce ! total en]gy-compon5ts ( matt] & gravitational field4 #ah4 ,! ,impulse-en]gy law = matt] z a consequ5ce ( ! field-equations4 ,if we multiply 7#ec7 ) g_/x1 we get 9 a way simil> to #ae1 rememb]+ t g g_/x vani%es1 ! equations t_/x - g_/x ,t #j or rememb]+ 7#ef7 7#eg7 ,t_/x g_/x ,t #j ,a comp>ison ) 7#da;b7 %[s t ^! equations = ! above *oice ( co-ord9ates 77-g7 #a7 ass]ts no?+ b ! vani%+ ( ! div]g5ce ( ! t5sor ( ! en]gy-compon5ts ( matt]4 ,physically ! appe>ance ( ! second t]m on ! left-h& side %[s t = matt] alone ! law ( cons]vation ( impulse & en]gy _c hold2 or c only hold :5 g's are con/ants2 i4e41 :5 ! field ( gravitation vani%es4 ,! second memb] is an expression = impulse & en]gy :i* ! gravitation-field ex]ts p] "t & p] volume ^u matt]4 ,?is comes out cle>] :5 in/ead ( 7#eg7 we write x 9 ! =m ( 7#dg74 7#eg;a7 ,t_/x - ,t4 ,! "r-h& side expresses ! int]action ( ! en]gy ( ! gravitational-field on matt]4 ,! field-equations ( gravitation conta9 ?us at ! same "t #d conditions :i* are to be satisfi$ by all mat]ial ph5om5a4 ,we get ! equations ( ! mat]ial ph5om5a completely :5 ! latt] is *>act]is$ by f\r o!r diff]5tial equations indep5d5t ( "o ano!r4 ,d4 ,,! 8,,mat]ial0 ,,ph5om5a4 ,! ,ma!matical auxili>ies develop$ "u #8,b' at once enables u to g5]alise1 accord+ to ! g5]alis$ !ory ( relativity1 ! physical laws ( matt] 7,hydrodynamics1 ,maxwell's ,electro-dynamics7 z !y lie already =mulat$ accord+ to ! special-relativity-!ory4 ,! g5]alis$ ,relativity ,pr9ciple leads u to no fur!r limitation ( possibilities2 b x enables u to "k exactly ! influ5ce ( gravitation on all processes )\t ! introduction ( any new hypo!sis4 ,x is ow+ to ?is1 t z reg>ds ! physical nature ( matt] 79 a n>r[ s5se7 no def9ite necess>y assumptions are to be introduc$4 ,! "q may lie op5 :e!r ! !ories ( ! electro-magnetic field & ! gravitational-field toge!r1 w =m a suffici5t basis = ! !ory ( matt]4 ,! g5]al relativity po/ulate c tea* u no new pr9ciple4 ,b by build+ up ! !ory x mu/ be %[n :e!r electro-magnetism & gravitation toge!r c a*ieve :at ! =m] alone did n succe$ 9 do+4 #ai4 ,eul]'s equations = frictionless adiabatic liquid4 ,let p & 1 be two scal>s1 ( :i* ! fir/ d5otes ! pressure & ! la/ ! d5sity ( ! fluid2 betwe5 !m !re is a relation4 ,let ! contrav>iant symmetrical t5sor ,t -g p dx_/ds dx_/ds 7#eh7 be ! contra-v>iant en]gy-t5sor ( ! liquid4 ,to x also belongs ! cov>iant t5sor 7#eh;a7 ,t -g p g dx_/ds g dx_/ds z well z ! mix$ t5sor 7#eh;b7 ,t - p g dx_/ds dx_/ds 4 ,if we put ! "r-h& side ( 7#eh;b7 9 7#eg;a7 we get ! g5]al hydrodynamical equations ( ,eul] accord+ to ! g5]alis$ relativity !ory4 ,?is 9 pr9ciple completely solves ! problem ( motion2 = ! f\r equations 7#eg;a7 toge!r ) ! giv5 equation betwe5 p & 1 & ! equation g dx_/ds dx_/ds #a1 are suffici5t1 ) ! giv5 values ( g1 = f9d+ out ! six unkn[ns p1 1 dx_/ds1 dx_/ds1 dx_/ds dx_/ds4 ,if g's are unkn[n we h also to take ! equations 7#ec74 ,!re are n[ #aa equations = f9d+ out #aj functions g1 s t ! numb] is m ?an suffici5t4 ,n[ x is be notic$ t ! equation 7#eg;a7 is already conta9$ 9 7#ec71 s t ! latt] only repres5ts 7#g7 indep5d5t equations4 ,?is indef9it5ess is due to ! wide fre$om 9 ! *oice ( co-ord9ates1 s t ma!matically ! problem is indef9ite 9 ! s5se t ?ree ( ! space-functions c be arbitr>ily *os54 #bj4 ,maxwell's ,electro-,magnetic field-equations4 ,let be ! compon5ts ( a cov>iant f\r-vector1 ! electro-magnetic pot5tial2 f x let u =m accord+ to 7#cf7 ! compon5ts ,f ( ! cov>iant six-vector ( ! electro-magnetic field accord+ to ! sy/em ( equations 7#ei7 ,f _/x - _/x4 ,f 7#ei71 x foll[s t ! sy/em ( equations 7#fj7 ,f_/x ,f_/x ,f_/x #j is satisfi$ ( :i* ! left-h& side1 accord+ to 7#cg71 is an anti-symmetrical t5sor ( ! ?ird k9d4 ,?is sy/em 7#fj7 conta9s ess5tially f\r equations1 :i* c be ?us writt53-- ,f_/x ,f_/x ,f_/x #j ,f_/x ,f_/x ,f_/x #j 7#fj;a7 ,f_/x ,f_/x ,f_/x #j ,f_/x ,f_/x ,f_/x #j4 ,?is sy/em ( equations corresponds to ! second sy/em ( equations ( ,maxwell4 ,we see x at once if we put ,f ,hx ,f ,ex 7#fa7 ,f ,hy ,f ,ey ,f ,hz ,f ,ez ,in/ead ( 7#fj;a7 we c !re=e write accord+ to ! usual notation ( ?ree-dim5sional vector-analysis3-- ,h_/t rot ,e #j 7#fj;b7 div ,h #j4 ,! fir/ ,maxwellian sy/em is obta9$ by a g5]alisation ( ! =m giv5 by ,m9k[ski4 ,we introduce ! contra-v>iant six-vector ,f by ! equation 7#fb7 ,f g g ,f1 & also a contra-v>iant f\r-vector ,j1 :i* is ! electrical curr5t-d5sity 9 vacuum4 ,!n rememb]+ 7#dj7 we c e/abli% ! sy/em ( equations1 :i* rema9s inv>iant = any sub/itution ) det]m9ant #a 7accord+ to our *oice ( co-ord9ates74 7#fc7 ,f_/x ,j ,if we put ,f ,hx ,f -,ex 7#fd7 ,f ,hy ,f -,ey ,f ,hz ,f -,ez :i* quantities become equal to ,hx 444 ,ex 9 ! case ( ! special relativity !ory1 & besides ,j#a ix 444 ,j#d we get in/ead ( 7#fc7 rot ,h - ,e_/t i 7#fc;a7 div ,e ,! equations 7#fj71 7#fb7 & 7#fc7 give ?us a g5]alisation ( ,maxwell's field-equations 9 vacuum1 :i* rema9s true 9 our *os5 sy/em ( co-ord9ates4 ,! en]gy-compon5ts ( ! electro-magnetic field4 ,let u =m ! inn]-product 7#fe7 ,k ,f ,j4 ,accord+ to 7#fa7 its compon5ts c be writt5 d[n 9 ! ?ree-dim5sional notation4 ,k ,ex i1 ,hx 7#fe;a7 -- -- -- ,k -- 7i1 ,e74 ,k is a cov>iant f\r-vector ^: compon5ts are equal to ! negative impulse & en]gy :i* are transf]r$ to ! electro-magnetic field p] unit ( "t1 & p] unit ( volume1 by ! electrical masses4 ,if ! electrical masses be free1 t is1 "u ! influ5ce ( ! electro-magnetic field only1 !n ! cov>iant f\r-vector ,k w vani%4 ,9 ord] to get ! en]gy compon5ts ,t ( ! electro-magnetic field1 we require only to give to ! equation ,k #j1 ! =m ( ! equation 7#eg74 ,f 7#fc7 & 7#fe7 we get fir/1 ,k ,f ,f_/x _/x 7,f ,f7 - ,f ,f_/x4 ,on acc\nt ( 7#fj7 ! second memb] on ! "r-h& side admits ( ! trans=mation-- ,f ,f_/x - ,f ,f_/x - g g ,f ,f_/x4 ,ow+ to symmetry1 ?is expression c also be writt5 9 ! =m -#a_/d g g ,f ,f_/x g g ,f_/x ,f1 :i* c also be put 9 ! =m - #a_/d _/x 7g g ,f ,f7 #a_/d ,f ,f _/x 7g g74 ,! fir/ ( ^! t]ms c be writt5 %ortly z - #a_/d _/x 7,f ,f71 & ! second aft] diff]5tiation c be trans=m$ 9 ! =m - ,f ,f g g_/x4 ,if we take all ! ?ree t]ms toge!r1 we get ! relation 7#ff7 ,k _/x - g g_/x :]e 7#ff;a7 -,f ,f #a_/d ,f ,f4 ,on acc\nt ( 7#cj7 ! equation 7#ff7 becomes equival5t to 7#eg7 & 7#eg;a7 :5 ,k vani%es4 ,?us 's are ! en]gy-compon5ts ( ! electro-magnetic field4 ,) ! help ( 7#fa7 & 7#fd7 we c easily %[ t ! en]gy-compon5ts ( ! electro-magnetic field1 9 ! case ( ! special relativity !ory1 give rise to ! well-kn[n ,maxwell-,poynt+ expressions4 ,we h n[ d$uc$ ! mo/ g5]al laws :i* ! gravitation-field & matt] satisfy :5 we use a co-ord9ate sy/em = :i* 7-g7 #a4 ,!reby we a*ieve an important simplification 9 all our =mulas & calculations1 )\t r5\nc+ ! conditions ( g5]al cov>iance1 z we h obta9$ ! equations ?r\gh a specialisation ( ! co-ord9ate sy/em f ! g5]al cov>iant-equations4 ,/ill ! "q is n )\t =mal int]e/1 :e!r1 :5 ! en]gy-compon5ts ( ! gravitation-field & matt] is def9$ 9 a g5]alis$ mann] )\t any specialisation ( co-ord9ates1 ! laws ( cons]vation h ! =m ( ! equation 7#ef71 & ! field-equations ( gravitation hold 9 ! =m 7#eb7 or 7#eb;a72 su* t on ! left-h& side1 we h a div]g5ce 9 ! usual s5se1 & on ! "r-h& side1 ! sum ( ! en]gy-compon5ts ( matt] & gravitation4 ,i h f\nd out t ?is is inde$ ! case4 ,b ,i am ( op9ion t ! communication ( my r compreh5sive "w on ?is subject w n pay1 = no?+ ess5tially new comes out ( x4 ,e4 #ba4 ,newton's !ory z a fir/ approximation4 ,we h already m5tion$ sev]al times t ! special relativity !ory is to be look$ ^u z a special case ( ! g5]al1 9 :i* g's h con/ant values 7#d74 ,?is signifies1 accord+ to :at has be5 said be=e1 a total neglect ( ! influ5ce ( gravitation4 ,we get "o important approximation if we consid] ! case :5 g's diff] f 7#d7 only by small magnitudes 7comp>$ to #a7 :]e we c neglect small quantities ( ! second & high] ord]s 7fir/ aspect ( ! approximation47 ,fur!r x %\ld be assum$ t )9 ! space-"t region consid]$1 g's at inf9ite di/ances 7us+ ! ^w inf9ite 9 a spatial s5se7 c1 by a suitable *oice ( co-ord9ates1 t5d to ! limit+ values 7#d72 i4e41 we consid] only ^? gravitational fields :i* c be reg>d$ z produc$ by masses di/ribut$ ov] f9ite regions4 ,we c assume t ?is approximation %\ld lead to ,newton's !ory4 ,= x h[ev]1 x is necess>y to treat ! fundam5tal equations f ano!r po9t ( view4 ,let u consid] ! motion ( a p>ticle accord+ to ! equation 7#df74 ,9 ! case ( ! special relativity !ory1 ! compon5ts dx_/ds1 dx_/ds1 dx_/ds1 c take any values4 ,?is signifies t any velocity v 77dx_/dx7 7dx_/dx7 7dx_/dx77 c appe> :i* is less ?an ! velocity ( light 9 vacuum 7v #a74 ,if we f9ally limit ourselves to ! consid]ation ( ! case :5 v is small comp>$ to ! velocity ( light1 x signifies t ! compon5ts dx_/ds1 dx_/ds1 dx_/ds1 c be treat$ z small quantities1 :]eas dx_/ds is equal to #a1 up to ! second-ord] magnitudes 7! second po9t ( view = approximation74 ,n[ we see t1 accord+ to ! fir/ view ( approximation1 ! magnitudes 's are all small quantities ( at lea/ ! fir/ ord]4 ,a glance at 7#df7 w also %[1 t 9 ?is equation accord+ to ! second view ( approximation1 we are only to take into acc\nt ^? t]ms = :i* #d4 ,by limit+ ourselves only to t]ms ( ! l[e/ ord] we get in/ead ( 7#df71 fir/1 ! equations3-- dx_/dt 1 :]e ds dx dt1 or by limit+ ourselves only to ^? t]ms :i* accord+ to ! fir/ /&-po9t are approximations ( ! fir/ ord]1 ,x mu/ be admitt$1 t ?is introduction ( ! en]gy-t5sor ( matt] _c be ju/ifi$ by means ( ! ,relativity-,po/ulate alone2 = we h 9 ! =ego+ analysis d$uc$ x f ! condition t ! en]gy ( ! gravitation-field %\ld ex]t gravitat+ action 9 ! same way z e o!r k9d ( en]gy4 ,! /ronge/ gr\nd = ! *oice ( ! above equation h[ev] lies 9 ?is1 t !y lead1 z _! consequ5ces1 to equations express+ ! cons]vation ( ! compon5ts ( total en]gy 7! impulses & ! en]gy7 :i* exactly correspond to ! equations 7#di7 & 7#di;a74 ,?is %all be %[n aft]w>ds4 #ag4 ,! laws ( cons]vation 9 ! g5]al case4 ,! equations 7#eb7 c be easily s trans=m$ t ! second memb] on ! "r-h& side vani%es4 ,we r$uce 7#eb7 ) ref]5ce to ! indices & & subtract ! equation s obta9$ aft] multiplication ) f 7#eb74 ,we obta91 7#eb;a7 _/x7g - g 7 -7t ,t7 we op]ate on x by _/x4 ,n[1 _/xx 7g7 - _/xx g g7g_/x g_/x - g_/x74 ,! fir/ & ! ?ird memb] ( ! r\nd bracket lead to expressions :i* cancel "o ano!r1 z c be easily se5 by int]*ang+ ! summation-indices 1 & 1 on ! "o h&1 & & 1 on ! o!r4 ,! second t]m c be trans=m$ accord+ to 7#ca74 ,s t we get1 7#ed7 _/xx 7g7 g_/xxx ,! second memb] ( ! expression on ! left-h& side ( 7#eb;a7 leads fir/ to - _/xx 7g7 or to #a_/d _/xx gg7 g_/x g_/x - g_/x74 ,! expression aris+ out ( ! la/ memb] )9 ! r\nd bracket vani%es accord+ to 7#bi7 on acc\nt ( ! *oice ( axes4 ,! two o!rs c be tak5 toge!r & give u on acc\nt ( 7#ca71 ! expression - g_/xxx ,s t rememb]+ 7#ed7 we h 7#ee7 _/xx 7g - g 7 #j4 id5tically4 ,f 7#ee7 & 7#eb;a7 x foll[s t 7#ef7 _/x 7t ,t7 #j ,f ! field equations ( gravitation1 x also foll[s t ! cons]vation-laws ( impulse & en]gy are satisfi$4 ,we see x mo/ simply foll[+ ! same reason+ :i* lead to equations 7#di;a72 only in/ead ( ! en]gy-compon5ts ( ! gravitational-field1 we are to introduce ! total en]gy-compon5ts ( matt] & gravitational field4 #ah4 ,! ,impulse-en]gy law = matt] z a consequ5ce ( ! field-equations4 ,if we multiply 7#ec7 ) g_/x1 we get 9 a way simil> to #ae1 rememb]+ t g g_/x vani%es1 ! equations t_/x - g_/x ,t #j or rememb]+ 7#ef7 7#eg7 ,t_/x g_/x ,t #j ,a comp>ison ) 7#da;b7 %[s t ^! equations = ! above *oice ( co-ord9ates 77-g7 #a7 ass]ts no?+ b ! vani%+ ( ! div]g5ce ( ! t5sor ( ! en]gy-compon5ts ( matt]4 ,physically ! appe>ance ( ! second t]m on ! left-h& side %[s t = matt] alone ! law ( cons]vation ( impulse & en]gy _c hold2 or c only hold :5 g's are con/ants2 i4e41 :5 ! field ( gravitation vani%es4 ,! second memb] is an expression = impulse & en]gy :i* ! gravitation-field ex]ts p] "t & p] volume ^u matt]4 ,?is comes out cle>] :5 in/ead ( 7#eg7 we write x 9 ! =m ( 7#dg74 7#eg;a7 ,t_/x - ,t4 ,! "r-h& side expresses ! int]action ( ! en]gy ( ! gravitational-field on matt]4 ,! field-equations ( gravitation conta9 ?us at ! same "t #d conditions :i* are to be satisfi$ by all mat]ial ph5om5a4 ,we get ! equations ( ! mat]ial ph5om5a completely :5 ! latt] is *>act]is$ by f\r o!r diff]5tial equations indep5d5t ( "o ano!r4 ,d4 ,,! 8,,mat]ial0 ,,ph5om5a4 ,! ,ma!matical auxili>ies develop$ "u #8,b' at once enables u to g5]alise1 accord+ to ! g5]alis$ !ory ( relativity1 ! physical laws ( matt] 7,hydrodynamics1 ,maxwell's ,electro-dynamics7 z !y lie already =mulat$ accord+ to ! special-relativity-!ory4 ,! g5]alis$ ,relativity ,pr9ciple leads u to no fur!r limitation ( possibilities2 b x enables u to "k exactly ! influ5ce ( gravitation on all processes )\t ! introduction ( any new hypo!sis4 ,x is ow+ to ?is1 t z reg>ds ! physical nature ( matt] 79 a n>r[ s5se7 no def9ite necess>y assumptions are to be introduc$4 ,! "q may lie op5 :e!r ! !ories ( ! electro-magnetic field & ! gravitational-field toge!r1 w =m a suffici5t basis = ! !ory ( matt]4 ,! g5]al relativity po/ulate c tea* u no new pr9ciple4 ,b by build+ up ! !ory x mu/ be %[n :e!r electro-magnetism & gravitation toge!r c a*ieve :at ! =m] alone did n succe$ 9 do+4 #ai4 ,eul]'s equations = frictionless adiabatic liquid4 ,let p & 1 be two scal>s1 ( :i* ! fir/ d5otes ! pressure & ! la/ ! d5sity ( ! fluid2 betwe5 !m !re is a relation4 ,let ! contrav>iant symmetrical t5sor ,t -g p dx_/ds dx_/ds 7#eh7 be ! contra-v>iant en]gy-t5sor ( ! liquid4 ,to x also belongs ! cov>iant t5sor 7#eh;a7 ,t -g p g dx_/ds g dx_/ds z well z ! mix$ t5sor 7#eh;b7 ,t - p g dx_/ds dx_/ds 4 ,if we put ! "r-h& side ( 7#eh;b7 9 7#eg;a7 we get ! g5]al hydrodynamical equations ( ,eul] accord+ to ! g5]alis$ relativity !ory4 ,?is 9 pr9ciple completely solves ! problem ( motion2 = ! f\r equations 7#eg;a7 toge!r ) ! giv5 equation betwe5 p & 1 & ! equation g dx_/ds dx_/ds #a1 are suffici5t1 ) ! giv5 values ( g1 = f9d+ out ! six unkn[ns p1 1 dx_/ds1 dx_/ds1 dx_/ds dx_/ds4 ,if g's are unkn[n we h also to take ! equations 7#ec74 ,!re are n[ #aa equations = f9d+ out #aj functions g1 s t ! numb] is m ?an suffici5t4 ,n[ x is be notic$ t ! equation 7#eg;a7 is already conta9$ 9 7#ec71 s t ! latt] only repres5ts 7#g7 indep5d5t equations4 ,?is indef9it5ess is due to ! wide fre$om 9 ! *oice ( co-ord9ates1 s t ma!matically ! problem is indef9ite 9 ! s5se t ?ree ( ! space-functions c be arbitr>ily *os54 #bj4 ,maxwell's ,electro-,magnetic field-equations4 ,let be ! compon5ts ( a cov>iant f\r-vector1 ! electro-magnetic pot5tial2 f x let u =m accord+ to 7#cf7 ! compon5ts ,f ( ! cov>iant six-vector ( ! electro-magnetic field accord+ to ! sy/em ( equations 7#ei7 ,f _/x - _/x4 ,f 7#ei71 x foll[s t ! sy/em ( equations 7#fj7 ,f_/x ,f_/x ,f_/x #j is satisfi$ ( :i* ! left-h& side1 accord+ to 7#cg71 is an anti-symmetrical t5sor ( ! ?ird k9d4 ,?is sy/em 7#fj7 conta9s ess5tially f\r equations1 :i* c be ?us writt53-- ,f_/x ,f_/x ,f_/x #j ,f_/x ,f_/x ,f_/x #j 7#fj;a7 ,f_/x ,f_/x ,f_/x #j ,f_/x ,f_/x ,f_/x #j4 ,?is sy/em ( equations corresponds to ! second sy/em ( equations ( ,maxwell4 ,we see x at once if we put ,f ,hx ,f ,ex 7#fa7 ,f ,hy ,f ,ey ,f ,hz ,f ,ez ,in/ead ( 7#fj;a7 we c !re=e write accord+ to ! usual notation ( ?ree-dim5sional vector-analysis3-- ,h_/t rot ,e #j 7#fj;b7 div ,h #j4 ,! fir/ ,maxwellian sy/em is obta9$ by a g5]alisation ( ! =m giv5 by ,m9k[ski4 ,we introduce ! contra-v>iant six-vector ,f by ! equation 7#fb7 ,f g g ,f1 & also a contra-v>iant f\r-vector ,j1 :i* is ! electrical curr5t-d5sity 9 vacuum4 ,!n rememb]+ 7#dj7 we c e/abli% ! sy/em ( equations1 :i* rema9s inv>iant = any sub/itution ) det]m9ant #a 7accord+ to our *oice ( co-ord9ates74 7#fc7 ,f_/x ,j ,if we put ,f ,hx ,f -,ex 7#fd7 ,f ,hy ,f -,ey ,f ,hz ,f -,ez :i* quantities become equal to ,hx 444 ,ex 9 ! case ( ! special relativity !ory1 & besides ,j#a ix 444 ,j#d we get in/ead ( 7#fc7 rot ,h - ,e_/t i 7#fc;a7 div ,e ,! equations 7#fj71 7#fb7 & 7#fc7 give ?us a g5]alisation ( ,maxwell's field-equations 9 vacuum1 :i* rema9s true 9 our *os5 sy/em ( co-ord9ates4 ,! en]gy-compon5ts ( ! electro-magnetic field4 ,let u =m ! inn]-product 7#fe7 ,k ,f ,j4 ,accord+ to 7#fa7 its compon5ts c be writt5 d[n 9 ! ?ree-dim5sional notation4 ,k ,ex i1 ,hx 7#fe;a7 -- -- -- ,k -- 7i1 ,e74 ,k is a cov>iant f\r-vector ^: compon5ts are equal to ! negative impulse & en]gy :i* are transf]r$ to ! electro-magnetic field p] unit ( "t1 & p] unit ( volume1 by ! electrical masses4 ,if ! electrical masses be free1 t is1 "u ! influ5ce ( ! electro-magnetic field only1 !n ! cov>iant f\r-vector ,k w vani%4 ,9 ord] to get ! en]gy compon5ts ,t ( ! electro-magnetic field1 we require only to give to ! equation ,k #j1 ! =m ( ! equation 7#eg74 ,f 7#fc7 & 7#fe7 we get fir/1 ,k ,f ,f_/x _/x 7,f ,f7 - ,f ,f_/x4 ,on acc\nt ( 7#fj7 ! second memb] on ! "r-h& side admits ( ! trans=mation-- ,f ,f_/x - ,f ,f_/x - g g ,f ,f_/x4 ,ow+ to symmetry1 ?is expression c also be writt5 9 ! =m -#a_/d g g ,f ,f_/x g g ,f_/x ,f1 :i* c also be put 9 ! =m - #a_/d _/x 7g g ,f ,f7 #a_/d ,f ,f _/x 7g g74 ,! fir/ ( ^! t]ms c be writt5 %ortly z - #a_/d _/x 7,f ,f71 & ! second aft] diff]5tiation c be trans=m$ 9 ! =m - ,f ,f g g_/x4 ,if we take all ! ?ree t]ms toge!r1 we get ! relation 7#ff7 ,k _/x - g g_/x :]e 7#ff;a7 -,f ,f #a_/d ,f ,f4 ,on acc\nt ( 7#cj7 ! equation 7#ff7 becomes equival5t to 7#eg7 & 7#eg;a7 :5 ,k vani%es4 ,?us 's are ! en]gy-compon5ts ( ! electro-magnetic field4 ,) ! help ( 7#fa7 & 7#fd7 we c easily %[ t ! en]gy-compon5ts ( ! electro-magnetic field1 9 ! case ( ! special relativity !ory1 give rise to ! well-kn[n ,maxwell-,poynt+ expressions4 ,we h n[ d$uc$ ! mo/ g5]al laws :i* ! gravitation-field & matt] satisfy :5 we use a co-ord9ate sy/em = :i* 7-g7 #a4 ,!reby we a*ieve an important simplification 9 all our =mulas & calculations1 )\t r5\nc+ ! conditions ( g5]al cov>iance1 z we h obta9$ ! equations ?r\gh a specialisation ( ! co-ord9ate sy/em f ! g5]al cov>iant-equations4 ,/ill ! "q is n )\t =mal int]e/1 :e!r1 :5 ! en]gy-compon5ts ( ! gravitation-field & matt] is def9$ 9 a g5]alis$ mann] )\t any specialisation ( co-ord9ates1 ! laws ( cons]vation h ! =m ( ! equation 7#ef71 & ! field-equations ( gravitation hold 9 ! =m 7#eb7 or 7#eb;a72 su* t on ! left-h& side1 we h a div]g5ce 9 ! usual s5se1 & on ! "r-h& side1 ! sum ( ! en]gy-compon5ts ( matt] & gravitation4 ,i h f\nd out t ?is is inde$ ! case4 ,b ,i am ( op9ion t ! communication ( my r compreh5sive "w on ?is subject w n pay1 = no?+ ess5tially new comes out ( x4 ,e4 #ba4 ,newton's !ory z a fir/ approximation4 ,we h already m5tion$ sev]al times t ! special relativity !ory is to be look$ ^u z a special case ( ! g5]al1 9 :i* g's h con/ant values 7#d74 ,?is signifies1 accord+ to :at has be5 said be=e1 a total neglect ( ! influ5ce ( gravitation4 ,we get "o important approximation if we consid] ! case :5 g's diff] f 7#d7 only by small magnitudes 7comp>$ to #a7 :]e we c neglect small quantities ( ! second & high] ord]s 7fir/ aspect ( ! approximation47 ,fur!r x %\ld be assum$ t )9 ! space-"t region consid]$1 g's at inf9ite di/ances 7us+ ! ^w inf9ite 9 a spatial s5se7 c1 by a suitable *oice ( co-ord9ates1 t5d to ! limit+ values 7#d72 i4e41 we consid] only ^? gravitational fields :i* c be reg>d$ z produc$ by masses di/ribut$ ov] f9ite regions4 ,we c assume t ?is approximation %\ld lead to ,newton's !ory4 ,= x h[ev]1 x is necess>y to treat ! fundam5tal equations f ano!r po9t ( view4 ,let u consid] ! motion ( a p>ticle accord+ to ! equation 7#df74 ,9 ! case ( ! special relativity !ory1 ! compon5ts dx_/ds1 dx_/ds1 dx_/ds1 c take any values4 ,?is signifies t any velocity v 77dx_/dx7 7dx_/dx7 7dx_/dx77 c appe> :i* is less ?an ! velocity ( light 9 vacuum 7v #a74 ,if we f9ally limit ourselves to ! consid]ation ( ! case :5 v is small comp>$ to ! velocity ( light1 x signifies t ! compon5ts dx_/ds1 dx_/ds1 dx_/ds1 c be treat$ z small quantities1 :]eas dx_/ds is equal to #a1 up to ! second-ord] magnitudes 7! second po9t ( view = approximation74 ,n[ we see t1 accord+ to ! fir/ view ( approximation1 ! magnitudes 's are all small quantities ( at lea/ ! fir/ ord]4 ,a glance at 7#df7 w also %[1 t 9 ?is equation accord+ to ! second view ( approximation1 we are only to take into acc\nt ^? t]ms = :i* #d4 ,by limit+ ourselves only to t]ms ( ! l[e/ ord] we get in/ead ( 7#df71 fir/1 ! equations3-- dx_/dt 1 :]e ds dx dt1 or by limit+ ourselves only to ^? t]ms :i* accord+ to ! fir/ /&-po9t are approximations ( ! fir/ ord]1 fracd#b xtaudt#b beg9bmatrix#ddtauendbmatrix 7tau #a1 #b1 #c7 fracd#b x#d;dt#b - beg9bmatrix#ddd;endbmatrix ,if we fur!r assume t ! gravitation-field is quasi-/atic1 i4e41 x is limit$ only to ! case :5 ! matt] produc+ ! gravitation-field is mov+ sl[ly 7relative to ! velocity ( light7 we c neglect ! diff]5tiations ( ! positional co-ord9ates on ! "r-h& side ) respect to "t1 s t we get 7#fg7 dx_/dt - g_/x 71 #a1 #b1 #c7 ,?is is ! equation ( motion ( a mat]ial po9t accord+ to ,newton's !ory1 :]e g_/ plays ! "p ( gravitational pot5tial4 ,! rem>kable ?+ 9 ! result is t 9 ! fir/-approximation ( motion ( ! mat]ial po9t1 only ! compon5t g ( ! fundam5tal t5sor appe>s4 ,let u n[ turn to ! field-equation 7#ec74 ,9 ?is case1 we h to rememb] t ! en]gy-t5sor ( matt] is exclusively def9$ 9 a n>r[ s5se by ! d5sity ( matt]1 i4e41 by ! second memb] on ! "r-h& side ( #eh 7#eh;a1 or #eh;b74 ,if we make ! necess>y approximations1 !n all compon5t vani% except 4 ,on ! left-h& side ( 7#ec7 ! second t]m is an inf9itesimal ( ! second ord]1 s t ! fir/ leads to ! foll[+ t]ms 9 ! approximation1 :i* are r int]e/+ = u3 fracp>tialp>tial x#a beg9bmatrixmunu#a;endbmatrix fracp>tialp>tial x#b beg9bmatrixmunu#b;endbmatrix fracp>tialp>tial x#c beg9bmatrixmunu#c;endbmatrix fracp>tialp>tial x#d beg9bmatrixmunu#d;endbmatrix ,by neglect+ all diff]5tiations ) reg>d to "t1 ?is leads1 :5 #d1 to ! expression - frac#ab 7 fracp>tial#b g#ddp>tial x#ba fracp>tial#b g#ddp>tial x#bb fracp>tial#b g#ddp>tial x#bc 7 - frac#ab ,v#b g#dd ,! la/ ( ! equations 7#ec7 ?us leads to 7#fh7 g 4 ,! equations 7#fg7 & 7#fh7 toge!r1 are equival5t to ,newton's law ( gravitation4 ,= ! gravitation-pot5tial we get f 7#fg7 & 7#fh7 ! exp4 7#fh;a47 -_/7#h7 d_/r :]eas ! ,newtonian !ory = ! *os5 unit ( "t gives -,k_/c d_/r1 :]e ,k d5otes usually ! gravitation-con/ant4 #f4g x #aj2 equat+ !m we get 7#fi7 #h,k_/c #a4hg x #aj4 #bb4 ,behavi\r ( measur+ rods & clocks 9 a /atical gravitation-field4 ,curvature ( light-rays4 ,p]ihelion-motion ( ! pa?s ( ! ,planets4 ,9 ord] to obta9 ,newton's !ory z a fir/ approximation we _h to calculate only g1 out ( ! #aj compon5ts g ( ! gravitation-pot5tial1 = t is ! only compon5t :i* comes 9 ! fir/ approximate equations ( motion ( a mat]ial po9t 9 a gravitational field4 ,we see h[ev]1 t ! o!r compon5ts ( g %\ld also diff] f ! values giv5 9 7#d7 z requir$ by ! condition g -#a4 ,= a heavy p>ticle at ! orig9 ( co-ord9ates & g5]at+ ! gravitational field1 we get z a fir/ approximation ! symmetrical solution ( ! equation3-- g - - 7x x7_/r 7 & #a1 #b1 #c7 7#gj7 g#d g#d #j 7 #a1 #b1 #c7 g #a - _/r4 is #a or #j1 accord+ z or n & r is ! quantity 7x x x74 ,on acc\nt ( 7#fh;a7 we h 7#gj;a7 ,m_/#d :]e ,m d5otes ! mass g5]at+ ! field4 ,x is easy to v]ify t ?is solution satisfies approximately ! field-equation outside ! mass ,m4 ,let u n[ inve/igate ! influ5ces :i* ! field ( mass ,m w h ^u ! metrical prop]ties ( ! field4 ,betwe5 ! l5g?s & times measur$ locally on ! "o h&1 & ! diff]5ces 9 co-ord9ates dx on ! o!r1 we h ! relation ds g dx dx4 ,= a unit measur+ rod1 = example1 plac$ p>allel to ! x axis1 we h to put ds -#a1 dx dx dx #j !n -#a gdx4 ,if ! unit measur+ rod lies on ! x axis1 ! fir/ ( ! equations 7#gj7 gives g -7#a _/r74 ,f bo? ^! relations x foll[s z a fir/ approximation t 7#ga7 dx #a - _/#br4 ,! unit measur+ rod appe>s1 :5 ref]r$ to ! co-ord9ate-sy/em1 %ort5$ by ! calculat$ magnitude ?r\gh ! pres5ce ( ! gravitational field1 :5 we place x radially 9 ! field4 ,simil>ly we c get its co-ord9ate-l5g? 9 a tang5tial position1 if we put = example ds -#a1 dx dx dx #j1 x r1 x x #j we !n get 7#ga;a7 -#a g dx -dx4 ,! gravitational field has no influ5ce ^u ! l5g? ( ! rod1 :5 we put x tang5tially 9 ! field4 ,?us ,euclidean geometry does n hold 9 ! gravitational field ev5 9 ! fir/ approximation1 if we conceive t "o & ! same rod indep5d5t ( its position & its ori5tation c s]ve z ! measure ( ! same ext5sion4 ,b a glance at 7#gj;a7 & 7#fi7 %[s t ! expect$ diff]5ce is mu* too small to be noticeable 9 ! measurem5t ( e>?'s surface4 ,we w\ld fur!r inve/igate ! rate ( go+ ( a unit-clock :i* is plac$ 9 a /atical gravitational field4 ,"h we h = a p]iod ( ! clock ds #a1 dx dx dx #j2 !n we h #a gdx dx #a_/7;g7 #a_/7a 7g - #a77 #a - 7g - #a7_/b or dx #a k_/#h d_/r4 ,!re=e ! clock goes sl[ly :at x is plac$ 9 ! neighb\rhood ( pond]able masses4 ,x foll[s f ?is t ! spectral l9es 9 ! light com+ to u f ! surfaces ( big />s %\ld appe> %ift$ t[>ds ! r$ end ( ! spectrum4 ,let u fur!r inve/igate ! pa? ( light-rays 9 a /atical gravitational field4 ,accord+ to ! special relativity !ory1 ! velocity ( light is giv5 by ! equation -dx - dx - dx dx #j2 ?us also accord+ to ! g5]alis$ relativity !ory x is giv5 by ! equation 7#gc7 ds g dx dx #j4 ,if ! direction1 i4e41 ! ratio dx 3 dx 3 dx is giv51 ! equation 7#gc7 gives ! magnitudes dx_/dx1 dx_/dx1 dx_/dx1 & ) x ! velocity1 77dx_/dx7 7dx_/dx7 7dx_/dx77 1 9 ! s5se ( ! ,euclidean ,geometry4 ,we c easily see t1 ) ref]5ce to ! co-ord9ate sy/em1 ! rays ( light mu/ appe> curv$ 9 case g's are n con/ants4 ,if n be ! direction p]p5dicul> to ! direction ( propagation1 we h1 f ,huyg5's pr9ciple1 t light-rays 7tak5 9 ! plane 71 n7 mu/ suff] a curvature _/n4 ,let u f9d out ! curvature :i* a light-ray suff]s :5 x goes by a mass ,m at a di/ance f x4 ,if we use ! co-ord9ate sy/em accord+ to ! above s*eme1 !n ! total b5d+ ,b ( light-rays 7reckon$ positive :5 x is concave to ! orig97 is giv5 z a suffici5t approximation by ,b - _/x dx :]e 7#gc7 & 7#gj7 gives 7-g_/g7 #a - _/#br 7#a x_/r74 ,! calculation gives ,b #b_/ ,,km_/#b4 ,a ray ( light j graz+ ! sun w\ld suff] a b5d+ ( #ag1 :]eas "o com+ by ,jupit] w\ld h a deviation ( ab\t #jb4 ,if we calculate ! gravitation-field to a great] ord] ( approximation & ) x ! correspond+ pa? ( a mat]ial p>ticle ( a relatively small 7inf9itesimal7 mass we get a deviation ( ! foll[+ k9d f ! ,kepl]-,newtonian ,laws ( ,planet>y motion4 ,! ,ellipse ( ,planet>y motion suff]s a sl[ rotation 9 ! direction ( motion1 ( am\nt 7#ge7 s #bd;a_/c7#a - e7 p] revolution4 ,9 ?is ,=mula #8a' signifies ! semi-major axis1 c1 ! velocity ( light1 measur$ 9 ! usual way1 e1 ! ecc5tricity1 1 ! "t ( revolution 9 seconds4 ,! calculation gives = ! planet ,m]cury1 a rotation ( pa? ( am\nt #dc p] c5tury1 correspond+ suffici5tly to :at has be5 f\nd by a/ronom]s 7,lev]ri]74 ,!y f\nd a residual p]ihelion motion ( ?is planet ( ! giv5 magnitude :i* c n be expla9$ by ! p]turbation ( ! o!r planets4 ,,notes ,note #a4 ,! fundam5tal electro-magnetic equations ( ,maxwell = /ation>y m$ia are3-- curl ,h #a_/;c 7,d_/t 7 7#a7 curl ,e -#a_/;c ,b_/t 7#b7 div ,d ,b ,h div ,b #j ,d k,e ,accord+ to ,h]tz & ,heaviside1 ^! require modification 9 ! case ( mov+ bodies4 ,n[ x is kn[n t due to motion alone !re is a *ange 9 a vector ,r giv5 by 7,r_/t7 due to motion u4 div ,r curl ,ru :]e u is ! vector velocity ( ! mov+ body & ,ru ! vector product ( ,r & u4 ,h5ce equations 7#a7 & 7#b7 become c curl ,h ,d_/t u div ,d curl ,vect4 ,du 7#aa7 & -c curl ,e ,b_/t u div ,b curl ,vect4 ,bu 7#ba7 :i* gives f9ally1 = #j & div ,b #j1 ,d_/t u div ,d c curl 7,h - #a_/;c ,vect4 ,du7 7#ab7 ,b_/t -c curl 7,e - #a_/;c ,vect4 u,b7 7#bb7 ,let u consid] a beam travell+ along ! x-axis1 ) app>5t velocity v 7i4e41 velocity ) respect to ! fix$ e!r7 9 m$ium mov+ ) velocity ux u 9 ! same direction4 ,!n if ! electric & magnetic vectors are proportional to ei,a7x - vt71 we h _/x i,a1 _/t -i,av1 _/y _/z #j1 uy uz #j ,!n ,dy_/t -c,hz_/x - u,dy_/z 444 7#aba7 & ,bz_/t -c,ey_/x - u,bz_/x 7#bba7 ,s9ce ,d ,,ke & ,b ,h1 we h i,av7,ey7 -ci,a7,hz u,,key7 7#abb7 i,av7,hz7 -ci,a7,ey u,hz7 7#bbb7 or v7,k - u7,ey c,hz 7#abc7 7v - u7,hz c,ey 7#bbc7 ,multiply+ 7#abc7 by 7#bbc7 ,k7v - u7 c ,h5ce 7v - u7 c_/k v v v u1 mak+ ,fresnelian convection co-effici5t simply unity4 ,equations 7#aba7 & 7#bba7 may be obta9$ m simply f physical consid]ations4 ,accord+ to ,heaviside & ,h]tz1 ! real seat ( bo? electric & magnetic pol>isation is ! mov+ m$ium itself4 ,n[ at a po9t :i* is fix$ ) respect to ! e!r1 ! rate ( *ange ( electric pol>isation is ,d_/t4 ,consid] a slab ( matt] mov+ ) velocity ux along ! x-axis1 !n ev5 9 a /ation>y field ( electro/atic pol>isation1 t is1 = a field 9 :i* ,d_/t #j1 !re w be "s *ange 9 ! pol>isation ( ! body due to its motion1 giv5 by ux7,d_/x74 ,h5ce we mu/ add ?is t]m to a purely temporal rate ( *ange ,d_/t4 ,do+ ?is we imm$iately arrive at equations 7#aba7 & 7#bba7 = ! special case consid]$ !re4 ,?us ! ,h]tz-,heaviside =m ( field equations gives unity z ! value = ! ,fresnelian convection co-effici5t4 ,x has be5 %[n 9 ! hi/orical introduction h[ ?is is entirely at v>iance ) ! obs]v$ optical facts4 ,z a matt] ( fact1 ,l>mor has %[n 7,ae!r & ,matt]7 t #a - #a_/ is n only suffici5t b is also necess>y1 9 ord] to expla9 exp]im5ts ( ! ,arago prism type4 ,a %ort summ>y ( ! electromagnetic exp]im5ts be>+ on ?is "q1 has already be5 giv5 9 ! introduction4 ,accord+ to ,h]tz & ,heaviside ! total pol>isation is situat$ 9 ! m$ium itself & is completely c>ri$ away by x4 ,?us ! electromagnetic effect outside a mov+ m$ium %\ld be proportional to ,k1 ! specific inductive capacity4 ,r[l& %[$ 9 #ahgf t :5 a *>g$ cond5s] is rapidly rotat$ 7! dielectric rema9+ /ation>y71 ! magnetic effect outside is proportional to ,k1 ! ,sp4 ,ind4 ,cap4 ,rntg5 7,annal5 d] ,physik #ahhh1 #ahij7 f\nd t if ! dielectric is rotat$ :ile ! cond5s] rema9s /ation>y1 ! effect is proportional to ,k - #a4 ,ei*5wald 7,annal5 d] ,physik #aijc1 #aijd7 rotat$ toge!r bo? cond5s] & dielectric & f\nd t ! magnetic effect 0 proportional to ! pot5tial diff]5ce & to ! angul> velocity1 b 0 completely indep5d5t ( ,k4 ,?is is ( c\rse q consi/5t ) ,r[l& & ,rntg54 ,blondlot 7,comptes ,r5dus1 #aija7 pass$ a curr5t ( air 9 a /eady magnetic field ,hy1 7,h ,hz #j74 ,if ?is curr5t ( air moves ) velocity ux along ! x-axis1 an electromotive =ce w\ld be set up along ! z-axis1 due to ! relative motion ( matt] & magnetic tubes ( induction4 ,a pair ( plates at z a1 w be *>g$ up ) d5sity ,dz ,,ke ,k4 us ,hy_/c4 ,b ,blondlot fail$ to detect any su* effect4 ,h4 ,a4 ,wilson 7,phil4 ,trans4 ,royal ,soc4 #aijd7 repeat$ ! exp]im5t ) a cyl9drical cond5s] made ( ebony1 rotat+ 9 a magnetic field p>allel to its own axis4 ,he obs]v$ a *ange proportional to ,k -- #a & n to ,k4 ,?us ! above set ( electro-magnetic exp]im5ts contradict ! ,h]tz-,heaviside equations1 & ^! mu/ be ab&on$4 ,p4 ,c4 ,m4 ,note #b4 ,lor5tz ,trans=mation4 ,lor5tz4 ,v]su* e9] !orie d] elektris*5 und optis*5 ,ers*e9ung5 im bewegt5 ,krp]n4 7,leid5--#ahie74 ,lor5tz4 ,!ory ( ,electrons 7,engli% edition71 pages #aig-bjj1 #bcj1 also notes #gc1 #hf1 pages #cah1 #cbh4 ,lor5tz want$ to expla9 ! ,mi*elson-,morley null-effect4 ,9 ord] to d s1 x 0 obvi\sly necess>y to expla9 ! ,fitzg]ald contraction4 ,lor5tz work$ on ! hypo!sis t an electron itself und]goes contraction :5 mov+4 ,he introduc$ new v>iables = ! mov+ sy/em def9$ by ! foll[+ set ( equations4 x 7x - ut71 y y1 z z1 t 7t - 7u_/c7x7 & = velocities1 us$ vx vx u1 vy vy1 vz vz & _/4 ,) ! help ( ! above set ( equations1 :i* is kn[n z ! ,lor5tz trans=mation1 he succe$$ 9 %[+ h[ ! ,fitzg]ald contraction results z a consequ5ce ( 8=tuit\s comp5sation ( oppos+ effects40 ,x %\ld be obs]v$ t ! ,lor5tz trans=mation is n id5tical ) ! ,e9/e9 trans=mation4 ,! ,e9/e9ian addition ( velocities is q diff]5t z also ! expression = ! 8relative0 d5sity ( electricity4 ,x is true t ! ,maxwell-,lor5tz field equations rema9 practically un*ang$ by ! ,lor5tz trans=mation1 b !y are *ang$ to "s slight ext5t4 ,"o m>k$ advantage ( ! ,e9/e9 trans=mation consi/s 9 ! fact t ! field equations ( a mov+ sy/em pres]ve exactly ! same =m z ^? ( a /ation>y sy/em4 ,x %\ld also be not$ t ! ,fresnelian convection coeffici5t comes out 9 ! !ory ( relativity z a direct consequ5ce ( ,e9/e9's addition ( velocities & is q indep5d5t ( any electrical !ory ( matt]4 ,p4 ,c4 ,m4 ,note #c4 ,see ,lor5tz1 ,!ory ( ,electrons 7,engli% edition71 #aha1 page #bac4 ,h4 ,po9c>e1 ,sur la dynamique #8electron1 ,r5diconti del circolo matematico di ,pal]mo #ba 7#aijf74 ,p4 ,c4 ,m4 ,note #d4 ,relativity ,!orem & ,relativity-,pr9ciple4 ,lor5tz %[$ t ! ,maxwell-,lor5tz sy/em ( electromagnetic field-equations rema9$ practically un*ang$ by ! ,lor5tz trans=mation4 ,?us ! electromagnetic laws ( ,maxwell & ,lor5tz c be def9itely prov$ 8to be indep5d5t ( ! mann] 9 :i* !y are ref]r$ to two coord9ate sy/ems :i* h a uni=m translatory motion relative to ea* o!r40 7,see 8,electrodynamics ( ,mov+ ,bodies10 page #e47 ,?us s f> z ! electromagnetic laws are conc]n$1 ! pr9ciple ( relativity c be prov$ to be true4 ,b x is n kn[n :e!r ?is pr9ciple w rema9 true 9 ! case ( o!r physical laws4 ,we c always proce$ on ! assumption t x does rema9 true4 ,?us x is always possible to con/ruct physical laws 9 su* a way t !y reta9 _! =m :5 ref]r$ to mov+ coord9ates4 ,! ultimate gr\nd = =mulat+ physical laws 9 ?is way is m]ely a subjective conviction t ! pr9ciple ( relativity is univ]sally true4 ,!re is no a priori logical necessity t x %\ld be s4 ,h5ce ! ,pr9ciple ( ,relativity 7s f> z x is appli$ to ph5om5a o!r ?an electromagnetic7 mu/ be reg>d$ z a po/ulate1 :i* we h assum$ to be true1 b = :i* we _c adduce any def9ite pro(1 until aft] ! g5]alisation is made & its consequ5ces te/$ 9 ! light ( actual exp]i5ce4 ,p4 ,c4 ,m4 ,note #e4 ,see 8,electrodynamics ( ,mov+ ,bodies10 p4 #e-h4 ,note #f4 ,field ,equations 9 ,m9k[ski's ,=m4 ,equations 7i7 & 7ii7 become :5 exp&$ into ,c>tesians3-- mz_/y - my_/z - ex_/ x mx_/z - mz_/x - ey_/ y 444 7#aa7 my_/x - mx_/y - ez_/ z & ex_/x ey_/y ez_/z 7#ba7 ,sub/itut+ x1 x1 x1 x & x1 y1 z1 & i2 & 1 1 1 = x1 y1 z1 i1 :]e i 7-#a74 ,we get1 mz_/x - my_/x - i7ex_/x7 x - mz_/x mx_/x - i7ey_/x7 y 444 7#ab7 my_/x - mx_/x - i7ez_/x7 z & multiply+ 7#ba7 by i we get iex_/x iey_/x iez_/x i 444 444 7#bb7 ,n[ sub/itute mx f -f & iex f -f my f -f iey f -f mz f -f iez f -f & we get f9ally3-- f_/x f_/x f_/x f_/x f_/x f_/x 444 7#c7 f_/x f_/x f_/x f_/x f_/x f_/x ,note #i4 ,on ! ,con/ancy ( ! ,velocity ( ,light4 ,page #ab--ref] also to page #f1 ( ,e9/e9's pap]4 ,"o ( ! two fundam5tal ,po/ulates ( ! ,pr9ciple ( ,relativity is t ! velocity ( light %\ld rema9 con/ant :e!r ! s\rce is mov+ or /ation>y4 ,x foll[s t ev5 if a radiant s\rce ,s move ) a velocity u1 x %\ld always rema9 ! c5tre ( sph]ical waves exp&+ outw>ds ) velocity c4 ,at fir/ sight1 x may n appe> cle> :y ! velocity %\ld rema9 con/ant4 ,inde$ accord+ to ! !ory ( ,ritz1 ! velocity %\ld become c u1 :5 ! s\rce ( light moves t[>ds ! obs]v] ) ! velocity u4 ,pr(4 de ,sitt] has giv5 an a/ronomical argum5t = decid+ betwe5 ^! two div]g5t views4 ,let u suppose !re is a d\ble /> ( :i* "o is revolv+ ab\t ! common c5tre ( gravity 9 a circul> orbit4 ,let ! obs]v] be 9 ! plane ( ! orbit1 at a great di/ance 4 ,illu/ration4 ,! light emitt$ by ! /> :5 at ! position ,a w be receiv$ by ! obs]v] aft] a "t1 _/7c u7 :ile ! light emitt$ by ! /> :5 at ! position ,b w be receiv$ aft] a "t _/7c - u74 ,let ,t be ! real half-p]iod ( ! />4 ,!n ! obs]v$ half-p]iod f ,b to ,a is approximately ,t - #bu_/c & f ,a to ,b is ,t #bu_/c4 ,n[ if #bu_/c be comp>able to ,t1 !n x is impossible t ! obs]vations %\ld satisfy ,kepl]'s ,law4 ,9 mo/ ( ! spectroscopic b9>y />s1 #bu_/c are n only ( ! same ord] z ,t1 b are mo/ly mu* l>g]4 ,= example1 if u #ajj km_/sec1 ,t #h days1 _/c #cc ye>s 7correspond+ to an annual p>allax ( #a71 !n ,t - #bu_/c #j4 ,! exi/5ce ( ! ,spectroscopic b9>ies1 & ! fact t !y foll[ ,kepl]'s ,law is !re=e a pro( t c is n affect$ by ! motion ( ! s\rce4 ,9 a lat] memoir1 reply+ to ! criticisms ( ,freundli* & ,gn?ick t an app>5t ecc5tricity occurs 9 ! motion proportional to ku1 u be+ ! maximum value ( u1 ! velocity ( light emitt$ be+ u c ku1 k #j ,lor5tz-,e9/e9 k #a ,ritz4 ,pr(4 de ,sitt] admits ! validity ( ! criticisms4 ,b he rem>ks t an upp] value ( k may be calculat$ f ! obs]vations ( ! d\ble /> -,aurigae4 ,= ?is />1 ! p>allax #jad1 e #jje1 u #aaj km_/sec1 ,t #cif1 #fe light-ye>s1 k is #jjb4 ,= an exp]im5tal pro(1 see a pap] by ,c4 ,majorana4 ,phil4 ,mag41 ,vol4 #ce1 p4 #afc4 ,m4 ,n4 ,s4 ,note #aj4 ,re/-d5sity ( ,electricity4 ,if is ! volume d5sity 9 a mov+ sy/em !n 7#a - u7 is ! correspond+ quantity 9 ! correspond+ volume 9 ! fix$ sy/em1 t is1 9 ! sy/em at re/1 & h5ce x is t]m$ ! re/-d5sity ( electricity4 ,p4 ,c4 ,m4 ,note #aa 7page #ag7 ,space-"t vectors ( ! fir/ & ! second k9d4 ,z we _h already occasion to m5tion1 ,somm]feld has1 9 two pap]s on f\r dim5sional geometry 7vide1 ,annal5 d] ,physik1 ,bd4 #cb1 p4 #gdi2 & ,bd4 #cc1 p4 #fdi71 translat$ ! ideas ( ,m9k[ski into ! language ( f\r dim5sional geometry4 ,in/ead ( ,m9k[ski's space-"t vector ( ! fir/ k9d1 he uses ! m expressive t]m #8f\r-vector1' !reby mak+ x q cle> t x repres5ts a direct$ quantity l a /raight l9e1 a =ce or a mom5tum1 & has got #d compon5ts1 ?ree 9 ! direction ( space-axes1 & "o 9 ! direction ( ! "t-axis4 ,! repres5tation ( ! plane 7def9$ by two /raight l9es7 is mu* m difficult4 ,9 ?ree dim5sions1 ! plane c be repres5t$ by ! vector p]p5dicul> to itself4 ,b t artifice is n available 9 f\r dim5sions4 ,= ! p]p5dicul> to a plane1 we n[ h n a s+le l9e1 b an inf9ite numb] ( l9es con/itut+ a plane4 ,?is difficulty has be5 ov]come by ,m9k[ski 9 a v elegant mann] :i* w become cle> lat] on4 ,mean:ile we (f] ! foll[+ extract f ! above m5tion$ "w ( ,somm]feld4 7,pp4 #gee1 ,bd4 #cb1 ,ann4 d4 ,physik47 8,9 ord] to h a bett] k ab\t ! nature ( ! six-vector 7:i* is ! same ?+ z ,m9k[ski's space-"t vector ( ! #bnd k9d7 let u take ! special case ( a piece ( plane1 hav+ unit area 7cont5ts71 & ! =m ( a p>allelogram1 b\nd$ by ! f\r-vectors u1 v1 pass+ ?r\gh ! orig94 ,!n ! projection ( ?is piece ( plane on ! xy plane is giv5 by ! projections ux1 uy1 vx1 vy ( ! f\r vectors 9 ! comb9ation x y uxvy - uyvx4 ,let u =m 9 a simil> mann] all ! six compon5ts ( ?is plane 4 ,!n six compon5ts are n all indep5d5t b are connect$ by ! foll[+ relation y z x l z x y l x y z l #j ,fur!r ! cont5ts ( ! piece ( a plane is to be def9$ z ! squ>e root ( ! sum ( ! squ>es ( ^! six quantities4 ,9 fact1 y z z x x y x l y l z l4 ,let u n[ on ! o!r h& take ! case ( ! unit plane 99 normal to 2 we c call ?is plane ! ,complem5t ( 4 ,!n we h ! foll[+ relations betwe5 ! compon5ts ( ! two plane3-- y z99 x l1 z x99 y l1 x y99 z l z l99 y x 444 ,! pro( ( ^! ass]tions is z foll[s4 ,let u991 v99 be ! f\r vectors def9+ 994 ,!n we h ! foll[+ relations3-- ux99 ux uy99 uy uz99 uz ul99 ul #j ux99 vx uy99 vy uz99 vz ul99 vl #j vx99 ux vy99 uy vz99 uz vl99 ul #j vx99 vx vy99 vy vz99 vz vl99 vl #j ,if we multiply ^! equations by vl1 ul1 vs1 & subtract ! second f ! fir/1 ! f\r? f ! ?ird we obta9 ux99 x l uy99 y l uz99 z l #j vx99 z l vy99 y l vz99 z l #j multiply+ ^! equations by vx99 4 ux991 or by vy99 4 uy991 we obta9 x z99 x l y z99 y l #j & x y99 x l z x99 z l #j f :i* we h y z99 3 x y99 3 z x99 x l 3 z l 3 y l ,9 a correspond+ way we h y z 3 x y 3 z x x l99 3 z l99 3 y l994 i4e4 i k99 7i k7 :5 ! subscript 7ik7 d5otes ! compon5t ( 9 ! plane conta9$ by ! l9es o!r ?an 7ik74 ,!re=e ! !orem is prov$4 ,we h 7 997 y z y z99 444 #b 7y z z l 4447 #j ,! g5]al six-vector f is compos$ f ! vectors 1 99 9 ! foll[+ way3-- f 99 991 & 99 d5ot+ ! cont5ts ( ! pieces ( mutually p]p5dicul> planes compos+ f4 ,! 8conjugate ,vector0 f99 7or x may be call$ ! complem5t ( f7 is obta9$ by int]*ang+ & 994 ,we h f99 99 99 ,we c v]ify t fy z99 fx l etc4 & f 991 7ff997 #b994 f & 7ff997 may be said to be inv>iants ( ! six vectors1 = _! values are indep5d5t ( ! *oice ( ! sy/em ( co-ord9ates4 ,m4 ,n4 ,s4 ,note #ab4 ,light-velocity z a maximum4 ,page #bc1 & ,electro-dynamics ( ,mov+ ,bodies1 p4 #ag4 ,putt+ v c - x1 & w c - 1 we get ,v 7#b;c - 7x 77_/7#a 7c - x77c - 7_/c7 7#b;c - 7x 77_/7c c - 7x 7c x_/c7 c 7#b;c - 7x 77_/7#b;c - 7x 7 x_/c7 ,?us v lt2 c1 s long z x #j4 ,?us ! velocity ( light is ! absolute maximum velocity4 ,we %all n[ see ! consequ5ces ( admitt+ a velocity ,w c4 ,let ,a & ,b be sep>at$ by di/ance l1 & let velocity ( a 8signal0 9 ! sy/em ,s be ,w c4 ,let ! 7obs]v+7 sy/em ,s h velocity v ) respect to ! sy/em ,s4 ,!n velocity ( signal ) respect to sy/em ,s is giv5 by ,w 7,w - v7_/7#a - ,wv_/c7 ,?us 8"t0 f ,a to ,b z measur$ 9 ,s1 is giv5 by l_/,w l7#a - ,wv_/c7_/7,w - v7 t 7#a7 ,n[ if v is less ?an c1 !n ,w be+ great] ?an c 7by hypo!sis7 ,w is great] ?an v1 i4e41 ,w v4 ,let ,w c & v c - 4 ,!n ,wv 7c 77c - 7 c 7 7c - 4 ,n[ we c always *oose v 9 su* a way t ,wv is great] ?an c1 s9ce ,wv is c if 7 7c - is #j1 t is1 if _/c2 :i* c always be satisfi$ by a suitable *oice ( 4 ,?us = ,w c we c always *oose 9 su* a way z to make ,wv c1 i4e41 - ,wv_/c negative4 ,b ,w - v is always positive4 ,h5ce ) ,w c1 we c always make t1 ! "t f ,a to ,b 9 equation 7#a7 8negative40 ,t is1 ! signal />t+ f ,a w rea* ,b 7z obs]v$ 9 sy/em ,s7 9 less ?an no "t4 ,?us ! effect w be p]ceiv$ be=e ! cause comm5ces to act1 i4e41 ! future w prec$e ! pa/4 ,:i* is absurd4 ,h5ce we conclude t ,w c is an impossibility1 !re c be no velocity great] ?an t ( light4 ,x is conceptually possible to imag9e velocities great] ?an t ( light1 b su* velocities _c occur 9 reality4 ,velocities great] ?an c1 w n produce any effect4 ,causal effect ( any physical type c nev] travel ) a velocity great] ?an t ( light4 ,p4 ,c4 ,m4 ,notes #ac & #ad4 ,we h d5ot$ ! f\r-vector by ! matrix 4 ,x is !n at once se5 t d5otes ! reciprocal matrix ,x is n[ evid5t t :ile ,a1 ,a 1 s ,! vector-product ( ! f\r-vector & s may be repres5t$ by ! comb9ation s s - ,x is n[ easy to v]ify ! =mula f ,af,a4 ,suppos+ = ! sake ( simplicity t f repres5ts ! vector-product ( two f\r-vectors 1 s1 we h f s s - s#aa ,a s,a - ,as,a ,as - s,a ,af,a4 ,n[ rememb]+ t g5]ally f 99994 ,:]e 1 99 are scal> quantities1 1 99 are two mutually p]p5dicul> unit planes1 !re is no difficulty 9 seem+ t f#a ,af,a4 ,note #ae4 ,! vector product 7wf74 7,p4 #cf74 ,?is repres5ts ! vector product ( a f\r-vector & a six-vector4 ,n[ z comb9ations ( ?is type are ( frequ5t occurr5ce 9 ?is pap]1 x w be bett] to =m an idea ( _! geometrical mean+4 ,! foll[+ is tak5 f ! above m5tion$ pap] ( ,somm]feld4 8,we c also =m a vectorial comb9ation ( a f\r-vector & a six-vector1 giv+ u a vector ( ! ?ird type4 ,if ! six-vector be ( a special type1 i4e41 a piece ( plane1 !n ?is vector ( ! ?ird type d5otes ! p>allelopip$ =m$ ( ?is f\r-vector & ! complem5t ( ?is piece ( plane4 ,9 ! g5]al case1 ! product w be ! geometric sum ( two p>allelopip$s1 b x c always be repres5t$ by a f\r-vector ( ! #a/ type4 ,= two pieces ( #c-space volumes c always be add$ toge!r by ! vectorial addition ( _! compon5ts4 ,s by ! addition ( two #c-space volumes1 we d n obta9 a vector ( a m g5]al type1 b "o :i* c always be repres5t$ by a f\r-vector 7loc4 cit4 p4 #gei74 ,! /ate ( affairs "h is ! same z 9 ! ord9>y vector calculus1 :]e by ! vector-multiplication ( a vector ( ! fir/1 & a vector ( ! second type 7i4e41 a pol> vector71 we obta9 a vector ( ! fir/ type 7axial vector74 ,! =mal s*eme ( ?is multiplication is tak5 f ! ?ree-dim5sional case4 ,let ,a 7,ax1 ,ay1 ,az7 d5ote a vector ( ! fir/ type1 ,b 7,by z1 ,bz x1 ,bx y7 d5ote a vector ( ! second type4 ,f ?is la/1 let u =m ?ree special vectors ( ! fir/ k9d1 namely-- ,bx 7,bx x1 ,bx y1 ,bx z7 ,by 7,by x1 ,by y1 ,by z7 7,bi k - ,bk i1 ,bi i #j74 ,bz 7,bz x1 ,bz y1 ,bz z7 ,s9ce ,bj j is z]o1 ,bj is p]p5dicul> to ! j-axis4 ,! j-compon5t ( ! vector-product ( ,a & ,b is equival5t to ! scal> product ( ,a & ,bj1 i4e41 7,a ,bj17 ,ax ,bj x ,ay ,bj y ,az ,bj z4 ,we see easily t ?is co9cides ) ! usual rule = ! vector-product2 e4 g41 = j x4 7,,abx7 ,ay ,bx y - ,az ,bz x4 ,correspond+ly let u def9e 9 ! f\r-dim5sional case ! product 7,pf7 ( any f\r-vector ,p & ! six-vector f4 ,! j-compon5t 7j x1 y1 z1 or l7 is giv5 by 7,pfj7 ,pxfj x ,pyfj y ,pwfj z ,pzfj l ,ea* "o ( ^! compon5ts is obta9$ z ! scal> product ( ,p1 & ! vector fj :i* is p]p5dicul> to j-axis1 & is obta9$ f f by ! rule fj 7fj x1 fj y1 fj z1 fj l7 fj j #j4 ,we c also f9d out "h ! geometrical significance ( vectors ( ! ?ird type1 :5 f 1 i4e41 f repres5ts only "o plane4 ,we replace by ! p>allelogram def9$ by ! two f\r-vectors ,u1 ,v1 & let u pass ov] to ! conjugate plane 991 :i* is =m$ by ! p]p5dicul> f\r-vectors ,u991 ,v994 ,! compon5ts ( 7,p7 are !n equal to ! #d ?ree-r[$ "u-det]m9ants ,dx ,dy ,dz ,dl ( ! matrix ,px ,py ,pz ,pl ,ux99 ,uy99 ,uz99 ,ul99 ,vx99 ,vy99 ,vz99 ,vl99 ,leav+ aside ! fir/ column we obta9 ,dx ,py7,uz99 ,vl99 - ,ul99 ,vz997 ,pz7,ul99 ,vy99 - ,uy99 ,vl997 ,pl7,uy99 ,vz99 - ,uz99 ,vy997 ,py z y99 ,pz99 l y ,pl 99y z4 ,py x y ,pz x z ,pl x l1 :i* co9cides ) 7,px7 accord+ to our def9ition4 ,examples ( ?is type ( vectors w be f\nd on page #cf1 w,f1 ! electrical-re/-=ce1 & #bwf991 ! magnetic-re/-=ce4 ,! re/-ray iw99 also belong to ! same type 7page #ci74 ,x is easy to %[ t -i w w w w ,:5 71 1 7 #j1 w i1 r$uces to ! ?ree-dim5sional vector 1 1 ,s9ce 9 ?is case1 w ,f en 7! electric =ce7 -iw f mx 7! magnetic =ce7 we h 77 ex ey ez mx my mz ,m4 ,n4 ,s4 ,note #af4 ,! electric-re/ =ce4 7,page #cg47 ,! f\r-vector w,f :i* is call$ by ,m9k[ski ! electric-re/-=ce 7elektris*e ,ruh-,kraft7 is v closely connect$ to ,lor5tz's ,pond]omotive =ce1 or ! =ce act+ on a mov+ *>ge4 ,if is ! d5sity ( *>ge1 we h1 :5 #a1 #a1 i4e41 = free space w ,f w ,f w ,f w ,f _/77#a - ,v_/c77 dx #a_/;c 7v h - v h7 ,n[ s9ce 7#a - ,v_/c7 ,we h dx #a_/;c 7v h - v h7 ,n4 ,b4--,we h put ! compon5ts ( e equival5t to 7dx1 dy1 dz71 & ! compon5ts ( m equival5t to hx hy hz71 9 accordance ) ! notation us$ 9 ,lor5tz's ,!ory ( ,electrons4 ,we h !re=e 71 1 7 7d #a_/;c vh71 i4e41 71 1 7 repres5ts ! =ce act+ on ! electron4 ,comp>e ,lor5tz1 ,!ory ( ,electrons1 page #ad4 ,! f\r? compon5t :5 multipli$ by repres5ts i-times ! rate at :i* "w is done by ! mov+ electron1 = i vxdx vydy vzdz vx vy vz 4 -7-#a7 times ! p[] possess$ by ! electron !re=e repres5ts ! f\r? compon5t1 or ! "t compon5t ( ! =ce-f\r-vector4 ,?is compon5t 0 fir/ introduc$ by ,po9c>e 9 #aijf4 ,! f\r-vector i,f99 has a simil> relation to ! =ce act+ on a mov+ magnetic pole4 ,m4 ,n4 ,s4 ,note #ag4 ,op]ator 8,lor0 7 #ab1 p4 #da74 ,! op]ation _/x _/x _/x _/x :i* plays 9 f\r-dim5sional me*anics a rle simil> to t ( ! op]ator 7i_/x1 j_/y1 k_/z 7 9 ?ree-dim5sional geometry has be5 call$ by ,m9k[ski #8,lor5tz-,op]ation' or %ortly #8lor' 9 hon\r ( ,h4 ,a4 ,lor5tz1 ! discov]] ( ! !orem ( relativity4 ,lat] writ]s h sometimes us$ ! symbol to d5ote ?is op]ation4 ,9 ! above-m5tion$ pap] 7,annal5 d] ,physik1 p4 #fdi1 ,bd4 #ch7 ,somm]feld has introduc$ ! t]ms1 ,div 7div]g5ce71 ,rot 7,rotation71 ,grad 7gradi5t7 z f\r-dim5sional ext5sions ( ! correspond+ ?ree-dim5sional op]ations 9 place ( ! g5]al symbol lor4 ,! physical significance ( ^! op]ations w become cle> :5 along ) ,m9k[ski's me?od ( treatm5t we also /udy ! geometrical me?od ( ,somm]feld4 ,m9k[ski beg9s "h ) ! case ( lor ,s1 :]e ,s is a six-vector 7space-"t vector ( ! #bnd k9d74 ,?is be+ a complicat$ case1 we take ! simpl] case ( lor s1 :]e s is a f\r-vector s1 s1 s1 s & s s s s s ,! foll[+ geometrical me?od is tak5 f ,somm]feld4 ,scal> ,div]g5ce--,let d5ote a small f\r-dim5sional volume ( any %ape 9 ! neighb\rhood ( ! space-"t po9t ,q1 d,s d5ote ! ?ree-dim5sional b\nd+ surface ( 1 n be ! out] normal to d,s4 ,let ,s be any f\r-vector1 ,pn its normal compon5t4 ,!n ,div ,s ,lim ,pnd,s_/4 #j ,n[ if = we *oose ! f\r-dim5sional p>allelopip$ ) sides 7dx1 dx1 dx1 dx71 we h !n ,div ,s s_/x s_/x s_/x s_/x lor ,s4 ,if f d5otes a space-"t vector ( ! second k9d1 lor f is equival5t to a space-"t vector ( ! fir/ k9d4 ,! geometrical significance c be ?us br\ght out4 ,we h se5 t ! op]ator #8lor' behaves 9 e respect l a f\r-vector4 ,! vector-product ( a f\r-vector & a six-vector is aga9 a f\r-vector4 ,!re=e x is easy to see t lor ,s w be a f\r-vector4 ,let u f9d ! compon5t ( ?is f\r-vector 9 any direction s4 ,let ,s d5ote ! ?ree-space :i* passes ?r\gh ! po9t ,q 7x1 x1 x1 x7 & is p]p5dicul> to s1 ,s a v small "p ( x 9 ! region ( ,q1 d is an elem5t ( its two-dim5sional surface4 ,let ! p]p5dicul> to ?is surface ly+ 9 ! space be d5ot$ by n1 & let fs n d5ote ! compon5t ( f 9 ! plane ( 7sn7 :i* is evid5tly conjugate to ! plane d4 ,!n ! s-compon5t ( ! vector div]g5ce ( f because ! op]ator lor multiplies f vectorially4 ,div fs ,lim 7 fs nd7_/,s4 s #j ,:]e ! integration 9 d is to be ext5d$ ov] ! :ole surface4 ,if n[ s is select$ z ! x-direction1 s is !n a ?ree-dim5sional p>allelopip$ ) ! sides dy1 dz1 dl1 !n we h ,div fx frac#a;dy dz dl dz4 dl4 fracp>tial fxyp>tial y dy dl dy fracp>tial fxyp>tial z dz dy dz fracp>tial fxyp>tial l dl fracp>tial fxyp>tial y fracp>tial fxyp>tial z fracp>tial fxyp>tial l & g5]ally ,div fj fj x_/x fj y_/y fj z_/z fj l_/l 7:]e fj1 j #j74 ,h5ce ! f\r-compon5ts ( ! f\r-vector lor ,s or ,div4 f is a f\r-vector ) ! compon5ts giv5 on page #db4 ,accord+ to ! =mulae ( space geometry1 ,dx d5otes a p>allelopip$ laid 9 ! 7y-z-l7 space1 =m$ out ( ! vectors 7,py ,pz ,pl71 7,uy99 ,uz99 ,ul997 7,vy99 ,vz99 ,vl9974 ,dx is !re=e ! projection on ! y-z-l space ( ! p>allelopip$ =m$ out ( ^! ?ree f\r-vectors 7,p1 ,u991 ,v9971 & c\ld z well be d5ot$ by ,dyzl4 ,we see directly t ! f\r-vector ( ! k9d repres5t$ by 7,dx1 ,dy1 ,dz1 ,dl7 is p]p5dicul> to ! p>allelopip$ =m$ by 7,p ,u99 ,v9974 ,g5]ally we h 7,pf7 ,,pd ,p99,d994 ,! vector ( ! ?ird type repres5t$ by 7,pf7 is giv5 by ! geometrical sum ( ! two f\r-vectors ( ! fir/ type ,,pd & ,p99,d994 ,m4 ,n4 ,s4 ,transcrib]'s ,notes3 ,! book's idiosyncratic spell+1 emphasis1 punctuation1 & symbology especially 9 ma!matical =mulas1 h be5 reta9$4 ,text t 0 9 italics is enclos$ by und]scores 7italics74 ,text t 0 9 bold face is enclos$ by equals signs 7bold74 ,footnotes h be5 mov$ to foll[ ! *apt]s 9 :i* !y are ref]5c$4