cov] ,relativity3 ,! ,special & ,g5]al ,!ory by ,alb]t ,e9/e9 ,au?oris$ ,translation by ,rob]t ,w4 ,lawson ,,alb]t ,,e9/e9 ,,ref]5ce ,,ar*ive ,,relativity3 ,,! ,,special ,,& ,,g5]al ,,!ory ,,by ,,alb]t ,,e9/e9 ,writt53 #aiaf 7?is revis$ edition3 #aibd7 ,s\rce3 ,relativity3 ,! ,special & ,g5]al ,!ory 7#aibj7 ,publi%]3 ,me?u5 ,co ,ltd ,fir/ ,publi%$3 ,decemb]1 #aiaf ,translat$3 ,rob]t ,w4 ,lawson 7,au?oris$ translation7 ,transcription_/,m>kup3 ,brian ,basg5 ,transcription to text3 ,gregory ,b4 ,newby ,?anks to3 ,e9/e9 ,ref]5ce ,ar*ive 7m>xi/s4org7 ,! ,e9/e9 ,ref]5ce ,ar*ive is onl9e at3 http3_/_/www4m>xi/s4org_/ref]5ce_/ar*ive _/e9/e9_/index4htm ,cont5ts ,preface ,"p ,i3 ,! ,special ,!ory ( ,relativity ,i4 ,physical ,mean+ ( ,geometrical ,propositions ,,ii4 ,! ,sy/em ( ,co-ord9ates ,,iii4 ,space & ,"t 9 ,classical ,me*anics ,,iv4 ,! ,galileian ,sy/em ( ,co-ord9ates ,v4 ,! ,pr9ciple ( ,relativity 79 ! ,re/rict$ ,s5se7 ,,vi4 ,! ,!orem ( ! ,addition ( ,velocities employ$ 9 ,classical ,me*anics ,,vii4 ,! ,app>5t ,incompatability ( ! ,law ( ,propagation ( ,light ) ! ,pr9ciple ( ,relativity ,,viii4 ,on ! ,idea ( ,"t 9 ,physics ,,ix4 ,! ,relativity ( ,simultaneity ,x4 ,on ! ,relativity ( ! ,conception ( ,di/ance ,,xi4 ,! ,lor5tz ,trans=mation ,,xii4 ,! ,behavi\r ( ,measur+-,rods & ,clocks 9 ,motion ,,xiii4 ,!orem ( ! ,addition ( ,velocities4 ,! ,exp]im5t ( ,fizeau ,,xiv4 ,! ,heuri/ic ,value ( ! ,!ory ( ,relativity ,,xv4 ,g5]al ,results ( ! ,!ory ,,xvi4 ,exp]i5ce & ! ,special ,!ory ( ,relativity ,,xvii4 ,m9k[ski's ,f\r-dim5sional ,space ,"p ,,ii3 ,! ,g5]al ,!ory ( ,relativity ,,xviii4 ,special & ,g5]al ,pr9ciple ( ,relativity ,,xix4 ,! ,gravitational ,field ,,xx4 ,! ,equality ( ,in]tial & ,gravitational ,mass z an ,argum5t = ! ,g5]al ,po/ulate ( ,relativity ,,xxi4 ,9 ,:at ,respects are ! ,f\ndations ( ,classical ,me*anics & ( ! ,special ,!ory ( ,relativity ,unsatisfactory8 ,,xxii4 ,a ,few ,inf]5ces f ! ,g5]al ,pr9ciple ( ,relativity ,,xxiii4 ,behavi\r ( ,clocks & ,measur+-,rods on a ,rotat+ ,body ( ,ref]5ce ,,xxiv4 ,euclidean & non-,euclidean ,cont9uum ,,xxv4 ,gaussian ,co-ord9ates ,,xxvi4 ,! ,space-,"t ,cont9uum ( ! ,special ,!ory ( ,relativity ,consid]$ z a ,euclidean ,cont9uum ,,xxvii4 ,! ,space-,"t ,cont9uum ( ! ,g5]al ,!ory ( ,relativity is ,n a ,euclidean ,cont9uum ,,xxviii4 ,exact ,=mulation ( ! ,g5]al ,pr9ciple ( ,relativity ,,xxix4 ,! ,solution ( ! ,problem ( ,gravitation on ! ,basis ( ! ,g5]al ,pr9ciple ( ,relativity ,"p ,,iii3 ,consid]ations on ! ,univ]se z a ,:ole ,,xxx4 ,cosmological ,difficulties ( ,newton's ,!ory ,,xxxi4 ,! ,possibility ( a 8,f9ite0 & yet 8,unb\nd$0 ,univ]se ,,xxxii4 ,! ,/ructure ( ,space ,accord+ to ! ,g5]al ,!ory ( ,relativity ,app5dices3 ,i4 ,simple ,d]ivation ( ! ,lor5tz ,trans=mation 7supplem5t>y to section ,,xi7 ,,ii4 ,m9k[ski's ,f\r-,dim5sional ,space 78,_w07 7supplem5t>y to section ,,xvii7 ,,iii4 ,! ,exp]im5tal ,confirmation ( ! ,g5]al ,!ory ( ,relativity ,,iv4 ,! ,/ructure ( ,space ,accord+ to ! ,g5]al ,!ory ( ,relativity 7supplem5t>y to section ,,xxxii7 ,v4 ,relativity & ! ,problem ( ,space ,note3 ,! fif? ,app5dix 0 add$ by ,e9/e9 at ! "t ( ! fifte5? re-pr9t+ ( ?is book2 & z a result is /ill "u copyright re/rictions s _c be add$ )\t ! p]mission ( ! publi%]4 ,,preface ,! pres5t book is int5d$1 z f> z possible1 to give an exact insight into ! !ory ( ,relativity to ^? read]s :o1 f a g5]al sci5tific & philosophical po9t ( view1 are int]e/$ 9 ! !ory1 b :o are n conv]sant ) ! ma!matical app>atus ( !oretical physics4 ,! "w presumes a /&>d ( education correspond+ to t ( a univ]sity matriculation exam9ation1 &1 despite ! %ortness ( ! book1 a fair am\nt ( pati5ce & =ce ( w on ! "p ( ! read]4 ,! au?or has sp>$ himself no pa9s 9 8 endeav\r to pres5t ! ma9 ideas 9 ! simple/ & mo/ intelligible =m1 & on ! :ole1 9 ! sequ5ce & connection 9 :i* !y actually orig9at$4 ,9 ! int]e/ ( cle>ness1 x appe>$ to me inevitable t ,i %\ld repeat myself frequ5tly1 )\t pay+ ! slighte/ att5tion to ! elegance ( ! pres5tation4 ,i adh]$ scrupul\sly to ! precept ( t brilliant !oretical physici/ ,l4 ,boltzmann1 accord+ to :om matt]s ( elegance ought to be left to ! tailor & to ! cobbl]4 ,i make no pret5ce ( hav+ )held f ! read] difficulties :i* are inh]5t to ! subject4 ,on ! o!r h&1 ,i h purposely treat$ ! empirical physical f\ndations ( ! !ory 9 a 8/ep-mo!rly0 fa%ion1 s t read]s unfamili> ) physics may n feel l ! w&]] :o 0 unable to see ! =e/ = ! trees4 ,may ! book br+ "s "o a few happy h\rs ( sugge/ive ?\ght6 ,decemb]1 #aiaf ,a4 ,,e9/e9 ,,"p ,i3 ,,! ,,special ,,!ory ,,( ,,relativity ,i4 ,,physical ,,mean+ ,,( ,,geometrical ,,propositions ,9 y\r s*ooldays mo/ ( y :o read ?is book made acqua9tance ) ! noble build+ ( ,euclid's geometry1 & y rememb]--p]haps ) m respect ?an love--! magnific5t /ructure1 on ! l(ty /aircase ( :i* y 7 *as$ ab\t = unc\nt$ h\rs by consci5ti\s tea*]s4 ,by reason ( our pa/ exp]i5ce1 y w\ld c]ta9ly reg>d ev]yone ) disda9 :o %\ld pron\nce ev5 ! mo/ out-(-!-way proposition ( ?is sci5ce to be untrue4 ,b p]haps ?is feel+ ( pr\d c]ta9ty w\ld leave y imm$iately if "s "o 7 to ask y3 8,:at1 !n1 d y mean by ! ass]tion t ^! propositions are true80 ,let u proce$ to give ?is "q a little consid]ation4 ,geometry sets out f c]ta9 conceptions su* z 8plane10 8po9t10 & 8/raight l9e10 ) :i* we are able to associate m or less def9ite ideas1 & f c]ta9 simple propositions 7axioms7 :i*1 9 virtue ( ^! ideas1 we are incl9$ to accept z 8true40 ,!n1 on ! basis ( a logical process1 ! ju/ification ( :i* we feel ourselves compell$ to admit1 all rema9+ propositions are %[n to foll[ f ^? axioms1 i4e4 !y are prov54 ,a proposition is !n correct 78true07 :5 x has be5 d]iv$ 9 ! recognis$ mann] f ! axioms4 ,! "q ( 8tru?0 ( ! individual geometrical propositions is ?us r$uc$ to "o ( ! 8tru?0 ( ! axioms4 ,n[ x has long be5 kn[n t ! la/ "q is n only unansw]able by ! me?ods ( geometry1 b t x is 9 itself entirely )\t mean+4 ,we _c ask :e!r x is true t only "o /raight l9e goes ?r\gh two po9ts4 ,we c only say t ,euclidean geometry deals ) ?+s call$ 8/raight l9es10 to ea* ( :i* is ascrib$ ! prop]ty ( be+ uniquely det]m9$ by two po9ts situat$ on x4 ,! concept 8true0 does n tally ) ! ass]tions ( pure geometry1 because by ! ^w 8true0 we are ev5tually 9 ! habit ( designat+ always ! correspond5ce ) a 8real0 object2 geometry1 h[ev]1 is n conc]n$ ) ! relation ( ! ideas involv$ 9 x to objects ( exp]i5ce1 b only ) ! logical connection ( ^! ideas among !mselves4 ,x is n difficult to und]/& :y1 9 spite ( ?is1 we feel con/ra9$ to call ! propositions ( geometry 8true40 ,geometrical ideas correspond to m or less exact objects 9 nature1 & ^! la/ are und\bt$ly ! exclusive cause ( ! g5esis ( ^? ideas4 ,geometry ought to refra9 f su* a c\rse1 9 ord] to give to its /ructure ! l>ge/ possible logical unity4 ,! practice1 = example1 ( see+ 9 a 8di/ance0 two m>k$ positions on a practically rigid body is some?+ :i* is lodg$ deeply 9 our habit ( ?\ght4 ,we are accu/om$ fur!r to reg>d ?ree po9ts z be+ situat$ on a /raight l9e1 if _! app>5t positions c be made to co9cide = obs]vation ) "o eye1 "u suitable *oice ( our place ( obs]vation4 ,if1 9 pursuance ( our habit ( ?\ght1 we n[ supplem5t ! propositions ( ,euclidean geometry by ! s+le proposition t two po9ts on a practically rigid body always correspond to ! same di/ance 7l9e-int]val71 indep5d5tly ( any *anges 9 position to :i* we may subject ! body1 ! propositions ( ,euclidean geometry !n resolve !mselves into propositions on ! possible relative position ( practically rigid bodies4#a ,geometry :i* has be5 supplem5t$ 9 ?is way is !n to be treat$ z a bran* ( physics4 ,we c n[ legitimately ask z to ! 8tru?0 ( geometrical propositions int]pret$ 9 ?is way1 s9ce we are ju/ifi$ 9 ask+ :e!r ^! propositions are satisfi$ = ^? real ?+s we h associat$ ) ! geometrical ideas4 ,9 less exact t]ms we c express ?is by say+ t by ! 8tru?0 ( a geometrical proposition 9 ?is s5se we und]/& its validity = a con/ruction ) rule & compasses4 #a ,x foll[s t a natural object is associat$ also ) a /raight l9e4 ,?ree po9ts ,a1 ,b & ,c on a rigid body ?us lie 9 a /raight l9e :5 ! po9ts ,a & ,c be+ giv51 ,b is *os5 su* t ! sum ( ! di/ances ,,ab & ,,bc is z %ort z possible4 ,?is incomplete sugge/ion w suffice = ! pres5t purpose4 ,( c\rse ! conviction ( ! 8tru?0 ( geometrical propositions 9 ?is s5se is f\nd$ exclusively on r incomplete exp]i5ce4 ,= ! pres5t we %all assume ! 8tru?0 ( ! geometrical propositions1 !n at a lat] /age 79 ! g5]al !ory ( relativity7 we %all see t ?is 8tru?0 is limit$1 & we %all consid] ! ext5t ( its limitation4 ,,ii4 ,,! ,,sy/em ,,( ,,co-,,ord9ates ,on ! basis ( ! physical int]pretation ( di/ance :i* has be5 indicat$1 we are also 9 a position to e/abli% ! di/ance betwe5 two po9ts on a rigid body by means ( measurem5ts4 ,= ?is purpose we require a 8di/ance0 7rod ,s7 :i* is to be us$ once & = all1 & :i* we employ z a /&>d measure4 ,if1 n[1 ,a & ,b are two po9ts on a rigid body1 we c con/ruct ! l9e jo9+ !m accord+ to ! rules ( geometry2 !n1 />t+ f ,a1 we c m>k (f ! di/ance ,s "t aft] "t until we rea* ,b4 ,! numb] ( ^! op]ations requir$ is ! num]ical measure ( ! di/ance ,,ab4 ,?is is ! basis ( all measurem5t ( l5g?4#b #b ,"h we h assum$ t !re is no?+ left ov] i4e4 t ! measurem5t gives a :ole numb]4 ,?is difficulty is got ov] by ! use ( divid$ measur+-rods1 ! introduction ( :i* does n dem& any fundam5tally new me?od4 ,e description ( ! sc5e ( an ev5t or ( ! position ( an object 9 space is bas$ on ! specification ( ! po9t on a rigid body 7body ( ref]5ce7 ) :i* t ev5t or object co9cides4 ,?is applies n only to sci5tific description1 b also to ev]yday life4 ,if ,i analyse ! place specification 8,trafalg> ,squ>e1 ,london0#c ,i arrive at ! foll[+ result4 ,! e>? is ! rigid body to :i* ! specification ( place ref]s2 8,trafalg> ,squ>e1 ,london0 is a well-def9$ po9t1 to :i* a "n has be5 assign$1 & ) :i* ! ev5t co9cides 9 space4#d #c ,i h *os5 ?is z be+ m famili> to ! ,engli% read] ?an ! 8,potzdamm] ,platz1 ,b]l910 :i* is ref]r$ to 9 ! orig9al4 7,r4 ,w4 ,l47 #d ,x is n necess>y "h to inve/igate fur!r ! significance ( ! expression 8co9cid5ce 9 space40 ,?is conception is suffici5tly obvi\s to ensure t diff]5ces ( op9ion are sc>cely likely to arise z to its applicability 9 practice4 ,?is primitive me?od ( place specification deals only ) places on ! surface ( rigid bodies1 & is dep5d5t on ! exi/5ce ( po9ts on ?is surface :i* are di/+ui%able f ea* o!r4 ,b we c free ourselves f bo? ( ^! limitations )\t alt]+ ! nature ( our specification ( position4 ,if1 = in/ance1 a cl\d is hov]+ ov] ,trafalg> ,squ>e1 !n we c det]m9e its position relative to ! surface ( ! e>? by erect+ a pole p]p5dicul>ly on ! ,squ>e1 s t x rea*es ! cl\d4 ,! l5g? ( ! pole measur$ ) ! /&>d measur+-rod1 comb9$ ) ! specification ( ! position ( ! foot ( ! pole1 supplies u ) a complete place specification4 ,on ! basis ( ?is illu/ration1 we are able to see ! mann] 9 :i* a ref9em5t ( ! conception ( position has be5 develop$4 7a7 ,we imag9e ! rigid body1 to :i* ! place specification is ref]r$1 supplem5t$ 9 su* a mann] t ! object ^: position we require is rea*$ by ! complet$ rigid body4 7b7 ,9 locat+ ! position ( ! object1 we make use ( a numb] 7"h ! l5g? ( ! pole measur$ ) ! measur+-rod7 in/ead ( designat$ po9ts ( ref]5ce4 7c7 ,we speak ( ! height ( ! cl\d ev5 :5 ! pole :i* rea*es ! cl\d has n be5 erect$4 ,by means ( optical obs]vations ( ! cl\d f diff]5t positions on ! gr\nd1 & tak+ into acc\nt ! prop]ties ( ! propagation ( light1 we det]m9e ! l5g? ( ! pole we %\ld h requir$ 9 ord] to rea* ! cl\d4 ,f ?is consid]ation we see t x w be advantage\s if1 9 ! description ( position1 x %\ld be possible by means ( num]ical measures to make ourselves indep5d5t ( ! exi/5ce ( m>k$ positions 7possess+ names7 on ! rigid body ( ref]5ce4 ,9 ! physics ( measurem5t ?is is atta9$ by ! application ( ! ,c>tesian sy/em ( co-ord9ates4 ,?is consi/s ( ?ree plane surfaces p]p5dicul> to ea* o!r & rigidly atta*$ to a rigid body4 ,ref]r$ to a sy/em ( co-ord9ates1 ! sc5e ( any ev5t w be det]m9$ 7= ! ma9 "p7 by ! specification ( ! l5g?s ( ! ?ree p]p5dicul>s or co-ord9ates 7x1 y1 z7 :i* c be dropp$ f ! sc5e ( ! ev5t to ^? ?ree plane surfaces4 ,! l5g?s ( ^! ?ree p]p5dicul>s c be det]m9$ by a s]ies ( manipulations ) rigid measur+-rods p]=m$ accord+ to ! rules & me?ods laid d[n by ,euclidean geometry4 ,9 practice1 ! rigid surfaces :i* con/itute ! sy/em ( co-ord9ates are g5]ally n available2 fur!rmore1 ! magnitudes ( ! co-ord9ates are n actually det]m9$ by con/ructions ) rigid rods1 b by indirect means4 ,if ! results ( physics & a/ronomy are to ma9ta9 _! cle>ness1 ! physical mean+ ( specifications ( position mu/ always be s\ght 9 accordance ) ! above consid]ations4#e #e ,a ref9em5t & modification ( ^! views does n become necess>y until we come to deal ) ! g5]al !ory ( relativity1 treat$ 9 ! second "p ( ?is book4 ,we ?us obta9 ! foll[+ result3 ,e description ( ev5ts 9 space involves ! use ( a rigid body to :i* su* ev5ts h to be ref]r$4 ,! result+ relation%ip takes = grant$ t ! laws ( ,euclidean geometry hold = 8di/ances20 ! 8di/ance0 be+ repres5t$ physically by means ( ! conv5tion ( two m>ks on a rigid body4 ,,iii4 ,,space ,,& ,,"t ,,9 ,,classical ,,me*anics ,! purpose ( me*anics is to describe h[ bodies *ange _! position 9 space ) 8"t40 ,i %\ld load my consci5ce ) grave s9s aga9/ ! sacr$ _s ( lucidity 7 ,i to =mulate ! aims ( me*anics 9 ?is way1 )\t s]i\s reflection & detail$ explanations4 ,let u proce$ to disclose ^! s9s4 ,x is n cle> :at is to be und]/ood "h by 8position0 & 8space40 ,i /& at ! w9d[ ( a railway c>riage :i* is travell+ uni=mly1 & drop a /one on ! embankm5t1 )\t ?r[+ x4 ,!n1 disreg>d+ ! influ5ce ( ! air resi/ance1 ,i see ! /one desc5d 9 a /raight l9e4 ,a p$e/rian :o obs]ves ! misde$ f ! footpa? notices t ! /one falls to e>? 9 a p>abolic curve4 ,i n[ ask3 ,d ! 8positions0 trav]s$ by ! /one lie 89 reality0 on a /raight l9e or on a p>abola8 ,moreov]1 :at is meant "h by motion 89 space08 ,f ! consid]ations ( ! previ\s section ! answ] is self-evid5t4 ,9 ! fir/ place we entirely %un ! vague ^w 8space10 ( :i*1 we mu/ hone/ly ackn[l$ge1 we _c =m ! slighte/ conception1 & we replace x by 8motion relative to a practically rigid body ( ref]5ce40 ,! positions relative to ! body ( ref]5ce 7railway c>riage or embankm5t7 h already be5 def9$ 9 detail 9 ! prec$+ section4 ,if in/ead ( 8body ( ref]5ce0 we ins]t 8sy/em ( co-ord9ates10 :i* is a useful idea = ma!matical description1 we are 9 a position to say3 ,! /one trav]ses a /raight l9e relative to a sy/em ( co-ord9ates rigidly atta*$ to ! c>riage1 b relative to a sy/em ( co-ord9ates rigidly atta*$ to ! gr\nd 7embankm5t7 x describes a p>abola4 ,) ! aid ( ?is example x is cle>ly se5 t !re is no su* ?+ z an indep5d5tly exi/+ trajectory 7lit4 8pa?-curve0#f1 b only a trajectory relative to a p>ticul> body ( ref]5ce4 #f ,t is1 a curve along :i* ! body moves4 ,9 ord] to h a complete description ( ! motion1 we mu/ specify h[ ! body alt]s its position ) "t2 i4e4 = e po9t on ! trajectory x mu/ be /at$ at :at "t ! body is situat$ !re4 ,^! data mu/ be supplem5t$ by su* a def9ition ( "t t1 9 virtue ( ?is def9ition1 ^! "t-values c be reg>d$ ess5tially z magnitudes 7results ( measurem5ts7 capable ( obs]vation4 ,if we take our /& on ! gr\nd ( classical me*anics1 we c satisfy ?is requirem5t = our illu/ration 9 ! foll[+ mann]4 ,we imag9e two clocks ( id5tical con/ruction2 ! man at ! railway-c>riage w9d[ is hold+ "o ( !m1 & ! man on ! footpa? ! o!r4 ,ea* ( ! obs]v]s det]m9es ! position on 8 own ref]5ce-body occupi$ by ! /one at ea* tick ( ! clock he is hold+ 9 8 h&4 ,9 ?is connection we h n tak5 acc\nt ( ! inaccuracy involv$ by ! f9it5ess ( ! velocity ( propagation ( light4 ,) ?is & ) a second difficulty prevail+ "h we %all h to deal 9 detail lat]4 ,,iv4 ,,! ,,galileian ,,sy/em ,,( ,,co-,,ord9ates ,z is well kn[n1 ! fundam5tal law ( ! me*anics ( ,galilei-,newton1 :i* is kn[n z ! law ( in]tia1 c be /at$ ?us3 ,a body remov$ suffici5tly f> f o!r bodies cont9ues 9 a /ate ( re/ or ( uni=m motion 9 a /raight l9e4 ,?is law n only says some?+ ab\t ! motion ( ! bodies1 b x also indicates ! ref]5ce-bodies or sy/ems ( coord9ates1 p]missible 9 me*anics1 :i* c be us$ 9 me*anical description4 ,! visible fix$ />s are bodies = :i* ! law ( in]tia c]ta9ly holds to a high degree ( approximation4 ,n[ if we use a sy/em ( co-ord9ates :i* is rigidly atta*$ to ! e>?1 !n1 relative to ?is sy/em1 e fix$ /> describes a circle ( imm5se radius 9 ! c\rse ( an a/ronomical day1 a result :i* is oppos$ to ! /atem5t ( ! law ( in]tia4 ,s t if we adh]e to ?is law we mu/ ref] ^! motions only to sy/ems ( coord9ates relative to :i* ! fix$ />s d n move 9 a circle4 ,a sy/em ( co-ord9ates ( :i* ! /ate ( motion is su* t ! law ( in]tia holds relative to x is call$ a 8,galileian sy/em ( co-ord9ates40 ,! laws ( ! me*anics ( ,galilei-,newton c be reg>d$ z valid only = a ,galileian sy/em ( co-ord9ates4 ,v4 ,,! ,,pr9ciple ,,( ,,relativity 7,,9 ,,! ,,re/rict$ ,,s5se7 ,9 ord] to atta9 ! greate/ possible cle>ness1 let u return to our example ( ! railway c>riage suppos$ to be travell+ uni=mly4 ,we call its motion a uni=m translation 78uni=m0 because x is ( con/ant velocity & direction1 8translation0 because al?\gh ! c>riage *anges its position relative to ! embankm5t yet x does n rotate 9 s do+74 ,let u imag9e a rav5 fly+ ?r\gh ! air 9 su* a mann] t its motion1 z obs]v$ f ! embankm5t1 is uni=m & 9 a /raight l9e4 ,if we 7 to obs]ve ! fly+ rav5 f ! mov+ railway c>riage4 we %\ld f9d t ! motion ( ! rav5 w\ld be "o ( diff]5t velocity & direction1 b t x w\ld /ill be uni=m & 9 a /raight l9e4 ,express$ 9 an ab/ract mann] we may say3 ,if a mass m is mov+ uni=mly 9 a /raight l9e ) respect to a co-ord9ate sy/em ,k1 !n x w also be mov+ uni=mly & 9 a /raight l9e relative to a second co-ord9ate sy/em ,k provid$ t ! latt] is execut+ a uni=m translatory motion ) respect to ,k4 ,9 accordance ) ! discussion conta9$ 9 ! prec$+ section1 x foll[s t3 ,if ,k is a ,galileian co-ord9ate sy/em4 !n e o!r co-ord9ate sy/em ,k is a ,galileian "o1 :51 9 relation to ,k1 x is 9 a condition ( uni=m motion ( translation4 ,relative to ,k ! me*anical laws ( ,galilei-,newton hold good exactly z !y d ) respect to ,k4 ,we advance a /ep f>!r 9 our g5]alisation :5 we express ! t5et ?us3 ,if1 relative to ,k1 ,k is a uni=mly mov+ co-ord9ate sy/em devoid ( rotation1 !n natural ph5om5a run _! c\rse ) respect to ,k accord+ to exactly ! same g5]al laws z ) respect to ,k4 ,?is /atem5t is call$ ! pr9ciple ( relativity 79 ! re/rict$ s5se74 ,z long z "o 0 conv9c$ t all natural ph5om5a 7 capable ( repres5tation ) ! help ( classical me*anics1 !re 0 no ne$ to d\bt ! validity ( ?is pr9ciple ( relativity4 ,b 9 view ( ! m rec5t developm5t ( electrodynamics & optics x became m & m evid5t t classical me*anics af=ds an insuffici5t f\ndation = ! physical description ( all natural ph5om5a4 ,at ?is juncture ! "q ( ! validity ( ! pr9ciple ( relativity became ripe = discussion1 & x did n appe> impossible t ! answ] to ?is "q might be 9 ! negative4 ,nev]!less1 !re are two g5]al facts :i* at ! outset speak v mu* 9 fav\r ( ! validity ( ! pr9ciple ( relativity4 ,ev5 ?\gh classical me*anics does n supply u ) a suffici5tly broad basis = ! !oretical pres5tation ( all physical ph5om5a1 /ill we mu/ grant x a consid]able measure ( 8tru?10 s9ce x supplies u ) ! actual motions ( ! heav5ly bodies ) a delicacy ( detail little %ort ( wond]ful4 ,! pr9ciple ( relativity mu/ !re=e apply ) great accuracy 9 ! doma9 ( me*anics4 ,b t a pr9ciple ( su* broad g5]ality %\ld hold ) su* exactness 9 "o doma9 ( ph5om5a1 & yet %\ld be invalid = ano!r1 is a priori n v probable4 ,we n[ proce$ to ! second argum5t1 to :i*1 moreov]1 we %all return lat]4 ,if ! pr9ciple ( relativity 79 ! re/rict$ s5se7 does n hold1 !n ! ,galileian co-ord9ate sy/ems ,k1 ,k1 ,k1 etc41 :i* are mov+ uni=mly relative to ea* o!r1 w n be equival5t = ! description ( natural ph5om5a4 ,9 ?is case we %\ld be con/ra9$ to believe t natural laws are capable ( be+ =mulat$ 9 a p>ticul>ly simple mann]1 & ( c\rse only on condition t1 f among/ all possible ,galileian co-ord9ate sy/ems1 we %\ld h *os5 "o 7,k#j7 ( a p>ticul> /ate ( motion z our body ( ref]5ce4 ,we %\ld !n be ju/ifi$ 7because ( its m]its = ! description ( natural ph5om5a7 9 call+ ?is sy/em 8absolutely at re/10 & all o!r ,galileian sy/ems ,k 89 motion40 ,if1 = in/ance1 our embankm5t 7 ! sy/em ,k#j !n our railway c>riage w\ld be a sy/em ,k1 relative to :i* less simple laws w\ld hold ?an ) respect to ,k#j4 ,?is dim9i%$ simplicity w\ld be due to ! fact t ! c>riage ,k w\ld be 9 motion 7i4e4 8really07) respect to ,k#j4 ,9 ! g5]al laws ( nature :i* h be5 =mulat$ ) ref]5ce to ,k1 ! magnitude & direction ( ! velocity ( ! c>riage w\ld necess>ily play a "p4 ,we %\ld expect1 = in/ance1 t ! note emitt$ by an organpipe plac$ ) its axis p>allel to ! direction ( travel w\ld be diff]5t f t emitt$ if ! axis ( ! pipe 7 plac$ p]p5dicul> to ?is direction4 ,n[ 9 virtue ( its motion 9 an orbit r\nd ! sun1 our e>? is comp>able ) a railway c>riage travell+ ) a velocity ( ab\t #cj kilometres p] second4 ,if ! pr9ciple ( relativity 7 n valid we %\ld !re=e expect t ! direction ( motion ( ! e>? at any mom5t w\ld ent] into ! laws ( nature1 & also t physical sy/ems 9 _! behavi\r w\ld be dep5d5t on ! ori5tation 9 space ) respect to ! e>?4 ,= ow+ to ! alt]ation 9 direction ( ! velocity ( revolution ( ! e>? 9 ! c\rse ( a ye>1 ! e>? _c be at re/ relative to ! hypo!tical sy/em ,k#j ?r\gh\t ! :ole ye>4 ,h[ev]1 ! mo/ c>eful obs]vations h nev] reveal$ su* anisotropic prop]ties 9 t]re/rial physical space1 i4e4 a physical non-equival5ce ( diff]5t directions4 ,?is is v p[]ful argum5t 9 fav\r ( ! pr9ciple ( relativity4 ,,vi4 ,,! ,,!orem ,,( ,,! ,,addition ,,( ,,velocities ,,employ$ ,,9 ,,classical ,,me*anics ,let u suppose our old fri5d ! railway c>riage to be travell+ along ! rails ) a con/ant velocity v1 & t a man trav]ses ! l5g? ( ! c>riage 9 ! direction ( travel ) a velocity w4 ,h[ quickly or1 9 o!r words1 ) :at velocity ,w does ! man advance relative to ! embankm5t dur+ ! process8 ,! only possible answ] seems to result f ! foll[+ consid]ation3 ,if ! man 7 to /& /ill = a second1 he w\ld advance relative to ! embankm5t ?r\gh a di/ance v equal num]ically to ! velocity ( ! c>riage4 ,z a consequ5ce ( 8 walk+1 h[ev]1 he trav]ses an additional di/ance w relative to ! c>riage1 & h5ce also relative to ! embankm5t1 9 ?is second1 ! di/ance w be+ num]ically equal to ! velocity ) :i* he is walk+4 ,?us 9 total he cov]s ! di/ance ,w v w relative to ! embankm5t 9 ! second consid]$4 ,we %all see lat] t ?is result1 :i* expresses ! !orem ( ! addition ( velocities employ$ 9 classical me*anics1 _c be ma9ta9$2 9 o!r words1 ! law t we h j writt5 d[n does n hold 9 reality4 ,= ! "t be+1 h[ev]1 we %all assume its correctness4 ,,vii4 ,,! ,,app>5t ,,incompatibility ,,( ,,! ,,law ,,( ,,propagation ,,( ,,light ,,) ,,! ,,pr9ciple ,,( ,,relativity ,!re is h>dly a simpl] law 9 physics ?an t accord+ to :i* light is propagat$ 9 empty space4 ,e *ild at s*ool kn[s1 or believes he kn[s1 t ?is propagation takes place 9 /raight l9es ) a velocity c #cjj1jjj km4_/sec4 ,at all ev5ts we "k ) great exactness t ?is velocity is ! same = all col\rs1 because if ?is 7 n ! case1 ! m9imum ( emission w\ld n be obs]v$ simultane\sly = diff]5t col\rs dur+ ! eclipse ( a fix$ /> by its d>k neighb\r4 ,by means ( simil> consid]ations bas$ on obs]vations ( d\ble />s1 ! ,dut* a/ronom] ,de ,sitt] 0 also able to %[ t ! velocity ( propagation ( light _c dep5d on ! velocity ( motion ( ! body emitt+ ! light4 ,! assumption t ?is velocity ( propagation is dep5d5t on ! direction 89 space0 is 9 itself improbable4 ,9 %ort1 let u assume t ! simple law ( ! con/ancy ( ! velocity ( light c 79 vacuum7 is ju/ifiably believ$ by ! *ild at s*ool4 ,:o w\ld imag9e t ?is simple law has plung$ ! consci5ti\sly ?\ghtful physici/ into ! greate/ intellectual difficulties8 ,let u consid] h[ ^! difficulties arise4 ,( c\rse we mu/ ref] ! process ( ! propagation ( light 7& inde$ e o!r process7 to a rigid ref]5ce-body 7co-ord9ate sy/em74 ,z su* a sy/em let u aga9 *oose our embankm5t4 ,we %all imag9e ! air above x to h be5 remov$4 ,if a ray ( light be s5t along ! embankm5t1 we see f ! above t ! tip ( ! ray w be transmitt$ ) ! velocity c relative to ! embankm5t4 ,n[ let u suppose t our railway c>riage is aga9 travell+ along ! railway l9es ) ! velocity v1 & t its direction is ! same z t ( ! ray ( light1 b its velocity ( c\rse mu* less4 ,let u inquire ab\t ! velocity ( propagation ( ! ray ( light relative to ! c>riage4 ,x is obvi\s t we c "h apply ! consid]ation ( ! previ\s section1 s9ce ! ray ( light plays ! "p ( ! man walk+ along relatively to ! c>riage4 ,! velocity ,w ( ! man relative to ! embankm5t is "h replac$ by ! velocity ( light relative to ! embankm5t4 w is ! requir$ velocity ( light ) respect to ! c>riage1 & we h w c - v4 ,! velocity ( propagation ot a ray ( light relative to ! c>riage ?us comes out small] ?an c4 ,b ?is result comes into conflict ) ! pr9ciple ( relativity set =? 9 ,section ,v4 ,=1 l e o!r g5]al law ( nature1 ! law ( ! transmission ( light 9 vacuo 9 vacuum mu/1 accord+ to ! pr9ciple ( relativity1 be ! same = ! railway c>riage z ref]5ce-body z :5 ! rails are ! body ( ref]5ce4 ,b1 f our above consid]ation1 ?is w\ld appe> to be impossible4 ,if e ray ( light is propagat$ relative to ! embankm5t ) ! velocity c1 !n = ?is reason x w\ld appe> t ano!r law ( propagation ( light mu/ necess>ily hold ) respect to ! c>riage--a result contradictory to ! pr9ciple ( relativity4 ,9 view ( ?is dilemma !re appe>s to be no?+ else = x ?an to ab&on ei!r ! pr9ciple ( relativity or ! simple law ( ! propagation ( light 9 vacuo4 ,^? ( y :o h c>efully foll[$ ! prec$+ discussion are almo/ sure to expect t we %\ld reta9 ! pr9ciple ( relativity1 :i* appeals s conv9c+ly to ! intellect because x is s natural & simple4 ,! law ( ! propagation ( light 9 vacuo w\ld !n h to be replac$ by a m complicat$ law con=mable to ! pr9ciple ( relativity4 ,! developm5t ( !oretical physics %[s1 h[ev]1 t we _c pursue ?is c\rse4 ,! epo*-mak+ !oretical inve/igations ( ,h4 ,a4 ,lor5tz on ! electrodynamical & optical ph5om5a connect$ ) mov+ bodies %[ t exp]i5ce 9 ?is doma9 leads conclusively to a !ory ( electromagnetic ph5om5a1 ( :i* ! law ( ! con/ancy ( ! velocity ( light 9 vacuo is a necess>y consequ5ce4 ,prom95t !oretical physici/s 7 !re=e m incl9$ to reject ! pr9ciple ( relativity1 9 spite ( ! fact t no empirical data _h be5 f\nd :i* 7 contradictory to ?is pr9ciple4 ,at ?is juncture ! !ory ( relativity ent]$ ! ar5a4 ,z a result ( an analysis ( ! physical conceptions ( "t & space1 x became evid5t t 9 reality !re is n ! lea/ incompatibilitiy betwe5 ! pr9ciple ( relativity & ! law ( propagation ( light1 & t by sy/ematically hold+ fa/ to bo? ^! laws a logically rigid !ory c\ld be arriv$ at4 ,?is !ory has be5 call$ ! special !ory ( relativity to di/+ui% x f ! ext5d$ !ory1 ) :i* we %all deal lat]4 ,9 ! foll[+ pages we %all pres5t ! fundam5tal ideas ( ! special !ory ( relativity4 ,,viii4 ,,on ,,! ,,idea ,,( ,,"t ,,9 ,,physics ,lightn+ has /ruck ! rails on our railway embankm5t at two places ,a & ,b f> di/ant f ea* o!r4 ,i make ! additional ass]tion t ^! two lightn+ fla%es occurr$ simultane\sly4 ,if ,i ask y :e!r !re is s5se 9 ?is /atem5t1 y w answ] my "q ) a decid$ 8,yes40 ,b if ,i n[ approa* y ) ! reque/ to expla9 to me ! s5se ( ! /atem5t m precisely1 y f9d aft] "s consid]ation t ! answ] to ?is "q is n s easy z x appe>s at fir/ sight4 ,aft] "s "t p]haps ! foll[+ answ] w\ld occur to y3 8,! significance ( ! /atem5t is cle> 9 itself & ne$s no fur!r explanation2 ( c\rse x w\ld require "s consid]ation if ,i 7 to be commission$ to det]m9e by obs]vations :e!r 9 ! actual case ! two ev5ts took place simultane\sly or n40 ,i _c be satisfi$ ) ?is answ] = ! foll[+ reason4 ,suppos+ t z a result ( +5i\s consid]ations an able meteorologi/ 7 to discov] t ! lightn+ mu/ always /rike ! places ,a & ,b simultane\sly1 !n we %\ld be fac$ ) ! task ( te/+ :e!r or n ?is !oretical result is 9 accordance ) ! reality4 ,we enc\nt] ! same difficulty ) all physical /atem5ts 9 :i* ! conception 8simultane\s0 plays a "p4 ,! concept does n exi/ = ! physici/ until he has ! possibility ( discov]+ :e!r or n x is fulfill$ 9 an actual case4 ,we ?us require a def9ition ( simultaneity su* t ?is def9ition supplies u ) ! me?od by means ( :i*1 9 ! pres5t case1 he c decide by exp]im5t :e!r or n bo? ! lightn+ /rokes occurr$ simultane\sly4 ,z long z ?is requirem5t is n satisfi$1 ,i all[ myself to be deceiv$ z a physici/ 7& ( c\rse ! same applies if ,i am n a physici/71 :5 ,i imag9e t ,i am able to atta* a mean+ to ! /atem5t ( simultaneity4 7,i w\ld ask ! read] n to proce$ f>!r until he is fully conv9c$ on ?is po9t47 ,aft] ?9k+ ! matt] ov] = "s "t y !n (f] ! foll[+ sugge/ion ) :i* to te/ simultaneity4 ,by measur+ along ! rails1 ! connect+ l9e ,,ab %\ld be measur$ up & an obs]v] plac$ at ! mid-po9t ,m ( ! di/ance ,,ab4 ,?is obs]v] %\ld be suppli$ ) an arrangem5t 7e4g4 two mirrors incl9$ at #ij7 :i* all[s him visually to obs]ve bo? places ,a & ,b at ! same "t4 ,if ! obs]v] p]ceives ! two fla%es ( lightn+ at ! same "t1 !n !y are simultane\s4 ,i am v pleas$ ) ?is sugge/ion1 b = all t ,i _c reg>d ! matt] z q settl$1 because ,i feel con/ra9$ to raise ! foll[+ objection3 8,y\r def9ition w\ld c]ta9ly be "r1 if only ,i knew t ! light by means ( :i* ! obs]v] at ,m p]ceives ! lightn+ fla%es travels along ! l5g? ,a ,m ) ! same velocity z along ! l5g? ,b ,m4 ,b an exam9ation ( ?is supposition w\ld only be possible if we already _h at our disposal ! means ( measur+ "t4 ,x w\ld ?us appe> z ?\gh we 7 mov+ "h 9 a logical circle40 ,aft] fur!r consid]ation y ca/ a some:at disda9ful glance at me--& rightly s--& y decl>e3 8,i ma9ta9 my previ\s def9ition nev]!less1 because 9 reality x assumes absolutely no?+ ab\t light4 ,!re is only "o dem& to be made ( ! def9ition ( simultaneity1 namely1 t 9 e real case x mu/ supply u ) an empirical decision z to :e!r or n ! conception t has to be def9$ is fulfill$4 ,t my def9ition satisfies ?is dem& is indisputable4 ,t light requires ! same "t to trav]se ! pa? ,a ,m z = ! pa? ,b ,m is 9 reality nei!r a supposition nor a hypo!sis ab\t ! physical nature ( light1 b a /ipulation :i* ,i c make ( my own freewill 9 ord] to arrive at a def9ition ( simultaneity40 ,x is cle> t ?is def9ition c be us$ to give an exact mean+ n only to two ev5ts1 b to z _m ev5ts z we c>e to *oose1 & indep5d5tly ( ! positions ( ! sc5es ( ! ev5ts ) respect to ! body ( ref]5ce#g 7"h ! railway embankm5t74 ,we are ?us l$ also to a def9ition ( 8"t0 9 physics4 ,= ?is purpose we suppose t clocks ( id5tical con/ruction are plac$ at ! po9ts ,a1 ,b & ,c ( ! railway l9e 7co-ord9ate sy/em7 & t !y are set 9 su* a mann] t ! positions ( _! po9t]s are simultane\sly 79 ! above s5se7 ! same4 ,"u ^! conditions we und]/& by ! 8"t0 ( an ev5t ! read+ 7position ( ! h&s7 ( t "o ( ^! clocks :i* is 9 ! imm$iate vic9ity 79 space7 ( ! ev5t4 ,9 ?is mann] a "t-value is associat$ ) e ev5t :i* is ess5tially capable ( obs]vation4 #g ,we suppose fur!r t1 :5 ?ree ev5ts ,a1 ,b & ,c occur 9 diff]5t places 9 su* a mann] t1 if ,a is simultane\s ) ,b1 & ,b is simultane\s ) ,c 7simultane\s 9 ! s5se ( ! above def9ition71 !n ! crit]ion = ! simultaneity ( ! pair ( ev5ts ,a1 ,c is also satisfi$4 ,?is assumption is a physical hypo!sis ab\t ! law ( propagation ( light2 x mu/ c]ta9ly be fulfill$ if we are to ma9ta9 ! law ( ! con/ancy ( ! velocity ( light 9 vacuo4 ,?is /ipulation conta9s a fur!r physical hypo!sis1 ! validity ( :i* w h>dly be d\bt$ )\t empirical evid5ce to ! contr>y4 ,x has be5 assum$ t all ^! clocks g at ! same rate if !y are ( id5tical con/ruction4 ,/at$ m exactly3 ,:5 two clocks arrang$ at re/ 9 diff]5t places ( a ref]5ce-body are set 9 su* a mann] t a p>ticul> position ( ! po9t]s ( ! "o clock is simultane\s 79 ! above s5se7 ) ! same position1 ( ! po9t]s ( ! o!r clock1 !n id5tical 8sett+s0 are always simultane\s 79 ! s5se ( ! above def9ition74 ,,ix4 ,,! ,,relativity ,,( ,,simultaneity ,up to n[ our consid]ations h be5 ref]r$ to a p>ticul> body ( ref]5ce1 :i* we h /yl$ a 8railway embankm5t40 ,we suppose a v long tra9 travell+ along ! rails ) ! con/ant velocity v & 9 ! direction indicat$ 9 ,fig #a4 ,p travell+ 9 ?is tra9 w ) a vantage view ! tra9 z a rigid ref]5ce-body 7co-ord9ate sy/em72 !y reg>d all ev5ts 9 ref]5ce to ! tra94 ,!n e ev5t :i* takes place along ! l9e also takes place at a p>ticul> po9t ( ! tra94 ,also ! def9ition ( simultaneity c be giv5 relative to ! tra9 9 exactly ! same way z ) respect to ! embankm5t4 ,z a natural consequ5ce1 h[ev]1 ! foll[+ "q arises3 image#jja ,are two ev5ts 7e4g4 ! two /rokes ( lightn+ ,a & ,b7 :i* are simultane\s ) ref]5ce to ! railway embankm5t also simultane\s relatively to ! tra98 ,we %all %[ directly t ! answ] mu/ be 9 ! negative4 ,:5 we say t ! lightn+ /rokes ,a & ,b are simultane\s ) respect to be embankm5t1 we mean3 ! rays ( light emitt$ at ! places ,a & ,b1 :]e ! lightn+ occurs1 meet ea* o!r at ! mid-po9t ,m ( ! l5g? ,a ,b ( ! embankm5t4 ,b ! ev5ts ,a & ,b also correspond to positions ,a & ,b on ! tra94 ,let ,m be ! mid-po9t ( ! di/ance ,a ,b on ! travell+ tra94 ,j :5 ! fla%es 7z judg$ f ! embankm5t7 ( lightn+ occur1 ?is po9t ,m naturally co9cides ) ! po9t ,m b x moves t[>ds ! "r 9 ! diagram ) ! velocity v ( ! tra94 ,if an obs]v] sitt+ 9 ! position ,m 9 ! tra9 did n possess ?is velocity1 !n he w\ld rema9 p]man5tly at ,m1 & ! light rays emitt$ by ! fla%es ( lightn+ ,a & ,b w\ld rea* him simultane\sly1 i4e4 !y w\ld meet j :]e he is situat$4 ,n[ 9 reality 7consid]$ ) ref]5ce to ! railway embankm5t7 he is ha/5+ t[>ds ! beam ( light com+ f ,b1 :il/ he is rid+ on ahead ( ! beam ( light com+ f ,a4 ,h5ce ! obs]v] w see ! beam ( light emitt$ f ,b e>li] ?an he w see t emitt$ f ,a4 ,obs]v]s :o take ! railway tra9 z _! ref]5ce-body mu/ !re=e come to ! conclusion t ! lightn+ fla% ,b took place e>li] ?an ! lightn+ fla% ,a4 ,we ?us arrive at ! important result3 ,ev5ts :i* are simultane\s ) ref]5ce to ! embankm5t are n simultane\s ) respect to ! tra91 & vice v]sa 7relativity ( simultaneity74 ,e ref]5ce-body 7co-ord9ate sy/em7 has its own p>ticul> "t2 unless we are told ! ref]5ce-body to :i* ! /atem5t ( "t ref]s1 !re is no mean+ 9 a /atem5t ( ! "t ( an ev5t4 ,n[ be=e ! adv5t ( ! !ory ( relativity x _h always tacitly be5 assum$ 9 physics t ! /atem5t ( "t _h an absolute significance1 i4e4 t x is indep5d5t ( ! /ate ( motion ( ! body ( ref]5ce4 ,b we h j se5 t ?is assumption is incompatible ) ! mo/ natural def9ition ( simultaneity2 if we disc>d ?is assumption1 !n ! conflict betwe5 ! law ( ! propagation ( light 9 vacuo & ! pr9ciple ( relativity 7develop$ 9 ,section ,,vii7 disappe>s4 ,we 7 l$ to t conflict by ! consid]ations ( ,section ,,vi1 :i* are n[ no long] t5able4 ,9 t section we conclud$ t ! man 9 ! c>riage1 :o trav]ses ! di/ance w p] second relative to ! c>riage1 trav]ses ! same di/ance also ) respect to ! embankm5t 9 ea* second ( "t4 ,b1 accord+ to ! =ego+ consid]ations1 ! "t requir$ by a p>ticul> occurr5ce ) respect to ! c>riage mu/ n be consid]$ equal to ! duration ( ! same occurr5ce z judg$ f ! embankm5t 7z ref]5ce-body74 ,h5ce x _c be cont5d$ t ! man 9 walk+ travels ! di/ance w relative to ! railway l9e 9 a "t :i* is equal to "o second z judg$ f ! embankm5t4 ,moreov]1 ! consid]ations ( ,section ,,vi are bas$ on yet a second assumption1 :i*1 9 ! light ( a /rict consid]ation1 appe>s to be arbitr>y1 al?\gh x 0 always tacitly made ev5 be=e ! introduction ( ! !ory ( relativity4 ,x4 ,,on ,,! ,,relativity ,,( ,,! ,,conception ,,( ,,di/ance ,let u consid] two p>ticul> po9ts on ! tra9 #h travell+ along ! embankm5t ) ! velocity v1 & inquire z to _! di/ance ap>t4 ,we already "k t x is necess>y to h a body ( ref]5ce = ! measurem5t ( a di/ance1 ) respect to :i* body ! di/ance c be measur$ up4 ,x is ! simple/ plan to use ! tra9 itself z ref]5ce-body 7co-ord9ate sy/em74 ,an obs]v] 9 ! tra9 measures ! int]val by m>k+ (f 8 measur+-rod 9 a /raight l9e 7e4g4 along ! floor ( ! c>riage7 z _m times z is necess>y to take him f ! "o m>k$ po9t to ! o!r4 ,!n ! numb] :i* tells u h[ (t5 ! rod has to be laid d[n is ! requir$ di/ance4 #h e4g4 ! middle ( ! fir/ & ( ! hundr$? c>riage4 ,x is a diff]5t matt] :5 ! di/ance has to be judg$ f ! railway l9e4 ,"h ! foll[+ me?od sugge/s itself4 ,if we call ,a & ,b ! two po9ts on ! tra9 ^: di/ance ap>t is requir$1 !n bo? ( ^! po9ts are mov+ ) ! velocity v along ! embankm5t4 ,9 ! fir/ place we require to det]m9e ! po9ts ,a & ,b ( ! embankm5t :i* are j be+ pass$ by ! two po9ts ,a & ,b at a p>ticul> "t t--judg$ f ! embankm5t4 ,^! po9ts ,a & ,b ( ! embankm5t c be det]m9$ by apply+ ! def9ition ( "t giv5 9 ,section ,,viii4 ,! di/ance betwe5 ^! po9ts ,a & ,b is !n measur$ by repeat$ application ( ! measur+-rod along ! embankm5t4 ,a priori x is by no means c]ta9 t ?is la/ measurem5t w supply u ) ! same result z ! fir/4 ,?us ! l5g? ( ! tra9 z measur$ f ! embankm5t may be diff]5t f t obta9$ by measur+ 9 ! tra9 itself4 ,?is circum/ance leads u to a second objection :i* mu/ be rais$ aga9/ ! app>5tly obvi\s consid]ation ( ,section ,,vi4 ,namely1 if ! man 9 ! c>riage cov]s ! di/ance w 9 a unit ( "t--measur$ f ! tra91--!n ?is di/ance--z measur$ f ! embankm5t is n necess>ily also equal to w4 ,,xi4 ,,! ,,lor5tz ,,trans=mation ,! results ( ! la/ ?ree sections %[ t ! app>5t incompatibility ( ! law ( propagation ( light ) ! pr9ciple ( relativity 7,section ,,vii7 has be5 d]iv$ by means ( a consid]ation :i* borr[$ two unju/ifiable hypo!ses f classical me*anics2 ^! are z foll[s3 7#a7 ,! "t-int]val 7"t7 betwe5 two ev5ts is indep5d5t ( ! condition ( motion ( ! body ( ref]5ce4 7#b7 ,! space-int]val 7di/ance7 betwe5 two po9ts ( a rigid body is indep5d5t ( ! condition ( motion ( ! body ( ref]5ce4 ,if we drop ^! hypo!ses1 !n ! dilemma ( ,section ,,vii disappe>s1 because ! !orem ( ! addition ( velocities d]iv$ 9 ,section ,,vi becomes invalid4 ,! possibility pres5ts itself t ! law ( ! propagation ( light 9 vacuo may be compatible ) ! pr9ciple ( relativity1 & ! "q arises3 ,h[ h we to modify ! consid]ations ( ,section ,,vi 9 ord] to remove ! app>5t disagreem5t betwe5 ^! two fundam5tal results ( exp]i5ce8 ,?is "q leads to a g5]al "o4 ,9 ! discussion ( ,section ,,vi we h to d ) places & times relative bo? to ! tra9 & to ! embankm5t4 ,h[ are we to f9d ! place & "t ( an ev5t 9 relation to ! tra91 :5 we "k ! place & "t ( ! ev5t ) respect to ! railway embankm5t8 ,is !re a ?9kable answ] to ?is "q ( su* a nature t ! law ( transmission ( light 9 vacuo does n contradict ! pr9ciple ( relativity8 ,9 o!r words3 ,c we conceive ( a relation betwe5 place & "t ( ! individual ev5ts relative to bo? ref]5ce-bodies1 su* t e ray ( light possesses ! velocity ( transmission c relative to ! embankm5t & relative to ! tra98 ,?is "q leads to a q def9ite positive answ]1 & to a p]fectly def9ite trans=mation law = ! space-"t magnitudes ( an ev5t :5 *ang+ ov] f "o body ( ref]5ce to ano!r4 ,be=e we deal ) ?is1 we %all introduce ! foll[+ incid5tal consid]ation4 ,up to ! pres5t we h only consid]$ ev5ts tak+ place along ! embankm5t1 :i* _h ma!matically to assume ! function ( a /raight l9e4 ,9 ! mann] indicat$ 9 ,section ,,ii we c imag9e ?is ref]5ce-body supplem5t$ lat]ally & 9 a v]tical direction by means ( a framework ( rods1 s t an ev5t :i* takes place any:]e c be localis$ ) ref]5ce to ?is framework4 ,simil>ly1 we c imag9e ! tra9 travell+ ) ! velocity v to be cont9u$ across ! :ole ( space1 s t e ev5t1 no matt] h[ f> (f x may be1 c\ld also be localis$ ) respect to ! second framework4 ,)\t committ+ any fundam5tal error1 we c disreg>d ! fact t 9 reality ^! frameworks w\ld cont9ually int]f]e ) ea* o!r1 ow+ to ! imp5etrability ( solid bodies4 ,9 e su* framework we imag9e ?ree surfaces p]p5dicul> to ea* o!r m>k$ out1 & designat$ z 8co-ord9ate planes0 78co-ord9ate sy/em074 ,a co-ord9ate sy/em ,k !n corresponds to ! embankm5t1 & a co-ord9ate sy/em ,k to ! tra94 ,an ev5t1 :]ev] x may h tak5 place1 w\ld be fix$ 9 space ) respect to ,k by ! ?ree p]p5dicul>s x1 y1 z on ! co-ord9ate planes1 & ) reg>d to "t by a "t value t4 ,relative to ,k1 ! same ev5t w\ld be fix$ 9 respect ( space & "t by correspond+ values x1 y1 z1 t1 :i* ( c\rse are n id5tical ) x1 y1 z1 t4 ,x has already be5 set =? 9 detail h[ ^! magnitudes are to be reg>d$ z results ( physical measurem5ts4 image#jjb ,obvi\sly our problem c be exactly =mulat$ 9 ! foll[+ mann]4 ,:at are ! values x1 y1 z1 t1 ( an ev5t ) respect to ,k1 :5 ! magnitudes x1 y1 z1 t1 ( ! same ev5t ) respect to ,k are giv58 ,! relations mu/ be s *os5 t ! law ( ! transmission ( light 9 vacuo is satisfi$ = "o & ! same ray ( light 7& ( c\rse = e ray7 ) respect to ,k & ,k4 ,= ! relative ori5tation 9 space ( ! co-ord9ate sy/ems indicat$ 9 ! diagram 7,fig4 #b71 ?is problem is solv$ by means ( ! equations3 image#jjc y y z z image#jjd ,?is sy/em ( equations is kn[n z ! 8,lor5tz trans=mation40#i #i ,a simple d]ivation ( ! ,lor5tz trans=mation is giv5 9 ,app5dix ,i4 ,if 9 place ( ! law ( transmission ( light we _h tak5 z our basis ! tacit assumptions ( ! old] me*anics z to ! absolute "* ( times & l5g?s1 !n in/ead ( ! above we %\ld h obta9$ ! foll[+ equations3 x x - vt y y z z t t ,?is sy/em ( equations is (t5 t]m$ ! 8,galilei trans=mation40 ,! ,galilei trans=mation c be obta9$ f ! ,lor5tz trans=mation by sub/itut+ an inf9itely l>ge value = ! velocity ( light c 9 ! latt] trans=mation4 ,aid$ by ! foll[+ illu/ration1 we c readily see t1 9 accordance ) ! ,lor5tz trans=mation1 ! law ( ! transmission ( light 9 vacuo is satisfi$ bo? = ! ref]5ce-body ,k & = ! ref]5ce-body ,k4 ,a light-signal is s5t along ! positive x-axis1 & ?is light-/imulus advances 9 accordance ) ! equation x ct1 i4e4 ) ! velocity c4 ,accord+ to ! equations ( ! ,lor5tz trans=mation1 ?is simple relation betwe5 x & t involves a relation betwe5 x & t4 ,9 po9t ( fact1 if we sub/itute = x ! value ct 9 ! fir/ & f\r? equations ( ! ,lor5tz trans=mation1 we obta93 image#jje f :i*1 by division1 ! expression x ct imm$iately foll[s4 ,if ref]r$ to ! sy/em ,k1 ! propagation ( light takes place accord+ to ?is equation4 ,we ?us see t ! velocity ( transmission relative to ! ref]5ce-body ,k is also equal to c4 ,! same result is obta9$ = rays ( light advanc+ 9 any o!r direction :atsoev]4 ,( cause ?is is n surpris+1 s9ce ! equations ( ! ,lor5tz trans=mation 7 d]iv$ con=mably to ?is po9t ( view4 ,,xii4 ,,! ,,behavi\r ,,( ,,measur+-,,rods ,,& ,,clocks ,,9 ,,motion ,place a metre-rod 9 ! x-axis ( ,k 9 su* a mann] t "o end 7! beg9n+7 co9cides ) ! po9t x #j :il/ ! o!r end 7! end ( ! rod7 co9cides ) ! po9t x #a4 ,:at is ! l5g? ( ! metre-rod relatively to ! sy/em ,k8 ,9 ord] to le>n ?is1 we ne$ only ask :]e ! beg9n+ ( ! rod & ! end ( ! rod lie ) respect to ,k at a p>ticul> "t t ( ! sy/em ,k4 ,by means ( ! fir/ equation ( ! ,lor5tz trans=mation ! values ( ^! two po9ts at ! "t t #j c be %[n to be image#jjf ! di/ance betwe5 ! po9ts be+ image#jjg ,b ! metre-rod is mov+ ) ! velocity v relative to ,k4 ,x !re=e foll[s t ! l5g? ( a rigid metre-rod mov+ 9 ! direction ( its l5g? ) a velocity v is image#jjh ( a metre4 ,! rigid rod is ?us %ort] :5 9 motion ?an :5 at re/1 & ! m quickly x is mov+1 ! %ort] is ! rod4 ,= ! velocity v c we %\ld h image#jji & = /ill great] velocities ! squ>e-root becomes imag9>y4 ,f ?is we conclude t 9 ! !ory ( relativity ! velocity c plays ! "p ( a limit+ velocity1 :i* c nei!r be rea*$ nor exce$$ by any real body4 ,( c\rse ?is feature ( ! velocity c z a limit+ velocity also cle>ly foll[s f ! equations ( ! ,lor5tz trans=mation1 = ^! became mean+less if we *oose values ( v great] ?an c4 ,if1 on ! contr>y1 we _h consid]$ a metre-rod at re/ 9 ! x-axis ) respect to ,k1 !n we %\ld h f\nd t ! l5g? ( ! rod z judg$ f ,k w\ld h be5 image#jaj ?is is q 9 accordance ) ! pr9ciple ( relativity :i* =ms ! basis ( our consid]ations4 ,a priori x is q cle> t we mu/ be able to le>n some?+ ab\t ! physical behavi\r ( measur+-rods & clocks f ! equations ( trans=mation1 = ! magnitudes z1 y1 x1 t1 are no?+ m nor less ?an ! results ( measurem5ts obta9able by means ( measur+-rods & clocks4 ,if we _h bas$ our consid]ations on ! ,galileian trans=mation we %\ld n h obta9$ a contraction ( ! rod z a consequ5ce ( its motion4 ,let u n[ consid] a seconds-clock :i* is p]man5tly situat$ at ! orig9 7x #j7 ( ,k4 t #j & t #a are two successive ticks ( ?is clock4 ,! fir/ & f\r? equations ( ! ,lor5tz trans=mation give = ^! two ticks3 t #j & image#jaa ,z judg$ f ,k1 ! clock is mov+ ) ! velocity v2 z judg$ f ?is ref]5ce-body1 ! "t :i* elapses betwe5 two /rokes ( ! clock is n "o second1 b image#jab seconds1 i4e4 a some:at l>g] "t4 ,z a consequ5ce ( its motion ! clock goes m sl[ly ?an :5 at re/4 ,"h also ! velocity c plays ! "p ( an unatta9able limit+ velocity4 ,,xiii4 ,,!orem ,,( ,,! ,,addition ,,( ,,velocities4 ,,! ,,exp]im5t ,,( ,,fizeau ,n[ 9 practice we c move clocks & measur+-rods only ) velocities t are small comp>$ ) ! velocity ( light2 h5ce we %all h>dly be able to comp>e ! results ( ! previ\s section directly ) ! reality4 ,b1 on ! o!r h&1 ^! results mu/ /rike y z be+ v s+ul>1 & = t reason ,i %all n[ draw ano!r conclusion f ! !ory1 "o :i* c easily be d]iv$ f ! =ego+ consid]ations1 & :i* has be5 mo/ elegantly confirm$ by exp]im5t4 ,9 ,section ,,vi we d]iv$ ! !orem ( ! addition ( velocities 9 "o direction 9 ! =m :i* also results f ! hypo!ses ( classical me*anics4 ,?is !orem c also be d$uc$ readily f ! ,galilei trans=mation 7,section ,,xi74 ,9 place ( ! man walk+ inside ! c>riage1 we introduce a po9t mov+ relatively to ! co-ord9ate sy/em ,k 9 accordance ) ! equation x wt ,by means ( ! fir/ & f\r? equations ( ! ,galilei trans=mation we c express x & t 9 t]ms ( x & t1 & we !n obta9 x 7v w7t ,?is equation expresses no?+ else ?an ! law ( motion ( ! po9t ) ref]5ce to ! sy/em ,k 7( ! man ) ref]5ce to ! embankm5t74 ,we d5ote ?is velocity by ! symbol ,w1 & we !n obta91 z 9 ,section ,,vi1 ,w v w 4 4 4 4 4 4 4 7,a74 ,b we c c>ry out ?is consid]ation j z well on ! basis ( ! !ory ( relativity4 ,9 ! equation x wt we mu/ !n express x & t 9 t]ms ( x & t1 mak+ use ( ! fir/ & f\r? equations ( ! ,lor5tz trans=mation4 ,in/ead ( ! equation 7,a7 we !n obta9 ! equation image#jac :i* corresponds to ! !orem ( addition = velocities 9 "o direction accord+ to ! !ory ( relativity4 ,! "q n[ arises z to :i* ( ^! two !orems is ! bett] 9 accord ) exp]i5ce4 ,on ?is po9t we are enlight5$ by a mo/ important exp]im5t :i* ! brilliant physici/ ,fizeau p]=m$ m ?an half a c5tury ago1 & :i* has be5 repeat$ s9ce !n by "s ( ! be/ exp]im5tal physici/s1 s t !re c be no d\bt ab\t its result4 ,! exp]im5t is conc]n$ ) ! foll[+ "q4 ,light travels 9 a motionless liquid ) a p>ticul> velocity w4 ,h[ quickly does x travel 9 ! direction ( ! arr[ 9 ! tube ,t 7see ! accompany+ diagram1 ,fig4 #c7 :5 ! liquid above m5tion$ is fl[+ ?r\gh ! tube ) a velocity v8 image#jad ,9 accordance ) ! pr9ciple ( relativity we %all c]ta9ly h to take = grant$ t ! propagation ( light always takes place ) ! same velocity w ) respect to ! liquid1 :e!r ! latt] is 9 motion ) ref]5ce to o!r bodies or n4 ,! velocity ( light relative to ! liquid & ! velocity ( ! latt] relative to ! tube are ?us kn[n1 & we require ! velocity ( light relative to ! tube4 ,x is cle> t we h ! problem ( ,section ,,vi aga9 be=e u4 ,! tube plays ! "p ( ! railway embankm5t or ( ! co-ord9ate sy/em ,k1 ! liquid plays ! "p ( ! c>riage or ( ! co-ord9ate sy/em ,k1 & f9ally1 ! light plays ! "p ( ! man walk+ along ! c>riage1 or ( ! mov+ po9t 9 ! pres5t section4 ,if we d5ote ! velocity ( ! light relative to ! tube by ,w1 !n ?is is giv5 by ! equation 7,a7 or 7,b71 accord+ z ! ,galilei trans=mation or ! ,lor5tz trans=mation corresponds to ! facts4 ,exp]im5t#aj decides 9 fav\r ( equation 7,b7 d]iv$ f ! !ory ( relativity1 & ! agreem5t is1 inde$1 v exact4 ,accord+ to rec5t & mo/ excell5t measurem5ts by ,zeeman1 ! influ5ce ( ! velocity ( fl[ v on ! propagation ( light is repres5t$ by =mula 7,b7 to )9 "o p] c5t4 #aj ,fizeau f\nd image#jae :]e image#jaf is ! index ( refraction ( ! liquid4 ,on ! o!r h&1 ow+ to ! smallness ( image#jag z comp>$ ) #a1 we c replace 7,b7 9 ! fir/ place by image#jah or to ! same ord] ( approximation by image#jai :i* agrees ) ,fizeau's result4 ,nev]!less we mu/ n[ draw att5tion to ! fact t a !ory ( ?is ph5om5on 0 giv5 by ,h4 ,a4 ,lor5tz long be=e ! /atem5t ( ! !ory ( relativity4 ,?is !ory 0 ( a purely electrodynamical nature1 & 0 obta9$ by ! use ( p>ticul> hypo!ses z to ! electromagnetic /ructure ( matt]4 ,?is circum/ance1 h[ev]1 does n 9 ! lea/ dim9i% ! conclusiv5ess ( ! exp]im5t z a crucial te/ 9 fav\r ( ! !ory ( relativity1 = ! electrodynamics ( ,maxwell-,lor5tz1 on :i* ! orig9al !ory 0 bas$1 9 no way opposes ! !ory ( relativity4 ,r has ! latt] be5 develop$ trom electrodynamics z an a/\nd+ly simple comb9ation & g5]alisation ( ! hypo!ses1 =m]ly indep5d5t ( ea* o!r1 on :i* electrodynamics 0 built4 ,,xiv4 ,,! ,,heuri/ic ,,value ,,( ,,! ,,!ory ,,( ,,relativity ,our tra9 ( ?\ght 9 ! =ego+ pages c be epitomis$ 9 ! foll[+ mann]4 ,exp]i5ce has l$ to ! conviction t1 on ! "o h&1 ! pr9ciple ( relativity holds true & t on ! o!r h& ! velocity ( transmission ( light 9 vacuo has to be consid]$ equal to a con/ant c4 ,by unit+ ^! two po/ulates we obta9$ ! law ( trans=mation = ! rectangul> co-ord9ates x1 y1 z & ! "t t ( ! ev5ts :i* con/itute ! processes ( nature4 ,9 ?is connection we did n obta9 ! ,galilei trans=mation1 b1 diff]+ f classical me*anics1 ! ,lor5tz trans=mation4 ,! law ( transmission ( light1 ! acceptance ( :i* is ju/ifi$ by our actual k1 play$ an important "p 9 ?is process ( ?\ght4 ,once 9 possession ( ! ,lor5tz trans=mation1 h[ev]1 we c comb9e ?is ) ! pr9ciple ( relativity1 & sum up ! !ory ?us3 ,e g5]al law ( nature mu/ be s con/itut$ t x is trans=m$ into a law ( exactly ! same =m :51 in/ead ( ! space-"t v>iables x1 y1 z1 t ( ! orig9al coord9ate sy/em ,k1 we introduce new space-"t v>iables x1 y1 z1 t ( a co-ord9ate sy/em ,k4 ,9 ?is connection ! relation betwe5 ! ord9>y & ! acc5t$ magnitudes is giv5 by ! ,lor5tz trans=mation4 ,or 9 brief3 ,g5]al laws ( nature are co-v>iant ) respect to ,lor5tz trans=mations4 ,?is is a def9ite ma!matical condition t ! !ory ( relativity dem&s ( a natural law1 & 9 virtue ( ?is1 ! !ory becomes a valuable heuri/ic aid 9 ! se>* = g5]al laws ( nature4 ,if a g5]al law ( nature 7 to be f\nd :i* did n satisfy ?is condition1 !n at lea/ "o ( ! two fundam5tal assumptions ( ! !ory w\ld h be5 disprov$4 ,let u n[ exam9e :at g5]al results ! latt] !ory has hi!rto ev9c$4 ,,xv4 ,,g5]al ,,results ,,( ,,! ,,!ory ,x is cle> f our previ\s consid]ations t ! 7special7 !ory ( relativity has gr[n out ( electrodynamics & optics4 ,9 ^! fields x has n appreciably alt]$ ! pr$ictions ( !ory1 b x has consid]ably simplifi$ ! !oretical /ructure1 i4e4 ! d]ivation ( laws1 &--:at is incomp>ably m important--x has consid]ably r$uc$ ! numb] ( indep5d5t hypo!ses =m+ ! basis ( !ory4 ,! special !ory ( relativity has r5d]$ ! ,maxwell-,lor5tz !ory s plausible1 t ! latt] w\ld h be5 g5]ally accept$ by physici/s ev5 if exp]im5t _h decid$ less unequivocally 9 its fav\r4 ,classical me*anics requir$ to be modifi$ be=e x c\ld come into l9e ) ! dem&s ( ! special !ory ( relativity4 ,= ! ma9 "p1 h[ev]1 ?is modification affects only ! laws = rapid motions1 9 :i* ! velocities ( matt] v are n v small z comp>$ ) ! velocity ( light4 ,we h exp]i5ce ( su* rapid motions only 9 ! case ( electrons & ions2 = o!r motions ! v>iations f ! laws ( classical me*anics are too small to make !mselves evid5t 9 practice4 ,we %all n consid] ! motion ( />s until we come to speak ( ! g5]al !ory ( relativity4 ,9 accordance ) ! !ory ( relativity ! k9etic en]gy ( a mat]ial po9t ( mass m is no long] giv5 by ! well-kn[n expression image#jbj b by ! expression image#jba ,?is expression approa*es inf9ity z ! velocity v approa*es ! velocity ( light c4 ,! velocity mu/ !re=e always rema9 less ?an c1 h[ev] great may be ! en]gies us$ to produce ! accel]ation4 ,if we develop ! expression = ! k9etic en]gy 9 ! =m ( a s]ies1 we obta9 image#jbb ,:5 image#jbc is small comp>$ ) unity1 ! ?ird ( ^! t]ms is always small 9 comp>ison ) ! second1 :i* la/ is alone consid]$ 9 classical me*anics4 ,! fir/ t]m mc#b does n conta9 ! velocity1 & requires no consid]ation if we are only deal+ ) ! "q z to h[ ! en]gy ( a po9t-mass2 dep5ds on ! velocity4 ,we %all speak ( its ess5tial significance lat]4 ,! mo/ important result ( a g5]al "* to :i* ! special !ory ( relativity has l$ is conc]n$ ) ! conception ( mass4 ,be=e ! adv5t ( relativity1 physics recognis$ two cons]vation laws ( fundam5tal importance1 namely1 ! law ( ! cons]vation ( en]gy & ! law ( ! cons]vation ( mass ^! two fundam5tal laws appe>$ to be q indep5d5t ( ea* o!r4 ,by means ( ! !ory ( relativity !y h be5 unit$ into "o law4 ,we %all n[ briefly consid] h[ ?is unification came ab\t1 & :at mean+ is to be atta*$ to x4 ,! pr9ciple ( relativity requires t ! law ( ! cons]vation ( en]gy %\ld hold n only ) ref]5ce to a co-ord9ate sy/em ,k1 b also ) respect to e co-ord9ate sy/em ,k :i* is 9 a /ate ( uni=m motion ( translation relative to ,k1 or1 briefly1 relative to e 8,galileian0 sy/em ( co-ord9ates4 ,9 contra/ to classical me*anics2 ! ,lor5tz trans=mation is ! decid+ factor 9 ! transition f "o su* sy/em to ano!r4 ,by means ( comp>atively simple consid]ations we are l$ to draw ! foll[+ conclusion f ^! premises1 9 conjunction ) ! fundam5tal equations ( ! electrodynamics ( ,maxwell3 ,a body mov+ ) ! velocity v1 :i* absorbs#aa an am\nt ( en]gy ,e#j 9 ! =m ( radiation )\t suff]+ an alt]ation 9 velocity 9 ! process1 has1 z a consequ5ce1 its en]gy increas$ by an am\nt image#jbd #aa ,e#j is ! en]gy tak5 up1 z judg$ f a co-ord9ate sy/em mov+ ) ! body4 ,9 consid]ation ( ! expression giv5 above = ! k9etic en]gy ( ! body1 ! requir$ en]gy ( ! body comes out to be image#jbe ,?us ! body has ! same en]gy z a body ( mass image#jbf mov+ ) ! velocity v4 ,h5ce we c say3 ,if a body takes up an am\nt ( en]gy ,e#j1 !n its in]tial mass increases by an am\nt image#jbg ! in]tial mass ( a body is n a con/ant b v>ies accord+ to ! *ange 9 ! en]gy ( ! body4 ,! in]tial mass ( a sy/em ( bodies c ev5 be reg>d$ z a measure ( its en]gy4 ,! law ( ! cons]vation ( ! mass ( a sy/em becomes id5tical ) ! law ( ! cons]vation ( en]gy1 & is only valid provid$ t ! sy/em nei!r takes up nor s5ds out en]gy4 ,writ+ ! expression = ! en]gy 9 ! =m image#jbh we see t ! t]m mc#b1 :i* has hi!rto attract$ our att5tion1 is no?+ else ?an ! en]gy possess$ by ! body#ab be=e x absorb$ ! en]gy ,e#j4 #ab ,z judg$ f a co-ord9ate sy/em mov+ ) ! body4 ,a direct comp>ison ( ?is relation ) exp]im5t is n possible at ! pres5t "t 7#aibj2 see,note1 p4 #dh71 ow+ to ! fact t ! *anges 9 en]gy ,e#j to :i* we c subject a sy/em are n l>ge en\gh to make !mselves p]ceptible z a *ange 9 ! in]tial mass ( ! sy/em4 image#jbg is too small 9 comp>ison ) ! mass m1 :i* 0 pres5t be=e ! alt]ation ( ! en]gy4 ,x is ow+ to ?is circum/ance t classical me*anics 0 able to e/abli% successfully ! cons]vation ( mass z a law ( indep5d5t validity4 ,note ,! equation ,e mc#b has be5 ?or\ghly prov$ "t & aga9 s9ce ?is "t4 ,let me add a f9al rem>k ( a fundam5tal nature4 ,! success ( ! ,f>aday-,maxwell int]pretation ( electromagnetic action at a di/ance result$ 9 physici/s becom+ conv9c$ t !re are no su* ?+s z in/antane\s actions at a di/ance 7n involv+ an int]m$i>y m$ium7 ( ! type ( ,newton's law ( gravitation4 ,accord+ to ! !ory ( relativity1 action at a di/ance ) ! velocity ( light always takes ! place ( in/antane\s action at a di/ance or ( action at a di/ance ) an inf9ite velocity ( transmission4 ,?is is connect$ ) ! fact t ! velocity c plays a fundam5tal role 9 ?is !ory4 ,9 ,"p ,,ii we %all see 9 :at way ?is result becomes modifi$ 9 ! g5]al !ory ( relativity4 ,,xvi4 ,,exp]i5ce ,,& ,,! ,,special ,,!ory ,,( ,,relativity ,to :at ext5t is ! special !ory ( relativity support$ by exp]i5ce8 ,?is "q is n easily answ]$ = ! reason already m5tion$ 9 connection ) ! fundam5tal exp]im5t ( ,fizeau4 ,! special !ory ( relativity has cry/allis$ out f ! ,maxwell-,lor5tz !ory ( electromagnetic ph5om5a4 ,?us all facts ( exp]i5ce :i* support ! electromagnetic !ory also support ! !ory ( relativity4 ,z be+ ( p>ticul> importance1 ,i m5tion "h ! fact t ! !ory ( relativity enables u to pr$ict ! effects produc$ on ! light rea*+ u f ! fix$ />s4 ,^! results are obta9$ 9 an exce$+ly simple mann]1 & ! effects indicat$1 :i* are due to ! relative motion ( ! e>? ) ref]5ce to ^? fix$ />s are f\nd to be 9 accord ) exp]i5ce4 ,we ref] to ! ye>ly movem5t ( ! app>5t position ( ! fix$ />s result+ f ! motion ( ! e>? r\nd ! sun 7ab]ration71 & to ! influ5ce ( ! radial compon5ts ( ! relative motions ( ! fix$ />s ) respect to ! e>? on ! col\r ( ! light rea*+ u f !m4 ,! latt] effect manife/s itself 9 a slight displacem5t ( ! spectral l9es ( ! light transmitt$ to u f a fix$ />1 z comp>$ ) ! position ( ! same spectral l9es :5 !y are produc$ by a t]re/rial s\rce ( light 7,doppl] pr9ciple74 ,! exp]im5tal argum5ts 9 fav\r ( ! ,maxwell-,lor5tz !ory1 :i* are at ! same "t argum5ts 9 fav\r ( ! !ory ( relativity1 are too num]\s to be set =? "h4 ,9 reality !y limit ! !oretical possibilities to su* an ext5t1 t no o!r !ory ?an t ( ,maxwell & ,lor5tz has be5 able to hold its own :5 te/$ by exp]i5ce4 ,b !re are two classes ( exp]im5tal facts hi!rto obta9$ :i* c be repres5t$ 9 ! ,maxwell-,lor5tz !ory only by ! introduction ( an auxili>y hypo!sis1 :i* 9 itself--i4e4 )\t mak+ use ( ! !ory ( relativity--appe>s extrane\s4 ,x is kn[n t ca?ode rays & ! s-call$ -rays emitt$ by radioactive sub/ances consi/ ( negatively electrifi$ p>ticles 7electrons7 ( v small in]tia & l>ge velocity4 ,by exam9+ ! deflection ( ^! rays "u ! influ5ce ( electric & magnetic fields1 we c /udy ! law ( motion ( ^! p>ticles v exactly4 ,9 ! !oretical treatm5t ( ^! electrons1 we are fac$ ) ! difficulty t electrodynamic !ory ( itself is unable to give an acc\nt ( _! nature4 ,= s9ce electrical masses ( "o sign repel ea* o!r1 ! negative electrical masses con/itut+ ! electron w\ld necess>ily be scatt]$ "u ! influ5ce ( _! mutual repulsions1 unless !re are =ces ( ano!r k9d op]at+ betwe5 !m1 ! nature ( :i* has hi!rto rema9$ obscure to u4#ac ,if we n[ assume t ! relative di/ances betwe5 ! electrical masses con/itut+ ! electron rema9 un*ang$ dur+ ! motion ( ! electron 7rigid connection 9 ! s5se ( classical me*anics71 we arrive at a law ( motion ( ! electron :i* does n agree ) exp]i5ce4 ,guid$ by purely =mal po9ts ( view1 ,h4 ,a4 ,lor5tz 0 ! fir/ to introduce ! hypo!sis t ! =m ( ! electron exp]i5ces a contraction 9 ! direction ( motion 9 consequ5ce ( t motion4 ! contract$ l5g? be+ proportional to ! expression image#jbi ,?is1 hypo!sis1 :i* is n ju/ifiable by any electrodynamical facts1 supplies u !n ) t p>ticul> law ( motion :i* has be5 confirm$ ) great precision 9 rec5t ye>s4 #ac ,! g5]al !ory ( relativity r5d]s x likely t ! electrical masses ( an electron are held toge!r by gravitational =ces4 ,! !ory ( relativity leads to ! same law ( motion1 )\t requir+ any special hypo!sis :atsoev] z to ! /ructure & ! behavi\r ( ! electron4 ,we arriv$ at a simil> conclusion 9 ,section ,,xiii 9 connection ) ! exp]im5t ( ,fizeau1 ! result ( :i* is =etold by ! !ory ( relativity )\t ! necessity ( draw+ on hypo!ses z to ! physical nature ( ! liquid4 ,! second class ( facts to :i* we h allud$ has ref]5ce to ! "q :e!r or n ! motion ( ! e>? 9 space c be made p]ceptible 9 t]re/rial exp]im5ts4 ,we h already rem>k$ 9 ,section ,v t all attempts ( ?is nature l$ to a negative result4 ,be=e ! !ory ( relativity 0 put =w>d1 x 0 difficult to become reconcil$ to ?is negative result1 = reasons n[ to be discuss$4 ,! inh]it$ prejudices ab\t "t & space did n all[ any d\bt to arise z to ! prime importance ( ! ,galileian trans=mation = *ang+ ov] f "o body ( ref]5ce to ano!r4 ,n[ assum+ t ! ,maxwell-,lor5tz equations hold = a ref]5ce-body ,k1 we !n f9d t !y d n hold = a ref]5ce-body ,k mov+ uni=mly ) respect to ,k1 if we assume t ! relations ( ! ,galileian trans=mation exi/ betwe5 ! co-ord9ates ( ,k & ,k4 ,x ?us appe>s t1 ( all ,galileian co-ord9ate sy/ems1 "o 7,k7 correspond+ to a p>ticul> /ate ( motion is physically unique4 ,?is result 0 int]pret$ physically by reg>d+ ,k z at re/ ) respect to a hypo!tical !r ( space4 ,on ! o!r h&1 all coord9ate sy/ems ,k mov+ relatively to ,k 7 to be reg>d$ z 9 motion ) respect to ! !r4 ,to ?is motion ( ,k aga9/ ! !r 78!r-drift0 relative to ,k7 7 attribut$ ! m complicat$ laws :i* 7 suppos$ to hold relative to ,k4 ,/rictly speak+1 su* an !r-drift ought also to be assum$ relative to ! e>?1 & = a long "t ! ef=ts ( physici/s 7 devot$ to attempts to detect ! exi/5ce ( an !r-drift at ! e>?'s surface4 ,9 "o ( ! mo/ notable ( ^! attempts ,mi*elson devis$ a me?od :i* appe>s z ?\gh x mu/ be decisive4 ,imag9e two mirrors s arrang$ on a rigid body t ! reflect+ surfaces face ea* o!r4 ,a ray ( light requires a p]fectly def9ite "t ,t to pass f "o mirror to ! o!r & back aga91 if ! :ole sy/em be at re/ ) respect to ! !r4 ,x is f\nd by calculation1 h[ev]1 t a slightly diff]5t "t ,t is requir$ = ?is process1 if ! body1 toge!r ) ! mirrors1 be mov+ relatively to ! !r4 ,& yet ano!r po9t3 x is %[n by calculation t = a giv5 velocity v ) ref]5ce to ! !r1 ?is "t ,t is diff]5t :5 ! body is mov+ p]p5dicul>ly to ! planes ( ! mirrors f t result+ :5 ! motion is p>allel to ^! planes4 ,al?\gh ! e/imat$ diff]5ce betwe5 ^! two times is exce$+ly small1 ,mi*elson & ,morley p]=m$ an exp]im5t involv+ int]f]5ce 9 :i* ?is diff]5ce %\ld h be5 cle>ly detectable4 ,b ! exp]im5t gave a negative result--a fact v p]plex+ to physici/s4 ,lor5tz & ,fitz,gerald rescu$ ! !ory f ?is difficulty by assum+ t ! motion ( ! body relative to ! !r produces a contraction ( ! body 9 ! direction ( motion1 ! am\nt ( contraction be+ j suffici5t to comp5sate = ! diff]5ce 9 "t m5tion$ above4 ,comp>ison ) ! discussion 9 ,section ,,xii %[s t also f ! /&po9t ( ! !ory ( relativity ?is solution ( ! difficulty 0 ! "r "o4 ,b on ! basis ( ! !ory ( relativity ! me?od ( int]pretation is incomp>ably m satisfactory4 ,accord+ to ?is !ory !re is no su* ?+ z a 8specially fav\r$0 7unique7 co-ord9ate sy/em to occasion ! introduction ( ! !r-idea1 & h5ce !re c be no !r-drift1 nor any exp]im5t ) :i* to demon/rate x4 ,"h ! contraction ( mov+ bodies foll[s f ! two fundam5tal pr9ciples ( ! !ory1 )\t ! introduction ( p>ticul> hypo!ses2 & z ! prime factor involv$ 9 ?is contraction we f9d1 n ! motion 9 itself1 to :i* we _c atta* any mean+1 b ! motion ) respect to ! body ( ref]5ce *os5 9 ! p>ticul> case 9 po9t4 ,?us = a co-ord9ate sy/em mov+ ) ! e>? ! mirror sy/em ( ,mi*elson & ,morley is n %ort5$1 b x is %ort5$ = a co-ord9ate sy/em :i* is at re/ relatively to ! sun4 ,,xvii4 ,,m9k[ski',s ,,f\r-,,dim5sional ,,space ,! non-ma!matician is seiz$ by a my/]i\s %udd]+ :5 he he>s ( 8f\r-dim5sional0 ?+s1 by a feel+ n unlike t awak5$ by ?\ghts ( ! occult4 ,& yet !re is no m common-place /atem5t ?an t ! _w 9 :i* we live is a f\r-dim5sional space-"t cont9uum4 ,space is a ?ree-dim5sional cont9uum4 ,by ?is we mean t x is possible to describe ! position ( a po9t 7at re/7 by means ( ?ree numb]s 7co-ord9ates7 x1 y1 z1 & t !re is an indef9ite numb] ( po9ts 9 ! neighb\rhood ( ?is "o1 ! position ( :i* c be describ$ by co-ord9ates su* z x#a1 y#a1 z#a1 :i* may be z ne> z we *oose to ! respective values ( ! co-ord9ates x1 y1 z1 ( ! fir/ po9t4 ,9 virtue ( ! latt] prop]ty we speak ( a 8cont9uum10 & ow+ to ! fact t !re are ?ree co-ord9ates we speak ( x z be+ 8?ree-dim5sional40 ,simil>ly1 ! _w ( physical ph5om5a :i* 0 briefly call$ 8_w0 by ,m9k[ski is naturally f\r dim5sional 9 ! space-"t s5se4 ,= x is compos$ ( individual ev5ts1 ea* ( :i* is describ$ by f\r numb]s1 namely1 ?ree space co-ord9ates x1 y1 z1 & a "t co-ord9ate1 ! "t value t4 ,! 8_w0 is 9 ?is s5se also a cont9uum2 = to e ev5t !re are z _m 8neighb\r+0 ev5ts 7realis$ or at lea/ ?9kable7 z we c>e to *oose1 ! co-ord9ates x#a1 y#a1 z#a1 t#a ( :i* diff] by an indef9itely small am\nt f ^? ( ! ev5t x1 y1 z1 t orig9ally consid]$4 ,t we h n be5 accu/om$ to reg>d ! _w 9 ?is s5se z a f\r-dim5sional cont9uum is due to ! fact t 9 physics1 be=e ! adv5t ( ! !ory ( relativity1 "t play$ a diff]5t & m indep5d5t rle1 z comp>$ ) ! space coord9ates4 ,x is = ?is reason t we h be5 9 ! habit ( treat+ "t z an indep5d5t cont9uum4 ,z a matt] ( fact1 accord+ to classical me*anics1 "t is absolute1 i4e4 x is indep5d5t ( ! position & ! condition ( motion ( ! sy/em ( co-ord9ates4 ,we see ?is express$ 9 ! la/ equation ( ! ,galileian trans=mation 7t t74 ,! f\r-dim5sional mode ( consid]ation ( ! 8_w0 is natural on ! !ory ( relativity1 s9ce accord+ to ?is !ory "t is robb$ ( its indep5d5ce4 ,?is is %[n by ! f\r? equation ( ! ,lor5tz trans=mation3 image#jcj ,moreov]1 accord+ to ?is equation ! "t diff]5ce t ( two ev5ts ) respect to ,k does n 9 g5]al vani%1 ev5 :5 ! "t diff]5ce t ( ! same ev5ts ) ref]5ce to ,k vani%es4 ,pure 8space-di/ance0 ( two ev5ts ) respect to ,k results 9 8"t-di/ance 0 ( ! same ev5ts ) respect to ,k4 ,b ! discov]y ( ,m9k[ski1 :i* 0 ( importance = ! =mal developm5t ( ! !ory ( relativity1 does n lie "h4 ,x is to be f\nd r 9 ! fact ( 8 recognition t ! f\r-dim5sional space-"t cont9uum ( ! !ory ( relativity1 9 its mo/ ess5tial =mal prop]ties1 %[s a pron\nc$ relation%ip to ! ?ree-dim5sional cont9uum ( ,euclidean geometrical space4#ad ,9 ord] to give due prom95ce to ?is relation%ip1 h[ev]1 we mu/ replace ! usual "t co-ord9ate t by an imag9>y magnitude image#jca proportional to x4 ,"u ^! conditions1 ! natural laws satisfy+ ! dem&s ( ! 7special7 !ory ( relativity assume ma!matical =ms1 9 :i* ! "t co-ord9ate plays exactly ! same role z ! ?ree space co-ord9ates4 ,=mally1 ^! f\r co-ord9ates correspond exactly to ! ?ree space co-ord9ates 9 ,euclidean geometry4 ,x mu/ be cle> ev5 to ! non-ma!matician t1 z a consequ5ce ( ?is purely =mal addition to our k1 ! !ory p]=ce ga9$ cle>ness 9 no mean measure4 #ad ,cf4 ! some:at m detail$ discussion 9 ,app5dix ,,ii4 ,^! inadequate rem>ks c give ! read] only a vague notion ( ! important idea contribut$ by ,m9k[ski4 ,)\t x ! g5]al !ory ( relativity1 ( :i* ! fundam5tal ideas are develop$ 9 ! foll[+ pages1 w\ld p]haps h got no f>!r ?an its long clo!s4 ,m9k[ski's "w is d\btless difficult ( access to anyone inexp]i5c$ 9 ma!matics1 b s9ce x is n necess>y to h a v exact grasp ( ?is "w 9 ord] to und]/& ! fundam5tal ideas ( ei!r ! special or ! g5]al !ory ( relativity1 ,i %all leave x "h at pres5t1 & rev]t to x only t[>ds ! end ( ,"p ,,ii4 ,,"p ,,ii3 ,,! ,,g5]al ,,!ory ,,( ,,relativity ,,xviii4 ,,special ,,& ,,g5]al ,,pr9ciple ,,( ,,relativity ,! basal pr9ciple1 :i* 0 ! pivot ( all our previ\s consid]ations1 0 ! special pr9ciple ( relativity1 i4e4 ! pr9ciple ( ! physical relativity ( all uni=m motion4 ,let z once m analyse its mean+ c>efully4 ,x 0 at all times cle> t1 f ! po9t ( view ( ! idea x conveys to u1 e motion mu/ be consid]$ only z a relative motion4 ,return+ to ! illu/ration we h frequ5tly us$ ( ! embankm5t & ! railway c>riage1 we c express ! fact ( ! motion "h tak+ place 9 ! foll[+ two =ms1 bo? ( :i* are equally ju/ifiable3 7a7 ,! c>riage is 9 motion relative to ! embankm5t1 7b7 ,! embankm5t is 9 motion relative to ! c>riage4 ,9 7a7 ! embankm5t1 9 7b7 ! c>riage1 s]ves z ! body ( ref]5ce 9 our /atem5t ( ! motion tak+ place4 ,if x is simply a "q ( detect+ or ( describ+ ! motion involv$1 x is 9 pr9ciple immat]ial to :at ref]5ce-body we ref] ! motion4 ,z already m5tion$1 ?is is self-evid5t1 b x mu/ n be confus$ ) ! mu* m compreh5sive /atem5t call$ 8! pr9ciple ( relativity10 :i* we h tak5 z ! basis ( our inve/igations4 ,! pr9ciple we h made use ( n only ma9ta9s t we may equally well *oose ! c>riage or ! embankm5t z our ref]5ce-body = ! description ( any ev5t 7= ?is1 too1 is self-evid5t74 ,our pr9ciple r ass]ts :at foll[s3 ,if we =mulate ! g5]al laws ( nature z !y are obta9$ f exp]i5ce1 by mak+ use ( 7a7 ! embankm5t z ref]5ce-body1 7b7 ! railway c>riage z ref]5ce-body1 !n ^! g5]al laws ( nature 7e4g4 ! laws ( me*anics or ! law ( ! propagation ( light 9 vacuo7 h exactly ! same =m 9 bo? cases4 ,?is c also be express$ z foll[s3 ,= ! physical description ( natural processes1 nei!r ( ! ref]5ce bodies ,k1 ,k is unique 7lit4 8specially m>k$ out07 z comp>$ ) ! o!r4 ,unlike ! fir/1 ?is latt] /atem5t ne$ n ( necessity hold a priori2 x is n conta9$ 9 ! conceptions ( 8motion0 & 8ref]5ce-body0 & d]ivable f !m2 only exp]i5ce c decide z to its correctness or incorrectness4 ,up to ! pres5t1 h[ev]1 we h by no means ma9ta9$ ! equival5ce ( all bodies ( ref]5ce ,k 9 connection ) ! =mulation ( natural laws4 ,our c\rse 0 m on ! foll[+ ,i9es4 ,9 ! fir/ place1 we />t$ out f ! assumption t !re exi/s a ref]5ce-body ,k1 ^: condition ( motion is su* t ! ,galileian law holds ) respect to x3 ,a p>ticle left to itself & suffici5tly f> remov$ f all o!r p>ticles moves uni=mly 9 a /raight l9e4 ,) ref]5ce to ,k 7,galileian ref]5ce-body7 ! laws ( nature 7 to be z simple z possible4 ,b 9 addition to ,k1 all bodies ( ref]5ce ,k %\ld be giv5 pref]5ce 9 ?is s5se1 & !y %\ld be exactly equival5t to ,k = ! =mulation ( natural laws1 provid$ t !y are 9 a /ate ( uni=m rectil9e> & non-rot>y motion ) respect to ,k2 all ^! bodies ( ref]5ce are to be reg>d$ z ,galileian ref]5ce-bodies4 ,! validity ( ! pr9ciple ( relativity 0 assum$ only = ^! ref]5ce-bodies1 b n = o!rs 7e4g4 ^? possess+ motion ( a diff]5t k9d74 ,9 ?is s5se we speak ( ! special pr9ciple ( relativity1 or special !ory ( relativity4 ,9 contra/ to ?is we wi% to und]/& by ! 8g5]al pr9ciple ( relativity0 ! foll[+ /atem5t3 ,all bodies ( ref]5ce ,k1 ,k1 etc41 are equival5t = ! description ( natural ph5om5a 7=mulation ( ! g5]al laws ( nature71 :atev] may be _! /ate ( motion4 ,b be=e proce$+ f>!r1 x ought to be po9t$ out t ?is =mulation mu/ be replac$ lat] by a m ab/ract "o1 = reasons :i* w become evid5t at a lat] /age4 ,s9ce ! introduction ( ! special pr9ciple ( relativity has be5 ju/ifi$1 e intellect :i* /rives aft] g5]alisation mu/ feel ! temptation to v5ture ! /ep t[>ds ! g5]al pr9ciple ( relativity4 ,b a simple & app>5tly q reliable consid]ation seems to sugge/ t1 = ! pres5t at any rate1 !re is little hope ( success 9 su* an attempt2 ,let u imag9e ourselves transf]r$ to our old fri5d ! railway c>riage1 :i* is travell+ at a uni=m rate4 ,z long z x is mov+ uni=mly1 ! occupant ( ! c>riage is n s5sible ( its motion1 & x is = ?is reason t he c )\t reluctance int]pret ! facts ( ! case z indicat+ t ! c>riage is at re/1 b ! embankm5t 9 motion4 ,moreov]1 accord+ to ! special pr9ciple ( relativity1 ?is int]pretation is q ju/ifi$ also f a physical po9t ( view4 ,if ! motion ( ! c>riage is n[ *ang$ into a non-uni=m motion1 z = in/ance by a p[]ful application ( ! brakes1 !n ! occupant ( ! c>riage exp]i5ces a correspond+ly p[]ful j]k =w>ds4 ,! ret>d$ motion is manife/$ 9 ! me*anical behavi\r ( bodies relative to ! p]son 9 ! railway c>riage4 ,! me*anical behavi\r is diff]5t f t ( ! case previ\sly consid]$1 & = ?is reason x w\ld appe> to be impossible t ! same me*anical laws hold relatively to ! non-uni=mly mov+ c>riage1 z hold ) ref]5ce to ! c>riage :5 at re/ or 9 uni=m motion4 ,at all ev5ts x is cle> t ! ,galileian law does n hold ) respect to ! non-uni=mly mov+ c>riage4 ,because ( ?is1 we feel compell$ at ! pres5t juncture to grant a k9d ( absolute physical reality to non-uni=m motion1 9 opposition to ! g5]al pr9ciple ( relativity4 ,b 9 :at foll[s we %all soon see t ?is conclusion _c be ma9ta9$4 ,,xix4 ,,! ,,gravitational ,,field 8,if we pick up a /one & !n let x g1 :y does x fall to ! gr\nd80 ,! usual answ] to ?is "q is3 8,because x is attract$ by ! e>?40 ,mod]n physics =mulates ! answ] r diff]5tly = ! foll[+ reason4 ,z a result ( ! m c>eful /udy ( electromagnetic ph5om5a1 we h come to reg>d action at a di/ance z a process impossible )\t ! int]v5tion ( "s int]m$i>y m$ium4 ,if1 = in/ance1 a magnet attracts a piece ( iron1 we _c be cont5t to reg>d ?is z mean+ t ! magnet acts directly on ! iron ?r\gh ! int]m$iate empty space1 b we are con/ra9$ to imag9e--aft] ! mann] ( ,f>aday--t ! magnet always calls into be+ some?+ physically real 9 ! space ar\nd x1 t some?+ be+ :at we call a 8magnetic field40 ,9 its turn ?is magnetic field op]ates on ! piece ( iron1 s t ! latt] /rives to move t[>ds ! magnet4 ,we %all n discuss "h ! ju/ification = ?is incid5tal conception1 :i* is inde$ a some:at arbitr>y "o4 ,we %all only m5tion t ) its aid electromagnetic ph5om5a c be !oretically repres5t$ mu* m satisfactorily ?an )\t x1 & ?is applies p>ticul>ly to ! transmission ( electromagnetic waves4 ,! effects ( gravitation also are reg>d$ 9 an analog\s mann]4 ,! action ( ! e>? on ! /one takes place indirectly4 ,! e>? produces 9 its surr\nd+ a gravitational field1 :i* acts on ! /one & produces its motion ( fall4 ,z we "k f exp]i5ce1 ! int5sity ( ! action on a body dimi%es accord+ to a q def9ite law1 z we proce$ f>!r & f>!r away f ! e>?4 ,f our po9t ( view ?is means3 ,! law gov]n+ ! prop]ties ( ! gravitational field 9 space mu/ be a p]fectly def9ite "o1 9 ord] correctly to repres5t ! dim9ution ( gravitational action ) ! di/ance f op]ative bodies4 ,x is some?+ l ?is3 ,! body 7e4g4 ! e>?7 produces a field 9 its imm$iate neighb\rhood directly2 ! int5sity & direction ( ! field at po9ts f>!r remov$ f ! body are !nce det]m9$ by ! law :i* gov]ns ! prop]ties 9 space ( ! gravitational fields !mselves4 ,9 contra/ to electric & magnetic fields1 ! gravitational field exhibits a mo/ rem>kable prop]ty1 :i* is ( fundam5tal importance = :at foll[s4 ,bodies :i* are mov+ "u ! sole influ5ce ( a gravitational field receive an accel]ation1 :i* does n 9 ! lea/ dep5d ei!r on ! mat]ial or on ! physical /ate ( ! body4 ,= in/ance1 a piece ( lead & a piece ( wood fall 9 exactly ! same mann] 9 a gravitational field 79 vacuo71 :5 !y />t (f f re/ or ) ! same initial velocity4 ,?is law1 :i* holds mo/ accurately1 c be express$ 9 a diff]5t =m 9 ! light ( ! foll[+ consid]ation4 ,accord+ to ,newton's law ( motion1 we h 7,=ce7 7in]tial mass7 x 7accel]ation71 :]e ! 8in]tial mass0 is a *>act]i/ic con/ant ( ! accel]at$ body4 ,if n[ gravitation is ! cause ( ! accel]ation1 we !n h 7,=ce7 7gravitational mass7 x 7int5sity ( ! gravitational field71 :]e ! 8gravitational mass0 is likewise a *>act]i/ic con/ant = ! body4 ,f ^! two relations foll[s3 image#jcb ,if n[1 z we f9d f exp]i5ce1 ! accel]ation is to be indep5d5t ( ! nature & ! condition ( ! body & always ! same = a giv5 gravitational field1 !n ! ratio ( ! gravitational to ! in]tial mass mu/ likewise be ! same = all bodies4 ,by a suitable *oice ( units we c ?us make ?is ratio equal to unity4 ,we !n h ! foll[+ law3 ,! gravitational mass ( a body is equal to its in]tial mass4 ,x is true t ?is important law _h hi!rto be5 record$ 9 me*anics1 b x _h n be5 int]pret$4 ,a satisfactory int]pretation c be obta9$ only if we recognise ! foll[+ fact3 ,! same quality ( a body manife/s itself accord+ to circum/ances z 8in]tia0 or z 8weight0 7lit4 8heav9ess074 ,9 ! foll[+ section we %all %[ to :at ext5t ?is is actually ! case1 & h[ ?is "q is connect$ ) ! g5]al po/ulate ( relativity4 ,,xx4 ,,! ,,equality ,,( ,,in]tial ,,& ,,gravitational ,,mass ,,z ,,an ,,argum5t ,,= ,,! ,,g5]al ,,po/ulate ,,( ,,relativity ,we imag9e a l>ge portion ( empty space1 s f> remov$ f />s & o!r appreciable masses1 t we h be=e u approximately ! conditions requir$ by ! fundam5tal law ( ,galilei4 ,x is !n possible to *oose a ,galileian ref]5ce-body = ?is "p ( space 7_w71 relative to :i* po9ts at re/ rema9 at re/ & po9ts 9 motion cont9ue p]man5tly 9 uni=m rectil9e> motion4 ,z ref]5ce-body let u imag9e a spaci\s *e/ resembl+ a room ) an obs]v] inside :o is equipp$ ) app>atus4 ,gravitation naturally does n exi/ = ?is obs]v]4 ,he mu/ fa/5 himself ) /r+s to ! floor1 o!rwise ! slighte/ impact aga9/ ! floor w cause him to rise sl[ly t[>ds ! ceil+ ( ! room4 ,to ! middle ( ! lid ( ! *e/ is fix$ ext]nally a hook ) rope atta*$1 & n[ a 8be+0 7:at k9d ( a be+ is immat]ial to u7 beg9s pull+ at ?is ) a con/ant =ce4 ,! *e/ toge!r ) ! obs]v] !n beg9 to move 8upw>ds0 ) a uni=mly accel]at$ motion4 ,9 c\rse ( "t _! velocity w rea* unhe>d-( values--provid$ t we are view+ all ?is f ano!r ref]5ce-body :i* is n be+ pull$ ) a rope4 ,b h[ does ! man 9 ! *e/ reg>d ! ,process8 ,! accel]ation ( ! *e/ w be transmitt$ to him by ! reaction ( ! floor ( ! *e/4 ,he mu/ !re=e take up ?is pressure by means ( 8 legs if he does n wi% to be laid out full l5g? on ! floor4 ,he is !n /&+ 9 ! *e/ 9 exactly ! same way z anyone /&s 9 a room ( a home on our e>?4 ,if he releases a body :i* he previ\sly _h 9 8 l&1 ! accel]tion ( ! *e/ w no long] be transmitt$ to ?is body1 & = ?is reason ! body w approa* ! floor ( ! *e/ ) an accel]at$ relative motion4 ,! obs]v] w fur!r conv9ce himself t ! accel]ation ( ! body t[>ds ! floor ( ! *e/ is always ( ! same magnitude1 :atev] k9d ( body he may happ5 to use = ! exp]im5t4 ,rely+ on 8 k ( ! gravitational field 7z x 0 discuss$ 9 ! prec$+ section71 ! man 9 ! *e/ w ?us come to ! conclusion t he & ! *e/ are 9 a gravitational field :i* is con/ant ) reg>d to "t4 ,( c\rse he w be puzzl$ = a mom5t z to :y ! *e/ does n fall 9 ?is gravitational field4 j !n1 h[ev]1 he discov]s ! hook 9 ! middle ( ! lid ( ! *e/ & ! rope :i* is atta*$ to x1 & he consequ5tly comes to ! conclusion t ! *e/ is susp5d$ at re/ 9 ! gravitational field4 ,ought we to smile at ! man & say t he errs 9 8 conclusion8 ,i d n believe we ought to if we wi% to rema9 consi/5t2 we mu/ r admit t 8 mode ( grasp+ ! situation violates nei!r reason nor kn[n me*anical laws4 ,ev5 ?\gh x is be+ accel]at$ ) respect to ! 8,galileian space0 fir/ consid]$1 we c nev]!less reg>d ! *e/ z be+ at re/4 ,we h ?us good gr\nds = ext5d+ ! pr9ciple ( relativity to include bodies ( ref]5ce :i* are accel]at$ ) respect to ea* o!r1 & z a result we h ga9$ a p[]ful argum5t = a g5]alis$ po/ulate ( relativity4 ,we mu/ note c>efully t ! possibility ( ?is mode ( int]pretation re/s on ! fundam5tal prop]ty ( ! gravitational field ( giv+ all bodies ! same accel]ation1 or1 :at comes to ! same ?+1 on ! law ( ! equality ( in]tial & gravitational mass4 ,if ?is natural law did n exi/1 ! man 9 ! accel]at$ *e/ w\ld n be able to int]pret ! behavi\r ( ! bodies ar\nd him on ! supposition ( a gravitational field1 & he w\ld n be ju/ifi$ on ! gr\nds ( exp]i5ce 9 suppos+ 8 ref]5ce-body to be 8at re/40 ,suppose t ! man 9 ! *e/ fixes a rope to ! inn] side ( ! lid1 & t he atta*es a body to ! free end ( ! rope4 ,! result ( ?is w be to /ret* ! rope s t x w hang 8v]tically0 d[nw>ds4 ,if we ask = an op9ion ( ! cause ( t5sion 9 ! rope1 ! man 9 ! *e/ w say3 8,! susp5d$ body exp]i5ces a d[nw>d =ce 9 ! gravitational field1 & ?is is neutralis$ by ! t5sion ( ! rope2 :at det]m9es ! magnitude ( ! t5sion ( ! rope is ! gravitational mass ( ! susp5d$ body40 ,on ! o!r h&1 an obs]v] :o is pois$ freely 9 space w int]pret ! condition ( ?+s ?us3 8,! rope mu/ p]=ce take "p 9 ! accel]at$ motion ( ! *e/1 & x transmits ?is motion to ! body atta*$ to x4 ,! t5sion ( ! rope is j l>ge en\gh to effect ! accel]ation ( ! body4 ,t :i* det]m9es ! magnitude ( ! t5sion ( ! rope is ! in]tial mass ( ! body40 ,guid$ by ?is example1 we see t our ext5sion ( ! pr9ciple ( relativity implies ! necessity ( ! law ( ! equality ( in]tial & gravitational mass4 ,?us we h obta9$ a physical int]pretation ( ?is law4 ,f our consid]ation ( ! accel]at$ *e/ we see t a g5]al !ory ( relativity mu/ yield important results on ! laws ( gravitation4 ,9 po9t ( fact1 ! sy/ematic pursuit ( ! g5]al idea ( relativity has suppli$ ! laws satisfi$ by ! gravitational field4 ,be=e proce$+ f>!r1 h[ev]1 ,i mu/ w>n ! read] aga9/ a misconception sugge/$ by ^! consid]ations4 ,a gravitational field exi/s = ! man 9 ! *e/1 despite ! fact t !re 0 no su* field = ! co-ord9ate sy/em fir/ *os54 ,n[ we might easily suppose t ! exi/5ce ( a gravitational field is always only an app>5t "o4 ,we might also ?9k t1 reg>dless ( ! k9d ( gravitational field :i* may be pres5t1 we c\ld always *oose ano!r ref]5ce-body su* t no gravitational field exi/s ) ref]5ce to x4 ,?is is by no means true = all gravitational fields1 b only = ^? ( q special =m4 ,x is1 = in/ance1 impossible to *oose a body ( ref]5ce su* t1 z judg$ f x1 ! gravitational field ( ! e>? 79 its entirety7 vani%es4 ,we c n[ appreciate :y t argum5t is n conv9c+1 :i* we br\ght =w>d aga9/ ! g5]al pr9ciple ( relativity at ! end ( ,section ,,xviii4 ,x is c]ta9ly true t ! obs]v] 9 ! railway c>riage exp]i5ces a j]k =w>ds z a result ( ! application ( ! brake1 & t he recognises1 9 ?is ! non-uni=mity ( motion 7ret>dation7 ( ! c>riage4 ,b he is compell$ by nobody to ref] ?is j]k to a 8real0 accel]ation 7ret>dation7 ( ! c>riage4 ,he might also int]pret 8 exp]i5ce ?us3 8,my body ( ref]5ce 7! c>riage7 rema9s p]man5tly at re/4 ,) ref]5ce to x1 h[ev]1 !re exi/s 7dur+ ! p]iod ( application ( ! brakes7 a gravitational field :i* is direct$ =w>ds & :i* is v>iable ) respect to "t4 ,"u ! influ5ce ( ?is field1 ! embankm5t toge!r ) ! e>? moves non-uni=mly 9 su* a mann] t _! orig9al velocity 9 ! backw>ds direction is cont9u\sly r$uc$40 ,,xxi4 ,,9 ,,:at ,,respects ,,are ,,! ,,f\ndations ,,( ,,classical ,,me*anics ,,& ,,( ,,! ,,special ,,!ory ,,( ,,relativity ,,unsatisfactory8 ,we h already /at$ sev]al times t classical me*anics />ts out f ! foll[+ law3 ,mat]ial p>ticles suffici5tly f> remov$ f o!r mat]ial p>ticles cont9ue to move uni=mly 9 a /raight l9e or cont9ue 9 a /ate ( re/4 ,we h also repeat$ly emphasis$ t ?is fundam5tal law c only be valid = bodies ( ref]5ce ,k :i* possess c]ta9 unique /ates ( motion1 & :i* are 9 uni=m translational motion relative to ea* o!r4 ,relative to o!r ref]5ce-bodies ,k ! law is n valid4 ,bo? 9 classical me*anics & 9 ! special !ory ( relativity we !re=e diff]5tiate betwe5 ref]5ce-bodies ,k relative to :i* ! recognis$ 8laws ( nature0 c be said to hold1 & ref]5ce-bodies ,k relative to :i* ^! laws d n hold4 ,b no p]son ^: mode ( ?\ght is logical c re/ satisfi$ ) ?is condition ( ?+s4 ,he asks3 8,h[ does x come t c]ta9 ref]5ce-bodies 7or _! /ates ( motion7 are giv5 priority ov] o!r ref]5ce-bodies 7or _! /ates ( motion78 ,:at is ! reason = ?is pref]5ce80 ,9 ord] to %[ cle>ly :at ,i mean by ?is "q1 ,i %all make use ( a comp>ison4 ,i am /&+ 9 front ( a gas range4 ,/&+ alongside ( ea* o!r on ! range are two pans s mu* alike t "o may be mi/ak5 = ! o!r4 ,bo? are half full ( wat]4 ,i notice t /eam is be+ emitt$ cont9u\sly f ! "o pan1 b n f ! o!r4 ,i am surpris$ at ?is1 ev5 if ,i h nev] se5 ei!r a gas range or a pan be=e4 ,b if ,i n[ notice a lum9\s some?+ ( blui% col\r "u ! fir/ pan b n "u ! o!r1 ,i cease to be a/oni%$1 ev5 if ,i h nev] be=e se5 a gas flame4 ,= ,i c only say t ?is blui% some?+ w cause ! emission ( ! /eam1 or at lea/ possibly x may d s4 ,if1 h[ev]1 ,i notice ! blui% some?+ 9 nei!r case1 & if ,i obs]ve t ! "o cont9u\sly emits /eam :il/ ! o!r does n1 !n ,i %all rema9 a/oni%$ & dissatisfi$ until ,i h discov]$ "s circum/ance to :i* ,i c attribute ! diff]5t behavi\r ( ! two pans4 ,analog\sly1 ,i seek 9 va9 = a real some?+ 9 classical me*anics 7or 9 ! special !ory ( relativity7 to :i* ,i c attribute ! diff]5t behavi\r ( bodies consid]$ ) respect to ! ref]5ce sy/ems ,k & ,k4#ae ,newton saw ?is objection & attempt$ to invalidate x1 b )\t success4 ,b ,e4 ,ma* recognis$ x mo/ cle>ly ( all1 & because ( ?is objection he claim$ t me*anics mu/ be plac$ on a new basis4 ,x c only be got rid ( by means ( a physics :i* is con=mable to ! g5]al pr9ciple ( relativity1 s9ce ! equations ( su* a !ory hold = e body ( ref]5ce1 :atev] may be its /ate ( motion4 #ae ,! objection is ( importance m especially :5 ! /ate ( motion ( ! ref]5ce-body is ( su* a nature t x does n require any ext]nal ag5cy = its ma9t5ance1 e4g4 9 ! case :5 ! ref]5ce-body is rotat+ uni=mly4 ,,xxii4 ,a ,,few ,,inf]5ces ,,f ,,! ,,g5]al ,,pr9ciple ,,( ,,relativity ,! consid]ations ( ,section ,,xx %[ t ! g5]al pr9ciple ( relativity puts u 9 a position to d]ive prop]ties ( ! gravitational field 9 a purely !oretical mann]4 ,let u suppose1 = in/ance1 t we "k ! space-"t 8c\rse0 = any natural process :atsoev]1 z reg>ds ! mann] 9 :i* x takes place 9 ! ,galileian doma9 relative to a ,galileian body ( ref]5ce ,k4 ,by means ( purely !oretical op]ations 7i4e4 simply by calculation7 we are !n able to f9d h[ ?is kn[n natural process appe>s1 z se5 f a ref]5ce-body ,k :i* is accel]at$ relatively to ,k4 ,b s9ce a gravitational field exi/s ) respect to ?is new body ( ref]5ce ,k1 our consid]ation also tea*es u h[ ! gravitational field influ5ces ! process /udi$4 ,= example1 we le>n t a body :i* is 9 a /ate ( uni=m rectil9e> motion ) respect to ,k 79 accordance ) ! law ( ,galilei7 is execut+ an accel]at$ & 9 g5]al curvil9e> motion ) respect to ! accel]at$ ref]5ce-body ,k 7*e/74 ,?is accel]ation or curvature corresponds to ! influ5ce on ! mov+ body ( ! gravitational field prevail+ relatively to ,k4 ,x is kn[n t a gravitational field influ5ces ! movem5t ( bodies 9 ?is way1 s t our consid]ation supplies u ) no?+ ess5tially new4 ,h[ev]1 we obta9 a new result ( fundam5tal importance :5 we c>ry out ! analog\s consid]ation = a ray ( light4 ,) respect to ! ,galileian ref]5ce-body ,k1 su* a ray ( light is transmitt$ rectil9e>ly ) ! velocity c4 ,x c easily be %[n t ! pa? ( ! same ray ( light is no long] a /raight l9e :5 we consid] x ) ref]5ce to ! accel]at$ *e/ 7ref]5ce-body ,k74 ,f ?is we conclude1 t1 9 g5]al1 rays ( light are propagat$ curvil9e>ly 9 gravitational fields4 ,9 two respects ?is result is ( great importance4 ,9 ! fir/ place1 x c be comp>$ ) ! reality4 ,al?\gh a detail$ exam9ation ( ! "q %[s t ! curvature ( light rays requir$ by ! g5]al !ory ( relativity is only exce$+ly small = ! gravitational fields at our disposal 9 practice1 its e/imat$ magnitude = light rays pass+ ! sun at graz+ incid5ce is nev]!less #a4g seconds ( arc4 ,?is ought to manife/ itself 9 ! foll[+ way4 ,z se5 f ! e>?1 c]ta9 fix$ />s appe> to be 9 ! neighb\rhood ( ! sun1 & are ?us capable ( obs]vation dur+ a total eclipse ( ! sun4 ,at su* times1 ^! />s ought to appe> to be displac$ outw>ds f ! sun by an am\nt indicat$ above1 z comp>$ ) _! app>5t position 9 ! sky :5 ! sun is situat$ at ano!r "p ( ! heav5s4 ,! exam9ation ( ! correctness or o!rwise ( ?is d$uction is a problem ( ! greate/ importance1 ! e>ly solution ( :i* is to be expect$ ( a/ronom]s4#af #af ,by means ( ! /> photographs ( two exp$itions equipp$ by a ,jo9t ,committee ( ! ,royal & ,royal ,a/ronomical ,societies1 ! exi/5ce ( ! deflection ( light dem&$ by !ory 0 fir/ confirm$ dur+ ! sol> eclipse ( #bi? ,may1 #aiai4 7,cf4 ,app5dix ,,iii47 ,9 ! second place our result %[s t1 accord+ to ! g5]al !ory ( relativity1 ! law ( ! con/ancy ( ! velocity ( light 9 vacuo1 :i* con/itutes "o ( ! two fundam5tal assumptions 9 ! special !ory ( relativity & to :i* we h already frequ5tly ref]r$1 _c claim any unlimit$ validity4 ,a curvature ( rays ( light c only take place :5 ! velocity ( propagation ( light v>ies ) position4 ,n[ we might ?9k t z a consequ5ce ( ?is1 ! special !ory ( relativity & ) x ! :ole !ory ( relativity w\ld be laid 9 ! du/4 ,b 9 reality ?is is n ! case4 ,we c only conclude t ! special !ory ( relativity _c claim an unlimit$ doma9 ( validity2 its results hold only s long z we are able to disreg>d ! influ5ces ( gravitational fields on ! ph5om5a 7e4g4 ( light74 ,s9ce x has (t5 be5 cont5d$ by oppon5ts ( ! !ory ( relativity t ! special !ory ( relativity is ov]?r[n by ! g5]al !ory ( relativity1 x is p]haps advisable to make ! facts ( ! case cle>] by means ( an appropriate comp>ison4 ,be=e ! developm5t ( electrodynamics ! laws ( electro/atics 7 look$ ^u z ! laws ( electricity4 ,at ! pres5t "t we "k t electric fields c be d]iv$ correctly f electro/atic consid]ations only = ! case1 :i* is nev] /rictly realis$1 9 :i* ! electrical masses are q at re/ relatively to ea* o!r1 & to ! co-ord9ate sy/em4 ,%\ld we be ju/ifi$ 9 say+ t = ?is reason electro/atics is ov]?r[n by ! field-equations ( ,maxwell 9 electrodynamics8 ,n 9 ! lea/4 ,electro/atics is conta9$ 9 electrodynamics z a limit+ case2 ! laws ( ! latt] lead directly to ^? ( ! =m] = ! case 9 :i* ! fields are inv>iable ) reg>d to "t4 ,no fair] de/9y c\ld be allott$ to any physical !ory1 ?an t x %\ld ( itself po9t out ! way to ! introduction ( a m compreh5sive !ory1 9 :i* x lives on z a limit+ case4 ,9 ! example ( ! transmission ( light j dealt )1 we h se5 t ! g5]al !ory ( relativity enables u to d]ive !oretically ! influ5ce ( a gravitational field on ! c\rse ( natural processes1 ! laws ( :i* are already kn[n :5 a gravitational field is abs5t4 ,b ! mo/ attractive problem1 to ! solution ( :i* ! g5]al !ory ( relativity supplies ! key1 conc]ns ! inve/igation ( ! laws satisfi$ by ! gravitational field itself4 ,let u consid] ?is = a mom5t4 ,we are acqua9t$ ) space-"t doma9s :i* behave 7approximately7 9 a 8,galileian0 fa%ion "u suitable *oice ( ref]5ce-body1 i4e4 doma9s 9 :i* gravitational fields are abs5t4 ,if we n[ ref] su* a doma9 to a ref]5ce-body ,k possess+ any k9d ( motion1 !n relative to ,k !re exi/s a gravitational field :i* is v>iable ) respect to space & "t4#ag ,! "* ( ?is field w ( c\rse dep5d on ! motion *os5 = ,k4 ,accord+ to ! g5]al !ory ( relativity1 ! g5]al law ( ! gravitational field mu/ be satisfi$ = all gravitational fields obta9able 9 ?is way4 ,ev5 ?\gh by no means all gravitationial fields c be produc$ 9 ?is way1 yet we may ent]ta9 ! hope t ! g5]al law ( gravitation w be d]ivable f su* gravitational fields ( a special k9d4 ,?is hope has be5 realis$ 9 ! mo/ beautiful mann]4 ,b betwe5 ! cle> vision ( ?is goal & its actual realisation x 0 necess>y to surm\nt a s]i\s difficulty1 & z ?is lies deep at ! root ( ?+s1 ,i d>e n )hold x f ! read]4 ,we require to ext5d our ideas ( ! space-"t cont9uum /ill f>!r4 #ag ,?is foll[s f a g5]alisation ( ! discussion 9 ,section ,,xx4 ,,xxiii4 ,,behavi\r ,,( ,,clocks ,,& ,,measur+-,,rods ,,on ,a ,,rotat+ ,,body ,,( ,,ref]5ce ,hi!rto ,i h purposely refra9$ f speak+ ab\t ! physical int]pretation ( space- & "t-data 9 ! case ( ! g5]al !ory ( relativity4 ,z a consequ5ce1 ,i am guilty ( a c]ta9 slov5l9ess ( treatm5t1 :i*1 z we "k f ! special !ory ( relativity1 is f> f be+ unimportant & p>donable4 ,x is n[ high "t t we rem$y ?is defect2 b ,i w\ld m5tion at ! outset1 t ?is matt] lays no small claims on ! pati5ce & on ! p[] ( ab/raction ( ! read]4 ,we />t (f aga9 f q special cases1 :i* we h frequ5tly us$ be=e4 ,let u consid] a space "t doma9 9 :i* no gravitational field exi/s relative to a ref]5ce-body ,k ^: /ate ( motion has be5 suitably *os54 ,k is !n a ,galileian ref]5ce-body z reg>ds ! doma9 consid]$1 & ! results ( ! special !ory ( relativity hold relative to ,k4 ,let u suppose ! same doma9 ref]r$ to a second body ( ref]5ce ,k1 :i* is rotat+ uni=mly ) respect to ,k4 ,9 ord] to fix our ideas1 we %all imag9e ,k to be 9 ! =m ( a plane circul> disc1 :i* rotates uni=mly 9 its own plane ab\t its c5tre4 ,an obs]v] :o is sitt+ ecc5trically on ! disc ,k is s5sible ( a =ce :i* acts outw>ds 9 a radial direction1 & :i* w\ld be int]pret$ z an effect ( in]tia 7c5trifugal =ce7 by an obs]v] :o 0 at re/ ) respect to ! orig9al ref]5ce-body ,k4 ,b ! obs]v] on ! disc may reg>d 8 disc z a ref]5ce-body :i* is 8at re/02 on ! basis ( ! g5]al pr9ciple ( relativity he is ju/ifi$ 9 do+ ?is4 ,! =ce act+ on himself1 & 9 fact on all o!r bodies :i* are at re/ relative to ! disc1 he reg>ds z ! effect ( a gravitational field4 ,nev]!less1 ! space-di/ribution ( ?is gravitational field is ( a k9d t w\ld n be possible on ,newton's !ory ( gravitation4#ah ,b s9ce ! obs]v] believes 9 ! g5]al !ory ( relativity1 ?is does n di/urb him2 he is q 9 ! "r :5 he believes t a g5]al law ( gravitation c be =mulat$--a law :i* n only expla9s ! motion ( ! />s correctly1 b also ! field ( =ce exp]i5c$ by himself4 #ah ,! field disappe>s at ! c5tre ( ! disc & increases proportionally to ! di/ance f ! c5tre z we proce$ outw>ds4 ,! obs]v] p]=ms exp]im5ts on 8 circul> disc ) clocks & measur+-rods4 ,9 do+ s1 x is 8 int5tion to arrive at exact def9itions = ! signification ( "t- & space-data ) ref]5ce to ! circul> disc ,k1 ^! def9itions be+ bas$ on 8 obs]vations4 ,:at w be 8 exp]i5ce 9 ?is ent]prise8 ,to />t )1 he places "o ( two id5tically con/ruct$ clocks at ! c5tre ( ! circul> disc1 & ! o!r on ! edge ( ! disc1 s t !y are at re/ relative to x4 ,we n[ ask ourselves :e!r bo? clocks g at ! same rate f ! /&po9t ( ! non-rotat+ ,galileian ref]5ce-body ,k4 ,z judg$ f ?is body1 ! clock at ! c5tre ( ! disc has no velocity1 :]eas ! clock at ! edge ( ! disc is 9 motion relative to ,k 9 consequ5ce ( ! rotation4 ,accord+ to a result obta9$ 9 ,section ,,xii1 x foll[s t ! latt] clock goes at a rate p]man5tly sl[] ?an t ( ! clock at ! c5tre ( ! circul> disc1 i4e4 z obs]v$ f ,k4 ,x is obvi\s t ! same effect w\ld be not$ by an obs]v] :om we w imag9e sitt+ alongside 8 clock at ! c5tre ( ! circul> disc4 ,?us on our circul> disc1 or1 to make ! case m g5]al1 9 e gravitational field1 a clock w g m quickly or less quickly1 accord+ to ! position 9 :i* ! clock is situat$ 7at re/74 ,= ?is reason x is n possible to obta9 a reasonable def9ition ( "t ) ! aid ( clocks :i* are arrang$ at re/ ) respect to ! body ( ref]5ce4 ,a simil> difficulty pres5ts itself :5 we attempt to apply our e>li] def9ition ( simultaneity 9 su* a case1 b ,i d n wi% to g any f>!r into ?is "q4 ,moreov]1 at ?is /age ! def9ition ( ! space co-ord9ates also pres5ts insurm\ntable difficulties4 ,if ! obs]v] applies 8 /&>d measur+-rod 7a rod :i* is %ort z comp>$ ) ! radius ( ! disc7 tang5tially to ! edge ( ! disc1 !n1 z judg$ f ! ,galileian sy/em1 ! l5g? ( ?is rod w be less ?an #a1 s9ce1 accord+ to ,section ,,xii1 mov+ bodies suff] a %ort5+ 9 ! direction ( ! motion4 ,on ! o!r h&1 ! measur+-rod w n exp]i5ce a %ort5+ 9 l5g?1 z judg$ f ,k1 if x is appli$ to ! disc 9 ! direction ( ! radius4 ,if1 !n1 ! obs]v] fir/ measures ! circumf]5ce ( ! disc ) 8 measur+-rod & !n ! diamet] ( ! disc1 on divid+ ! "o by ! o!r1 he w n obta9 z quoti5t ! famili> numb] #c4ad 4 4 41 b a l>g] numb]1#ai :]eas ( c\rse1 = a disc :i* is at re/ ) respect to ,k1 ?is op]ation w\ld yield exactly4 ,?is proves t ! propositions ( ,euclidean geometry _c hold exactly on ! rotat+ disc1 nor 9 g5]al 9 a gravitational field1 at lea/ if we attribute ! l5g? #a to ! rod 9 all positions & 9 e ori5tation4 ,h5ce ! idea ( a /raight l9e also loses its mean+4 ,we are !re=e n 9 a position to def9e exactly ! co-ord9ates x1 y1 z relative to ! disc by means ( ! me?od us$ 9 discuss+ ! special !ory1 & z long z ! co-ord9ates & times ( ev5ts h n be5 def9$1 we _c assign an exact mean+ to ! natural laws 9 :i* ^! occur4 #ai ,?r\gh\t ?is consid]ation we h to use ! ,galileian 7non-rotat+7 sy/em ,k z ref]5ce-body1 s9ce we may only assume ! validity ( ! results ( ! special !ory ( relativity relative to ,k 7relative to ,k a gravitational field prevails74 ,?us all our previ\s conclusions bas$ on g5]al relativity w\ld appe> to be call$ 9 "q4 ,9 reality we mu/ make a subtle det\r 9 ord] to be able to apply ! po/ulate ( g5]al relativity exactly4 ,i %all prep>e ! read] = ?is 9 ! foll[+ p>agraphs4 ,,xxiv4 ,,euclidean ,,& ,,non-,,euclidean ,,cont9uum ,! surface ( a m>ble table is spread out 9 front ( me4 ,i c get f any "o po9t on ?is table to any o!r po9t by pass+ cont9u\sly f "o po9t to a 8neighb\r+0 "o1 & repeat+ ?is process a 7l>ge7 numb] ( times1 or1 9 o!r words1 by go+ f po9t to po9t )\t execut+ 8jumps40 ,i am sure ! read] w appreciate ) suffici5t cle>ness :at ,i mean "h by 8neighb\r+0 & by 8jumps0 7if he is n too p$antic74 ,we express ?is prop]ty ( ! surface by describ+ ! latt] z a cont9uum4 ,let u n[ imag9e t a l>ge numb] ( little rods ( equal l5g? h be5 made1 _! l5g?s be+ small comp>$ ) ! dim5sions ( ! m>ble slab4 ,:5 ,i say !y are ( equal l5g?1 ,i mean t "o c be laid on any o!r )\t ! ends ov]lapp+4 ,we next lay f\r ( ^! little rods on ! m>ble slab s t !y con/itute a quadrilat]al figure 7a squ>e71 ! diagonals ( :i* are equally long4 ,to ensure ! equality ( ! diagonals1 we make use ( a little te/+-rod4 ,to ?is squ>e we add simil> ones1 ea* ( :i* has "o rod 9 common ) ! fir/4 ,we proce$ 9 l mann] ) ea* ( ^! squ>es until f9ally ! :ole m>ble slab is laid out ) squ>es4 ,! arrangem5t is su*1 t ea* side ( a squ>e belongs to two squ>es & ea* corn] to f\r squ>es4 ,x is a v]itable wond] t we c c>ry out ?is bus9ess )\t gett+ into ! greate/ difficulties4 ,we only ne$ to ?9k ( ! foll[+4 ,if at any mom5t ?ree squ>es meet at a corn]1 !n two sides ( ! f\r? squ>e are already laid1 &1 z a consequ5ce1 ! arrangem5t ( ! rema9+ two sides ( ! squ>e is already completely det]m9$4 ,b ,i am n[ no long] able to adju/ ! quadrilat]al s t its diagonals may be equal4 ,if !y are equal ( _! own accord1 !n ?is is an especial fav\r ( ! m>ble slab & ( ! little rods1 ab\t :i* ,i c only be ?ankfully surpris$4 ,we mu/ exp]i5ce _m su* surprises if ! con/ruction is to be successful4 ,if ev]y?+ has really gone smoo?ly1 !n ,i say t ! po9ts ( ! m>ble slab con/itute a ,euclidean cont9uum ) respect to ! little rod1 :i* has be5 us$ z a 8di/ance0 7l9e-int]val74 ,by *oos+ "o corn] ( a squ>e z 8orig90 ,i c *>act]ise e o!r corn] ( a squ>e ) ref]5ce to ?is orig9 by means ( two numb]s4 ,i only ne$ /ate h[ _m rods ,i mu/ pass ov] :51 />t+ f ! orig91 ,i proce$ t[>ds ! 8"r0 & !n 8upw>ds10 9 ord] to arrive at ! corn] ( ! squ>e "u consid]ation4 ,^! two numb]s are !n ! 8,c>tesian co-ord9ates0 ( ?is corn] ) ref]5ce to ! 8,c>tesian co-ord9ate sy/em0 :i* is det]m9$ by ! arrangem5t ( little rods4 ,by mak+ use ( ! foll[+ modification ( ?is ab/ract exp]im5t1 we recognise t !re mu/ also be cases 9 :i* ! exp]im5t w\ld be unsuccessful4 ,we %all suppose t ! rods 8exp&0 by 9 am\nt proportional to ! increase ( temp]ature4 ,we heat ! c5tral "p ( ! m>ble slab1 b n ! p]iph]y1 9 :i* case two ( our little rods c /ill be br\ght into co9cid5ce at e position on ! table4 ,b our con/ruction ( squ>es mu/ necess>ily come into disord] dur+ ! heat+1 because ! little rods on ! c5tral region ( ! table exp&1 :]eas ^? on ! out] "p d n4 ,) ref]5ce to our little rods--def9$ z unit l5g?s--! m>ble slab is no long] a ,euclidean cont9uum1 & we are also no long] 9 ! position ( def9+ ,c>tesian co-ord9ates directly ) _! aid1 s9ce ! above con/ruction c no long] be c>ri$ out4 ,b s9ce !re are o!r ?+s :i* are n influ5c$ 9 a simil> mann] to ! little rods 7or p]haps n at all7 by ! temp]ature ( ! table1 x is possible q naturally to ma9ta9 ! po9t ( view t ! m>ble slab is a 8,euclidean cont9uum40 ,?is c be done 9 a satisfactory mann] by mak+ a m subtle /ipulation ab\t ! measurem5t or ! comp>ison ( l5g?s4 ,b if rods ( e k9d 7i4e4 ( e mat]ial7 7 to behave 9 ! same way z reg>ds ! influ5ce ( temp]ature :5 !y are on ! v>iably heat$ m>ble slab1 & if we _h no o!r means ( detect+ ! effect ( temp]ature ?an ! geometrical behavi\r ( our rods 9 exp]im5ts analog\s to ! "o describ$ above1 !n our be/ plan w\ld be to assign ! di/ance "o to two po9ts on ! slab1 provid$ t ! ends ( "o ( our rods c\ld be made to co9cide ) ^! two po9ts2 = h[ else %\ld we def9e ! di/ance )\t our proce$+ be+ 9 ! highe/ measure grossly arbitr>y8 ,! me?od ( ,c>tesian coord9ates mu/ !n be disc>d$1 & replac$ by ano!r :i* does n assume ! validity ( ,euclidean geometry = rigid bodies4#bj ,! read] w notice t ! situation depict$ "h corresponds to ! "o br\ght ab\t by ! g5]al po/ulate ( relativity 7,section ,,xxiii74 #bj ,ma!maticians h be5 confront$ ) our problem 9 ! foll[+ =m4 ,if we are giv5 a surface 7e4g4 an ellipsoid7 9 ,euclidean ?ree-dim5sional space1 !n !re exi/s = ?is surface a two-dim5sional geometry1 j z mu* z = a plane surface4 ,gauss und]took ! task ( treat+ ?is two-dim5sional geometry f fir/ pr9ciples1 )\t mak+ use ( ! fact t ! surface belongs to a ,euclidean cont9uum ( ?ree dim5sions4 ,if we imag9e con/ructions to be made ) rigid rods 9 ! surface 7simil> to t above ) ! m>ble slab71 we %\ld f9d t diff]5t laws hold = ^! f ^? result+ on ! basis ( ,euclidean plane geometry4 ,! surface is n a ,euclidean cont9uum ) respect to ! rods1 & we _c def9e ,c>tesian co-ord9ates 9 ! surface4 ,gauss indicat$ ! pr9ciples accord+ to :i* we c treat ! geometrical relation%ips 9 ! surface1 & ?us po9t$ out ! way to ! me?od ( ,riemann ( treat+ multi-dim5sional1 non-,euclidean cont9uum4 ,?us x is t ma!maticians long ago solv$ ! =mal problems to :i* we are l$ by ! g5]al po/ulate ( relativity4 ,,xxv4 ,,gaussian ,,co-,,ord9ates image#jcc ,accord+ to ,gauss1 ?is comb9$ analytical & geometrical mode ( h&l+ ! problem c be arriv$ at 9 ! foll[+ way4 ,we imag9e a sy/em ( arbitr>y curves 7see ,fig4 #d7 drawn on ! surface ( ! table4 ,^! we designate z u-curves1 & we indicate ea* ( !m by means ( a numb]4 ,! ,curves u #a1 u #b & u #c are drawn 9 ! diagram4 ,betwe5 ! curves u #a & u #b we mu/ imag9e an inf9itely l>ge numb] to be drawn1 all ( :i* correspond to real numb]s ly+ betwe5 #a & #b4 ,we h !n a sy/em ( u-curves1 & ?is 8inf9itely d5se0 sy/em cov]s ! :ole surface ( ! table4 ,^! u-curves mu/ n int]sect ea* o!r1 & ?r\gh ea* po9t ( ! surface "o & only "o curve mu/ pass4 ,?us a p]fectly def9ite value ( u belongs to e po9t on ! surface ( ! m>ble slab4 ,9 l mann] we imag9e a sy/em ( v-curves drawn on ! surface4 ,^! satisfy ! same conditions z ! u-curves1 !y are provid$ ) numb]s 9 a correspond+ mann]1 & !y may likewise be ( arbitr>y %ape4 ,x foll[s t a value ( u & a value ( v belong to e po9t on ! surface ( ! table4 ,we call ^! two numb]s ! co-ord9ates ( ! surface ( ! table 7,gaussian co-ord9ates74 ,= example1 ! po9t ,p 9 ! diagram has ! ,gaussian co-ord9ates u #c1 v #a4 ,two neighb\r+ po9ts ,p & ,p on ! surface !n correspond to ! co-ord9ates ,p3 u1 v ,p3 u du1 v dv1 :]e du & dv signify v small numb]s4 ,9 a simil> mann] we may indicate ! di/ance 7l9e-int]val7 betwe5 ,p & ,p1 z measur$ ) a little rod1 by means ( ! v small numb] ds4 ,!n accord+ to ,gauss we h ds#b g#aa;du#b #b;g#ab;du dv g#bb;dv#b1 :]e g#aa1 g#ab1 g#bb1 are magnitudes :i* dep5d 9 a p]fectly def9ite way on u & v4 ,! magnitudes g#aa1 g#ab & g#bb1 det]m9e ! behavi\r ( ! rods relative to ! u-curves & v-curves1 & ?us also relative to ! surface ( ! table4 ,= ! case 9 :i* ! po9ts ( ! surface consid]$ =m a ,euclidean cont9uum ) ref]5ce to ! measur+-rods1 b only 9 ?is case1 x is possible to draw ! u-curves & v-curves & to atta* numb]s to !m1 9 su* a mann]1 t we simply h3 ds#b du#b dv#b ,"u ^! conditions1 ! u-curves & v-curves are /raight l9es 9 ! s5se ( ,euclidean geometry1 & !y are p]p5dicul> to ea* o!r4 ,"h ! ,gaussian coord9ates are simply ,c>tesian ones4 ,x is cle> t ,gauss co-ord9ates are no?+ m ?an an association ( two sets ( numb]s ) ! po9ts ( ! surface consid]$1 ( su* a nature t num]ical values diff]+ v slightly f ea* o!r are associat$ ) neighb\r+ po9ts 89 space40 ,s f>1 ^! consid]ations hold = a cont9uum ( two dim5sions4 ,b ! ,gaussian me?od c be appli$ also to a cont9uum ( ?ree1 f\r or m dim5sions4 ,if1 = in/ance1 a cont9uum ( f\r dim5sions be suppos$ available1 we may repres5t x 9 ! foll[+ way4 ,) e po9t ( ! cont9uum1 we associate arbitr>ily f\r numb]s1 x#a1 x#b1 x#c1 x#d1 :i* are kn[n z 8co-ord9ates40 ,adjac5t po9ts correspond to adjac5t values ( ! coord9ates4 ,if a di/ance ds is associat$ ) ! adjac5t po9ts ,p & ,p1 ?is di/ance be+ measurable & well def9$ f a physical po9t ( view1 !n ! foll[+ =mula holds3 ds#b g#aa;dx#ab #b;g#ab;dx#a;dx#b 4 4 4 4 g#dd;dx#db1 :]e ! magnitudes g#aa1 etc41 h values :i* v>y ) ! position 9 ! cont9uum4 ,only :5 ! cont9uum is a ,euclidean "o is x possible to associate ! co-ord9ates x#a 4 4 x#d4 ) ! po9ts ( ! cont9uum s t we h simply ds#b dx#ab dx#bb dx#cb dx#db4 ,9 ?is case relations hold 9 ! f\r-dim5sional cont9uum :i* are analog\s to ^? hold+ 9 our ?ree-dim5sional measurem5ts4 ,h[ev]1 ! ,gauss treatm5t = ds#b :i* we h giv5 above is n always possible4 ,x is only possible :5 suffici5tly small regions ( ! cont9uum "u consid]ation may be reg>d$ z ,euclidean cont9ua4 ,= example1 ?is obvi\sly holds 9 ! case ( ! m>ble slab ( ! table & local v>iation ( temp]ature4 ,! temp]ature is practically con/ant = a small "p ( ! slab1 & ?us ! geometrical behavi\r ( ! rods is almo/ z x ought to be accord+ to ! rules ( ,euclidean geometry4 ,h5ce ! imp]fections ( ! con/ruction ( squ>es 9 ! previ\s section d n %[ !mselves cle>ly until ?is con/ruction is ext5d$ ov] a consid]able portion ( ! surface ( ! table4 ,we c sum ?is up z foll[s3 ,gauss inv5t$ a me?od = ! ma!matical treatm5t ( cont9ua 9 g5]al1 9 :i* 8size-relations0 78di/ances0 betwe5 neighb\r+ po9ts7 are def9$4 ,to e po9t ( a cont9uum are assign$ z _m numb]s 7,gaussian coord9ates7 z ! cont9uum has dim5sions4 ,?is is done 9 su* a way1 t only "o mean+ c be atta*$ to ! assignm5t1 & t numb]s 7,gaussian coord9ates7 :i* diff] by an indef9itely small am\nt are assign$ to adjac5t po9ts4 ,! ,gaussian coord9ate sy/em is a logical g5]alisation ( ! ,c>tesian co-ord9ate sy/em4 ,x is also applicable to non-,euclidean cont9ua1 b only :51 ) respect to ! def9$ 8size0 or 8di/ance10 small p>ts ( ! cont9uum "u consid]ation behave m ne>ly l a ,euclidean sy/em1 ! small] ! "p ( ! cont9uum "u our notice4 ,,xxvi4 ,,! ,,space-,,"t ,,cont9uum ,,( ,,! ,,special ,,!ory ,,( ,,relativity ,,consid]$ ,,z ,a ,,euclidean ,,cont9uum ,we are n[ 9 a position to =mulate m exactly ! idea ( ,m9k[ski1 :i* 0 only vaguely indicat$ 9 ,section ,,xvii4 ,9 accordance ) ! special !ory ( relativity1 c]ta9 co-ord9ate sy/ems are giv5 pref]5ce = ! description ( ! f\r-dim5sional1 space-"t cont9uum4 ,we call$ ^! 8,galileian co-ord9ate sy/ems40 ,= ^! sy/ems1 ! f\r co-ord9ates x1 y1 z1 t1 :i* det]m9e an ev5t or--9 o!r words--a po9t ( ! f\r-dim5sional cont9uum1 are def9$ physically 9 a simple mann]1 z set =? 9 detail 9 ! fir/ "p ( ?is book4 ,= ! transition f "o ,galileian sy/em to ano!r1 :i* is mov+ uni=mly ) ref]5ce to ! fir/1 ! equations ( ! ,lor5tz trans=mation are valid4 ,^! la/ =m ! basis = ! d]ivation ( d$uctions f ! special !ory ( relativity1 & 9 !mselves !y are no?+ m ?an ! expression ( ! univ]sal validity ( ! law ( transmission ( light = all ,galileian sy/ems ( ref]5ce4 ,m9k[ski f\nd t ! ,lor5tz trans=mations satisfy ! foll[+ simple conditions4 ,let u consid] two neighb\r+ ev5ts1 ! relative position ( :i* 9 ! f\r-dim5sional cont9uum is giv5 ) respect to a ,galileian ref]5ce-body ,k by ! space co-ord9ate diff]5ces dx1 dy1 dz & ! "t-diff]5ce dt4 ,) ref]5ce to a second ,galileian sy/em we %all suppose t ! correspond+ diff]5ces = ^! two ev5ts are dx1 dy1 dz1 dt4 ,!n ^! magnitudes always fulfill ! condition4#ba #ba ,cf4 ,app5dixes ,i & ,,ii4 ,! relations :i* are d]iv$ !re = ! co-ord9ates !mselves are valid also = co-ord9ate diff]5ces1 & ?us also = co-ord9ate diff]5tials 7indef9itely small diff]5ces74 dx#b dy#b dz#b - c#b;dt#b dx#b dy#b dz#b - c#b;dt#b4 ,! validity ( ! ,lor5tz trans=mation foll[s f ?is condition4 ,we c express ?is z foll[s3 ,! magnitude ds#b dx#b dy#b dz#b - c#b dt#b1 :i* belongs to two adjac5t po9ts ( ! f\r-dim5sional space-"t cont9uum1 has ! same value = all select$ 7,galileian7 ref]5ce-bodies4 ,if we replace x1 y1 z1 image#jcd by x#a1 x#b1 x#c1 x#d1 we also obta9 ! result t ds#b dx#ab dx#bb dx#cb dx#db4 is indep5d5t ( ! *oice ( ! body ( ref]5ce4 ,we call ! magnitude ds ! 8di/ance0 ap>t ( ! two ev5ts or f\r-dim5sional po9ts4 ,?us1 if we *oose z "t-v>iable ! imag9>y v>iable image#jce in/ead ( ! real quantity t1 we c reg>d ! space-"t cont9tium--accordance ) ! special !ory ( relativity--z a 8,euclidean0 f\r-dim5sional cont9uum1 a result :i* foll[s f ! consid]ations ( ! prec$+ section4 ,,xxvii4 ,,! ,,space-,,"t ,,cont9uum ,,( ,,! ,,g5]al ,,!ory ,,( ,,relativity ,,is ,,n ,a ,,euclidean ,,cont9uum ,9 ! fir/ "p ( ?is book we 7 able to make use ( space-"t co-ord9ates :i* all[$ ( a simple & direct physical int]pretation1 & :i*1 accord+ to ,section ,,xxvi1 c be reg>d$ z f\r-dim5sional ,c>tesian co-ord9ates4 ,?is 0 possible on ! basis ( ! law ( ! con/ancy ( ! velocity ( light4 ,b accord+ to ,section ,,xxi ! g5]al !ory ( relativity _c reta9 ?is law4 ,on ! contr>y1 we arriv$ at ! result t accord+ to ?is latt] !ory ! velocity ( light mu/ always dep5d on ! co-ord9ates :5 a gravitational field is pres5t4 ,9 connection ) a specific illu/ration 9 ,section ,,xxiii1 we f\nd t ! pres5ce ( a gravitational field invalidates ! def9ition ( ! coord9ates & ! "t1 :i* l$ u to our objective 9 ! special !ory ( relativity4 ,9 view ( ! resu,its ( ^! consid]ations we are l$ to ! conviction t1 accord+ to ! g5]al pr9ciple ( relativity1 ! space-"t cont9uum _c be reg>d$ z a ,euclidean "o1 b t "h we h ! g5]al case1 correspond+ to ! m>ble slab ) local v>iations ( temp]ature1 & ) :i* we made acqua9tance z an example ( a two-dim5sional cont9uum4 ,j z x 0 !re impossible to con/ruct a ,c>tesian co-ord9ate sy/em f equal rods1 s "h x is impossible to build up a sy/em 7ref]5ce-body7 f rigid bodies & clocks1 :i* %all be ( su* a nature t measur+-rods & clocks1 arrang$ rigidly ) respect to "o ano!r1 %all indicate position & "t directly4 ,su* 0 ! ess5ce ( ! difficulty ) :i* we 7 confront$ 9 ,section ,,xxiii4 ,b ! consid]ations ( ,sections ,,xxv & ,,xxvi %[ u ! way to surm\nt ?is difficulty4 ,we ref] ! f\r-dim5sional space-"t cont9uum 9 an arbitr>y mann] to ,gauss co-ord9ates4 ,we assign to e po9t ( ! cont9uum 7ev5t7 f\r numb]s1 x#a1 x#b1 x#c1 x#d 7co-ord9ates71 :i* h n ! lea/ direct physical significance1 b only s]ve ! purpose ( numb]+ ! po9ts ( ! cont9uum 9 a def9ite b arbitr>y mann]4 ,?is arrangem5t does n ev5 ne$ to be ( su* a k9d t we mu/ reg>d x#a1 x#b1 x#c1 z 8space0 co-ord9ates & x#d1 z a 8"t0 co-ord9ate4 ,! read] may ?9k t su* a description ( ! _w w\ld be q inadequate4 ,:at does x mean to assign to an ev5t ! p>ticul> co-ord9ates x#a1 x#b1 x#c1 x#d1 if 9 !mselves ^! co-ord9ates h no significance8 ,m c>eful consid]ation %[s1 h[ev]1 t ?is anxiety is unf\nd$4 ,let u consid]1 = in/ance1 a mat]ial po9t ) any k9d ( motion4 ,if ?is po9t _h only a mom5t>y exi/5ce )\t duration1 !n x w\ld to describ$ 9 space-"t by a s+le sy/em ( values x#a1 x#b1 x#c1 x#d4 ,?us its p]man5t exi/5ce mu/ be *>act]is$ by an inf9itely l>ge numb] ( su* sy/ems ( values1 ! co-ord9ate values ( :i* are s close toge!r z to give cont9uity2 correspond+ to ! mat]ial po9t1 we ?us h a 7uni-dim5sional7 l9e 9 ! f\r-dim5sional cont9uum4 ,9 ! same way1 any su* l9es 9 our cont9uum correspond to _m po9ts 9 motion4 ,! only /atem5ts hav+ reg>d to ^! po9ts :i* c claim a physical exi/5ce are 9 reality ! /atem5ts ab\t _! enc\nt]s4 ,9 our ma!matical treatm5t1 su* an enc\nt] is express$ 9 ! fact t ! two l9es :i* repres5t ! motions ( ! po9ts 9 "q h a p>ticul> sy/em ( co-ord9ate values1 x#a1 x#b1 x#c1 x#d1 9 common4 ,aft] mature consid]ation ! read] w d\btless admit t 9 reality su* enc\nt]s con/itute ! only actual evid5ce ( a "t-space nature ) :i* we meet 9 physical /atem5ts4 ,:5 we 7 describ+ ! motion ( a mat]ial po9t relative to a body ( ref]5ce1 we /at$ no?+ m ?an ! enc\nt]s ( ?is po9t ) p>ticul> po9ts ( ! ref]5ce-body4 ,we c also det]m9e ! correspond+ values ( ! "t by ! obs]vation ( enc\nt]s ( ! body ) clocks1 9 conjunction ) ! obs]vation ( ! enc\nt] ( ! h&s ( clocks ) p>ticul> po9ts on ! dials4 ,x is j ! same 9 ! case ( space-measurem5ts by means ( measur+-rods1 z a little consid]ation w %[4 ,! foll[+ /atem5ts hold g5]ally3 ,e physical description resolves itself into a numb] ( /atem5ts1 ea* ( :i* ref]s to ! space-"t co9cid5ce ( two ev5ts ,a & ,b4 ,9 t]ms ( ,gaussian co-ord9ates1 e su* /atem5t is express$ by ! agreem5t ( _! f\r co-ord9ates x#a1 x#b1 x#c1 x#d4 ,?us 9 reality1 ! description ( ! "t-space cont9uum by means ( ,gauss co-ord9ates completely replaces ! description ) ! aid ( a body ( ref]5ce1 )\t suff]+ f ! defects ( ! latt] mode ( description2 x is n ti$ d[n to ! ,euclidean "* ( ! cont9uum :i* has to be repres5t$4 ,,xxviii4 ,,exact ,,=mulation ,,( ,,! ,,g5]al ,,pr9ciple ,,( ,,relativity ,we are n[ 9 a position to replace ! provisional =mulation ( ! g5]al pr9ciple ( relativity giv5 9 ,section ,,xviii by an exact =mulation4 ,! =m !re us$1 8,all bodies ( ref]5ce ,k1 ,k1 etc41 are equival5t = ! description ( natural ph5om5a 7=mulation ( ! g5]al laws ( nature71 :atev] may be _! /ate ( motion10 _c be ma9ta9$1 because ! use ( rigid ref]5ce-bodies1 9 ! s5se ( ! me?od foll[$ 9 ! special !ory ( relativity1 is 9 g5]al n possible 9 space-"t description4 ,! ,gauss co-ord9ate sy/em has to take ! place ( ! body ( ref]5ce4 ,! foll[+ /atem5t corresponds to ! fundam5tal idea ( ! g5]al pr9ciple ( relativity3 8,all ,gaussian co-ord9ate sy/ems are ess5tially equival5t = ! =mulation ( ! g5]al laws ( nature40 ,we c /ate ?is g5]al pr9ciple ( relativity 9 /ill ano!r =m1 :i* r5d]s x yet m cle>ly intelligible ?an x is :5 9 ! =m ( ! natural ext5sion ( ! special pr9ciple ( relativity4 ,accord+ to ! special !ory ( relativity1 ! equations :i* express ! g5]al laws ( nature pass ov] into equations ( ! same =m :51 by mak+ use ( ! ,lor5tz trans=mation1 we replace ! space-"t v>iables x1 y1 z1 t1 ( a 7,galileian7 ref]5ce-body ,k by ! space-"t v>iables x1 y1 z1 t1 ( a new ref]5ce-body ,k4 ,accord+ to ! g5]al !ory ( relativity1 on ! o!r h&1 by application ( arbitr>y sub/itutions ( ! ,gauss v>iables x#a1 x#b1 x#c1 x#d1 ! equations mu/ pass ov] into equations ( ! same =m2 = e trans=mation 7n only ! ,lor5tz trans=mation7 corresponds to ! transition ( "o ,gauss co-ord9ate sy/em into ano!r4 ,if we desire to adh]e to our 8old-"t0 ?ree-dim5sional view ( ?+s1 !n we c *>act]ise ! developm5t :i* is be+ und]gone by ! fundam5tal idea ( ! g5]al !ory ( relativity z foll[s3 ,! special !ory ( relativity has ref]5ce to ,galileian doma9s1 i4e4 to ^? 9 :i* no gravitational field exi/s4 ,9 ?is connection a ,galileian ref]5ce-body s]ves z body ( ref]5ce1 i4e4 a rigid body ! /ate ( motion ( :i* is s *os5 t ! ,galileian law ( ! uni=m rectil9e> motion ( 8isolat$0 mat]ial po9ts holds relatively to x4 ,c]ta9 consid]ations sugge/ t we %\ld ref] ! same ,galileian doma9s to non-,galileian ref]5ce-bodies also4 ,a gravitational field ( a special k9d is !n pres5t ) respect to ^! bodies 7cf4 ,sections ,,xx & ,,xxiii74 ,9 gravitational fields !re are no su* ?+s z rigid bodies ) ,euclidean prop]ties2 ?us ! fictiti\s rigid body ( ref]5ce is ( no avail 9 ! g5]al !ory ( relativity4 ,! motion ( clocks is also influ5c$ by gravitational fields1 & 9 su* a way t a physical def9ition ( "t :i* is made directly ) ! aid ( clocks has by no means ! same degree ( plausibility z 9 ! special !ory ( relativity4 ,= ?is reason non-rigid ref]5ce-bodies are us$1 :i* are z a :ole n only mov+ 9 any way :atsoev]1 b :i* also suff] alt]ations 9 =m ad lib4 dur+ _! motion4 ,clocks1 = :i* ! law ( motion is ( any k9d1 h[ev] irregul>1 s]ve = ! def9ition ( "t4 ,we h to imag9e ea* ( ^! clocks fix$ at a po9t on ! non-rigid ref]5ce-body4 ,^! clocks satisfy only ! "o condition1 t ! 8read+s0 :i* are obs]v$ simultane\sly on adjac5t clocks 79 space7 diff] f ea* o!r by an indef9itely small am\nt4 ,?is non-rigid ref]5ce-body1 :i* might appropriately be t]m$ a 8ref]5ce-mollusc01 is 9 ! ma9 equival5t to a ,gaussian f\r-dim5sional co-ord9ate sy/em *os5 arbitr>ily4 ,t :i* gives ! 8mollusc0 a c]ta9 compreh5sibility z comp>$ ) ! ,gauss co-ord9ate sy/em is ! 7really unju/ifi$7 =mal ret5tion ( ! sep>ate exi/5ce ( ! space co-ord9ates z oppos$ to ! "t co-ord9ate4 ,e po9t on ! mollusc is treat$ z a space-po9t1 & e mat]ial po9t :i* is at re/ relatively to x z at re/1 s long z ! mollusc is consid]$ z ref]5ce-body4 ,! g5]al pr9ciple ( relativity requires t all ^! molluscs c be us$ z ref]5ce-bodies ) equal "r & equal success 9 ! =mulation ( ! g5]al laws ( nature2 ! laws !mselves mu/ be q indep5d5t ( ! *oice ( mollusc4 ,! great p[] possess$ by ! g5]al pr9ciple ( relativity lies 9 ! compreh5sive limitation :i* is impos$ on ! laws ( nature 9 consequ5ce ( :at we h se5 above4 ,,xxix4 ,,! ,,solution ,,( ,,! ,,problem ,,( ,,gravitation ,,on ,,! ,,basis ,,( ,,! ,,g5]al ,,pr9ciple ,,( ,,relativity ,if ! read] has foll[$ all our previ\s consid]ations1 he w h no fur!r difficulty 9 und]/&+ ! me?ods lead+ to ! solution ( ! problem ( gravitation4 ,we />t (f on a consid]ation ( a ,galileian doma91 i4e4 a doma9 9 :i* !re is no gravitational field relative to ! ,galileian ref]5ce-body ,k4 ,! behavi\r ( measur+-rods & clocks ) ref]5ce to ,k is kn[n f ! special !ory ( relativity1 likewise ! behavi\r ( 8isolat$0 mat]ial po9ts2 ! latt] move uni=mly & 9 /raight l9es4 ,n[ let u ref] ?is doma9 to a r&om ,gauss coord9ate sy/em or to a 8mollusc0 z ref]5ce-body ,k4 ,!n ) respect to ,k !re is a gravitational field ,g 7( a p>ticul> k9d74 ,we le>n ! behavi\r ( measur+-rods & clocks & also ( freely-mov+ mat]ial po9ts ) ref]5ce to ,k simply by ma!matical trans=mation4 ,we int]pret ?is behavi\r z ! behavi\r ( measur+-rods1 clocks & mat]ial po9ts "u ! influ5ce ( ! gravitational field ,g4 ,h]eupon we introduce a hypo!sis3 t ! influ5ce ( ! gravitational field on measur+-rods1 clocks & freely-mov+ mat]ial po9ts cont9ues to take place accord+ to ! same laws1 ev5 9 ! case :]e ! prevail+ gravitational field is n d]ivable f ! ,galileian special case1 simply by means ( a trans=mation ( co-ord9ates4 ,! next /ep is to inve/igate ! space-"t behavi\r ( ! gravitational field ,g1 :i* 0 d]iv$ f ! ,galileian special case simply by trans=mation ( ! coord9ates4 ,?is behavi\r is =mulat$ 9 a law1 :i* is always valid1 no matt] h[ ! ref]5ce-body 7mollusc7 us$ 9 ! description may be *os54 ,?is law is n yet ! g5]al law ( ! gravitational field1 s9ce ! gravitational field "u consid]ation is ( a special k9d4 ,9 ord] to f9d out ! g5]al law-(-field ( gravitation we /ill require to obta9 a g5]alisation ( ! law z f\nd above4 ,?is c be obta9$ )\t caprice1 h[ev]1 by tak+ into consid]ation ! foll[+ dem&s3 7a7 ,! requir$ g5]alisation mu/ likewise satisfy ! g5]al po/ulate ( relativity4 7b7 ,if !re is any matt] 9 ! doma9 "u consid]ation1 only its in]tial mass1 & ?us accord+ to ,section ,,xv only its en]gy is ( importance = its effect 9 excit+ a field4 7c7 ,gravitational field & matt] toge!r mu/ satisfy ! law ( ! cons]vation ( en]gy 7& ( impulse74 ,f9ally1 ! g5]al pr9ciple ( relativity p]mits u to det]m9e ! influ5ce ( ! gravitational field on ! c\rse ( all ^? processes :i* take place accord+ to kn[n laws :5 a gravitational field is abs5t i4e4 :i* h already be5 fitt$ into ! frame ( ! special !ory ( relativity4 ,9 ?is connection we proce$ 9 pr9ciple accord+ to ! me?od :i* has already be5 expla9$ = measur+-rods1 clocks & freely mov+ mat]ial po9ts4 ,! !ory ( gravitation d]iv$ 9 ?is way f ! g5]al po/ulate ( relativity excels n only 9 its beauty2 nor 9 remov+ ! defect atta*+ to classical me*anics :i* 0 br\ght to light 9 ,section ,,xxi2 nor 9 int]pret+ ! empirical law ( ! equality ( in]tial & gravitational mass2 b x has also already expla9$ a result ( obs]vation 9 a/ronomy1 aga9/ :i* classical me*anics is p[]less4 ,if we conf9e ! application ( ! !ory to ! case :]e ! gravitational fields c be reg>d$ z be+ weak1 & 9 :i* all masses move ) respect to ! coord9ate sy/em ) velocities :i* are small comp>$ ) ! velocity ( light1 we !n obta9 z a fir/ approximation ! ,newtonian !ory4 ,?us ! latt] !ory is obta9$ "h )\t any p>ticul> assumption1 :]eas ,newton _h to introduce ! hypo!sis t ! =ce ( attraction betwe5 mutually attract+ mat]ial po9ts is inv]sely proportional to ! squ>e ( ! di/ance betwe5 !m4 ,if we increase ! accuracy ( ! calculation1 deviations f ! !ory ( ,newton make _! appe>ance1 practically all ( :i* mu/ nev]!less escape ! te/ ( obs]vation ow+ to _! smallness4 ,we mu/ draw att5tion "h to "o ( ^! deviations4 ,accord+ to ,newton's !ory1 a planet moves r\nd ! sun 9 an ellipse1 :i* w\ld p]man5tly ma9ta9 its position ) respect to ! fix$ />s1 if we c\ld disreg>d ! motion ( ! fix$ />s !mselves & ! action ( ! o!r planets "u consid]ation4 ,?us1 if we correct ! obs]v$ motion ( ! planets = ^! two influ5ces1 & if ,newton's !ory be /rictly correct1 we ought to obta9 = ! orbit ( ! planet an ellipse1 :i* is fix$ ) ref]5ce to ! fix$ />s4 ,?is d$uction1 :i* c be te/$ ) great accuracy1 has be5 confirm$ = all ! planets save "o1 ) ! precision t is capable ( be+ obta9$ by ! delicacy ( obs]vation atta9able at ! pres5t "t4 ,! sole exception is ,m]cury1 ! planet :i* lies ne>e/ ! sun4 ,s9ce ! "t ( ,lev]ri]1 x has be5 kn[n t ! ellipse correspond+ to ! orbit ( ,m]cury1 aft] x has be5 correct$ = ! influ5ces m5tion$ above1 is n /ation>y ) respect to ! fix$ />s1 b t x rotates exce$+ly sl[ly 9 ! plane ( ! orbit & 9 ! s5se ( ! orbital motion4 ,! value obta9$ = ?is rot>y movem5t ( ! orbital ellipse 0 #dc seconds ( arc p] c5tury1 an am\nt ensur$ to be correct to )9 a few seconds ( arc4 ,?is effect c be expla9$ by means ( classical me*anics only on ! assumption ( hypo!ses :i* h little probability1 & :i* 7 devis$ solely = ?is purponse4 ,on ! basis ( ! g5]al !ory ( relativity1 x is f\nd t ! ellipse ( e planet r\nd ! sun mu/ necess>ily rotate 9 ! mann] indicat$ above2 t = all ! planets1 ) ! exception ( ,m]cury1 ?is rotation is too small to be detect$ ) ! delicacy ( obs]vation possible at ! pres5t "t2 b t 9 ! case ( ,m]cury x mu/ am\nt to #dc seconds ( arc p] c5tury1 a result :i* is /rictly 9 agreem5t ) obs]vation4 ,ap>t f ?is "o1 x has hi!rto be5 possible to make only two d$uctions f ! !ory :i* admit ( be+ te/$ by obs]vation1 to wit1 ! curvature ( light rays by ! gravitational field ( ! sun1#bb & a displacem5t ( ! spectral l9es ( light rea*+ u f l>ge />s1 z comp>$ ) ! correspond+ l9es = light produc$ 9 an analog\s mann] t]re/rially 7i4e4 by ! same k9d ( atom74#bc ,^! two d$uctions f ! !ory h bo? be5 confirm$4 #bb ,fir/ obs]v$ by ,edd+ton & o!rs 9 #aiai4 7,cf4 ,app5dix ,,iii74 #bc ,e/abli%$ by ,adams 9 #aibd4 7,cf4 p4 #acb7 ,,"p ,,iii3 ,,consid]ations ,,on ,,! ,,univ]se ,,z ,a ,,:ole ,,xxx4 ,,cosmological ,,difficulties ,,( ,,newton',s ,,!ory ,"p f ! difficulty discuss$ 9 ,section ,,xxi1 !re is a second fundam5tal difficulty att5d+ classical cele/ial me*anics1 :i*1 to ! be/ ( my k1 0 fir/ discuss$ 9 detail by ! a/ronom] ,seelig]4 ,if we pond] ov] ! "q z to h[ ! univ]se1 consid]$ z a :ole1 is to be reg>d$1 ! fir/ answ] t sugge/s itself to u is surely ?is3 ,z reg>ds space 7& "t7 ! univ]se is inf9ite4 ,!re are />s ev]y:]e1 s t ! d5sity ( matt]1 al?\gh v v>iable 9 detail1 is nev]!less on ! av]age ev]y:]e ! same4 ,9 o!r words3 ,h[ev] f> we might travel ?r\gh space1 we %\ld f9d ev]y:]e an att5uat$ sw>m ( fix$ />s ( approrimately ! same k9d & d5sity4 ,?is view is n 9 h>mony ) ! !ory ( ,newton4 ,! latt] !ory r requires t ! univ]se %\ld h a k9d ( c5tre 9 :i* ! d5sity ( ! />s is a maximum1 & t z we proce$ outw>ds f ?is c5tre ! gr\p-d5sity ( ! />s %\ld dim9i%1 until f9ally1 at great di/ances1 x is succe$$ by an inf9ite region ( empt9ess4 ,! /ell> univ]se ought to be a f9ite isl& 9 ! inf9ite ocean ( space4#bd #bd ,pro(--,accord+ to ! !ory ( ,newton1 ! numb] ( 8l9es ( =ce0 :i* come f inf9ity & t]m9ate 9 a mass m is proportional to ! mass m4 ,if1 on ! av]age1 ! mass d5sity #j is con/ant ?r\gh\t ! univ]se1 !n a sph]e ( volume ,v w enclose ! av]age mass #j,v4 ,?us ! numb] ( l9es ( =ce pass+ ?r\gh ! surface ,f ( ! sph]e into its int]ior is proportional to #j,v4 ,= unit area ( ! surface ( ! sph]e ! numb] ( l9es ( =ce :i* ent]s ! sph]e is ?us proportional to #j,v_/,f or to #j,r4 ,h5ce ! int5sity ( ! field at ! surface w\ld ultimately become inf9ite ) increas+ radius ,r ( ! sph]e1 :i* is impossible4 ,?is conception is 9 itself n v satisfactory4 ,x is /ill less satisfactory because x leads to ! result t ! light emitt$ by ! />s & also individual />s ( ! /ell> sy/em are p]petually pass+ out into inf9ite space1 nev] to return1 & )\t ev] aga9 com+ into int]action ) o!r objects ( nature4 ,su* a f9ite mat]ial univ]se w\ld be de/9$ to become gradually b sy/ematically impov]i%$4 ,9 ord] to escape ?is dilemma1 ,seelig] sugge/$ a modification ( ,newton's law1 9 :i* he assumes t = great di/ances ! =ce ( attraction betwe5 two masses dim9i%es m rapidly ?an w\ld result f ! inv]se squ>e law4 ,9 ?is way x is possible = ! mean d5sity ( matt] to be con/ant ev]y:]e1 ev5 to inf9ity1 )\t inf9itely l>ge gravitational fields be+ produc$4 ,we ?us free ourselves f ! di/a/eful conception t ! mat]ial univ]se ought to possess some?+ ( ! nature ( a c5tre4 ,( c\rse we pur*ase our emancipation f ! fundam5tal difficulties m5tion$1 at ! co/ ( a modification & complication ( ,newton's law :i* has nei!r empirical nor !oretical f\ndation4 ,we c imag9e innum]able laws :i* w\ld s]ve ! same purpose1 )\t our be+ able to /ate a reason :y "o ( !m is to be pref]r$ to ! o!rs2 = any "o ( ^! laws w\ld be f\nd$ j z little on m g5]al !oretical pr9ciples z is ! law ( ,newton4 ,,xxxi4 ,,! ,,possibility ,,( ,a 8,,f9ite0 ,,& ,,yet 8,,unb\nd$0 ,,univ]se ,b speculations on ! /ructure ( ! univ]se also move 9 q ano!r direction4 ,! developm5t ( non-,euclidean geometry l$ to ! recognition ( ! fact1 t we c ca/ d\bt on ! inf9it5ess ( our space )\t com+ into conflict ) ! laws ( ?\ght or ) exp]i5ce 7,riemann1 ,helmholtz74 ,^! que/ions h already be5 treat$ 9 detail & ) unsurpassable lucidity by ,helmholtz & ,po9c>1 :]eas ,i c only t\* on !m briefly "h4 ,9 ! fir/ place1 we imag9e an exi/5ce 9 two dim5sional space4 ,flat be+s ) flat implem5ts1 & 9 p>ticul> flat rigid measur+-rods1 are free to move 9 a plane4 ,= !m no?+ exi/s outside ( ?is plane3 t :i* !y obs]ve to happ5 to !mselves & to _! flat 8?+s0 is ! all-inclusive reality ( _! plane4 ,9 p>ticul>1 ! con/ructions ( plane ,euclidean geometry c be c>ri$ out by means ( ! rods e4g4 ! lattice con/ruction1 consid]$ 9 ,section ,,xxiv4 ,9 contra/ to ours1 ! univ]se ( ^! be+s is two-dim5sional2 b1 l ours1 x ext5ds to inf9ity4 ,9 _! univ]se !re is room = an inf9ite numb] ( id5tical squ>es made up ( rods1 i4e4 its volume 7surface7 is inf9ite4 ,if ^! be+s say _! univ]se is 8plane10 !re is s5se 9 ! /atem5t1 because !y mean t !y c p]=m ! con/ructions ( plane ,euclidean geometry ) _! rods4 ,9 ?is connection ! individual rods always repres5t ! same di/ance1 indep5d5tly ( _! position4 ,let u consid] n[ a second two-dim5sional exi/5ce1 b ?is "t on a sph]ical surface in/ead ( on a plane4 ,! flat be+s ) _! measur+-rods & o!r objects fit exactly on ?is surface & !y are unable to leave x4 ,_! :ole univ]se ( obs]vation ext5ds exclusively ov] ! surface ( ! sph]e4 ,are ^! be+s able to reg>d ! geometry ( _! univ]se z be+ plane geometry & _! rods )al z ! realisation ( 8di/ance08 ,!y _c d ?is4 ,= if !y attempt to realise a /raight l9e1 !y w obta9 a curve1 :i* we 8?ree-dim5sional be+s0 designate z a great circle1 i4e4 a self-conta9$ l9e ( def9ite f9ite l5g?1 :i* c be measur$ up by means ( a measur+-rod4 ,simil>ly1 ?is univ]se has a f9ite area t c be comp>$ ) ! area1 ( a squ>e con/ruct$ ) rods4 ,! great *>m result+ f ?is consid]ation lies 9 ! recognition ( ! fact t ! univ]se ( ^! be+s is f9ite & yet has no limits4 ,b ! sph]ical-surface be+s d n ne$ to g on a _w-t\r 9 ord] to p]ceive t !y are n liv+ 9 a ,euclidean univ]se4 ,!y c conv9ce !mselves ( ?is on e "p ( _! 8_w10 provid$ !y d n use too small a piece ( x4 ,/>t+ f a po9t1 !y draw 8/raight l9es0 7arcs ( circles z judg$ 9 ?ree dim5sional space7 ( equal l5g? 9 all directions4 ,!y w call ! l9e jo9+ ! free ends ( ^! l9es a 8circle40 ,= a plane surface1 ! ratio ( ! circumf]5ce ( a circle to its diamet]1 bo? l5g?s be+ measur$ ) ! same rod1 is1 accord+ to ,euclidean geometry ( ! plane1 equal to a con/ant value 1 :i* is indep5d5t ( ! diamet] ( ! circle4 ,on _! sph]ical surface our flat be+s w\ld f9d = ?is ratio ! value image#jcf i4e4 a small] value ?an 1 ! diff]5ce be+ ! m consid]able1 ! great] is ! radius ( ! circle 9 comp>ison ) ! radius ,r ( ! 8_w-sph]e40 ,by means ( ?is relation ! sph]ical be+s c det]m9e ! radius ( _! univ]se 78_w071 ev5 :5 only a relatively small "p ( _! worldsph]e is available = _! measurem5ts4 ,b if ?is "p is v small inde$1 !y w no long] be able to demon/rate t !y are on a sph]ical 8_w0 & n on a ,euclidean plane1 = a small "p ( a sph]ical surface diff]s only slightly f a piece ( a plane ( ! same size4 ,?us if ! sph]ical surface be+s are liv+ on a planet ( :i* ! sol> sy/em occupies only a negligibly small "p ( ! sph]ical univ]se1 !y h no means ( det]m9+ :e!r !y are liv+ 9 a f9ite or 9 an inf9ite univ]se1 because ! 8piece ( univ]se0 to :i* !y h access is 9 bo? cases practically plane1 or ,euclidean4 ,x foll[s directly f ?is discussion1 t = our sph]e-be+s ! circumf]5ce ( a circle fir/ increases ) ! radius until ! 8circumf]5ce ( ! univ]se0 is rea*$1 & t x !nce=w>d gradually decreases to z]o = /ill fur!r increas+ values ( ! radius4 ,dur+ ?is process ! area ( ! circle cont9ues to increase m & m1 until f9ally x becomes equal to ! total area ( ! :ole 8_w-sph]e40 ,p]haps ! read] w wond] :y we h plac$ our 8be+s0 on a sph]e r ?an on ano!r clos$ surface4 ,b ?is *oice has its ju/ification 9 ! fact t1 ( all clos$ surfaces1 ! sph]e is unique 9 possess+ ! prop]ty t all po9ts on x are equival5t4 ,i admit t ! ratio ( ! circumf]5ce c ( a circle to its radius r dep5ds on r1 b = a giv5 value ( r x is ! same = all po9ts ( ! 8worldsph]e02 9 o!r words1 ! 8_w-sph]e0 is a 8surface ( con/ant curvature40 ,to ?is two-dim5sional sph]e-univ]se !re is a ?ree-dim5sional analogy1 namely1 ! ?ree-dim5sional sph]ical space :i* 0 discov]$ by ,riemann4 its po9ts are likewise all equival5t4 ,x possesses a f9ite volume1 :i* is det]m9$ by its 8radius0 7#bb,r#c74 ,is x possible to imag9e a sph]ical space8 ,to imag9e a space means no?+ else ?an t we imag9e an epitome ( our 8space0 exp]i5ce1 i4e4 ( exp]i5ce t we c h 9 ! movem5t ( 8rigid0 bodies4 ,9 ?is s5se we c imag9e a sph]ical space4 ,suppose we draw l9es or /ret* /r+s 9 all directions f a po9t1 & m>k (f f ea* ( ^! ! di/ance r ) a measur+-rod4 ,all ! free end-po9ts ( ^! l5g?s lie on a sph]ical surface4 ,we c specially measure up ! area 7,f7 ( ?is surface by means ( a squ>e made up ( measur+-rods4 ,if ! univ]se is ,euclidean1 !n ,f #dr#b2 if x is sph]ical1 !n ,f is always less ?an #dr#b4 ,) increas+ values ( r1 ,f increases f z]o up to a maximum value :i* is det]m9$ by ! 8_w-radius10 b = /ill fur!r increas+ values ( r1 ! area gradually dim9i%es to z]o4 ,at fir/1 ! /raight l9es :i* radiate f ! />t+ po9t div]ge f>!r & f>!r f "o ano!r1 b lat] !y approa* ea* o!r1 & f9ally !y run toge!r aga9 at a 8c\nt]-po9t0 to ! />t+ po9t4 ,"u su* conditions !y h trav]s$ ! :ole sph]ical space4 ,x is easily se5 t ! ?ree-dim5sional sph]ical space is q analog\s to ! two-dim5sional sph]ical surface4 ,x is f9ite 7i4e4 ( f9ite volume71 & has no b\nds4 ,x may be m5tion$ t !re is yet ano!r k9d ( curv$ space3 8elliptical space40 ,x c be reg>d$ z a curv$ space 9 :i* ! two 8c\nt]-po9ts0 are id5tical 7indi/+ui%able f ea* o!r74 ,an elliptical univ]se c ?us be consid]$ to "s ext5t z a curv$ univ]se possess+ c5tral symmetry4 ,x foll[s f :at has be5 said1 t clos$ spaces )\t limits are conceivable4 ,f among/ ^!1 ! sph]ical space 7& ! elliptical7 excels 9 its simplicity1 s9ce all po9ts on x are equival5t4 ,z a result ( ?is discussion1 a mo/ int]e/+ "q arises = a/ronom]s & physici/s1 & t is :e!r ! univ]se 9 :i* we live is inf9ite1 or :e!r x is f9ite 9 ! mann] ( ! sph]ical univ]se4 ,our exp]i5ce is f> f be+ suffici5t to enable u to answ] ?is "q4 ,b ! g5]al !ory ( relativity p]mits ( our answ]+ x ) a mod]ate degree ( c]ta9ty1 & 9 ?is connection ! difficulty m5tion$ 9 ,section ,,xxx f9ds its solution4 ,,xxxii4 ,,! ,,/ructure ,,( ,,space ,,accord+ ,,to ,,! ,,g5]al ,,!ory ,,( ,,relativity ,accord+ to ! g5]al !ory ( relativity1 ! geometrical prop]ties ( space are n indep5d5t1 b !y are det]m9$ by matt]4 ,?us we c draw conclusions ab\t ! geometrical /ructure ( ! univ]se only if we base our consid]ations on ! /ate ( ! matt] z be+ some?+ t is kn[n4 ,we "k f exp]i5ce t1 = a suitably *os5 co-ord9ate sy/em1 ! velocities ( ! />s are small z comp>$ ) ! velocity ( transmission ( light4 ,we c ?us z a r\gh approximation arrive at a conclusion z to ! nature ( ! univ]se z a :ole1 if we treat ! matt] z be+ at re/4 ,we already "k f our previ\s discussion t ! behavi\r ( measur+-rods & clocks is influ5c$ by gravitational fields1 i4e4 by ! di/ribution ( matt]4 ,?is 9 itself is suffici5t to exclude ! possibility ( ! exact validity ( ,euclidean geometry 9 our univ]se4 ,b x is conceivable t our univ]se diff]s only slightly f a ,euclidean "o1 & ?is notion seems all ! m probable1 s9ce calculations %[ t ! metrics ( surr\nd+ space is influ5c$ only to an exce$+ly small ext5t by masses ev5 ( ! magnitude ( our sun4 ,we might imag9e t1 z reg>ds geometry1 our univ]se behaves analog\sly to a surface :i* is irregul>ly curv$ 9 its individual p>ts1 b :i* n[h]e dep>ts appreciably f a plane3 some?+ l ! rippl$ surface ( a lake4 ,su* a univ]se might fitt+ly be call$ a quasi-,euclidean univ]se4 ,z reg>ds its space x w\ld be inf9ite4 ,b calculation %[s t 9 a quasi-,euclidean univ]se ! av]age d5sity ( matt] w\ld necess>ily be nil4 ,?us su* a univ]se c\ld n be inhabit$ by matt] ev]y:]e2 x w\ld pres5t to u t unsatisfactory picture :i* we portray$ 9 ,section ,,xxx4 ,if we are to h 9 ! univ]se an av]age d5sity ( matt] :i* diff]s f z]o1 h[ev] small may be t diff]5ce1 !n ! univ]se _c be quasi-,euclidean4 ,on ! contr>y1 ! results ( calculation indicate t if matt] be di/ribut$ uni=mly1 ! univ]se w\ld necess>ily be sph]ical 7or elliptical74 ,s9ce 9 reality ! detail$ di/ribution ( matt] is n uni=m1 ! real univ]se w deviate 9 individual p>ts f ! sph]ical1 i4e4 ! univ]se w be quasi-sph]ical4 ,b x w be necess>ily f9ite4 ,9 fact1 ! !ory supplies u ) a simple connection#be betwe5 ! space-expanse ( ! univ]se & ! av]age d5sity ( matt] 9 x4 #be ,= ! radius ,r ( ! univ]se we obta9 ! equation image#jcg ,! use ( ! ,c4,g4,s4 sy/em 9 ?is equation gives #b_/k #a4jh x #ajbg2 is ! av]age d5sity ( ! matt] & k is a con/ant connect$ ) ! ,newtonian con/ant ( gravitation4 ,,app5dices ,,app5dix ,i ,,simple ,,d]ivation ,,( ,,! ,,lor5tz ,,trans=mation 7,,supplem5t>y ,,to ,,section ,,xi7 ,= ! relative ori5tation ( ! co-ord9ate sy/ems indicat$ 9 ,fig4 #b1 ! x-axes ( bo? sy/ems p]man5tly co9cide4 ,9 ! pres5t case we c divide ! problem into p>ts by consid]+ fir/ only ev5ts :i* are localis$ on ! x-axis4 ,any su* ev5t is repres5t$ ) respect to ! co-ord9ate sy/em ,k by ! abscissa x & ! "t t1 & ) respect to ! sy/em ,k by ! abscissa x & ! "t t4 ,we require to f9d x & t :5 x & t are giv54 ,a light-signal1 :i* is proce$+ along ! positive axis ( x1 is transmitt$ accord+ to ! equation x ct or x - ct #j 4 4 4 4 4 7#a74 ,s9ce ! same light-signal has to be transmitt$ relative to ,k ) ! velocity c1 ! propagation relative to ! sy/em ,k w be repres5t$ by ! analog\s =mula x - ct #j 4 4 4 4 4 7#b7 ,^? space-"t po9ts 7ev5ts7 :i* satisfy 7#a7 mu/ also satisfy 7#b74 ,obvi\sly ?is w be ! case :5 ! relation 7x - ct7 7x - ct7 4 4 4 7#c74 is fulfill$ 9 g5]al1 :]e indicates a con/ant2 =1 accord+ to 7#c71 ! disappe>ance ( 7x - ct7 involves ! disappe>ance ( 7x - ct74 ,if we apply q simil> consid]ations to light rays :i* are be+ transmitt$ along ! negative x-axis1 we obta9 ! condition 7x ct7 7x ct7 4 4 4 7#d74 ,by add+ 7or subtract+7 equations 7#c7 & 7#d71 & introduc+ = conv5i5ce ! con/ants a & b 9 place ( ! con/ants & :]e image#jch & image#jci we obta9 ! equations image#jdj ,we %\ld ?us h ! solution ( our problem1 if ! con/ants a & b 7 kn[n4 ,^! result f ! foll[+ discussion4 ,= ! orig9 ( ,k we h p]man5tly x #j1 & h5ce accord+ to ! fir/ ( ! equations 7#e7 image#jda ,if we call v ! velocity ) :i* ! orig9 ( ,k is mov+ relative to ,k1 we !n h image#jdb ,! same value v c be obta9$ f equations 7#e71 if we calculate ! velocity ( ano!r po9t ( ,k relative to ,k1 or ! velocity 7direct$ t[>ds ! negative x-axis7 ( a po9t ( ,k ) respect to ,k4 ,9 %ort1 we c designate v z ! relative velocity ( ! two sy/ems4 ,fur!rmore1 ! pr9ciple ( relativity tea*es u t1 z judg$ f ,k1 ! l5g? ( a unit measur+-rod :i* is at re/ ) ref]5ce to ,k mu/ be exactly ! same z ! l5g?1 z judg$ f ,k1 ( a unit measur+-rod :i* is at re/ relative to ,k4 ,9 ord] to see h[ ! po9ts ( ! x-axis appe> z view$ f ,k1 we only require to take a 8snap%ot0 ( ,k f ,k2 ?is means t we h to ins]t a p>ticul> value ( t 7"t ( ,k71 e4g4 t #j4 ,= ?is value ( t we !n obta9 f ! fir/ ( ! equations 7#e7 x ax ,two po9ts ( ! x-axis :i* are sep>at$ by ! di/ance x #a :5 measur$ 9 ! ,k sy/em are ?us sep>at$ 9 our in/antane\s photograph by ! di/ance image#jdc ,b if ! snap%ot be tak5 f ,k7t #j71 & if we elim9ate t f ! equations 7#e71 tak+ into acc\nt ! expression 7#f71 we obta9 image#jdd ,f ?is we conclude t two po9ts on ! x-axis sep>at$ by ! di/ance #a 7relative to ,k7 w be repres5t$ on our snap%ot by ! di/ance image#jde ,b f :at has be5 said1 ! two snap%ots mu/ be id5tical2 h5ce x 9 7#g7 mu/ be equal to x 9 7#g;a71 s t we obta9 image#jdf ,! equations 7#f7 & 7#g;b7 det]m9e ! con/ants a & b4 ,by ins]t+ ! values ( ^! con/ants 9 7#e71 we obta9 ! fir/ & ! f\r? ( ! equations giv5 9 ,section ,,xi4 image#jdg ,?us we h obta9$ ! ,lor5tz trans=mation = ev5ts on ! x-axis4 ,x satisfies ! condition x#b - c#bt#b x#b - c#bt#b 4 4 4 4 4 4 7#h;a74 ,! ext5sion ( ?is result1 to include ev5ts :i* take place outside ! x-axis1 is obta9$ by reta9+ equations 7#h7 & supplem5t+ !m by ! relations image#jdh ,9 ?is way we satisfy ! po/ulate ( ! con/ancy ( ! velocity ( light 9 vacuo = rays ( light ( arbitr>y direction1 bo? = ! sy/em ,k & = ! sy/em ,k4 ,?is may be %[n 9 ! foll[+ mann]4 ,we suppose a light-signal s5t out f ! orig9 ( ,k at ! "t t #j4 ,x w be propagat$ accord+ to ! equation image#jdi or1 if we squ>e ?is equation1 accord+ to ! equation x#b y#b z#b - c#bt#b #j 4 4 4 4 4 7#aj74 ,x is requir$ by ! law ( propagation ( light1 9 conjunction ) ! po/ulate ( relativity1 t ! transmission ( ! signal 9 "q %\ld take place--z judg$ f ,k--9 accordance ) ! correspond+ =mula r ct or1 x#b y#b z#b - c#bt#b #j 4 4 4 4 4 4 7#aj;a74 ,9 ord] t equation 7#aj;a7 may be a consequ5ce ( equation 7#aj71 we mu/ h x#b y#b z#b - c#bt#b 7x#b y#b z#b - c#bt#b7 7#aa74 ,s9ce equation 7#h;a7 mu/ hold = po9ts on ! x-axis1 we ?us h #a4 ,x is easily se5 t ! ,lor5tz trans=mation really satisfies equation 7#aa7 = #a2 = 7#aa7 is a consequ5ce ( 7#h;a7 & 7#i71 & h5ce also ( 7#h7 & 7#i74 ,we h ?us d]iv$ ! ,lor5tz trans=mation4 ,! ,lor5tz trans=mation repres5t$ by 7#h7 & 7#i7 /ill requires to be g5]alis$4 ,obvi\sly x is immat]ial :e!r ! axes ( ,k be *os5 s t !y are spatially p>allel to ^? ( ,k4 ,x is also n ess5tial t ! velocity ( translation ( ,k ) respect to ,k %\ld be 9 ! direction ( ! x-axis4 ,a simple consid]ation %[s t we are able to con/ruct ! ,lor5tz trans=mation 9 ?is g5]al s5se f two k9ds ( trans=mations1 viz4 f ,lor5tz trans=mations 9 ! special s5se & f purely spatial trans=mations4 :i* corresponds to ! replacem5t ( ! rectangul> co-ord9ate sy/em by a new sy/em ) its axes po9t+ 9 o!r directions4 ,ma!matically1 we c *>act]ise ! g5]alis$ ,lor5tz trans=mation ?us3 ,x expresses x1 y1 x1 t1 9 t]ms ( l9e> homog5e\s functions ( x1 y1 x1 t1 ( su* a k9d t ! relation x#b y#b z#b - c#bt#b x#b y#b z#b - c#bt#b 7#aa;a74 is satisficd id5tically4 ,t is to say3 ,if we sub/itute _! expressions 9 x1 y1 x1 t1 9 place ( x1 y1 x1 t1 on ! left-h& side1 !n ! left-h& side ( 7#aa;a7 agrees ) ! "r-h& side4 ,,app5dix ,,ii ,,m9k[ski',s ,,f\r-,,dim5sional ,,space 78,,_w07 7,,supplem5t>y ,,to ,,section ,,xvii7 ,we c *>act]ise ! ,lor5tz trans=mation /ill m simply if we introduce ! imag9>y image#jca 9 place ( t1 z "t-v>iable4 ,if1 9 accordance ) ?is1 we ins]t image#jej & simil>ly = ! acc5t$ sy/em ,k1 !n ! condition :i* is id5tically satisfi$ by ! trans=mation c be express$ ?us3 x#ab x#bb x#cb x#db x#ab x#bb x#cb x#db 7#ab74 ,t is1 by ! a=e-m5tion$ *oice ( 8coord9ates10 7#aa;a7 see ! end ( ,app5dix ,,ii is trans=m$ into ?is equation4 ,we see f 7#ab7 t ! imag9>y "t co-ord9ate x#d1 ent]s into ! condition ( trans=mation 9 exactly ! same way z ! space co-ord9ates x#a1 x#b1 x#c4 ,x is due to ?is fact t1 accord+ to ! !ory ( relativity1 ! 8"t0 x#d1 ent]s into natural laws 9 ! same =m z ! space co ord9ates x#a1 x#b1 x#c4 ,a f\r-dim5sional cont9uum describ$ by ! 8co-ord9ates0 x#a1 x#b1 x#c1 x#d1 0 call$ 8_w0 by ,m9k[ski1 :o also t]m$ a po9t-ev5t a 8_w-po9t40 ,f a 8happ5+0 9 ?ree-dim5sional space1 physics becomes1 z x 71 an 8exi/5ce0 9 ! f\r-dim5sional 8_w40 ,?is f\r-dim5sional 8_w0 be>s a close simil>ity to ! ?ree-dim5sional 8space0 ( 7,euclidean7 analytical geometry4 ,if we introduce into ! latt] a new ,c>tesian co-ord9ate sy/em 7x#a1 x#b1 x#c7 ) ! same orig91 !n x#a1 x#b1 x#c1 are l9e> homog5e\s functions ( x#a1 x#b1 x#c :i* id5tically satisfy ! equation x#ab x#bb x#cb x#ab x#bb x#cb ,! analogy ) 7#ab7 is a complete "o4 ,we c reg>d ,m9k[ski's 8_w0 9 a =mal mann] z a f\r-dim5sional ,euclidean space 7) an imag9>y "t coord9ate72 ! ,lor5tz trans=mation corresponds to a 8rotation0 ( ! co-ord9ate sy/em 9 ! f\r-dim5sional 8_w40 ,,app5dix ,,iii ,,! ,,exp]im5tal ,,confirmation ,,( ,,! ,,g5]al ,,!ory ,,( ,,relativity ,f a sy/ematic !oretical po9t ( view1 we may imag9e ! process ( evolution ( an empirical sci5ce to be a cont9u\s process ( induction4 ,!ories are evolv$ & are express$ 9 %ort compass z /atem5ts ( a l>ge numb] ( individual obs]vations 9 ! =m ( empirical laws1 f :i* ! g5]al laws c be asc]ta9$ by comp>ison4 ,reg>d$ 9 ?is way1 ! developm5t ( a sci5ce be>s "s resemblance to ! compilation ( a classifi$ catalogue4 ,x is1 z x 71 a purely empirical ent]prise4 ,b ?is po9t ( view by no means embraces ! :ole ( ! actual process2 = x slurs ov] ! important "p play$ by intuition & d$uctive ?\ght 9 ! developm5t ( an exact sci5ce4 ,z soon z a sci5ce has em]g$ f its initial /ages1 !oretical advances are no long] a*iev$ m]ely by a process ( arrangem5t4 ,guid$ by empirical data1 ! inve/igator r develops a sy/em ( ?\ght :i*1 9 g5]al1 is built up logically f a small numb] ( fundam5tal assumptions1 ! s-call$ axioms4 ,we call su* a sy/em ( ?\ght a !ory4 ,! !ory f9ds ! ju/ification = its exi/5ce 9 ! fact t x correlates a l>ge numb] ( s+le obs]vations1 & x is j "h t ! 8tru?0 ( ! !ory lies4 ,correspond+ to ! same complex ( empirical data1 !re may be sev]al !ories1 :i* diff] f "o ano!r to a consid]able ext5t4 ,b z reg>ds ! d$uctions f ! !ories :i* are capable ( be+ te/$1 ! agreem5t betwe5 ! !ories may be s complete t x becomes difficult to f9d any d$uctions 9 :i* ! two !ories diff] f ea* o!r4 ,z an example1 a case ( g5]al int]e/ is available 9 ! prov9ce ( biology1 9 ! ,d>w9ian !ory ( ! developm5t ( species by selection 9 ! /ruggle = exi/5ce1 & 9 ! !ory ( developm5t :i* is bas$ on ! hypo!sis ( ! h]$it>y transmission ( acquir$ *>act]s4 ,we h ano!r in/ance ( f>-rea*+ agreem5t betwe5 ! d$uctions f two !ories 9 ,newtonian me*anics on ! "o h&1 & ! g5]al !ory ( relativity on ! o!r4 ,?is agreem5t goes s f>1 t up to ! pres5t we h be5 able to f9d only a few d$uctions f ! g5]al !ory ( relativity :i* are capable ( inve/igation1 & to :i* ! physics ( pre-relativity days does n also lead1 & ?is despite ! pr(\nd diff]5ce 9 ! fundam5tal assumptions ( ! two !ories4 ,9 :at foll[s1 we %all aga9 consid] ^! important d$uctions1 & we %all also discuss ! empirical evid5ce app]ta9+ to !m :i* has hi!rto be5 obta9$4 7a7 ,motion ( ! ,p]ihelion ( ,m]cury ,accord+ to ,newtonian me*anics & ,newton's law ( gravitation1 a planet :i* is revolv+ r\nd ! sun w\ld describe an ellipse r\nd ! latt]1 or1 m correctly1 r\nd ! common c5tre ( gravity ( ! sun & ! planet4 ,9 su* a sy/em1 ! sun1 or ! common c5tre ( gravity1 lies 9 "o ( ! foci ( ! orbital ellipse 9 su* a mann] t1 9 ! c\rse ( a planet-ye>1 ! di/ance sun-planet gr[s f a m9imum to a maximum1 & !n decreases aga9 to a m9imum4 ,if in/ead ( ,newton's law we ins]t a some:at diff]5t law ( attraction into ! calculation1 we f9d t1 accord+ to ?is new law1 ! motion w\ld /ill take place 9 su* a mann] t ! di/ance sun-planet exhibits p]iodic v>iations2 b 9 ?is case ! angle describ$ by ! l9e jo9+ sun & planet dur+ su* a p]iod 7f p]ihelion--close/ proximity to ! sun--to p]ihelion7 w\ld diff] f #cfj4 ,! l9e ( ! orbit w\ld n !n be a clos$ "o b 9 ! c\rse ( "t x w\ld fill up an annul> "p ( ! orbital plane1 viz4 betwe5 ! circle ( lea/ & ! circle ( greate/ di/ance ( ! planet f ! sun4 ,accord+ also to ! g5]al !ory ( relativity1 :i* diff]s ( c\rse f ! !ory ( ,newton1 a small v>iation f ! ,newton-,kepl] motion ( a planet 9 its orbit %\ld take place1 & 9 su* away1 t ! angle describ$ by ! radius sun-planet betwe5 "o p]helion & ! next %\ld exce$ t correspond+ to "o complete revolution by an am\nt giv5 by image#jea 7,n4,b4--,"o complete revolution corresponds to ! angle #b 9 ! absolute angul> measure cu/om>y 9 physics1 & ! above expression giv5 ! am\nt by :i* ! radius sun-planet exce$s ?is angle dur+ ! int]val betwe5 "o p]ihelion & ! next47 ,9 ?is expression a repres5ts ! major semi-axis ( ! ellipse1 e its ecc5tricity1 c ! velocity ( light1 & ,t ! p]iod ( revolution ( ! planet4 ,our result may also be /at$ z foll[s3 ,accord+ to ! g5]al !ory ( relativity1 ! major axis ( ! ellipse rotates r\nd ! sun 9 ! same s5se z ! orbital motion ( ! planet4 ,!ory requires t ?is rotation %\ld am\nt to #dc seconds ( arc p] c5tury = ! planet ,m]cury1 b = ! o!r ,planets ( our sol> sy/em its magnitude %\ld be s small t x w\ld necess>ily escape detection4#bf #bf ,especially s9ce ! next planet ,v5us has an orbit t is almo/ an exact circle1 :i* makes x m difficult to locate ! p]ihelion ) precision4 ,9 po9t ( fact1 a/ronom]s h f\nd t ! !ory ( ,newton does n suffice to calculate ! obs]v$ motion ( ,m]cury ) an exactness correspond+ to t ( ! delicacy ( obs]vation atta9able at ! pres5t "t4 ,aft] tak+ acc\nt ( all ! di/urb+ influ5ces ex]t$ on ,m]cury by ! rema9+ planets1 x 0 f\nd 7,lev]ri]3 #ahei2 & ,newcomb3 #ahie7 t an unexpla9$ p]ihelial movem5t ( ! orbit ( ,m]cury rema9$ ov]1 ! am\nt ( :i* does n diff] s5sibly f ! above m5tion$ #dc seconds ( arc p] c5tury4 ,! unc]ta9ty ( ! empirical result am\nts to a few seconds only4 7b7 ,deflection ( ,light by a ,gravitational ,field image#jeb ,9 ,section ,,xxii x has be5 already m5tion$ t accord+ to ! g5]al !ory ( relativity1 a ray ( light w exp]i5ce a curvature ( its pa? :5 pass+ ?r\gh a gravitational field1 ?is curvature be+ simil> to t exp]i5c$ by ! pa? ( a body :i* is project$ ?r\gh a gravitational field4 ,z a result ( ?is !ory1 we %\ld expect t a ray ( light :i* is pass+ close to a heav5ly body w\ld be deviat$ t[>ds ! latt]4 ,= a ray ( light :i* passes ! sun at a di/ance ( sun-radii f its c5tre1 ! angle ( deflection 77 %\ld am\nt to image#jec ,x may be add$ t1 accord+ to ! !ory1 half ( ?is deflection is produc$ by ! ,newtonian field ( attraction ( ! sun1 & ! o!r half by ! geometrical modification 78curvature07 ( space caus$ by ! sun4 ,?is result admits ( an exp]im5tal te/ by means ( ! photographic regi/ration ( />s dur+ a total eclipse ( ! sun4 ,! only reason :y we mu/ wait = a total eclipse is because at e o!r "t ! atmosph]e is s /rongly illum9at$ by ! light f ! sun t ! />s situat$ ne> ! sun's disc are invisible4 ,! pr$ict$ effect c be se5 cle>ly f ! accompany+ diagram4 ,if ! sun 7,s7 7 n pres5t1 a /> :i* is practically inf9itely di/ant w\ld be se5 9 ! direction ,d#a1 z obs]v$ front ! e>?4 ,b z a consequ5ce ( ! deflection ( light f ! /> by ! sun1 ! /> w be se5 9 ! direction ,d#b1 i4e4 at a some:at great] di/ance f ! c5tre ( ! sun ?an corresponds to its real position4 ,9 practice1 ! "q is te/$ 9 ! foll[+ way4 ,! />s 9 ! neighb\rhood ( ! sun are photograph$ dur+ a sol> eclipse4 ,9 addition1 a second photograph ( ! same />s is tak5 :5 ! sun is situat$ at ano!r position 9 ! sky1 i4e4 a few mon?s e>li] or lat]4 ,z comp>$ ) ! /&>d photograph1 ! positions ( ! />s on ! eclipse-photograph ought to appe> displac$ radially outw>ds 7away f ! c5tre ( ! sun7 by an am\nt correspond+ to ! angle a4 ,we are indebt$ to ! ,briti% ,royal ,society & to ! ,royal ,a/ronomical ,society = ! inve/igation ( ?is important d$uction4 ,undaunt$ by ! fir/ _w w> & by difficulties ( bo? a mat]ial & a psy*ological nature ar\s$ by ! w>1 ^! societies equipp$ two exp$itions--to ,sobral 7,brazil71 & to ! isl& ( ,pr9cipe 7,we/ ,africa7--& s5t sev]al ( ,brita9's mo/ celebrat$ a/ronom]s 7,edd+ton1 ,cott+ham1 ,crommel91 ,davidson71 9 ord] to obta9 photographs ( ! sol> eclipse ( #bi? ,may1 #aiai4 ,! relative discrepancies to be expect$ betwe5 ! /ell> photographs obta9$ dur+ ! eclipse & ! comp>ison photographs am\nt$ to a few hundr$?s ( a millimetre only4 ,?us great accuracy 0 necess>y 9 mak+ ! adju/m5ts requir$ = ! tak+ ( ! photographs1 & 9 _! subsequ5t measurem5t4 ,! results ( ! measurem5ts confirm$ ! !ory 9 a ?or\ghly satisfactory mann]4 ,! rectangul> compon5ts ( ! obs]v$ & ( ! calculat$ deviations ( ! />s 79 seconds ( arc7 are set =? 9 ! foll[+ table ( results3 image#jed 7c7 ,displacem5t ( ,spectral ,l9es ,t[>ds ! ,r$ ,9 ,section ,,xxiii x has be5 %[n t 9 a sy/em ,k :i* is 9 rotation ) reg>d to a ,galileian sy/em ,k1 clocks ( id5tical con/ruction1 & :i* are consid]$ at re/ ) respect to ! rotat+ ref]5ce-body1 g at rates :i* are dep5d5t on ! positions ( ! clocks4 ,we %all n[ exam9e ?is dep5d5ce quantitatively4 ,a clock1 :i* is situat$ at a di/ance r f ! c5tre ( ! disc1 has a velocity relative to ,k :i* is giv5 by v r1 :]e repres5ts ! angul> velocity ( rotation ( ! disc ,k ) respect to ,k4 ,if v#j1 repres5ts ! numb] ( ticks ( ! clock p] unit "t 78rate0 ( ! clock7 relative to ,k :5 ! clock is at re/1 !n ! 8rate0 ( ! clock 7v7 :5 x is mov+ relative to ,k ) a velocity v1 b at re/ ) respect to ! disc1 w1 9 accordance ) ,section ,,xii1 be giv5 by image#jee or ) suffici5t accuracy by image#jef ,?is expression may also be /at$ 9 ! foll[+ =m3 image#jeg ,if we repres5t ! diff]5ce ( pot5tial ( ! c5trifugal =ce betwe5 ! position ( ! clock & ! c5tre ( ! disc by 1 i4e4 ! "w1 consid]$ negatively1 :i* mu/ be p]=m$ on ! unit ( mass aga9/ ! c5trifugal =ce 9 ord] to transport x f ! position ( ! clock on ! rotat+ disc to ! c5tre ( ! disc1 !n we h image#jeh ,f ?is x foll[s t image#jei ,9 ! fir/ place1 we see f ?is expression t two clocks ( id5tical con/ruction w g at diff]5t rates :5 situat$ at diff]5t di/ances f ! c5tre ( ! disc4 ,?is result is also valid f ! /&po9t ( an obs]v] :o is rotat+ ) ! disc4 ,n[1 z judg$ f ! disc1 ! latt] is 9 a gravitational field ( pot5tial 1 h5ce ! result we h obta9$ w hold q g5]ally = gravitational fields4 ,fur!rmore1 we c reg>d an atom :i* is emitt+ spectral l9es z a clock1 s t ! foll[+ /atem5t w hold3 ,an atom absorbs or emits light ( a frequ5cy :i* is dep5d5t on ! pot5tial ( ! gravitational field 9 :i* x is situat$4 ,! frequ5cy ( an atom situat$ on ! surface ( a heav5ly body w be some:at less ?an ! frequ5cy ( an atom ( ! same elem5t :i* is situat$ 9 free space 7or on ! surface ( a small] cele/ial body74 ,n[ - ,k 7,m_/r71 :]e ,k is ,newton's con/ant ( gravitation1 & ,m is ! mass ( ! heav5ly body4 ,?us a displacem5t t[>ds ! r$ ought to take place = spectral l9es produc$ at ! surface ( />s z comp>$ ) ! spectral l9es ( ! same elem5t produc$ at ! surface ( ! e>?1 ! am\nt ( ?is displacem5t be+ image#jfj ,= ! sun1 ! displacem5t t[>ds ! r$ pr$ict$ by !ory am\nts to ab\t two million?s ( ! wave-l5g?4 ,a tru/wor?y calculation is n possible 9 ! case ( ! />s1 because 9 g5]al nei!r ! mass ,m nor ! radius r are kn[n4 ,x is an op5 "q :e!r or n ?is effect exi/s1 & at ! pres5t "t 7#aibj7 a/ronom]s are work+ ) great zeal t[>ds ! solution4 ,ow+ to ! smallness ( ! effect 9 ! case ( ! sun1 x is difficult to =m an op9ion z to its exi/5ce4 ,:]eas ,grebe & ,ba*em 7,bonn71 z a result ( _! own measurem5ts & ^? ( ,ev]%$ & ,s*w>zs*ild on ! cyanog5 b&s1 h plac$ ! exi/5ce ( ! effect almo/ beyond d\bt1 :ile o!r inve/igators1 p>ticul>ly ,/4 ,john1 h be5 l$ to ! opposite op9ion 9 consequ5ce ( _! measurem5ts4 ,mean displacem5ts ( l9es t[>ds ! less refrangible end ( ! spectrum are c]ta9ly reveal$ by /ati/ical inve/igations ( ! fix$ />s2 b up to ! pres5t ! exam9ation ( ! available data does n all[ ( any def9ite decision be+ arriv$ at1 z to :e!r or n ^! displacem5ts are to be ref]r$ 9 reality to ! effect ( gravitation4 ,! results ( obs]vation h be5 collect$ toge!r1 & discuss$ 9 detail f ! /&po9t ( ! "q :i* has be5 engag+ our att5tion "h1 9 a pap] by ,e4 ,freundli* entitl$ 8,zur ,prfung d] allgeme95 ,relativitts-,!orie0 7,die ,naturwiss5s*aft51 #aiai1 ,no4 #ce1 p4 #ebj3 ,julius ,spr+]1 ,b]l974 ,at all ev5ts1 a def9ite decision w be rea*$ dur+ ! next few ye>s4 ,if ! displacem5t ( spectral l9es t[>ds ! r$ by ! gravitational pot5tial does n exi/1 !n ! g5]al !ory ( relativity w be unt5able4 ,on ! o!r h&1 if ! cause ( ! displacem5t ( spectral l9es be def9itely trac$ to ! gravitational pot5tial1 !n ! /udy ( ?is displacem5t w furni% u ) important in=mation z to ! mass ( ! heav5ly bodies4#bg #bg ,! displacem5t ( spectral l9es t[>ds ! r$ end ( ! spectrum 0 def9itely e/abli%$ by ,adams 9 #aibd1 by obs]vations on ! d5se companion ( ,sirius1 = :i* ! effect is ab\t ?irty times great] ?an = ! ,sun4 ,r4,w4,l4--translator ,,app5dix ,,iv ,,! ,,/ructure ,,( ,,space ,,accord+ ,,to ,,! ,,g5]al ,,!ory ,,( ,,relativity 7,,supplem5t>y ,,to ,,section ,,xxxii7 ,s9ce ! publication ( ! fir/ edition ( ?is little book1 our k ab\t ! /ructure ( space 9 ! l>ge 78cosmological problem07 has _h an important developm5t1 :i* ought to be m5tion$ ev5 9 a popul> pres5tation ( ! subject4 ,my orig9al consid]ations on ! subject 7 bas$ on two hypo!ses3 7#a7 ,!re exi/s an av]age d5sity ( matt] 9 ! :ole ( space :i* is ev]y:]e ! same & diff]5t f z]o4 7#b7 ,! magnitude 78radius07 ( space is indep5d5t ( "t4 ,bo? ^! hypo!ses prov$ to be consi/5t1 accord+ to ! g5]al !ory ( relativity1 b only aft] a hypo!tical t]m 0 add$ to ! field equations1 a t]m :i* 0 n requir$ by ! !ory z su* nor did x seem natural f a !oretical po9t ( view 78cosmological t]m ( ! field equations074 ,hypo!sis 7#b7 appe>$ unavoidable to me at ! "t1 s9ce ,i ?\ght t "o w\ld get into bottomless speculations if "o dep>t$ f x4 ,h[ev]1 already 9 ! 'tw5ties1 ! ,russian ma!matician ,fri$man %[$ t a diff]5t hypo!sis 0 natural f a purely !oretical po9t ( view4 ,he realiz$ t x 0 possible to pres]ve hypo!sis 7#a7 )\t introduc+ ! less natural cosmological t]m into ! field equations ( gravitation1 if "o 0 ready to drop hypo!sis 7#b74 ,namely1 ! orig9al field equations admit a solution 9 :i* ! 8_w radius0 dep5ds on "t 7exp&+ space74 ,9 t s5se "o c say1 accord+ to ,fri$man1 t ! !ory dem&s an expansion ( space4 ,a few ye>s lat] ,hubble %[$1 by a special inve/igation ( ! extra-galactic nebulae 78milky ways071 t ! spectral l9es emitt$ %[$ a r$ %ift :i* increas$ regul>ly ) ! di/ance ( ! nebulae4 ,?is c be int]pret$ 9 reg>d to our pres5t k only 9 ! s5se ( ,doppl]'s pr9ciple1 z an expansive motion ( ! sy/em ( />s 9 ! l>ge--z requir$1 accord+ to ,fri$man1 by ! field equations ( gravitation4 ,hubble's discov]y c1 !re=e1 be consid]$ to "s ext5t z a confirmation ( ! !ory4 ,!re does arise1 h[ev]1 a /range difficulty4 ,! int]pretation ( ! galactic l9e-%ift discov]$ by ,hubble z an expansion 7:i* c h>dly be d\bt$ f a !oretical po9t ( view71 leads to an orig9 ( ?is expansion :i* lies 8only0 ab\t #aji ye>s ago1 :ile physical a/ronomy makes x appe> likely t ! developm5t ( individual />s & sy/ems ( />s takes consid]ably long]4 ,x is 9 no way kn[n h[ ?is incongruity is to be ov]come4 ,i fur!r want to rem>k t ! !ory ( exp&+ space1 toge!r ) ! empirical data ( a/ronomy1 p]mit no decision to be rea*$ ab\t ! f9ite or inf9ite "* ( 7?ree-dim5sional7 space1 :ile ! orig9al 8/atic0 hypo!sis ( space yield$ ! closure 7f9it5ess7 ( space4 ,k co-ord9ate sy/em x1 y two-dim5sional co-ord9ates x1 y1 z ?ree-dim5sional co-ord9ates x1 y1 z1 t f\r-dim5sional co-ord9ates t "t ,i di/ance v velocity ,f =ce ,g gravitational field