On the formulation of the laws of Nature with five homogeneous coordinates

Transcription

1 Über de Formuerung der Naturgesetze mt fünf homogenen Koordnaten Te I: Kasssche Theore Ann d Phys 18 (1933) On the formuaton of the aws of Nature wth fve homogeneous coordnates Part I: cassca theory By W Pau 1 Introducton Defnton of tensors n homogeneous coordnates 3 Metrc Connecton wth nhomogeneous coordnates Decomposton of homogeneous tensors nto affne parts 4 Covarant dfferentaton by means of the three-ndex symbos Γ 5 Parae dspacement of λ vectors Geodetc nes 6 Curvature 7 The form of the aws of Nature Varatona prncpe Appendx: Generazaton of the Ansatz for the Γ λ 1 Introducton It s we nown that Kauza and Ken have succeeded n formuatng fed aws of gravtaton and eectrcty (at east n the absence of charges and ponderabe masses) n a unfed way and n addton presentng the aw of moton for a charged mass pont as a geodetc ne n the fve-dmensona contnuum Ths achevement of puttng the foundatons of physcs on a geometrca footng stands n star contrast wth the foowng shortcomngs whch eep many theoretcans from acceptng the dea of a ffth dmenson: 1 It must be formuated n such a way that the component g of the fvedmensona metrc tensors sha not depend upon the ffth coordnate x 5 e they sha be functons of ony the frst four coordnates By means of ths addtona condton - the so-caed cynder condton dsturbs the genera covarance of the theory and the ffth coordnates appears to be an artfca appendage that must utmatey dsappear The frst ten components of g namey g wth = 1 4 can be dentfed wth the ordnary metrc tensor and the four extra g 5 ( = 1 4) components can be dentfed wth the eectromagnetc potentas (up to a numerca factor) Snce a physca nterpretaton of the component g 00 can be attaned one must arbtrary set g 00 = 1 whch represents a new non-nvarant condton Ths reduces the number of equatons for the g from 15 whch s expressed by the vanshng of the contracted curvature tensor to the correct number 14 namey the 10 equatons for the g ( = 1 4) and the 4 Maxwe equatons (There then exst 5 dfferenta denttes between them) An essenta advance was then made by Ensten and Mayer 1 ) In ths theory the ntroducton of a ffth coordnate was competey absent aong wth any cynder Transated by DH Dephench 1 ) A Ensten and W Mayer Ber Ber (1931 pp 541

2 W Pau On the formuaton of the aws of Nature etc I condton; rather every pont of the four-dmensona contnuum was assocated wth a fve-dmensona vector space A partcuar reatonshp between these vectors s then posed axomatcay n the form of ther parae dspacement and how t reates to the ordnary Remannan parae dspacement of the ordnary vectors of the parae dspacement The eectromagnetc potentas are competey absent from ths theory but ony the fed strengths Ths theory aso has certan forma mperfectons that were not present n the Kauza- Ken formuaton but appeared for the frst tme n t namey: 1 The fed equatons can not be derved from a varatona prncpe From ths one nfers that such fed equatons that satsfy mportant dfferenta denttes (whch a varatona prncpe automatcay guarantees) seem a bt artfca whch s aso expressed n the form of the fed equatons 1 ) The frst system of Maxwe equatons (whch s equvaent to the possbty of dervng the fed strengths from potentas) does not foow from the assumed structure of space n ths theory but must be postuated expcty (n whch t can generay be connected wth the curvature) These mperfectons are not very momentous n themseves but one woud not e to renounce the advantages that were aready present n the Ken-Kauza theory Here we now ntroduce another way of presentng the Ken-Kauza theory namey the projectve way In ths presentaton the contnuum s regarded as four-dmensona but as n projectve geometry fve homogeneous coordnates (whch w be denoted by = 1 5) w be ntroduced; e a coordnates that dffer by the same factor beong to the same pont of the contnuum A tensors (or as one prefers to say n ths case: projectors ) must then be homogeneous functons (of varous dfferng degrees) of the coordnates Ths way of regardng thngs was frst apped to physcs by Veben and Hoffmann ) and ndeed to the nterpretaton of the Ken-Kauza theory However these authors choose a formuaton that as a consequence of an unnecessary specazaton of the coordnate system dstngushes the ffth coordnate n a manner that s competey smar to the usua one that Ken-Kauza ntroduced by the cynder condton and that therefore dd not enter nto the atter theory n an essenta way The advance made by van Dantzg 3 ) was to thoroughy examne projectors wth homogeneous coordnates aong wth ther covarant dfferentaton by means of the three-ndex symbos Γ λ geodetc nes and the nvarant form: g of the metrc that was ntroduced wth the fundamenta tensor g = g Fnay Schouten and van Dantzg 4 ) gave a formuaton to fed theory wth the hep of ths genera projectve dfferenta geometry (here we w next dscuss ony the cassca part 1 ) Cf the transton from P ιp to U ιp n oc ct 5 eq (41) ) O Veben and B Hoffmann Phys Rev 36 pp Summary by O Veben Reatvtätstheore Bern 1933 Further terature s contan theren 3 ) D van Dantzg Math Ann 106 pp Confer the geodetc nes n partcuar; Amst Proc 35 pp 54 and ) JA Schouten and D van Dantzg Zetschr f Phys 78 pp ; furthermore Ann Math (3) 34 pp Ctatons to the earer wors can be found n these wors

3 W Pau On the formuaton of the aws of Nature etc I 3 that corresponds to the absence of matera partces) that combnes a of the advantages of Ken-Kauza and Ensten-Mayer whe avodng a of the dsadvantages The crcumstance that a projectors depend homogeneousy on the coordnates renders any partcuar cynder condton superfuous Furthermore the condton g 55 = 1 of the Ken-Kauza theory appears here n the nvarant form: g = ±1 and can be stated as a normazaton of the g The estabshment of ths condton by the varaton: δ g = 0 further reduces the number of fed equatons n a natura way to ther correct number The scaar curvature appears n the acton ntegra as a Lagrangan functon and s then vares wth the g Fnay one does not need to ntroduce the eectromagnetc potentas themseves nto the theory but they can be repaced (up to a numerca factor) by: = g The vadty of the frst system of Maxwe equatons s then sef-evdent We sha preserve these essenta resuts of Schouten and van Dantzg here Therefore as far as the three-ndex symbos Γ λ are concerned we tae the poston that the symmetry requrement: Γ λ = Γ λ (whch s satsfed for Ken-Kauza and Veben-Hoffmann) s the most natura one for these quanttes Together wth the requrement that the covarant dervatves of the g t then eads to the usua expresson: Γ λ = 1 g g g g + λ λ More generay Schouten and van Dantzg ntroduced asymmetrc Γ λ Snce ths ncreases the number of possbe mathematca nvarants wthout any beneft to the physca nterpretaton (the partcuar grounds for another specazaton of the genera Γ that Schouten and van Dantzg mantaned do not seem vad to us) we sha λ forego ths gratutous generazaton here and menton t ony ncdentay (cf the Appendx) The connecton between projectors and ordnary tensors n nhomogeneous coordnates has a far-reachng forma anaogy wth the connecton between tensors on fve-spaces and those on the four-spaces of Ensten-Mayer In the form presented n 4 and 6 ths anaogy s partcuar obvous We must apoogze for the fact that the cassca part of the theory w be represented so comprehensvey athough t devates from the wor of Schouten and van

4 W Pau On the formuaton of the aws of Nature etc I 4 Dantzg n severa essenta ponts On the one hand ths s done n order to have a certan foundaton foowng secton and on the other hand n order to mae understandabe those specazed facts that are famar from genera reatvty but not the extended one that s dscussed here Defnton of tensors wth homogeneous coordnates The descrpton of an n-dmensona contnuum by n + 1 homogeneous coordnates 1 n+1 can naturay permt ony such transformatons that mae the new coordnates homogeneous functons of the frst degree of the od coordnates 1 ) That s: must satsfy: = f ( 1 n+1 ) f ( 1 n+1 ) = f ( 1 n+1 ) n whch s an arbtrary functon of the Accordng to the Euer homogenety condton ths requrement s equvaent to: (1) = In order for us to now proceed to the defnton of vectors we next defne a contravarant vector a (a covarant vector b resp) by the transformaton aws: () a = a (3) b = b such that: (4) a b = a b = c s a scaar For ths reason vector feds and scaar feds that are defned by homogeneous coordnates w be subjected to a further requrement that specfes not ther behavor under coordnate transformatons but ther dependence on the coordnates n a fxed system of reference Namey we w aow ony those feds whose components are homogeneous functons of the coordnates Ths requrement s nvarant under the homogeneous coordnate transformatons (1) that are consdered here and from ths the degree of the homogeneous functons that represent the fed components remans nvarant snce the coordnate transformatons are homogeneous of frst degree For the sae of ater appcatons the components of a contravarant vector are aways homogeneous of degree +1 those of a covarant vector are of degree 1 and 1 ) Here and n the seque Gree ndces sha aways range from 1 to n + 1 and Latn ones from 1 to n Later correspondng to the four-dmensona spacetme contnuum we sha set n = 4 n + 1 = 5

5 W Pau On the formuaton of the aws of Nature etc I 5 scaars are of degree 0 From (4) ths conventon s usuay n harmony wth the constructon of scaars by contracton The foowng are then vad: (a) (3a) (4a) a b c = a = b = 0 It s a characterstc stuaton n the theory of homogeneous coordnates that these coordnates themseves (not ther dfferentas) defne a (contravarant) vector fed Then accordng to (1) they satsfy n fact the necessary condtons () for a contravarant vector fed [whereas reatons (a) are obvousy satsfed dentcay] By contrast the dfferentas d may not be used as a vector fed snce they are not homogeneous functons of frst degree n the coordnates We can now easy generaze the defnton of vector feds and scaar feds to arbtrary tensor feds A genera mxed tensor: satsfes the transformaton aw: T σ (5) T σ = σ σ T σ Furthermore t w be assumed that the components of the tensor fed T are σ homogeneous functons of the and aso of degree r whch s equa to the dfference between the number of upper and ower ndces of T One then has: σ (5a) a T σ a = rt σ It shoud be remared that the voume eement: d 1 d d n+1 (as opposed to the dfferentas d themseves) can possby vansh If t s mutped by the transformaton: (6) = n whch r s an arbtrary homogeneous functon of nu degree of the such that:

6 W Pau On the formuaton of the aws of Nature etc I 6 (6a) = 0 and smary for n+1 Proof: The asserton s equvaent to the statement that under the transformaton (6) the determnant: D () = = δ + has the vaue n+1 or the statement that: 1 f () = n 1 + D () = 1 δ + has the vaue 1 In order to prove ths we consder: f (t) f ( t 1 ) = δ + to be a functon of t Ths functon has the foowng propertes: One then has: whch vanshes from (6a); a) f (0) = 1 f (1) = f () b) f (0) = 0 f (0) = c) f (t 1 + t ) = f (t 1 ) f(t ) 1 1 m δ + t 1 t 0 = t 1 whch foows drecty from the defnton of f (t) and the mutpcaton aws for determnants From ths t further foows that: f ( t) = f(t) f (0) hence from (b): f ( t) = 0 f(t) = const = f(0) = 1 Thus: D () = n+1 f(1) = n+1 QED The possbty of defnng a meanngfu voume eement from homogeneous coordnates s mportant for the physca appcatons snce t aows one to defne varatona prncpes 3 Metrc Connecton wth nhomogeneous coordnates Decomposton of homogeneous tensors nto affne parts We ntroduce a metrc tensor g n our space that satsfes the usua symmetry condton g = g and dstngushes the quadratc form: g = F

7 W Pau On the formuaton of the aws of Nature etc I 7 Correspondng to our conventon g s homogeneous of degree and the scaar F s homogeneous of nu degree Physcs gves us a specazaton of ths Ansatz n whch the scaar F s set equa to a constant Ths constant can then be normazed to +1 or 1 such that we can set: (7) g = ε (ε = +1 or 1) One can aso express ths n such a way that ony the quotents g /F n whch F = g sha enter nto the equatons thus defned Ths specazaton s anaogous to the conventon that g 55 = 1 n the Ken-Kauza theory Ths specazaton may seem arbtrary from a geometrca standpont but t s n any event an advantage of the use of homogeneous coordnates that the specazaton can be formuated n an nvarant way The metrc tensor can therefore be used to rase or ower tensor ndces n an nvarant manner For physca appcatons t s essenta for us to possess a method for dervng the ordnary tensors n n nhomogeneous coordnates x from the homogeneous tensors n the n + 1 homogeneous coordnates such that the x are arbtrary homogeneous functons of nu degree of the : (8) x = f ( 1 n+1 ) The dervatves: satsfy the homogenety condton: = x (9) = x = 0 and transform under arbtrary pont transformatons of the x e a contravarant vector and e a covarant vector under homogeneous transformaton of the n whch (9) remans nvarant throughout Thus the covarant vector a accordng to: can be used to frst assocate every covarant vector a wth a (10) a = a n whch from (9): (10a) a = 0 (whch s expressed by the overbar on a) and second to assocate every contravarant vector a wth a contravarant vector a accordng to: (11) a = a n whch a vectors of the form w be assocated wth the nu vector We can arrange that the quotent / +1 can be expressed n terms of the x unquey The n

8 W Pau On the formuaton of the aws of Nature etc I 8 reatons (9) are then the ony dependences that exst between the the ony souton of the equaton: v = 0 of the specfed form s: v = Any speca a covarant vector that satsfes the reaton (10a) namey: a = 0 and we have that can then aso be unquey assocated wth a vector a that satsfes (10) Namey f one sets: a = a then one must have: (1) = δ From the statements above the new coeffcents are unquey determned by these equatons precsey up to an addtona term accordng to: = + whch from (10a) actuay annus the vector a Furthermore accordng to (1) the vector a s by way of: a = a assocated wth a set of vectors of the form: a + (a ) that a satsfy equaton (11) Up t now the matrx g was not used n the coeffcents and at a Now we can mae the defnton of the unque by assocatng the same a for whch one has: (13a) a a g = 0 wth a vectors of the form a + wth a fxed and arbtrary If we correspondngy estabsh the unquey by the condton that enters nto (1): (1a) g = 0

9 W Pau On the formuaton of the aws of Nature etc I 9 then we obtan the unque assocaton of a wth a accordng to: (13) a = a ( a = 0 ) and the unque assocaton of a wth a accordng to: (14) a = a n whch s assocated wth the nu vector From (11) (14) and the fact that a = g a and consderaton of (1a) t foows that: a = ( g ) a If one then demands that when a = g a the assocated vectors a a must satsfy the metrc reaton: a = g a then t foows that the metrc tensor n nhomogeneous coordnates s: (15) g = g We now compute the sums: d = whch defnes a mxed tensor of ran two n the space of Next from (1) we have: hence: hence: d = ( d δ ) = 0 d = ( d δ ) = 0 d = δ + = 0 = d = δ + d = 0 + ε = 0 = ε accordng (7) Thus fnay one has: (16) d The vector: satsfes the equatons: = = δ ε a = d a a = 0 a = a = a

10 W Pau On the formuaton of the aws of Nature etc I 10 hence we have: (17a) and ewse: (17b) a = d a = a a = d a = a Fnay from (15) and the fact that: t foows that: or: g = d d σ g σ = ( g = g δ ε ) ( δ ε σ ) g σ g = g ε (18) g = g + ε = g + ε One further confrms by way of: (18a) g = g + ε = g + ε the reaton: g g σ = d σ + ε σ = δ σ We now have the means to characterze each tensor unquey by affne parts Thus f a s unquey determned by means of the and through a = a and a = a then we have: (19a) a = a + ε a = a + ε a = d a + ε a just as a s determned through a = (19b) a = σ a and a = a accordng to: a + ε a = a + ε a = d a + ε a Lewse a tensor T of the second ran s characterzed by the four affne quanttes: and: T = T T = T (0) T = T (0) T = T (0)(0) For a symmetrc tensor T T s symmetrc and the vectors T (0) and T (0) are dentca; t s thus characterzed by the a symmetrc tensor a vector and a scaar n nhomogeneous coordnates For the fundamenta tensor g the vector g (0) vanshes n partcuar accordng to (1a) If T s sew-symmetrc then the scaar T (0)(0) s equa to nu

11 W Pau On the formuaton of the aws of Nature etc I 11 In genera a tensor wth N ndces gves rse to N affne tensors between whch certan near dependences can exst when the orgna tensor possesses a symmetry property We add a remar about the possbty of defnng a vector that satsfes one of these requrements from the dfferentas d whch do themseves satsfy the homogenety condton accordng to: (0) d = d d = d ε ( d ) In fact by repacng wth one has: d = d d( ) = d ( d + d) = d d = d snce w be annued by d Moreover one has: x (0a) d = d hence: (0b) d = dx = d x = dx A curve x = f (r) w be represented n homogeneous coordnates n the form: (1) = F (r) = f (r) n whch s an arbtrary homogeneous functon of nu degree n the Then one has: (1a) d dr = df dr = df dr whereas d /dr has no smpe meanng 1 ) 4 Covarant dfferentaton by means of the three-ndex symbo Γ As usua the covarant dfferenta cacuus w be ntroduced by means of a Γ λ - fed for whch one demands that: (a) a = a σ ; + Γλ a λ λ 1 ) The deveopments that were gven here are argey anaogous to the wor of Ensten and Mayer that was cted n The quantty that was denoted by Σ n that wor s denoted by d n ours and furthermore our coordnates pay the same roe as the vector A of Ensten-Mayer

12 W Pau On the formuaton of the aws of Nature etc I 1 (b) b ; = b λ Γ b λ The second equaton foows from the frst one by the requrement that: (a b ) ; a ; b + a b ; = ( a b ) The dfferenta cacuus s extended n a we-nown way for arbtrary mxed tensors by the demand that for two tensors A and B the product rue of ordnary dfferentaton s st vad: (A B ) ; = A ; B + B ; A Ths then yeds: A A ; = + ( ) n whch the sum has exacty as many summands as the tensor has ndces and ndeed: an upper ndex A corresponds to a summand + Γ λ A a ower ndex A corresponds to a summand Γ A λ Snce a ; s a tensor when a s a vector (from whch the tensor character of b ; and genera A ; foows) the Γ λ must obey the transformaton aws: λ (3) λ Γ = Γ λ λ + λ λ The ast term dsturbs the tensor character of the Γ λ Snce the homogenety requrement for the A ; foows from the one for the A the Γλ must be homogeneous of degree 1 as the genera demands From (3) t foows that part of Γ that s sew-symmetrc n the ndces λ whch s therefore determned by: λ (4) S λ s a tensor Further tensors (projectors resp) Π = 1 ( Γλ Γ λ ) and Q are defned by: (5a) a ; = Π a (5b) b ; = Π b (the atter reaton foows from the frst one due to the fact that:

13 W Pau On the formuaton of the aws of Nature etc I 13 (a b ) ; = 0 ) and by: (6) ; = Q From the defnton (a) ((b) resp) of a ; (b ; resp) t foows that: (7) λ Π = δ + Γ λ (8) Q λ = δ + Γ hence: (9) λ 1 ( Π Q ) = S λ λ A n between the Γ λ and the metrc w be gven here by the condton: (I) g ; = From ths and the fact that: g Γ g Γ g λ λ λ λ = 0 λ (30) Γ gλ = Γ (30a) S λ g λ (1) mpes the foowng reaton: (31) Γ = 1 g g g + λ Γ = g λ Γ = S S λ = g λ S + (S + S + S ) A n between the metrc and the λ Γ and the (and the ) s then produced by the foowng requrement: When: a = 0 hence a = a one must have: (II) a ; = a ; n whch a ; s defned n the norma way n terms of the Chrstoffe symbos nm accordng to: a a m ; = a x m

14 W Pau On the formuaton of the aws of Nature etc I 14 = = 1 g m m g g g + x x x n nm n m m n We can formuate the requrement (II) n a dfferent way when we defne a covarant dervatve ; accordng: (3) ; Ths mpes the product rue: ( a ) ; = λ m + Γλ m a ; + ; a b ; = ( b ) ; = ; b + b ; Lewse t foows that: (3a) λ m ; = + Γ λ m a ; = ( a ) ; = ; a + a ; ( b ) ; = b ; + ; b Thus (II) s equvaent to: (II ) m; or: (II ) = 0 ; = 0 By antsymmetrzaton n the and λ t then foows from the atter equaton that: (33) m λ S λ = 0 e the affne part of S λ must vansh However S λ s then aready determned by S λ and λ S λ whch can be expressed by way of Π dervatves 1 ) From (5) (6) t now foows that: and Q for a gven g and ts 1 ) The Π and Q (wth the ndces owered by means of the g ) yed ony the condtons: [the atter foows from (7)] and: Otherwse Π and Q are arbtrary Π = Π Q = Q = 0 Π = Q ( = ; )

15 W Pau On the formuaton of the aws of Nature etc I 15 a ; = a; + ε ( a ); ( a = a a = d a ) (34) a a ; = a + ε Π a + ε Q a + εaq and ewse: b (34a) b ; = b ; ε Π b + ε Q b εbq + In addton the g determne the Γ λ unquey by way of the feds Π and Q Obvousy our requrements are st much too genera to be usefu to physcs The smpest case that comes nto consderaton s the one for whch: (III) S = 0 e n whch the Γ λ are assumed to be symmetrc n and as one does n Remannan geometry Ths assumpton seems to be the most natura to us We woud e dscuss the more genera case that Schouten examned ater on as an appendx ( 7) If we accept the severe restrcton (III) then one has n partcuar: (31 ) Γ = Γ = 1 One then has: g g g + Γ λ λ = 1 g g g λ + λ λ Due to the homogenety condton one further has: λ g = g λ hence due to the fact that: (8 ) Π = g + Γ λ λ (9 ) Q = g + Γ λ λ one has n our case: (35) Q = Π = 1 g g λ λ λ λ If for the sae of what foows we ntroduce the fundamenta sew-symmetrc tensor: (36) = then we have: g = g λ λ λ

16 W Pau On the formuaton of the aws of Nature etc I 16 (35a) Q = Π = 1 = 1 Furthermore we have: = λ g λ + λ gλ = λ g λ + ( λ g λ ) + λ g λ hence due to (7): (37) = 0 If we then defne the sew-symmetrc tensor: (38) = = then one has conversey: (38a) = Accordng to (36) aso satsfes the reatons: (36a) + + = 0 From (18) n whch we have used (37) we can nfer the foowng: (39) + + = 0 In order to smpy the notaton we woud e assert that f three ndces are encosed wthn bracets n an expresson then what that means s that these ndces w be cyccay permuted and added together n the expresson One then wrtes eg (36a) and (39) as: = 0 = 0 resp [ ] [ ] Now t foows from (38a) and (36a) that: ( ) 0 = We have substtuted [ ] = 1 ( ) + x [ ] [ ] x = n the atter expresson Now however one has:

17 W Pau On the formuaton of the aws of Nature etc I 17 x x x x = x x = x x and t foows that: ( ) [ ] = 0 Thus one has: = = 0 x [ ] x [ ] from whch one mmedatey nfers that: = 0 x [ ] e one nfers the vadty of (39) There thus exsts a covarant vector f such that: f f (40) = x x From (36) (38a) and (40) t foows that: or: and snce: one has: f f x x f f x x n whch we have set: (41) f = From ths t utmatey foows that: x x = = ( f ) ( f ) = 0 f (f = 0) (4) = f + ε 1 F og F = f F + ε = 0 = 0

18 W Pau On the formuaton of the aws of Nature etc I 18 n whch the sgn ε s ncuded from whch: (4a) F = F hence F s homogeneous of frst degree Our metrc s then exacty as genera that t s (for a gven ) characterzed by g and the sew-symmetrc tensor that satsfes (39) One s tempted to dentfy the atter up to a proportonaty factor wth the tensor F = F for the eectromagnetc fed strength If κ s the Ensten gravtaton constant then: (43) f = κ F c has the dmensons of ength such that we can set: (44) = r f = r κ F c n whch r s a rea dmensoness numerca factor Wth: (44) may be extended to: F = 0 (44a) = r f = r κ F c It s satsfyng that wth ths geometrca meanng for the eectromagnetc fed strength the frst system of Maxwe equaton s automatcay satsfed As for the potenta ϕ whch s defned by: Φ Φ (45) F = x x ϕ ϕ κ f = ϕ = Φ x x c from (40) t can be dentfed wth f up to the factor 1/r: (46) f = r ϕ = r c κ Φ hence wth: (47) ϕ = ϕ ϕ ϕ (45a) f = ϕ = 0

19 W Pau On the formuaton of the aws of Nature etc I 19 (46a) κ f = rϕ = r Φ c f eϕ s defned ony up to an addtve gradent As Schouten has remared the theory can however be formuated n such a way that f andϕ do not appear expcty but the we-defned vector aways appears n pace of the -defned vector f 5 Parae dspacement of vectors Geodetc nes One can defne the parae dspacement of a vector a (b resp) aong a curve wth the hep of the modfed coordnate dfferentas that were defned by (0) (0b): namey: (48) or: (48a) d = d d = dx ( ) δ a = d a ; = dx a ; a λ = dx ; + ; Γλ a ( ) δb = d b ; = dx b ; b dx b λ = ; ; Γ λ resp From (34) one can aso wrte ths as: (49) δa =( a ; εa Q ) (49a) δb =( b ; b ε Q ) + dx + ε da + dx + ε db resp upon ntroducng the Remannan parae dspacement: R a δ = a ; dx R b δ = b ; dx and substtuton of Q = Q = 0: (50) δa R = δ a + εaq dx dx + ε da R (50a) δb = δ b + εbq dx dx + ε db

20 W Pau On the formuaton of the aws of Nature etc I 0 We now defne geodetc nes by sayng that the vector a s parae dspaced aong them n the drecton assocated wth: (51) dx dτ = a = a and postuate that the vector a does not change under ths moton We sha then have: (5) δ a dτ = d dτ a ; = 0 when (51) s true The atter s equvaent wth: (51a) a = dx dτ = ε a such that from (50) the requrement (5) decomposes nto: or: (53) and: (54) R δ dx dτ dτ + ε a Q m n d x dx dx + dτ mn dτ dτ da dτ = 0 dx = 0 dτ = ε a Q a = const dx dτ If we now ntroduce the heretofore unused reaton (35) ((35a) resp) and tae account of (44) then we have: (53a) m n d x dx dx + dτ mn dτ dτ = ε a Q m m dx = 1 εar dτ κ F c m m dx dτ From the sew-symmetry of F t foows from ths that: g dx dτ such that we can normaze to: dx (55) g dτ dx dτ = const dx dτ = 1

21 W Pau On the formuaton of the aws of Nature etc I 1 (The negatve sgn corresponds to the tmee character of the curve) Comparng ths wth the word nes of charged mass ponts shows that they corresponds qute neaty wth the generazed geodetc nes that we consder here when we set the arbtrary ntegraton constant a to: c e (56) ε a = κ r mc n whch e and m mean the charge and mass of the mass pont If we ntroduce the mpuse vector: (57) p = m g a dx = a = m g dτ + c e κ r mc then one has: δ p (58) dτ = 0 by: We further remar that n many wors the geodetc nes that satsfy (51) are defned a a ; = dx dτ + εa a ; = a δ dτ + ε a a ; = 0 nstead of by (5) snce that seems ess natura to us Furthermore from (4) and (46a) the vector p that s gven by (57) dffers by a gradent from: and not as one mght perhaps expect from: dx m g dτ + e c Φ We sha come bac to ths ater ( 7) dx m g dτ + e Φ c 6 Curvature When Γ λ s symmetrc n λ as we woud e to assume here the curvature tensor P σ s defned by: (59) ; ; a σ ; ; a σ = P σ a Snce: (a b ) ; ; σ (a b ) ; σ ; = (a ; ; σ a ; ; σ ) b a (b ; ; σ b ; ; ) = 0

22 W Pau On the formuaton of the aws of Nature etc I t foows that: (59a) b ; ; σ b ; σ ; = P σ b The defnton of P σ s sew-symmetrc n σ and carryng out the cacuatons yeds: Γ Γ (60) P σ = + Γ Γ Γ Γ σ tensor τ τ τ τ σ We woud now e to express the P σ n terms of the correspondng Remann R m In order to do ths t s more convenent to compute the expresson b ; ; σ b ; ; σ by usng the fact that: (59 ) Snce: R R ; ; m ; m; t foows: b b = b R m R b ; b; m b ; ; σ = b; ; m + b; R σ ; σ and snce from (3a) the symmetry of ; σ n and σ foows from that of the Γ λ we obtan from(59 ) : (61) b ; ; σ b ; σ ; = m R σ m b On the other hand from the fact that: one has: b = b b ; = ; b + b ; b ; ; σ b ; σ ; = ( ; ; σ ; σ ; ) b + ( b ; ; σ b ; σ ; ) Tang nto account (61) and (59a) t foows: (6) P σ On the other hand from (59) one has: R m σ m = ; ; σ ; ; σ ; σ ; = P σ hence from (36) and (35a): (63) P σ 1 = ( ; ; ) σ σ R ; ; σ

23 W Pau On the formuaton of the aws of Nature etc I 3 We can now compute the rght-hand sde of (6) f we now ; From(II ) however ; and: However one has: and: s determned f we now: ; ; ; = ; = 1 ; = P = 1 so one has: ε (64) ; = ( ) and t ewse foows that: (64a) ; = 1 ( + ) From ths one further fnds that: (65) ; ; σ ; σ ; { = ε ( + ) 1 4 σ σ σ + ( ) ( ) ε ε ; σ σ ; ; σ σ ; τ + ( ) 1 4 στ σ τ } From (6) (63) and (65) a bref cacuaton fnay gves the utmate formua for the curvature tensor: (66) m P σ = σ R m ε = ( σ + σ σ ) ε + ( ; σ σ ; ) ε { ( ; σ σ ; ) ( ; σ σ ; )} τ ( στ σ τ )( )} Of partcuar nterest to physcs s the contracted curvature tensor: P σ = P σ

24 W Pau On the formuaton of the aws of Nature etc I 4 whch s symmetrc n and σ n the case of a symmetrc Γ λ that satsfes (I) By the use of: α ;α = Π Π = 1 ( ) = 1 ( ) + = 0 we get: ε α P = P = R + α (67) ε α α 1 σ ( ; α + ; α ) + α 4 We thus extract from ths: ε r (67a) P ; = P = R + (67b) P (0) = 1 r P = ; Furthermore: (68) P = g P = R + 4 ε σ σ (69) P = P = 1 4 σ σ r 7 The form of the aws of Nature Varatona prncpe For physcs the utmate goa for the appcaton of the theory of the group of homogeneous coordnate transformatons and ts covarant structure s to derve the aws of gravtatona feds and eectromagnetc feds n a unfed way Thus n the cassca part of the theory we restrct ourseves to the case of the absence of charge and mass In order to reduce the number of fed aws we next propose the noton of actua tensors (projectors) Ths noton s such that t s constructed ony from the g and the Γ λ as we as the dervatves of the Γ λ wthout the expct ncuson of the and the or the and aso wthout anythng expcty enterng nto the Γ λ but the dervatves of the g The smpest form of the aws of Nature woud then be the one that smpy expresses the vanshng of an actua projector However one thus aways obtans one equaton too many and t s thus necessary to cast the fed aws n the foowng modfed form: One must have: (70) K = F

25 W Pau On the formuaton of the aws of Nature etc I 5 n whch K s an actua tensor and ndeed a symmetrc tensor of the second ran whereas F s a yet-to-be-determned scaar hence a homogeneous functon of nu degree n the λ From (70) one mmedatey fnds that: (71) F = K K such that we can wrte: (7) K = K Upon mutpyng by and and then performng the assocated contractons one derves an dentty and now (7) ncudes ony 14 ndependent equatons nstead of 15 The necessty of ths dentty s most cosey connected wth the reaton (7) for the g In order to satsfy the genera covarance and to aow for suffcent generaty n the souton of the g -fed (7) must satsfy fve further dfferenta denttes that we can pose by anaogy wth genera reatvty theory n the form: (73) K ; 0 Ths s permssbe because: ( K ) ; = K + K ; + K ; = 0 snce the ndvdua terms a vansh We now decompose eq (7) and the denttes (73) nto correspondng equatons for the nhomogeneous tensors Wth the fact that: (74) K = K K (0) = K (7) becomes: (75a) K = 0 (75b) K (0) = 0 Snce (71) and (74) means the same thng as: (74a) K = then t foows that: K + ε K (0) + ε K (0) + K K ; = ; K + K (0); + ( ; + ; ) K + ε ; K (0) + ε K (0) ; + e K (0) From (64) one has:

26 W Pau On the formuaton of the aws of Nature etc I 6 ε ; = ( ) ε ; = ( ) Thus one has: ε ε K ; = K = K = 0 and t foows that: ; = 0 ; = 1 (75) ; K = ( ; (0) ) K ε K ε K (0); Thus (73) spts nto two denttes: + (76a) K ; ε K (0) 0 (76b) K (0); 0 In the case of the presence of matter as we sha see part II these denttes ead to the theorem of the conservaton of energy mpuse and charge whch are summarzed n one tensor equaton n the homogeneous coordnates It s now worth pontng out that one obtans equatons of the form (7) wth the denttes (73) when one starts wth a varatona prncpe: (77) δ L g d (1) d (5) = 0 wth the suppementary condton: (78) δ(g ) = δg = 0 n whch L refers to an actua scaar and g refers to the absoute vaue of the determnant of the g As usua the varatons of the g and the g sha vansh on the boundary α Next f one ets: (79) δ L g d (1) d (5) ± K δg g d (1) d (5) wthout regard for the suppementary condton then from the fact that: δg = δg the fed equatons read e: (70) K = F

27 W Pau On the formuaton of the aws of Nature etc I 7 n whch F s a yet-to-be-determned Lagrange mutper whch woud have been prevousy ntroduced nto (70); furthermore K s an actua tensor when L s an actua scaar Fnay K aso satsfes the denttes (73) as one nows n whch one performs the varatons of the g under an nfntesma coordnate transformaton: from whch: = + ε ξ δg ξ ξ ξ = e g + g g and the varaton of the acton ntegra vanshes dentcay We further remar that from (18a) for fxed coordnates n and under the assumpton that the suppementary condton (78) s vad one has: (80) δg = δ g + g δ If one aso fxes the coordnates x then one has moreover: (81a) hence from the fact that: (81b) δ = = x δ = 0 δ = 0 δ = 0 δ = δ δ = ε ( δ ) t foows from the vadty of (78) on account of (74) that: (8) δ L g d (1) d (5) = ± {K ε K (0) δ } g d (1) d (5) We st have to estabsh the scaar L and thus the K If we demand that L sha ncude no dervatves of the Γ λ hgher than the frst ones then L s unquey dentfed wth the curvature scaar (83) L = P Therefore by means of the contracted curvature tensor P that s defned by (67) one has: (84) K = P 1 g P

28 W Pau On the formuaton of the aws of Nature etc I 8 n anaogy wth the reatvstc theory of gravtaton Independenty of the varatona prncpe ths K s the ony actua tensor that ncudes no dervatves of the Γ λ that are hgher than the frst ones and aso satsfes the denttes (73) One thus has: (84a) K = R 1 g ε 1 R + r g rs 4 (84b) K (0) = 1 K ; rs If we equate ths wth the fed equatons from the combned Ensten theory of gravtaton and Maxwe theory of eectrodynamcs whch n the present case of nonexstent charge and matter read e: (85a) R 1 g κ 1 rs R + F Fr gf Frs c 4 = 0 and: (85b) F ; = 0 then we see that get agreement between these equatons when the number r that was ntroduced n (44) accordng to: = r f = r c κ F satsfes the equaton: (86) εr = 1 Snce r s rea t ewse foows from ths that: (86a) ε = + 1 (86b) r = ± We further remar that (84a) can then aso be wrtten as: (84c) K (0) = 1 F ; r cκ The determnaton of e and r can aso come about by the requrement whch s equvaent to (85a) that the expresson: be dentca wth the ordnary form: L = P = R + 4 ε rs rs

29 W Pau On the formuaton of the aws of Nature etc I 9 κ 1 (87) L = R + c F rs F rs We have acheved our goa The second system of Maxwe equatons and the gravtatona equatons met together nto a snge system that s drecty connected wth the curvature The frst system of Maxwe equatons [eq (39)] foows drecty from the assumed structure of space Furthermore the aw of moton for a charged mass pont can be nterpreted as the generazed equaton for geodetc nes Wth ths we can cose part one but for the sae of competeness we woud e to refer to a generazaton of the Ansatz for the Γ λ that s due to Schouten and van Dantzg Appendx: Generazaton of the Ansatz for the Γ λ Schouten and van Dantzg have shown that the most genera Ansatz for the Γ λ whose consequences are n harmony wth physcs s: In pace of (III) and (35) use the postuate: (III ) p Π = 1 Q = q 1 n whch p and q are numerca coeffcents From (9) one further has: (88) S λ λ = 1 4 (p q) Therefore from (9 ) and (36) we nfer from (31) that: Q = q 1 = 1 + (S λ + S λ + S λ ) λ or: (q 1) 1 = 1 4 (p q)( + ) + S λ λ hence: (89) S λ λ = (q 1) 1 From(II ) and (3a) t then foows that: (90) S λ λ m n = 0 hence: (91) S λ = ε 1 4 (p q) ( λ M λ ) + ε (q 1) 1 λ

30 W Pau On the formuaton of the aws of Nature etc I 30 For the geodetc nes t further foows that: δ a dτ δ dx + εa = 0 dτ dτ a = const and: m n d x dx dx dx + = ε aq dτ mn dτ dτ dτ m dx = ε aq m = 1 m dτ ε a q r κ dx F m c dτ hence: (9) ε a = c e κ rq mc and wth: (93) dx e c p = m g + dτ m κ rq (94) δ dτ = 0 Consequenty: (95) dx e 1 F δ + Φ d q c + τ F = 0 for a certan choce of F Schouten posed the partcuar requrement that Φ sha appear n ths expresson wth the coeffcent 1 whch aso entas that: q = However we woud e do wthout ths competey snce such a demand seems to us to be n no way mperatve One can defne the curvature tensor P σ by way of (60) when one preserves the order of the ndces n the Γ λ By a engthy cacuaton one then fnds for the curvature tensor nstead of (68) the expresson: (96) P = R + 4 ε (q + p pq) σ σ On the bass of the varatona prncpe: δ P g d (1) d (5) = 0 wth the suppementary condton: δ (g ) = 0

31 W Pau On the formuaton of the aws of Nature etc I 31 one fnds the fed equatons: K = 0 K (0) = 0 and n pace of (84a) and (84b) the foowng expressons appear: (97a) and: K = R + 1 g ε R + (q 1 + p pq)? r g rs 4 4 (97b) K (0) = 1 (q + p pq) ; rs The atter foows from (8); one shoud observe that (84) s no onger vad (96) mpes the foowng condton: εr (98) (q + p pq) = 1 fro whch one nfers that ε has the same sgn as (q + p pq) From (98) one can aso wrte (97b) as: 1 κ (99) K (0) = ε F ; r c As a crtque of the generazed Znsatz for the Γ λ t must be remared that the curvature scaar n ths case s not the ony actua scaar snce: J = g S S σ σ can aso come nto consderaton whch agrees wth: σ σ up to a generay non-vanshng numerca factor An arbtrary Lagrangan functon can be represented as a near combnaton of J and P For that reason we woud e to propose that the orgna Ansatz S λ = 0 s the most natura one On the other hand the Ensten-Mayer theory can be characterzed by p = 0 snce ony a ; = a ; appes to t Zürch Physca Insttute of the ETH (Receved 15 Juy 1933)