The Theory of Special Relativity. (and its role in the proof of Fermat s Theorem w.r.t. the Binomial Expansion)

Transcription

1 The Theory of Speil Reltivity (d its role i the proof of Fermt s Theorem w.r.t. the Biomil Epsio) Updted: 5/9/7 9:7 AM PST Flmeo Chuk Keyser (Chrles H. Keyser) 3/6/7 :8 AM PST BuleriChk@ol.om (Alwys refresh your browser to mke sure you hve the ltest versio. This is still work i progress, lthough I believe the fudmetl oepts to be orret tht sid, there re umber of ples where there re errors, d I m leig those up ow. (I did lot of uttig d pstig..) If yoe hs questios, I will be hppy to respod vi emil There re more topis I wt to ddress i otet.) Tht is, hek your time stmp i the Updted lie bove. Note: This Doumet reples the origil doumet The Reltivisti Uit Cirle I wrote ddressig this subjet (the origil hs topis tht re ot iluded here, but I thik this is lerer. I will be updtig this doumet diretly)

2 Tble of Cotets Itrodutio Summry The se > The se > Speil Reltivity Fil Stte Alysis Iitil Stte Alysis Iitil d Fil Sttes with respet to Dimesios Pythgore Triples d Fermt s Theorem Mwell s Result, Reltivity, d the Biomil Epsio The Iitil d Fil Sttes of the Uiverse Commet Colusios Iitil Proof of Fermt s Lst Theorem Prologue The Fudmetl Equtio of the Speil Theory of Reltivity The Three Solutios of the Fudmetl Equtio of STR Physis Iterprettios Mthemtil Iterprettio A Seod System b Glile Spe-Time Glileo Vs STR The Deostrutio of Bet Time-Diltio

3 Spe-Cotrtio Spe-Time Coordite Reltio to STR (Digrm) The Reltivisti Uit Cirle (Fil Stte Solutio) The Four Qudrts Itertio i the umber system {} The retio of the ple (,y) Retur to the Uit Cirle Desity d reltio to Spi Compriso of Iitil Stte d Fil Stte Represettio of v Let There Be Light (The Cretio of the Uiverse) Iterprettio of the Biomil Epsio Reltivisti Rottio i the system {} Positive Rottio Qudrts d 3 Qudrts d 4 Negtive Rottio Reltivisti Spi (=) The seod rel umber system {b} The Defiitio of Multiplitio The ple of itegers Group Multiplitio Fermt s Theorem Proof Fermt s Theorem Pythgore Triples,,, '

4 Fermt s Theorem d Presburger Arithmeti Positive Rel Numbers over the sme set Defiitio of Multiplitio Reltivisti Additio of Veloities Iitil Stte Solutio (The Cotiuum) Euler s Formul d the Biomil Epsio Reltivisti Cosequees The Hyperboli Futios (Iitil Stte Ivrit) Completeess of the Rel Number system Sigle-Vlued Rel Number Systems Philosophy Subjetivity Itrodutio to Eistei (The Chrteriztio of Mthemtil d Physil Costts) The Oe Soure Mss model of Qutum Mehis The Two Soure Mss model of Speil Reltivity The Cretio d Destrutio Futios Appedies A. Fermt s Proof from Reltivity

5 Supplemetl Topis Vetor Commuttio d Spi Divisio by Zero The Loretz Trsform d the Time Diltio equtio Proof of Fermt s Theorem The reltivisti Loretz Fore The Equtio of the Uiverse CPT Ivrie The Geerl Theory of Reltivity /r Physil Lws Newto s Lw of Grvity Ampere s Lw (TBD) Coulomb s Lw (TBE) Eletromgeti Fore Lw (TBD) The Eistei/Bergm derivtio of the Loretz Trsform Additiol Ppers I hve writte Referees

6 Itrodutio The fudmetl equtio of the Speil Theory of reltivity ' v ' umbers osistig of the field vribles, v,, t ' for positive rel for the sigle set of rel umbers is show tht the metri ot be iteger if subsets of ilude multiplitive elemets with the sme metri, d the set of tht iludes powers of elemets of two these subsets re hrterized by the Biomil Epsio, whih is omplete for the rel umbers for. It is lso show tht the equtio of irle is ot vlid for subsets of sie elimitig multiplitive elemets requires the itrodutio of the omple umber, whih is ot member of The Biomil Epsio is the ivoked for the se where it is show tht Fermt s Epressio ot be vlid withi the set for positive itegers, sie there is remider ftor tht ot be elimited by subtrtio or by ivotio of omple umbers, sie the vribles do ot ommute s i the se Eistei s origil theory pplied oly to positive rel umbers for Speil Reltivity. The Geerl Theory iluded omple umber i four-dimesiol spe i the Mikowski metri by ttemptig to ilude the three dimesios of spe-time (whih re irrelevt to the Speil Theory). (The Mikowski metri is set to Theory. r y z the it beomes (,) orrespodig to the Speil However, the positive hlf ple of the reltivisti uit irle be hrterized s retio vetor spe sie there re o destrutors, the multiplitive elemets ot be removed by the presriptio y ( iy)( iy) for reltivisti mometum d similrly for 4 4 y 4 y ( y ) Fermt s Epressio is vlid beuse the egtive hlf ple is ot iluded i the Biomil Epsio (whih is -dimesiol epsio for two omils i the sme ple), d therefore the positive terms i rem(, y, ) ot be destroyed by their outerprts i the egtive hlf ple. I prtiulr, the propositio tht Pythgore triples belog to the set b,, is flse; rther, they belog to the set tht iludes egtive umber (e.g.,, b,, b ). (Pythgore Triples require the egtive umber s bove, d so re ot i the set of positive rel umbers Tht is, for, b, but rther, b,, b positive itegers

7 ,,, b, b, b Pythgore Triples relly be hrterized s Jedi mid trik - the ide tht d b re positive for ll rel itegers i the set, b, is illusio). This is lso true for (gi, by the Biomil Epsio). Tht is, for, b, but rther, b,,, b, rem(, b, ), rem(, b, ) Ad lso for the rel umbers: For z y,, y, z positive rel umbers d, y, z,, y, z, rem(, y, ), rem(, y, ) rel umber d for omple umbers: The Biomil Epsio (Wikipedi) It relly is Jedi mid trik. Or I hve t log lst beome Jedi This is true for both d, sie the itegers re subset withi the rel umbers. The ull vetor provides ommuttivity betwee two positive rel umbers (d is represettive of equl d opposite fores o the horizotl is i the fil sttes of two prtiles b, The ple p q, y p q the refers to the produt ( y ) o the upper hlf of the ple, where p d q for the Biomil Epsio (d thus Fermt s Epressio). Thus for Speil Reltivity, the bsis set is hrterized by,, the Mikowski spe by (,,, ) (ivokig three retor dimesios si( ) d sigle destrutor dimesios, wheres the Dir vetor spe (,,, ) hrterized by the mtri dds the lower hlf of the reltivisti uit irle s destrutor opertors si( ) so the reltivisti uit irle does ideed hrterize irle where y is vlid whe egtive rel umbers re iluded.

8 It should be emphsized tht first order opertors hrterize mometum, d seod order opertors hrterize eergy, so to destroy eergy, omple opertors re eessry s i the Puli d Dir formultios ( )( ) d ( )( ) z iz iz z z y iz y iz y z for the omple is. This is beuse the itertios betwee third is z i the spe (, y, z ) must be elimited for the three dimesios eperieed eperimetlly, so tht the three ples, y, z, d y, z re idepedet i three-dimesiol vetor spe. This hrteriztio the forms the foudtio of the Stdrd Model of Qutum Field Theory, where the reltivisti fil stte i three dimesios is hrterized by,,, - the three dimesiol ull vetor, d i four dimesios by,,, otherwise kow s the Higgs Boso, where Qutum Trivility represets true Blk Hole (there re o virtul photos hrterized by v for y of the four dimesios i the mthemtil model. Summry The sigle vlued futio m Crtesi set m defies multiplitio for ll positive rel umbers i the where m,, d re i the set of rel umbers defied by the positive rel vrible so tht m,,, is omplete uder the opertio of multiplitio *, idited by jutpositio of vribles * * d the vribles ommute uder multiplitio. m(, ):, m is the sigle vlued futio mppig the two- dimesiol Crtesi produt ito itself, where, so m Similrly, the futio m ( v, '): v, ' m v m v ' v v. is omplete uder the opertio *, so tht the pir vt v m implies tht v, ' so m ' ' v The power futio f (, ) v, so tht,,, ' tke to the power of d their produts ommute (e.g. The sigle vlued futio m m v v where eh ostituet is is lso positive rel umber uder the sum opertio, so tht m m m m is omplete for v v v,,, ' uder the opertios,*, where it is uderstood tht ll elemets of the set ommute uder the opertios, d the set of positive rel umbers is omplete uder these opertios. (he opertios d * re

9 biry opertios reltig idepedet elemets of, d the ottio multiplitio opertio is rried out o idetil elemets of, idites tht the ) The the Biomil Epsio is omplete i the set v k umber, where,,, ',,, for the positive rel k k m mv m mv m mv rem( m, mv, ), where k k! is positive rel iteger for itegers k k!( k)! hrterizes ll terms i the epsio tht re ot d k m or m. The the futio is ot omplete for y pir ( m, m ) m m v v, d rem ( m, mv, ) v sie it does ot ilude the itertio produt rem ( m, mv, ). Tht is, for to be vlid, rem ( m, mv, ) for ll m k k d mv uder the opertios m m, v,, ',, k v, k d the set is omplete for the positive rel umbers. However, (, v, ) rem m m for ll, v,, ',, k m m v ot be vlid d i prtiulr, so the epressio where b (Fermt s Epressio), where, b, d re positive itegers withi the set This ostitutes the proof of Fermt s Theorem, whih sttes positive itegers withi the set b for of rel umbers.,, b,, Note tht i the Biomil Epsio rem ( m, mv, ) implies tht m, m i whih se m v or m The se = For the se, or both (i whih se ) m m m m m m, where the ftor of rises from the v v v, property tht m d m ommute uder both dditio d multiplitio. The itertio v (multiplitio) term mm be elimited usig the omple ojugte multiplitio opertio v (redefied s *), where, * ( m im )( m im ) m m where i d the v v v v

10 itertio produt mm v mvm is dded d subtrted by the ojugtio opertio. However, this opertio is ot withi the set of positive rel umbers, so set of positive itegers where ( b) ( ib)( ib). The se > k The ojugtio opertio ot be used for sie the elemets i m m *, d i prtiulr the k v re ot k k k k iterhgeble; tht is m mv mv m so the itertio produts ot be dded d k k k k subtrted for eh term. However, they still ommute, sie m m m m still positive rel umbers withi the set ojugtio i the se. d re v v. Thus rem ( m, mv, ) ot be elimited eve by Speil Reltivity The epressio be hrterized s the fudmetl equtio of the Speil Theory of Reltivity * ' v ' for the se of the Biomil Epsio, where be hrterized s iitil stte, v ' s stte of hge, d ' s the fil stte. (Note tht if there is either iitil stte or fil stte.), where the multiplitive itertio produt v hs bee elimited from ' ojugtio opertio *. The equtio ' v '. Solvig for ' ( m m ) m m m m by the omple hs three solutios; v v v v ',,,,, : ', ',, vlid for ll, v,, ' uder the opertios ' v. The Fil Stte ivrit: Note tht d ',, ' v 3. The Iitil Stte ivrit: '

11 Sie the system v,,, ' uder the opertios ', ', k, oly be iteger if, i whih se the iitil d fil sttes re the sme ' ' so tht d where is iteger. ' The ot be iteger for, Ad ot be iteger uless, Therefore, the system hrterized by the fudmetl equtio of Speil Reltivity oly be ' iteger if ' whih mes tht v d ' the fil stte. so tht the iitil stte is equl to Fil Stte Alysis Cosider the se, r r so tht where the ftor of rises from the ft tht d ommute uder multiplitio, d d ommute uder dditio; i.e. d ) The for vish. If (the reltivisti uit irle the fil stte), the term must the so the equtio is idepedet of However, if i i the the equtio forms the reltivisti uit irle where os d si so tht os si d the itertio term sios is dded d subtrted from the equtio with the help f the omple uit i.

12 It is istrutive to osider the Biomil Epsio: rem (,, ), where ot be iteger uless so tht., Fil Stte Alysis + + +

13 Iitil Stte Alysis Iitil Stte Alysis

14 I the se of the iitil stte r equtio beomes. r I terms of vetors, this equtio is tully so tht r r i i j j k k, so tht,. For the r, so for d, d r r r r Settig r resets the the iitil stte to be the fil stte for (t whih poit it beome other iitil stte if there is further hge represeted by higher order of i the Biomil Epsio. Note tht s ireses with (i.e., s v the effet of the iitil oditio beomes less d less. The Cotiuum is hieved oly t (e.g. reltivisti mss)

15 Iitil d Fil Sttes with respet to Dimesios Fil Stte Alysis Iitil Stte Alysis + + +

16 Note tht dividig the epressio for the Iitil stte f ( ) by results i the epressio for the fil stte: whih is the i reltivisti uit irle, whih is oly vlid if the iitil stte is equl to the fil stte. The I both ses, whih mes tht i i,,, (, v,, ') oly be rel i oe dimesio for f i. Sie os os,, This umber for ll lso pplies to the ull vetor, so tht,,, (,,, ') v Epdig the epressio for the iitil stte by the Biomil Epsio results i: rem,, where so tht for positive rel umber so tht the fudmetl equtio of Speil Reltivity is oly vlid for ll rel umbers i oe dimesio for the fil stte equl to the iitil stte. I prtiulr, this is lso true for the rel positive sigle vlued umber, where ' ',, where i so tht For seod positive rel umber y where v y,,, ' d y y j j,,, y (, v,, ') is modeled by the STR where the fil stte is equl to the y y y iitil stte so tht y y y y y The i v j for the rel umber pir (, v) (, y) d if orthogol (i.e., idepeet), i v j v. If ot orthogol the the Biomil Epsio pplies. For the se, the y ( iy)( iy) so tht y ( iy)( iy), where the itertio term y i y y y is ot hs bee elimited by omple ojugtio (whih mes tht the metri y omplete for the set, y, sie the produt y is ot iluded )(i.e., the metri is tht of Presburger rithmeti, whih lso is verifitio of Gödel s Theorem ). This ot be omplished, where y y rem(, y, ) for sie y y uless k. k k

17 This lysis is the reso tht Fermt s Theorem is vlid, d is idepedet of the iteger property. Pythgore Triples d Fermt s Theorem (Note tht for Pythgore Triples i the se b ( ib)( ib) but tht, b b b, where the ltter is vlid without omple ojugtio. For b b rem(, b, ) so tht for, (,, ), rem b (i.e., must vish) whih oly be the se for or b, so tht for two positive itegers b, Fermt s Epressio b is vlid for the speil se of Mwell s Result, Reltivity, d the Biomil Epsio, positive iteger. Mwell hieved his formul for by pplyig the dot d ross produts vi Stokes d Grees Spe-Time itegrls to re d volume iterprettios of hrge d urret vi Guss s d Ampere s (lssil) lws. I doig so, he lierized spe d time by elimitig the full itertios betwee hrge d urret. Eistei s ttempt to use differetil geometry to restore the o-lierities i oept led to his field equtios i spe d time, where elertio is iterpreted s urvture modifitio to the orthogol Glileo/Desrtes/Newto spe-time frme. However, suh guge hge relly results i hge to, (Fil stte lysis) or, (Iitil Stte lysis), where the full olierity) is represeted by the Biomil Theorem (whih is model of the itegrted elemets of the Christoffel symbols i the Geerl Theory). By delrig to be ivrit, he relegted the observtio of the Uiverse to those we mke i the prkig lot, (o the surfe of this plet, i this solr system, i this gly) whih is true to everyoe eept osmologists; yet eve Eistei persisted (who dmitted he ws serhig for the ture of God.) Just s the metri y is iomplete with respet to the rel umbers, sie it does t ilude the itertio term rem (, y, ) d so o multiplitive ftors of the form Biomil epsio, the epressio k k y i the is iomplete beuse it does t oti y dditive terms but is oly multiplitio o sigle bsis elemet where the iitil stte is equl to the fil stte for ' Cosider the epsio

18 r r results i where (Mwell) so tht for r r r r Compre this equtio with d ote tht Mwell s vlue (the itertio term) hs bee elimited. where Tht is, i i But the m m, m

19 Fil Stte Ivrit ' ', : is the urret to hrge rtio. For m d m', this is equivlet to the reltivisti equtio m' m so tht if there hs bee o hge to the rest mss (of light); i.e., the lssil mgeti field B v orrespodig to reltivistilly. (Biomil Epsio) rem(,, ) rem,, rem,, ( ) Iitil Stte Ivrit, r : (Biomil Epsio) rem(,, ) rem,, r r r r, where,, rem,,, r

20 The Iitil d Fil Sttes of the Uiverse Let be the stte of photo-equivlet system (Uiverse) hrterized by its iitil d fil sttes, ' where m h. Let ' be the bsis for the system, where the iitil stte is tke to be ivrit, d be the bsis for the system, whe the fil stte is tke to be ivrit. ' i i ii i ii rem( i, ii, ) ' i ri r i i i i rem(, i, ) i ri r i i i i The for k suh systems of order, k k k k k k k k r r i i i i i i so tht k k k k k k k k i i rem,,, k r r k k k r d where k or k, k or both, i whih se whih is divisio by zero (tht is, the dimesio r is udefied), so tht for the uiverse to eist, either ' ' or must eist (either iitil or fil stte must eist, or there is o uiverse). (If the the itil d fil sttes re the sme, of ourse.) Whih is tully true is questio of religio, ïve solipsism or ïve relism (The Uiverse does t mesure itself), sie d ilude ll possible eistig sttes ( imgied s positive rel umbers) i w.r.t. the Biomil Theorem, where the Biry elemets re the positive rel umbers systems tht model the iitil d fil sttes (the Wow d Ow of physis, where together with our subjetive imgied/imposed) oordite systems omprise the To of physis, where To ( Wow) Ow Re mwow, Ow,

21 (Note tht the rtio d is defiitio of the derivtive osistet with the Fil Stte of the Uiverse, d d is logous to the Christoffel symbols of differetil geometry) Commet ' The reso tht the proof of Fermt s Theorem works is beuse i the Fil Stte solutio ' for the se the rest mss d perturbtio v ' ommute; tht is whih is whih is mere questio of lbelig for v,,, ' rel umbers, with o physil iterprettio. This is ot true if the iitil stte beuse ow there is sig (prity) distitio betwee ' d v ' For the se the epsio to the Biomil Theorem is refletio of the requiremet tht d ' (i.e., y ) re ot idepedet if both ddititive ( y) ( y ) d multiplitive y y re required to defie the positive rel umbers. For the se it is possible to irumvet this requiremet by omple ojugtio ( iy)( iy) y s otrsted with ( y) y y y y but i the higher powers of the Biomil Epsio the vribles re ot iterhgeble i rem (, y, ) d so ot be elimited by ojugtio. Tht is, i the equtio ( iy)( iy) y there is o distitio betwee whih is the iitil stte d whih is the perturbtio if the fil stte is tke to be the bsis for the dimedio. If the iitil stte is tke s the bsis for the dimesio, the two dimesios re required, but the fil stte lso iludes multiplitive itertio betwee the iitil stte d the perturbtio y m m. If it does ot, the Fermt s epressio is the bsis for Pressburger rithmeti tht v does ot ilude multiplitio, d so is iomplete (whih is true eve for, sie ojugtio is ' required for lierity) so tht ( ) (os si ) i the se whih does ot hold for ' the se where hyperboli futio re required for lierity.

22 Colusios. The lysis of the rel umber system i terms of the Fudmetl Equtio of Speil Reltivity shows there be oly oe bsis for eh omplete rel umber system. If rel umber is re-ified (i.e. is ivrit), the reltioships withi eh dimesio re desribed i terms of multiples of its bsis vetor. 3. For distit ( re-ified)) rel umber systems, the reltios betwee ivrits betwee dimesios is hrterized by the Biomil Theorem (for two systems) d by the Multiomil Theorem for multiple systems. 4. Clulus d lssil urvture does ot pply to reified systems, either i GTR or STR for more th oe dimesio.. Cosider the Biomil epsio of ( h) d ote tht lim( h) l; the substitute rel ( re-ified ) umber for the uit iteger. b. Wiles proof ot be orret, sie it postultes ellipti urves t the strt; these urves ot be elimited (by ojugtio) i the oe dimesio, d eve for the elimitio requires the itrodutio of omple umbers ( subtrtio ), so ot be relevt to system of positive re-ified rel umbers. At the very lest, it must be isomorphi to the Biomil Epsio.. The lssil desriptio of the derivtive is epressio tht re-ifies rel umber vribles: h f ( h) f () f '() lim h h, d so is tutology i two dimesios, orrespodig the ivrie of the Fudmetl Theory of Reltivity i the se tht the iitil stte is equl to the fil stte. d. Fermt s Epressio: b is epressio i two dimesios (reple b by h d tke the limit s i (4.) bove. The Biomil Epressio is ( b) b rem (, b, ) rem (, b, ). For Fermt s epressio to be vlid rem (, b, ) whih is oly true i the limit b. Fermt s Epressio be reovered from the Biomil Theorem by omple ojugtio: ( i )( i ) oly by settig, i whih se the equtio is tutology i oe dimesio (i.e., b ). e. This is the reso lulus ot be pplied (eept s pproimtio) to either QFT or GRT, sie it requires tkig the derivtive of (ostt, re-ified ) rel umber i the fil lysis,

23 5. The fudmetl equtio of Speil Reltivity: ' v ' desribes the trsitio from iitil stte to fil stte vi perturbtio for positive defiite rel umber system, but does ot ilude the itertio betwee the iitil stte d the perturbtio, d so is iomplete for v the fil stte eists.. There re three solutios to the fudmetl equtio of STR. Solutio i terms of slig ftor v ',, b. Solutio i terms of fil stte ivrit ',,, '. If, either the iitil or, ' os si, where fil represets the re of the reltivisti uit irle, where os, d si. Solutio i terms of iitil stte ivrit osh sih, os, si, d, b. If the iitil stte is equl to the fil stte d, there is o perturbtio, sie v, v

24 6. The Biomil Theorem i both the iitil d fil stte hrteriztios is give by k k! ( y) y, k k k k!( k)!, where y rem (, y, ), d rem (, y, ) iludes ll terms i or y does iludes the itertios.. Cse The Biomil Theorem the se. ; tht is, si os... ; tht is, sih,. iitil., ; tht is, set of ll rel umbers v sigle dimesio tht re ot whih is the uit bsis for,,, '. The set of ll rel umbers is omplete for the i for the se d. The itertios be elimited by omple ojugtio: * i i * i i This oly be pplied for sie, d, re first order ( i k ) d therefore ommute d be iterhged w.r.t. the omple elemet. However, for, k k k k k k k k the terms i eh se ommute), but k k k d k k k k k k k d (so

25 k k k k, so ot be iterhged, so the term rem (, y, ) k k ot be elimited by ojugtio i either the iitil or fil stte represettio. e. I lssil Qutum Mehis, h d h, respetively, d h h i h h the Shrödiger equtio ; h h, whih implies the QM wve equtio is irrelevt to lssil mehis d eletromgetism (where there is o light/mss itertio), d where represets mss d represets hrge i both the iitil d fil ses. If there is o hrge, the the mss ' m is ivrit. '. For the fil stte lysis, this is the equivlet of ' i j. However, i j, so tht represets the itertio betwee the elemets (, ).. For the iitil stte lysis, this is the equivlet of i j. However, i j, so tht represets the itertio betwee the elemets (, ). 3. d so tht The iitil stte lysis d the fil stte lysis for the sigle vrible i the rel umber system v,,, ' is omplete. The iteger for equivlet systems is defied by i, j, k,... i j k..., i, i j k...,, so tht i, j, k,... i i j k..., i i

26 The itegers re subset, d ll itegers represeted by re iluded i the set, so the set is omplete i the sigle dimesio i A similr lysis for the sigle vrible y i the rel umber system v y,,, ', d is y therefore omplete. The itegers b b y re subset, d ll itegers represeted by re iluded i the set, so the set b is omplete i the sigle dimesio y yi The the Biomil Epsio ( b) b rem (, b, ) is omplete for the two sets, whih iludes the itertio term rem b (, b ) (, y ) (,, ) rem(, y, ). For the se the itertio term b be elimited by ojugtio, so tht b ( ib)( ib) oly if ojugtio is pplied, where ( b) b b, whih is vlid for Pythgore Triples. However, the set represeted by bis ot omplete, sie it does ot ilude the itertio terms b b The set represeted by r b i b j b k is omplete, where r r b b i i b j j b i j k so tht d b b r b b b b where is iteger, but r r i b j b k so tht is ot iteger uless or b, i whih se r, d b or, respetively. Therefore, b is ot omplete; for it to be vlid the remider term rem (, b,) b i ( b) b b would hve to vish, whih requires the itrodutio of omple umbers. I the se, the remider term ever vish, d rem (, b, ) i ll ses. Sie! b ( b) b,, where is iteger oly if k k k k!( k)! k k k b whih oly hppe for k k, so tht (or symmetrilly for b, so Fermt s epressio itegers iy y b ever be true i system with two omplete sets of b y, sie it requires omple ojugtio ( i ) where i or for the se d the pplitio of i ot be elimite rem (, y, ) for i y se, sie the vribles i tht term ot be ehged, eve though they ommute.

27 For the geerl se of the Biomil Epsio, k k ( ) k k, so tht k k, k k so tht, so oly oe bsis iteger is defied per dimesio. I the se of h (where h s hrterizes spi, (s opposed to hrge to mss rtio, or the urret to stti hrge rtio), the epsio beomes k k h h k k where h ; i.e. whih mes tht h m, d h, hrterizig the (whether represets subjetive time is up to the imgitio; rell tht oe of the prmeters v,,, ' re prmeterized i physil terms sie oe re ivrit (uless delred so by Eistei).

28 The ovetiol defiitio of the derivtive for the Theory of Speil Reltivity for the se should therefore be evisioed s f f f f '( ) f ( h) f ( ) lim, where h h or where both require tht f f f os si f os '( ) lim lim or si si ( ) ( ) '( ) lim f f, sih f f( ) f(osh sih ) f(osh ) '( ) lim lim where ( ) (i.e., ivrit Blk Hole ); tht is, lim lim. f ' f '( )

29 Iitil Proof of Fermt s Lst Theorem Fermt s Theorem: b for b,, d positive itegers. (The presumptio tht the Fermt s Theorem hs t bee proved seems to me to be either sm by Mth Deprtmets worldwide of Jedi mid trik. If either the Jedi or the Mth Deprtmets uderstd reltivity, the they eed to red this pdf ) The Biomil Theorem The Biomil Epsio is give by the equtio! k k!( k!) epressio k k ( y) y k k, is positive iteger tke to be the biomil oeffiiet. I use the k k Rem(, y, ) y ( y ) to idite those terms i the k k Biomil Epsio ot equl to either or The Biomil Epsio pplies to itegers s well s rel umbers (i ft, it ws first formulted this wy), so usig d b, the epressio is give by: k k k b k y. Settig this epressio equl to rbitrry iteger so tht k k b ( b) b Re m(, b, ), where k k k k Rem(, b, ) b ( b ) m, d where m (is positive iteger) k k proves Fermt s Theorem by ispetio, sie b b (Fermt s Epressio) ( b) b rem (, b, ) (Biomil Epsio) rem (, b, ) lwys (iludig ) Q.E.D. (uless the Biomil Epsio is flse)

30 (The reltivisti lysis shows tht positive multiplitio is defied by the Biomil Epsio for rel umbers, d is thus true for ll positive rel umbers).

31 A Brief Prologue Clerk Mwell (together with Mihel Frdy) hd produed epressio for the speed of light s where d re the eperimetl permittivity d permeblilty ostts from Ampere s d Coulomb s fore lws, with the reltio derived by usig the oept of displemet urret d symmetry reltios i Grees d Stokes represettio of spe-time res d volume reltioships. The filure of the M-M eperimet to detet y hge i with respet to positio o the surfe of the erth, t vrious spets of erth orbit d rottios (s well s rottios of the pprtus led Loretz to produe trsformtio (the Loretz Trsform) o spe d time i two dimesios: ' ( vt) v t' ( t ) where,, v whih is solutio to the equtio ( vt) ( t) together with the ssumptio tht t ' t ' s wy of outig for the ull result by otrtio of the rm direted ito the supposed flow of the ether. If oe uses the ltter ssumptio with respet to these equtios by substitutig t ' ' d t i the first equtio, oe rrives t the sigle equtio: t t vt v t ' ( ) ( ) or ' vt This ws ustisfyig theoretilly t the time, beuse o-oe ould thik of reso why suh rm would otrt, but it did provide osistey with the ull results. The itrodutio of the Speil Theory of Reltivity by Albert Eistei provided lier solutio tht ws osistet with Mwell s symmetries i spe-time s well s the ull results of the MM eperimet, d osistet with the wve oepts of Qutum Theory vi Plk s ostt. I order to do this, Eistei ltered the Loretz trsform, d elimited Mwell s equtios by providig solutio tht ws vlid for y vlue of i reltio to y motio through ether by isistig tht the reltio v be preserved i terms of wht he lled iertil frmes. Agi, it is importt to stress the oept of ierti s otrsted with the Glile/Newtoi oepts of veloity d mss i Crtesi oordite system.

32 Eistei s solutio ws to lter the Loretz trsform by settig v i the umertor (elimitig Newto s mometum P mv deomitor: m mv s lssil motio i spe d time), but retiig eergy i the v This results i the so-lled time-diltio equtio, t' t ; however, it should be oted tht sie d m re ot ivolved epliitly, t ot be iterpreted i terms of the Newto/Glileo model. He lso elimited y sptil hge by settig ', so tht the result be purely ddressed i terms of the set of field vribles, v, t, t ' for sigle system The sigle system (The Fudmetl Equtio of STR) Eistei mde two fudmetl ssumptios. The first is tht the speed of light be idepedet of y veloity of either, so tht they ould be relted i oordites system i two dimesios ( v, ) The seod ssumptio ws tht suh lw hd to be vlid for y reltio of the two vribles, whih be omplished by the slrs d ', so tht ew oordite system (, v ') is the estblished, with d v ' idepedet of eh other (e.g. o orthogol es). Sie two dimesios re itrodued, they be relted by the Fudmetl Equtio of Speil Reltivity: ' v ' (.) It is istrutive to thik of this equtio s tht of reltig iitil stte s to d idepedet hge of stte ( s v ') resultig i fil stte s ' without y otetul iterprettio if the set of positive vribles, v,, ', v,, ' tht s s s for sigle positive rel umber system, so (.) (w.r.t. STR, the vribles i the set v,,, ' re sometimes lled proper to distiguish them from spe-time vribles i Glile oordites.) However, the vribles d v be iterpreted s widget retio rtes, with d ' represetig widget retio times for geerl reltio ivolvig widget hge from iitil oditio ( rest widget) to fil widget ' vi perturbtio ( v ') with the uderstdig tht the fil stte the beome iitil stte of seod proess. (Note: widget either be rel or figmets of the imgitio. For itegers, the widget must be ivrit. Wheeler refers to the widgets s bits ).

33 The Three Solutios of the Fudmetl Equtio of STR There re three possible solutios of equtio (.):. Solvig for ' whih gives the time diltio equtio v ',, (.3). Fil Stte solutio: Iterpretig the equtio i terms of the fil stte by dividig both sides by ', with the result:, whih is the equtio of the reltivisti uit irle. (.4) Ay irese i y of the prmeters v,,, ' will irese the vlue of the fil stte of the widget represeted by the ostt, so the umber desity of the system will irese. 3. Iitil Stte solutio: Iterpretig the equtio i terms of the iitil stte by dividig both sides r r by, with the result: where the r is the resultt of the vetor hrteriztio., (.5) Ay irese i y of the prmeters v,,, ' will irese the vlue of the iitil stte of the widget represeted by the ostt, so the totl umber represeted by the system will irese by the ftor. Note tht if, v, the ', d the Iitil stte is the sme s the Fil stte ( ). Therefore, t v, there is o hge i the widget, d the totl system is hrterized by '. If there is o widget. Physis Iterprettios I the followig, it is importt to remid the reder tht both mss d hrge re positive i this otet. Mss d Chrge

34 If represets hrge (i.e., eletromgeti light ) d represets mss, the iresig will irese both their vlues, d represets the hrge to mss rtio. If there is o hrge, there is o light d the totl system is mss (, )). If there is o mss, there is oly light,. If there is o light d o mss, there is othig. (Mss d Mss hge) If represets the iitil mss of grvittig stte d represets hge to tht stte, the the grvittiol hge will be give by m' m. If v there is o hge m' m whih represets the iitil stte. If there is o iitil stte, there is othig. (Light d Light Chge) h Here the iitil stte is represeted by Plk s ostt where E h h, E h, so tht E' h' d E h, ' so h is the iitil eergy of light for ' E m h,, v Ay hge will be will be give by E ' m ' h, where the hge i y of the prmeters v,,, ' will hge the eergy of the system (hrterized s sigle photo, or photo-equivlet mss of eletro. Tht is, if there is o light. Mthemtil Iterprettio The subsript o idites tht it serve s vetor bsis for the system, sie possible field vribles s slrs (iludig itegers where i for ll Therefore hrterizes oditio of o hge so tht hrterizes the stte of our lol system t y give ple d lotio of where we re s observers ; i.e., our lol istteous positio i spe-time where there is o hge v eve though we eist; i.e.. If, there is o iitil stte, so o hge reltive to the oeistet iitil stte, so o fil stte i terms of the oeistet iitil stte d reltive hge. I prtiulr ote tht hgig or y of the other field vribles i the set v,,, ' does ot hge the form of the equtios (.) through (.5), sie they re vlid for ll possible vlues of the field vribles (positive rel umbers). These solutios will be disussed i the followig setios, d form the foudtio of Qutum Field Theory. *

35 A Seod System b A seod idepedet system b be reted from idepedet set of field vribles, v,, ', v,, ', resultig i ew bsis vetor b b b b b b m m mi where, of ourse, m be equl to or ot; the umber sets for eh oeffiiet re omplete idepedetly. The if the Iitil d Fil sttes of the system b re the sme ( b ) b, but the field vribles my be ompletely differet from tht of system. The two idepedet systems the be hrterized i the two-dimesiol vetor Crtesi ple ( b, ) with bsis vetors (, b) ( i, j). Glile Spe-Time v I Glile frme, the reltio betwee time d diste is give by the veloity v, with t Newto s lws of motio (mometum d kieti eergy) give by P mv d b E mv v, whih re osistet with the rules of differetitio d itegrtio of the Clulus. The Glile oordites be hrterized by Glile spe time digrm : The Glile oept of veloity ws iorported by Newto i his Lws of Motio:

36 P mv d E mv, d re the foudtio of Clssil kiemtis. Glileo vs STR The Speil Theory of Reltivity (STR) iterprets the oordites i terms of widget mss d eergy, rther th time d spe, with referee to iertil frmes, rther th spe-time frmes. If the Glile spe vribles from v vt d v t re set equl i (e.g.) v v t t t, v, tv t, but it must be uderstood tht ppers s seod t t v v v mv order qutity for y rbitrry mss m. It is therefore turl to ssoite m m, m v ' v d m' m from whih the reltivisti eergy equtio follows: m' P m Therefore, d ' pper s slig ftors o widget mss : m ' ' m, so tht is v iterpreted s hge i mss desity determied by, where v d re to be iterpreted i terms of mss retio rtes rther th Glile spe-time speed. A hge i desity will me tht photo trvelig through medium t speed v other th vuum will trvel shorter diste th photo i vuo t, or equivletly, photo will tke loger time to trvel the sme diste s photo i vuo t, but the reltio is futio of rther th beig simply lier. This iterprettio is show i the digrm below, with otrtio d diltio iresig with v Tht is, the medium beomes more dese s ireses, d, d s dereses, the photo will tke less time to trvel the diste v

37 I order to uderstd Legth Cotrtio d Time-Diltio, it is istrutive to deostrut i terms of its Glile spe-time ompoets v ',, v t t v i the time-diltio equtio v v The Deostrutio of Time-Diltio ( v ) v Settig v, is equivlet to holdig ostt d llowig the reltive time to vry so tht t t t v t. The time diltio equtio the beomes ' t, t,, t tv t, tv where it is ow uderstood tht d ' refers to the mss retio times with m d m m, where is sometimes referred to s proper legth d s proper time, ' ' t ' t t v whih mes the desity t ow vries with time tv t Uder this iterprettio, the t t t time diltio is tully t s Glile rtio, whih llows mss to vry s t m ' ' m t t t t Spe-Cotrtio ( t v t ) v t v The rtio lso be preserved by settig tv t d llowig the reltive vritio of v so tht ',,, v The spe-otrtio v v Is ow see s otrtio L. t v t v v v Note tht t, so tht the rtio is preserved i the produt. t t t t v v v

38 For,, whih emphsizes the oept tht is eergy epressio i two t t dimesios whe relted to spe d time.

39 Spe-Time Coordite Reltio to STR Diltio, Cotrtio

40 This digrm (I reted this some time go, s I ws lerig how to do digrms) shows plot of s ' v futio of, where, symptotilly., so tht ireses from, d The futio is o-lier beuse the reltioship is tht of eergy, d ot of mometum.

41 Fil Stte Solutio (The Reltivisti Uit Cirle) Dividig both sides of equtio (.) by the fil stte s' ' results i the equtio (.3): from : whih be hrterized s the reltivisti uit irle, where rottes w si os However, for the Biomil Epsio, the result is: r (.6) I prtiulr, ote tht i j, but tht i j k d d 4 whe ll four qudrts re tke ito out.

42 Note tht the rdius remis ostt s ireses (so ireses d dereses, suggestig tht irese i represets irese i desity w.r.t. uity (, ), iresig towrd ifiite desity s., As this hppes, ote tht the fil stte represeted by ' '. shriks s ', so the totl re of the widget i the fil stte pprohes. If ', if, there is either iitil stte or fil stte; there is othig there. If ', there is oly the iitil stte. It is lso importt to uderstd tht i this pproh, there re o restritios o the vlue of, sie the iterprettio is oly i terms of widget retio

43 The Four Qudrts of the Reltivisti Uit Cirle The restritio to the upper right qudrt of the reltivisti irle be removed if it is uderstood tht eh qudrt is idetil represettio of the positive umber set v,,, ', so eh qudrt represets oe fourth of the totl irle. Therefore, the desriptor should ow be relbeled to reple the origil, where 4 (old) ow refers to the sigle reltivisti irle 4 hrterized by ll four qudrts. Therefore,, the sigle reltivisti irle (osistig of positive elemets oly) is hrterized by the sigle positive uit vetor, where it is represeted i mtri ottio by Positive rel umbers (iludig itegers) re the represeted idividully by the epressio i where it is uderstood tht i is the uit vetor refereig the system d is uderstood i the preset otet. These four sttes be desigted by : i,,3,4 to desigte the seprte qudrts for the system 4, i i i 4 i 4, where 4 i, i so tht ll four 4 4 qudrts rotte eqully, d os si i os i si i i 3 4

44 All four of the qudrts will be idetil (i.e., 3 4 ) sie they ll re geerted from the sme set of positive rel umbers v,,, '. Therefore, the lier prt of the reltivisti uit irle is hrterized by ' ', However, the full hrteriztio of the irle is give by, ' ' where the sum over ll four qudrts is iluded i the itertio term. I the Reltivisti Uit Cirle, d re orthogol, the vetor reltio i two dimesios is give by r i j os i si j, r, (.7) where is uderstood to hrterize the gle betwee the orthogol uit bsis vetors i i d j j i two dimesios. I oe dimesio,, this equtio redues to oe dimesio: os, r i i r i, where (.8) r r ros r ', (.9) r So tht for r, this epressio beomes

45 The i i is the bsis vetor for sigle dimesio i the umber system, (.) where the oeffiiet of the vetor spe i relevt to the RUC is hrterized by the sigle field vrible (rel umber) i i I the subspe i i the sigle umber system i, hrterized by the four proper field vribles i, v,, ', v,, ' reltivisti uit irle. (.) i terms of the The uit vetor thus hrterizes the stte whe whih is the oditio tht v, d thus ' so tht ' idepedet of. This is the se whe the Iitil stte is equl to the fil stte s s i equtio (.). I prtiulr, gi it is emphsized tht the oditio ' mes there is o umber system t ll (i.e., o iitil stte or fil stte, so o hge stte (i.e., v v ' ). The the fil stte for the sigle system (hrterized by v is give by v i i i i,, v the omplete set of rel umbers (s widgets) from '., where the fil stte represets (.3) ' to

46 The se whe represets the hge bsis i the umber system, where j si j () j, I this se,, but v. However, this stte is virtul sie there is o iitil stte, s outlied bove. Whe the gle, the uit rdius rottes, d while the reltive vlues of the rdius r is ivrit sie os i si j mes tht d my hge, r si os is ivrit, where osi si j is iteger oly if for, showig tht the system is ideed reltivisti model for sigle umber system represeted by the positive qudrt of the reltivisti uit irle. At, the umber system o loger eists (i sese it hs bee destroyed by the rottio by w rottio from - where the iitil stte equls the fil stte to ). 4 As ltertive, osider the se whe rottes lokwise from. At,, so the umber system does ot eist, but the hge vetor is t mimum v j j, v. As rottes, si dereses d ireses util the fil stte t whih poit, d the umber system osistig of the set of rel umbers from to hs bee reted. Therefore, rottio withi the upper qudrt of the uit irle be hrterized s positive widget retio or destrutio of the set i the rge of to depedig o the diretio of rottio of.

47 Itertio i the umber system {} For the reltivisti uit irle, if there is o itertio betwee the prmeters d ; i.e. betwee si d os the i vetor termiology they re orthogol i j d osi si j with the omplete system hrterized t,, v, For, v, d the irle plus itertio for the sigle qudrt is,, os si v v, If ' the system does ot eist, sie there is o iitil oditio; However, withi the rel umber system the symmetry of the irle represeted by, be distorted i terms of the field vribles, v, t, t ' if equtio (.) is llowed to hge i terms of, so miig is llowed betwee si d os. This will be the se if two rdii re llowed i the system whih will hge the reltive vlues d for eh rdius, but will still remi withi the rel umber h system This be omplished by speifyig two gles for eh of the rdii, so tht but os si os si, so the bsis of the stte is still preserved s the uit iteger. The eh pir of rdii will hve its ow vlues,,,, where they be ombied ito the overll desriptio of the reltivisti irle for the system by

48 where r iteger uless, i whih se r r r r r d r d r, but either r or r is This is omplished by speifyig to sets of field vribles withi the set,,, ' v t t This will yield differet set of vlues where ' ' ' v ' v ' Dividig by will rete ew bsis iteger ' for the set ' vribles re still withi, but ' ' ' v ' v ' ' ' () ' d ' d, v, t, t ' d (sie ll the field will ot be iteger, sie re both itegers withi the set where d so r r, so tht ' ' r r therefore ot iteger., d is () Sie d re both uit itegers withi the sme set, they both be multiplied by idepedet rel vribles withi the sme set i d y y j (where d y hve the sme vlue s bove for r d r s bsis vetors for the subsets, but re distiguished by the lol uit vetors (i.e., withi the globl set ) for eh vrible. Suppose ' ( i, y j) ( y) i i y j j y where,, y y y i j rem(, y,) d y y i the Biomil Epsio ( y) y y where the subsripts hve bee dropped i the ltter se, sie the oeffiiets d y re lredy uderstood wr.t. their respetive lol bsis vetors i, y j withi the set. Compre ( y) y y with

49 Fermt s epressio: y, where rem (, y,) y whih oly be true if y, whih oly be true if so tht d y re prllel (i.e., o the sme umber lie if they hve ommo origi t the poit i so tht y i i yi i y y I the se of Pythgore triples, where ' o the bsis b for b,, positive itegers, this mes tht b oly our for b, b b or either or b, ll of whih re otrditios. The solutio is give if imgiry umbers re used, so the itertio (multiplitio) term b is reted d destroyed by the proess ' ( b)( b) ' b ' where, so b is ot rel umber d the set, y, sie i Tht mes the epressio b ot be i the set b ( b) to geerte it from the elemets geerl epressio y,, y The retio of the ple, y The idepedet elemets d b d y rete ple, where positios i the ple re represeted by the produts y y. sie it requires the egtive umber. Note tht this is lso true for the y of rel umber pirs, where, where the elemets d y ommute, d oly be elimited by itroduig omple umber for the se This mes the Biomil Epsio for : y y if y y is vlid withi is lso iluded i, so tht the produt y y y y so tht retio produt of the iitil oditio s ; otherwise does ot osist of rel umbers where multiplitio is ssumed i the Agi, ote tht the bsis elemet for the ple ' ot be iteger if there re two idepedet rdii withi the reltivisti uit irle. Eh of these rdii be hrterized s subset withi of rel umbers, so eh be desigted by its ow subsript, b eh withi the origil set where is the subset where hrterizes the iitil oditio, d the seod subset

50 b hrterizes the hge oditio v ', where the oditio b mes tht does ot eist, sie does ot eist.

51 Retur to the Uit irle Cosider the epsio ( ) for, so ll the umbers, v,, ', v,, ' i the system re positive rel umbers. The re of the trigle i the upper left qudrt is give by A ( ). A ( ) The re of this trigle ireses s rottes w from deresig from d iresig from. At 4, os i si j (.77..) i (.77..) j , d A ( ) As otiues to rotte w, eists t,,,, to, with 4 4 os si 4 4, where so tht otiues to derese d ireses, util the system o loger Notie, however, tht is orthogol to so tht lwys, but si, Tht is, the ross produt lwys eists uless if the implies the set does ot eist. h d re ot orthogol, but

52 Cse I this se,, but v v v' ', so tht is uffeted by the idividul slig vribles d ' However if does t eist, the either does the system (the system is omplete t. I prtiulr, there is o multiplitive slig produt for the first order terms (, ) sie i j () If is iterpreted s mss d is iterpreted s hrge, the the system is ll hrge d o mss i this se, it is model of ll hge d o iitil oditio. If represets the fil stte v (o hrge ) the is the iitil stte divided by the mss, where it is uderstood tht the iitil stte is equl to the fil stte for v so tht the m, v Tht is the stte where ' i equtio (.). If the mss ow ireses, ' d v ' will both hge with, v but the rtio, v will remi ivrit for. As the mss ireses, the iitil ' ' stte dereses, d so does the fil stte; there is either fil stte or iitil stte t Therefore mss hge is proportiol to both d reltive to, where hge is diretly proportiol to d the rest mss is iversely proportiol to fil stte) ivrit. Note tht the otrtio of m m sie m for (the m', m m' m' whih is iversely proportiol to v is iversely proportiol to the spe-time otrtio whih ireses mss ( v ) reltive to t Cosider equtio (.) where the first order multiplitive term ( vt ') hs bee iluded ( ') ( v ') ( ) vt ' (.4) Dividig by ' s before results i the equtio

53 ( ') ( v ') ( ) vt ' (.5) vt ' v t ' ' ', where the r.h.s. is ow the equtio of the reltivisti uit irle + the itertio term (.6) '. For v,, this term vishes, but for, v ' For y gle other th or, so tht ' ' Settig s the referee widget retio rte, ommuttivity so tht i j, llowig for i the umber system. Without this first order of ommuttivity term, the system is iomplete with respet to the rel umbers, whih requires both first order multiplitio d ommuttivity. With the dditio of the ommuttive term, the umber system is ow omplete for At, v b with the dditio of multiplitive iverse whih is lwys vlid sie the v deomitor i ever vishes. Commuttivity of rel umbers i the umber system {} The opertio i j k d j i k i j j i k k, where where the ull vetor hrterizes the ft tht rel umbers ommute:, so the ross produt d the ull vetor represet rel umber multiplitio i the system.

54 I prtiulr, ote tht the multiplitive produt is the oeffiiet of the ull vetor, d so is ot member of ( i, j ), but rther of the three-dimesiol vetor spe i, j, for the set of rel umbers i the rge. (Note tht the proess of ddig k to k is logous to osiderig the positive hlf ple isted of just the sigle qudrt, sie si lwys d os( ) os, so if left d right rottios from to, re iluded, the two sets of rel umbers o the two vetors re reted, so tht d s before. Differet rottios of the two vetors my result i differet vlues of d i the two qudrts t y give stte of the two s, but ll of the resultts re still iluded i the set of positive rel umbers for the system.) I prtiulr, there is oly oe iteger i the umber system i the umber system for the se, v where ', '. If the itertio is iluded, the omplete rel umber system is the set,, uit iteger whe, v. whih is the Without the itertive term, the equtio of the Uit Cirle is (.7) Ad with the itertive term, the equtio for the omplete umber system is ow (.8) With the itertive term oly if re orthogol (i.e., ) The itertive term lso be mde to vish by imgitively ddig d subtrtig the itertio term i the epressio

55 i i, i (.9)

56 Compriso of Iitil Stte d Fil Stte Represettio of v,,, ' Note tht the rtio of the iitil sttes d the fil sttes remis the sme i the uit vetor, but tht its ompoets hge together. Fil Stte Iitil Stte I both se, remis the sme; for the Fil Stte t, d for the Iitil stte, t. Note tht for reltivisti irle the res of the Fil stte trigle is A d tht of the Iitil stte trigle is A, but there re four qudrts to the irle, so the totl re of the itertios re 4( ) d 4( ) for the Fil stte d Iitil stte respetively, so the result is osistet with the Biomil Epsio for the se, where: ' Fil Stte : '

57 ' d Iitil Stte : ' os si osh sih where eh represets the sme set of rel umbers v Let There Be Light The Cretio of the Uiverse,,,, ' I the followig digrms, the itertios dded by the Biomil Epsios re idited i blue for the Fil Stte lysis, d red for the Iitil Stte lysis. Note tht the Fil Stte is reltio betwee trigoometri futios, d tht the Iitil Stte is reltio betwee hyperboli futios (see Hyperboli Futios below).

58 Fil Stte Alysis v At the poit (,,) othig eists, sie, m ' v ' Note tht the rtio ssumes tht both v d re widget retio rtes (so the equtio is dimesiolly osistet), whih hve bee re-med light i the setio title. Therefore v is light retio rte reltive to. As ireses from zero, the eter of the irle shifts from stte of othig to somethig. If durig the proess, oly the uit vetor i is reted, so there is o reltivisti irle, oly vetor i oe dimesio. However, for, the digrm grows i three dimesios; the o- itertig reltivisti uit irle hrterized by remis ivrit

59 The equl d opposite spi itertio is hrterized by. i the Biomil Epsio for Iitil Stte Alysis Iitil Stte Alysis

60 The Iitil stte Is hrterized by r k k r r i i j j ( ) r r Note tht this hrteriztio is osistet with tht the fil stte ( ), r, r ' : ' where otrts s the eter of the irle moves to the left, with s,,, keepig i mid tht so tht log with..

61 Desity d reltio to Spi It is ler tht the uit bsis elemet represets miimum of iitil oditio i the rel umber system v,,, ' where, d. Ay rottio of will irese os, to the vlue os si,,, idetilly i ll four positive qudrts, both iside d outside the uit irle. The both d be hrterized s ireses i desity from uit desity s ireses towrds with w rottio, d similrly dereses from to uit desity with lokwise rottio. Withi the uit irle s the fil oditio, this mes tht if represets the metri i the dimesio of the rel umber system where ',, the represets otrtio of the reltive metri (whih is ot to be iterpreted eessrily s legth i Glile spe-time, but rther s irese i desity of whtever physil prmeter is represeted by. Outside the uit irle s the iitil oditio, ' Spi represeted by the itertio betwee d with respet to the reltivisti uit irle, where ( ) withi the uit irle s the fil oditio. Similrly, outside the uit irle s iitil oditio similr rottio will represet diltio of the metri, gi represetig irese i desity of whtever physil prmeter is represeted by the produt, whih represets spi i the equtio ( ) outside the uit irle s the iitil oditio. The ftor of is beuse eh qudrt otributes ftor of the resultig spis s (iside) or (outside) with 4 d s 4, respetively.

62 Iterprettio of the Biomil Epsio The epoet i the Biomil Epsio rem(,, ) represets the equl umber of positive widgets represeted by d i the totl system, with rem(,, ) represetig the itertio of subsets of the widgets for eh power. For emple, i the se 3, 3 3 ( rr) , rem (,,3) 3 3 where d be idetified s gulr mometum (respetively) i system osistig of eter of mss 3 3, where () 3 3 r r Note tht 3, d, 3 3 hrterize grdiets for d respetively i the itertio term, d thus orrespod to Christoffel symbols i Differetil Geometry (d thus GTR), where is idetified with dd for d with d d for ', respetively. ' The se 4 for the Biomil Epsio is lso importt, s will be ddressed i the forthomig disussio of the Puli d Dir mtries. t

63 Reltivisti Rottio i the system {} Positive rottio The positive rottio here defied s lokwise rottio hrterizes irese i i eh qudrt from i where it is uderstood tht the idies re yli i ( 4) I I 3 I I 4 Qudrts d 3 For qudrts i,3 the positive (w) rottio orrespods to derese i (d irese i ) from the iitil stte t, d resolvig t, t the et qudrt i, where it is uderstood tht i, so tht pproimtio sigs re used t the iitil stte to idite i tht is udefied t i i these qudrts (orrespodig to ifiite mss, uless (e.g., m h orrespodig to the zero stte where there is othig i the Uiverse, ot eve light).

64 Positive rottio i qudrts d the orrespod to 3 si i, os i, i,3 where 3 d 3 3. At, 3, the bsis vetors d 3 oly eist s hge vetors where, v, but is udefied Qudrts d 4 For qudrts i,4 the positive (w) rottio orrespods to derese i (d irese i ) from the iitil stte t i,, i, i,4 i, i, i, i, 4 d resolvig t i t the et qudrt i, where it is uderstood tht i, so tht pproimtio sigs re used t the fil stte to idite tht is udefied i i t i these qudrts (orrespodig to ifiite mss, uless (e.g., mh orrespodig to the zero stte where there is othig i the Uiverse, ot eve light). Positive rottio i qudrts d the orrespod to 4 d 4 os i, si i, i,3 where 3. At 4, the uit bsis vetors d 4 re relized. The rottio,,3 i qudrts (,3) hs the opposite effet o d s the rottio,,4 qudrts (,4), so the rdius is ivrit durig the rottios, eept where it is udefied t 3 d, d where d represets the uit vetor represets the uit vetor os, i os, Agi it is emphsized tht eh qudrt yields 4 the re of eept i the se of, i whih se the uit vetors for, together represet.

65 The totl re of the RUC for,,,,,3,4 is i i i i os 4 i i i i, where Negtive Rottio i i i,. 3 4 A w rottio be defied s, where v si( ) si( ), whih defies the set, v,, ', v,, ' This rottio will ot ffet, so tht is uffeted, but it llows distitio betwee the sets v,,, ' d, v,, ' (For Physis, this suggests the mtri v q, where uit hrge 3 q v mtri). 3 q q hrterizes the Puli

66 Cosider the equtio ( ), d ote tht this is iosistet with tht of the reltivisti uit irle beuse of the term where d re both positive (this term is iterpreted s spi i physis for resos tht will beome obvious.). However, if the equtio is iterpreted s ( )( ) where ( ) it beomes, so tht the itertio hs bee removed by the equl d opposite hge (gi, iterpreted s hrge (or photo-equivlet spi) i Physis). (The eessity of hvig to use egtive to remove the itertive produt is yet other proof of Fermt s Theorem, where,where ot be elimited uless the egtive elemet is itrodued, violtig the ssumptio of positive rel umbers i. Sie the egtive hrge is irrelevt to the bsis vetors for eh set, y y j y b is lso positive, so there is o distitio betwee them tht wy. The i is positive d ( ) y y rem (, y, ), rem (, y, ), so y, gi provig Fermt s Theorem. Reltivisti Spi (=) If two idepedet prtiles i the sets d y hve the sme vlue so tht s y, the the itertive term is give by tht s h, where h is Plk s ostt, the y y s If the itertive term is qutized so h s, whih is the spi of qutized prtile (eletro or photo), d is resposible for the hyperfie splittig observed i tomi spetr, d represets the itertio of qutized eletro (i.e. photo-equivlet eletro) with its surroudig field withi sigle orbit i the tom. Note the releve to the Biomil Epsio for ( ) ( y) ( y ) y ( y ) y I terms of the reltivisti uit irle, the equtio is so tht s, s determied by the hrge to mss rtio i tht iterprettio. h h, s so spi is

67 The seod rel umber system {b} Cosider the lysis for the umber system pplied to seod rel umber system b, where d v, b, so tht the bsis vetor b b os from the set of field vribles b, v,, ', v,, ' whih re idepedet of those of i Crtesi spe, b b b b b b If the itertio is iluded, the omplete rel umber system b is the set,, whih is the uit iteger b whe b, v, where ow rottes w from b b to b The iteger b t - v, b is the oly iteger i the set of rel umbers b

68 The Defiitio of Multiplitio The reltivisti uit irle together with the Biomil Epsio be tke s defiitio of multiplitio betwee rel umbers. Withi the sigle rel umber system v fil oditios re give by,,, ', the two bses hrterizig the iitil d ' ', (.) Ad, (.) Neither of these equtios be vlid for multiplitio is defied by d both re equl to d otherwise, so respetively, where the ftor of two rises from the ft tht ll four qudrts of the reltivisti uit irle hve to be tke ito out. Similrly, for powers of epsio, we hve rem(,, ) where,, rem d similrly for, rem,, where rem multiplitio for both ses. Sie these equtios hold for,, defies for ll rel umbers, the the equtios lk the opertio of multiplitio eve i the sme umber system. Therefore, Fermt s epressio is ot true for y pir of rel umbers i system where multiplitio is defied, sie the produts re idetilly equl to zero. d

69 Tht is, ( y) y rem (, y, ) hrterizes multiplitio betwee two systems, v,, ', y, v,, ' y d y does ot (there is o multiplitio betwee d y. Sie this result is true for ll, y yy, Fermt s Theorem is prove (the itegers re subset of the rel umbers).

70 The ple of itegers Sie the umber systems d b re idepedet, the two itegers d b form orthogol system, b, where the vetor bsis be represeted by, i j Sie slr multiplitio is ow iluded, eh of these system be multiplied by d rel umber i their respetive systems, where the vribles re represeted by (, y), y prtiulr, if the umbers re itegers, by, b, b iteger vribles over their respetive sets b, d i b b where ow d b represet Sie the sets d b re disjut, the set ( b, ) is lled biomil. If there is o itertio betwee the two sets of itegers, the b, meig there re o multiplitive terms of the form b b, whih mes tht b Tht is, d b re restrited to their respetive umber lies s oeffiiets of the uit vetors. Note tht this epressio oly be geerted ( b)( b) b, whih requires the iteger b b b,, (Note: To elimite reltivisti eergy itertio, oe must use omple umbers so tht ib ib b, whih is implemeted i both the Puli d Dir Mtries i vrious wys.) However, if there is itertio, the the Biomil Epsio for the se results i the epressio ( b) b b b rem (, b,) where the system of idepedet itegers is ow omplete. For the iomplete system (with the elimitio of b b ) b, if is iteger, d the system is the set of b,, of Pythgore Triples, where. However, if the system is omplete, it iludes multiplitive produts, so the Biomil Epsio holds. However, ( b) b b b rem b (,,) i b j b () b b

71 ot be iteger eve if b is iteger.

72 For, the Biomil Epsio is give by ( b) b rem (, b, ) where rem(, b, ) osists of ll the terms i the epsio tht re ot or Formlly, the Biomil Epsio (Wikipedi) is give by b k k! ( b) b, k k k k!( k)! Note tht with the eeptio of ( k) d b ( k ), ll the terms i the epsio re of the form p b p pq q b the se p pq p pq, where the terms p pq pq p b b ommute, but the terms b do ot for the se (s ws the se i first order where b Tht is, the terms i rem(, b, ) for ever be elimited, sie orthogol to p is ever b i p q b, eve though the produts form oeffiiets of the ull vetor, so tht ( b) i b j rem(, b, ), where eh term i (,, ) positio ( p p q, b ) o the iteger ple. Pythgore Triples rem b is loted t the Although it might pper tht right trigle formed by Pythgore triple is two dimesiol i ture, the proess tully results from outig reltivisti uit irles; i 3,4,5 triple, the equtio i vetor form is tully stte where,, i i i 5 5 i where eh i represets irle i it iitil, where eh set is formed from its ow set seprte rel systems for eh iteger re idepedet. The tully represets i v,,, ' so the i 5 v,,, ' idividul rel umber systems with the bove oditios. The oeffiiets seem to be idepedet beuse the rel umber sets re idepedet. Thus itegers do ot itert with eh other, so tht 3 4. The tul vlues of Pythgore triples the deped o the umber bse (s i to the bse ) of the idividul system i whih they re represeted; systems i differet umber bses will hve differet vlues d symmetry groupigs. b

73 The ft tht ((3,4) pper to be idepedet i 3,4,5 right trigle is beuse the uit bses for eh reltivisti irle re idepedet; the symmetry rises from prtitioig the uit elemets (estblished from the bse ). A Pythgore futio would the be reltioship betwee umber systems of differet bses, whih would estblish the vlue of eh symmetry i eh bse (d eve reltive to eh other). Pythgore triples stisfy the Biomil Theorem for : ( b) b b 3, b4 3 (4) [3 4] The the triple b,, is Pythgore triple if is iteger. Pythgore triples be geerted by the equtio ( ib)( ib) b Defiitio of Group (Group Multiplitio) I bstrt lgebr, this mes tht eh elemet, v,, ' of group is withi idividul set of rel umbers v,,, ' where,, (i.e., the symbol s represettive of the group is the uit elemet v,,, '. Cosider multiplitio betwee two group elemets b b. The defie group elemet suh, where, b, tht b b G Defie seod group elemet d b. The, b,, d G d ( b) b b b b b bb, so tht The d ( b)( b) G defies group multiplitio for ll group elemets, b,, d The if b b b d b

74 Sie osists of oly produts b, but ot sums, the both d b d d ot represet group multiplitio for ll elemets, b,, d, d The b b ( b) G, so the opertios d re iomplete, d must be repled by d (diret sums d produts), d so hrterize vetor ompoets i two dimesios.

75 Fermt s Theorem Fermt s epressio is b Fermt s Theorem sys the epressio ot be true for positive itegers, b,, for (.3) p pq pq p without multiplitive terms suh s b b or b b, d is therefore iomplete for the idepedet sets of itegers ( b, ) whih require the bsis vetors, b.

76 Proof Fermt s Theorem A rel positive vrible, v,, ' be ssiged to ll possible vlues th stisfy equtio. ; i.e., where ( ) os si (, v,, ') rges over vlid for ll (, v,, ') The sme lysis be performed to hrterize idepedet seod set y, v,, ' tht b b, so b( y) yb b yb os b si b yb b b vlid for ll (, v,, ') b where y equtio. d b, d the sme for third idepedet vrible r v The remig ( ) d y ( y), the spe z f, y, r futio osistig of ll rel positive ombitios of, y d r. I prtiulr, ote tht r r d y b b.,,, ' stisfyig r is positive rel sigle-vlued Sie the vribles re idepedet, the diret sum d outer produt be used to defie ordiry dditio d multiplitio betwee ll the elemets of eh idepedet set: y r ( y) r d y y y r r r r Let z y y rem(, y, r) re ll positive umbers where rem (, y, r) osists of ll elemets of the Biomil Epsio where rem (, y, ) d rem (, y, ) y. The r r r z y, d i prtiulr whe,, y r re itegers. Thus Fermt s Theorem is prove, sie the reltivisti uit irle is omplete for ll positive rel umbers i, y d r. The Biomil Theorem (Wikipedi), where the geerl epressio is give by

77 ( y) y y kk kk, where the oeffiiet of k k k k! formul. k k!( k)! k y k is give by the Sie itegers re subset of the rel umbers, this epressio be restrited to itegers i the followig form: ( b) ( b) ( b) i b j rem(, b, ), where rem(, b, ) iludes everythig i tht is ot or b, d i j The Biomil Epsio is omplete, but rem(, b, ) vishes oly if or b, i whih se b or respetively. This emple of Qutum Trivility for the se of two distit Plk s ostts (two differet photo eergies) where the oe remiig elemet is the Higgs Boso. Note tht is iteger eve for, but tht for it ot beuse of the uless d b re o the sme umber lie i oe dimesio, i whih se the epressios bb d b b re tutologies, sie the l.h.s. d the r.h.s. refer to the sme sigle umber i eh equtio. Fermt s epressio represets the metri of rithmeti, whih oly iludes dditio d powers (of idepedet elemets), d so is emple of rithmeti system tht is osistet, but ot omplete. If the system does ilude iteger multiplitio, the the system is omplete, but is ot osistet with Gödel umberig, whih does ot ilude the i its syt (i.e., does ot ilude slr multiplitio, but oly prtitioig outs ito dditive groups i oe dimesio). I prtiulr, Fermt s Theorem is reltioship betwee groups, where where i, i, i i, i, b, so tht Fermt s Epressio d b b b does ot ilude multiplitio of itegers represeted by v d b v,,, ',,, ', so ot b be omplete i the set of ll itegers v,,, ' whih ilude ll possible sets of positive rel umbers. The Biomil Epsio is omplete for the groups d b, d sie rem (, b, ) for ll groups represeted by the vribles d b, Fermt s Theorem is proved.

78 Fermt s Theorem d Presburger Arithmeti Note tht the multiplitio for ot be removed by usig first-order omple umbers o the itertio produts for d b s it for, where both elemets i the produt re first order. b i( rem(, b, ) b i( rem(, b, ) b ( rem(, b, ) ( (,, ) b rem b b b Cotrsted with ( ib)( ib) ib ib b b for the se Therefore, the Fermt epressio represets Presburger rithmeti ()Wikipedi) osistig of dditio oly), i whih multiplitio is elimited by omple umbers, so the Presburger rithmeti ot be omplete i terms of rel umbers (iludig itegers), d so the epressio ot be true for y set of rel umbers stisfyig the Biomil Epsio. This lso proves Gödel s Theorem, sie his syttil elemets re represeted oly by prime umbers s positive itegers, so tht if the Biomil Epsio is omplete d osistet, the Fermt s Epressio ot be omplete for y two idepedet elemets b, or, y hrterized by, where d ( ) ( ) rem(,, ), where,, v,, ', y, y, y whe slr multiplitio (itertio) is iluded

79 Positive Rel Numbers over the sme set Let, y Let d y ' The for the fil stte, (, v,, '), y (, v,, ') y d y, epressig the iitil d fil sttes, respetively., y, d for the fil stte, ' y, ' y y so tht solvig for y y os si os y So tht os oly if,, d Defiitio of Multiplitio The y y si, ( ) lwys i the reltivisti uit irle. be tke s the defiitio of multiplitio betwee positive rel umbers d y i the sme set,,, ' v, where y y y iludes ll four qudrts of the reltivisti uit irle, d is vlid for ll positive vlues of d y y withi the sigle rel umber system (, v,, ') (, v,, ') where the four prmeters ( v,,, ') vry idepedetly for the vribles d y over the set. This result the results i the Biomil Epsio (Wikipedi) by epdig to powers of : k k k k! y y y, kk kk k k!( k)!, d Where y y rem(, y, ) y rem(, y, ) y, rem(, y, ) whih proves Fermt s Theorem for his epressio y ( y) for ll positive rel umbers, ot just itegers. Tht is, z y iy iy so the set of umbers tht llows this epressio ot be omposed of positive rel umber loe; it must ilude omple umbers, whih llow for elimitio of the itertive elemets.

80 Reltivisti Additio of Veloities (d reltio to Newtoi Eergy) The reltivisti veloity multiplitio formul is usully writte s: V r u v uv Sie the two positive qudrts of the reltivisti uit irle hrterize two idepedet sets of rel umbers d y where ordiry multiplitio betwee is the reltio betwee them i the fil stte where y y y, i terms of the RUC, this is equivlet to multiplyig hge sttes withi eh qudrt: b u v y Vr, where for v or vy, Vr y or Vr, respetively. It hs uv y lredy bee show tht i the Glile oordite system (where i this otet, t represet Glile oordites) i the sigle umber system. The t lim, y y y V v v v v y r y y y y (.4), orrespodig to ordiry dditio of veloities betwee the omplete sets d b, where v v v v i the sets b y b y ( v vy) v vy vvy If there is o itertio betwee re idepedet ( v v y ), the the term vv y vishes, so tht v d v y (so they v v. The for sigle ivrit (Newtoi) mss m, m mv mv y the kieti eergies dd, d m m m v m v, y The the Newtoi veloity reltig to eergy is give by Newtoi eergy of two prtiles, eh of mss v y so ( v ) d the totl m is give by E m v y

81 Note, however, tht reltivisti hge i veloity is ttmout to elertio, d the Newtoi lysis does ot ddress the orrespodig hge i m, mb,, whih the must be osidered ifiitesiml i the limit, so the effet must be hrterized for bot reltivisti veloities very lose to, so tht, d therefore i the limit s. Therefore, prtilly o mss is ivolved, d the effet is due mostly to hrge, whih is why Eistei itrodued differetils to model grvity withi the sigle rel umber system. The f '( ) lim f( b) f( ) b b where b b b is the tget to the geodesi t, f ( ) b b light (s hrge) hs ifiitesiml but rel mss. f ( ) b t the limit, d b i the GTR reltivisti theory of mss, d b Tht is, if, is iterpreted s the hrge to mss rtio for eh photo-equivlet b systems d b, the Note the logy with df ( ) dm b, f '( b) f '( mb ). d dq b b, f ( mh) f ( h) df ( m) f '( b) f '( mb ) lim( ) where h is Plk s h h d h ostt hrterizig the tio ( eergy ) of photo-equivlet eletro b The dditio of reltivisti veloities orrespods to ordiry multiplitio, sie y Vr v vy y b, y vv y

82 Iitil Stte Solutio (The Cotiuum) For the iitil stte solutio to the equtio (.), both sides of the equtio re divided by the iitil stte s t, whih results i equtio (.5);. v Sie., the trigle eteds to the right with iresig, but However, the equtio is still iosistet with the Biomil Epsio, hrterized by As before, the itertio term be oly removed by the omple ojugte: i i

83 For the geerl Biomil Epsio, the epressio beomes: rem(,, ), where gi rem(,, ) so tht. Note tht this will pper s power epsio i with Biomil Coeffiiets (but oly if oe igores the powers of uity, whih idite the umber of uit itertios for eh term, d thus is misledig i tht iterprettio).

84 Euler s Formul d the Biomil Epsio i Euler s equtio is e os isi The epressio i i os si os si os si elimites the itertio term by omple ojugtio, ompred to the Biomil Epsio for : (os si ) os si os si Reltivisti Cosequees ' Cse (ivrit fil stte) ' This equtio elimites itertio betwee os d si ( d respetively), where i i os si os si os si The Biomil Epsio for of this equtio is give by: os si os si os si si os Note tht the itertio re i eh qudrt of the reltivisti uit irle is outs for the ftor of, where 4 Higher orders of the Biomil Epsio re give by rem os si os si (os,si, ) A, whih Cse From the Biomil Epsio ( ) of

85 The itertio term is elimited i the ezpressio: i i, whih orrespods to the reltivisti epressio, where The time diltio equtio rises beuse of the elimitio of the itertio produts d for the se i the Biomil Epsio, where, re orthogol (i.e., idepedet ) for ' d, re orthogol (, ) for. ' This mkes the lier form Speil Theory of Reltivity osistet with Mwell s equtios for both eletromgetism d reltivisti mss idividully. but ot for both (whih re relted by the Loretz fore d the vetor potetil A). Note tht the oditios o the vetor potetil for the Eletromgeti Potetil re tht the ovrit d otrvrit derivtives multiply the ull vetor (i.e., re equl d opposite. These derivtives orrespod to the spi itertio terms i the Biomil Epsio (for ) for both the iitil d fil ivrit oditios. For higher orders of itertio, the oditios o the vetor potetil ot hold, sie the derivtives do ot ommute. However, Mwell s equtios hieve lierity due to the use of dot d ross produts i the Stokes d Gree s formultio with respet to res d volumes i terms of hrge d urret oly, d QFT hieves lierity through the elimitio of the itertio terms mm through the use of the v epressio m im m im m m whih is epressio of the oept tht the v v v perturbtio does ot ffet the rest mss (the iitil oditio). The Biomil Epsio (for ) iludes this itertio, d is epressed s m mv m mv mmv where v the fil result s o-lierity. For higher orders of itertio, the epressio beomes m m m m rem( m, m, ) v v v m sles m d is physilly to be iluded i

86 The Hyperboli Futios (Iitil Stte Ivrit) Agi, it is emphsized tht i this otet, ll four qudrts of the reltivisti digrm re positive, so the itertio term is tully whe the reltivisti uit irle is tke s the iitil stte., so tht, so tht,, i (Note tht orrespodig to the reltivisti uit irle) where i ) The Hyperboli Futio (Wikipedi) hrteriztio of the Speil Theory of reltivity hrterized i terms of r where r is the rdius of the hge equivlet irle epded from to. e e e e osh r r e e r r r r e e e e sih r r e e For the stte r r r r, r e e e e sih e e e e sih, e e e e osh e e e e osh r r

87 osh rsih r osh r,, v, sih r Notie tht the oly differee betwee osh d sih is the sig i the umertor betwee d e ' whih will determie whether the irese is due to hge i the totl eergy ' sih iitil oditio or the itertio eergy where r, r so tht where osh d r e osh sih ( ) I terms of the reltivisti irle below, ( )r, so tht iteger multiples of orrespod to multiples of omplete rottios of the irle; ( ) r. I prtiulr, ote tht r for r i rdis, where the irumferee ssumes the role of reltivisti (positive) mometum. I this otet, r tkes o the otet of eergy. Iitil Stte ( ) ( ), r r

88 Iitil Stte

89 Iitil Stte Alysis I terms of the Biomil Epsio, the results re (i terms of the vetor resultt r):