1 Einstein s vision I: Classical unified field equations for gravity and electromagnetism

Transcription

1 1 Eisei s visio I: Classical uified field equaios for graviy ad elecromageism usig Riemaia quaerios bsrac The equaios goverig graviy ad elecromageism show boh profoud similariies ad uambiguous differeces lber Eisei worked o uify graviy ad elecromageism, maily by ryig o geeralize Riemaia geomery Hamilo s quaerios are a 4-dimesioal opological algebraic field relaed o he real ad complex umbers equipped wih a saic Euclidea 4-basis Riemaia quaerios as defied herei explicily allow for dyamic chages i he basis vecors The equivalece priciple of geeral relaiviy which applies oly o mass is geeralized because for ay Riemaia quaerio differeial equaio, he chai rule meas a chage could be caused by he poeial ad/or he basis vecors The Maxwell equaios are geeraed usig a quaerio poeial ad operaors Uforuaely, he algebra is complicaed The uified force field proposed is modeled o a simplificaio of he elecromageic field sregh esor, beig formed by a quaerio differeial operaor acig o a poeial, Box* * This geeraes a eve, aisymmeric-marix force field for elecriciy ad a odd, aisymmeric-marix force field for mageism, where he eve field coserves is sig if he order of he differeial ad he poeial are reversed ulike he odd field Gauge symmery is broke for massive paricles by he eve, symmeric-marix erm, which is ierpreed as beig due o graviy I esor aalysis, a differeial operaor acig o he field sregh esor creaes he Maxwell equaios The uified field equaios for a isolaed source are geeraed by acig o he uified force field wih a addiioal differeial operaor, Box* Box* * 4 pi J This coais a quaerio represeaio of he Maxwell equaios, a classical lik o he quaum haroov-bohm effec, ad dyamic field equaios for graviy Vacuum ad zero e curre soluios o he uified field equaios are discussed The field equaios coserve boh elecric charge desiy ad mass desiy Uder a Lorez rasformaio, he graviaioal ad elecromageic fields are Lorez ivaria ad Lorez covaria respecively, bu here are residual erms whose meaig is o clear presely addiioal cosrai is required for gauge rasformaios of a massive field (PCS:11-g) Eisei s visio usig quaerios Three of he four kow forces i physics have bee uified via he sadard model:he elecromageic, he weak, ad he srog forces The holdou remais graviy, he firs force characerized mahemaically by Isaac Newo The parallels bewee graviy ad elecromageism are evide Newo s law of graviy ad Coulomb s law are iverse square laws Boh forces ca be aracive, bu Coulomb s law ca also be a repulsive force logsadig goal of moder physics is o explai he similariies ad differeces bewee graviy ad elecromageism lber Eisei had a specific idea for how o formulae a accepable uified field heory (see Fig 1, ake from Pais, Suble is he Lord he sciece ad life of lber Eisei, Claredo Pres, 198) Oe uusual aspec of Eisei s view was ha he believed he uified field would lead o a ew foudaio for quaum mechaics, a idea which is o shared by some of oday s hikers (S Weiberg, Dreams of a fial heory, Paheo Books, New York, 199) Mos of Eisei s effors over 4 years were direced i a search o geeralize Riemaia differeial geomery i four dimesios To a degree which has pleasaly surprised he auhor, Eisei s visio o uify graviy ad elecromageism has bee followed The cosrucio of a ew 4-dimesioal geomery is dicaed by isighs garered from physics Eves i spaceime are composed of a scalar for ime ad a 3-vecor for space The four-dimesioal opological algebraic field of quaerios has he same srucure, so quaerios will be he sarig poi of his effor Laws of physics are expressed i a coordiae-idepede way The sum or differece of wo quaerios ca oly be defied if he wo quaerios i quesio share he same 4-basis Riemaia quaerios make coordiaeidepedece explici I special relaiviy, regios i spaceime are delimied by he ligh coe, where he e chage i 3-space is equal o he e chage i ime The pariy bewee chages i 3-space ad ime is cosruced io he 1

2 defiiio of a Riemaia quaerio I geeral relaiviy, he field equaios make he meric a dyamic variable The basis vecors of Riemaia quaerios ca be dyamic, so he meric ca be dyamic The dyamic aure of he basis vecors leads o he geeral equivalece priciple, whereby ay law, eve hose i elecromageism, ca be he resul of a chage i referece frame Physical laws are he resul of simple Riemaia quaerio differeial equaios Firs-order Riemaia quaerio differeial equaios creae force fields for graviy, elecriciy, ad mageism Secod-order differeial equaios creae dyamic field equaios for graviy, he Maxwell equaios for elecromageism, ad a classical couerpar o he haraov-bohm effec of quaum mechaics Third-order differeial equaios creae coservaio laws Homogeeous soluios o he secod order differeial equaios are relaed o gauge symmery The secod paper i his series of hree ivesigaes a uified force law, wih a focus o a paricular soluio which may elimiae he eed for dark maer o explai he mass disribuio ad velociy profile for spiral galaxies The hird paper develops a ew approach o quaerio aalysis The equaios of he firs wo papers are recas wih he ew defiiio of a quaerio derivaive, resulig i a quaum uified field ad force heory Eves i spaceime ad quaerios eve i spaceime is cosidered by he auhor as he fudameal form of iformaio i physics Eves have srucure There are four degrees of freedom divided io wo dissimilar pars:ime is a scalar, ad space is a 3-vecor This srucure should be refleced i all he mahemaics used o describe paers of eves For his reaso, his paper focuses exclusively o quaerios, he 4-dimesioal umber where he erms scalar ad vecor where firs used Hamilo s quaerios, alog wih he far beer kow real ad complex umbers, ca be added, subraced, muliplied ad divided Techically, hese hree umbers are he oly fiie-dimesioal, associaive, opological, algebraic fields, up o a isomorphism (L S Poryagi, Topological groups, raslaed from he Russia by Emma Lehmer, Priceo Uiversiy Press, 1939) Properies of hese umbers are summarized i he able below by dimesio, if oally ordered, ad if muliplicaio commues: Number Dimesios Toally Ordered Commuaive Real 1 Yes Yes Complex No Yes Quaerios 4 No No Hamilo s quaerios have a Euclidea 4-basis composed of 1, i, j, ad k The rules of muliplicaio were ispired by hose for complex umbers: 1ˆ 1, iˆ jˆ kˆ ijk -1 Quaerios also have a real 4x4 marix represeaio: q, x, y, z x y z x z y y z x z y x lhough wrie i Caresia coordiaes, quaerios ca be wrie i ay liearly-idepede 4-basis because marix algebra provides he ecessary echiques for chagig he basis Therefore, like esors, a quaerio equaio is idepede of he chose basis Oe could view quaerios as esors resriced o a 4-dimesioal algebraic field For he sake of cosisecy, all rasformaios are also cosraied o he same divisio algebra This cosrai migh firs appear oo resricive sice for example i elimiaes simple marices for row permuaios Sice quaerios are a algebraic field, here ecessarily exiss a combiaio of quaerios ha achieves he acio of a permuaio The eed for cosisecy will overrule coveiece Laws i physics are idepede of coordiae sysems To make he coordiae idepedece explici, ampliudes ad basis vecors will be separaed usig a ew oaio Cosider a quaerio 4-fucio, (a, a 1, a, a 3), ad a arbirary 4-basis, Iha (iha, iha 1, iha, iha 3) I spaceime, he lie ha divides causaliy is defie by he ligh coe O he ligh coe, he oal chage i 3-space over he chage i ime is equal o oe Physics herefore idicaes pariy bewee he oal 3-vecor ad he scalar, isead of weighig all four equally coordiaeidepede Riemaia quaerio is defied o be Iha (a iha /3, a 1 iha 1/3, a iha /3, a 3 iha 3/3)

3 The scalig facor of a hird for he 3-vecor plays a vial role i he defiiio of a regular fucio i he hird paper of his series The equivalece priciple of geeral relaiviy assers, wih experimes o back i up, ha he ierial mass equals he graviaioal mass acceleraed referece frame ca be idisiguishable from he effec of a mass desiy No correspodig priciple applies o elecromageism, which depeds oly o he elecromageic field esor buil from he poeial Wih Riemaia quaerios, he 4-ui vecor does o have o be saic, as illusraed by akig he ime derivaive of he firs erm ad usig he chai rule: a î i ˆ i a a i î i The ui vecor for ime, iha, ca chage over a ifiiely small amou of ime, i y chage i a quaerio poeial fucio could be due o coribuios from a chage i poeial, he iha da /di erm, ad/or a chage i he basis, he a diha /di erm Is his mahemaical propery relaed o physics? Cosider Gauss law wrie wih Riemaia quaerios: ˆ i 9 e i ˆ i e 9 î i 4, 1,, 3 The divergece of he elecric field migh equal he source, or equivalely, he divergece of he basis vecors The geeral equivalece priciple as defied here meas ha ay measureme ca be due o a chage i he poeial ad/or a chage i he basis vecors The geeral equivalece priciple is applicable o boh graviy ad elecromageism Merics ad quaerio producs The heories of special ad geeral relaiviy dicae he disace bewee eves i spaceime lhough fudameally differe i heir mahemaical srucure, ieria is a lik bewee he wo Special relaiviy dicaes he rasformaio rules for observers who chage heir ieria, assumig he sysem observed does o chage The field equaios of geeral relaiviy deail he chages i disace due o a sysem chagig is ieria from he vacuum o a o-zero eergy desiy quaerio produc ecessarily coais iformaio abou he meric, bu also has iformaio i he 3-vecor This addiioal iformaio abou quaerio producs will sugges a provocaive lik bewee merics ad ieria cosise wih boh special ad geeral relaiviy Mos srucures i Naure do o rasform like a scalar ad a 3-vecor Quaerio producs muliply wo 4-basis vecors, ad hose producs will rasform differely The rules of quaerio muliplicaio mirror hose of complex umbers Isead of he imagiary umber i, here is a ui 3-vecor for each quaerio playig a aalogous role The differece is ha ui 3-vecors do o all have o poi i he same direcio Based o he agle bewee hem, wo differe ui 3-vecors have boh a do ad cross produc The do ad cross producs compleely characerize he relaioship bewee he wo ui vecors Compare he produc of muliplyig wo complex umbers (a, bi) ad (c, di): a, bi c, di wih wo quaerios, (a, B ac bd, ad bc, i ) ad (c, D a, BÎ c, DÎ ac BDÎ Î, adî i ), BcÎ BDÎ Î Complex umbers commue because hey do o have a cross produc i he resul If he order of quaerio muliplicaio is reversed, he oly he cross produc would chage is sig Quaerio muliplicaio does o commue due o he behavior of he cross produc If he cross produc is zero, he quaerio muliplicaio has all of he properies of complex umbers If, o he oher had, he oly value of a quaerio produc is equal o he cross produc, he muliplicaio is ai-commuaive Idividually, he mahemaical properies of commuig ad ai-commuig 3

4 algebras are well kow quaerio produc is he superposiio of hese wo ypes of algebras ha forms a divisio algebra Several seps are required o square of he differece of wo Riemaia quaerios o form a measure of disace Firs, he basis of he wo quaerios mus be shared I makes o sese o subrac somehig i spherical coordiaes from somehig i Caresia coordiaes The basis does o have o be cosa, oly shared Every quaerio commues wih iself, so he cross produc is zero There are seve uique pairs of basis vecors i a square: da î Î, d 3 da î d ˆ I î, da 9 d 3 The sigs were chose o be cosise wih Hamilo s quaerio algebra The four square basis vecors i muˆ defie he meric If he basis vecors are o cosa, he he meric is dyamic Defie a 3-rope o be he hree oher erms, which have he form i I Noice ha he 3-rope sars i oe ime-space locaio ad will have a ozero legh if i eds up a a differe locaio ad ime Wih quaerio producs, he 3-rope is a aural compaio o a meric for iformaio abou disace I special relaiviy, if he ieria of he observer bu o he sysem is chaged, he meric is ivaria The 3-rope is covaria, because i is kow how i chages Give he uiliy of dualiy, a complemeary hypohesis o he ivaria meric of special relaiviy would propose he if he ieria of he sysem bu o he observer is chaged, here exiss a choice of basis vecors such ha he 3-rope is ivaria bu he meric chages i a kow way This could be wrie i algebraically usig he followig rule: ˆ i 1 ˆ, î I ˆ I 1 If he magiude of he ime ad 3-space basis vecors are iversely relaed, he magiude of he produc of he ime basis vecor wih each 3-space basis vecor will be cosa eve if he basis vecors hemselves are dyamic This hypohesis assers he exisece of such a basis, bu ha paricular basis does o have o be used Hamilo had he freedom o use he rule foud i he above equaio, bu made he more obvious choice of i ˆ -I ˆ The exisece of a basis where he 3-rope is cosa despie a chage i he ieria of he sysem will have o be reaed as provisioal i his paper I he secod paper of his series, a meric wih his propery will be foud ad discussed ˆ I Physically releva differeial equaios Is here a raioal way o cosruc physically releva quaerio equaios? The mehod used here will be o mimic he esor equaios of elecromageism The elecromageic field sregh esor is formed by a differeial operaor acig o a poeial The Maxwell equaios are formed by acig o he field wih aoher differeial operaor The Lorez 4-force is creaed by he produc of a elecric charge, he elecromageic field sregh esor, ad a 4- velociy This paer will be repeaed sarig from a asymmeric field o creae he same field ad force equaios usig quaerio differeials ad poeials The challege i his exercise is i he ierpreaio, o see how every erm coecs o esablished laws of physics s a firs sep o cosrucig differeial equaios, examie how he differeial operaor (d/d, Del) acs o a poeial fucio (phi, ):,,, x For he sake of clariy, he oaio iroduced for Riema quaerios has bee suppress, so he reader is ecouraged o recogize ha here are also a parallel se of erms for chages i he basis vecors The previous equaio is a complee assessme of he chage i he 4-dimesioal poeial/basis, ivolvig wo ime derivaives, he divergece, 4

5 he gradie ad he curl all i oe uified field heory should accou for all coceivable forms of chage i a 4- dimesioal poeial/basis, as is he case here Quaerio operaors ad poeials have o bee used o express he Maxwell equaios The reaso ca be foud i he previous equaio, where he sig of he divergece of is opposie of he curl of I he Maxwell equaios, he divergece ad he curl ivolvig he elecric ad mageic field are all posiive May ohers, eve i Maxwell s ime, have used complex-valued quaerios for he ask because he exra imagiary umber ca be used o ge he sigs correc However, complex-valued quaerios are o a algebraic field The orm, ˆ xˆ yˆ zˆ, for a o-zero quaerio could equal zero if he values of, x, y, ad z were complex This paper ivolves he cosrai of workig exclusively wih 4-dimesioal algebraic fields Therefore, o maer how saluary he work wih complex-valued quaerios, i is o releva o his paper The reaso o hope for uificaio usig quaerios ca be foud i a aalysis of symmery provided by lber Eisei: The physical world is represeed as a four-dimesioal coiuum If i his I adop a Riemaia meric, ad look for he simples laws which such a meric ca saisfy, I arrive a he relaivisic graviaio heory of empy space If I adop i his space a vecor field, or he aisymmeric esor field derived from i, ad if I look for he simples laws which such a field ca saisfy, I arrive a he Maxwell equaios for free space [eisei1934] The four-dimesioal coiuum could be viewed as a echical cosrai ivolvig opology Foruaely, quaerios do have a opological srucure sice hey have a orm Naure is asymmeric, coaiig boh a symmeric meric for graviy ad a aisymmeric esor for elecromageism Wih his i mid, rewrie ou he real 4x4 marix represeaio of a quaerio: q, x, y, z x y z x z y y z x z y x The scalar compoe ( i represeaio above) ca be represeed by a symmeric 4x4 marix, ivaria uder rasposiio ad cojugaio (hese are he same operaios for quaerios) The 3-vecor compoe (x, y ad z i he represeaio above) is off-diagoal ad ca be represeed by a aisymmeric 4x4 marix, because akig he raspose will flip he sigs of he 3-vecor Quaerios are asymmeric i heir marix represeaio, a propery which is criical o usig hem for uifyig graviy ad elecromageism Recreaig he Maxwell equaios Maxwell speculaed ha his se of equaios migh be expressed wih quaerios someday (J C Maxwell, Treaise o Elecriciy ad Mageism, Dover repri, hird ediio, 1954) The divergece, gradie, ad curl were iiially developed by Hamilo durig his ivesigaio of quaerios For he sake of logical cosisecy, ay sysem of differeial equaios, such as he Maxwell equaios, ha depeds o hese ools mus have a quaerio represeaio The Maxwell equaios are gauge ivaria How ca his propery be buil io a quaerio expressio? Cosider a commo gauge such as he Lorez gauge, dphi/d div I quaerio parlace, his is a quaerio-scalar formed from a differeial quaerio acig o a poeial To be ivaria uder a arbirary gauge rasformaio, he quaerio-scalar mus be se o zero This ca be doe wih he vecor operaor, (q-q*)/ Search for a combiaio of quaerio operaors ad poeials ha geerae he Maxwell equaios: Vecor Vecor B,, E B 5

6 , 4 J This is mpere s law ad he o moopoles vecor ideiy (assumig a simply-coeced opology) y choice of gauge will o make a coribuio due o he vecor operaor If he vecor operaor was o used, he he gradie of he symmeric-marix force field would be liked o he elecromageic source equaio, mpere s law Geerae he oher wo Maxwell equaios: Vecor Vecor!, E, B 4, E " This is Gauss ad Faraday s law gai, if he vecor operaor had o bee used, he ime derivaive of he symmericmarix force field would be associaed wih he elecromageic source equaio, Gauss law To specify he Maxwell equaios compleely, wo quaerio equaios are required, jus like he 4-vecor approach lhough successful, he quaerio expressio is uappealig for reasos of simpliciy, cosisecy ad compleeess complicaed collecio of sums or differeces of differeial operaors acig o poeials - alog wih heir cojugaes - is required There is o obvious reaso his combiaio of erms should be ceral o he aure of ligh Oe moivaio for he search for a uified poeial field ivolves simplifyig he above expressios Whe a quaerio differeial acs o a fucio, he divergece always has a sig opposie he curl The opposie siuaio applies o he Maxwell equaios Of course he sigs of he Maxwell equaios cao be chaged However, i may be worh he effor o explore equaios wih sig coveios cosise wih he quaerio algebra, where he operaors for divergece ad curl were coceived Iformaio abou he chage i he poeial is explicily discarded by he vecor operaor Jusificaio comes from he plea for gauge symmery, esseial for he Maxwell equaios The Maxwell equaios apply o massless paricles Gauge symmery is broke for massive fields More iformaio abou he poeial migh be used i uificaio of elecromageism wih graviy gauge is also marix symmeric, so i could provide a complee picure cocerig symmery Oe uified force field from oe poeial field For massless paricles, he Maxwell equaios are sufficie o explai classical ad quaum elecrodyamic pheomea i a gauge-ivaria way To uify elecromageism wih graviy, he gauge symmery mus be broke, opeig he door o massive paricles Because of he cosrais imposed by quaerio algebra, here is lile freedom o choose he gauge wih a simple quaerio expressio I he sadard approach o he elecromageic field, a differeial 4-vecor acs o a 4-vecor poeial i such a way as o creae a aisymmeric secod-rak esor The uified field hypohesis proposed ivolves a quaerio differeial operaor acig o a quaerio poeial:, x This is a aural suggesio wih his algebra The aisymmeric-marix compoe of he uified field has he same elemes as he sadard elecromageic field esor Defie he elecric field E as he eve erms, he oes ha will o chage sigs if he order of he differeial operaor ad he poeial are reversed The mageic field B is he curl of, he odd erm The jusificaio for proposig he uified force field hypohesis ress o he presece of he elecric ad mageic fields 6

7 I some ways, he above equaio looks jus like he old idea of combiig a scalar gauge field wih he elecromageic field sregh esor, as Gupa did i 195 i order o quaize he Maxwell equaios He cocluded ha alhough useful because i is wrie i maifesly relaivisic form, o ew resuls beyod he Maxwell equaio are obaied Examie jus he gauge coribuio o he Lagragia for his uified field: L # 1 %$, 1 3 x x, 1 y 3 y, 1 3 z z Take he derivaive of he Lagragia wih respec o he gauge variables: L & ' & x' By Noeher s heorem, his coserved curre idicaes a symmery of he Lagragia This is why he proposal ivolves ew physics The gauge is a dyamic variable cosraied by he Lagragia quaerio poeial fucio has four degrees of freedom represeed by he scalar fucio phi ad he 3-vecor fucio cig o his wih oe[or more] differeial operaors does o chage he degrees of freedom Isead, he age spaces of he poeial will offer more suble views o he rules for how poeials chage The hree classical force fields, g, E, ad B, deped o he same quaerio poeial, so here are oly four degrees of freedom Wih seve compoes o he hree classical force fields, here mus be hree cosrais bewee he fields Two cosrais are already familiar The elecric ad mageic field form a vecor ideiy via Faraday s law ssumig spaceime is simply coeced, he o moopoles equaio is aoher ideiy ew cosrai arises because boh he force fields for graviy ad elecriciy are eve I will be show subsequely how he eve force fields ca parially cosrucively or desrucively ierfere wih each oher Uified Field equaios I he sadard approach o geeraig he Maxwell equaios, a differeial operaor acs o he elecromageic field sregh esor uified field hypohesis for a isolaed source is proposed which ivolves a differeial quaerio operaor acig o he uified field: 4, J, ",,, This secod order se of four parial differeial equaios has four ukows so his is a complee se of field equaios Rewrie he equaios above i erms of he classical force fields: 4, g J E g x B B E xe B, The uified field equaios coai hree of he four Maxwell equaios explicily:gauss law, he o mageic moopoles law, ad mpere s law Faraday s law is a vecor ideiy, so i is sill rue implicily Therefore, a subse of he uified field equaios coais a quaerio represeaio of he Maxwell equaios The jusificaio for ivesigaig he uified field equaio hypohesis is due o he presece of he Maxwell equaios 7

8 There is a very simple relaioship bewee Faraday s law ad he equaio above ll ha eeds o be doe is o subrac wice he ime derivaive of he mageic field from boh sides Wha does his do o he 4-vecor curre desiy J? Now here is a curre ha rasforms like a pseudo-curre desiy, makig he proposal more complee The volume iegral of his pseudo-curre desiy is he oal mageic flux: The uified field equaio posulaes a pseudo 3-vecor curre composed of he differece bewee he ime derivaive of he mageic field ad he curl of he elecric field The haroov-bohm effec depeds o he oal mageic flux o creae chages see i he eergy specrum[aharoov1959] The volume iegral of he ime derivaive of he mageic field is a measure of he oal mageic flux The pseudo-curre desiy is quie uusual, rasformig differely uder space iversio ha he elecric curre desiy Oe migh imagie ha a Lorez rasformaio would shif his pseudo-curre desiy io a pseudo-charge desiy This does o happe however, because he vecor ideiy ivolvig he divergece of a curl sill applies The haroov-bohm pheomeo, firs viewed as a purely quaum effec, may have a classical aalogue i he uified field equaios k B dv e ) ( B c The cosas used here were chose o sugges a coecio o he haroov-bohm (Y haroov ad D Bohm, Sigificace of elecromageic poeials i he quaum heory, Phys Rev, 115: , 1959) The haroov- Bohm effec depeds o he oal mageic flux o creae chages see i he eergy specrum pseudo-curre desiy is uusual, rasformig differely uder space iversio ha he elecric curre desiy Oe migh imagie ha a Lorez rasformaio would shif his pseudo-curre desiy io a pseudo-charge desiy This does o happe however, because he vecor ideiy ivolvig he divergece of a curl sill applies The haroov-bohm pheomeo, firs viewed as a purely quaum effec, may have a classical aalogue i he uified field equaios The field equaios ivolvig he graviaioal force field are dyamic ad deped o four dimesios This makes hem likely o be cosise wih special relaiviy Sice hey are geeraed alogside he Maxwell equaios, oe ca reasoably expec he differeial equaios will share may properies, wih he oes ivolvig he symmeric-marix graviaioal force field beig more symmeric ha hose of he elecromageic couerpar The uified source ca be defied i erms of more familiar charge ad curre desiies by separaely seig he graviy or elecromageic field equal o zero I hese cases, he source is due oly o elecriciy or mass respecively This leads o coecios bewee he uified source, mass, ad charge: J J m iff J J e E B J B iff g I would be icorrec - bu almos rue - o say ha he uified charge ad curre are simply he sum of he hree: mass, elecric charge, ad he haraov-bohm pseudo-curre (or oal mageic flux over he volume) These erms cosrucively ierfere wih each oher, so hey may o be viewed as beig liearly idepede Up o four liearly idepede uified field equaios ca be formulaed differe se could be creaed by usig he differeial operaor wihou akig is cojugae: 4 J g g E E B, x B B xe, This is a ellipic equaio Sice he goal of his work is a complee sysem of field equaios, his may ur ou o be a advaage ellipic equaio combied wih a hyperbolic oe migh more fully describe graviaioal ad 8

9 elecromageic waves from sources Ulike he firs se of field equaios, he cross erms desrucively ierfere wih each oher The ellipic field equaio agai coais hree of four Maxwell equaios explicily:gauss law, he o mageic moopoles vecor ideiy ad Faraday s law This ime, mpere s law looks differe To be cosise wih mpere s law, agai a pseudo-curre mus be icluded This may be he differeial form of a classical haroov-bohm effec The oly erm ha does o chage bewee he wo field equaios is he oe ivolvig he dyamic graviaioal force This migh be a clue for why his force is oly aracive Soluios o he uified field equaios ll he soluios ha have bee worked ou for he Maxwell equaios will work wih he uified field equaios For example, if he poeial is saic, he scalar equaio for hyperbolic field equaio is he Poisso equaio The uified equaios are more iformaive, sice ay poeial which is a soluio o he scalar Poisso equaio will also characerize he correspodig curre The field equaios of geeral relaiviy ad he Maxwell equaios boh have vacuum soluios,such as plae wave soluios The uified field equaios do o have such a soluio, oher ha a cosa Give hisorical radiio, his may seem like a deadly flaw However, i may be somehig ha is required for a fial ad complee heory I a uified field heory, he graviaioal par may be zero while he elecrical par is o, ad visa versa No-zero soluios are worh explorig iverse square poeial plays a impora role i boh graviy ad elecromageism Examie he scalar field ivolvig he iverse ierval squared: 1 x y z, 4 3 x y z x y z 3, This poeial solves he Maxwell equaios i he Lorez gauge: 1 x y z, The o-zero par may have everyhig o do wih graviy plae wave soluio does exis,bu o for a pure vacuumisead,a plae wave soluio exiss wih he cosrai ha he e curre is zero for he ellipical field equaios The field equaios of geeral relaiviy ad he Maxwell equaios boh have vacuum soluios vacuum soluio for he uified field equaio is appare for he ellipical field equaios: e K * R +-,, e K * R +-, The uified field equaio will evaluae o zero if Scalar c, K The dispersio relaio is a ivered disace, so i will deped o he meric The same poeial ca also solve he hyperbolic field equaios uder differe cosrais ad resulig dispersio equaio (o show) There were wo reasos for o icludig he cusomary imagiary umber i i he expoeial of he poeial Firs, i was o ecessary Secod, i would have creaed a complex-valued quaerio, ad herefore is ouside he domai of his paper The impora hig o realize is ha vacuum soluios o he uified field equaios exis whose dispersio equaios deped o he meric This is a idicaio ha uifyig graviy ad elecromageism is a appropriae goal 9

10 Coservaio Laws Coservaio of elecric charge is implici i he Maxwell equaios Is here also a coserved quaiy for he graviaioal field? Examie how he differeial operaor acs o he uified field equaio: g E g, E B B Noice ha he graviaioal force field oly appears i he quaerio scalar The elecromageic fields oly appear i he 3-vecor This geeraes wo ypes of cosrais o he sources No chage i he elecric source applies o he quaerio scalar No chage i he graviaioal source applies o he 3-vecor Scalar Scalar Vecor J e J B J m e J m J B J e m J m The firs equaio is kow as he coiuiy equaio, ad is he reaso ha elecric charge is coserved For a differe ierial observer, his will appear as a coservaio of elecric curre desiy There is o source erm for he haraov-bohm curre, ad subsequely o coservaio law The 3-vecor equaio is a cosrai o he mass curre desiy, ad is he reaso mass curre desiy is coserved For a differe ierial observer, he mass desiy is coserved Trasformaios of he uified force field The rasformaio properies of he uified field promise o be more iricae ha eiher graviy or elecromageism separaely Wha migh be expeced o happe uder a Lorez rasformaio? Graviy ivolves mass ha is Lorez ivaria, so he field ha geeraes i should be Lorez ivaria The elecromageic field is Lorez covaria However, a rasformaio cao do boh perfecly The reaso is ha a Lorez rasformaio mixes a quaerio scalar wih a 3-vecor If a rasformaio lef he quaerio scalar ivaria ad he 3-vecor covaria, he wo would effecively o mix The effec of uificaio mus be suble, sice he rasformaio properies are well kow experimeally Cosider a boos alog he x-axis The graviaioal force field is Lorez ivaria ll he erms required o make he elecromageic field covaria uder a Lorez rasformaio are prese, bu covariace of he elecromageic fields requires he followig residual erms: / / Residual, 1 1 x z y x, y z x 1 1 x, his ime, he correc ierpreaio of he residual erm is uclear Mos imporaly, i was show earlier ha charge is coserved These erms could be a velociy-depede phase facor If so, i migh provide a es for he heory The mechaics of he Lorez rasformaio iself migh require careful re-examiaio whe so sricly cofied o quaerio algebra For a boos alog he x-axis, if oly he differeial rasformaio is i he opposie direcio, he he elecromageic field is Lorez covaria wih he residual erm residig wih he graviaioal field The meaig of his observaio is eve less clear Oly relaively recely has DeLeo bee able o represe he Lorez group usig real quaerios (S De Leo, Quaerios ad special relaiviy, J Mah Phys, 37(6): , 1996) The delay appears odd sice he ierval of special relaiviy is he scalar of he square of he differece bewee wo eves I he real 4x4 marix represeaio, he ierval is a quarer of he race of he square Therefore, ay marix 1

11 wih a race of oe ha does o disor he legh of he scalar ad 3-vecor ca muliply a quaerio wihou effecig he ierval Oe such class is 3-dimesioal, spaial roaios operaor ha adds ohig o he race bu disors he leghs of he scalar ad 3-vecor wih he cosrai ha he differece i leghs is cosa will also suffice These are booss i a ierial referece frame Booss plus roaios form he Lorez group Three ypes of gauge rasformaios will be ivesigaed: a scalar, a 3-vecor, ad a quaerio gauge field Cosider a arbirary scalar field rasformaio of he poeial: 3 / 54 The elecromageic fields are ivaria uder his rasformaio addiioal cosrai o he gauge field is required o leave he graviaioal force field ivaria, amely ha he scalar gauge field solves a homogeeous ellipical equaio From he perspecive of his proposal, he freedom o choose a scalar gauge field for he Maxwell equaios is due o he omissio of he graviaioal force field Trasform he poeial wih a arbirary 3-vecor field: 3 / 6 This ime he graviaioal force field is ivaria uder a 3-vecor gauge field rasformaio ddiioal cosrais ca be placed o he 3-vecor gauge field o preserve a chose elecromageic ivaria For example, if he differece bewee he wo elecromageic fields is o remai ivaria, he he 3-vecor gauge field mus be he soluio o a ellipical equaio Oher classes of ivarias could be examied The scalar ad 3-vecor gauge fields could be combied o form a quaerio gauge field This gauge rasformaio would have he same cosrais as hose above o leave he fields ivaria Is here ay such gauge field? The quaerio gauge field ca be represeed he followig way: 3 / 6 If a force field is creaed by hiig his gauge rasformaio wih a differeial operaor, he he gauge field becomes a uified field equaio Sice vacuum soluios have bee foud for hose equaios, a quaerio gauge rasformaio ca leave he field ivaria Fuure direcios The fields of graviy ad elecromageism were uified i a way cosise wih Eisei s visio, o his echique The guidig priciples were simple bu uusual:geerae expressios familiar from elecromageism usig quaerios, srivig o ierpre ay exra erms as beig due o graviy The firs hypohesis abou he uified field ivolved oly a quaerio differeial operaor acig o a poeial, o exra erms added by had I coaied he ypical poeial represeaio of he elecromageic field, alog wih a symmeric-marix force field for graviy The secod hypohesis cocered a uified field equaio formed by acig o he uified field wih oe more differeial operaor ll he Maxwell equaios are icluded explicily or implicily ddiioal erms suggesed he iclusio of a classical represeaio of he haraov-bohm effec Four liearly idepede uified field equaios exis, bu oly he hyperbolic ad ellipic cases were discussed large family of vacuum soluios exiss, ad will require fuure aalysis o appreciae To work wihi he guidelies of his paper, oe should avoid soluios represeed by complex-valued quaerios Why did his approach work? The hypohesis ha iiiaed his lie of research was ha all eves i spaceime could be represeed by quaerios, o maer how he eves were geeraed This is a broad hypohesis, aempig o reach all areas covered by physics Based o he equaios preseed i his paper, a logical srucure ca be cosruced, sarig from eves (see Fig ) se of eves forms a paer ha ca be described by a poeial The chage i a poeial creaes a field The chage i field creaes a field equaio The erms ha do o chage uder differeiaio of a field equaio form coservaio laws 11

12 ckowledgemes Thaks for may producive discussios ProfGuido Sadri 1