Generalized electromagnetic energy-momentum tensor and scalar curvature of space at the location of charged particle

Transcription

1 Generalized eleromagnei energy-momenum ensor and salar urvaure of spae a he loaion of harged parile A.L. Kholmeskii 1, O.V. Missevih and T. Yarman 3 1 Belarus Sae Universiy, Nezavisimosi Avenue, 0030 Minsk, Belarus Insiue for Nulear roblems, Belarus Sae Universiy, 11 Bobruiskaya Sree, 0030 Minsk, Belarus 3 Okan Universiy, Akfira, Isanbul, Turkey and Savronik, Eskisehir, Turkey khol13@yahoo.om Absra We onsider he Einsein equaion, where he ommon eleromagnei energy momenum ensor is replaed by is generalized equivalen as suggesed in our earlier paper (A.L. Kholmeskii e al. hys. Sr. 83, (011)). Now we show ha wih his new eleromagnei energy-momenum ensor, he salar urvaure a he loaion of harges is signifianly alered in omparison wih he ommon resul, and i even may hange is sign. Some impliaions of he obained resuls are disussed. Keywords: eleromagnei energy momenum ensor, Einsein equaion, salar urvaure of spae ACS numbers: 0.5.dg, 1.0. q 1. Inroduion In he presen onribuion we address o he Einsein equaion and refer o our reen finding [1] ha he ommon eleromagnei energy-momenum (EMEM) ensor is inomplee and is relevan only for he desripion of eleromagnei (EM) fields in he regions free of harged pariles. For omplee sysems harged pariles along wih heir EM fields, an addiional erm should be inrodued in he sruure of EMEM ensor, whih essenially influenes is general properies. In pariular, he rae of he new (generalized) EMEM ensor is no longer vanishing and hus we ineviably mus onlude ha he EM field should influene he salar urvaure of spae unlike he ommon resul. A loser look a his problem shows ha in he spaial regions free of harged pariles he salar urvaure of spae is deermined by he maer ensor as before. However, a he loaion of harges he onribuion of he generalized EMEM ensor subsanially influenes he salar urvaure and may even hange is sign in omparison wih he ommon value (seion ). The resul obained is disussed in seion 3.. Einsein equaion wih he modified eleromagnei energy-momenum ensor We sar wih he Einsein equaion 1 Rν g ν R = T ν (, ν =0 3) (1) where R ν is he Rii ensor, g is he meri ensor, R = g ν Rν is he salar urvaure of spae, and T ν is he energy-momenum ensor, whih represens he sum of he maer par and EM field par. For a sysem of N poin-like harged pariles he maer ensor has he form

2 T ν maer = dx dx N ν i i mi i= 1 d dτ, () (where m i being he mass densiy of parile i, and τ is he proper ime), while he EMEM ensor is ommonly wrien as T ν 1 γ ν 1 ν γα = F F γ + g Fγα F EM π, (3) ν ν ν where F = A A is he ensor of EM field, and A is he four-poenial. The rae of he EMEM ensor (3) is equal o zero and hene we arrive a he ommon resul ha EM field does no affe he salar urvaure [], i.e. ( ) ( ) ) R ( ) = R = Tmaer + T EM = T maer. () However, in our reen paper [1] we have shown ha he expression for he EMEM ensor (3) is inomplee, and i is appliable only o he spaial regions free of harged pariles. In he general ase, when he sysem in quesion inludes boh pariles and heir EM fields, eq. (3) has o be generalized o he form ν ( A j ) pr ν 1 γ 1 ν γα T EM F F ν 1 = γ + g Fγα F, (5) π where he subsrip pr (proper) signifies ha he four-poenial omponens A and foururren omponens j are aken for he same soure parile ν (pariles). In ref. [1], he inompleeness of eq. (3) has been demonsraed in wo differen ways: - via he analysis of oyning heorem for a bound (veloiy-dependen) EM field; - via he gauge modifiaion of he anonial EMEM ensor o he symmeri form, assuming he presene of harges in he spaial region onsidered. Conerning hese ways, we refer he readers o he original work [1] for more deails. Now we only menion ha in he analysis of oyning heorem we found is equivalen presenaion for he ase of purely bound EM field, whih, however, anno be reprodued wih he ommon EMEM ensor (3) and wih is any gauge modifiaion. In onras, he ensor (5) via is appropriae gauge ransformaion (named in ref. [1] as gauge renormalizaion ) yields boh forms of he oninuiy equaions menioned above. On he seond way (he gauge ransformaion of he anonial EMEM ensor o he symmeri form) he inompleeness of EMEM ensor (3) follows from he ommon appliaion of he homogeneous Maxwell equaions in suh a ransformaion (e.g. []), whih, however are relevan only for he spaial regions free of harges. Using insead he general non-homogeneous Maxwell equaions, we finally arrived o he EMEM ensor (5). One an add ha he equaion (5) an be approahed alernaively saring from he definiion of he energy-momenum ensor given by Hilber [1]. Furher, we noie ha he EMEM ensor (5) onains he four-poenial in is las (addiional) erm and hus, i seems o be gauge-dependen. In his respe we have o menion ha he four-divergene of ime-like omponens of his erm is equal o zero [1], i.e. 0 A j = ( ) 0. Therefore, he ensor (5) yields he sandard oyning heorem, like he ommon EMEM ensor (3). Furhermore, as shown in ref. [1], via he operaion of gauge renormalizaion, he infinie (for poin-like harges) erm ( A j ) pr 1 ν is merged o he oal observed mass of harged parile M, so ha he moional equaion derived wih he ensor (5) does no inlude he fourpoenial and finally is gauge-independen. We poin ou ha he possibiliy o inlude he erm 1 ν ( A j ) pr o he oal mass of harged parile in he orre mahemaial way (i.e. gauge

3 ransformaion of EMEM ensor (5)) is also imporan for he analysis presened below. ν So, as we have menioned above, he oal energy-momenum ensor T of any marosopi sysem of harged pariles has o be undersood as a sum of maer par and field par, i.e. ν ν ν T = T maer + T EM, (6) ν where for a sysem of poin-like pariles T maer is defined by eq. (). Now we noie an imporan propery of EMEM ensor (5): in onras o he onvenional EMEM ensor (3), is rae is no equal o zero due o he onribuion ( A j ) pr 1 ν. Hene using eq. (5) we ge he possibiliy o desribe he EM mass onribuion ino he oal mass of harged parile whih is no viable using he sandard ensor (3). The laer saemen issues from he known fa ha any mass ensor mus have he non-vanishing rae [3]. The EM mass ensor an be inrodued by analogy wih he maer ensor () as ν ν dx dx ( TEM ) = mem (7) mass d dτ (where m EM is EM mass densiy), o whih is added he oinaré sresses ensor ν ν dx dx T = m, (8) d dτ where m is he negaive mass densiy assoiaed wih he energy of "oinaré sresses" neessary for he sabiliy of he lassial eleron []. Thus he mehanial (maer) ensor () is modified o he form ν ν dx dx Tmaer = ( m + m ), (9) d dτ and he oal (observable) mass densiy of harged parile beomes he sum of hree omponens: m = m + m + mem, (10) The deailed analysis of eah of he omponens of eq. (10) for he lassial eleron has been arried ou in ref. []. In pariular, i has been found in [] ha he raio M : M : M : M = : 3: :1 (11) EM for any power funion, desribing he harge disribuion inside he lassial eleron (here he apial leers sands for he orresponding mass omponen). Of ourse, in furher appliaion of he lassial esimaion (11) one has o ake ino aoun is ondiional haraer. I is only imporan ha all he mass omponens of his equaion have he same order of magniude, where M, M, M EM are posiive, while M is negaive. Thus he non-vanishing rae of he generalized EMEM ensor (5) allows us o inrodue he onep of EM mass of lassial harged parile in he non-onradiory way and, wha is imporan for he purpose of he presen paper, he fa of non-vanished rae of his ensor should be aouned in he soluion of Einsein equaion (1). In pariular, wih he EMEM ensor (5), he expression for salar urvaure of spae beomes 1 R = ( Tmaer ) ( A j ) (1) pr insead of eq. (). In order o analyze his equaion, we expliily alulae he erm A j = ρϕ j A, ( ) ( ) pr ( ) pr pr where ρ is he harge densiy and ϕ is he eleri poenial. For a sysem of N harged parile his equaion reads: ( A j ) pr = N k = 1 ρ ϕ N ( k ) ( k ) ( k ) ( k ) k = 1 j A. (13) Suh a speial form of eq. (13), where all quaniies are evaluaed for eah fixed harge, signifies 3

4 ha we do no have a righ o go from disree disribuion of harges haraerized by he harge densiies ρ (k=1 N) o any averaged oninuous harge disribuion. (Here we reall ha his ( k ) operaion is ommonly used in general relaiviy heory wih he inroduion of oninuous averaged disribuion of maer). Therefore, we onlude ha in any spaial poin free of harged pariles, he rae of EMEM ensor is equal o zero due o he equaliies ρ =0, j =0 for any k. 1 ν Hene he addiion of he erm ( A j ) pr o he ommon EMEM ensor does no affe he salar urvaure in any free spaial poin, where he ommon Einsein equaion (1) remains in fore. Thus, in omparison wih ommon general heory of relaiviy, he salar urvaure is modified only wihin a region defined by parile s harge radius due o he non-vanishing rae of he omponen (13) of EMEM ensor, and his resul seems insignifian in osmology. Neverheless, i is worh o oninue he analysis of eq. (13), implying is possible usefulness o he lassial limis of quanum graviy heories. In suh an analysis we will deal wih a single harged parile (N=1), e.g. wih he lassial eleron. Furher on i is onvenien o express he quaniies in eq. (13) referred o he moving eleron via he proper harge densiy ρ 0 and proper eleri poenial ϕ 0 fixed in is res frame, assuming a he momen any gauge, where he four-poenial is vanishing a he infiniy. Then we ge 0v ρ = γρ 0, ϕ = γϕ0, j = γ j 0 = γρ0v, A = γϕ, (1) where v is he veloiy of he eleron, and γ is is Lorenz faor. Hene, subsiuing eqs. (1) ino eq. (13), we obain 1 v ( A j ) = γ ρ0ϕ0 ρ0ϕ0 = ρ 0ϕ0. (15) pr We menion ha he produ ρ 0 ϕ 0 is always posiive regardless of he sign of harge (eleron, posiron), and for he reasonable models of he lassial elerons (see e.g. []), i represens he finie quaniy, whose pariular value depends on he adoped disribuion of harge inside he lassial eleron. I is also ineresing o noie ha he erm (15) responsible for he onribuion of generalized EMEM ensor (5) ino he salar urvaure, does no depend on he Lorenz faor γ. Nex we alulae he rae of he maer ensor (9): dx dx ( ) ( ) ( ) ( ) ( m + m ) Tmaer = m + m γ m m γ m m v = + + =, (16) d dτ γ where we have aken ino aoun ha for a moving parile dτ=d/γ. Subsiuing eqs. (15), (16) ino eq. (1), we derive he expression for spaial urvaure a he loaion of lassial eleron: ( m + m ) R = ρ 0ϕ0. (17) γ Furher on we involve he equaion obained in ref. [] for he densiy of oinaré sresses mass: ρ0ϕ0 m =. (18) In a rough approximaion (whih is anyway allowed in he lassial approah o parile physis), where we assume he homogeneous disribuion of m, m and m EM inside he eleron, he relaionship (11) an be adoped for he orresponding mass densiies, oo, i.e. m : m : m : m = : 3: :1. (19) EM ( k ) ( k )

5 Then eqs. (18) and (19) allow us o express he produ ρ 0 ϕ 0 via he densiy of oal observable mass of he lassial eleron as ρ 0 ϕ 0 = m, (0) and also o ge he relaionships m = m, m = 3m. (1) Hene, ombining (17), (18), (0) and (1), we derive he expression for spaial urvaure a he loaion of lassial eleron in gauge independen form: 1 R = m 1. () γ We menion ha he numerial oeffiien of eq. () is, of ourse, model-dependen, and an be hanged via a proper variaion of he raios of he mass densiy omponens (19). However, his irumsane is no so imporan, beause in any ase he lassial analysis is jus he approximaion. Anyway, we poin ou ha he Lorenz faor γ always exeeds uniy, and hus he salar urvaure a he loaion of he lassial eleron is always posiive. Hene i dereases he visible size of he eleron in omparison wih he ase R=0. We noie ha he salar urvaure a he loaion of eleron, being alulaed wih he ommon expression for he symmeri EMEM ensor (3) (whose rae is equal o zero), is ompleely deermined by he maer ensor onribuion and equal o m R =, γ whih is always negaive and does inrease he visible size of eleron. In he presen onribuion we skip he lassial analysis of hadrons, whih ours even more ondiional ha ha for he lassial eleron. Anyway, he onribuion of he las erm in rhs of eq. (5) o he salar urvaure should be aken ino aoun for he hadrons, oo. In pariular, we anno exlude ha he sign of he salar urvaure migh be posiive, oo, jus like for he eleron. 3. Conlusion Having modified he EMEM ensor (3) o he form (5), whih overs he general ase of he presene of boh pariles and fields, we subsiued his new (generalized) EMEM ensor (5) ino he Einsein equaion (1). In he spaial regions free of harged pariles, he rae of generalized EMEM ensor is equal o zero, jus like wih he ommon EMEM ensor. Hene we onluded ha EM field does no affe he salar urvaure of spae ouside he harged pariles. Moreover, sine ouside harged pariles he erm ( A j ) pr 1 ν is equal o zero, is addiion o he ommon EMEM ensor (3) does no affe he soluions of Einsein equaion in suh spaial regions. Thus he generalized EMEM ensor (5) we inrodued in ref. [1], does no hange anyhing in general relaiviy heory impliaions. 1 ν However, a he loaion of an eleri harge he addiional erm ( A j ) pr beomes signifian, and alers he salar urvaure inside he harged pariles. In pariular, for he lassial eleron (or posiron) he salar urvaure no only hanges is value, bu he sign, oo. We hope ha he resuls obained in his paper ould be useful for a es of lassial limis of quanum heories of graviaion and, perhaps, for invesigaion of earlies sages of he Universe expansion. 5

6 Referenes 6 [1] A.L. Kholmeskii, O.V. Missevih, T. Yarman, hys. Sr. 83, (011) [] L.D. Landau, E.M. Lifshiz, The Classial Theory of Fields nd edn (ergamon ress, New York, 196) [3] L.D. Landau, L.. iaevskii, E.M. Lifshiz, Elerodynamis of Coninuous Media nd edn (ergamon ress, New York, 198) [] A.L. Kholmeskii, Found. hys. Le. 19, 696 (006)