Could be explained the origin of dark matter and dark energy through the. introduction of a virtual proper time? Abstract

Transcription

1 Could be explaned the orgn o dark matter and dark energy through the ntroduton o a vrtual proper tme? Nkola V Volkov Department o Mathemats, StPetersburg State Eletrotehnal Unversty ProPopov str, StPetersburg, Russa e-mal: nkvolkov4@gmalom Abstrat I we ntrodue a vrtual proper tme n the spae-tme metr, then any physal eld s omplemented by ts own vrtual eld Ths vrtual eld has an energy-momentum and a massve n the presene o eld soures In ths artle we onsder the above phenomenon or the eletromagnet (Maxwell`s eld whose own vrtual eld s salar-eletr Ths vrtual salar-eletr eld s massve n the presene o eletr harges and urrents In the ase o gravtatonal eld ts massve vrtual eld has an energy-momentum and manests tsel n gravtatonal nteratons Suh massve vrtual eld ould explan the orgn o dark matter and dark energy Introduton In Mnkowsk spae wth the metr = dt dx the proper tme τ s determned by the equalty = dτ For a movng partle the proper tme τ s measured by the lok whh move wth ths partle and at rest relatve to t [] Thereore, n a movng nertal reerene system (urther, rs the proper tme τ s not observable and not measured dretly by the lok o tme t We onsder the system o two expressons gven above or the nvarant nterval s Thereby, we pass rom Mnkowsk spae to the our-dmensonal spae-tme wth the double metr In ths our-dmensonal bmetr spae-tme all varables depend not only on the physal oordnates t, x,y,z, but are also dependent on vrtual proper tme τ

2 Now movng rs nludes the lok o vrtual proper tme τ that s separated rom the lok o physal tme t The vrtual proper tme τ s not observable tme n a movng rs We aept by denton that the lok o tme τ s synhronzed wth the lok o tme t n rs at rest Wth nluson o a vrtual proper tme n the metr o Mnkowsk spae the physal eletromagnet (Maxwell's eld s omplemented by ts vrtual salar-eletr eld In the plane salar-eletromagnet wave the physal eletromagnet wave has a transverse polarzaton and the vrtual salar-eletr wave has a longtudnal polarzaton The vrtual salar-eletr eld s massve n the presene o eletr harges and urrents Then the massless photons o eletromagnet eld wth spn and two proetons ± are beomng the massve photons o salar-eletromagnet eld wth spn and three proetons, ± Ths result s physally equvalent to what we have n the ase o spontaneous breakng o the gauge U( - symmetry or Abelan vetor eld [,3,4] The massve salar-eletromagnet eld may also explan the orgn o the eletron sel-energy We use the ollowng abbrevatons: the rs - the nertal reerene system, the t -lok - the lok o tme t, SEM - salar-eletromagnet, SE - salar-eletr We assume that the ndes,, k take on the values,, 3;,β, γ, ν, λ take on the values,,, 3; take on the values,,, 3,

3 3 I 4-dmensonal bmetr pseudo-euldean spae-tme V 4 4-dmensonal bometr pseudo-euldean spae o -vetors V 4 Spae V 4 V - 4-dmensonal pseudo-euldean lnear spae onsstng o 4-vetors x = ( x, x 4 s = V4 wth the metr ( d ( dx ( dx Spae V V - -dmensonal lnear spae onsstng o -vetors (salars x wth the metr ( d ( dx s = V 3 Spae V V - -dmensonal pseudo-euldean lnear spae onsstng o -vetors = ( = ( wth the metr ( d ( dx ( dx ( dx x x, x x, x, x 4 Spae V 4 А 4 s = + V x = x x V suh V - 4-dmensonal lnear spae onsstng o -vetors (, x x that ( = x and that later we wll all 4 vetors B V4 - pseudo-euldean spae wth the double metr (bmetr whh s the system ( ( dx ( dx ( = =, V4 V4 ( ( dx ( = =, or V4 V ( ( ( ( dx dx + dx ( = =, V4 V ( ( dx ( = = V4 V

4 4 It s the latter orm o the double metr we all the anonal orm o the metr n V4 sne V nludes the spae V 4 Denton For the 4 vetor x ( x, x = the 4 vetor x s alled the base part and s denoted x = x, the vetor (salar base x s alled the own part and s denoted x = x Thus, the 4 vetor x = ( x, x = ( x x own base, own 4-dmensonal bmetr pseudo-euldean spae-tme V 4 The double metr n V 4 Let the 4-vetor (, (, x x x t = = x V4, where 4 V - 4-dmensonal bas spaetme (Mnkowsk spae wth the metr = dx dx = dt dx At eah pont A ( t, x we deal only wth the physal (observable oordnates t and x Let the -vetor (salar x = τ V, where τ s the proper tme That s, the metr n V : = dτ Then the -vetor x = ( x, x = ( t, x, τ V, where V = V V - -dmensonal spae-tme wth the metr 4 ( = = dx dx = dt x dτ V d + Denton 4-dmensonal bmetr pseudo-euldean spae-tme V4 s the lnear spae onsstng o 4 -vetors x, or whh a the double metr n the proetve = dx dx = dt dx, = dx dx = dτ ;

5 b the double metr n the anonal orm s t τ d = dx dx = d dx d, + = dx dx = dτ Inertal reerene system n the spae-tme V 4 In eah movng rs there s the lok o vrtual proper tme τ that s separated rom the lok o physal tme t The rate and dreton o tme onde or the τ - and t -loks n eah rs where the t -lok at rest Corollares Sne the spae-tme V4 s our-dmensonal, then the vrtual proper tme τ s not observable n a movng rs β s = x = ± τ Here and elsewhere the sgn ± orrespon to the orward / bakward dreton o the vrtual proper tme τ γ I τ, then the nterval s s always tmelke, that s, > s δ The event n V4 s dened by the pont A ( t, x, τ Thus, n the movng rs not all omponents o A ( t, x, τ orrespondng to the event are physal (observable 3 Transormaton group n V 4 In the spae-tme V4 somorph to Mnkowsk spae V4 as a ontnuous transormaton group o omponents x o the 4 vetor x ( x, x = we examne the Ponare group or, n a speal ase, the 6-parametr Lorentz group The omponent s Lorentz nvarant x = τ

6 6 Remarks The last statement about transormaton o omponents x remans vald also or any 4 vetor a ( a, a = suh that aa = a Here, the base part a base = a, the own part a own = a β For any 4-vetor a there s the ouple 4 -vetors a ( a, a = suh that ± ± aa ( ± a = and onversely On physal reasons, the sgns o a and a must be ondent Thus, there s a one-to-one orrespondene between a and a ( a, a = 3 Invarant systems or 4 vetors n the spae-tme V 4 Respet to transormatons o the Lorentz group we have the nvarant expressons wrtten below n the orm o systems: the 4 vetor x = ( x, x = ( t, x, τ x x = nv, xx =, e nv x x = x ; the 4 vetor o veloty u (the 4 -veloty dx dt u ( u, u,, v = = =± u =±,, Here, dτ ε ε s = = = t, ε = ( v /, d dx ± dτ ± ε d d v = x dt Then uu =, uu =, e uu = Corollary I a, then a = au s the 4 vetor a ( a, a =, suh that aa = a or = a a a

7 7 3 the momentum 4 vetor (the 4 momentum o a massve pont partle ± = =± = ± m mv (, (,, p mu p p m ε ε, p p = m, pp = m, e p p = m Here, m s the vrtual mass o a movng partle By value m ondes wth the physal mass m o a partle at rest 4 the energy-momentum ± p = m u =± ( E, p, E Here, E = m, ε m v p =, ε E = m, respetvely: the physal energy, 3-momentum, the vrtual sel-energy o a movng partle By value the vrtual sel-energy E = m ondes wth the physal sel-energy E = m o a partle at rest E 4 p m, = = m 4 E, e E 4 p E m + = Remark The -aeleraton w ( w, w du = =, w =, and the -ore dp, ( = =, =, are not 4 -vetors, sne, n general ase, ww and 4 The mass urrent 4 -vetor and the energy-momentum 4 -tensor o a pont partle n the spae-tme V 4 The mass urrent 4 -vetor o a movng pont partle 4 λ λ m = ρmu = m δ ( x x ( ϑ ε( ϑ u ( ϑ dϑ, where

8 8 4 λ λ ( ( 4 ( ( ( 4 λ λ δ x x ϑ = δ x x ϑ δ x x ( ϑ, δ ( x x ( ϑ dx =, ρm = mεδ( x x ( t δ( τ τ ( t s the physal mass densty Let the energy-momentum 4 tensor o a massve partle T = u ν ν m m Trae o = = T ( m Tm, Tm ν T m : T T ρ m =, where T = L = ρ m m m T = u =, or whh the onservaton equaton m m m T m = and T m = The value T = u = u or m m Tm = = ρ% s not the 4 -vetor ε m m m T u and s usually alled the momentum densty o a partle The moment 4 -vetor o a partle 3 P = Tm d x = mu = p s alled the 4 -momentum ν ν ν For the symmetr energy-momentum 4 tensor o a partle T = ( T T m base, own the base part T = T = u s the symmetr 4 tensor, ν β β base m m the own part T ν ( own Tm, Tm = are two equal 4 -vetors By analogy wth the 4 -vetor = there are x or the 4 -vetor T ( m Tm, Tm the nvarant equaltes: T T = T, T T = T Respetvely, or the 4 -tensor β β γ = γ, β T T T T <, ν T m there are the nvarant equaltes: ν ν γ = γ, ν T T T T < The harge urrent 4 -vetor o a massve pont partle The harge urrent 4 -vetor o a partle wth the harge physal e and vrtual mass m e = ρeu or e e = = m m T e, where the vrtual harge densty ρe = ρm m e m m

9 9 I we assume the postve dreton o tme τ and t, then the harge urrent 4 -vetor o a partle = ρu = (, = ( ρ,, ρ %, where ρ s the vrtual harge densty, ρ ρ % = s the physal harge densty, = ρсu = ρ% v Here, ε = or % ρ = ρ The equaton o urrent ontnuty = = That s, ρ = = Then the physal harge Q = ρ% dv = ρ dv, 3 dv = ε d = d x V II The salar-eletromagnet eld n the spae-tme V 4 -potental o the SEM-eld x =, = φ A ф s the ν Let the salar-eletromagnet potental A ( ( A A (,, λ -vetor x V, V = V4 V, but not the 4 -vetor x ν V4 That s, the salar potental ф s a vrtual nvarant, but the nequalty A A A or ф φ A takes plae respet to the transormatons o the Lorentz group (o boosts and spatal rotatons ν In the ase o a massve SEM-eld wth soures the 4-potental ( A x learly depen on the vrtual proper tme τ, е A Thus, the massve SEM-eld wth the ν -potental ( A x s onsdered n 4 ν V, where the 4 -vetor x ( x, x In the ase o a massless SEM-eld wthout soures the -potental = A does not depend on the vrtual proper tme τ, that s, A = Thus, the massless SEM-eld s onsdered atually n 4 V and s gven by the -potental ( A x, x V4 The theory o a massless SEM-eld s nvarant respet to gauge transormatons o the potental A : A A = A, where : = In what ollows we wll use manly the Heavsde-Lorentz system o unts, where

10 h = = e = 4π, -tensor o the SEM-eld strengths F = A A = ( EH,,C, Э ν ν ν Ex Ey Ez C Ex Hz Hy Эx = Ey Hz Hx Э y, where Ez Hy Hx Э z C Эx Эy Эz the physal eletr eld A E = grad φ, the physal magnet eld H = rot A, t the vrtual eletr eld A Э grad ф, the vrtual salar eld = C = ф t φ base own For the antsymmetr -tensor F ( F, F base = the base part F = F = ( E H ν ν ν ν β, =, own s the antsymmetr 4-tensor o the physal EM-eld, the own part F ( F, F where F = ( C, Э, = F whh s not observable dretly n a movng rs ν, are two opposte -vetors o the vrtual SE-eld By analogy wth the 4 -vetor A, or the 4 -tensor Fν we have the nequaltes: β β γ γ, β F F F F <, e H E C Э, F F ν ν γ FγF, ν < In the general ase, γ γ F F F, е С Э The equalty takes plae n the speal ase or a plane SEM-wave 3 Transormaton o the vrtual SE-eld strengths The physal EM- and vrtual SE- el transorm ndependently under the Lorentz group As a result o boosts the vrtual SE-eld transorms as the 4-vetor ( C, V VЭ = ( С VЭ, = + ( + С C + ε Э Э, where ε ε + ε F = Э : = V, V = V

11 4 The rst unon o the SEM-eld equatons F + F + F =, е νλ ν λ λ ν H rot E =, dv H =, t H rot Э =, grad С = E Э t The harge urrent -vetor ( ( ρ ρ, %,,, where = = s physal, = ρ s vrtual, but ρu Thus, or s not the 4 -vetor From ths, % ρ ρ 6 Lagrangans o the massve SEM-eld wth soures The ull Lagrangan L = L + L n t, where ν L = L SEM = FνF + м AA = ( E H + Э C + м ( A + ф 4 φ, м ths s the vrtual mass o a quantum SE-eld, n t = L A Sne L L + L, e LSEM = LSE + L EM, then the own part o Lagrangan L: = own base L = L + L = + м = n t own own own FF AA A ( Э C ( A + ф = + м φ ρф, the base part o Lagrangan L : L = L + L n t base base base Obvously, Lagrangan β = FβF A = ( Ε H ( ρ A 4 L base = L EM s gauge nvarant, but own = SE %φ L L s not 7 The seond unon o equatons o the massve SEM-eld wth soures Elmnaton o the nrared dvergenes Proa equatons or the massve SEM-eld wth soures are obtaned rom the Lagrangans L and L own as the system

12 + м =, ν ν F A F + м A = From ths, the equatons: C dve + м φ = ρ% +, E + м = + + Э rot H A, t C = м φ, Э = м A In the derene o these equatons we obtan : β β F =, that s, dv ρ E = %, rot = + E t H Also, rom the system we obtan the equaton: F + м A =, that s, C dv Э + м ф = ρ t Corollary As ollows rom the equatons o the massve SEM-eld, the vrtual SE-eld as the unobservable own part o SEM-eld vares n tme τ and, thereore, s massve n the presene o eld soures The small vrtual mass м o quantum SE-eld protets rom the nrared atastrophe n QED [] The physal EM-eld as the observable base part o SEMeld s massless and long-range 8 Wave equatons or a massve SEM-eld wth soures Usng the Stukelberg && Lagrangans wth the nteraton term ν ( Fν F A м AA A 4 = + L, ( = + L м, own F F A AA A we obtan the system o SEM-eld equatons + м =, ν ν F A F + м A =, wth the ondton A =

13 3 That s equvalent to the system o wave equatons or the -potental A ( м I A =, where ν I = ν, ( м + =, wth the ondton A A = Sne ν ν γ = γ +, then rom the system we get the equatons : W A =, ( W м A =, where γ W = γ Then the system o wave equatons or the SEM-eld strengths ( м = I F ν J ν, where ν ν ν J =, ( м β + = F From here, the wave equatons or EM-eld strengths : β β W F = J, ( м β + F = = The wave equatons or SE-eld strengths : ( м I F J That s, or the strengths Э and С o the vrtual massve SE-eld: ( I м Э = grad ρ +, ( м I С = ρ% ρ 9 The equaton o urrent ontnuty Conserved harges t From the SEM-eld equatons t ollows the equaton o urrent ontnuty = together wth the ondton = Thereore, n V4 the physal harge λ 3 3 ( x x ρ % x, the vrtual harge λ 3 3 Q = ( x d x = ρ d x, where Q = d = d Q > Q, sne ρ d d ρ % =, are onserved n tme: Q =, Q = ε dt dτ The anonal energy-momentum tensor o the massve SEM-eld From the ull Lagrangan L o the massve SEM-eld we an obtan the energymomentum tensor T ν ν ν ( T T =, or n the matrx orm base, own

14 4 T ν β w g v T T ˆσ S h T T v u R = = All equaltes below are gven wth an auray to terms that dsappear upon ntegraton over 3 d x n V 4 The base part o the energy-momentum tensor omponents: the energy densty ( = T ν base β = T has the physal w ( E H Э C м ( A ф = = = T + φ + T T, EM SE g - the momentum densty 3-vetor, { } [ ] { C } { = T = = T T } g EH Э, S - the energy lux densty 3-vetor (the Poyntng vetor, = { T } = [ ] the stress 3-tensor ˆσ { T } = ν The own part o the energy-momentum tensor Town ( T, T EM SE S EH CЭ, = has the omponents: u ( C + м ( ф = T = + + = SE E H Э φ A L L, EM vrtual: v h = Э H + C E, R = { T } = [ Э H] + CE = T = ЭE, { T } = [ ] Trae o the energy-momentum tensor o the massve SEM-eld ( + C ( + ф = + = = T T T E H Э м φ A L SEM Corollary The vrtual SE-eld brngs n the negatve ontrbuton to the energy-momentum o the SEM-eld Thus, or hydrogen atom n an external eld the vrtual SE-eld shts the energy levels and ths lea to two addtonal amendments

15 Equatons o the vrtual SE-waves The plane SEM-wave We an obtan the equatons o the SEM-waves rom the equatons or a massve SEM- eld, when = м = In partular, the equatons o the vrtual SE-waves: dv C Э =, rot Э = t, grad С = Э t Here below the dot above denotes the derentaton wth respet to tme t Let the propagaton dreton o the plane SEM-wave n //Ox We an nd rs n whh A = φ = onst Then E A &, H = rot A = Thus, E = [ Hn ], H = [ n E], = E H, n E =, n H =, EH = That s, the physal EM-waves are transverse Further, Э = grad ф = n ф &, C = ф & Thus, Э = n C, С = n Э, // Э n, C = Э, = Э E, Э H = That s, the vrtual SE-waves are longtudnal Sne H = E, C = Э, then T = and the energy densty o the plane SEM-wave T = E Э The Poyntng vetor S = [ EH] CЭ = S EM SSE= ( E Э = T n n That s, T = n S The equaton o moton o the harged pont partle n an external massve SEM-eld ( =, where F м AA ν ν ν ν ν - the harge urrent -vetor, - the -ore atng on the partle wth the physal harge e and vrtual mass m Ths pont partle moves n an external massve SEM-eld n the orward dreton o tme τ and tme t Other hand, d du = T m = ρm, where ρ m - the physal mass densty o a partle (see I4 and T m - the momentum densty 4 -vetor o a partle ν ν Hene, = E + ρc м ( AA, = E Э + [ H] м ( ν ur %, ρ ρ Aν A

16 6 ( AA = ρ% м However, ν Э C ν du = ρ m = Thereore, ν ν ν = ν F м ( ν We an see that ( AA ν A A = Then, ν F = ρ = ν Э % C, e ρ % C = Э or С = v Э On the other hand, the -ore atng on a movng harge rom an external massve SEM-eld wth the energy-momentum tensor T ν, s equal ν = νt From the equalty = ollows that = = That s, ν νt = 3 The eld orgn o the eletron vrtual mass The onsderaton o only the physal EM-eld annot explan the orgn o mass, selenergy and momentum o an eletron The stablty o an eletron annot be aheved only through physal eletromagnet ores [6,7] It should also take nto aount the vrtual massve SE-eld In the spae-tme V4 the vrtual mass m o a movng eletron (see I3 has the orgn o a massve SEM-eld and s explaned by the presene o the vrtual sel-sem-eld o an eletron The latter orrespon to the nonzero base part o the energy-momentum vetor o the massve SEM-eld, e, T = base T Sne the momentum densty o the vrtual sel-sem-eld o an eletron { T } [ ] h = = ( ЭH + CE, where =, then the 3-momentum И 3 = h d x = ( [ ЭH ] + C E d 3 x In rs, where the eletron at rest, C =, H = From the transormaton o the SEM-eld strengths (see II3 t ollows that С = V Э, H = [ VE ]

17 7 = = 3 Then, И V ЭE d x mv, where the vrtual mass o an eletron m 3 = dx ЭE, T = ЭE Thus, the vrtual sel-energy m = Э E d x 3 Also, we have proved the equalty 3 3 T d x = m d x, where m - the vrtual mass urrent 4-vetor (see I4 Remark By value the vrtual sel-energy o a movng eletron = m = Э E d x E 3 ondes wth the physal sel-energy o an eletron at rest = = E, e wth 3 E m d x the energy o the eletrostat eld [8,9] Reerenes Landau L D, Lshtz E M: Course o Theoretal Physs, Pergamon, Oxord, 97, Vol : The Classal Theory o Fel, Ch Chang T-P, L L-F: Gauge Theory o Elementary Partle Physs, Oxord,, Ch8 3 Ryder L H: Quantum Feld Theory, Cambrdge Unversty Press, Cambrdge, 98,Ch8 4 Kane G: Modern Elementary Partle Physs, Addson Wesley, 987, Ch8 Itzykson C, Zuber J-B: Quantum Feld Theory, MGraw Hll, New York, 98, Ch3 6 Feynman R P, Leghton R B, San M: The Feynman Letures on Physs, Addson Wesley, 964, Vol, Ch8 7 Eddngton AS: The Mathematal Theory o Relatvty, Cambrdge Unversty Press, Cambrdge, 94, Ch6 8 Jakson JD: Classal Eletrodynams, Wley, New York, 96, Ch7 9 Tamm IE: Fundamentals o the Theory o Eletrty, Mr Publsh, Mosow, 979, Ch

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