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A New Set of Maxwell-Lorentz Equations and Rediscovery of Heaviside-Maxwellian (Vector) Gravity from Quantum Field Theory

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A New Set of Maxwell-Lorentz Equations and Rediscovery of Heaviside-Maxwellian (Vector) Gravity from Quantum Field Theory Noname manuscript No. (will be inserted by the editor) A New Set of Maxwell-Lorentz Equations and Rediscovery of Heaviside-Maxwellian (Vector) Gravity from Quantum Field Theory Harihar Behera · Niranjan Barik Received: date / Accepted: date Abstract We show that if we start with the free Dirac Keywords Maxwell-Lorentz Equations Gravito- · Lagrangian, and demand local phase invariance, assum- magnetism Speed of Gravitational Waves (GWs) · · ing the total phase coming from two independent con- Attraction in Vector Gravity Energy of GWs · tributions associated with the charge and mass degrees of freedom of charged Dirac particles, then we are forced to introduce two massless independent vector fields for charged Dirac particles that generate all of electrody- 1 Introduction namics and gravitodynamics of Heaviside’s Gravity of 1893 or Maxwellian Gravity and specify the charge and Many field theorists, like Gupta [1], Feynman [2], Low mass currents produced by charged Dirac particles. From [3], Padmanabhan [4], Zee [5] and Gasperini [6] and this approach we found: (1) a new set of Maxwell-Lorentz Straumann [7] have rejected spin-1 vector theory of equations, (2) two equivalent sets of gravito-Maxwell- gravity on the ground that if gravitation is described Lorentz equations (3) a gravitational correction to the as a spin-1 theory like Maxwell’s electromagnetic the- standard Lagrangian of electrodynamics, which, for a ory, then two static masses of same sign will repel each neutral massive Dirac particle, reduces to the Lagrangian orther analogous the case in electromagnetism where for gravitodynamics, (4) attractive interaction between two static charges of same sign repel each other, while two static like masses, contrary to the prevalent view according to Newton’s gravitational theory - two static of many field theorists and (5) gravitational waves em- masses of same sign attract each other. However, here anating from the collapsing process of self gravitat- we show that this not true, if one considers appropri- ing systems carry positive energy and momentum in ate field equations of vector gravity derived here in a the spirit of Maxwell’s electromagnetic theory despite novel application of the well establisshed principle of the fact that the intrinsic energy of static gravitoelec- local phase (or gauge) invariance of field theory to mas- tromagnetic fields is negative as dictated by Newton’s sive Dirac fields. Subscribing to Feynman’s view [2] that gravitational law and its time-dependent extensions to “space-time curvature is not essential to physics”, and Heaviside-Maxwellian Gravity (HMG). Fundamental con- adopting Minkoskian space-time here we show that if we start with the free Dirac Lagrangian, and demand lo- arXiv:1810.04791v2 [physics.gen-ph] 2 Mar 2020 ceptual issues in linearized Einstein’s Gravity are also discussed. cal phase invariance, considering the total phase coming from two independent contributions associated with the charge and mass degrees of freedom of charged Dirac H. Behera particles, then we are forced to introduce two mass- BIET Degree College and BIET Higher Secondary School, less independent vector fields for charged Dirac parti- Dhenkanal-759001, Odisha, India cles that generate all of electromagnetism and gravielec- E-mail: [email protected] tromagnetism of Heaviside’s Gravity (HG)1[8,9,10,11, N. Barik Department of Physics, Utkal University, Vani Vihar, 1 Heaviside had speculated a gravitational analogue of Bhubaneswar-751004, Odisha, India Lorentz force law with a sign error that is corrected in this E-mail: [email protected] work. 2 Harihar Behera, Niranjan Barik 2 12,13,14,15] of 1893 or Maxwellian Gravity(MG) [16] ers. The flat space-time symmetric metric tensor ηαβ = αβ and specify the charge and mass currents produced by η is a diagonal matrix with diagonal elements η00 = α charged Dirac particles. Our new approach naturally 1, η11 = η22 = η33 = 1, space-time 4-vector x = x = − α α renders a gravitodynamics correction to the standard (ct, x) and xα = (ct, x), 4-velocity dx /dτ =x ˙ = − − Lagrangian of quantum electrodynamics, which, for a (cγ , uγ ) is the 4-velocity with γ = (1 u2/c2) 1/2, u u u − neutral massive Dirac particle, reduces to the Lagrangian and τ is the proper time along the particle’s world-line, α of quantum gravitodynamics. The resulting spin-1 vec- energy momentum four vector p = (p0, p) = (E/c, p), tor gravity is shown to produce attractive interaction ∂ (∂/c∂t, ) , ∂α (∂/c∂t, ), the D’Alembertian α ≡ ∇ ≡ −∇ between two static like masses, contrary to the preva- operator is = ∂ ∂α = ∂2/c2∂t2 2, where Ein- α − ∇ lent view. In the present approach, we also found a new stein’s convention of sum over repeated indices is used. set of Maxwell-Lorentz equations (n-MLEs) of electro- dynamics physically equivalent to the standard Maxwell- Lorentz equations (s-MLEs). The n-MLEs and s-MLEs 2 Consequences of Local Phase Invariance for are listed in Table 1 for comparison. Similarly, our present Charge and Mass Degrees of Freedom findings of the gravitational Maxwell-Lorentz equations It is well known that the free Dirac Lagrangian density (g-MLEs) of HG and MG along with n-MLEs are listed for a Dirac particle of rest-mass m in the Table-2, which exactly match with the recent 0 µ 2 results obtained by Behera [17] following Schwinger’s = i~cψγ ∂µψ m0c ψψ (1) inference of s-MLEs within Galileo-Newtonian physics, L − is invariant under the transformation if the speed of gravitational waves in vacuum cg = c, the speed of light in vacuum. ψ eiθψ (global phase transformation) (2) → where θ is any real number. This is because under global Table 1 Standard Maxwell-Lorentz Equations (s-MLEs) and − phase transformation eq. (2), ψ e iθψ which leaves new Maxwell-Lorentz Equations (n-MLEs) in SI units. → ψψ in (1) unchanged as the exponential factors cancel s-MLEs n-MLEs out. But eq. (1) is not invariant under the following ∇· E = ρe/ǫ0 ∇· E = ρe/ǫ0 ∇· B = 0 ∇· B = 0 transformation 1 ∂E 1 ∂E ∇ × B = + µ0j − 2 ∇ × B = − µ0j − 2 e c ∂t e c ∂t ψ eiθ(x)ψ (local phase transformation) (3) ∇ × − ∂B ∇ × ∂B E = ∂t E = + ∂t → dp × dp − × µ dt = q [E + u B] dt = q [E u B] where θ is now a function of space-time x(= x ), be- B = + ∇ × Ae B = −∇ × Ae cause the factor ∂µψ in (1) now picks up an extra term E = − ∇φ − ∂Ae E = − ∇φ − ∂Ae e ∂t e ∂t from the derivative of θ(x): ∂ ψ ∂ eiθ(x)ψ = i (∂ θ) eiθψ + eiθ∂ ψ (4) µ → µ µ µ Table 2 Gravito-Maxwell-Lorentz Equations (g-MLEs) of so that under local phase transformation, Heaviside Gravity (HG) and Maxwellian Gravity (MG), 2 ′ µ where µ0g = 4πG/c . = ~c (∂ θ) ψγ ψ. (5) L → L L− µ g-MLEs of HG g-MLEs of MG Now suppose that the phase θ(x) is made up of two ∇· g = −4πGρ0 = −ρ0/ǫ0g ∇· g = − 4πGρ0 = −ρ0/ǫ0g ∇· b = 0 ∇· b = 0 parts: 1 ∂g 1 ∂g ∇ × b = + µ0 j − 2 ∇ × b = − µ0 j + 2 g g c ∂t g g c ∂t θ(x)= θ (x)+ θ (x), (6) ∇ × ∂b ∇ × − ∂b 1 2 g = + ∂t g = ∂t dp dp 0 − × 0 × dt = m [g u b] dt = m [g + u b] which come from two independent contributions. Then − ∇ × ∇ × b = Ag b = + Ag (6) becomes g = − ∇φ − ∂Ag g = − ∇φ − ∂Ag g ∂t g ∂t ′ = ~c (∂ θ ) ψγµψ ~c (∂ θ ) ψγµψ (7) L → L L− µ 1 − µ 2 Units and Notations: Here we use SI units so that For a charged Dirac particle of charge q and mass m0, the paper can easily be understood by general read- we can re-write ′ in eq. (7) as L 2 ′ Which looks mathematically different from Heaviside’s = ~c (∂ θ ) ψγµψ ~c (∂ θ ) ψγµψ L L− µ 1 − µ 2 Gravity due to some differences in the sign of certain terms. ~ ~ But HG and MG are shown here to represent a single physical µ = + ∂µ θ1 q + ∂µ θ2 m0 cψγ ψ theory called Heaviside-Maxwellian Gravity (HMG) by cor- L − q −m0 rect representations of their respective field and force equa- = + jµ∂ λ (x)+ jµ∂ λ (x), (8) tions. L e µ 1 g µ 2 Title Suppressed Due to Excessive Length 3 µ where Ag must be mass-less (m1 =0= m2), otherwise the µ µ invariance will be lost for these two independent fields. je = qc(ψγ ψ) = 4-charge-current density, (9) The complete Lagrangian density then becomes µ µ j = m0c(ψγ ψ) = 4-mass-current density, (10) g = [i~cψγµ∂ ψ m c2ψψ] + + , (20) L µ − 0 Le Lg and λ (x) and λ (x), respectively stands for 1 2 where ~ ~ κ1 µν µ λ1(x)= θ1(x), and λ2(x)= θ2(x). (11) e = F Fµν je Aeµ and (21) − q −m0 L 4 − κ2 µν µ g = f fµν jg Agµ . (22) In terms of λ1 and λ2 then, under the local phase trans- L 4 − formation The equations of motion of these new fields can be ob- ′ − i tained using the Euler-Lagrange equations: ψ ψ = e ~ [qλ1(x)+m0λ2(x)]ψ, (12) → ′ µ µ β ∂ e ∂ e β ∂ g ∂ g = + je ∂µλ1 + jg ∂µλ2. (13) ∂ L = L and ∂ L = L . (23) L → L L β α α β α α ∂(∂ Ae ) ∂Ae ∂(∂ Ag ) ∂Ag Now, we demand that the complete Lagrangian be in- variant under local phase transformations. Since, the A bit calculation (see for example, Jackson [18]) yields free Dirac Lagrangian density (1) is not locally phase ∂ e ∂ g Lα = jeα and Lα = jgα.(24) invariant, we are forced to add something to swallow ∂Ae − ∂Ag − up or nullify the extra term in eq.
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