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

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arXiv:1810.04791v2 [physics.gen-ph] 2 Mar 2020 eta susi ierzdEnti’ rvt r also are Gravity Einstein’s discussed. linearized in issues ceptual to con- Fundamental extensions (HMG). Gravity Newton’s time-dependent Heaviside-Maxwellian by its and dictated gravitoelec- law as static gravitational negative of is energy fields intrinsic tromagnetic despite the in that theory fact momentum electromagnetic gravitat- the Maxwell’s and of self energy spirit of positive the carry process systems collapsing ing em- the waves gravitational view from (5) prevalent and anating the theorists to field contrary many of masses, like static between two interaction attractive (4) gravitodynamics, Lagrangian the a for to for reduces particle, which, Dirac massive electrodynamics, neutral the of to correction Lagrangian gravitational standard gravito-Maxwell- a (3) of equations sets Lorentz equivalent two Maxwell-Lorentz of (2) set new equations, a (1) found: we From approach particles. this Dirac and charged charge by the produced currents specify mass and of Gravity Gravity Maxwellian Heaviside’s or electrody- 1893 of of gravitodynamics all and generate namics that for particles fields Dirac vector independent charged massless forced two are we introduce then to particles, degrees Dirac charged mass of and con- freedom charge of independent the two with from associated coming tributions phase total the assum- invariance, ing phase local demand and Lagrangian, Abstract date Accepted: / date Received: Behera Harihar of Field Rediscovery Quantum and Theory from Equations Gravity Maxwell-Lorentz (Vector) of Heaviside-Maxwellian Set New A eateto hsc,UklUiest,Vn Vihar, Vani University, [email protected] Utkal E-mail: India Physics, Odisha, Bhubaneswar-751004, of Department Barik N. School, Secondary [email protected] Higher E-mail: BIET India and Odisha, Dhenkanal-759001, College Degree BIET Behera H. wl eisre yteeditor) the by inserted be (will No. manuscript Noname eso hti esatwt h reDirac free the with start we if that show We · iajnBarik Niranjan oet oc a ihasg ro hti orce nthis in corrected is that error sign a with work. law force Lorentz Keywords rmgeimo evsd’ rvt (HG) Gravity Heaviside’s gravielec- of parti- and mass- tromagnetism electromagnetism Dirac of two charged all generate introduce for that cles fields to Dirac vector forced charged independent are less of we freedom then of particles, degrees mass and the charge with associated contributions independent coming two phase from total the lo- considering demand invariance, if and phase Lagrangian, cal that Dirac free show the with we start we here and space-time physics”, to Minkoskian essential adopting not is that curvature [2] view “space-time Feynman’s mas- to to Subscribing theory of fields. field Dirac principle of sive a invariance establisshed gauge) in (or well phase here the local derived appropri- of gravity considers vector application one novel of if equations true, field here not However, ate this other. that each show attract we sign static same two while of - other, masses theory each gravitational repel Newton’s sign to where same according electromagnetism of charges in each static repel case two will the sign same analogous of the- orther masses electromagnetic described static Maxwell’s two is like then ory, gravitation theory of if spin-1 theory that a and vector as ground [6] spin-1 the Gasperini on rejected gravity and have [5] [7] Zee Low Straumann [2], [4], Feynman Padmanabhan [1], Gupta [3], like theorists, field Many Introduction 1 magnetism trcini etrGravity Vector in Attraction 1 evsd a pcltdagaiainlaaou of analogue gravitational a speculated had Heaviside · awl-oet Equations Maxwell-Lorentz pe fGaiainlWvs(GWs) Waves Gravitational of Speed · nryo GWs of Energy · Gravito- 1 [8,9,10,11, · 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 electrodynamics 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 invariant 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. (13). To this end, we ∂ ∂ e = κ F and g = κ f .(25) suppose βL α 1 αβ βL α 2 αβ ∂(∂ Ae ) − ∂(∂ Ag ) − ~ µ 2 µ µ = [i cψγ ∂µψ m0c ψψ] je Aeµ jg Agµ (14) From eqs. (23)-(25) we get the equations of motion of L − − − the new fields as where A and A are some new fields which interact eµ gµ 1 with the charge and mass current densities and change ∂βF = j . (26) αβ κ eα in coordination with the local phase transformation of 1 β 1 ψ according to the rule ∂ fαβ = jgα. (27) κ2 A A + ∂ λ and A A + ∂ λ . (15) eµ → eµ µ 1 gµ → gµ µ 2 The ‘new, improved’ Lagrangian (14) is now locally 2.1 Maxwell’s Fields from Charge Degree of Freedom phase invariant. But this was ensured at the cost of introducing two new vector fields that couples to ψ For classical fields, the 4-charge-current density in eq. through the last terms in eq. (14). But the eq. (14) is (9) is represented by devoid of ‘free’ terms for the fields A and A (hav- α eµ gµ je = (cρe, je), jeα = (cρe, je) (28) ing the dimensions of velocity: [L][T ]−1). Since these − are independent vectors, we look to the Proca-type La- where je = ρev, with ρe = electric charge density. For grangians for these fields [18]: static charge distributions, the current density jeα = je0 = cρe; it produces a time-independent - static - κ m c 2 free = 1 F µν F + κ 1 AµA (16) field, given by (26): Le 4 µν 01 ~ e eµ 2 0 free κ2 µν m2c µ ✚❃ = f f + κ A A (17) 1 ∂F✚00 ∂F01 ∂F02 ∂F03 ρec Lg 4 µν 02 ~ g gµ ✚ = ✚c ∂t − ∂x − ∂y − ∂z κ1 where κ1,κ2,κ01, and κ02 are some dimensional constants 2 ∂(cF01) ∂(cF02) ∂(cF03) ρec to be determined, m1 and m2 are the mass of the free or + + = . (29) ∂x ∂y ∂z − κ1 fields Aeµ and Agµ respectively. But there is a problem here, for whereas Eq. (29) gives us Coulomb field (E) as expressed in the Gauss’s law of electrostatics, viz., µν µ ν ν µ F = (∂ A ∂ A ) or Fµν = (∂µAeν ∂ν Aeµ) e − e − ∂E ∂E ∂E ρ (18) E = x + y + z = e (in SI units) (30) ∇ · ∂x ∂y ∂z ǫ0 µν µ ν ν µ f = (∂ Ag ∂ Ag ) or fµν = (∂µAgν ∂ν Agµ) − − (ǫ = electric permittivity of vacuum), if we make the (19) 0 following identifications: µ are invariant under the transformation eqs. (15), Ae Aeµ Ex Ey Ez 2 and AµA are not. Evidently, the new fields Aµ and F01 = , F02 = , F03 = & κ1 = ǫ0c . (31) g gµ e c c c − 4 Harihar Behera, Niranjan Barik

With the above value of κ1, ﬁxed by Coulomb’s law not For reference, we note the ﬁeld strength tensor with two by us, eqs. (21) and (26) become contravariant indices: 0 Ex Ey Ez 2 − c − c − c ǫ0c µν µ Ex e = F Fµν je Aeµ, (32)  c 0 Bz By  L − 4 − −  Ey 1  c Bz 0 Bx β  −  ∂ Fαβ = 2 jeα = µ0jeα. (33)  Ez  − ǫ0c −  c By Bx 0   −     For SME From the anti-symmetry property of Fαβ (Fαβ = Fβα), αβ αγ δβ  − F = η Fγδη =  (38) it follows form the results (31) that | {z }  0 Ex Ey Ez − c − c − c Ex  0 Bz By Ex Ey Ez  c − F10 = , F20 = , F30 = & Fαα =0. (34)  Ey − c − c − c  Bz 0 Bx  c −   Ez   By Bx 0  The other elements of F can be obtained as follows.  c −  αβ    For NME For α = 1, i.e. je1 = jex, eq.(33) gives us  −  αβ From eq. (33) and the anti-symmetry| {z property} of F , µ j = µ j it follows that jα is divergence-less: − 0 e1 0 ex e ✟✟✯ 0 α 1 ∂(ρec) ∂ρe 0 1✟ 2 3 ∂ j =0= + je = je + . (39) = ∂ F10 +✟∂ F11 + ∂ F12 + ∂ F13 α e c ∂t ∇ · ∇ · ∂t 1 ∂Ex ∂F12 ∂F13 This is the continuity equation expressing the local con- = −c2 ∂t − ∂y − ∂z servation of electric charge.

1 ∂Ex Equation (33) gives us two in-homogeneous equations 2 + ( B) (For SME) = − c ∂t ∇× x (35) 1 ∂Ex of SME and NME. The very definition of Fαβ in eq. ( 2 ( B) (For NME) − c ∂t − ∇× x (18), automatically guarantees us the Bianchi identity: where F = B and F = B for the standard ∂αFβγ + ∂βFγδ + ∂γFαβ =0, (40) 12 − z 13 y Maxwell’s Equations (SME); F = B and F = B 12 z 13 − y (where α,β,γ are any three of the integers 0, 1, 2, 3), for a possible form of New Maxwell’s Equations (NME). from which two homogeneous equations emerge natu- This way, we determined all the elements of the anti- rally: symmetric ‘field strength tensor’ Fαβ: B = 0 (For both SME and NME) (41) ∇ · ∂B ∂t (For SME) 0 Ex/c Ey/c Ez/c − E =  (42)  Ex/c 0 Bz By  ∇× − −  ∂B   + ∂t (For NME)  Ey/c Bz 0 Bx − −    The Bianchi identity (40) may concisely be expressed  Ez/c By Bx 0   − −  by the zero divergence of a dual field-strength tensor   F αβ F =  For SME (36) e , viz., αβ   0 Ex/c Ey/c Ez/c F αβ F αβ | {z } ∂α e = 0, where e is defined by (43)  Ex/c 0 Bz By   − −  Ey/c Bz 0 Bx − −  0 Bx By Bz   − −E − Ey  Ez/c By Bx 0 z   αβ 1 αβγδ Bx 0 c c − −  F = ǫ Fγδ =  −  (44)   e B Ez 0 Ex  For NME 2 y c c  −Ey Ex  Bz 0  | {z }  c − c  and the Ampère-Maxwell law of SME and NME:   For SME and the totally anti-symmetric fourth rank tensor ǫαβγδ 1 ∂E | {z } + µ0je + c2 ∂t (For SME) (called Levi-Civita Tensor) is defined by B = (37) ∇×   1 ∂E +1 for α =0,β =1,γ =2,δ =3,and  µ j 2 (For NME) − 0 e − c ∂t αβγδ  any even permutation  ǫ =  where the magnetic field, B is generated by charge cur-  1 for any odd permutation − rent and time-varying electric field E. 0 if any two indices are equal.    Title Suppressed Due to Excessive Length 5

(45) In three dimensional form the equations of motion (55b), take the following forms: F αβ The dual field-strength tensor e for the NME can be obtained from eq. (44) by substitution B B, → − dp q [E + u B] (For SME) with E remaining the same. = × (56) dt (q [E u B] (For NME) Eq. (41) suggests that B can be defined as the curl of − × a vector function A (say). If we define dE e = qu E (For both SME and NME) (57) dt · + A (For SME) B = ∇× e (46) ( Ae (For NME) −∇× 2.2 Maxwell-like Fields from Mass Degree of Freedom then using these definitions in (42), we find For classical fields the 4-current mass density or 4-momentum ∂Ae density in (10) is represented by E + = 0 (For SME and NME), (47) ∇× ∂t α jg = (cρ0, jg), jgα = (cρ0, jg) (58) which is equivalent to say that the vector quantity in- − side the parentheses of eq. (47) can be written as the where jg = ρ0v, with ρ0 = proper mass density. For gradient of a scalar potential, Ae0: static mass distributions, the current density jgα = jg0 = cρ0. It produces a time-independent - static - ∂Ae E = Ae0 (For SME and NME). (48) field, given by eq. (27). By establishing its correspon- − ∇ − ∂t dence with Newtonian gravitostatic field g dictated by In relativistic notation, eqs. (46) and (48) become g = 4πGρ , as was done for the Coulomb field in ∇ · − 0 the previous section, we obtain: αβ α β β α F = ∂ Ae ∂ Ae , (49) − c2 κ = , G is Newton’s gravitational constant. (59) (as they must, because of their common origin) where 2 4πG Aα = (A , A ) = (φ /c, A ). (50) With this value of κ (fixed by Newton’s law, g = e e0 e e e 2 ∇ · 4πGρ , not by us, just as the value of κ was fixed by − 0 1 In terms of this 4-potential, the in-homogeneous eqs. Coulomb’s law in eq. (30)), eqs. (22) and (27) turned (33) of SME and NME read: out as ∂ ∂βAα ∂α(∂ Aβ) = µ jα. (51) c2 ǫ c2 β e − β e 0 e = f µν f jµA = 0g f µν f jµA Lg 16πG µν − g gµ 4 µν − g gµ Under the Lorenz condition, (60) β 4πG ∂βAe = 0, (52) ∂βf = j = µ j . αβ c2 gα 0g gα the in-homogeneous equations (51) simplify to the fol- (61) lowing equations: where we have introduced two new constants ǫ0g and β α α α µ0g such that ∂β∂ Ae = Ae = µ0je (For SME & NME). (53) 1 4πG 1 ǫ0g = and µ0g = 2 = c = , The relativistic Lagrangian (not Lagrangian density) 4πG c ⇒ √ǫ0gµ0g for a single particle of proper mass m0 and electric (62) charge q moving in the external field of SME and NME, in complete analogy with the electromagnetic case where is written as −1/2 c = (ǫ0µ0) . Therefore ǫ0g may be called the gravitic or gravito-electric permittivity of free space and ǫ0g αβ dxα dxβ dxα α Le = m0 η + q Ae (x) (54) may be called the gravito-magnetic permeability of free − " r dτ dτ dτ # space. Now following the methods adopted in the pre- Using the Lagrangian (54) in Euler-Lagrange equations, vious section for discovering electromagnetic theory, we one obtains the co-variant equation of motion of a charged get the following results for gravito-electromagnetic (GEM) particle in electromagnetic field: theory or what we call Heaviside-Maxwellian Gravity. α α The Bianchi identity for HMG: dx˙ q αβ dp q αβ (a) = F uβ, (b) = F pβ. (55) dτ m0 dτ m0 ∂αfβγ + ∂βfγδ + ∂γ fαβ =0. (63) 6 Harihar Behera, Niranjan Barik

The gravitational analogues of eqs. (54)-(55) are The eq. (70) represents the gravito-Faraday’s law for MG and HG. Eq. (69) suggests that b can be defined dxα dxβ dxα αβ α as the curl of a vector function Ag (say). If we define Lg = m0 η + m0 Ag (x) (64) − " r dτ dτ dτ # + A (For MG) b = ∇× g (71) α α Ag (For HG) du˙ αβ dp αβ ( (a) = f uβ, (b) = f pβ. (65) −∇× dτ dτ then using these definitions in eq. (70), we get The anti-symmetric ‘field strength tensor’ fαβ of what ∂Ag we call Maxwellain Gravity (MG) and Heaviside Grav- g + = 0 (For both MG and HG). (72) ∇× ∂t ity (HG): So the vector quantity inside the parentheses of eq. (72) 0 gx/c gy/c gz/c is written as the gradient of a scalar potential, Ag0:  gx/c 0 bz by  A  − − (For MG) ∂ g  g = Ag0 (For both MG and HG). (73)  gy/c bz 0 bx ∂t   − ∇ − − −   gz/c by bx 0  In relativistic notation, eqs. (71) and (73) become − −  fαβ =   (66) αβ α β β α  f = ∂ Ag ∂ Ag , where (74)  −  0 gx/c gy/c gz/c α Ag = (Ag0, Ag) = (φg /c, Ag). (75)  gx/c 0 bz by   − − (For HG) In terms of this 4-potential, the in-homogeneous eqs.  g /c b 0 b  y z x  (61) of MG and HG read: − −   gz/c by bx 0  − −  β α α β 4πG α α   ∂β∂ Ag ∂ (∂βAg ) = jg = µ0gjg . (76) and the gravito-Ampère-Maxwell law of MG and HG: − − c2 −

4πG 1 ∂g Under gravito-Lorenz condition, c2 jg + c2 ∂t (For MG) − β b = (67) ∂βAg = 0, (77) ∇×  4πG 1 ∂g  + 2 j 2 (For HG) the in-homogeneous eqs. (76) simplify to the following c g − c ∂t equations: where b is named as gravitomagnetic field, which is β α α α generated by gravitational charge (or mass) current and ∂β∂ A = A = µ0gj (For MG & HG). (78) g g − g time-varying gravitational or gravitoelectric field g. The Before concluding this section we wish to note that the αβ field strength tensor f is obtained as: proper acceleration of a particle in the fields of HMG 0 gx gy gz is independent of its rest mass, m0 is a natural conse- − c − c − c gx quence of (65). This is the relativistic generalization of  0 bz by  c − Galileo’s law of Universality of Free Fall (UFF) - known  gy  bz 0 bx  c −  to be true both theoretically and experimentally since  gz   by bx 0  Galileo’s time. It states that all (non-spinning) particles  c −    of whatever rest mass, moving with same proper veloc-  (For MG)  αβ αβ αγ δβ  ity dxβ/dτ in a given gravitational field f , experience f = η fγδη = | {z } (68)  the same proper acceleration. In three dimensional form  0 gx gy gz − c − c − c the equations of motion (65), take the following forms: gx 0 b b  c z y   gy −  b 0 b dp m0 [g + u b] (For MG)  c z x  = × (79)  g −  z  dt m0 [g u b] (For HG)  c by bx 0  (  −  − ×   dE  (For HG) = m u g (For both MG and HG). (80)  0  dt · The two homogeneous| equations{z follow from} the Bianchi It is to be noted that the gravito-Lorentz force law identity eq. (63) as: originally speculated by Heaviside by electromagnetic b = 0 (For both MG and HG) (69) analogy was of MG-type in (79). The two basic sets of ∇ · Lorentz-Maxwell-like Equations (ME) of gravity pro- ducing the same physical effects are given in Table 2. ∂b (For MG) They represent a single vector gravitational theory, which g = − ∂t (70) ∇× ∂b ( + ∂t (For HG) we call Heaviside-Maxwellian Gravity (HMG). Title Suppressed Due to Excessive Length 7

3 Discussions to have gone unnoticed thus far, and therefore offered no contradiction to his hypothesis thatcg = c. Surpris- ingly, Heaviside seemed to be unaware of the long his- The analogies and peculiar differences between New- tory of measurements of the precession of Mercury’s ton’s law of gravitostatics and Coulomb’s law of elec- orbit as noted by McDonald [15], who reported Heavi- trostatics, noted by M. Faraday [19] in 1832, have been side’s gravitational equations (in our present notation) largely investigated since the nineteenth century, focus- as given in Table 1 under the head Maxwellian Grav- ing on the possibility that the motion of masses could ity (MG) - a name coined by Behera and Naik [16],3, produce a magnetic-like field of gravitational origin - who obtained these equations demanding the Lorentz the gravitomagnetic field. After the null experimental invariance of physical laws. It is to be noted that with- results on the measurement of gravitomagnetic field by out the correction of Heaviside’s speculative gravito- M. Faraday in 1849 and then again in 1859 [19], J. C. Lorentz force law the effect the gravitomagnetic field Maxwell [20] tried to formulate a field theory of gravity of the spinning Sun on the precession of a planet’s analogous to electromagnetic theory in 1865 but aban- orbit has the opposite sign to the observed effect as doned it because he was dissatisfied with his results: rigtly noted by McDonald [15] and Iorio and Corda [25]. the potential energy of a static mass distribution al- Apart from Maxwell and Heaviside, prior attempts to ways negative, but he felt this should be re-expressible modify Newton’s theory of gravitation were made by as an integral over field energy density which, which Lorentz in 1900 [26] and Poincarè[27] in 1905. There being the square of the gravitational field intensity, is was a good deal of debate concerning Lorentz-covariant positive [15]. We note that Maxwell did a miscalcula- theory of gravitation in the years leading up to Ein- tion, if one does the actual calculation analogous to stein’s publication of his work in 1915 [28]. For an overview electrostatic field energy [21], a negative sign comes be- of research on gravitation from 1850 to 1915, the reader fore the square of gravitational field intensity. Later may see Roseveare [29], Renn et al. [30]. Walter [31] Holzmüller [22] and and Tisserand [23,24] unsuccess- in ref. [30] discussed the Lorentz-covariant theories of fully attempted to explain the advance of Mercury’s gravitation where no mention of Heaviside’s Gravity is perihelion through Weber’s electrodynamics. In 1893, seen. However, the success of Einstein’s gravitation the- Heaviside [8,9,10,11,12,13,14,15] proposed a self con- ory, described in General Relativity (see for instance sistent theory of gravitomagnetism and gravitational [28,32,33,34,35,36,37,38,39]), led to the abandonment wave (GW) by writing down a set of g-MLEs (except of these old efforts. It must be noted that Einstein for a sign error in the gravito-Lorentz force law), which was unaware of Heaviside’s work on gravity, otherwise predict transverse gravitational waves propagating in his confidence in the correctness of Newtonian Grav- vacuum at some finite speed c according to Heaviside- g ity would not have been shaken as he stated before the Poynting’s theorem, analogous to the electromagnetic 1913 congress of natural scientists in Vienna [40], viz., case. To complete the dynamic picture, in a subsequent paper (Part II) [9,10,11,12,13,14] Heaviside speculated For before Maxwell, electromagnetic processes were a gravitational analogue of Lorentz force law, in the traced back to elementary laws that were fash- form that comes under g-MLEs of MG in Table 1, to ioned as closely as possible on the pattern of calculate the effect of the b field (particularly due to Newton’s force law. According to these laws, elec- the motion of the Sun through the cosmic aether) on trical masses, magnetic masses, current elements, Earth’s orbit around the Sun. Recently Behera [17] (fol- etc., are supposed to exert on each other actions- lowed Galileo-Newtonian Relativistic approach) and here at-a-distance that require no time for their prop- we found the correct form of Heaviside’s speculative agation through space. Then Hertz showed 25 gravito-Lorentz force as shown in Table 1, following years ago by means of his brilliant experimen- two independent approaches. This correction ensures tal investigation of the propagation of electrical that in both HG and MG, like mass currents (parallel force that electrical effects require time for their currents) should repel each other and unlike mass cur- propagation. In this way he helped in the victory rents (anti-parallel currents) should attract each other of Maxwell’s theory, which replaced the unmedi- in their gravitomagnetic interaction - opposite to the ated action-at-a-distance by partial differential case of electromagnetism where like electric currents at- equations. After the un-tenability of the theory of tract each other and unlike electric currents repel each action at distance had thus been proved in the do- other in their magnetic interaction. Heaviside also cal- 3 culated the precession of Earth’s orbit around the Sun Who relying on McDonald’s [15] report of HG, stated that MG is same as HG. This should not be taken for granted with- by considering his speculative force law of MG-type in out a proof because a sign difference in some vector quantities Table 1 and concluded that this effect was small enough or equations has different physical meaning/effect. 8 Harihar Behera, Niranjan Barik

main of electrodynamics, conﬁdence in the cor- 3.2 GRMG of Forward, Braginsky et al. and Thorne rectness of Newton’s action-at-a-distance theory (GRMG-FBT): of gravitation was also shaken. The conviction had to force itself through that Newton’s law of In the weak gravity and small velocity approximations gravitation does not embrace gravitational phe- of GR, the following linear gravito-Maxwell-Lorentz equa- nomena in their totality any more than Coulomb’s tions may be obtained following Forward [64], Bragin- law of electrostatics and magnetostatics embraces sky et al. [65] and Thorne [66] by neglecting the non- the totality of electromagnetic phenomena. linear terms:

Further Heaviside’s work would have played the same g = 4πGρ0, (81a) ∇ · − role on equal footing as Maxwell’s electromagnetic the- 4πG 1 ∂g H = 4 (ρ v) + , (81b) ory did in the development of special relativity. How- ∇× − c2 0 c2 ∂t ever, after Sciama’s consideration [41] of MG, in 1953 H = 0, (81c) to explain the origin of inertia, there have been several ∇ · ∂H studies on vector gravity, see [14,17,42,43,44,45,46,47, g = (81d) 48,49,50,51,52,53,54,55] and other references therein. ∇× − ∂t The g-MLEs obtained here corroborate the g-LMEs ob- (81e) tained by several authors using a variant of classical methods: (a) Schwinger’s Galileo-Newtonian Relativis- dv m = m g + m v H (82) tic approach to get the SMLEs [17,?], (b) Special Rel- 0 dt 0 0 × ativitic approaches to gravity [16,52,53,54], (c) modi- where ρ0 is the density of rest mass, v is the veloc- fication of Newton’s law on the basis of the principle ity of ρ0. Thorne [66] noted that the only differences of causality [14,49] , (d) some axiomatic methods [50, from Maxwell’s equations are (i) the minus signs before 51] common to electromagnetism and gravitoelectro- the source terms (terms with ρ0 in (81a) and (ρ0v) in magnetism and also (e) a specific linearization scheme (81b), which cause gravity to be attractive rather than of General Relativity (GR) in the weak field and slow repulsive; (ii) a factor 4 in the strength of H, presum- motion approximation [56]. However, in the context of ably due to gravity being associated with a spin-2 field GR several versions of linearized approximations exist, rather than spin-1; (iii) the replacement of charge den- which are not isomorphic and predict different values sity by mass density times Newton’s gravitation con- of speed of gravity cg in vacuum as explicitly shown by stant G and (iv) the replacement of charge current den- Behera [53]. This is one of the limitations of GR. MG sity by Gρ0v, where v is the velocity of ρ0. In empty of GR origin will be denoted as GRMG below. Out of a space (ρ0 = 0), the field eqs. (81a)-(81d) reduce to the number of linearized versions of GR considered in [53], following equations here we pick out only 4 versions for our discussion on the value of cg below for explicit comparison and other g =0, (83a) ∇ · purpose. 4 ∂g H = , (83b) ∇× c2 ∂t H = 0, (83c) ∇ · ∂H 3.1 On the Speed of Gravitational Waves (cg) g = (83d) ∇× − ∂t It is interesting to note that our theoretical prediction Now taking the curl of eq. (83d) and utilizing eqs. on the value of cg = c precisely agree with a remarkably (83a) and (83b), we get the wave equation for the field precise measurement of the value of cg = c with devia- g in empty space and taking the curl of eq. (83b) and −15 tions smaller than a few parts in 10 coming from the utilizing eqs. (83c) and (83d), we get the wave equation combination of the gravitational wave event GW170817 for the field H as [56], observed by the LIGO/Virgo Collaboration, and 4 ∂2g 1 ∂2g of the gamma-ray burst GRB 170817A [57]. This pre- 2g 2g 0 2 2 = 2 2 = , (84a) cise measurement of cg has dramatic consequences on ∇ − c ∂t ∇ − cg ∂t the viability of several theories of gravity [58,59,60,61, 4 ∂2H 1 ∂2H 2H 2H 0 62,63] that have been intensively studied in the last few 2 2 = 2 2 = , (84b) ∇ − c ∂t ∇ − cg ∂t years because many of them generically predict c = c. g 6 However, here we discuss below some linearized versions where cg = c/2, contrary to the recent experimental of GR which predict the value of c = c and also c = c. data [56,57]. g 6 g Title Suppressed Due to Excessive Length 9

3.2.1 GRMG of Ohanian and Ruffini (GRMG-OR) GRMG-PS-M in eq. (86a)-(86d), but the gravito-Lorentz force is In the Non-relativistic limit and Newtonian Gravity dv correspondence of GR, from Ohanian and Ruffini [38] m = m g + m v b. (87) 0 dt 0 0 × (Sec. 3.4 of [38]) one gets the gravito-Maxwell-Lorentz equations as The field equations of GRMG-UG yield cg = c in vacuum. Note the absence of the factor of 4 in grvaito- g = 4πGρ0, (85a) Lorentz equation. The Maxwell-Lorentz equations of ∇ · − 1 ∂H GRMG-UG match with the non-relativistic limit of our g = , (85b) ∇× −2 ∂t findings here. H =0, (85c) Thus the reader can now realize that the predictions on ∇ · 16πG 4 ∂g the speed of gravity in the weak field and slow motion H = j + , (85d) approximation of GR are not unique, but the value of ∇× − c2 c2 ∂t dv cg is uniquely and unambiguously fixed at cg = c in m0 = m0[g + v H] (85e) the present field theoretical findings of HMG or our dt × previous findings [16,53]. It is interesting to note that where ρ0 is the (rest) mass density, j = ρ0v is the the existence of gravitational waves has recently been momentum density and the gravitational displacement detected [69,70,71,72] and also the existence of the term in gravito-Ampère-Maxwell law in eq. ((85d)) was gravitomagnetic field generated by mass currents has recently added by Behera [53] to make the gravito- been confirmed by experiments [73,74,75,76,77,78,79]. Ampère law of Ohanian and Ruffini self consistent with These are being considered as new confirmation tests of the equation of continuity of rest mass. Without this GR [69,70,71,72,73,74,75,76,79]. The explanations for added term there can not be gravitational waves in experimental data on gravitational waves and the grav- vacuum. The wave equations for the g and H fields of itomagnetic field within the framework of HMG are be- GRMG-OR, in vacuum now obtainable from eqs. (85a)- ing explored by the authors, since the explanations for (85d) yield cg = c/√2. the (a) perihelion advance of Mercuty (b) gravitational bending of light and (c) the Shapiro time delay within 3.2.2 GRMG of Pascual-Sànchez and Moore the vector theory of gravity exist in the literature [46, (GRMG-PS-M): 47,48,80]. Recently Hilborn [81] following an electromagnetic analogy, calculated the wave forms of grav- In some linearized scheme of GR, Pascual-Sànchez [67] itational radiation emitted by orbiting binary objects obtained the following gravito-Maxwell-Lorentz equa- that are very similar to those observed by the Laser In- tions which match with Moore’s findings [68]: terferometer Gravitational-Wave Observatory (LIGO- VIRGO) gravitational wave collaboration in 2015 up to g = 4πGρ , (86a) ∇ · − 0 the point at which the binary merger occurs. Hilborn’s ∂b calculation produces results that have the same depen- g = , (86b) ∇× − ∂t dence on the masses of the orbiting objects, the or- H =0, (86c) bital frequency, and the mass separation as do the re- ∇ · 4πG 1 ∂g sults from the linear version of general relativity (GR). b = j + , (86d) ∇× − c2 c2 ∂t But the polarization, angular distributions, and overall dv power results of Hilborn differ from those of GR. Very m = m [g + 4v H] (86e) 0 dt 0 × recently we have reported an undergraduate level explanation of the Gravity Probe B experimental results j v where = ρ0 . The waves equations in vacuum that (of NASA and Stanford University) [77,78,79] using the emerge from eqs. (86a)-(86d) give us c = c. But note a g HMG [82]. factor of 4 in the gravitomagnetic force term in eq.(86e), which defies correspondence principle. 3.3 Does GR satisfy the correspondence principle? 3.2.3 GRMG of Ummarino-Gallerati (GRMG-UG): By deducing Newtonian Gravity (NG) from GR, all In another linearized approximations of GR, recently texts books on GR teach us that GR does satisfy the Ummarino and Gallerati [55] derived the following gravito- correspondence principle by which a more sophisticated Lorentz-Maxwell equations from Einstein’s GR. The theory should reduce to a theory of lesser sophistica- gravito-Maxwell’s equations are the same as those of tion by imposing some conditions; Misner, Thorne and 10 Harihar Behera, Niranjan Barik

Wheeler [32] in a boxed item (Box 17.1, page-412) of where, their book “Gravitation” have put much emphasis on dΨg it by giving a host of examples. In the light of our find- Id = ǫ0g , (89) − dt ings on HMG here and in [53] we see that GR defies the Ψ = g ds = gravitoelectric flux, (90) correspondence principle (cp) in its true sense: GRMG g · ✟✟ SRMG N(R)MG NG, where SRMG, N(R)MG I ⇔ ⇔ ⇔ and NG stands for Special Relativistic Maxwellian Grav- Ic is the conduction current of mass and Id is the dis- ity, Non-relativistic or Newtonain MG and Newtonian placement current. Gravity respectively. Consider a closed surface enclosing a volume. Suppose some mass is entering the volume and some mass also leaving the volume. If no mass is accumulated inside the 3.4 Misner, Thorne and Wheeler on HMG and volume, total mass going into the volume in any time is Experimental Tests of HMG equal to the total mass leaving it during the same time. The conduction current of mass is continuous. Misner,Thorne and Wheeler (MTW)[32], in their “Ex- If mass is accumulated inside the volume, as in the case ercises on flat space-rime theories of gravity”, have con- coalescence of two massive objects such as two neutron sidered a possible vector theory of gravity within the stars or any massive objects, this continuity breaks. framework of special relativity. They considered a La- However, if we consider the conduction mass current grangian density of the form (60) and found it to be plus the gravitational displacement current, the total deficient in that there is no bending of light, perihe- current is still continuous. Any loss of conduction mass lion advance of Mercury and gravitational waves carry current Ic appears as gravitational displacement cur- negative energy in vector theory of gravity. As regards rent Id. This can be shown as follows. the classical tests of the GR, we have noted before that Suppose a total conduction mass current I1 goes into the explanation of these tests exist in the literature [46, the volume and a total conduction mass current I2 goes 47,48,80]. But the issue of energy and momentum car- out of it. The mass going into the volume in a time dt ried by gravitational waves is far from clear yet, even is I1dt and that coming out is I2dt. The mass accumu- within the framework of GR. In the community of gen- lated inside the volume is eral relativists, there is no unanimity of opinion on the d dminside = I dt I dt or (minside)= I I (91) energy carried by gravitational waves. For instance, one 0 1 − 2 dt 0 1 − 2 finds references in the literature on GR which describes From Gauss’s law: (not in the gravito-electromagnetic approach) the radi- inside ation from a gravitating system as carrying away energy m0 Ψg = g ds = (92) [32,83], bringing in energy [84], carrying no energy [85] · − ǫ0g or having an energy dependent on the coordinate sys- I tem used [85]. However, in the gravito-electromagnetic From eqs. (89),(91) and (92) we get: approach to gravitational waves we briefly show that I I = I or I = I + I . (93) gravitational waves carry positive energy in accordance 1 − 2 d 1 2 d with the continuity equation or gravitational Heaviside- Thus total conduction current going into the volume Poynting’s theorem in spite of the fact that intrinsic en- is equal to the total current (conduction + displace- ergy of static gravitoelectromagnetic fields is negative. ment) going out of it. Note that since Ψg is negative, Id is positive, which carries positive field momentum 3.4.1 Gravito-Maxwell Displacement Current and and energy as no actual mass is moving in such cur- Continuity of Gravitoelectric Current rent. So gravitational collapse always leads to positive field energy and momentum coming out in the form of Let us recall that in Maxwell’s theory the displacement gravitational radiation. current is responsible for electromagnetic waves carrying energy and momentum in accordance with the con- 3.4.2 Gravitational Heaviside-Poynting’s theorem tinuity equation. Similar things occur in gravitoelectromagnetic theory under discussion. To see this consider The forms of the laws of conservation of energy and the integral form of gravito-Ampére-Maxwell Equation momentum are important results to establish for the of MG: gravitoelectromagetic field. Following the methods of electromagnetic theory, we obtained the following law b dl = µ (I + I ), (88) · − 0g c d of conservation of energy expressed by what we call I Title Suppressed Due to Excessive Length 11

Heaviside-Poynting’s theorem as Heaviside first consid- unambiguous prediction on the spin of graviton is dif- ered such a law (with a wrong sign for the Poynting vec- ficult to get [53]. tor) in his theory of gravity. The mathematical form of this theorem, in the form of a differential conservation law, is obtained for MG as 3.6 Attraction Between Static Like Masses ∂ u | | + S = j g (94) It is frequently overlooked that the interaction between ∂t ∇ · · two static (positive) masses, in a linear gravitational 1 2 2 2 1 where u = ǫ0g(g + c b ), S = (g b). (95) theory such as the MG or linearized vesions of GR listed | | 2 µ0g × earlier here, is definitely attractive. This fact was clearly Note that the sign before the source term j is positive, understood and stated by Sciama [41] and Thorne [66], whereas in the electromagnetic case the sign is negative. who attributed this attaction to the sign before the This is because the source terms in the field equations source terms of gravito-Maxwell equations, but rarely of MG has opposite sign to that in standard Maxwell’s recognized. However, to see it explicitly, let us find equations. The integration of j g over a fixed volume the static interaction between two neutral point (posi- · is the total rate of doing work by the fields in that vol- tive) masses at rest within the framework of Maxwellian ume, which is always positive for self gravitating sys- Gravity, following two approaches: (1) a classical ap- tems. So the right hand side of the differential energy proach by Shapiro and Teukolsky [88] and (B) Feyn- conservation law in eq. (94) is positive. The vector S, man’s [2] quantum field theoretical approach already representing the energy flow, is the Heaviside-Poynting described for the electrostatic case as follows. vector. The work done per unit time per unit volume by the fields (j g) is a conversion of gravitoelectromagnetic · energy into mechanical or heat energy. Thus there is a 3.6.1 Classical Approach of Shapiro and Teukolsky decrease in field energy density = ∂u/∂t. Since field [88]. − energy density u = u , we have ∂u/∂t = ∂ u /∂t, −| | − | | which is positive. So positive energy flux of field energy For a neutral particle having gravitational charge mg = must come out of systems collapsing under self gravity. m0 at rest at the origin, the 4-current densities: 0 3 jg = m0cδ (x), jg = 0 (96) 0 3.5 On the spin of graviton: spin 1 or spin 2? In eq. (77), we put Ag = φg/c, Ag = 0 (97) to get Following the usual procedures of electrodynamics (see, 2 2 3 3 for instance [86]) for obtaining the spin of photon, the φg = µ0gc m0δ (x) =4πGm0δ (x). (98) spin of graviton (a quantum of gravitational wave carry- ∇ ing energy and momentum) in the framework of HMG This is nothing but the Poisson’s equation for gravita- can be shown to be 1 in the unit of ~. Regarding the idea tional potential of a point mass at rest at origin. Us- of spin-2 graviton, Wald [33](see p.76) noted that the ing Green’s Function, the potential at a distance r for linearized Einstein’s equations in vacuum are precisely a central point particle having gravitational mass m0 the equations written down by Fierz and Pauli [87], in (i.e., the fundamental solution) is 1939, to describe a massless spin-2 field propagating in Gm φ (r)= 0 , (99) flat space-time. Thus, in the linear approximation, gen- g − r eral relativity reduces to the theory of a massless spin-2 which is equivalent to Newton’s law of universal grav- field which undergoes a non-linear self- interaction. It itation. The interaction energy of two point particles should be noted, however, that the notion of the mass ′ having gravitational charges m0 = M1 and m0 = M2 and spin of a field require the presence of a flat back separated by a distance r is ground metric ηab which one has in the linear approximation but not in the full theory, so the statement that, g GM1M2 U12 = M2φg = , (100) in general relativity, gravity is treated as a mass-less − r spin-2 field is not one that can be given precise mean- which is negative for like gravitational charges and pos- ing outside the context of the linear approximation [33]. itive for unlike gravitational charges, if they exist. With Even in the context of linear approximations, the orig- M1 at rest at the origin, the force on another stationary inal idea of spin-2 graviton gets obscured due to the gravitational charge M2 at a distance r from origin is several faces of non-isomorphic Gravito-Maxwell equa- g GM1M2 g tions seen in the literature from which a unique and F = M2 φg(r)= ˆr = F . (101) 21 − ∇ − r2 − 12 12 Harihar Behera, Niranjan Barik

of eq. (104) as

4πG 1 ′ ′ ′ ′ j j0 j j j j j j c2 ω2/c2 κ2 g0 g − g1 g1 − g2 g2 − g3 g3 − 4πG ′ ′ ′ ′ = j j j j j j j j (105) ω2 c2κ2 g0 g0 − g1 g1 − g2 g2 − g3 g3 − The conservation of proper mass, which states that the four divergence of proper mass current is zero, in momentum-space becomes simply the restriction Fig. 1 Feynman Diagram α k jgα =0. (106)

This force is attractive, if M1 and M2 are of same sign In the coordinate system we have chosen, this restric- and repulsive if they are of opposite sign, unlike the tion connects the third and the zeroth component of case of electrical interaction between two static electric the currents by charges. ω ω j κj =0, or j = j . (107) c g0 − g3 g3 κc g0

3.6.2 Quantum Field Theoretical Approach of If we insert this expression for jg3 into eq. (105), we get Feynman [2]. ′ 4πG ′ 4πG ′ ′ j Aα = j j j j + j j Analogous to the case of electromagnetism, the source gα g −κ2c2 g0 g0 − ω2 c2κ2 g1 g1 g2 g2 − of gravito-electromagnetism4 is the the vector current (108) α α jg , which is related to vector potential Ag by the rela- tion Now we can give interpretation to the two terms in eq. (108). The zeroth component of the current is simply α 4πG 1 α the mass density; in the situation where we have sta- Ag = 2 2 jg . (102) c k tionary masses, it is the only on-zero component of cur- 1 α α rent. The first term is independent of frequency; when In electromagnetism: Ae = µ0 2 je . (103) − k we take the inverse Fourier transform to convert this Here we have taken Fourier transforms and used the to a space-interaction, we find that it represents an in- momentum-space representation. The D’Alembertian op- stantaneously acting Newton potential. β 2 erator (∂β∂ ) in eq. (78) is simply k in momentum- ′ − − 4πG ′ Gm m ′ space. As in electromagnetism the calculation of am- (F.T.) 1 j j = 0 0 δ(t t ). (109) −κ2c2 g0 g0 − r − plitudes in gravito-electromagnetism is made with the help of propagators connecting currents in the manner This is always the leading term in the limit of small as symbolized by Feynman diagrams as that in Figure velocities. The term appears instantaneous, but this is 1. The amplitudes for such processes are generally com- only due to the separation we have made into two terms puted as a function of relativistic invariants restricting is not manifestly co-variant. The total interaction is re- the answer as demanded by rules of momentum and ally an invariant quantity; the second term represents energy conservation. As in electromagnetism, the guts corrections to the instantaneous Newtonian interaction. ′ of gravito-electromagnetism are contained in the spec- The force in eq. (109) is attractive, if m0 and m0 are of ification of the interaction between a mass current and the same sign and repulsive if they are of the opposite α the field as jgαAg ; in terms of the sources, this becomes sign - the reverse case of electrical interaction between an interaction between two currents: two static electric charges. Besides the above two different approaches, one may adopt Zee’s [5] path-integral approach to get at the same conclusion if one uses our ′ 4πG ′ 1 j Aα = j jα. (104) equations (60) and (78). gα g c2 gα k2 g In our choice of coordinates and units kα = (ω/c, 0, 0,κ), 4 Conclusion k = (ω/c, 0, 0, κ) and Aα is given by eq. (75). Then α − g the current-current interaction when the exchanged par- We have arrived at Maxwell-Lorentz electrodynamics ticle has a momentum kα is given by the right hand side using the principle of local phase invariance applied to 4 This term is coined because of the analogy of electromag- the free Dirac Lagrangian by considering the phase as- netism with HMG. sociated with the electric charge of a Dirac particle and Title Suppressed Due to Excessive Length 13

Coulomb’s law corresponding to the static part of the which we named as Heaviside-Maxwellian Gravito- Dirac particle charge density in its classical limit. Free electromagnetism or Gravity (HMG), Dirac Lagrangian density eq. (1) when combined with (4) the gravito-Lorentz-Maxwell’s equations of MG de- µ µ eq. (32) with je = qc(ψγ ψ) as in eq. (9) one obtains rived here using the well established principle of the Lagrangian density for quantum electrodynamics local phase (or gauge) invariance of field (partic- - charged Dirac fields (electrons and positrons) inter- ularly Dirac’s spinor field: a quantum field) theory acting with Maxwell’s fields (photons). This is truly perfectly match with those obtained from a variant a breathtaking accomplishment as Griffiths [85] states other established methods of study or principles of it; because the requirement of local phase invariance classical physics: (a) Schwinger’s formalism based associated with the charge of the Dirac particle, ap- on Galilio-Newtonian relativity if cg = c, (b) special plied to the free Dirac Lagrangian density, generates relativistic approaches of different types, (c) prin- all of electrodynamics and specifies the charge current ciples of causality, (d) some axiomatic approaches produced by charged Dirac particles. From 1820 when common to electromagnetism and gravitoelectromag- Oersted discovered magnetic effects of electric current, netism and (e) in some specific linearization method through Faraday’s discovery of electromagnetic induc- of general relativity, tion in 1831 to Maxwell’s synthesis of all experimen- (5) Galileo’s Law of Universality of Free Fall is a con- tal laws of electromagnetism and prediction of electro- sequence of HMG, not an initial assumption as in magnetic waves and their subsequent observation by Einstein’s General Relativity (GR), Hertz in 1887, people took almost 70 years to under- (6) our prediction of an unambiguous and unique value stand the nature of classical electromagnetic phenom- of speed of gravitational waves (cg = c), which agree ena. But the principle of local phase invariance led us very well with recent experimental data, unlike the to arrive at the Maxwell’s equations almost with no the ambiguous and non-unique value of cg obtain- time in comparison with the 70 years. This shows the able from different linearized versions of GR, predictive power of the principle of local phase invari- (7) possible existence of spin-1 graviton, in contrast with ance regarding the nature of the fields and their inter- the idea of spin-2 graviton within GR - an idea not actions with their sources, which we apply here to the well founded in the GR, mass degree of freedom of the same charged particle in (8) that the spin-1 vector gravity of HMG denomination exploring the nature of gravitational field and its in- produces attractive interaction between like static teraction with its sources. Inspired by this successful masses contrary to the prevalent view of the field story of local phase invariance and out of scientific cu- theorists, riosity, here in this work we applied the same principle (9) a brief discussion on the issue of negative/positive of local phase (gauge) invariance of field theory to the energy of gravitational waves both in HMG and Lagrangian of a free Dirac charged particle in flat space- GR and two theoretical demonstrations that grav- time and explored the question, ‘What would result if itational waves emanating from the collapsing pro- the total phase comes from two independent contribu- cess of self gravitating systems carry positive energy tions associated with the charge and mass degrees of and momentum in the spirit of Maxwell’s electro- freedom of charged Dirac particle?’ As a result of this magnetic theory despite the fact that the intrinsic study we found two independent vector fields one de- energy of static gravitoelectromagnetic fields is neg- scribing Maxwell’s theory and the other is a rediscovery ative as dictated by Newton’s gravitational law and of Heaviside’s Gravity of 1893. The important findings its time-dependent extensions to HMG, of this curious study include: (10) a gravitational correction to the standard Lagrangian of electrodynamics, which, for a neutral massive Dirac (1) a new set of Maxwell-Lorentz Equations (n-MLEs) particle, reduces to the Lagrangian for gravitody- of electromagnetism which is physically equivalent namics and to the standard set of equations; these n-MLEs has (11) our mention of the works of some other researchers also been found by the 1st author using Schwinger’s which correctly explain some crucial test of GR, viz., non-relativistic formalism, (a) non-Newtonain perihelion advance of planetary (2) a field theoretical derivation of the field equations of orbits including Mercury,(b) gravitational bending Heaviside’s Gravity (HG) and Maxwellian Gravity of light, (c) Shapiro time delay and (d) gravitational (MG) as well as their respective Lorentz force laws wave forms of recently detected gravitational waves, in which we found a correction to the Heaviside’s in a non-GR approach but using some aspects of speculative gravito-Lorentz force, HMG. (3) our findings that HG and MG are mere two different mathematical representations of a single theory 14 Harihar Behera, Niranjan Barik

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