arXiv:1709.06876v2 [physics.gen-ph] 13 Dec 2017 aia Behera Theory Harihar Field Quantum and from Relativity Gravity Special (Vector) Heaviside-Maxwellian Attractive b a 1 2 stesadr e yrdc forpeetstudy. present our of byproduct a Lorentz- effects - of physical set set standard same the new the as a having note equations also Maxwell’s velocity- We with mass mass. assump- gravitational dependent initial of equality any the without of HMG, tion fall of free consequence of a universality view of is prevalent law Galileo’s is the theorists. to HMG field of masses contrary spin-1 like condition, between This static force law. under attractive force produce a to Lorentz corrected shown the also ana- We gravitational of speculative vacuum. logue Heaviside’s in in light error of sign equal speed be to the found to uniquely is Heaviside- vacuum grav- in as of waves here speed itational the name which in we (HMG), Gravity that Maxwellian gravity the- vector physical two of single represent ory a to of field shown representations respective are mathematical their they of to terms However, due some other equations. in each difference from sign different a look Grav- which Maxwellian (MG), and ity 1893 of of (HG) equations Gravity field Heaviside fundamental the parti- rediscovered we Dirac cles, neutral Dirac electrically the massive to for applied Lagrangian theory phase field local quantum of (b) invariance and laws physical of Lorentz-invariance Abstract later inserted be should acceptance and receipt of date the wl eisre yteeditor) the by No. inserted manuscript be C (will J. Phys. Eur. -al [email protected] e-mail: hsc eatet ITHge eodr col Govindp School, Secondary Higher BIET Department, Physics eateto hsc,UklUiest,Vn ia,Bhuba Vihar, Vani University, Utkal Physics, of Department -al [email protected] e-mail: dpigtoidpnetapoce (a) approaches independent two Adopting a,1 .Barik N. , b,2 ewr710,Oih,India Odisha, neswar-751004, r hnaa-501 dsa India Odisha, Dhenkanal-759001, ur, 3 n aprn [4] Gasperini and [3] Eecsso a pc-ietere fgaiy,sug- gravity”, of their theories space-rime in flat (MTW)[5], attrac- on is Wheeler “Exercises sign and same to Misner,Thorne the tive. according of - masses gravitational between while static interaction the sign, - theory same gravitational the Newton’s of be- sources interactions like static tween repulsive theory in- produce field vector-like will if teractions vector then that theory, a ground electromagnetic by the Maxwell’s described on gravity is of gravitation theory vector spin-1 ayfil hoit,lk ut 1,Fymn[2] Feynman [1], Gupta like theorists, field Many Introduction 1 h aesg,wie-a swl nw h ttcgravita- static attrac the is - sign same known the well of is tive. masses as between - interaction while tional sign, inter sourc same vector-like between interactions the reason: immedi- static simple repulsive be very produce to a tions has for however, discarded theory. speculation, ately vector electromagnetic attractive appropriate the an with an Such analogy by close in represented field, gravitati be relativistic the might the to and interaction correspond four-vector, corre- vec- might a electromagnetic potential of potential the component Newton Coulomb of the the component potential, as fourth tor electric the way among for- to same force sponds close the electrostatic the In Coulomb static by charges. among the suggested force and Newton be the masses between to existing seems analogy mal generalizatio answer of natural kind Which A theory. appropriate an Newtons requires of thus eralization regime relativistic the need in no ity simpler is the there until fails; theories inadequate.” 2 consider 2 complicated found if to are more 4 need ones the perhaps we out and work that 2, to and so e 0 that forces, spins found attractive only also to is prop lead it a conversely, fact spins theories; is in 1 odd-spin is spin This all the attract. of of un-likes consequence and one repel, likes because is gravthat them, but of There fields, one spin-1 electrodynamics. two be of can’t as existence the same forbid to the nothing essentially be would 1 npg 7 aprn oe:“ orc ecito fgrav- of description correct “A noted: Gasperini 27, page On theory spin-1 “A noted: Feynman [2], ref. of 30 page On 2 onm e,hv rejected have few, a name to , 1 Zee , sof es onal gen- erty ven ac- ity n? - 2 gested an action functional for a possible vector theory ing the resulting theory here as Heaviside-Maxwellian of gravity within the framework of special relativity and Gravity (HMG), (iii) a correction to gravitational ana- asked the reader to find it to be deficient in that there logue of the Lorentz force law speculated by Heavi- is no bending of light, incorrect value for the perihelion side (iv) Suggestion of a Lagrangian that reproduces advance of Mercury and gravitational waves carry neg- all of HMG with gravitational waves carrying positive ative energy in a vector theory. Nevertheless, there have energy. In Section 4, we follow the usual procedure of been several studies on vector gravitational field theory quantum electrodynamics (in flat space-time) starting (reviewed here in Section 2) ever since Maxwell’s [6] first with the free Dirac Lagrangian, the requirement of lo- unsucessful attempt in 1865 and later Heaviside’s [7,8, cal phase invariance now applied to massive electrically 9,10,11,12,13] successful theoretical formulation of the neutral Dirac particles having rest mass m0 to find fundamental field equations of a vector gravitational a Lagrangian that generates all of gravitodynamcis of theory, called Heaviside Gravity (HG), which we de- HMG and specifies the current produced by massive rive here following two independent approaches: (a) us- Dirac particles. Spin-1 graviton is described in Section ing the Lorentz invariance of physical laws and (b) us- 5; while in Section 6, we show the attraction between ing the principle of local gauge invariance of quantum two static (positive) masses in the frame-work of HMG. field theory as applied to a massive electrically neutral In Section 6, we note our conclusions. Dirac spin-1/2 Fermion. However, Heaviside’s specula- tive gravitational analogue of the Lorentz force law had a sign error, whose correction we report for the first 2 Vector Gravity: A Brief Review time in this paper through our derivation. Alongside, using the above two approaches we also derived the fun- By recognizing the striking structural similairy of New- damental equations of Maxwellian Gravity (MG) [14] ton’s law of gravitational interaction between two which we show to be physically equivalent to HG de- masses and Coulomb’s law of electrical (or magnetic) in- spite the appearance of some sign differences in certain teraction between two charges (or magnetic poles) and terms of their respective equations. Because of our es- also their fundamental differences, J. C.Maxwell [6], in tablishment of the equivalence between HG and MG, sect. 82 of his great 1865 paper, A Dynamical Theory we named the resulting vector theory here as Heaviside- of the Electromagnetic Field, made a note on the at- Maxwellian Gravity (HMG). Since the explanations of traction of gravitation, in which he considered whether the classical tests of general relativity (GR) pointed Newtonian gravity could be extended to a form similar out by MTW within the framework of vector gravity to the form of electromagnetic theory - a vector field now exist in the literature [15,16,17], the main aim of theory - where the fields in a medium possess intrinsic this paper is to show the attractive interaction between energy. As a first step in this line of thought, Maxwell two static (positive) masses in a vector theory of grav- calculated the intrinsic energy Ug of the static gravi- ity, contrary to the prevalent view of the field theo- tational field at any place around gravitating bodies: rists. Moreover, we suggest a Lagrangian (density) for ′ U = C C g2d3x (1) this vector field theory of gravity in which gravitational g − waves carry positive energy. ZAll space ′ This paper is organized as follows. Section 2 details a where C and C are two positive constants and g is the review of vector gravitational theory. In Section 3, the gravitational field intensity at the place. If we assume 3 4 fundamental equations of HMG are derived using the that energy is essentially positive , as Maxwell did , Lorentz invariance of physical laws by adopting Behera then the constant C must have a value greater than ′ 2 and Naik’s approach to Maxwellian gravity (MG)[14], C g , where g is the greatest value of the gravitational wherein Galileo’s law of universality of free fall is a con- field at any place of the universe: and hence at any sequence of the theory, without any initial assumption 3Which is not true if one considers gravitostatic field energy of the equality of gravitational mass with velocity de- only. In fact following the electrostatic field energy calculation pendent inertial mass - whose violation is demonstrated (see for example, Griffiths’s Introduction to Electrodynamics) one obtains U = − 1 g2d3x. Thus one can set in a relativistic thought experiment that resolves Ed- g 8πG RAll space C = 0 and C′ = 1 in Eq. (1). The value of U calculated dington’s “gravitational mass ambiguity” [18] (stated 8πG g by this field theoretical method by using (1) with C = 0 and here in Sec.3). The new findings in Section 3, not ex- ′ 1 C = 8πG , for a spherical body of mass M, radius R with plicitly shown by Behera and Naik [14] are (i) the rel- uniform mass density within the body’s volume, turns out as 3 GM 2 ativistic rediscovery of Heaviside Gravity (HG) of 1893 Ug = − 5 R , which is the correct Newtonian (non-field- [7,8,9,10,11,12,13], (ii) the establishment of the phys- theoretic) result. 4 ical equivalence of HG with MG[14] and thereby nam- By stating, “As energy is essentially positive it is impossible for any part of space to have negative intrinsic energy.” 3 place where g = 0, the intrinsic energy must have an this paper, the correct gravito-Lorentz force law for HG | | enormously great value. Being dissatisfied with this should be of the form: result, Maxwell, concluded his note on gravitation by HG dv stating, “As I am unable to understand in what way a F = m0 = m0g m0v b (corrected). (4) gL dt − × medium can possess such properties, I can not go any further in this direction in searching for the cause of However, Heaviside by considering Eq. (3) calculated gravitation”. the precession of Earth’s orbit around the Sun and The first written record of a vector gravitational concluded that this effect was small enough to have theory was made by Oliver Heaviside [7,8,9,10,11,12, gone unnoticed thus far, and therefore offered no 13] in 1893. Studying by electromagnetic analogy, he contradiction to his hypothesis that gravitational found a set of four field equations for gravity akin to effects propagate at the speed of light. Surprisingly, Maxwell’s equations of electromagnetism representing Heaviside seemed to be unaware of the long history what we call Heaviside Gravity (HG). The gravita- of measurements of the precession of Mercury’s orbit tional field equations of HG, as we recently notice as noted by McDonald [19], who reported Heaviside’s in Heaviside’s original work, appear in the following gravitational equations (in our present notation) as Maxwellian form (written here in our notation). given below under the name Maxwellian Gravity.

Field Equations of Heaviside Gravity (HG): Field Equations of Maxwellian Gravity (MG):

g = 4πGρ = ρ /ǫ , (2a) g = 4πGρ0 = ρ0/ǫ0g, (5a) ∇ · − 0 − 0 0g ∇ · − − 4πG 1 ∂g 1 ∂g 4πG 1 ∂g 1 ∂g b = j + = µ j + , (5b) c2 c2 ∂t 0g c2 ∂t b = 2 j 2 = µ0gj 2 , (2b) ∇× − g g − g ∇× cg − cg ∂t − cg ∂t b = 0, (5c) b = 0, (2c) ∇ · ∇ · ∂b g = . (5d) ∂b ∇× − ∂t g = . (2d) ∇× ∂t with cg and the gravito-Lorentz force law as stated where in Eq. (2e) and Eq. (3) respectively. The vector grav- itational theory, represented by the Eqs. (3) and (5) 1 4πG 1 has been named as Maxwellian Gravity (MG) by Be- ǫ0g = , µ0g = 2 cg = (2e) 4πG cg ⇒ √ǫ0gµ0g hera and Naik [14]5 in honor of J. C. Maxwell for his first attempt in this direction. Behera and Naik [14] with cg representing the the speed of gravitational obtained these equations from relativistic considera- waves in vacuum, which might well be the speed of light tions, which will be revisited in this paper to obtain c in vacuum as Heaviside thought it, ρ0 is the ordinary some new results, viz., (a) derivation the HG equa- (rest) mass density, j = ρ0v is the mass current den- tions form special relativity, (b) establishment of the sity (v is velocity) and by electromagnetic analogy, b is physical equivalence of HG and MG and (c) finding the called the gravitomagnetic field, the Newtonian gravi- correct gravito-Lorentz force law (4) for HG. Without tational field g is called the gravitoelectric field, ǫ0g is this correction, the effect the gravitomagnetic field of called the gravitelectric (or gravitic) permitivity of vac- the spinning Sun on the precession of a planet’s orbit cum and µ0g is called the gravitomagnetic permeability has the opposite sign to the observed effect as noted of vacuum. To complete the dynamic picture, in a sub- in refs. [19,20]. Heaviside also considered, the propa- sequent paper (Part II) [7,8,9,10,11,12,13] Heaviside gation gravitational waves carrying energy momentum speculated a gravitational analogue of Lorentz force law in terms of gravitational analogue of electromagnetic in the following form: Heaviside-Poynting’s theorem. Apart from Maxwell and Heaviside, prior attempts to HG dv FgL = m0 = m0g + m0v b (speculated), (3) build a relativistic theory of gravitation were based on dt × 5 to calculate the effect of the b field (particularly due Who relying on McDonald’s [19] report of HG, stated that MG is same as HG. This should not be taken for granted with- to the motion of the Sun through the cosmic aether) out a proof because a sign difference in some vector quantities on Earth’s orbit around the Sun. As will be shown in or equations has different physical meanings. 4 an application of Maxwell’s equations were made by existence of a second gravitational field called gravita- Lorentz in 1900 [21] and Poincar`e[22] in 1905. There tional vortex (here called gravito-magnetic field) and was a good deal of debate concerning Lorentz-covariant assumed cg = c by electromagnetic analogy [36]. In theory of gravitation in the years leading up to Ein- 1980, Cattani [37] considered linear equations for the stein’s publication of his work in 1915 [5,23]. For an gravitational field by introducing a new field by calling overview of research on gravitation from 1850 to 1915, it the Heavisidian field which depends on the veloci- see Roseveare [24], Renn et al. [25]. Walter [26] in ref. ties of gravitational charges in the same way as a mag- [25] discussed the Lorentz-covariant theories of gravi- netic field depends on the velocities of electric charges tation. However, the success of Einstein’s gravitation and shown that a gravitational field may be written theory, described in many books [4,5,18,23,24,27,28, with linear co-variant equations in the same way as 29,30,31], led to the abandonment of these old efforts. for the electromagnetic field. Cattani’s equations dif- It seems, Einstein was unaware of Heaviside’s work on fer from some important formulae of general relativity gravity, otherwise his remark on Newton’s theory of such as the gravitational radiation, Coriolis force by gravity would have been different than what he made a factor of 4. In 1982, Singh [15] considered a vector before the 1913 congress of natural scientists in Vienna gravitational theory having formal symmetry with the [32],viz., electromagnetic theory and explained the (a) preces- After the un-tenability of the theory of action at sion of the perihelion of a planet (b) bending of light distance had thus been proved in the domain of in the gravitational field and (c) gravitational red-shift electrodynamics, confidence in the correctness of by postulating the self-interaction between a particle Newton’s action-at-a-distance theory of gravita- velocity and its vector potential. In 2004, Flanders and tion was shaken. One had to believe that New- Japaridze [16] axiomatically used the field equations of ton’s law of gravity could not embrace the phe- MG and special relativity to explain the photon deflec- nomena of gravity in their entirety, any more tion and perihelion advance of Mercury in the gravita- than Coulomb’s law of electrostatics embraced tional field of the Sun. Borodikhin [17] explained the the theory of electromagnetic processes. perihelion advance of Mercury, gravitational deflection However, in 1953, Sciama[33] hypothetically adopted of light as well as Shapiro time delay by postulating a 6 vector theory of gravity in flat space-time that is noth- MG (by assuming gravitational mass mg = m0, the rest mass - a measure of inertia of a body at rest) to ing but MG. Borodikhin also showed that in a vector explain the origin of inertia, calling it a toy model the- theory of gravity, there exists a model for an expanding ory of gravity which differs from general relativity (GR) Universe. Jefimenko [13,38] also deduced the equations principally in three respects: (a) It enables the amount of MG by extending Newton’s gravitational theory to of matter in the universe to be estimated from a knowl- time-dependent sources and fields and using the causal- edge of the gravitational constant, (b) The principle ity principle. Jefimenko assumed cg = c and postulated of equivalence is a consequence of the theory, not an a gravito-Lorentz force. Recently, Heras [39], by recog- initial axiom and (c) It implies that gravitation must nizing the general validity of the axiomatic approach be attractive. However, he concluded his paper men- to Maxwell’s equations of electromagnetic theory, used tioning three limitations of such a theory: (i) It is in- those axioms to derive only the field equations (leav- complete because the relativistic form of Newton’s law ing out gravito-Lorentz force law) of MG, where the must be derived from a tensor potential7, not from a in-variance of gravitational charge (or mass) is consid- vector potential, (ii) It is difficult to give a consistent ered. Other recent derivations of MG equations from relativistic discussion of the structure of the universe different approaches include the works of Nyambuya as a whole and (iii) It is also difficult to describe the [40], Sattinger [41], Vieira and Brentan [42]. The his- motion of light in a gravitational field. Carstoiu [34, torical objections of several researchers, starting from 35,36], in 1969, rediscovered Heaviside’s gravitational J. C. Maxwell [6] upto Misner, Thorne and Wheeler equations in the form of Eqs. (5) (in our present nota- (MTW, Sec.7.2)[5], concerning negative energy density tion as per the report of Brilloiun [36]) assuming the of gravitational field (‘Maxwell’s Enigma’ as Sattinger puts it) in a linear Lorentz invariant field theory of grav- 6The mass that appears in the Newton’s law of gravitostat- ity are also refuted by Sattinger [41], who considered ics is called the gravitational mass, which in analogy with negative field energy density for MG in agreement with Coulomb’s law of electrostatics may be regarded as the grav- itational charge of a body. the result reported by Behera and Naik [5]. In the dis- 7 2 This thought comes to anyone who believes in mg = E/c , cussion on the Dark Matter problem, Sattinger further where E is the relativistic energy (that includes the rest en- noted: “The Maxwell-Heaviside equations of gravitation 2 ergy E0 = m0c ) which may not be true as will be shown constitute a linear, relativistic correction to Newton’s later. 5 equations of motion; they interpolate between Newton’s The most serious objection against the Newto- and Einstein’s theories of gravitation, and are there- nian Law as an exact law was that it had become fore a natural mathematical model on which to build a ambiguous. The law refers to the product of the dynamical theory of galactic structures”. However, in masses of the two bodies; but the mass depends this work the gravitational energy density for free fields on the velocity- a fact unknown in Newton’s is fixed positive by choice to address the objection of days. Are we to take the variable mass, or the MTW (Sec. 7.2)[5] without any inconsistency with the mass reduced to rest? Perhaps a learned judge, field equations HG or MG. A detailed discussion on the interpreting Newton’s statement like a last will energy-momentum of gravitational field, which has a and testament, could give a decision; but that is long history, is left out here for another paper. scarcely the way to settle an important point in In the context of General Relativity, several authors scientific theory. have obtained different Lorentz-Maxwell-type equa- tions for gravity following different linearization pro- In his construction of a relativistic theory of gravity cedures leading to different versions that are not iso- popularly known as General Relativity (GR), Einstein morphic and have several serious limitations as seen in has taken a decision in favor of the equality of m with the recent report of Behera [43]. mg. For a theoretical justification of this decision, Ein- stein by writing Newton’s equation of motion in a gravi- tational field (in our present mathematical notation) as 3 HMG Form Special Relativity dv m = mgg (6) The fundamental equations of HG and MG will be dt derived here using special theory of relativity (SR), (wrongly!) inferred from it ([23], pp. 57): wherein we will make a correction to Heaviside’s spec- ulative gravitational analogue of the Lorentz force law It is only when there is numerical equality be- and establish the physical equivalence of HG and MG. tween the inertial and gravitational mass that With the establishment of SR and the equivalence of the acceleration is independent of the nature of mass and energy, the meaning of the inertial mass and the body. gravitational mass became ambiguous, because SR sug- This inference is often expressed in one of the two ways: gests two inertial mass-energy concepts: (1) the Lorentz (S1) that the particle’s motion is mass independent, or invariant rest-mass m = E /c2 (E = rest-energy, 0 0 0 (S2) that the particle’s inertial mass m = its gravita- which is the sum total of all forms of energy in the tional mass m . rest frame of a body or particle) and (2) the mass at- g The two statements (S1) and (S2) are sometimes tributed to the relativistic energy m = E/c2 (E = sum used interchangeably as the weak equivalence princi- of all forms of energy at rest and motion) which is not ple (WEP) in the literature [27,28,29,30,31]. This use Lorentz-invariant. The qualitative distinction that ex- of terminology is rather confusing, as the two state- isted between two inertial mass concepts in Newtonian ments are logically independent [45]. They happen to mechanics became quantitatively distinct and clear in coincide in the context of Galileo-Newtonian physics SR. Now, one fundamental question arises, “What form where m = m = m but may diverge in the context of of mass (or energy) should represent the gravitational 0 g special relativity where m = m and Einstein’s wrong mass8 (m ) in a relativistic version of Newtonian grav- 6 0 g inference of m = m = m from a non-relativistic Eq. ity?” In any construction of a field theory of gravity 0 6 g (6), where m = m and m = m is a condition for compatible with SR and the correspondence principle 0 g 0 Galileo’s law of Universality of Free Fall to be true. To by which a relativistic theory gravity is reducible to explore this possibility, to get new insights for making Newtonian gravity, a decision on which form of “mass” Newtonian gravity compatible with the SR, to regard (or energy) is the source of gravity has to be taken. Such old problems from a new angle, we re-examined [14] a decision, as Price [44] has rightly pointed out, will be an often cited [46,47] Salisbury-Menzel’s [48,49,50,51] crucial not only to the resolution of the ambiguity men- thought experiment (SMTE) from a new perspective as tioned above but also to the issue of the nonlinear na- discussed in the following subsection containing some ture of gravity. One of the Eddington’s [18] four reasons new thoughts and results not explicitly revealed in [14]. to feel dissatisfied with Newton’s Law of gravitation is Before that the authors would like to remark that per- appropriate here to quote: haps Einstein, himself, was not satisfied with his above 8 In Newtonian physics mg = m0 for Galileo’s law of Univer- inference of mg = m, as we can sense from his another sality of Free Fall to be true. statement on the equality of mg with m [52,53]: 6

The proportionality between the inertial and the said particle system from the point of view of an gravitational masses holds for all bodies with- observer in another inertial frame S, in uniform rela- out exception, with the (experimental) accuracy tive motion with respect to the S′-frame. To simplify achieved thus far, so that we may assume its the investigation, let the relative velocity v of S and S′- general validity until proved otherwise. frame be along a common X/X′-axis with correspond- ing planes parallel as usual. Since the particles are at The last three words of Einstein’s above state- rest in S′-frame, both of them have the same uniform ment,‘until proved otherwise’, show that he was very velocity v relative to the S-frame. Let the position vec- cautious and not very confident of what he was stating. tor of the particle No.1 with respect to the particle No.2 Based on the experimental results available up to 1993, as observed in the S-frame be r and the angle between Mashhoon [47] noted that the observational evidence v and r be θ. for the principle of equivalence of gravitational and in- For an observer in the S-frame, the force of electric ertial masses was not yet precise enough to reflect the origin on either particle (say on particle No.1 due to wave nature of matter and radiation in their interac- particle No.2) is no more simply a Coulomb force, but tions with gravity (see other references on equivalence a Lorentz force, viz., principle in [14,47]). F = q E + q v B (9) L 1 2 1 × 2

3.1 Re-Examination of SMTE to Show m0 = mg where 2 q2(1 β )r Consider a system of two non-spinning point-like E2 = − , (β = v/c) (10) 3 2 2 3/2 charged particles with charges q1 and q2 and respec- 4πǫ0r 1 β sin θ 2 2 − tive rest masses m01 (= E01/c ) and m02 (= E02/c ) ′
such that they are at rest in an inertial frame S under 2 v E2 (q2v) r (1 β ) equilibrium condition due to a mutual balance of the B2 = × = × − c2 2 3 2 2 3/2 ′ 4πǫ0c r 1 β sin θ force of Coulombic repulsion (FC ) and the Newtonian − ′ 2 gravitostatic attraction (FN ) between them. Our aim is µ0 (q2v) r (1 β )
= × − 3/2 (11) to investigate the condition of equilibrium of this two- 4π r3 1 β2 sin2 θ particle system (realizable in a Laboratory by taking − 1/2
two perfectly identical spherical metallic spheres hav- r′ 1 β2 sin2 θ ing requisite masses and charges so that they are in r = − . (12) (1 β2)1/2 equilibrium) in different inertial frames in relative mo- −
tion. For our re-examination purpose, suppose that the What about the force of gravitational interaction as particles are positively charged and they are in empty observed in the S-frame? It can not simply be a New- space. Let the particle No.2 be positioned at the origin tonian force but something else, otherwise the particle of S′-frame and r′ be the position vector of the particle system will not remain in equilibrium in the S-frame. No.1 with respect to the particle No.2. In this S′-frame Such a situation will amount to a violation of the princi- the condition of equilibrium is fulfilled by ple of relativity in special relativity. A null force should ′ ′ remain null in all inertial frames. Therefore, a new force ′ ′ q1q2r Gm01m02r law of gravity has to be invoked so that the equilibrium FC + FN = ′3 ′3 = 0, (7) 4πǫ0r − r is maintained in accordance with the principle of rel- where r′ = r′ and other symbols have their usual ativity (Lorentz invariance of physical laws). Let this | | meanings. From Eq. (7) we get new unknown force be represented by FgL such that the equilibrium condition in S-frame is satisfied as: q1q2 m01m02 = Gm01m02 = (ǫ0g =1/4πG). (8) 4πǫ 4πǫ FgL + FL = 0 = FgL = FL. (13) 0 0g ⇒ −

Eq. (8) represents the condition of equilibrium, in terms Taking into account the Eqs. (9)-(12), FgL in Eq. (13) of the charges and rest masses (or rest energies) of the can be expressed as: particles, under which an equilibrium can be ensured 2 ′ q1q2 1 β r in the S -frame. For example, if each metallic sphere is FgL = − −6 − 3 2 2 3/2 given a charge of 1 10 Coulomb, then the rest mass 4πǫ0r 1 β sin
θ × 4 − of each sphere should be 1.162 10 kg, to fulfill the µ q q v (v r) 1 β2 × 0 1 2 × ×
− equilibrium condition (8) in a laboratory experiment. 3/2 . (14) − 4π r3 1 β2 sin2 θ
Now, let us investigate the problem of equilibrium of −
7

Now, using Eq.(8), we can eliminate q1q2 from Eq. (14) of the physical laws. These findings are in conformity to get the expression for FgL in terms of m01,m02 and with Poinca`re’s [54] remark that if equilibrium is to be G as: a frame-independent condition, it is necessary for all Gm m 1 β2 r forces of non-electromagnetic origin to have precisely 01 02 − FgL = 3/2 the same transformation law as that of the Lorentz- − r3 1 β2sin2 θ
− force. Now, following Rosser’s [55] approach to classical 2 G m01m02v (v r) 1 β electromagnetism via relativity, Behera and Naik [14] ×
× − 2 3/2 and Behera [38] had obtained the field equations of MG − c r3 1 β2 sin2 θ
− as given by the Eqs. (5). 1 m m 1 β2 r 01 02 −
On the other hand, if one retains the definition of g2 as = 3/2 − 4πǫ0g r3 1 β2 sin2 θ
in Eq. (19) and redefines b2 as − 2 µ0g m01m02v (v r) 1 β 2 × ×
− v g2 µ0g (m02v) r (1 β ) 3/2 , (15) − 4π 3 2 2 b2 = ×2 = × − 3/2 r 1 β sin θ
− c 4π r3 1 β2 sin2 θ − − where
µ (m v) r 0g 02 × (when β << 1)
, (21) 1 4πG 1 ≃ 4π r3 ǫ = , µ = = c = . (16) 0g 4πG 0g c2 ⇒ ǫ µ √ 0g 0g then the gravito-Lorentz force law (18) must be of the By comparing the quantities in Eq. (16) with that of form as in Eq. (4) so as to describe the same physics (or −1/2 electromagnetic theory, viz., c = (ǫ0µ0) , we imme- physical effects) as implied by Eqs. (18)-(20) or their diately find that, gravitational waves, if they exist, must source Eq. (15). The field equations that are consis- have a wave velocity cg in vacuum: tent with the equations (19), (21) and (4) can again be obtained following Rosser [55]. These represent Heav- 1 cg = = c. (17) iside’s gravitational field equations as originally pro- √ǫ0gµ0g posed by him [7,9,10,11,12,13], now written in our Now, Eq. (15) may be rearranged to the following form present notation and convention as in the Eqs. (2). The to represent the Gravito-Lorentz force law of special equation of continuity relativistic Maxwellian Gravity (SRMG): ∂ρ j + 0 = 0 (22) FMG = m g + m v b (For MG) (18) ∇ · ∂t gL 01 2 01 × 2 where follows from the in-homogeneous equations of HG and 2 MG. In vacuum (where ρ0 = 0, j = 0), the field equa- 1 m02(1 β )r g2 = − 3/2 tions of HG and MG give us the wave equations for the − 4πǫ0g r3 1 β2 sin2 θ − g and b fields: 1 m r 02 (when β
<< 1), (19) ≃− 4πǫ r3 1 ∂2g 0g 2g = (23a) ∇ c2 ∂t2 2 2 v g µ (m v) r (1 β ) 2 1 ∂ b b = × 2 = 0g 02 × − b = (23b) 2 2 3/2 ∇ c2 ∂t2 c − 4π r3 1 β2 sin2 θ − µ (m v) r which show that the wave velocity of gravitational 0g 02 × (when β <<
1). (20) ≃− 4π r3 waves in vacuum cg = c. Alternatively, after recognizing our new findings from Eqs. (18)-(20) are in complete formal analogy with the above thought experiment, especially m = m and the Eqs. (9)-(11) of classical electromagnetism in its g 0 c = c from Eq. (17), one may follow the following pro- relativistic version. Thus, from the requirement of g cedure to arrive at the field equations of MG and HG. the frame-independence of the equilibrium condition, we not only obtained a gravitational analogue of the Lorentz-force law expressed by Eq. (18) but also un- 3.1.1 Alternative derivation of Field Equations of MG expectedly found the Lorentz-invariant rest mass as and HG the gravitational analogue of the electric charge by electromagnetic analogy. From this analysis, the grav- We take for granted the Gauss’s law of gravitostatics itational charge (or rest mass) invariance may be in- (2a) and the equation of continuity (22) as valid laws terpreted as a consequence of the Lorentz-invariance of physics. To establish a link between the Eqs. (2a) 8 and (22), we take the time derivative of Eq. (2a) and on the precession of a planet’s orbit has the opposite write the result as sign to the observed effect as noted by McDonald [19] and Iorio [20], who did not trace the cause of this er- ∂ρ0 1 ∂g = (24) ror. As per our present relativistic study, we found HG ∂t −4πG∇ · ∂t   and MG to represent the same physical phenomena, the From the equation of continuity (22) and the Eq. (24), sign differences in some terms in their equations are at- we get tributed to the definitions of some physical quantities. Thus, HG and MG are mere two mathematical repre- 1 ∂g ∂g j = j ǫ =0. (25) sentations of a single vector theory of gravity named ∇ · − 4πG ∂t ∇ · − 0g ∂t     here as Heaviside-Maxwellian Gravity (HMG). 2 Since b = 0 for both HG and MG, b can be Now we multiply Eq. (25) by µ0g =4πG/c , as defined ∇ · in Eq. (17), to obtain the equation: defined as the curl of some vector function, say Ag. If we define, 1 ∂g µ0gj =0. (26) ∇ · − c2 ∂t Ag (For HG)   b = −∇× (32) (+ Ag (For MG) The quantity inside the parenthesis of Eq. (26) is a ∇× vector whose divergence is zero. Since ( X)=0 then using these definitions in Eqs. (31), we find ∇ · ∇× for any vector X, the vector inside the parenthesis of ∂A Eq. (26) can be expressed as the curl of some other g + g = 0 (For both MG and HG), (33) ∇× ∂t vector, say b. Mathematically speaking, the Eq. (26)   admits of two independent solutions: which is equivalent to say that the vector quantity in-

1 ∂g side the parentheses of Eq. (33) can be written as the +µ0gj 2 (For HG) c ∂t gradient of a scalar potential, φg: b = − 1 ∂g (27) ∇× ( µ0gj + 2 (For MG) − c ∂t ∂A g = φ g (For both MG and HG). (34) Thus, we arrived at the Eq. (2b) of HG and Eq. (5b) of − ∇ g − ∂t MG. In vacuum (j = 0), the Eqs. (27) become Substituting the expression for g given by Eq. (34) 1 ∂g and the expression for b defined by Eq. (32) in the c2 ∂t (For HG) b = − 1 ∂g (28) in-homogeneous field Eqs. (2a)-(2b) of HG and (5a)- ∇× + 2 (For MG) ( c ∂t (5b) of MG, we get the following expressions for their Taking the curl of the Eqs. (28) we get in-homogeneous equations in terms of scalar and vector potentials as 1 ∂ 2 c2 ∂t ( g) (For HG) 2 ( b) b = − ∇× (29) 2 1 ∂ φg ρ0 1 ∂ φg = (For both MG and HG), (35) ∇ ∇· −∇ (+ 2 ( g) (For MG) 2 2 c ∂t ∇× ∇ − c ∂t ǫ0g The Eqs. (29) will reduce to the wave equation (23b) for the b field, if the following conditions: 2 2 1 ∂ Ag Ag 2 2 = µ0gj (For both MG and HG), b = 0 (For both HG and MG) (30) ∇ − c ∂t ∇ · (36)

∂b if the following gravitational Lorenz gauge condition, + ∂t (For HG) g = ∂b (31) ∇× ( (For MG) 1 ∂φg − ∂t Ag + = 0 (For both MG and HG), (37) ∇ · c2 ∂t are satisfied. Thus, we arrived at the Eqs. (2c)-(2d) of HG and Eqs. (5c)-(5d) of MG by imposing the condition is imposed. These will determine the generation of grav- of existence of gravitational waves in vacuum. This way itational waves by prescribed gravitational mass and we found the correctness of the original gravitational mass current distributions. Particular solutions of Eq. field equations found by Heaviside and as seen in [7,9, (35) and Eq. (36) in vacuum are 10,11,12,13] and corrected Heaviside’s gravito-Lorentz ′ ′ 1 ρ0(r ,t ) ′ force law to the form as given in Eq. (4) so as to be con- φg(r,t)= ′ dv and (38) −4πǫ0g r r sistent with his field equations (2a)-(2d)). It is due an Z | − | ′ ′ error in the sign in the gravitomagnetic force term that µ j(r ,t ) ′ A (r,t)= 0g dv , (39) the effect of gravitomagnetic filed of the spinning Sun g − 4π r r′ Z | − | 9 where t′ = t r r′ /c is the retarded time and dv′ one can rewrite the field equations of MG as: − | − | ′ is an elementary volume element at r . Thus, we saw that retardation in gravity is possible in flat space-time β β 4πG ∂ fαβ = ∂ (∂αAgβ ∂βAgα)= jα = µ0gjα, in the same procedure as we adopt in electrodynam- − c2 ics. Hence, we have reasons to strongly disagree with (41) those who believe in Rohrlich’s conclusion [56]: “Be- cause the Newtonian theory is entirely static, retarda- ∂ f + ∂ f + ∂ f =0, (42) tion is not possible until the correction due to deviations α βγ β γδ γ αβ from Minkowski space is considered”. Before passing to where α, β, γ are any three of the integers 0, 1, 2, 3; α the next section, the authors wish to note that using and the Gravito-Lorenz condition: ∂ Agα = 0. Now, our present approach to the discovery of HG and MG, the relativistic gravito-Lorentz force law of MG takes one can obtain a new mathematical form of Lorentz- the following form Maxwell’s equations, which are physically equivalent to d2xα dx the standard Lorentz-Maxwell’s equations. These new αβ β 2 = f , (43) form of Lorentz-Maxwell’s equations are noted in the dτ dτ following box. where τ is the proper time along the particle’s world- line and f αβ is given by New Form of Lorentz-Maxwell Equations gx gy gz of Electrodynamics: 0 c c c gx − − − αβ αγ δβ c 0 bz by f = η fγδη =  gy −  , (44) b 0 b c z − x E = ρe/ǫ0,  gz b b 0  ∇ ·  c − y x  1 ∂E   Bnew = µ0je 2 , αβ ∇× − − c ∂t where the flat space-time metric tensor ηαβ = η is Bnew = 0, represented by symmetric diagonal matrix with ∇ · ∂Bnew E = , η =1, η = η = η = 1. (45) ∇× ∂t 00 11 22 33 − where The relativistic equation of motion (43) is indepen- 1 dent of the mass of the particle moving in an external c = αβ √ǫ0µ0 gravito-electromagnetic (GEM) field f . Thus we saw that the motion of a particle in an external GEM field Fnew = q (E v B ) L − × new can be independent of its mass without any postula- Bnew = Anew tion on the equality of gravitational mass with frame- −∇× dependent inertial mass. Equation (43) is the relativis- ∂Anew E = φe tic generalization of Galileo’s law of Universality of Free −∇ − ∂t Fall (UFF) expressed through the non-relativistic equa- tions of motion (6) and known to be true both theoret- ically and experimentally since Galileo’s time. Now, if we introduce the energy momentum four vector: 3.2 Lorentz co-variant formulation of MG α p = (p0, p)= m0(U0, U) (46) In the Lorentz co-variant formulation, by introducing α α the space-time 4-vector x = (ct, x), proper (or rest) where p0 = E/c and U = (γc, γv) is the 4-velocity, α −1/2 mass current density 4-vector j = (ρoc, j), jα = γ = 1 v2/c2 is the Lorentz factor, then with α − (ρoc, j), Agα = (φg/c, Ag) and Ag = (φg /c, Ag); this pα we can re-write Eq. (43) as − α −
∂α (∂/c∂t, )& ∂ (∂/c∂t, ) and second-rank ≡ ∇ ≡ −∇ α anti-symmetric gravitational field strength tensor fαβ dp αβ = f pβ. (47) for MG9 dτ

gx gy gz Thus, the fields fαβ of MG couple to the energy- 0 c c c gx momentum 4-vector of all particles of whatever rest c 0 bz by fαβ = ∂αAgβ ∂βAgα = − gy −  , (40) masses they have, provided mg = m0 holds exactly. − bz 0 bx − c −  gz b b 0  It is to be noted that the equation of motion (43) holds − c − y x    only in an inertial frame. Appropriate modifications are 9Here we left out the case for HG, which the reader may try. necessary for its application in non-inertial frames, as is 10 done in non-relativistic physics by introducing pseudo- 4 HMG Form Local Gauge Invariance forces. One can verify that the equations of motion of the fields It is well known that the free Dirac Lagrangian den- of MG can be obtained using the Euler-Lagrange equa- sity10 tions of motion: = i~cψγµ∂ ψ m c2ψψ (in SI units) (54) L µ − 0 β ∂ MG ∂ MG ∂ Lβ α = L α , (48) is invariant under the transformation ∂(∂ Ag ) ∂Ag ψ eiθψ (global phase transformation) (55) where the Lagrangian density for MG is chosen as → 2 where θ is any real number. This is because under global c µν µ − MG = f fµν + j Agµ. (49) phase transformation (55) ψ e iθψ which leaves ψψ L − 16πG → in Eq. (54) unchanged as the exponential factors cancel The negative sign before the first term on the right out. But the Lagrangian density (54) is not invariant hand side of Eq. (49) is fixed by choice to fulfill our under the following transformation requirement that the corresponding free Hamiltonian (or, better, energy densities) be positive and definite. ψ eiθ(x)ψ (local phase transformation) (56) → where θ is now a function of space-time x = xµ = 3.3 Original analysis of SMTE with assumption of (ct, x), because the factor ∂µψ in (54) now picks up m = m = m / 1 v2/c2 an extra term from the derivative of θ(x): g 0 − p ∂ ψ ∂ eiθ(x)ψ = i (∂ θ) eiθψ + eiθ∂ ψ (57) In the original analysis of SMTE [48] Salisbury and µ → µ µ µ Menzel (SM) axiomatically used flat space-time and as-   2 2 so that under local phase transformation, sumed mg = m = m0/ 1 v /c for their thought − ′ experimental demonstration of gravito-magnetic field ~ µ p = c (∂µθ) ψγ ψ. (58) (they called it Gyron field) and the gravitational ana- L → L L− logue of Lorentz force law. From the analysis of their For massive particles (m0 = 0), we can re-write the 6 ′ transformed Lagrangian density in Eq. (58) as results, one can find that in the slow motion approx- L imation, if the gravito-Lorentz force law is written in ′ = ~c (∂ θ) ψγµψ the following form L L− µ ~ µ µ SM dv = ∂µ θ m0cψγ ψ = j ∂µλ(x) F = m = m g + m v b, then (50) L− m0 L− gL 0 dt 0 0 ×    (59) 8πG 1 µSM = while ǫSM = , (51) where 0g c2 0g 4πG µ µ which yields j = m0c(ψγ ψ) = 4-current momentum density,

− (60) SM SM SM 1/2 cg = µ0g µ0g = c/√2. (52) and λ(x) stands for On the other hand
if one considers cSM = c, then Eq. g ~ ~c (50) has to be written in the following form: λ(x)= θ(x)= θ(x). (61) m0 m0c dv FSM = m = m g +2m v b. (53) In terms of λ, then, gL 0 dt 0 0 × ′ = jµ∂ λ, (62) We designate this type of gravity as linearized version L → L L− µ of non-linear special relativistic MG (SRGM-N) in flat under the local transformation space-time. The origins of the non-linearity of SRGM- im0λ(x)/~ N, the appearance of the spurious value of cg = c/√2 or ψ e ψ. (63) a factor of “2” in the gravitomagnetic force term (due → Now, we demand that the complete Lagrangian be in- to a supposed value of cg = c) are all now traced to the adoption of Einstein’s doubtful postulate on the equal- variant under local phase transformations. Since, the ity of gravitational mass with the velocity dependent 10We adopt SI units in this paper for clarity to the general inertial mass. reader. 11 free Dirac Lagrangian density (54) is not locally phase Using Eqs. (71) and (72) in Euler-Lagrange Eq. (70), invariant, we are forced to add something to swallow we get the equations of motion of the new field as up or nullify the extra term in Eq. (62). Specifically, we 1 ∂βf = j . (73) suppose αβ κ α µ 2 µ = [i~cψγ ∂µψ m0c ψψ] + j Agµ, (64) Eqs. (73) express the generation of fαβ fields by the 4- L − current momentum density associated with the proper where Agµ is some new field, which changes (in coor- (or rest) mass of neutral massive Dirac particles. How- dination with the local phase transformation of ψ ac- ever, for classical fields, the 4-current momentum den- cording to the rule sity is represented by

Agµ Agµ + ∂µλ. (65) jα = (cρ , j), j = (cρ , j) (74) → 0 α 0 − This ‘new, improved’ Lagrangian is now locally invari- where j = ρ0v, with ρ0 = proper mass density. For ant. But this was ensured at the cost of introducing static mass distributions, the current density jα = j0 = a new vector field that couples to ψ through the last cρ0. It produces a time-independent - static - field, given term in Eq. (64). But the Eq. (64) is devoid of a ‘free’ by Eqs. (73): term for the field Agµ (having the dimensions of veloc- − 0 ity: [L][T ] 1) itself. Since it is a vector, we look to the 1 ∂f✚✚❃ ∂f ∂f ∂f ρ c ✚00 01 02 03 = 0 (75) Proca-type Lagrangian [57]: ✚c ∂t − ∂x − ∂y − ∂z κ 2 α κ µν mac µ where we use [∂α (∂/c∂t, )& ∂ (∂/c∂t, )]. = f fµν + κ0 Ag Agµ (66) ≡ ∇ ≡ −∇ Lfree − 4 ~ Multiplying Eq. (75) by c we get   where κ > 0 (positive) and κ0 are some dimensional ∂(cf ) ∂(cf ) ∂(cf ) ρ c2 01 + 02 + 03 = 0 . (76) constants and ma is the mass of the free field Agµ. But ∂x ∂y ∂z − κ there is a problem here, for whereas Eq. (76) gives us Newton’s gravitational field (g) as f µν = (∂µAν ∂ν Aµ) or f = (∂ A ∂ A ) expressed in the Gauss’s law of gravitostatics (5a), viz., g − g µν µ gν − ν gµ (67) ∂gx ∂gy ∂gz g = + + = 4πGρ , (77) ∇ · ∂x ∂y ∂z − 0 µ is invariant under (65), Ag Agµ is not. Evidently, the if we make the following identifications: new field must be mass-less (ma = 0), otherwise the in- variance will be lost. The negative sign before κ in Eq. g g g c2 f = x , f = y , f = z and κ = . (78) (66) is fixed by choice to fulfill our requirement that 01 c 02 c 03 c 4πG the corresponding free Hamiltonian (or, better, energy With these findings, we write Eq. (73) as densities) be positive and definite. The complete La- 4πG grangian density then becomes ∂βf = j , (79) αβ c2 α µ 2 = [i~cψγ ∂µψ m0c ψψ] + new (68) L − L which is applicable to Dirac current density (60) as where well as classical current density (74). From the anti- symmetry property of f αβ (f αβ = f βα), it follows κ µν µ − new = f fµν + j Agµ. (69) form the results (78) that L − 4 gx gy gz The equation of motion of this new field can be obtained f = , f = , f = and f =0. (80) 10 − c 20 − c 30 − c αα using the Euler-Lagrange equations of motion: The other elements of fαβ can be obtained as follows. ∂ new ∂ new ∂β L = L . (70) For α = 1, i.e. j1 = jx, Eq. (79) gives us ∂(∂βAα) ∂Aα − g g 4πG 4πG j = j A bit calculation (see for example, Jackson [57]) yields − c2 x c2 1 ✟✯ 0 0 1✟ 2 3 ∂ new = ∂ f10 +✟∂ f11 + ∂ f12 + ∂ f13 Lβ α = κfαβ, (71) ∂(∂ Ag ) 1 ∂g ∂f ∂f (81) = x 12 13 and −c2 ∂t − ∂y − ∂z 1 ∂gx ∂ 2 + ( b) (For MG) new = − c ∂t ∇× x L α = jα. (72) 1 ∂gx ∂A ( 2 ( b) (For HG) g − c ∂t − ∇× x 12 where f = b and f = b for Maxwellian Gravity (where α,β,γ are any three of the integers 0, 1, 2, 3), 12 − z 13 y (MG); f = b and f = b for Heaviside Gravity from which two homogeneous equations emerge natu- 12 z 13 − y (HG). This way, we determined all the elements of the rally: anti-symmetric ‘field strength tensor’ fαβ: b = 0 (For both MG and HG) (87) ∇ · gx gy gz 0 c c c gx ∂b  0 bz by  (For MG)  − c − (For MG) ∂t  gy −  c bz 0 bx g = (88) − −  ∇×   gz   ∂b  c by bx 0   + (For HG) − −  ∂t fαβ =   (82)  The Bianchi identity (86) may concisely be expressed  gx gy gz  0 c c c by the zero divergence of a dual field-strength tensor gx F αβ  0 bz by , viz.,  − c − (For HG)  gy  c bz 0 bx αβ − −  ∂αF = 0, (89)  gz   c by bx 0  − −  F αβ   where is defined by and the Gravito-Amp`ere-Maxwell law of MG and HG: 0 b b b − x − y − z 4πG 1 ∂g 1 b 0 g /c g /c 2 j + 2 (For MG) F αβ αβγδ x z y − c c ∂t = ǫ fγδ =  −  (90) b = (83) 2 by gz/c 0 gx/c ∇×  −  4πG 1 ∂g bz gy/c gx/c 0   + 2 j 2 (For HG)  −  c c ∂t   − For MG where b is named as gravitomagnetic field, which is and the totally anti-symmetric| fourth{z rank tensor}ǫαβγδ generated by gravitational charge (or mass) current and (known as Levi-Civita Tensor) is defined by time-varying gravitational or gravitoelectric field g. For reference, we note the field strength tensor with two +1 for α =0,β =1,γ =2,δ =3,and contravariant indices:  any even permutation ǫαβγδ = g gy g  0 x z  1 for any odd permutation − c − c − c − gx  0 bz by  0 if any two indices are equal.  c − (For MG)  gy   bz 0 bx  (91)  c −    gz   c by bx 0  αβ  −  The dual field-strength tensor F for HG can be ob- f αβ = ηαγ f ηδβ =   γδ  tained from Eq. (90) by substitution b b, with g  gx gy gz → −  0 remaining the same. − c − c − c gx  0 bz by  In terms of this 4-potentials,  c − (For HG)  gy  bz 0 bx α  c −  A = (φg/c, Ag), (92)  gz   c by bx 0   −    (84) the in-homogeneous equations (79) of MG and HG read:  4πG αβ ∂ ∂βAα ∂α(∂ Aβ) = jα. (93) From Eq. (79) and the anti-symmetry property of f , β g − β g − c2 it follows that jα is divergence-less: Under Gravito-Lorenz condition, 1 ∂(ρ c) ∂ρ α 0 0 β ∂αj =0= + j = j + . (85) ∂ A = 0, (94) c ∂t ∇ · ∇ · ∂t β g This is the continuity equation expressing the local con- the in-homogeneous Eqs. (93) simplify to the equations: servation of proper mass or (proper energy). β α  α 4πG α Equation (79) gives us two in-homogeneous equations ∂β∂ A = A = j (For MG & HG), (95) g g − c2 of MG and HG. The very definition of fαβ in Eq. (67), automatically guarantees us the Bianchi identity: where 2  α 1 ∂ 2 ∂ f + ∂ f + ∂ f =0, (86) = ∂α∂ = (96) α βγ β γδ γ αβ c2 ∂t2 − ∇ 13 is the D’Alembertian operator. In Maxwell’s theory which means that the polarization three vector (ǫ) is of electrodynamics, the equation corresponding to Eq. perpendicular to the direction of propagation. So, we (95) is say that a free graviton is transversely polarized. Since there are two linearly independent three-vectors per-  α α A = µ0je (in SI units) (97) pendicular to p; for instance, if p points in the z direc- tion, we might choose where µ0 is the permeability of vacuum and the elec- tromagnetic 4-vector potential Aα and the electric 4- ǫ(1) = ( 1, 0, 0), ǫ(2) = (0, 1, 0). (104) α − − current vector je are respectively represented by Instead of four independent solutions for a given mo- α α A = (φe/c, A), je = (cρe, je) (98) mentum, we are left with only two. A massive particle of spin s admits 2s + 1 different spin orientations, but with the symbols having their usual meanings. The a mass-less particle has only two, regardless of its spin crucial sign difference between the equations (95) and (except for s = 0, which has only one). Along its direc- (97) will explain why two like masses attract each tion of motion, it can only have m =+s or m = s; s s − other under static conditions, while two like charges its helicity, in other words, can only be +1 or 1. − repel each other under static conditions as we shall see. Thus for a graviton we write Since the fundamental field equations are the same for −(i/~)p.x (s) MG and HG, they represent the same physical thing Agα(x) = Ne ǫα (105) and any sign difference in some particular terms arise where s = 1, 2 for two spin states (polarizations). The due to particular definitions which will not change the (s) nature of physical interactions. Hence, in what follows, polarization vectors ǫα satisfy the momentum space what we call MG is to be understood as HMG. Gravito-Lorenz condition (102). They are orthogonal in the sense that

(1)∗ (2)α ǫα ǫ =0 (106) 5 The Graviton and normalized

α α∗ In Quantum Gravitodynamics (QGD), A becomes the ǫ ǫα = 1. (107) g − wave function of the graviton. Free graviton satisfies Eq. In the Newton gauge, A = 0, the polarization three- (95) with jα = 0, ∇· g vectors obey the completeness relation Aα = 0. (99) ∗ g ǫ(s)ǫ(s) = δ pˆ pˆ . (108) i j ij − i j s=1,2 If we consider the vacuum plane-wave solutions of Eq. X (99) with four momentum p = (E/c, p), then Regarding the idea of spin-2 graviton, Wald ([31], − ~ pp.76) noted that the linearized Einstein’s equations Aα(x) = Ne (i/ )p.xǫα(p), (100) g in vacuum are precisely the equations written down where N is a normalization factor and ǫα(p) is the po- by Fierz and Pauli [58], in 1939, to describe a mass- larization vector, which characterizes the spin of the less spin-2 field propagating in flat space-time. Thus, graviton. Substitution of Eq. (100) into Eq. (99), yields in the linear approximation, general relativity reduces a constraint of pα: to the theory of a massless spin-2 field which undergoes a non-linear self- interaction. It should be noted, how- pαp = 0, or E = p c (101) ever, that the notion of the mass and spin of a field α | | require the presence of a flat back ground metric ηab which is as required for a mass-less particle. which one has in the linear approximation but not in the α Now we notice that ǫ has 4-components, but they are full theory, so the statement that, in general relativity, not all independent. The Gravito-Lorenz condition Eq. gravity is treated as a mass-less spin-2 field is not one (94) demands that that can be given precise meaning outside the context of the linear approximation [31]. Even in the context lin- pαǫ = 0. (102) α ear approximations, the original idea of spin-2 graviton In the Newton gauge, A = 0 (the analogue of gets obscured due to the several faces of non-isomorphic ∇ · g Coulomb gauge), we get Gravito-Maxwell equations seen in the literature from which a unique and unambiguous prediction on the spin ǫ0 =0, ǫ p =0 (103) of graviton is difficult to get as shown in Ref. [43]. · 14

6 Attraction Between Like Masses 6.1 Lagrangian For Quantum Gravitodynamics

Let us find the static interaction between two point According to the present study, the final expression for (positive) masses at rest, following a classical approach the Lagrangian density for quantum gravitodynamics [59] within the framework of Maxwellian Gravity as fol- (QGD) of neutral massive Dirac fields interacting with lows. For a particle having gravitational charge mg = fields of Maxwellian Gravity (spin-1 gravitons) in flat m0 at rest at the origin, the 4-current densities can be space-time turns out (in SI units) as shown to be [59]: c2 0 3 = [i~cψγµ∂ ψ m c2ψψ] f f µν +jµA , j = m0cδ (x), j = 0. (109) LQGD µ − 0 −16πG µν gµ (115) In Eq. (95), we can therefore set where jµ = m c(ψγµψ) and A are the solutions of A0 = φ /c, A = 0, (110) 0 gµ g g g the Eq. (95). Note that the sign of the free-field terms µν where fµν f determine the sign of the fee Hamiltonians (or, better, energy densities) being positive and definite in 2φ =4πGm δ3(x). (111) Eq. (115). ∇ g 0 In this work, the gravitational energy density for free This is nothing but the Poisson’s equation for gravita- fields is fixed positive by choice to address the objection tional potential of a point mass at rest at origin. Us- of MTW (Sec. 7.2)[5] without any inconsistency with ing Green’s Function, the potential at a distance r for the field equations of MG. Our quantum field theoret- a central point particle having gravitational mass m0 ical derivation of MG (assuming the positive energy (i.e., the fundamental solution) is carried by freely propagating fields) corroborates all the suggested or derived linear vector gravitational Gm0 φg(r)= , (112) equations in flat space-time reviewed in sect. 2. This − r means that the field equations of MG have rooms which is equivalent to Newton’s law of universal gravita- for both positive and negative energy solutions. This tion. The interaction between two point particles hav- is because, the Lagrangian density for a particular ′ ing gravitational charges m0 and m0 separated by a system is not unique; one can always multiply by a L distance r is constant, or add a constant - or for that matter the ′ µ µ ′ Gm m divergence of an arbitrary function (∂µM , where M U = m φ = 0 0 , (113) 12 0 g − r is any function of φi and ∂µφi); such terms cancel out when we apply the Euler-Lagrange equations, which is negative for like gravitational charges and posi- so they do not affect the field equations [60]. For tive for un-like gravitational charges, if they exist. With instance, we can multiply equation (49) by 1 to ′ − m0 at rest at the origin (designated as mass 1), the force obtain another Lagrangian density = , ′ LMG −LMG on another stationary gravitational charge m0 (desig- which would imply negative energy for the fields - nated as mass 2) at a distance r from origin is probably these fields are static (non-propagating) fields ′ of gravitostatics/gravito-magnetostatics. However, ′ Gm0m0 F21 = m0 φg(r)= ˆr = F12. (114) the issue of this positive vs negative energy solutions − ∇ − r2 − and their physical interpretations/implications in the ′ This force is attractive, if m0 and m0 are of same sign context of gravitation is far from clear and is being and repulsive if they are of opposite sign - the reverse investigated by the authors. case of electrical interaction between two static electric charges. In stead of the above classical approach, one may fol- low Feynman’s [2] detailed quantum field theoretical Conclusions approach using our Eq. (95) to arrive at our above con- clusion. This is possible because of a difference in sign Following two independent approaches involving (a) the as seen in the Eqs. (95) and (97) here. Zee’s [3] path- Lorentz-invariance of physical laws under special rela- integral approach may also be used to arrive at the tivity and (b) the principle of local gauge invariance of same conclusion. The fact that a vector gravitational quantum field theory applied to a massive electrically theory can produce attractive interaction was transpar- neutral Dirac spin 1/2 particle, we rediscovered spin- ently clear to Sciama [33]. 1 Heaviside-Maxwellian Gravity, in which like masses 15 are shown to attract each other under static condi- 15. A. Singh, Experimental Tests of the Linear Equations tions, contrary to the standard view of field theorists. for the Gravitational Field, Lettere Al Nuovo Cimento, 34, We also suggested a Lagrangian density in which the 193-196 (1982). 16. W. D. Flanders and G. S. 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