Eur. Phys. J. C (2017) 77:822 https://doi.org/10.1140/epjc/s10052-017-5386-4

Comment

Comments on of Ummarino and Gallerati in “Superconductor in a weak static gravitational field” vs other versions

Harihar Beheraa Department, BIET Higher Secondary School, Govindpur, Dhenkanal 759001, Odisha, India

Received: 12 September 2017 / Accepted: 14 November 2017 / Published online: 2 December 2017 © The Author(s) 2017. This article is an open access publication

Abstract Recently reported [Eur. Phys. J. C., 77, 549 ∇·Eg =−4πGρg, (1) (2017). https://doi.org/10.1140/epjc/s10052-017-5116-y] ∇·Bg = 0, (2) gravitoelectromagnetic equations of Ummarino and Gallerati 4πG 1 ∂Eg (UG) in their linearized version of (GR) are ∇×Bg =− jg + , (3) c2 c2 ∂t shown to match with (a) our previously reported special rela- g g ∂ tivistic Maxwellian equations in the non-relativistic Bg ∇×Eg =− , (4) limit and with (b) the non-relativistic equations derived here, ∂t when the cg (an undetermined parameter of and the equation of motion of a particle moving with non- the theory here) is set equal to c (the in vac- relativistic , v, is given by Gravito- law: uum). Seen in the light of our new results, the UG equations satisfy the Correspondence Principle (cp), while many other dv = Eg + v × Bg. (5) versions of linearized GR equations that are being (or may be) dt used to interpret the experimental data defy the cp. Such new In the above equations ρg = ρ0 is the (rest) density, findings assume significance and relevance in the contexts jg = ρgv mass , the speed of gravitational of recent detection of gravitational waves and the gravito- waves in vacuum cg = c (the speed of light in vacuum) magnetic field of the spinning and their interpretations. in [1], Eg is the usual Newtonian gravitational field (called Being well-founded and self-consistent, the equations may gravitoelectric field) and Bg is the gravitational analogue of be of interest and useful to researchers exploring the phe- magnetic induction field (called gravito-magnetic field). The nomenology of gravitomagnetism, gravitational waves and Eg and Bg fields are related to gravitoelectric scalar potential the novel interplay of gravity with different states of matter φg and gravitomagnetic vector potential Ag as in flat space-time like UG’s interesting work on supercon- ∂Ag ductors in weak gravitational fields. E =−∇φ − , (6) g g ∂t Bg =∇×Ag. (7) 1 Introduction Ummarino and Gallerati [1] derived these equations from Einstein’s GR by linearization procedure in the weak field In a recent interesting theoretical study on the interplay and slow motion approximations. We name the Eqs. (1Ð7) of superconductivity and weak static gravitational field, to represent the General Relativistic Maxwellian Gravity of Ummarino and Gallerati [1] concluded that the reduction Ummarino-Gallerati as (GRMG-UG) to differentiate it from of the gravitational field in a superconductor, if it exists, is other existing (or future) formulations that (may) result from a transient phenomenon and depends strongly on the param- different methods of study. For instance, in this communica- eters that characterize the superconductor. The gravitational tion we show how one can derive these equations in a non- equations used by the authors in their study are represented relativistic approach without invoking the space-time curva- by the following Gravito-Maxwell Equations: ture and the linearization schemes of GR. By adopting, in essence, the non-relativistic approach of Schwinger et al. [2] a e-mail: [email protected] for their derivation of Maxwell’s equations and the Lorentz 123 822 Page 2 of 19 Eur. Phys. J. C (2017) 77 :822 force law in the electromagnetic case, we derive the fun- literature as new crucial tests of general relativity having no damental equations of Time-Dependent Galileo-Newtonian Galileo-Newtonian counterpart. For instance, Ciufolini et al. Gravitodynamics of moving bodies (called here as Non- [10], in their report of measurement of the Lense-Thirring Relativistic Maxwellian Gravity or NRMG in short) within effect4 stated: the Galileo-Newtonian relativity physics by combining the Newton’s law of has a formal counterpart following ingredients: in Coulomb’s law of electrostatics; however, Newton’s the- ory has no phenomenon formally analogous to . (a) The validity of Newton’s laws of gravitostatics, On the other hand, Einstein’s theory of gravitation predicts (b) The validity of the equation of continuity that expresses that the force generated by a current of electrical charge, the law of conservation of mass,1 and described by Ampère’s law, should also have a formal coun- (c) Postulating the existence of gravitational waves trav- terpart “force” generated by a current of mass. eling in free space with a finite velocity cg (which is an undetermined parameter of the theory but whose value may be fixed in measurement of physical quan- 2 Non-relativistic Maxwellian gravity (NRMG) tities involving cg or by comparing the field equations 5 with those obtainable from more advanced theories). In Galileo-Newtonian physics, gravitational mass, mg,is the source of Newtonian gravitational field, Eg, which obeys As will be shown here, our derived equations of NRMG the following two equations, viz., match with the Eqs. (1Ð7) of GRMG-UG, if c = c. Further, g ∇·E =−4πGρ and (8) if c = c, the equations of NRMG also match with those g g g ∇× = , of Special Relativistic Maxwellian Gravity (SRMG) [3]2 in Eg 0 (9) flat space-time, where c = c is a natural outcome of the g where ρg = ρ0 is the (positive) rest mass density (to be theory there (here discussed in Sect. 3). From McDonald’s shown shortly in this section and also in the next section in [4] report of little known Heaviside’s Gravity (HG) [4,5] the relativistic case) and G is Newton’s universal gravita- 3 of 1893, we find that the equations of NRMG also match tional constant having the value 6.674 × 10−11 N · m2/kg2. with those of HG in which Heaviside thought cg might be In an inertial frame, the gravitational force on a point parti- equal to c. These findings assume significance and relevance cle having gravitational mass mg in a gravitational field Eg in the contexts of recent experimental detection of gravita- is expressed as tional waves [6Ð9], gravito-magnetic field of the spinning Earth using the orbital data of two laser-ranged satellites Fg = mgEg =−mg∇φg, (10) (LAGEOS and LAGEOS II) [10Ð14] and the Gavity Probe B (GP-B) experimental results [15Ð17] vindicating Einstein’s where φg is at the position of mg. Fur- GR, because all of these results are being interpreted in the ther, the mass that appears in Newton’s law of motion, which Newton actually wrote in The Principia, is of the fol- 1 Schwinger et al. [2] derived (b) using the Galileo-Newton principle lowing form (valid only in inertial frames): of relativity ( at rest and masses with a common velocity viewed by a co-moving observer are physically indistinguishable). Here, we d will use this relativity principle for deriving the gravitational analogue (mv) = F = Net force on m, (11) of the Lorentz force law. dt 2 We named our relativistic gravity as Maxwellian Gravity since J. C. is called the inertial mass, m, which is a measure of inertia Maxwell [J.C Maxwell, Phil. Trans. Roy. Soc. London 155, 459Ð512 of a body’s resistance to any change in its state of motion (1865), sec. 82: Note on the Attraction of Gravitation] first tried to con- = struct a field theory of gravity analogous to Classical Electromagnetic represented by its velocity v or linear p mv. theory and did not work on the matter further as “He was dissatisfied If m does not change with time, then the most powerful and with his results because the potential of a static configuration is profound Eq. (11) of Newton in classical physics, whose always negative but he felt this should be re-expressible as an integral form survived special relativistic revolution, takes the famil- over field energy density which, being the square of the gravitational v<< field, is positive [3,4]. Maxwell concluded his note on the attraction of iar non-relativistic ( c)form, gravitation with the statement,“As I am unable to understand in what dv way a medium can possess such properties, I cannot go any further in m = ma = F, (12) this direction in searching for the cause of gravitation”. dt 3 Reproduced in (1.) O. Heaviside, Electromagnetic Theory, Vol. 1, 4 455Ð465. The Electrician Printing and Publishing Co., London, (1894); This effect and the precessing spin axis of on board gyroscopes of the (2.) O. Heaviside, Electromagnetic Theory, Appendix B, pp. 115Ð118. GP-B experiment etc. can also be understood in terms of gravitomag- Dover, New York, (1950)(See also the quotation in the Introduction of netic effects. this book.); (3.) O. Heaviside, Electromagnetic Theory, Vol. 1, 3rd Ed. 5 The mass that appears in Newton’s law of Gravitation is called the 455Ð466. Chelsea Publishing Company, New York, N. Y., (1971). gravitational mass of a body. 123 Eur. Phys. J. C (2017) 77 :822 Page 3 of 19 822 where a is the acceleration of m as measured in an inertial his theory. Despite his great efforts, Newton could not offer a frame. In Galileo-Newtonian physics, inertia is of two types, plausible mechanism for resolving the problem of action-at- viz., inertia of rest and inertia of motion. Accordingly, iner- a-distance and left the problem for others to resolve as evi- tial mass is of two types: rest mass (m0 = a measure of inertia dent from his 3rd letter to Bentley, dated February 5, 1692/3, of a body at rest) and inertial6 mass (m = a measure of inertia wherehewrote[18]: of a body in motion). Thus a qualitative distinction between It is inconceivable, that inanimate brute Matter should, two types of inertial masses exists in Newtonian physics, but without the Mediation of something else, which is not there is no quantitative distinction, i.e., m = m in Newto- 0 material, operate upon, and affect other Matter with- nian world of physics.7 With this understanding, if the only out mutual Contact, as it must be, if Gravitation in the force acting on the particle is the gravitational force, then the Sense of Epicurus, be essential and inherent in it. And equation of motion of a particle of rest mass m and gravi- 0 this is one Reason why I desired you would not ascribe tational mass m can be obtained from Eqs. (10) and (12)as g innate Gravity to me. That Gravity should be innate, m inherent and essential to Matter, so that one Body may dv = g . Eg (13) act upon another at a Distance thro’ a Vacuum, without dt m0 the Mediation of any thing else, by and through which Equation (13) shows us that if m = m (or ρ = ρ for g 0 g 0 their Action and Force may be conveyed from one to continuous mass distribution), then any particle of whatever another, is to me so great an Absurdity, that I believe rest mass will fall with the same acceleration in a given grav- no Man who has in philosophical Matters a compe- itational field. This is the law of universality of free fall Ð tent Faculty of thinking, can ever fall into it. Gravity known to be true since the time of Galileo. In view of these must be caused by an Agent acting constantly accord- observations, in this section term ‘mass’ is to be understood ing to certain Laws; but whether this Agent be material as the rest mass which represents the gravitational charge (or or immaterial, I have left to the Consideration of my mass) of a particle. Readers. In Newtonian physics, mass obeys a local conservation law expressed by the equation of continuity: In our attempt to resolve Newton’s action-at-a-distance prob- lem within his domain of physics, we first wish to explore ∂ a system where the Gauss’s law of gravitostatics (8) and the ∇·j0(r, t) + ρ0(r, t) = 0. (14) ∂t equation of continuity (14) work peacefully or co-exist simul- taneously but now with the restrictions in (15) removed by where j0 = ρ0(r, t)v is the rest mass current density. With the following conditions: ρg = ρ0, the three laws expressed in Eqs. (8), (9) and (14) have their individual indisputable validity in Newto- ∇· ( , ) = , nian Physics. But they are simultaneously valid only for the j0 r t 0 (16a) systems or situations where ∂ ρ (r, t) = 0, (16b) ∂t 0 ∇·j (r, t) = 0, (15a) ∂ 0 E (r, t) = 0. (16c) ∂ ∂t g ρ (r, t) = 0, (15b) ∂t 0 ∂ For this purpose, we introduce the time dependence in (8)by Eg(r, t) = 0, (15c) taking its time derivative and then write the result as ∂t remain valid (here in (15) the time dependence of the phys- ∂ρg ∂ρ0 1 ∂Eg = =− ∇· . (17) ical quantities is made explicit). The Eqs. (8Ð15) describe ∂t ∂t 4πG ∂t the familiar Galileo-Newtonian gravitodynamics. Newton’s gravitational force law (10) implies “action-at-a-distance” Ð Now using Eq. (17)inEq.(14) we obtain the gravitational force seems to act instantaneously at a dis-   1 ∂Eg tance Ð which Newton considered as an absurd element of ∇· j − = 0. (18) 0 4πG ∂t 6 Despite the fact that rest mass is also inertial mass, we are reluctant to add one more adjective to mass in motion. The quantity inside the parenthesis of (18) is a vector whose 7 However, this is not the case in relativistic world of physics. But we divergence is zero. Since ∇·(∇×X) = 0 for any vector X, are least concerned about this in the non-relativistic situation that we the vector inside the parenthesis of (18) can be expressed as want to study in this section. In the next section the equality of mg = m0 will be clearly demonstrated in a thought experiment without violating the curl of some other vector, say h. Mathematically speak- Galileo’s law of universality of free fall in the relativistic case also. ing, the Eq. (18) admits of two independent solutions: 123 822 Page 4 of 19 Eur. Phys. J. C (2017) 77 :822   ∂ ∂2 1 Eg 2 1 E ∇×h =± j0 − . (19) ∇ E − = 0, (24a) 4πG ∂t c2 ∂t2 1 ∂2B ∇2B − = 0. (24b) Except for the sign ambiguity in (19) (to be clear up soon), c2 ∂t2 we now realize that we have just arrived at (in some form) the gravitational analogue of the Ampère–Maxwell law in Before we explore analogous wave equations for the Eg and classical electrodynamics: h fields, we note that one of the solutions in (19) would cor- respond to the reality and the other might be a mathematical ∂E possibility having no or new physical significance in which ∇×H =+j +  or (20a) e 0 ∂t we are not presently interested in. We should note that Nature ∂E does not behave in two different ways at the same place and ∇×B =+μ j +  μ , (20b) 0 e 0 0 ∂t time - A physical quantity has to take a unique value at a par- ticular place and time to be real. For this reason and for our where H is called the magnetic field, B = μ0H is called the present purpose, we have to choose one of the two solutions magnetic induction field in vacuum, E represents the electric that may correspond to the reality. Which one to choose? We field, je = ρev = the current density (ρe can answer this question by following the rule of study by being electric ), 0 and μ0 respectively repre- analogy. A close look at Maxwell’s in-homogeneous Eqs. sents the electric permitivity and the magnetic permeability (22a, 22b) suggests us: of free space or vacuum and they are related to the speed of electromagnetic wave in vacuum, c , by the relation (a) that the source terms (ρe) and (je) are of the same sign (b) that the analogue of 0 in gravity is 0g: 1 c = √ (21)   0μ0 1 1  g = or  g =− (25) 0 4πG 0 4πG and c is a universal constant of nature. In fact, there are four field equations in electrodynamics which are collectively which we may call the gravito-electric permitivity of vac- known as Maxwell’s equations: uum or whatever better name one may assign to it, and

ρ (c) to introduce a new constant μ as ∇·E =+ e (22a) 0g 0   1 ∂E 4πG 4πG ∇×B =+μ j + (22b) μ = or μ =− (26) 0 e 2 ∂ 0g 2 0g 2 c t cg cg ∇·B = 0 (22c) ∂B which is the gravitational analogue of μ0 and cg is a new ∇×E =− (22d) ∂t universal constant for vacuum having the dimension of velocity (which will turn out as the speed of gravitational Maxwell’s Eqs. (22aÐ22d) form the basis of all classi- waves in vacuum, if they exist) such that we get a relation cal electromagnetic phenomena including the production an analogous to (21)as and transmission of electromagnetic waves carrying energy 1 and momentum even in vacuum. When combined with the c = √ . g  μ (27) Lorentz force equation, 0g 0g

Accepting the suggestion (a), we are naturally led to choose FL = qE + qv × B (23) the solution (19) having a negative sign before j0 as the real solution in view of Eq. (8), where the source term ρ has and Newton’s 2nd law of motion (11), these equations pro- 0 a negative sign before it. Now, we multiply8 μ with our vide a complete description of classical dynamics of interact- 0g chosen Eq. (19) to get ing charged particles and electromagnetic fields. Do analo- gous phenomena occur in gravitational physics? For instance, 1 ∂Eg ∇×B =∇×(μ h) =−μ j + , (28) Maxwell’s equations predict the existence of electromagnetic g 0g 0g 0 2 ∂ cg t waves that travel through vacuum (where ρe = 0 and je = 0) at a universal speed c = 3 × 108 m/s and the wave equations 8 We can multiply a positive scalar quantity with a vector equation for the fields E and B in vacuum are obtainable from (22aÐ without altering the physical content of that equation as we only re- 22d)as scaling the those vectors in some new units. 123 Eur. Phys. J. C (2017) 77 :822 Page 5 of 19 822

μ = π / 2 = where we considered ( 0g 4 G cg) and defined Bg uniform motion with a common velocity v with respect to S- μ0gh as the gravitomagnetic induction field. Now taking the frame. Here we use the Galileo-Newton curl of (28) and using the vector identity ∇×(∇×B) = (masses at rest and masses with a common velocity viewed ∇(∇·B) −∇2B, we get by a co-moving observer are physically indistinguishable) and insist that physical laws are the same in the two inertial 1 ∂ | | << = ∇(∇·B ) −∇2B =−μ ∇×j + (∇×E ). (29) frames. Further, we assume that v cg c. To catch g g 0g 0 2 ∂ g cg t up with the moving masses one would have to move with velocity v. Accordingly, the time derivative in the co-moving = In vacuum (where j0 0)Eq.(29) will reduce to a wave system, in which the masses are at rest, is the sum of explicit equation for the Bg field, viz., time dependent and co-ordinate dependent contributions,

2 1 ∂ Bg d ∂ ∇2B − = 0, (30) = + ·∇ g 2 ∂ 2 v (36) cg t dt ∂t provided the following two conditions: so, in going from static system to uniformly moving system, we make the replacement ∇·Bg = 0 and (31) ∂Bg ∂ d ∂ ∇×Eg =− (32) −→ = + v ·∇. (37) ∂t ∂t dt ∂t are fulfilled. Now the taking curl of the Eq. (32) and using (28) The equation for constancy of gravitational field in Eq. (15c) we get the following equations for vacuum (where j0 = 0) becomes, in the moving system 2 ∂ 1 ∂ Eg ∇(∇·E ) −∇2E =− ∇×B =− . (33) ∂ ∂ g g ∂ g 2 ∂ 2 Eg dEg Eg t cg t 0 = −→ 0 = = + (v ·∇)E . (38) ∂t dt ∂t g Since ρ = 0 =∇·E in vacuum, this equation yields a 0 g The vector identity for constant v is wave equation for the Eg field:

2 ∇×(v × Eg) = v(∇·Eg) − (v ·∇)Eg. (39) 1 ∂ Eg ∇2E = . (34) g c2 ∂t2 g Now using Gauss’s law of gravitostatics (35a) in the above Thus, we notice that we have arrived at identity we get producing Gravito-Maxwell equations representing what call (ρ0v) here as ∇×(v × Eg) =− − (v ·∇)Eg 0g (40) Non-relativistic Maxwellian gravity (NRMG): j0 =− − (v ·∇)Eg. 0g ∇·Eg =−ρ0/0g (35a) 1 ∂Eg Again, from Eqs. (38) and (40), we get ∇×B =−μ j + (35b) g 0g 0 c2 ∂t g ∂E ∇×( × ) =−j0 + g . ∇·Bg = 0 (35c) v Eg (41) 0g ∂t ∂Bg ∇×Eg =− (35d) ∂t −2 Multiplication of equation (41)bycg gives us To complete the dynamical picture, we explore the gravita-   tional the analogue of Lorentz force law (23) below adopting v × Eg 1 ∂Eg ∇× =−μ j + . (42) 2 0g 0 2 ∂ the logic and methods of Schwinger et al. [2] in their deriva- cg cg t tion of the Lorentz force law.  Consider two inertial frames S and S having a relative Equation (42) will agree with the Eq. (35b) of NRMG, only velocity v between them. Let all the masses are in static when arrangement in one of these frames, say S, which is moving with velocity v. This way we introduce the time dependence v × Eg B = . (43) ρ g 2 of and Eg in the simplest way by assuming all masses are in cg 123 822 Page 6 of 19 Eur. Phys. J. C (2017) 77 :822

Now consider the vector identity for constant v particle of mass m0, when that particle moves with some non-relativistic velocity v in given Eg and Bg field? For this ∇×(v × Bg) = v(∇·Bg) − (v ·∇)Bg =−(v ·∇)Bg (44) purpose, consider two coordinate systems, one in which the particle is at rest (co-moving coordinate system) and one in wherewehaveused∇·Bg = 0. Moreover, in the co-moving which it moves at velocity v. Suppose, in the later coordinate system where the masses are at rest Ð static Ð the Bg field system, the gravitational (or gravito-electric) and gravito- should also not change with time: magnetic fields are given by Eg and Bg, respectively. In the co-moving frame, the force on the particle is dBg ∂Bg = + (v ·∇)B = 0. (45) dt ∂t g = eff, Fg m0Eg (53) From (44)and (45) we get eff where Eg is the gravitational (or gravito-electric) field in ∂Bg this frame. In transforming to the co-moving frame, all the =∇×(v × Bg). (46) ∂t other masses Ð those responsible for Eg and Bg Ð have been given an additional counter velocity −v.FromEq.(50), we Again, the vector identity then infer that (+v × Bg) has the character of an additional gravito-electric field in the co-moving frame. Hence, the sug- ∇2E = E (∇·E ) −∇×(∇×E ) (47) eff g g g g gested Eg is gives us the left hand side of the wave Eq. (34)forEg in eff = + × , Eg Eg v Bg (54) vacuum (i.e. outside the mass distribution where ∇·Eg = 0) as leading to the gravitational analogue of the Lorentz force law 2 that we call GravitoÐLorentz force law: ∇ Eg =−∇×(∇×Eg). (48)  F = m0 Eg + v × Bg . (55) By means of the Eq. (35b) of NRMG and Eq. (46), the right gL = side of the wave Eq. (34) becomes (j0 0 outside the charge This gravito-Lorentz force law when used in Newton’s 2nd distribution) law of motion, 2 1 ∂ Eg ∂ dv =+ (∇×Bg) m = F , (56) c2 ∂t2 ∂t (49) 0 gL g   dt =+∇× ∇×(v × B ) . g we get the following equation of motion of NRMG

Equations (48, 49) show that the wave equation for Eg (34) dv = E + v × B . (57) will hold if dt g g

Eg =−v × Bg. (50) Since ∇·Bg = 0, Bg can be defined as the curl of some vector function, say Ag: But this cannot be completely correct since as v → 0 ⇒ Eg → 0. No gravitostatics ! However, all that is necessary is Bg =∇×Ag, (58) that the curl of this Eg in (50) should be valid: where Ag is the vector potential for NRMG. Using Eq. (58) ∇×Eg =−∇×(v × Bg) (51) in the GravitoÐFaraday law of NRMG (35d), we get the fol- lowing expression for the Eg in terms of φg and Ag as or, if we use Eq. (46), ∂Ag Eg =−∇φg − . (59) ∂Bg ∂t ∇×E =− . (52) g ∂t We now recognize that the equations of NRMG correspond to This is consistent with gravitostatics since it generalizes ∇× the Eqs. (1Ð7) of GRMG-UG provided we set cg = c.Inthis Eg = 0 to time-dependent situation. derivation, cg is an undetermined parameter of the theory, Now we are ready to address the question: What replaces whose value may be obtained from some more advanced the equation F = m0Eg to describe the force on a point theory or from experiments. 123 Eur. Phys. J. C (2017) 77 :822 Page 7 of 19 822

Substituting the expression for Eg given by Eq. (59) and traveling at a finite speed. This is what Heaviside had done the expression for Bg defined by (58) in the in-homogeneous in 1893 [4,5]. May be Einstein had not seen Heaviside’s field field Eqs. (35a, 35b) of NRMG, we get the following expres- equations when he was working on his relativistic theory of sions for the in-homogeneous Eqs. (35a, 35b) in terms of gravity. Had Einstein seen Heaviside’s field equations, his potentials (φg, Ag) as remark on Newton’s theory of gravity would have been dif- 2 ferent than what he made before the 1913 congress of natural 1 ∂ φg ρ ∇2φ − = 0 , (60) scientists in Vienna [21],viz., g 2 ∂ 2  cg t 0g 2 After the un-tenability of the theory of action at distance 1 ∂ Ag ∇2A − = μ j , (61) had thus been proved in the domain of electrodynamics, g 2 ∂ 2 0g 0 cg t confidence in the correctness of Newton’s action-at-a- if the following gravitational Lorenz gauge condition, distance theory of gravitation was shaken. One had to believe that Newton’s law of gravity could not embrace 1 ∂φg the phenomena of gravity in their entirety, any more ∇·A + = 0, (62) g 2 ∂ cg t than Coulomb’s law of electrostatics embraced the the- ory of electromagnetic processes. is imposed. These will determine the generation of gravi- tational waves by prescribed gravitational mass and mass current distributions. Particular solutions of (60) and (61)in 3 Special relativistic Maxwellian gravity (SRMG) vacuum are   1 ρ0(r , t )  With the establishment of (SR) theory and φg(r, t) =− dv and (63) 4π |r − r| the equivalence of mass and energy, the meaning of the 0 g μ j (r, t) inertial mass and gravitational mass became ambiguous, A (r, t) =− 0g 0 dv, g π | − | (64) because SR suggests two inertial mass-energy concepts: (1) 4 r r 2 the Lorentz invariant rest-mass m0 = E0/c (E0 =rest-    where t = t−|r−r |/cg is the retarded time and dv is an ele- energy, which is the sum total of all forms of energy in the  mentary volume element at r . Thus, we saw that retardation rest frame of a body or particle) and (2) the mass attributed to in gravity is possible in Newtonian space and time in the same the relativistic energy m = E/c2 (E = sum of all forms of procedure as we adopt in electrodynamics. Hence, we have energy at rest and motion) which is not Lorentz-invariant. The reasons to strongly disagree with Rohrlich’s conclusion [19]: qualitative distinction that existed between two inertial mass “Because the Newtonian theory is entirely static, retardation concepts in Newtonian mechanics became quantitatively dis- is not possible until the correction due to deviations from tinct and clear in SR. Now, one fundamental question arises, is considered”. According to the present “What form of mass (or energy) should be the source of grav- field theoretical view, gravitation, like and ity (mg) in a relativistic version of Newtonian gravity?” In all other fundamental interactions, acts locally through fields: any construction of a field theory of gravity compatible with A mass at one point produces a field, and this field acts SR and the correspondence principle by which a relativistic on whatever masses with which it comes into contact [20]. theory gravity is reducible to Newtonian gravity, a decision Because of the finite propagation speed of the fields, gravita- on which form of “mass” (or energy) is the source of gravity tional effects/information propagate at finite speed, that is the has to be taken. Such a decision, as Price [22] has rightly cause behind retarded interaction. However, in case of static pointed out, will be crucial not only to the resolution of the or quasi-static mass distributions, retardation effects are neg- ambiguity mentioned above but also to the issue of the non- ligible and hence no distinction can be made between local linear nature of gravity. One of the Eddington’s [23] four interaction and action-at-a-distance. By introducing the con- reasons to feel dissatisfied with Newton’s Law of gravitation cept of physical fields carring energy and momentum,9 one is appropriate here to quote: can address Newton’s “action-at-a-distance” problem within Newton’s world of physics by extending his field equations The most serious objection against the Newtonian Law to time-dependent fields, sources and searching for the con- as an exact law was that it had become ambiguous. ditions for the existence of gravitational waves in free space The law refers to the product of the masses of the two bodies; but the mass depends on the velocity- a fact unknown in Newton’s days. Are we to take the variable mass, or the mass reduced to rest? Perhaps a learned judge, interpreting Newton’s statement like a last will 9 is the material Newton was serching for: the field is material and testament, could give a decision; but that is scarcely because it possesses an energy density [20]. the way to settle an important point in scientific theory. 123 822 Page 8 of 19 Eur. Phys. J. C (2017) 77 :822

In his construction of a relativistic theory of gravity popularly that we may assume its general validity until proved known as General Relativity (GR), Einstein has taken a deci- otherwise. sion in favor of the equality of m with m . For a theoretical g The last three words of Einstein’s above statement,‘until justification of this decision, Einstein by writing Newton’s proved otherwise’, show that he was very cautious and not equation of motion in a gravitational field (in our present very confident of what he was stating. Based on the exper- mathematical notation) as imental results available up to 1993, Mashhoon [30] noted that the observational evidence for the principle of equiva- dv m = m E (65) lence of gravitational and inertial masses is not yet precise dt g g enough to reflect the wave nature of matter and radiation in their interactions with gravity (see other references on equiv- (wrongly!) inferred from it [24]: alence principle in [3,30]). It is only when there is numerical equality between the inertial and gravitational mass that the acceleration is 3.1 Re-examination of SMTE to show m0 = mg independent of the nature of the body. This inference is often expressed in one of the two ways: Consider a system of two non-spinning point-like charged particles with charges q1 and q2 with respective rest masses m (= E /c2) and m (= E /c2) such that they are (S1) that the particle’s motion is mass independent, or 01 01 02 02 at rest in an inertial frame S under equilibrium condition (S2) that the particle’s inertial mass m = its gravitational due to a mutual balance of the force of Coulombic repul- mass mg.   sion (FC ) and the Newtonian gravitostatic attraction (FN ) between them. Our aim is to investigate the condition of equi- The two statements (S1) and (S2) are sometimes used inter- librium of this two-particle system (realizable in a Laboratory changeably as the weak (WEP) in the by taking two perfectly identical spherical metallic spheres literature [25Ð27]. This use of terminology is rather con- having requisite masses and charges so that they are in equi- fusing, as the two statements are logically independent [28]. librium) and in different inertial frames in relative motion. They happen to coincide in the context of Galileo-Newtonian For our re-examination purpose, suppose that the particles = = physics where m0 m mg but may diverge in the context are positively charged and they are in empty space. Let the =   of special relativity where m m0 and Einstein’s wrong particle No. 2 be positioned at the origin of S -frame and r = = inference of m0 m mg from a non-relativistic Eq. (65), be the position vector of the particle No. 1 with respect to the =  which is exactly the Eq. (13)ofNRMG,wherem m0 and particle No. 2. In this S -frame the condition of equilibrium = mg m0 is a condition for Galileo’s law of Universality of is fulfilled by Free Fall to be true. To explore this possibility, to get new insights for making Newtonian gravity compatible with the     q1q2r Gm01m02r SR, to regard old problems from a new angle, we re-examined F + F = − = 0 (66) C N 4π r 3 r 3 [3] an often cited [29,30] Salisbury-Menzel’s [31]10 thought 0 experiment (SMTE) from a new perspective as discussed in where r  =|r| and other symbols have their usual mean- the following subsection. Before that the author would like ings. From (66) we get to remark that perhaps Einstein, himself, was not satisfied with his above inference of mg = m, as we can sense from q1q2 = = m01m02 ( = / π ) his another statement on the equality of mg with m [32,33]: Gm01m02 0g 1 4 G (67) 4π0 4π0g The proportionality between the inertial and gravita- tional masses holds for all bodies without exception, Equation (67) represents the condition of equilibrium, in with the (experimental) accuracy achieved thus far, so terms of the charges and rest masses (or rest ) of the particles, under which an equilibrium can be ensured in  10 The correct special relativistic result is obtained by P. Lorrain, Grav- the S -frame. For example, if each metallic sphere is given −6 itational analogue of the magnetic force, Nature, 257 161Ð162 (1975). a charge of 1 × 10 Coulomb, then the rest mass of each See the response to P. Lorrain by D. H. Menzel, W. W. Salisbury, Nature sphere should be 1.162 × 104 kg, to fulfill the equilibrium 257 162 (1975). Present author is not satisfied with this response because condition (67) in a laboratory experiment. = = / − v2/ 2 Menzel and Salisbury did not prove mg m m0 1 c , Now, let us investigate the problem of equilibrium of the = / − v2/ 2 although they said m m0 1 c has the experimental valid- said particle system from the point of view of an observer ity. A general relativistic analysis of SMTE is done by A. R. Khan, R. F. O’Connell, Gravitational analogue of magnetic force, Nature, 261 in another inertial frame S, in uniform relative motion with  480Ð481 (1976). respect to the S -frame. To simplify the investigation, let 123 Eur. Phys. J. C (2017) 77 :822 Page 9 of 19 822   2 the relative velocity v of S and S -frame be along a com- Gm01m02 1 − β r F =−  /  gL 3/2 mon X X -axis with corresponding planes parallel as usual. r 3 1 − β2 sin2 θ   Since the particles are at rest in S -frame, both of them have 2 G m01m02v × (v × r) 1 − β the same uniform velocity v relative to the S-frame. Let the −  2 3/2 position vector of the particle No. 1 with respect to the par- c r 3 1 − β2 sin2 θ  ticle No. 2 as observed in the S-frame be r and the angle 2 1 m01m02 1 − β r θ =−  between v and r be . π 3/2 4 0g r 3 1 − β2 sin2 θ For an observer in the S-frame, the force of electric origin  2 on either particle (say on particle No. 1 due to particle No. μ m01m02v × (v × r) 1 − β − 0g  , 3/2 (74) 2) is no more simply a Coulomb force, but a Lorentz force, 4π r 3 1 − β2 sin2 θ viz., where FL = q1E2 + q1v × B2 (68) 1 4πG 1  = ,μ = ⇒ c = √ . 0g 0g 2 (75) where 4πG c 0gμ0g

( − β2) By comparing the quantities in Eq. (75) with those in Eqs. = q2 1 r ,(β = / ) E2  / v c (69) 3 2 2 3 2 (25Ð27), we immediately find that 4π0r 1 − β sin θ × ( ) × ( − β2) = v E2 = q2v r 1 1 B2  / cg = √ = c. (76) c2 2 3 2 2 3 2  μ 4π0c r 1 − β sin θ 0g 0g μ ( ) × ( − β2) = 0 q2v r 1  / (70) Now, (74) may be rearranged to the following form to repre- 4π 3 2 2 3 2 r 1 − β sin θ sent the GravitoÐLorentz force law of SRMG:  1/2 r 1 − β2 sin2 θ = . r  / (71) FgL = m01Eg2 + m01v × Bg2 (77) 1 − β2 1 2 where What about the force of gravitational interaction as observed in the S-frame? It can not simply be a Newtonian force but 1 m (1 − β2)r E =−  02 something else, otherwise the particle system will not remain g2 π 3/2 4 0g r 3 1 − β2 sin2 θ in equilibrium in the S-frame. Such a situation will amount to a violation of the principle of relativity in special relativity. − 1 m02r ( β<< ), 3 when 1 (78) A null force should remain null in all inertial frames. There- 4π0g r fore, a new force law of gravity has to be invoked so that v × E μ (m v) × r (1 − β2) B = g2 =− 0g 02 the equilibrium is maintained in accordance with the prin- g2 2 3/2 c 4π r 3 1 − β2 sin2 θ ciple of relativity (Lorentz invariance of physical laws). Let μ0g (m02v) × r this new unknown force be represented by FgL such that the − (when β<<1). (79) π 3 equilibrium condition in S-frame is satisfied as: 4 r Equations (77Ð79) are in complete formal analogy with + = ⇒ =− . FgL FL 0 FgL FL (72) the Eqs. (68Ð70) of classical electromagnetism in its rela- tivistic version. Thus, from the requirement of the frame- Taking into account the Eqs, (68)Ð(71), FgL in Eq. (72) can independence of the equilibrium condition, we not only be expressed as: obtained a gravitational analogue of the Lorentz-force law  expressed by Eq. (77) but also unexpectedly found the 2 q1q2 1 − β r Lorentz-invariant rest mass as the gravitational analogue =−  FgL 3/2 4π r 3 1 − β2 sin2 θ of the electric charge by electromagnetic analogy. From 0  2 this analysis, the gravitational charge (or rest mass) invari- μ q1q2v × (v × r) 1 − β − 0  . (73) ance may be interpreted as a consequence of the Lorentz- π 3/2 4 r 3 1 − β2 sin2 θ invariance of the physical laws. These findings are in confor- mity with Poinca`re’s [34] remark that if equilibrium is to be Now, using Eq. (67), we can eliminate q1q2 from Eq. (73)to a frame-independent condition, it is necessary for all forces get the expression for FgL in terms of m01, m02 and G as: of non-electromagnetic origin to have precisely the same 123 822 Page 10 of 19 Eur. Phys. J. C (2017) 77 :822 transformation law as that of the Lorentz-force.Now,fol- The relativistic equation of motion (83) is independent of lowing Rosser’s [35] approach to classical electromagnetism the mass of the particle moving in an external gravito- via relativity one can obtain the field equations of SRMG as electromagnetic (GEM) field f αβ . Thus we saw that the represented in Eqs. (35aÐ35d) with cg = c,fromtheEqs. motion of a particle in an external GEM field can be inde- (77Ð79). Alternatively, after recognizing our new findings pendent of its mass without any postulation on the equality from the above thought experiment, especially mg = m0 of gravitational mass with frame-dependent inertial mass. and cg = c, one may follow the procedure we followed in Equation (83) is the relativistic generalization of Galileo’s NRMG to arrive at desired gravitational wave producing field law of Universality of Free Fall (UFF) expressed through equations. It is to be noted that the Lorrain’s (See Lorrain in the non-relativistic equations of motion (13) and (57) and footnote 10 of this paper) exact special relativistic derivation known to be true both theoretically and experimentally since of gravitational analogue of the magnetic force from SMTE Galileo’s time. matches with our SRMG results. Now, if we introduce the energy momentum four vector:

α 3.2 Lorentz co-variant formulation of SRMG p = (p0, p) = m(U0, U) (86)

α α In Lorentz co-variant formulation, by introducing the space- where p0 = E/c and U is the 4-velocity, then with this p time 4-vector xα = (ct, x), proper (or rest) mass current den- we can re-write equation (83)as α = (ρ , ) sity 4-vector j oc j0 and second-rank anti-symmetric α dp αβ gravitational field strength tensor fαβ : = f pβ (87) dτ ⎛ ⎞ / / / 0 Egx cEgy cEgz c fαβ ⎜ ⎟ Thus, in SRMG the fields couple to the energy- ⎜−Egx/c 0 −Bgz Bgy ⎟ fαβ = ∂α Aβ − ∂β Aα = , momentum 4-vector of all particles of whatever rest masses ⎝−E /cB 0 −B ⎠ gy gz gx they have, provided mg = m0 holds exactly. It is to be noted − / − Egz c Bgy Bgx 0 that the equation of motion (83 or 87) holds only in an inertial (80) frame. Appropriate modifications are necessary for its appli- cation in non-inertial frames, as is done in non-relativistic one can rewrite the field equations of SRMG as: physics by introducing pseudo-forces.  β β 4πG ∂ fαβ = ∂ ∂α Aβ − ∂β Aα = jα = μ jα, (81) c2 0g 3.3 Original analysis of SMTE with assumption of 2 2 ∂α fβγ + ∂β fγδ + ∂γ fαβ = 0, (82) mg = m = m0/ 1 − v /c where α, β, γ are any three of the integers 0, 1, 2, 3; α In the original analysis of SMTE [31] Salisbury and Men- jα = (ρ c, −j ) and j = (ρ c, j ); Gravito-Lorenz con- o 0 o 0 zel (SM) axiomatically used flat space-time and assumed ∂α = = (φ , − ) α = (φ , ) dition: Aα 0; Aα g Ag and A g Ag 2 2 mg = m = m0/ 1 − v /c for their thought experimen- with φ = scalar potential and A = vector potential; ∂α ≡ g g tal demonstration of gravito-magnetic field (they called it (∂/c∂t, ∇) & ∂α ≡ (∂/c∂t, −∇). In this convention, the rel- Gyron field) and the gravitational analogue of Lorentz force ativistic gravito-Lorentz force law takes the following form law. From the analysis of their results, one can find that in 2 α the slow motion approximation, if the gravito-Lorentz force d x αβ dxβ = f (83) law is written in the following form dτ 2 dτ where τ is the proper time along the particle’s world-line and SM = dv = + × , FgL m0 m0Eg m0v Bg then (88) f αβ is given by dt 8πG 1 ⎛ ⎞ μSM = while SM = , (89) 0g 2 0g π 0 −Egx/c −Egy/c −Egz/c c 4 G ⎜ ⎟ αβ αγ δβ ⎜Egx/c 0 −Bgz Bgy ⎟ which yields f = η fγδη = ⎝ ⎠ Egy/cBgz 0 −Bgx   −1/2 √ Egz/c −Bgy Bgx 0 SM = μSMμSM = / . cg 0g 0g c 2 (90) (84) SM = αβ On the other hand if one considers cg c, then equation where the flat space-time ηαβ = η is repre- (88) has to be written in the following form: sented by symmetric diagonal matrix with dv η = ,η = η = η =− . FSM = m = m E + 2m v × B . (91) 00 1 11 22 33 1 (85) gL 0 dt 0 g 0 g 123 Eur. Phys. J. C (2017) 77 :822 Page 11 of 19 822

We designate this type of gravity as linearized version of non- assumed cg = c by electromagnetic analogy [39]. In 1980, linear SRMG (SRGM-N) in flat space-time. The origins of Cattani [40] considered linear equations for the gravitational the non-linearity of (SRGM-N),√ the appearance of the spuri- field by introducing a new field by calling it the Heavisidian ous value of cg = c/ 2 or a factor of “2” in the gravitomag- field which depends on the of gravitational charges netic force term (due to a supposed value of cg = c)areall in the same way as a magnetic field depends on the veloc- now traced to the adoption of Einstein’s doubtful postulate ities of electric charges and shown that a gravitational field on the equality of gravitational mass with the velocity depen- may be written with linear co-variant equations in the same dent inertial mass. As such, SRMG-N does not correspond way as for the electromagnetic field. Cattani’s equations dif- either to the NRMG when cg = c or to the non-relativistic fer from some important formulae of general relativity such limit of SRMG. as the gravitational radiation, Coriolis force by a factor of 4. In 1982, Singh [41] considered a vector gravitational the- ory having formal symmetry with the electromagnetic theory 4 Discussions and explained the (a) precession of the perihelion of a planet (b) bending of light in the gravitational field and (c) gravi- Several authors have suggested or obtained different Maxwell- tational red-shift by postulating the self-interaction between type equations for gravity following different approaches. a particle velocity and its vector potential. In 2004, Flanders Some without using the formalism of GR [36Ð50] and some and Japaridze [42] axiomatically used the field equations of using the formalism of GR in the weak field and linearized SRMG (albeit without reference to [3]) and special relativ- approximations. These are discussed in two separate sections ity to explain the deflection and perihelion advance below. The two subsequent sections concern discussions on of Mercury in the gravitational field of the . Borodikhin Spin-1 Vector Gravity vs Spin-2 Tensor Gravity and remarks [43] explained the perihelion advance of Mercury, gravita- on little known Heaviside’s work on gravity. tional deflection of light as well as by postulating a vector theory of gravity in flat space-time that 4.1 Maxwellian Gravity of others without GR is nothing but SRMG. Borodikhin also showed that in a vec- tor theory of gravity, there exists a model for an expanding Sciama[36], in 1953, hypothetically adopted SRMG (assum- Universe. Jefimenko [44,45] also deduced the equations of ing mg = m0) to explain the origin of inertia, calling it a NRMG by extending Newton’s gravitational theory to time- theory of gravity which differs from GR princi- dependent sources and fields and using the causality prin- pally in three respects: (a) It enables the amount of mat- ciple. Jefimenko assumed cg = c and postulated a gravito- ter in the universe to be estimated from a knowledge of the Lorentz force. Recently, Heras [46], by recognizing the gen- , (b) The principle of equivalence is eral validity of the axiomatic approach to Maxwell’s equa- a consequence of the theory, not an initial axiom and (c) tions of electromagnetic theory, used those axioms to derive It implies that gravitation must be attractive. However, he only the field equations (leaving out gravito-Lorentz force concluded his paper mentioning three limitations of such a law) of SRMG, where the in-variance of gravitational charge theory: (i) It is incomplete because the relativistic form of is considered. Other recent derivations of SRMG equations Newton’s law must be derived from a tensor potential,11 not from different approaches include the works of Nyambuya from a vector potential, (ii) It is difficult to give a consis- [47], Sattinger [48], Vieira and Brentan [49]. The historical tent relativistic discussion of the structure of the universe objections of several researchers, starting from Maxwell (see as a whole12 and (iii) It is also difficult to describe the footnote 2 of this paper) upto Misner, Thorne and Wheeler motion of light in a gravitational field.13 Carstoiu [37Ð39], in (MTW, Sect. 7.2) [54], concerning negative energy density 1969, rediscovered Heaviside’s gravitational Eqs. (35aÐ35d) of gravitational field (‘Maxwell’s Enigma’ as Sattinger puts (in our present notation as per the report of Brilloiun [39]) it) in a linear Lorentz invariant field theory of gravity are assuming the existence of a second gravitational field called also refuted by Sattinger [48], who considered negative field gravitational vortex (here called gravito-magnetic field) and energy density for SRMG in agreement with our result [3]. In the discussion on the Dark Matter problem, Sattinger fur- 11 2 This thought comes to anyone who believes in mg = E/c ,where ther noted: “The Maxwell-Heaviside equations of gravitation E is the relativistic energy, which may not be true as per our findings constitute a linear, relativistic correction to Newton’s equa- discussed here. tions of motion; they interpolate between Newton’s and Ein- 12 An interesting topic of research not yet fully explored. As matter of stein’s theories of gravitation, and are therefore a natural scientific curiosity, one may explore the Universe from the new per- mathematical model on which to build a dynamical theory of spective of a vector gravitational theory. galactic structures”. Our unique, important and new findings 13 It is a deep question involving the interaction of two fundamental fields which is beyond the scope of this paper but needs further inves- reported and discussed here, again confirmed by our very tigation. recent work on “Attractive Heaviside-Maxwellian (vector) 123 822 Page 12 of 19 Eur. Phys. J. C (2017) 77 :822

Gravity from Quantum Field Theory” [50] (where gravita- Bg = ∇ × Ag (94) tional energy density for free fields is fixed positive by choice 3 ∂φg ∇ · A + = 0 (For Lorenz-type gauge) to address the objection of MTW (Sect. 7.2) [54] without g c ∂t any inconsistency with the field Eqs. (81Ð83)ofSRMG), (95) corroborate all the above suggested or derived linear vector gravitational equations in flat space-time which are seen to where the number 3 in the Lorenz-type gauge above is the satisfy the correspondence principle (cp) in its correct sense: GR value for some PPN parameters used in [52]. For the | | < 5 << Newtonian Gravity ⇔ NRMG ⇔ SRMG. This means that source and particle of velocities v0 10 cm/sec c, Braginsky et al. [52] approximated the gravitational force the field equations of SRMG have room for both positive and 15 negative energy solutions Ð a discussion of which is left out (with a typographical error in eqn. (3.10) , p.2054, [52], here. corrected here) on a unit mass,   F 1 v2 1  4.2 Maxwellian gravity From GR (GRMG) = + ( γ + ) 0 E + v × H , 1 2 1 2 g 0 g (96) m0 2 c c Different MaxwellÐLorentz-type equations for gravity obtained from GR by different researchers following different lin- where the PPN parameter γ 1 in GR. In empty space earization procedures are not isomorphic [51] as seen below: (ρ0 = 0) with no radiation pressure (p = 0), if we consider some contain non-linear terms and do not satisfy the cp Coulomb-Newton Gauge (∇ · Ag = 0), the field Eqs. (92a, from the perspective of our present new NRGM results sub- 92d) reduce to the following equations stantiated by the cp-respecting SRMG results. Some sam- ∇· = , ples of this type of General Relativistic Maxwellian Gravity Eg 0 (97a) (GRMG) are listed below for discussion. 4 ∂Eg ∇×H =+ , (97b) g c ∂t 4.2.1 GRMG of Braginsky et al. and Forward (GRMG-BF) ∇·Hg = 0, (97c) 1 ∂Hg Braginsky et al. [52], following Forward [53] and Misner, ∇×Eg =− . (97d) c ∂t Thorne and Wheeler [54], reported the following Maxwell- type equations of GR in their parametrized-post-Newtonian Now taking the curl of (97d) and utilizing Eqs. (97a) and 14 (PPN) formalism as : (97b), we get the wave equation for the field Eg in empty   space as v2  3p ∇·E =− πGρ + + + g 4 0 1 2 2 2 2 2 2 c c ρ0c 4 ∂ E 1 ∂ E ∇2E − g = ∇2E − g = 0, 2 g 2 2 g 2 2 (98) 3 ∂ φg c ∂t c ∂t + (92a) g c2 ∂t2 = / 16πG 4 ∂Eg where cg c 2. Similarly, the wave equation for the field ∇×H =− (ρ v) + (92b) g c 0 c ∂t Hg can be obtained by taking the curl of Eq. (97b) and uti- lizing Eqs. (97c) and (97d): ∇·Hg = 0 (92c) ∂ 1 Hg 2 2 ∇×E =− (92d) 4 ∂ Hg 1 ∂ Hg g ∂ ∇2H − = ∇2H − = 0, (99) c t g 2 ∂ 2 g 2 ∂ 2 c t cg t where we have put the values of PPN parameters as appro- = / priate for GR, ρ0 is the density of rest mass in the local rest where again we find cg c 2. This result is against the spe- frame of the matter, v is the ordinary (co-ordinate velocity) cial relativistic (as well as the gauge field theoretic) expecta- velocity of the rest mass relative to the PPN frame,  is the tion that the speed of gravitational waves (if they exist) should specific internal energy (energy per unit rest mass) and p is be equal to the speed of light in any Lorentz-covariant field the radiation pressure and φg is the electric-type scalar poten- theory of gravity and has escaped the attention of the authors tial. In terms of (φg) and magnetic-type gravitational vector [52]. It is to be noted that the odd factor of 4 is responsible potential (Ag), Braginsky et al. [52] wrote (in our present for this strange result. Thus GRMG-BF formulation is defec- notation) tive not only for yielding cg = c/2 for gravitational waves in vacuum but also for the Goravito-Lorentz force law not satis- 1 ∂Ag E =−∇φ − , (93) fying satisfying the correspondence principle as judged from g g c ∂t

  2  14 E g H H 15 F = + 1 ( γ + ) v E + 1 v × H Here we use the notation g for and g for in [52]. Viz.: m 1 2 2 1 c2 g c g . 123 Eur. Phys. J. C (2017) 77 :822 Page 13 of 19 822 the the non-relativistic GravitoÐLorentz force of NRMG or ∇·g =−4πGρ (105a) of the SRMG. Further equation of continuity does not follow 0 1 ∂b from GRMG-BF field equations because of the existence of ∇×g =− ∂ (105b) some non-linear terms in Eq. (92a). 2 t ∇·b = 0 (105c) 16πG 4.2.2 GRMG of Harris (GRMG-H) ∇×b =− j (105d) c2 0 Instead of using the PPN formalism of GRMG-BF, Harris where ρ is the (rest) mass density, j is the momentum den- [55] derived a set of gravitational equations for slowly mov- 0 0 sity. The Eq. (105d) (representing the gravito-Ampère law) ing particles in weak gravitational fields starting from the is valid for time independent field [20], i.e., ∂g/∂t = 0, equations of GR. The resulting equations have some resem- which in view of Eq. (105a) is equivalent to ∂ρ /∂t = 0. blance to those in electromagnetism: 0 Since the divergence of curl of any vector is identically zero, the divergence of Eq. (105d) gives us ∇·j = 0. Thus ∇· =− π ρ , 0 Eg 4 G 0 (100a) Eq. (105d) has the same limitation as that of the Ampère’s π ∂ 16 G 4 Eg law of electromagnetism. Therefore, it needs a correction ∇×Hg =− (ρ0v) + , (100b) c c ∂t like Maxwell’s correction to the Amp‘ere’s law. While the ∇·Hg = 0, (100c) conditions ∇·j0 = 0 and ∂ρ0/∂t = 0 are valid for steady- ∇· = 1 ∂Hg state problems, the general situation, where j0 0 and ∇×Eg =− . (100d) ∂ρ /∂ = 2c ∂t 0 t 0, is given by the continuity equation for mass and mass current or momentum density: The GravitoÐLorentz force equation of Harris is of the fol- ∂ρ lowing form ∇·j + 0 = . 0 ∂ 0 (106)   t dv 1 1 ∂φg m = m E + (v × H ) + m v . (101) The Eqs. (105a) and (105d) will be consistent with the con- 0 0 g g 0 2 ∂ dt c 2c t tinuity Eq. (106), if we make the following Maxwell-like correction to the gravito-Ampère law (105d): The field Eg is related to the scalar potential (φg) and vector potential (Ag)as 16πG 4 ∂g ∇×b =− j + . (107) c2 0 c2 ∂t 1 ∂Ag 2E =−∇φ − . (102) g g c ∂t Further, without the above correction to the Eq. (105d), there can not be gravitational waves. Now, the corrected self- In empty space (where ρ0 = 0), the field equations (100a- consistent field Eqs. (105aÐ105c, 107) yield transverse grav- 100d) give us the following wave equations for the fields itational waves; the wave equations for the g and b fields of (Eg, Bg): GRMG-OR, in vacuum, take the following forms:

∂2 ∂2 2 ∂2g 1 ∂2g 2 2 Eg 2 1 Eg ∇2g − =∇2g − = 0, ∇ E − =∇ E − = 0, (103a) 2 2 2 2 (108a) g 2 ∂ 2 g 2 ∂ 2 c ∂t c ∂t c t cg t g ∂2 ∂2 ∂2 ∂2 2 2 b 2 1 b 2 2 Hg 2 1 Hg ∇ b − =∇ b − = 0, (108b) ∇ Hg − =∇ Hg − = 0, (103b) 2 ∂ 2 2 ∂ 2 2 ∂ 2 2 ∂ 2 c t cg t c t cg t √ √ where c = c/ 2. This means that the transverse gravita- where c = c/ 2. So GRMG-H is also defective like g g tional waves originating from slowly varying fields and weak GRMG-BF. sources travel through vacuum√ not at the light speed but at a reduced speed cg = c/ 2. Thus GRMG-OR will have√ 4.2.3 GRMG of Ohanian and Ruffini (GRMG-OR) exact correspondence with NRMG provided cg = c/ 2. However, GRMG-OR will not correspond to the slow motion In the Non-relativistic limit and Newtonian Gravity core- approximation of SRMG where cg = c in vacuum even at spondence of GR, Ohanian and Ruffini [20] (Sect. 3.4 of relativistic motion of the fields and sources. Now, if we define [20]) obtained the following equations from GR: a new filed for GRMG-OR as dv b = g + v × b (104) b˜ = , (109) dt 2 123 822 Page 14 of 19 Eur. Phys. J. C (2017) 77 :822 then the GravitoÐLorentz force law of GRMG-OR will take discussed here. It is to be noted that Ciubotariu [61] consid- the form: ered Peng’s [62] version of GRMG equations16 in the predic- tion of absorption of gravitational waves, while Minter et al. dv = g + 2v × b˜ (110) [63] considered GRMG-PS-M version in their investigation dt on the question of the existence of mirrors for gravitational waves. and the field Eqs. (105aÐ105c, 107) will take the form

4.2.5 Maxwellian Gravity of Mashhoon (MG-Mashhoon) ∇·g =−4πGρ0 (111a) ∂b˜ ∇×g =− Mashhoon [64,65], in his general linear solution of the gravi- ∂ (111b) t tational field equations of Einstein, obtained following grav- ˜ ∇·b = 0 (111c) itational analogues of Maxwell’s equations: 8πG 2 ∂g ∇×b˜ =− j + (111d) c2 0 c2 ∂t ∇·E = 4πGρ , (115a)  g 0 π ∂ ˜ 1 4 G 1 Eg Again these equations yield wave equations for the g and b ∇× Bg = (ρ0v) + , (115b) √ 2 c c ∂t fields for which c = c/ 2. Further, if we define B = b/4,   g g 1 the redefined field equations (not written here) yield√ wave ∇· Bg = 0, (115c) 2 equations for the g and Bg fields for which cg = c/ 2in   vacuum, while the gravito-Lorentz force law of GRMG-OR 1 ∂ 1 ∇×Eg =− Bg . (115d) takes the following form c ∂t 2 dv By defining Eg and Bg fields in terms of scalar and vector = g + 4v × Bg. (112) dt potentials (φg, Ag) as   The defects of GRMG-OR are apparent from the perspectives 1 ∂ 1 of cp violation and the suspected value of c in Einstein’s Eg =−∇φg − Ag , Bg = ∇ × Ag, (116) g c ∂t 2 linearized equations. he wrote the Lagrangian, L, for the motion of a 4.2.4 GRMG of Pascual-Sànchez, Moore, (GRMG-PS-M): of rest mass m0 (to linear order in φg and Ag)as

Following Huei [56], Wald [57] and Ohanian and Ruffini   1 v2 2 [58], Pascual-Sànchez [59], by using some approximation 2 L =−m0c 1 − of GR, obtained the following set of LorentzÐMaxwell-like c2   (117) gravitomagnetic equations that match with Moore’s [60] v2 2m + m γ 1 + φ − 0 γ v · A , equations from GR in the the weak field and slow motion 0 c2 g c g approximation: where γ = (1 − v2/c2)−1/2 is the Lorentz factor. The equa- dv  m = m E + 4v × B . (113) tion of motion, dp/dt = F, where p = γ mv is the kinetic 0 dt 0 g g momentum, is expressed as ∇· =− π ρ , Eg 4 G 0 (114a) dp v =−m E − 2m × B , (118) 4πG 1 ∂Eg 0 g 0 g ∇×B =− (ρ v) + , (114b) dt c g c2 0 c2 ∂t ∂ /∂ = v/ φ ∇·Bg = 0, (114c) if Ag t 0 and F is expressed to lowest order in c, g and A . ∂Bg g ∇×Eg =− . (114d) In empty space, the field Eqs. (115aÐ115d)giveuswave ∂t equations for Eg and Bg fields with cg = c. But because of the Although the field Eqs. (114aÐ114d) match with those of factor of 2 appearing in the relativistic Gravito-Lorentz force GRMG-UG [1], SRMG where cg = c exactly and with law (118), it does not correspond to the cp respecting special NRMG equations conditionally when cg = c as shwon here, the Gravito-Lorentz force, containing a factor of 4 in the grav- 16 Which, may be of GRMG-PS-M type as the author could not get the itomagnetic interaction, does not satisfy the cp in the sense ref. [62] and Ciubotariu. 123 Eur. Phys. J. C (2017) 77 :822 Page 15 of 19 822 relativistic Gravito-Lorentz force law of SRMG expressed in investigate the role of gravitoelectromagnetism in different equations (77, 83). fields of study, ranging from classical physics to quantum physics [1,3] and to cosmology [36,43] even quantum cos- dp v mology to test the validity (or the domain of validity) of the =−m E − m × B , (119) dt 0 g 0 c g proposed gravitoelectromagnetic theory from experimental point of view. written here in Mashhoon’s convention for the gravitostatics Basically, we offer a new pack of beautiful cards (or a Gauss’s law (115a) and Gravito-Lorentz force Law (118). beautifully simple, self-consistent toy model theory of grav- Further, in the non-relativistic case also, the Eq. (118) does ity) to play with, if one likes. However, it remains to be seen = not match with our non-relativistic result here, if cg c. how the recently observed phenomena concerning gravito- In our above discussions, we found that the speed of grav- magnetism and gravitational waves may be interpreted with itational waves in different linearized versions of Einstein’s our new findings with cg = c, since these phenomena could equations is not unique; it depends on the thought of a sci- well be explained by NRMG just by√ adjusting the undeter- entist concerning the mangagement of the spurious factor of mined parameter to cg = c/2orc/ 2 or by cp-violating = × 4 by splitting it 4 2 2 and moving a factor of 2 to some SRMG without invoking the curved space-time concept of 17 other place of the equations for consistency. Eddington Einstein and keeping other experimental parameters intact or [66] in his textbook “The Mathematical Theory of Relativ- by SRMG (cg = c with normal gravito-Lorentz force) and ity, which Einstein suggested was “the finest presentation varying other parameters of the experiment in a self consis- of the subject in any language”, rightly found and said that tent way, or re-analyzing the sources of possible theoretical weak-field solutions of the wave equation obtained from Ein- and experimental errors in the interpretation of the experi- stein’s field equations were just coordinate changes which we mental data. The author aims to address these questions in can “propagate” with the speed of thought. Apart from this future, if his odd situation permits and wishes the young defect of yielding a non-unique value of cg, the linearized minds do this, if they can. theory of gravitational waves has its limits because the linear approximation is not valid for sources where gravitational 4.3 Spin-1 vector gravity vs spin-2 tensor gravity self-energy is not negligible [67], as in the case of merg- ing of highly compact objects like the Neutron stars and the Many quantum field theorists, like Gupta [71], Feynman ‘so called’ Black Holes. It is important to note that the cur- [72],19 Zee [73] and Gasperini [74],20 to name a few, have rent experimental data on observation of gravitational waves rejected spin-1 vector theory of gravity on the ground that [6Ð9] and grvitomagnetic phenomena are being (or may be) explained by using any one of these cp-defying linearized Footnote 18 continued versions of GR. Further, these generally perceived general general description of gravitational waves in GR is different [67]. How- relativistic phenomena are being interpreted as having no ever, in spite of the interesting work by Mead [68], attempting to provide counterpart in the Newtonian world, which we found not a non-general relativistic explanation of the generation and detection of gravitational waves, which Isi et al. [69] referred to, a proper description to be true and satisfactory. The author wishes to stress that of gravitational waves within a purely gravitoelectromagnetic formal- just by connecting the Gauss Law of gravitostatics and equa- ism is still far from being established. tion of continuity in a consistent way, one gets gravitational 19 On page 30 of ref. [72], one finds: A spin-1 theory would be essen- analogue of Ampère–Maxwell law, which is being hailed tially the same as electrodynamics. There is nothing to forbid the exis- as one of the most important predictions of GR (where the tence of two spin-1 fields, but gravity can’t be one of them, because one consequence of the spin 1 is that likes repel, and un-likes attract. This mathematical trees seem to obscure the physical forest). Any is in fact a property of all odd- spin theories; conversely, it is also found talk of gravitomagnetism has long been the prerogative of that even spins lead to attractive forces, so that we need to consider only general relativists. With our present report, even undergrad- spins 0 and 2, and perhaps 4 if 2 fails; there is no need to work out the uates, untrained in the mathematical gymnastics of general more complicated theories until the simpler ones are found inadequate. 20 relativity, can now talk and think of the generation, transmis- On page 27, Gasperini noted: A correct description of gravity in the 18 relativistic regime thus requires an appropriate generalization of New- sion and detection of gravitational waves [68Ð70] akin to tons theory. Which kind of generalization? A natural answer seems to electromagnetic waves as matter of scientific curiosity and be suggested by the close formal analogy existing between the Newton force among static masses and the Coulomb electrostatic force among electric charges. In the same way as the Coulomb potential corresponds 17 = × 2 Such a scheme my be made general by thinking 4 2x x with x to the fourth component of the electromagnetic vector potential, the being any non-zero real number to get other sorts of spurious results in Newton potential might correspond to the component of a four-vector, a self-consistent way. and the relativistic gravitational interaction might be represented by 18 Which, in the framework of gravito-electromagnetic theory, cannot an appropriate vector field, in close analogy with the electromagnetic be just any other wave, but the waves whose nature is dictated by equa- theory. Such an attractive speculation, however, has to be immediately tions of gravitodynamics: Gravito-Maxwell Equations. By contrast, the discarded for a very simple reason: vector-like interactions produce repulsive static interactions between sources of the same sign, while - 123 822 Page 16 of 19 Eur. Phys. J. C (2017) 77 :822 if gravitation is described by a vector field theory like however,that the notion of the mass and spin of a field require Maxwell’s electromagnetic theory as discussed here, then the presence of a flat back ground metric ηab which one has vector-like interactions will produce repulsive static inter- in the linear approximation but not in the full theory, so the actions between sources of the same sign, while - accord- statement that, in general relativity, gravity is treated as a ing to Newton’s gravitational theory - the static gravitational mass-less spin-2 field is not one that can be given precise interaction between masses of the same sign is attractive. meaning outside the context of the linear approximation [57]. However, in one of our recent work [50] on quantum field Even in the context linear approximation, the original idea theoretical rediscovery of Heaviside-Maxwellian gravity (in of spin-2 gets obscured due to the several faces of flat space-time) (same as SRMG), we have shown that this Gravito-Maxwell equations seen here. This may be seen as not true because of the existence of a fundamental difference another limitation of GR for not making a unique and unam- in the sign before the source term in the in-homogeneous biguous prediction on the spin of graviton. field equations of SRMG and relativistic Maxwell’s electro- magnetism (RMEM) as seen below in SI units: 4.4 Little known Heaviside’s work on gravity

 μ =−μ μ,  μ = μ μ Ag 0g j0 A 0 je (120) Heaviside’s work on gravity, which McDonald [4] called a low velocity, weak-field approximation to general rela- μ 1 ∂2 μ μ tivity  = ∂ ∂μ = −∇2 A j , is little known and has not received as much atten- where c2 ∂t2 , g and 0 respec- tively represents the 4-potential and 4-mass current density tion as it deserves. This is because, in many leading papers μ μ and books exploring gravitomagnetic phenomena and grav- of SRMG while A and je respectively represents the 4- potential and 4-charge current density of RMEM. This fact itational waves, one finds rare or no mention of Heavi- that a vector gravitational theory, as proposed first by Heavi- side’s name, although Heaviside [5] predicted gravitomag- side and later rediscovered by many others following differ- netic effects and considered the necessity of possible exis- ent appoaches, implies attractive interaction between static tence of gravitational waves (for which there must be some masses was transparently clear to Sciama [36]. In spin-1 wave equations first written down by him) almost 20 years SRMG, contrary to the electromagnetic cases, like masses before Einstein’s prediction of gravitational waves [76,77]. (or gravitational charges) should attract and unlike masses Further, it is more surprising not to find Heaviside’s name in (if they exist) should repel each other under static condi- the Scientific Background on the Nobel Prize in Physics 2017 tions, while like (parallel) mass currents repel and un-like [78] when the 2017 Physics Nobel Prize was declared to be mass currents attract each other [3]. Similarly, there should awarded to Rainer Weiss, Barry C. Barish and Kip S. Thorne be attraction between like gravitomagnetic poles and repul- for (their) decisive contributions to the LIGO detector and sion between un-like gravitomagnetic poles: opposite to the the observation of gravitational waves. Brillouin [39], in his case in electromagnetism where like magnetic poles repel final remark on Carstoiu’s [37,38] suggestions for gravity and un-like magnetic poles attract each other [3]. Following waves aptly stated, “It is very strange that such an impor- Dirac’s scheme of predicting the spin of tant paper had been practically ignored for so many years, an electron, in our previous work [3], we have shown that but the reader may remember that Heaviside was the forgot- the gravitomagnetic moment of a Dirac (spin 1/2) fermion ten genius of physics, abandoned by everybody except a few μ = h¯ faithful friends.” is exactly equal to its spin : sg 2 , which can just be inferred from the magnetic moment of a qh¯ Dirac (spin 1/2) fermion, μse = , by replacing the elec- 2m0 5 Conclusions tric charge q with the gravitational charge m0 of the Dirac fermion as per SRMG. However, in GR, μsg value is not unique as found by different authors and noted in [3]. Following Schwinger’s non-relativistic formalism of clas- Regarding the idea of spin-2 graviton, Wald [57] noted sical electrodynamics, here we derived the fundamental that the linearized Einstein’s equations in vacuum are pre- equations of Non-Relativistic Maxwellian Gravity (NRMG), cisely the equations written down by Fierz and Pauli [75], in which matches with Heaviside’s Gravity of 1893 and offers a 1939, to describe a massless spin-2 field propagating in flat plausible mechanism for resolving the problem of action-at- space-time. Thus, in the linear approximation, general rela- a-distance in Newtonian gravity, within Galileo-Newtonian tivity reduces to the theory of a massless spin-2 field which domain of physics by demanding the existence of gravi- undergoes a non-linear self- interaction. It should be noted, tational waves propagating in vacuum at a non-zero finite speed cg, whose value has to be determined from experiments on measurable quantities involving c or from some more Footnote 20 continued g as is well known - the static gravitational interaction between masses advanced theory. Then, following an independent special rel- of the same sign is attractive. ativistic approach, we re-discovered NRMG in its relativistic 123 Eur. Phys. J. C (2017) 77 :822 Page 17 of 19 822 version and named it Special Relativistic Maxwellian Gravity origin of another factor of “2” or “1” in linearized GR may (SRMG), where cg = c comes out naturally and the equality be attributed to the adoption of the notion of space-time cur- of gravitational mass mg with Lorentz-invariant rest mass m0 vature and on how one chooses to define the gravitomagnetic is clearly demonstrated (resolving Eddington’s “mass ambi- field in terms of the potentials (that is, the perturbations in guity”) in the re-examination of an old thought experiment the metric) in different linearization schemes. Moreover, our aimed at finding the natural conditions of equilibrium of a two findings in no way affect the main conclusions of Ummarino particle system having requisite masses and electric charges and Gallerati’s paper on “Superconductor in a weak static in two inertial frames in relative motion such that the equi- gravitational field”. librium remains frame-independent. Most importantly the Acknowledgements The author is extremely grateful to the anony- equality mg = m0 emerges as a consequence of the Lorentz- invariance of physical laws and the Law of Universality of mous referee and author’s PhD supervisor Prof. Gautam Mukhopad- hyay, Department of Physics, IIT Bombay, Mumbai, India, for their Free Fall emerges as a consequence of mg = m0, not an many valuable suggestions for the improvement of the manuscript. The initial assumption in SRMG. Both NRMG and SRMG obey author thanks his wife, Pramila, for her constant support and encourage- correspondence principle (cp) in its true sense: Newtonian ment in this endeavor. His sincere thanks are also due to Prof. Niranjan Gravity ⇔ NRMG ⇔ SRMG. These flat space-time ver- Barik, Physics Department, Utkal University, Vanivihar, Bhubaneswar, Odisha, India, for his constant motivation to the author to pursue things sions of Maxwellian gravity matches with those considered from new perspectives in consonance with the scientific philosophy by several authors either for explaining some GR tests or for of Einstein and Infeld (in: The Evolution of Physics):“To raise new their derivation of SRMG following different approaches. questions, new possibilities, to regard old problems from a new angle, By the way, we also considered a non-linear form of SRMG requires creative imagination and marks real advance in science.” (SRMG-N) in flat , where the non-linearity arises Open Access This article is distributed under the terms of the Creative 2 2 due to the initial axiom of mg = m0/ 1 − v /c . The non- Commons Attribution 4.0 International License (http://creativecomm relativistic linearized version of SRMG-N does not corre- ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, c = c and reproduction in any medium, provided you give appropriate credit spond to NRMG when g or the non-relativistic version to the original author(s) and the source, provide a link to the Creative of SRMG. Thus, SRMG-N seems to defy the cp. Further, we Commons license, and indicate if changes were made. noted several versions of General Relativistic Maxwellian Funded by SCOAP3. Gravity (GRMG), including Ummarino and Gallerati’s ver- sion, which seem to defy the cp: Newtonian Gravity ⇔ NRMG ⇔ SRMG ¨⇔¨ GRMG; although they are being or may be (rightly or wrongly) employed to explain the experi- mental data on gravitational waves and the whole other class References of gravitomagnetic effects predicted by GR. While SRMG = 1. G.A. Ummarino, A. Gallerati, Superconductor in a weak static unambiguously fixes the exact value of cg c and spin of gravitational field. Eur. Phys. J. C. 77, 549 (2017) graviton sg = 1 uniquely, their values in GR are ambiguous 2. J. Schwinger, L.L. DeRaad Jr., K.A. Milton, W.-Y. Tsai, Classical and non-unique. However, the author leaves it for the con- Electrodynamics (Perseus Books, Reading, 1998), pp. 8Ð12 sideration of the readers to decide which version of gravi- 3. H. Behera, P.C. Naik, Gravitomagnetic moments and dynamics of Dirac (spin 1/2) Fermions in flat space-time Maxwellian Gravity. toelctromagnetism or Maxwellian Gravity is to be taken into Int. J. Mod. Phys. A 19, 4207Ð4229 (2004) consideration not only in the interpretation, theoretical as 4. K.T. McDonald, Answer to question #49. Why c for gravitational well as experimental error analysis of recent experimental waves? Am. J. Phys. 65, 591Ð592 (1997) data on the detection of gravitomagnetic field generated by 5. O. Heaviside, A gravitational and electromagnetic analogy. The Electrician 31, 281Ð282 (1893) mass-energy currents and the very recent detection of gravi- 6. B.P.Abbott et al., GW170814: a three-detector observation of grav- tational waves but also in the search for the interplay of gravi- itational waves from a binary coalescence. Phys. Rev. tational fields with other fields/states of matter in nature. The Lett. 119, 141101 (2017) author wishes to make an important remark that none of the 7. B.P. Abbott et al., GW170104: observation of a 50-solar-mass binary black hole coalescence at redshift 0.2. Phys. Rev. Lett. 118, authors who come up with a factor of 4 or 2 in their Lorentz- 221101 (2017) Maxwell-like solutions of GR have made an error in their 8. B.P. Abbott et al., GW151226: observation of gravitational waves calculations; the spurious factor of 4 or 2 (surprisingly 1 in from a 22-solar-mass binary black hole coalescence. Phys. Rev. GRMG-UG formulation), “really does” follow from GR or Lett. 116, 241103 (2016) 9. B.P. Abbott et al., Observation of gravitational waves from a binary rather from its basic building blocks: Einstein’s initial axioms black hole merger. Phys. Rev. Lett. 116, 061102 (2016) 2 of (i) mg = E/c and (ii) space-time curvature, taken as 10. I. Ciufolini, E. Pavlis et al., Test of general relativity and measure- inputs to the whole mathematical structure of GR. In our dis- ment of the lense-thirring effect with two earth satellites. Science cussion of non-linear SRMG (SRMG-N) in flat space-time, 279, 2100Ð2103 (1998) 11. I. Ciufolini, E.C. Pavlis, A confirmation of the general relativistic we have shown that a factor of 2 originates form the adop- prediction of the lensethirring effect. Nature 431, 958Ð960 (2004) 2 tion of mg = E/c in flat (Minkowski) spacetime, hence the 12. I. Ciufolini, Dragging of inertial frames. Nature 449, 41Ð47 (2007) 123 822 Page 18 of 19 Eur. Phys. J. C (2017) 77 :822

13. L. Iorio, How accurate is the cancellation of the first even zonal har- 43. V.N. Borodikhin, Vectortheory of gravity. Gravit. Cosmol. 17, 161Ð monic of the geopotential in the present and future LAGEOS-based 165 (2011) Lense-Thirring tests? Gen. Rel. Gravit. 43, 1697Ð1706 (2011) 44. O. Jefimenko, Gravitation and Cogravitation: Developing New- 14. L. Iorio, The impact of the orbital decay of the LAGEOS satellites ton’s Theory of Gravitation to its Physical and Mathematical Con- on the frame-dragging tests. Adv. Space Res. 57, 493498 (2016) clusion (Electret Scientific Company, Star City, 2006) 15. C.W.F. Everitt et al., : final results of a space exper- 45. O. Jefimenko, Causality, Electromagnetic Induction, and Gravita- iment to test general relativity. Phys. Rev. Lett. 106, 221101 (2011) tion: A Different Approach to the Theory of Electromagnetic and 16. C.M. Will, Finally, results from gravity probe B. Physics 4,43 Gravitational Fields, 2nd edn. (Electret Scientific Company, Star (2011) City, 2000) 17. C.W.F. Everitt et al., The gravity probe B test of general relativity. 46. J.A. Heras, An axiomatic approach to Maxwell’s equations. Eur. J. Class. Quant Gravit. 32, 224001 (2015) Phys. 37, 055204 (2016) 18. I.B. Cohen, ’s Papers and Letters on Natural Philos- 47. G.G. Nyambuya, Fundamental physical basis for MaxwellÐ ophy, 2nd edn. (Mass, London, Cambridge, 1978), pp. 302Ð303 Heaviside gravitomagnetism. J. Mod. Phys. 6, 1207Ð1219 (2015) 19. F. Rohrlich, Causality, the Coulomb field, and Newton’s law of 48. D.H. Sattinger, Gravitation and special relativity. J. Dyn. Differ. gravitation. Am. J. Phys. 70, 411Ð414 (2002) Equ. 27, 1007Ð1025 (2015) 20. H.C. Ohanian, R. Ruffini, Gravitation and Spacetime, 3rd edn. 49. R.S. Vieira, H. B. Brentan, Covariant theory of gravitation in the Cambridge University Press, New York (2013) framework of special relativity (2016). arXiv:1608.00815 21. R. Torreti, Relativity and Geometry, 130 (Dover Pub. Inc., New 50. H. Behera, N. Barik, Attractive HeavisideÐMaxwellian (vector) York, 1996) gravity from quantum field theory (2017). arXiv:1709.06876 22. R.H. Price, General relativity primer. Am. J. Phys. 50, 300Ð329 51. S.J. Clark, R.W. Tucker, Gauge symmetry and gravitomagnetism. (1982). (see p. 303) Class. Quant. Gravit. 17, 4125Ð4157 (2000) 23. A.S. Eddington, Space Time and Gravitation (Cambridge Univer- 52. V.L. Braginski, C.M. Caves, K.S. Thorne, Laboratory experiments sity Press, Cambridge, 1953), pp. 93Ð94 to test relativistic gravity. Phys. Rev. D 15, 2047 (1977) 24. A. Einstein, The Meaning of Relativity (New Age International (P) 53. R.L. Forward, General relativity for the experimentalist. Proc. IRE Ltd., Publishes, New Delhi, 2006), p. 57 49, 892Ð904 (1961) 25. I. Ciufolini, J.A. Wheeler, Gravitation and Inertia (Princeton Uni- 54. C.W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation, 1066–1095 versity Press, Princeton, New Jersey, 1995) (W. H. Freeman and Co., San Francisco, 1973) 26. C.M. Will, Theory and Experiment in Gravitational Physics, 2nd 55. E.G. Harris, Analogy between general relativity and electromag- edn. (Cambridge University Press, Cambridge, 1993) netism for slowly moving particles in weak gravitational fields. 27. W. Rindler, Essential Relativity: Special, General and Cosmolog- Am. J. Phys. 59, 421Ð425 (1991) ical, 2nd edn. (Springer, New York, 1977) 56. P. Huei, On calculation of magnetic-type gravitation and experi- 28. A. Herdegen, J. Wawrzycki, Is Einsteins equivalence principle valid ments. Gen. Relat. Gravit. 15, 725Ð735 (1983) for a quantum particle? Phys. Rev. D 66, 044007 (2002) 57. R.M. Wald, General Relativity (The University of Chicago Press, 29. V. de Sabbata, C. Sivaram, Spin and Torsion in Gravitation (World Chicago, 1984) Scientific, Singapore, 1994) 58. H.C. Ohanian, R. Ruffini, Gravitation and Spacetime (W.W.Norton 30. B. Mashhoon, On the gravitational analogue of Larmor’s theorem. Company, New York, 1994) Phys. Lett. A 173, 347Ð354 (1993) 59. J.-F. Pascual-Sàanchez, The harmonic gauge condition in the grav- 31. W.W. Salisbury, D.H. Menzel, Gyron Field-gravitational analogue itomagnetic equations. Nuovo Cimento B 115, 725Ð732 (2000). of magnetic force. Nature 252, 664Ð665 (1974) arXiv:gr-qc/0010075 32. A. Einstein, Jahrbuch der Radioaktivi¬tat und. Elektronik 4, 443Ð 60. T.A. Moore, A General Relativity Workbook (University Science 444 (1907) Books, Mill Valley, 2013), p. 409 33. J. Mehra, Einstein, Hilbert and the Theory of Gravitation: Histor- 61. C.D. Ciubotariu, Absorption of gravitational waves. Phys. Lett. A ical Origins of General Relativity Theory (D. Reidel Publishing I58, 27Ð30 (1991) Co., Dordrecht, 1974), p. 2 62. H. Peng, Gravitoelectric and gravitomagnetic waves and the T 34. H. Poinca`re, Rend, Circ. Mat. Palermo 21, 165 (1906); See: A. L. gauge. Preprint UAHDP-661 (1986). Presented at GR11, Stock- Harvey, Am. J. Phys. 33, 449 (1965), pp. 452 holm (1986) 35. W.G.V. Rosser, Classical Electromagnetism via Relativity: An 63. S. Minter, K. Wegter-McNelly, R.Y. Chiao, Do mirrors for grav- Alternative Approach to Maxwell’s Equations (Butterworths, Lon- itational waves exist? Phys. E Low Dimen. Syst. Nanostruct. 42, don, 1968) 234Ð255 (2010). arxiv:0903.0661 36. D.W. Sciama, On the origin of inertia. Monthly Notice R. Astron. 64. B. Mashhoon, Gravitoelectromagnetism. In:Proceedings of the Soc 113, 34Ð42 (1953) XXIII Spanish Relativity Meeting on Reference Frames and Grav- 37. J. Carstoiu, Les deux champs de gravitation et propagation des itomagnetism, edited by J.-F. Pascual-Sánchez, L. Floría, A. San ondes gravifiques. Compt. Rend. 268, 201Ð263 (1969) Miguel, and F. Vicente, 121Ð132, World Scientific, Singapore 38. J. Carstoiu, Nouvelles remarques sur les deux champs de grav- (2001). https://arxiv.org/abs/gr-qc/0011014 itation et propagation des ondes gravifiques. Compt. Rend. 268, 65. B. Mashhoon, Gravitoelectromagnetism: A Brief Review, In: Iorio, 261Ð264 (1969) L. (ed.) The Measurement of Gravitomagnetism: A Challenging 39. L. Brillouin, Relativity Reexamined (Academic Press, New York, Enterprise, p. 29. NOVA, Hauppauge (2007). https://arxiv.org/pdf/ 1970), pp. 101Ð103 gr-qc/0311030.pdf 40. D.D. Cattani, Linear equations for the gravitational field. Nuovo 66. A.S. Eddington, The Mathematical Theory of Relativity (Cam- Cimento B Serie 11(60B), 67Ð80 (1980) bridge University Press, Cambridge, 1930), p. 130 41. A. Singh, Experimental tests of the linear equations for the gravi- 67. I. Ciufolini, V. Gorini, Gravitational waves, theory and experiment tational field. Lettere Al Nuovo Cimento 34, 193Ð196 (1982) (an overview), Chapter 1: in Gravitational Waves,EditedbyI. 42. W.D. Flanders, G.S. Japaridze, Photon deflection and precession of Ciufolini et al., IOP Publishing, Bristol and Philadelphia (2001), the periastron in terms of spatial gravitational fields. Class. Quant. p. 2 Gravit. 21, 1825Ð1831 (2004) 68. C. Mead, Gravitational Waves in G4v (2015). https://authors. library.caltech.edu/59770/ 123 Eur. Phys. J. C (2017) 77 :822 Page 19 of 19 822

69. M. Isi, A.J. Weinstein, C. Mead, M. Pitkin, Detecting beyond- 75. M. Fierz, W. Pauli, On relativistic wave equations for particles Einstein polarizations of continuous gravitational waves. Phys. of arbitrary spin in an electromagnetic field. Proc. R. Soc. Lond. Rev. D 91, 082002 (2015). arXiv:1502.00333 A173, 211Ð232 (1939) 70. Watch the seminar talk of C. Mead in YouTube: https://www. 76. A. Einstein, Näherungsweise Integration der Feldgleichungen der youtube.com/watch?v=XdiG6ZPib3c Gravitation, Sitzungber. Preuss. Akad. Wiss. Berlin, part 1, 688 71. S.N. Gupta, Einstein’s and other theories of gravitation. Rev. Mod. (1916) Phys. 29, 334 (1957) 77. A. Einstein, Über Gravitationswellen, Sitzungber. Preuss. Akad. 72. R.P. Feynman, F.B. Morinigo, W.G. Wagner, Feynman Lectures on Wiss. Berlin, part 1, 154 (1918) Gravitation, Addison-Wesley Pub. Co., Reading p. 29Ð35 (1995) 78. https://www.nobelprize.org/nobel_prizes/physics/laureates/2017/ 73. A. Zee, Quantum Field Theory in a Nutshell, 2nd edn. (Princeton advanced.html. Retrieved on 4 Oct 2017 University Press, Princeton, 2010), pp. 32Ð36 74. M. Gasperini, Theory of Gravitational Interactions (Springer, Milan, 2013)

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