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Attractive Heaviside-Maxwellian (Vector) Gravity from Special Eur. Phys. J. C manuscript No. (will be inserted by the editor) Attractive Heaviside-Maxwellian (Vector) Gravity from Special Relativity and Quantum Field Theory Harihar Beheraa,1, N. Barikb,2 1Physics Department, BIET Higher Secondary School, Govindpur, Dhenkanal-759001, Odisha, India 2Department of Physics, Utkal University, Vani Vihar, Bhubaneswar-751004, Odisha, India the date of receipt and acceptance should be inserted later Abstract Adopting two independent approaches (a) 1 Introduction Lorentz-invariance of physical laws and (b) local phase invariance of quantum field theory applied to the Dirac Many field theorists, like Gupta [1], Feynman [2]1, Zee Lagrangian for massive electrically neutral Dirac parti- [3] and Gasperini [4]2, to name a few, have rejected cles, we rediscovered the fundamental field equations of spin-1 vector theory of gravity on the ground that if Heaviside Gravity (HG) of 1893 and Maxwellian Grav- gravitation is described by a vector field theory like ity (MG), which look different from each other due to Maxwell’s electromagnetic theory, then vector-like in- a sign difference in some terms of their respective field teractions will produce repulsive static interactions be- equations. However, they are shown to represent two tween sources of the same sign, while - according to mathematical representations of a single physical the- Newton’s gravitational theory - the static gravitational ory of vector gravity that we name here as Heaviside- interaction between masses of the same sign is attrac- Maxwellian Gravity (HMG), in which the speed of grav- tive. Misner,Thorne and Wheeler (MTW)[5], in their itational waves in vacuum is uniquely found to be equal “Exercises on flat space-rime theories of gravity”, sug- to the speed of light in vacuum. We also corrected a 1On page 30 of ref. [2], Feynman noted: “A spin-1 theory sign error in Heaviside’s speculative gravitational ana- would be essentially the same as electrodynamics. There is logue of the Lorentz force law. This spin-1 HMG is nothing to forbid the existence of two spin-1 fields, but gravity shown to produce attractive force between like masses can’t be one of them, because one consequence of the spin 1 is under static condition, contrary to the prevalent view that likes repel, and un-likes attract. This is in fact a property of all odd-spin theories; conversely, it is also found that even of field theorists. Galileo’s law of universality of free fall spins lead to attractive forces, so that we need to consider is a consequence of HMG, without any initial assump- only spins 0 and 2, and perhaps 4 if 2 fails; there is no need tion of the equality of gravitational mass with velocity- to work out the more complicated theories until the simpler dependent mass. We also note a new set of Lorentz- ones are found inadequate.” 2On page 27, Gasperini noted: “A correct description of grav- Maxwell’s equations having the same physical effects ity in the relativistic regime thus requires an appropriate gen- as the standard set - a byproduct of our present study. eralization of Newtons theory. Which kind of generalization? A natural answer seems to be suggested by the close for- mal analogy existing between the Newton force among static arXiv:1709.06876v2 [physics.gen-ph] 13 Dec 2017 masses and the Coulomb electrostatic force among electric charges. In the same way as the Coulomb potential corre- sponds to the fourth component of the electromagnetic vec- tor potential, the Newton potential might correspond to the component of a four-vector, and the relativistic gravitational interaction might be represented by an appropriate vector field, in close analogy with the electromagnetic theory. Such an attractive speculation, however, has to be immedi- ately discarded for a very simple reason: vector-like interac- tions produce repulsive static interactions between sources of the same sign, while - as is well known - the static gravita- ae-mail: [email protected] tional interaction between masses of the same sign is attrac- be-mail: [email protected] tive. 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.
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