Covariant theory of gravitation in the framework of special relativity

Abstract

In this work, we study the magnetic effects of gravity in the framework of special relativity. Imposing covariance of the gravitational force with respect to the Lorentz transformations, we show from a thought experiment that a magnetic-like force must be present whenever two or more bodies are in motion. The exact expression for this gravitomagnetic force is then derived purely from special relativity and the consequences of such a covariant theory are developed. For instance, we show that the gravitomagnetic fields satisfy a system of differential equations similar to the Maxwell equations of electrodynamics. This implies that the gravitational waves spread out with the speed of light in a flat spacetime, which is in agreement with the recent results concerning the gravitational waves detection. We also propose that the vector potential can be associated with the interaction momentum in the same way as the scalar potential is usually associated with the interaction energy. Other topics are also discussed, for example, the transformation laws for the fields, the energy and momentum stored in the gravitomagnetic fields, the invariance of the gravitational mass and so on. We remark that is not our intention here to propose an alternative theory of gravitation but, rather, only a first approximation for the gravitational phenomena, so that it can be applied whenever the gravitational force can be regarded as an ordinary effective force field and special relativity can be used with safety. To make this point clear we present briefly a comparison between our approach and that based on the (linearized) Einstein's theory. Finally, we remark that although we have assumed nothing from the electromagnetic theory, we found that gravity and electricity share many properties in common -these similarities, in fact, are just a requirement of special relativity that must apply to any physically acceptable force field.

A Novel Covariant Approach to Gravito-Electromagnetism

“…Interest in the analogies between gravitation and electromagnetism, also called gravito-electromagnetism (GEM), has increased in recent years. These relationships were observed and reported during the second half of the nineteenth century, and we recommend [1,2] for historical review and references. Various approaches to GEM have recently been proposed, and we quote a non-exhaustive list of papers concerning, by way of example, gravitomagnetic effects [3][4][5][6], the relation of GEM to special relativity [2,7], tidal tensors [8,9], weak-field approximation [10][11][12], the Lorentz violation [13,14], teleparallel gravity [15,16], the Mashhoon-Theiss effect [17], quantum gravity [18,19], gravitational waves [20,21], the relation of GEM to electro-dynamics in curved spacetime [22,23], gravitational field of astrophysical objects [24,25], the Sagnac effect [26,27], torsion gravity [28], the Schrödinger-Newton equation [29], non-commutative geometry [30], spin-gravity coupling [31], gravity and thermodynamics [32], the Casimir effect [33], gauge transformations [34] and, quantum field gravity [35,36].…”

“…The existence of the constraint (40) indicates that our model is not equivalent to the Heaviside gravity formulation of [36], and it again different from electromagnetism. We notice that the force law in (40) fits exactly with (2), and the flipped signal may be obtained from redefining b in (3). However, as we have already pointed out, it is not possible to consider an exact match between both of these expressions because (2) comprises the µ = 0 component of a presently unknown third rank tensor.…”

“…This force law is inspired by (2), which is obtained from Einstein's equations, and it is possible to flip the signal of p × g, a possibility that is entertained in Section V. The aim of the present article is to examine the consistency of the above proposal. The model may be considered as self-interacting because the time variation of gravity also contributes to gravity itself, however in Section IV we show that linear Maxwell-like field equations may be obtained from (5).…”

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

“…Some without using the formalism of GR [36][37][38][39][40][41][42][43][44][45][46][47][48][49][50] and some using the formalism of GR in the weak field and linearized approximations. These are discussed in two separate sections below.…”

“…Recently, Heras [46], by recognizing the general validity of the axiomatic approach to Maxwell's equations of electromagnetic theory, used those axioms to derive only the field equations (leaving out gravito-Lorentz force law) of SRMG, where the in-variance of gravitational charge is considered. Other recent derivations of SRMG equations from different approaches include the works of Nyambuya [47], Sattinger [48], Vieira and Brentan [49]. The historical objections of several researchers, starting from Maxwell (see footnote 2 of this paper) upto Misner, Thorne and Wheeler (MTW, Sect.…”