Before Maxwell there was Newton. “Standing on the shoulders of men like Brahe and Kepler”, Newton gave us the law of Universal Gravitation. As Einstein said, Newton’s gravity law “… supplies us with the actual motions of the heavenly bodies with a delicacy of detail little short of wonderful.”
Coulomb provided us with the equation for the force between two electric charges, much as Newton had done for two masses. Thanks to the insights of Ampere et al, however, it became clear that there is more to the interaction of electric charges than the static situation described by Coulomb. For it turned out that the charge-charge interaction is not inverse square when the charges are in motion.
Now according to Maxwell there is a second “field” complementing the electric field, and the total force experienced by a moving charge is influenced by the charge’s velocity in that second, “B” field. Lorentz elegantly specified the net magnetic force acting on a charge in his ubiquitous force law, F=q(v X B).
The parallels between mass-mass interactions and charge-charge interactions (when charges and masses are at rest) quite naturally prompted the question “should there not be an analogue to Maxwell’s B field in mass-mass interactions?” And if there is, then there is the possibility that Newton’s gravitational force might not fully describe mass-mass interactions.
In a simple thought experiment(1), the author suggested to Professor Edward Purcell (Harvard) that there must be an “Orthogravic” field in the case of moving masses, quite as Maxwell defines B in his field equations. Just as magnetic effects are ordinarily much weaker than electric effects, the “orthogravic” effects could be expected to be much smaller than the already-faint force of gravity. Nevertheless, certain subtle effects like the annual precession of the orbit of Mercury suggested that an analogue to the Lorentz force law might be in order.
But the idea of gravitational analogues to Maxwell’s equations had at least one difficulty: it is readily shown that the electric field of a charge, q, has electric and magnetic field energy densities proportional to E2 and B2. In this regard there is a rather elegant “book keeping” mechanism: given two like-sign charges, separated in space, positive work must be expended to force the charges closer together. And this work always equals the rise in the field energy of the net electric field!
By contrast, when two masses are allowed to move closer together the work expended is negative! Book keeping indicates that the net gravitational field energy should decrease!
This difficulty was rather elegantly solved by Professor Philip Morse’s suggestion that the book keeping would conserve energy only if the gravitational field is mathematically imaginary!
The author subsequently enlarged this idea(1), suggesting that gravitational mass (mgrav) is also imaginary, and that there is an imaginary analogue to Maxwell’s B field (dubbed the O field). And of course the analogue to the Lorentz force law contains an m(v X O) term, but with one important difference: whereas the “right hand rule” must be used to determine the direction of the Lorentz magnetic force when q moves, a “left hand rule” must be used to determine the direction of the “orthogravic” force acting on a moving mass.
On the Internet, articles dealing with mass-mass interactions have used the word “gravitomagnetic” instead of “orthogravic”. In deference to experiments by NASA et al, the word “gravitomagnetic” is now used by the author.
The precession of Mercury’s orbit has been predicted(2) to be the observed 43 arcseconds per century by assuming that the dipolar O field of the spinning Sun perturbs the closed elliptic orbit predicted by Newtonian gravity. Also, it has been shown that stars in spiral galaxies are theoretically acted on by centric gravitomagnetic forces in addition to gravitational ones. Modeling of such galaxies has shown that the peripheral stars should shoot off into intergalactic space if subjected to gravity alone. Augmenting gravity with the gravitomagnetic force may be just what is required to hold the peripheral stars in their circular paths around the galactic center. (The relatively small gravitomagnetic forces have hitherto been ascribed to “dark matter”, which by definition can never be detected.)
On a final note, it would be remiss not to mention gravitomagnetic theory’s suggestion that photons, although chargeless, have inertial and gravitational mass. (How Einstein could (a) have written m=E/c2 in general, and (b) E=hf in the case of photons, and then (c) subsequently have taken the position that photons are massless is, in the author’s mind, one of the great mysteries of modern physics!)
In gravitomagnetic theory photons, in the presence of ambient gravitational fields, travel in paths consistent with Newton’s second law. The difference between photons and massive, sub-lightspeed particles, is that the magnitude of a given photon’s acceleration does not vary, although its velocity may.
In general, gravitomagnetic theory holds that space-time is “flat”, which is to say it is not warped by large, gravitating masses. Furthermore, by Newton3, photons exert small gravitational and gravitomagnetic forces on other particles with mass (including other photons) … a possibility of possible interest in the analysis of dark holes, quasars, and other astronomical phenomena.
George R. Dixon, June 15, 2021.
- See When Gravity Balances the Lorentz Force, www.physics-by-dixon.com.
- See On the Spinning Sun-Engendered Mercury Orbital Precession of .43 Arc Seconds per Year, www.physics-by-dixon.com.