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Title: Foundations of Gravitomagnetic Theory.

 Author: George Russell Dixon. Retired S/W Engineer and Database Analyst.

 Education:

BA, Hamilton College, 1963, Special student courses in physics and mathematics at MIT. Cornell, and the Univ. of Colorado. Present occupation: administrator of and contributor to  the physics and math website www.grdmax.net.

Abstract.

A relatively simple alternative to General Relativity is presented for analyzing mass-mass interactions and the motion of photons in a gravitational field. Section 2 introduces a set of substitutions that can be made in electromagnetic equations, in order to solve many problems in the realm of mass-mass interactions. Subsequent sections use the substitutions to solve several interesting problems.   

1. Introduction.

The failure of Newton’s Law of Universal Gravitation to completely specify mass-mass interactions (planet-satellite, star-planet. star-star, etc.) has stimulated much thought, as has the apparent bending of rays of electromagnetic radiation by an ambient gravitational field. Subtle perturbations of Mercury’s orbit were puzzling for decades after their discovery. These and other effects, not fully explained by Newtonian theory, are explained by a set of substitutions and rules helpful to students of  Gravitomagnetic Theory (or GMT for short).

2. Rules and Substitutions in Electromagnetism.

Any electromagnetic equation involving only positive charge can be changed to an analogous gravogravitomagnetic equation by the following rules and substitutions.

Substitute:

i|m_{gravitational}| for q.           (2.1)              (Note that i|m_{gravitational}| is mathematically imaginary.)    

G for 1/4pe₀.                       (2.2)          (G is real.)  

i|g| for E.                             (2.3)          (i|g| is imaginary.)

i|O| for B.                            (2.4)          (i|O| is imaginary and stands for “gravitomagnetic”.)

Rules:

Force, inertial mass, kinematic variables and position variables are real in both regimes and require no substitutions.

Inertial mass (as used in Newton 2) is real and, in sub-light-speed cases is a function of particle speed: m_inertial = m_o/(1-v²/c²)^1/2. Its rest value equals the magnitude of gravitational mass.

Every sub-light-speed particle has speed-dependant inertial mass and constant gravitational mass.

Every sub-light-speed  particle also has charge. (“Uncharged particles like neutrons may have zero net charge, but they are composed of charged particles (quarks)).

Particles that always travel at the speed of light (e.g. photons) have no charge.

Particles that always travel at the speed of light have no rest mass, but they do have inertial and gravitational mass.

Examples Of the Substitution Rules in Use:

The energy density in an electric field is u_E=e_oE²/2. Hence the energy density in a gravitational field is u_g=|ig|²/8pG. |ig|² is negative real. The energy density in a magnetic field is u_B=B²/2m_o. Hence the energy density in a gravitomagnetic field is u_O=c²|iO|²/8pG. |iO|² is negative real.

3. A Simple Thought Experiment.

Let us imagine that two discrete spheres are in equilibrium in deep space, far from other gravitating objects. Each sphere consists of a solid mass of lead with a layer of positive charge “sprayed” on its surface. The spheres are initially at rest in inertial frame K, and the proportions of lead and charge are such that the total force on each sphere, attributable to the other sphere, is zero. According to Newton 2 the spheres should remain at rest in frame K. And according to the Lorentz transformation the spheres should mutually move with a constant velocity v=-u relative to any other inertial frame K’.

But according to Maxwell, in K’ each shell of charge will engender a magnetic field and, according to the Lorentz force law, the other sphere will generally experience a magnetic force in addition to the omnipresent electric force. For example if K’ is such that the mutual sphere velocity v’ is perpendicular to a line between the two spheres’ centers, then the magnetic force on either sphere will be oppositely directed to and less than the electric force. In K’ the total electromagnetic force on a sphere will be less than the purely electric force in K.

Now If the force of the mass-mass interaction is the same in both frames (as Newton stipulates), then each sphere should experience a nonzero net force in K’,  and should (according to Newton 2) accelerate in K’. But of course this does not occur. Evidently in K’ the net mass-mass and charge-charge interactive forces must be equal but oppositely directed, quite as they are in K.

In explaining this equality we might suppose that in K’ there is an analogous field to the magnetic field, and that there is a Lorentzian type force between the masses in K’ just as there is a magnetic force between the charges. Let us refer to this hypothetical field as the “gravitomagnetic” field, and symbolize it with the letter “O”. In order to explain the equilibrium in K’ we should theoretically write

F’_q-q=q(E’ + v x B’),    (3.1)

F’_m-m=m_grav(g’ + v x O’).    (3.2)

F’_total=m_grav(g’ + v x O’) + q(E’ + v x B’) = 0.  (3.3)

An important difference between  these two equations is that, according to GMT, m_grav, g’ and O’ are mathematically imaginary.

4. Photons and Gravity.

Every photon of electromagnetic radiation theoretically has momentum of magnitude |p|=h/l. In GMT every photon accordingly has mass, m= h/lc. (Since all photons propagate at speed c, this mass is not a rest mass.) According to GMT a photon experiences a force in an ambient gravitational field. If the photon’s velocity is perpendicular to the ambient g field, then Newton 2 indicates a radial acceleration (but constant speed). If g is parallel to the photon’s velocity, then there is a change in l, but speed again remains a constant c. The radial acceleration theoretically explains the bending of light rays as they pass through the gravitational field of a massive body such as the Sun. The change in l has been measured for light emitted "upward" in the Earth’s g field and is referred to as “Red Shift.”

Note that Newton 3 implies that photons themselves have g fields. They theoretically exert a force on the Sun and other massive objects, and can be significant contributors to the total gravitational field of a black hole.

5. The Sun’s Gravitomagnetic Field and Mercury’s Orbit.

More than a century after Newton published his Principia , le Verrier noted that non-elliptic irregularities in Mercury’s orbit could not be wholly ascribed to gravitation from the other planets and the Sun. He suggested that there must be an unseen planet (dubbed “Vulcan”) to account for Mercury’s wobble. But Vulcan was never observed, and for many years the whole matter remained a mystery. It is postulated in GMT that the irregularity is caused by the dual rules that (a) the gravitational mass is independent of speed, but (b) the inertial mass is a function of speed.

It is also clear by analogy to Maxwell’s equations that the spinning Sun must engender a bipolar gravitomagnetic field in the surrounding space. This field is not easily computed since the Sun is not a homogeneous sphere. But the field’s direction and magnitude at points in Mercury’s orbit can be closely estimated. And application of the GMT version of the Lorentz force law shows that Mercury must experience a gravitomagnetic force that opposes the gravitational force of the Sun. (n.b.: Mercury’s gravitational mass and the Sun’s gravitomagnetic field are both Imaginary, and hence a left-hand rule must be used to determine the direction of the gravitomagnetic force in the force law F_gravmag=m_gravv x O.)

One can estimate that the Sun is homogeneous and calculate Mercury’s orbit using a computer program. When this is done it is found that the orbit “precesses” approximately .43 arc seconds per year, which agrees with the astronomical observations.

6. Spiral Galaxies and Gravitomagnetism.

A spiral galaxy can be modeled as a spinning disc of mass. It theoretically engenders an O field which, at interior points, points in the same direction as the disc’s angular velocity. A constituent star thus experiences a gravitomagnetic force that points inward, augmenting the gravitational force. This is consistent with the astronomical conclusion that gravity alone cannot explain why the fastest moving peripheral stars do not spiral away into intergalactic space. Of course the gravitomagnetic force is much smaller than the gravitational force. But it may, in a given case, be just what is needed for the galaxy to hold together.

7. Gravogravitomagnetic Waves and Field Energy.

Gravogravitomagnetic field energy is negative. Negative work must be done to bring two masses in from infinite separation. The energy in a Gravogravitomagnetic wave is accordingly negative; the g and O vectors in such waves are imaginary. And the gravogravitomagnetic analogue to the Poynting vector is negative.

We have little experience with waves with negative energies because all particles (except for light-speed ones) have electric charge, and the inertia of a charged particle far overshadows that of an uncharged particle. But the negative values of gravitational fields is necessary. For the field energy density of a gravitational field is proportional to g², which is negative since g is imaginary.    

8. Conclusions.

Gravitomagnetic fields, and attendant gravitomagnetic forces, may explain many small discrepancies in mass-mass interactive forces computed using Newton’s Law of Universal Gravitation and, more generally, the fields of a point charge with known past motion. In general the net force experienced by a mass, in the fields of another mass, should be calculated by an application of the Lorentz analogue, F=m(g + v x O), where m, g and O are imaginary.

A vast trove of electromagnetic equations can be transformed to analogous gravogravitomagnetic equations by using the transformations in Section 2. And as Richard Feynman pointed out, the same equations have the same solutions. The substitutions in Section 2 make a host of new equations available for use by astronomers and astrophysicists.