Roots of Gravitomagnetic Theory (GMT)


1. Introduction.

 Before the rise and fall of mighty Rome, there were intellectual giants in the world of the ancient Greeks. They had names like Euclid, Democritus  and Archimedes. Their insights into the natural world and its rules were many, profound and prescient. Following Rome’s fall there was a long period of intellectual silence known as the dark ages. It ended with the ideas of men like Galileo, Copernicus, and Kepler. These insights culminated in a set of definitions and rules formulated by Sir Isaac Newton. In four laws he described the behavior of most phenomena in the inanimate world as it was perceived during his time. These four laws, along with (a) four pivotal equations or “laws” suggested later by James Clerk Maxwell, and (b) a force law by Hendrik Lorentz, provide much of the framework for Gravito-Magnetic Theory (or “GMT”). Some details of this framework are provided in the following sections.

 2. Newton’s Laws.

 2.1 Newton’s First Law and Frames of Reference.

 Newton’s First Law states that “Objects in motion tend to stay in motion, and objects at rest tend to stay at rest unless an outside force acts upon them”. Newton’s First Law essentially distinguishes inertial frames of reference from non-inertial frames. By implication his laws apply only from the perspective of inertial frames of reference.

All of the rules/equations of GMT are also cast in inertial frames of reference, as are Maxwell’s equations and the Lorentz force law.

2.2 Newton’s Second Law and Force.

Newton’s Second Law states “The rate of change of the momentum of a body is directly proportional to the net force acting on it, and the direction of the change in momentum takes place in the direction of the net force.” Newton set the proportionality constant in the associated equation to unity, thereby defining an equality between the unit of force and the units of mass, space and time:

F = 1 dP/dt = d(mv)/dt = ma + v(dm/dt).       (2.2.1)

In cases where v<<c (the speed of light in any inertial frame), m is practically constant at m0 (the so-called rest mass), and it is convenient to say more simply that

F = ma.      (2.2.2)

More generally, however, an object’s mass is a function of its speed relative to one’s selected inertial frame:

m(v) = m0/(1-v2/c2)1/2.           (2.2.3)

The variable 1/(1-v2/c2)1/2 is by convention abbreviated as g(v)= 1/(1-v2/c2)1/2. Thus Eq. 2.2.3 is usually written as m(v)=gm0.

The value of F, used in Eq. 2.2.1, must be found by application of a suitable Force Law. Newton’s Law of Universal Gravitation (discussed below) is an example of a force law.

2.3 Newton’s Third Law and “Self Forces”.

Newton’s Third Law states that “To every action (force applied) there is an equal but opposite reaction (equal force applied in the opposite direction). The source of the reaction force was something of an enigma for many years. However, it eventually became clear that, when an electrically charged object is accelerated, components in its electric field are induced right at the object, such that the object experiences a force in its own fields. This induced “self-force” is presumably passed back through to the external, accelerating agent.

2.4 Newton’s Gravitation Force Law.

Newton theorized that every material object in the Universe attracts every other material object with a gravitational force that is proportional to the product of the object’s masses and inversely proportional to the distance between them:

Fgrav = Gm1m2r/r3.                  (2.4.1)

“G” is the gravitational constant, and in the mks system of units is equal to G=6.67408 × 10-11 nt meter2 kilogram-2. The gravitational force is always attractive, whereas Eq. 2.4.1 suggests that (given positive masses m1 and m2) it is repulsive. GMT meets this objection by defining gravitational mass to be mathematically imaginary. Consequently m1m2 is real and negative.

For many years it was believed that this “action-at-a-distance” force immediately adjusts to accommodate changes in r. However, later developments in field theory suggested that the “action-at-a-distance forces” are actually attributable to changes in the fields, such changes propagating from one gravitating object to another at the speed of light.

3. Maxwell’s Equations.

3.1. Electric Charge and the Electromagnetic Field.

In 1872 James Clerk Maxwell published “A Treatise on Electricity and Magnetism”. Included in this treatise were four equations that, like Newton’s laws, would revolutionize mankind’s model of the physical world. These four equations mathematically specified how the two field vectors E and B (the Electric and Magnetic fields) relate to one another and to electric charge at rest and/or in motion. Although there is only one sign of mass (positive) in the Universe as we know it, there are two signs of charge: positive and negative.

3.2. Coulomb’s Law.

Before Maxwell, Coulomb had discovered that like charges repel and oppositely signed charges attract with a force proportional to the product of the charges and inversely proportional to the square of the distance between them:

Felectric = (1/4pe0) q1q2r/r3,              (3.2.1)

Note in this equation that 1/4pe0 is a constant like G in Newton’s gravitation law (Eq. 2.4.1). It has the dimensions nt m2 coul-2. The two laws are mathematically identical. But in GMT gravitational masses are imaginary whereas, in Maxwellian theory electric charges are mathematically real.

4. Foundations of Gravitomagnetic Theory.

4.1. A Simple Thought Experiment.

Let us imagine that we have fashioned two distinct spherical shells of charge and mass, each shell consisting of a layer of charge superimposed upon a layer of mass. Shell A and B both have the same sign of charge. Thus according to Coulomb they would repel one another. However, we superimpose a shell of mass on A and B which is just what is required to balance the electric force of repulsion. In the inertial rest frame K the two shells are therefore in equilibrium. That is, regardless of their orientation in K, r (the distance between their centers) remains constant in time, provided there is a cohesion of a charge-mass pair. This cohesion is part of gravito-magnetic theory.

Now viewed from any other inertial frame K’, the distance r’ between the shells should also be constant in time. Equilibrium should be observed in all inertial frames. But according to electromagnetic theory (and experiment) there should be magnetic (as well as electric) forces in frames K’. The total electromagnetic force between the shells of charge will be different, viewed from K’. Yet equilibrium is observed in every inertial frame! The observed equilibrium can most understandably be attributed to a mass-mass interactive force that is equal and oppositely directed to the electromagnetic forces in every inertial frame. In seeking to explain this force, GMT theorizes a companion “gravitomagnetic” imaginary vector (dubbed O) to the real electromagnetic vector, B. In GMT a moving mass engenders a non-zero imaginary O field, quite as a moving charge engenders a real magnetic field vector, B. And in our experiment the sum of the gravitational plus the gravitomagnetic force is equal and oppositely directed to the sum of the electric plus the magnetic force in all inertial frames.

4.2. GMT and Maxwell.

Just as Newton’s law of gravitation can be mapped to Coulomb’s law by making suitable substitutions, so can the electromagnetic (or Lorentz) force law be mapped to a mass-mass force law by substituting

i |mgravitational| for q.  i=(-1)1/2,    

|minertial| for q

G for 1/4pe0 (and/or G for moc2/4p),

g/i for E,

O/i for B.

Most of GMT is simply the equations of Maxwellian/Lorentzian electromagnetic theory with these substitutions. The vector “O” stands for “gravitomagnetic field.” “g” stands for “gravitational field”.

5. Particle Models.

In GMT all elementary particles are modeled as having mass. Mass comes in two types: (1) Real or inertial mass (e.g. Newton;s Second Law of mechanics), and (2) Imaginary or gravitational mass (e.g. Newton’s Law of Universal Gravitation). Inertial mass is mathematically real; gravitational mass is mathematically imaginary. The choice of which type of mass to use, in a given instance, depends upon the context.

A particle’s (real) inertial mass is always equal to the magnitude of its (imaginary) gravitational mass.

Most elementary particles also have electric charge.

5.1. Particles With No Charge.

Only photons and gluons (and possibly gravitons) have no charge. In the vacuum of “empty” or inter-particle space these particles are measured to propagate at the speed of light relative to every inertial frame! They have non-zero energies and hence, by E=mc2, they have non-zero masses. For example, the energy of a photon is

E = hn,        (5.1)

where n is the frequency of the associated “carrier” wave. Assuming E=mc2, The photon’s inertial mass is

m = hn/c2 = h/lc.         (5.2)

According to Newton’s gravitational force law (and by the Equivalence Principle thought experiment where an elevator accelerates in gravity-free space), a photon is affected by an ambient gravitational field. If the photon propagates normal to g, its trajectory is curved. If it propagates parallel to g, its frequency and wavelength vary in time.

In GMT a photon theoretically engenders a gravitational field. Thus by the Gravitation Force Law, photons and gluons attract other particles.  The theory has many interesting ramifications. For example, it suggests that a significant fraction of a black hole’s net gravitational field is attributable to trapped photons.

5.2. Particles With Charge.

Aside from photons and gluons and possibly gravitons, all the rest of the elementary particles (e.g. electrons) have both mass and charge, and they propagate at sub-light speeds relative to every inertial frame. These particles have electromagnetic fields in addition to the grav-gravitomagnetic fields specified by GMT.

Given a knowledge of a charged particle’s past motion, its electromagnetic field vectors at any other particle can theoretically be solved. The same is true for the gravitational and gravitomagnetic field vectors of a point mass. Application of the Lorentzian force law produces the grav-gravitomagnetic force experienced by a second mass. Thence the second mass’ motion can be determined. Again the possibilities are many. For example, this exercise (conducted with the aid of a computer) produces an open, precessing orbit for a planet orbiting a spinning star.

In addition to having gravitational fields, particles that spin also have gravitomagnetic fields (even when their centers are at a fixed position). Similar remarks apply to electric charges. The total energies of particles equals  (1) their grav-gravitomagnetic field energies, plus (2) their electromagnetic field energies (where applicable), plus (3) their kinetic energies.

6. Concluding Remarks.

GMT is only in its infancy. As Richard Feynman pointed out, every theory must ultimately agree with physical experiment. GMT suggests many experiments with particles and the larger entities (e.g. stars, planets and even galaxies) consisting of multiple particles.