**Roots of Gravitomagnetic Theory (GMT)
**

__1.
Introduction.__

__2.
Newton’s Laws.
__

*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.

__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 d

In cases where v<<c
(the speed of light in *any* inertial
frame), m is practically constant at m_{0} (the so-called rest mass),
and it is convenient to say more simply that

** F** = m

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

m(v) = m_{0}/(1-v^{2}/c^{2})^{1/2}.
(2.2.3)

The variable
1/(1-v^{2}/c^{2})^{1/2} is by convention abbreviated as g(v)= 1/(1-v^{2}/c^{2})^{1/2}.
Thus Eq. 2.2.3 is usually written as m(v)=gm_{0}.

The value of
** F**, used in Eq. 2.2.1, must be
found by application of a suitable

__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:

**F**_{grav}
= Gm_{1}m_{2}** r**/r

“G” is
the gravitational constant, and in the mks system of units is equal to G=6.67408 × 10^{-11} nt meter^{2} 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 m_{1}m_{2} 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

__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:

__F___{electric}
= (1/4pe_{0})
q_{1}q_{2}** r**/r

Note in this equation that 1/4pe_{0}
is a constant like G in Newton’s gravitation law (Eq. 2.4.1). It has the
dimensions nt m^{2} 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,

__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 |m_{gravitational}| for q.
i=(-1)^{1/2},

|m_{inertial}| for q

G for 1/4pe_{0} (and/or G for m_{o}c^{2}/4p),

** g/i **for

** O/i** for

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=mc^{2},
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=mc^{2},
The photon’s inertial mass is

m = hn/c^{2} = 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

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.