Title: Foundations of Gravitomagnetic
Author: George Russell Dixon. Retired S/W Engineer and Database Analyst.
Author: George Russell Dixon. Retired S/W Engineer and Database Analyst.
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.
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
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
i|mgravitational| for q.
(Note that i|mgravitational| is mathematically imaginary.)
G for 1/4pe0.
(2.2) (G is real.)
i|g| for E.
(i|g| is imaginary.)
i|O| for B.
(i|O| is imaginary and stands for “gravitomagnetic”.)
inertial mass, kinematic variables and
position variables are real in both regimes and require no substitutions.
mass (as used in Newton 2) is real and, in sub-light-speed cases is a function
of particle speed: minertial = mo/(1-v2/c2)1/2.
Its rest value equals the magnitude of
sub-light-speed particle has speed-dependant inertial mass and constant
sub-light-speed particle also has
charge. (“Uncharged particles like neutrons may have zero net charge, but they
are composed of charged particles (quarks)).
that always travel at the speed of light (e.g. photons) have no charge.
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
energy density in an electric field is uE=eoE2/2.
Hence the energy density in a gravitational field is ug=|ig|2/8pG. |ig|2 is negative real. The energy density in a magnetic
field is uB=B2/2mo.
Hence the energy density in a gravitomagnetic field is uO=c2|iO|2/8pG. |iO|2
is negative real.
3. A Simple Thought Experiment.
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
relative to any other inertial frame K’.
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.
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.
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
+ v x B’),
+ v x O’).
+ v x O’) +
+ v x B’)
= 0. (3.3)
important difference between these
two equations is that, according to GMT, mgrav, g’ and O’
are mathematically imaginary.
4. Photons and Gravity.
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.”
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.
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.
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 Fgravmag=mgravv
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
6. Spiral Galaxies and
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 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
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
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.
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.