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Particles, Newton’s Laws and Gravitomagnetic Theory

In this article Newton’s three laws of mechanics and his law of universal gravitation, applied to particles, are discussed in the context of gravitomagnetic theory.

Newton 1.

Newton’s 1st law of mechanics  essentially distinguishes inertial frames of reference from non-inertial frames. By implication his other laws apply only from the perspective of inertial frames. All of the rules/equations of gravitomagnetic theory are also cast in inertial frames of reference, as are Maxwell’s equations and the Lorentz force law (from which much of gravitomagnetic theory has evolved).

Newton 3.

Newton’s 3rd law states that in general every particle in nature can experience two kinds of force: (1) action forces, as a consequence of the particle being acted upon by others in its environment, and (2) reaction forces, as a consequence of the particle being acted upon by its own acceleration-induced gravitomagnetic fields. The general idea is that these reaction (or “self”) forces are passed through to the particles exerting the action forces. The law states that whenever one of these forces is present, so is the other. Furthermore, these two forces are always equal in magnitude and oppositely directed.  

Newton 2.

Newton’s 2nd law states that whenever an action force acts on a particle, the particle’s time rate of change of P, its “quantity of motion”, (P = mv), changes proportionately. He set the proportionality constant to unity, thereby defining an equality between the unit of force and the units of mass, space and time:

F = dP/dt = m0a + v(dm/dt).

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

F = ma.

Newton’s Law of Universal Gravitation.

Newton finished his theory with his Law of Universal Gravitation. The idea in this case is that every particle has an intrinsic quantity of matter or mass, m. Given two particles, separated by a displacement r from the acting to the reacting particle, each particle experiences a gravitational force toward the other:

F1 = Gm1m2r1,2/r1,23,

F2 = Gm1m2r2,1/r2,13.

Since the force is always attractive, the particle masses are assumed to be mathematically imaginary in gravitomagnetic theory. On the other hand, the force in Newton’s 2nd law is real, and it must accordingly  be that the particle’s inertial mass is real. Thus in gravitomagnetic theory every particle has an imaginary gravitational mass, mgrav, and a real inertial mass, minert. For any given particle, the inertial mass equals the magnitude of the gravitational mass:

minert = |mgrav|.

The Mechanics Paradigm.

The general paradigm for analyzing the behavior of a given particle is to calculate the imaginary gravitational and gravitomagnetic fields (g and O), at the particle and ascribable to all other particles. Then the real gravitational and gravitomagnetic forces  ascribable to the other particles can be calculated using the gravitomagnetic version of Lorentz force law,

F = m(g + v x O).

Note that since m and O are imaginary, a left hand rule must be used to determine the direction of the gravitomagnetic force.

Newton’s gravitational force law is replaced by much more difficult equations for g and O (particularly for g). The equations for calculating these field vectors are provided in the companion article,“On the Interactions of Point Masses”, provided on this site.