When Gravity Balances the Lorentz Force

G.R.Dixon, engineer, with kids

1. Gravitomagnetic forces.

Figure 1 illustrates two resting, hollow spheres of positive charge, viewed from the perspective of inertial frame, K. The center of q_b is at (0,0,0) and the center of q_a is at (0,R.0). The electric force on q_a is away from q_b and has the magnitude F_a=q_bq_a/4πε_oR². Similarly for q_b. Since they are permanently at rest, some agent must hold them at a constant distance from one another so that the total force on each charge is zero.

Viewed from any other inertial frame, the charges move at a common, constant velocity, which among other things means that (a) the distance between them is unchanging in every inertial frame, and (b) the total force on each charge is zero in every inertial frame. 

Of course if frame K’ moves to the right relative to K (so that the charges move to the left), then either charge experiences a magnetic force toward the other one, in addition to a repellent  electric force that is greater than the electrostatic force in K. And viewed from K’ the net electromagnetic force on either charge is somewhat more than the electrostatic force in K. (This fact is consistent with the way forces in general transform relativistically.)⁽¹⁾

Figure 1

Two spherical shells of positive charge, held at rest in  frame K

Let us now replace the constraining agent by filling the two spheres with non-conducting neutral matter at a density such that the gravitational attractive forces in frame K precisely cancel the electrostatic repulsive ones. We know that the electromagnetic force experienced by each charge varies from frame to frame. But we also know that the total force on each charge must be zero in every inertial frame. In effect, then, the force of the solid spheres of matter must transform from frame to frame precisely as the electromagnetic force does.

In gravitomagnetic theory it is assumed that the mass-mass interactions have the same formulae as charge-charge interactions have in Maxwell’s equations and the Lorentz force law. For example, the gravitational field of m_a in Fig. 1 everywhere points away from m_a. But the gravitational force on m_b (i.e, m_bg_a) must point toward m_a. (And similarly m_ag_b must point toward m_b.) Gravitomagnetic theory solves this requirement by stipulating that gravitational mass and the gravitational field are mathematically imaginary. This being the case, m_bg_a points toward m_a as required. Similarly for m_ag_b.

But what about frame K’? According to Maxwell and Lorentz, viewed from that frame there is a magnetic force on q_a in addition to the electric force. In order for equilibrium to prevail, there must be a mass-engendered force in K’ which is analogous to the charge-engendered magnetic force. This theoretical force is dubbed the “gravitomagnetic” force. And the field that causes it is the gravitomagnetic field, symbolized by the letter O’. Like g’, O’ is imaginary. And like q(v’ X B’), m(v’ X O’) is real. But because m and O’ are imaginary, the gravitomagnetic force is repulsive. (Note that we must use a left hand rule to obtain the direction of m(v’ X O’).)

2. Gravitomagnetic Substitutions and Rules.

Any electromagnetic equation involving only positive charge can be changed to an analogous gravo-gravitomagnetic equation by the following rules and substitutions:

Substitute:

m for q.                    Comment: m is mathematically imaginary     

G for 1/4πε₀.  

g for E.                     Comment: g is mathematically imaginary 

O for B.                    Comment: O is mathematically imaginary 

Rules:

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

Gravitational mass, like electric charge, is invariant under a Lorentz transformation. But inertial mass (as used in Newton 2) is real and is a function of particle speed: m_inertial = m_o/(1-v²/c²)^1/2. Its rest value equals the magnitude of gravitational mass: m_o=|m|.

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

Every sub-light-speed particle has charge. (“Uncharged” particles such as neutrons may have zero net charge, but they are composed of charged particles (quarks) whose summed charge equals zero).

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.

Example of the Substitution Rules in Use:

The energy density in an electric field is u_E=ε_oE²/2. Hence the energy density in a gravitational field is u_g=g²/8πG. Since g is imaginary, u_g is negative real. 

The energy density in a magnetic field is u_B=B²/2μ_o. Or, since 1/μ_o=ε_oc², the energy density in a gravitomagnetic field is u_O=c²O²/8πG. Like u_g, u_O is negative real.

————————————————————

Last updated on February 3, 2021.

1. See Special Relativity, A.P.French, Table 8-1.