On Charge/Charge Mass/Mass Interaction Analogs

G.R.Dixon, 7/4/2009

Background/Acknowledgement.

In 1972 I took an undergraduate course in theoretical physics at MIT. The lecturer was the late Philip Morrison. He was a wonderful communicator. I made an office appointment with him and ran the following thought experiment by him.

Two identical particles, each of mass m and charge q, are at rest in inertial frame K. q and m are sized such that

.

(Clearly in MKS units q<<m.)

At any separation, d, the particles are in equilibrium and remain at rest. Of course the electric charges repel and the masses attract.

Regardless of the particle orientations in K, the Lorentz transformation indicates that they move with a common, constant velocity in every other inertial frame. In brief, the particles are in equilibrium in every inertial frame; the __net__ force on each particle is always zero.

Now the electromagnetic (Lorentz) force on either charge generally __varies__ from frame to frame. For example, in frame K (the rest frame) there is no magnetic force, whereas in K’ there may be a nonzero magnetic force. The only way the __total__ force on each particle can be zero in every frame would appear to be if the mass/mass interaction varies, from frame to frame, exactly as the charge/charge interaction does. This requirement suggests a parallel set of Maxwellian equations and a Lorentzian force law for the mass/mass interactions.

Well, Professor Morrison was clearly interested. He suggested that the Eotvos experiment demonstrated the equality of inertial and gravitational mass. But it wasn’t clear to me that the equality doesn’t break down at high speeds (with the inertial mass increasing relativistically with increasing speed, while the gravitational mass, like electric charge, remains invariant and independent of speed). We didn’t settle anything in that office session, but in the next class lecture he digressed and mentioned how the work expended to change a static configuration of __repelling__ electric charges always equals the change in the electric field energy. And he explained that the only way such an agreement could occur in the case of __attracting__ masses would be if the gravitational field were mathematically __imaginary__, so that its square (and hence its energy density) would always be __negative__.

I have toyed with this idea off and on through the years, and have discussed it with others. For the most part General Relativists have rejected it and I, knowing little about General Relativity, haven’t strenuously argued with them. But it did turn out that one highly respected contributor to the General Relativity literature __did__ feel that the "gravitomagnetic" concept might have its place when the masses are not very great.

In any case, most recently I briefly mentioned the charge/charge mass/mass parallel idea on MySpace.com/physics forum, and a fellow blogger asked me to elucidate. That is the grounds for the present article. I am placing it here on the Maxwell Society website, where he and others can view it, because it is too large for a MySpace blog.

G.R.Dixon, July 4, 2009.

1. Introduction.

From practically the moment Coulomb published his charge/charge force law,

,

others noted the mathematical sameness to the older, Newtonian mass/mass law:

.

Subsequent developments by Maxwell and Lorentz indicated that there is more to charge/charge interactions than Coulomb’s action-at-a-distance law. And the thought experiment discussed at the beginning of this article suggests that there may also be more to mass/mass interactions than Newton’s gravitational law. (General Relativists would of course agree.)

In this article we postulate, in a mostly qualitative way, parallels between charge/charge and mass/mass interactions. We begin by stating Maxwell’s equations, and their suggested parallels in the mass/mass domain.

2. Maxwellian Equations.

Maxwell (chge/chge)

Maxwellian Parallels (mass/mass)

E_{A}, or "E Analog," is the gravitational field. B_{A} or "B Analog" is the gravitomagnetic field. r_{m} is gravitational mass density.

Notes.

1. In the Maxwellian parallels, __E___{A}, r_{m}, __B___{A}, __j___{m} are mathematically imaginary. Thus, for example, __E___{A}^{2} is real __negative__.

2. g_{o} = 1/4pG.

3. The energy density of the gravitational field is

.

4. Mass is a complex quantity. Its real part is the inertial mass (m_{I}) of Newtonian mechanics:

,

Its gravitational, imaginary part (m_{g}), like charge, does not vary with speed.

3. Examples.

Examples of parallels between charge/charge and mass/mass interactions follow. Several of the charge/charge cases are elaborated upon in other articles featured on this site. Links to those articles are embedded where appropriate.

1. The Gravitational Field of a Resting, Point Mass.

2. The Force Between 2 Resting, Point Masses is Real and Attractive.

3. The Energy Density in the Gravitational Field is Negative.

Consequence: If 2 masses are allowed to move closer together, the net gravitational field energy becomes more negative.

4. The Gravitational Field Energy of a Solid Sphere of Gravitational Mass is Negative.

, a = sphere’s radius

5. If the Radius of a Sphere of Gravitational Mass is Decreased, then |U| Increases.

That is, U becomes more negative. The work done by the external agent, who allows the radius to decrease, is negative and equals the change in U.

6. Particles in Equilibrium in One Inertial Frame Must be in Equilibrium in All Inertial Frames.

Reference: "When Gravity Balances the Lorentz Force"

For example, given an inertial frame moving in any direction relative to rest frame K, the Lorentz transformation indicates that the particles must move with a common, constant velocity relative to K’. By Newton, then, it must generally be true for any particle that

.

7. B-Analog (gravitomagnetic) Forces in Galaxies.

Reference: "On the Constant Dimensions of Rotating Galaxies"

A rotating disc of stars has a __B___{A} field that points in the same direction as the galactic angular velocity vector at points in the disc. Thus the B-Analog force,

,

acting on a star that is r distant from the disc center, points toward the galaxy’s center and adds to the gravitational force. The motions of the galaxy’s stars cannot be explained solely by the gravitational force.

8. Perihelion Precession in Planet Orbits.

Reference: "Orbital Precession in an Inverse Square Force Field"

The sun rotates and theoretically has a dipolar __B___{A} field that points opposite to its angular velocity vector, in its external equatorial plane. Planets orbiting the sun in its equatorial plane experience combined attractive gravitational and repulsive gravitomagnetic forces. The net effect is to cause the axes of planetary elliptic orbits to precess in time.

9. Inertia.

Reference: "On the Non-Relativistic Electrodynamics of an Isolated Spherical Shell of Charge"

As a charge accelerates, its own time-varying fields induce an __E__ field component right at the charge. If the charge is positive, then this __E__ field component points opposite to __a__. The constant of proportionality is the charge’s electromagnetic mass. As a mass accelerates, there is an induced __E___{A} right at the mass, and this field points in the *same* direction as __a__. The mass experiences a self, inertial force that points opposite to __a__.

10. Maxwell/Lorentz Work Equally Well in All Inertial Frames.

Reference: "Learning to Love the Lorentz Transformation"

Given a very light, negatively charged satellite in inertial frame K, orbiting in a circle around a much more massive, fixed, positively charged central body, the satellite’s motion relative to K’ can be determined either (1) by applying the Lorentz transformation to space-time points in K, or (2) by applying the electrodynamics of Maxwell/Lorentz (with the dependence of m_{I} factored in). The same is expected if the central body and satellite are uncharged. That is, it should be possible to compute the motion in K’ by application of the Lorentz transformation or by applying the Maxwell/Lorentz Analogs. Among other things, it is demonstrated in both cases that length contraction and time dilation lie implicit within the electrodynamics of Maxwell/Lorentz and/or the electrodynamical analogs.