Some Further Thoughts About Neutral Matter Interactions
Abstract. Discusses the analogies between mass/mass and charge/charge interactions.
1. Inertial Mass … Slightly More than Additive?
Let us assume that there is a solid sphere of charge q, at rest or moving at a constant velocity. The net interactive electric force between all the infinitesimal charge increments comprising the sphere is zero.
If the charge is accelerated, then additional, acceleration-induced components of the electric field result in a net electric force on the sphere of charge. Regardless of the charge’s sign, this net force points opposite to the acceleration. Indeed in non-relativistic cases it is proportional to the acceleration, and the constant of proportionality is commonly referred to as the charge’s electromagnetic mass:
Let us now assume that a second, equal charge q’ can be overlaid on the first charge. The two charges are accelerated equally. q experiences the force in Eq. 1_1, and q’ experiences the same force in its own acceleration-induced electric field. However, q experiences not only its own acceleration-induced field; it also experiences the acceleration-induced field of q’. Similarly for q’. The net force experienced by q is thus twice the force in Eq. 1_1, and similarly for q’. The net force acting on the combined charges, or on the sphere of charge with magnitude 2q, is not twice the force in Eq. 1_1; it is four times as much. Doubling the charge has quadrupled the inertial reaction force. In general, a charge’s electromagnetic mass is proportional to the charge squared.
The acceleration-induced components of an accelerated charge’s electric field are not limited to the space occupied by the charge. They also exist in the space surrounding the charge. Thus a second charge, placed next to the first, will experience the first charge’s acceleration-induced field as well as its own acceleration-induced field. In effect the inertia of the two side-by-side charges will be greater than twice the inertia of either charge alone.
We know now that "neutral matter" is invariably composed of charged particles. Atoms are comprised of electrons and protons and "uncharged" neutrons. But even the neutrons are thought now to be composed of charged quarks. How is it, then, that attaching a second neutron or atom to an identical first one only doubles the mass? It is because each neutron and atom contains zero net charge. And the acceleration-induced electromagnetic fields of the positive and negative constituent charges effectively sum to zero "outside of" each atom/neutron. So far as acceleration-induced, interactive electric forces go, they act almost exclusively on the charged particles within each atom. Indeed the neutral atom’s inertia is predominantly electromagnetic, but the powerful electromagnetic fields are generally limited to the atom’s interior. Newton of course had little or no knowledge of the charged particles that comprise all bodies of neutral matter, nor of the powerful electromagnetic forces at work within such bodies. What he did know was that proximal neutral bodies are not entirely unaware of each other; they attract one another weakly.
Fig. 1_1 shows two resting positive charges, attached by glass threads to two spheres of uncharged lead. Things have been sized so that the four bodies are in equilibrium in inertial frame K (their rest frame). The repulsive electrostatic force between the charges is precisely counteracted by the attractive gravitational force between the masses.
Uncharged Bodies and Charges in Equilibrium
If this system is viewed from frame K’, which moves to the right with constant velocity v relative to frame K, then relative to K’ everything moves to the left with the one speed v. If nothing else the charges and lead balls are at rest relative to one another in K’, as they are in K. Evidently equilibrium prevails in K’ as it does in K.
Now in K’ each charge has both an electric and a magnetic field. At either charge, the other charge’s magnetic field points perpendicular to the plane of the diagram and to the charge’s velocity. Thus in K’ each charge experiences a magnetic force in addition to the usual electric force. And it is clear that the magnetic force is attractive whereas the electric force is again repulsive. In short, the total Lorentz force in K’ is still repulsive but it is less than the electrostatic force in K. Since the four objects are still in equilibrium, the force of attraction between the lead balls must be less by an identical factor.
Altering the orientation of the charges and lead balls in frame K (e.g. rotating the system around an axis perpendicular to the figure’s plane) has no affect on the electrostatic and gravitational forces in that frame. However, in K’ alteration of the orientation in K may result in different Lorentz forces in K’. Yet here again equilibrium must prevail in K’ as in K. In general the gravitational force of attraction between the lead balls must, for every orientation, change in the same manner as the electromagnetic forces do. Indeed to the extent there is one unique transformation for d(mv)/dt, all forces that equate to d(mv)/dt must transform identically between inertial frames.
One might wonder if, in the realm of neutral matter interactions, there might be a field analogous to electromagnetism’s magnetic field, and an interactive force analogous to the magnetic force between charges. Such considerations might make us wonder whether masses are truly additive in the case of neutral bodies. To the extent the great preponderance of a neutral body’s inertia can be attributed to acceleration-induced electromagnetic forces within the body’s atoms, and to the extent atoms do not electromagnetically sense the existence of other nearby atoms, the inertial masses would be additive. But to the extent there might be acceleration-induced components to the gravitational field, not only right at an atom but reaching out and affecting other atoms in the environment, things might not be quite linear.
The force of gravity is of course minuscule compared to the electromagnetic forces. Any non-linearity (if it exists at all) would accordingly be relatively slight. But it is nonetheless interesting that the masses of neutral bodies might not be perfectly additive.
2. Some Ramifications of an Imaginary Gravitational Field.
Work must be done by some agent when two like-signed charges are pushed closer together. That is, the energy of the two-charge system is increased, and this increase rather elegantly equates to the gained energy in the electric field. Similar reasoning suggests that the energy of two attracting bodies of neutral matter must decrease if an agent allows them to move closer together.
As discussed in the last section, there are reasons to contemplate certain parallels between charge/charge and matter/matter interactions. A good starting point is the question, "How can two bodies of ‘positive’ matter attract one another, and not repel (as two like-signed charges do)?" A suggested solution might be that gravitational mass is mathematically imaginary. And the gravitational field engendered by a given body of "neutral" matter is also imaginary. Although the gravitational field vector of "positive" body A would point away from the body, "positive" body B would experience a gravitational force containing an i2=-1 term, and would accordingly be attracted to body A.
Assuming the energy in the gravitational field is proportional to the field squared (as the energy in the electric field is), gravitational field energy would always be negative. The total energy in the net field of bodies A and B would accordingly decrease if the bodies were allowed to move closer together.
The net, acceleration-induced gravitational "self" force of an accelerated body of neutral matter would point in the same direction as the acceleration! This would be an absurd result were it not for the fact that the neutral body’s inertia is all but entirely attributable to interactive electromagnetic forces within the constituent atoms (said forces acting between the constituent charged particles that comprise those atoms). This electromagnetic effect always points opposite to the acceleration. The gravitational effects, which point in the same direction as the acceleration, are much smaller than the electromagnetic effects. Consequently the inertia of two bodies of neutral matter, allowed to come into close proximity to one another, should be only very slightly less than twice the inertia of the individual bodies when they are infinitely separated.
The suggested analogue to the magnetic field of electromagnetism would also be imaginary. Thus the two lead balls in Fig. 1_1 would experience a small, magnetic analogue repulsive force in frame K’, in addition to the ever-present gravitational attractive force. In brief, the charges and lead balls are in equilibrium in frame K’ as they are in frame K.
Of course such parallels suggest that a single object of neutral matter (e.g. an atom) would emit waves when the object is made to oscillate, quite as a charge emits electromagnetic radiation. Since the relatively minuscule gravitational part of the overall inertial reaction force always points in the same direction as the driving force, it feebly "assists" the driving agent at all times and the energy in the waves would be negative. However, when incident upon other neutral bodies these imaginary waves would exert perfectly real forces!
Although the gravitational inertial effects are relatively slight, it might be possible to demonstrate that the inertia of two sheets of lead (for example) is very slightly less than twice the inertia of either sheet alone, when the sheets are in contact and are forced to accelerate as a single unit.