On the Constant Dimensions of Rotating Galaxies
A topic of speculation in present day cosmology is the more or less static diameters of galaxies. Both Newton’s Law of Universal Gravitation and Einstein’s General Relativity Theory evidently fail to explain how these objects are able to maintain their shape. For given their rates of rotation, these massive systems should theoretically disperse with the passage of time. "Something extra" is theoretically needed to hold them together. "Dark" (or unseen) matter and/or black holes have both been suggested. Yet another candidate is tentatively suggested below.
Fig. 1 shows two small, like-sign charges that are permanently at rest in inertial frame K. The electrostatic forces of repulsion are counteracted by (shall we say) gravitational forces of attraction.
Charges and Neutral Matter in Equilibrium
Now Newton’s Law of Universal Gravitation and Coulomb’s Law are both inverse square. Mathematically the equilibrium in Fig. 1 can be explained by assuming that the gravitational masses and the gravitational field are imaginary, say
The gravitational force, mgravitationalg, would contain an i2=-1 term and be attractive.
Viewed from inertial frame K’ let us say that the system (and all of its parts) moves to the left with constant speed u. Equilibrium prevails in K’ as in K. But in K’ each charge experiences an attractive magnetic force toward the other charge, in addition to the repulsive electric force. According to both Maxwell/Lorentz and the general force transformation, the repulsive force between the charges is less in K’ than it is in K. Thus the attractive force between the masses must be less by the same factor.
It is noteworthy that other orientations in K must also yield equilibrium in K’, even though the Lorentz force in K’ will vary with changes in the orientation in K.
In any case the requirement is that the mass/mass interactive force transform quantitatively as the charge/charge interactive (Lorentz) force transforms. An obvious suggestion is that there be a mass/mass interaction analogous to the magnetic force in Maxwell/Lorentz. For example, we might speculate that there is a nonzero O’ field in K’, quite as there is a nonzero B’ field. Like the gravitational field, this O’ field might be supposed to be mathematically imaginary. Viewed from K’, the B’ field at q1 points into the page (Fig. 1), as does the O’ field at m1. Thus q1u x B’ points down, but im1u x O’ would point up.
If a spiral-armed galaxy is approximated as a uniform disc of uncharged matter, lying in the xz-plane and with an angular velocity (or momentum) in the positive y-direction, then at all points in the disc the theoretical O field would point in the positive y-direction. (Outside, in the xz-plane, O would point in the negative y-direction.) Circling in this field, the disc’s matter would theoretically experience a centripetal, magnetic-like force in addition to the centripetal gravitational force. The suggestion here is that this additional force might be sufficient to hold the galaxy together --- a feat that Newtonian gravity and General Relativity theory both reportedly fall short of accomplishing.