On the nature of the force acting on a charged classical particle deviated from its geodesic path in a gravitational field

In general relativity the gravitational field is a manifestation of spacetime curvature and unlike the electromagnetic field is not a force field. A particle falling in a gravitational field is represented by a geodesic worldline which means that no force is acting on it. If the particle is at rest in a gravitational field, however, its worldline is no longer geodesic and it is subjected to a force. The nature of that force is an open question in general relativity. The aim of this paper is to outline an approach toward resolving it in the case of classical charged particles which was initiated by Fermi in 1921. General relativity provides a consistent no-force explanation of gravitational interaction of bodies following geodesic paths. However, it is silent on the nature of the very force we regard as gravitational the force acting upon a body deviated from its geodesic path due to its being at rest in a gravitational field. In both special relativity (in flat spacetime) and general relativity (in curved spacetime) a particle offering no resistance to its motion is represented by a geodesic worldline. As the non-resistant motion of a particle is regarded as inertial a particle whose worldline is geodesic is moving by inertia. In both special and general relativity a particle whose worldline is not geodesic is prevented from moving by inertia and therefore is subjected to an inertial force. Hence a particle supported in a gravitational field is deviated from its geodesic path (i.e. prevented from moving by inertia) which means that the force acting on it is not gravitational but inertial in origin. The mass causing the spacetime curvature determines the shape of the particle’s geodesic worldline, and in general which reference frames are inertial [7], but the force arising when the particle is deviated from its geodesic path originates neither from that mass nor from the distant masses (as Mach proposed). This force has the same origin as the force acting on a test particle prevented from following a geodesic path in an empty spacetime. It should be stressed that in general relativity the force acting on a particle deviated from its geodesic path due to its being at rest in a gravitational field is non-gravitational in origin. As Rindler put it ”ironically, instead of explaining inertial forces as gravitational... in the spirit of Mach, Einstein explained gravitational forces as inertial” [8]. This is the reason why ”there is no such thing as the force of gravity” in general relativity [9]. Here it will be shown that a corollary of general relativity that the propagation of light in a gravitational field is anisotropic in conjunction with the classical electromagnetic mass theory [1][6] sheds some light on the nature of the force acting on a classical charged particle deviated from its geodesic path. Consider a classical electron [10] at rest in the non-inertial reference frame N of an observer supported in the Earth’s gravitational field. Following Lorentz [4] and Abraham [5] we assume that the electron charge is uniformly distributed on a spherical shell. The repulsion of the charge elements of an electron in uniform motion in flat spacetime cancels out exactly and there is no net force acting on the electron. As we shall see below, however, the average anisotropic velocity of light in N (i) gives rise to a self-force acting on an electron deviated from its geodesic path by disturbing the balance of the mutual repulsion of its charge elements, and (ii) makes a free electron fall in N with an acceleration g in order to balance the repulsion of its charge elements. No force is acting upon a falling electron (whose worldline is geodesic) but if it is prevented from falling (i.e. deviated from its geodesic path) the average velocity of light with respect to it becomes anisotropic and disturbs the balance of the mutual repulsion of the elements of its charge