In the 19
th century, electromagnetic radiation/waves were discovered [
4]. The phenomena were explained by the wave equation derived from Maxwell’s equations. Then, Maxwell and many contemporaries believed in local interaction (retarded) instead of action at a distance [
5]. In Maxwell-Lorentz electrodynamics, the interaction between charged particles is mediated by fields. One particle emits a field, which propagates at light speed. The field reaches and interacts with another particle, causing action.
In the early 20
th century, Einstein introduced special relativity with the assumption of constant light speed in vacuum [
6]. In this theory, no information or force can travel faster than the speed of light. Thus interaction of matter needs to obey locality, i.e. either retarded action at a distance or field mediated local interaction. This has become the accepted view in mainstream physics today, replacing the concept of instantaneous action at a distance. Retarded interaction was then introduced into physical laws, including gravitational theory [
7], and Weber’s electrodynamics [
8].
Nevertheless, there persist valid arguments both for and against the two theories, instantaneous action at a distance and retarded interaction [
9]. Field theory (retarded interaction) has the distinct advantage of explaining most of the electromagnetic wave phenomena. However, it is not without its drawbacks [
10], particularly in terms of the non-straightforward treatment of energy and momentum conservation among interacting particles [
11,
12]. Whereas instantaneous action at a distance has the distinct advantage of obeying Newton’s third law, thus naturally obeys energy and momentum conservation without seeking contributions from fields, although it does not directly account for the propagation of waves (signal/energy).
In earlier studies, attempts were made to reconcile both locality and instantaneous action at a distance [
2]. For instance, the direct action theory was introduced [
13], aiming to maintain consistency with wave propagation. However, this approach assumes both retarded and advanced field propagation, and it does not assign properties to the field. Despite creating an appearance of direct action at a distance, it is not strictly instantaneous. Another approach involves starting with Maxwell’s equations and derives Weber’s force under certain conditions [
14], resulting in a force that has an apparent instantaneous action at a distance. Also, instantaneous action at a distance has been seen as a representation from specific gauge choices of electromagnetic potentials [
15].
In this paper, we try to shed some light on this fundamental question from a new perspective. We first introduce Weber’s force law and the postulate of vacuum polarization. Subsequently, we consider instances of Weber’s force at play in both empty space and within a polarizable vacuum. The latter case demonstrates the locality of interaction within Weber’s electrodynamics given the vacuum polarization postulate. Finally, we discuss some potential implications of this new approach.
Weber’s Electrodynamics
Weber extended Coulomb’s static force law to the dynamic regime by incorporating the effect of charge velocity and acceleration [
16]. As an alternative to Maxwell-Lorentz electrodynamics, Weber’s electrodynamics has been successfully applied to a number of electric phenomena [
17], extended to high velocity particles [
18,
19], and used to derive an electric wave equation in the rest frame of the media (polarizable vacuum) [
20]. Compared to Maxwell-Lorentz electromagnetism, Weber’s electrodynamics gives a much simpler form for particle-particle interaction forces.
According to Weber’s electrodynamics, two charge particles interact with each other, determined by their relative position, velocity and acceleration. The force exerted by one particle on the other is given by [
21]
Where and are two electrical charges, is the force that charge exerts on charge , is the distance between the two charges, is the unit vector pointing from charge to charge , and are velocity and acceleration of charge relative to charge , is the dielectric constant and is the speed of light.
Some view Weber’s electrodynamics as an instantaneous action at a distance [
16], even though the concept of retarded action has previously been successfully incorporated into a Weberian framework [
8]. Nevertheless, in this paper, we opt to retain the instantaneous action of Weber’s electrodynamics.
Vacuum Polarization
In quantum electrodynamics, vacuum polarization describes a process in which a background electromagnetic field produces virtual electron–positron pairs [
22,
23]. Similar and different to this well established concept, the vacuum was recently postulated to be a “sea” filled with positive-negative charge pairs [
20]. In the absence of an external electric field, the positive-negative charge pairs can fully overlap each other (
Figure 1) rendering the vacuum electrically neutral or unpolarized. However, the introduction of an external electric field prompts a relative displacement between positive and negative charges (
Figure 1), resulting in vacuum polarization. It was demonstrated how the vacuum polarization postulate [
24] can directly lead to electric wave propagation within Weber’s electrodynamics [
20]. In this paper, we explore how instantaneous action at a distance may behave differently under this polarizable vacuum concept.
In this paper we choose a reference frame in which the vacuum is stationary. The displacement, velocity and acceleration of negative charge has the same value as that of positive charge within each pair, but with opposite direction.
where
,
, and
are displacement, velocity, and acceleration of positive charge,
,
, and
are displacement, velocity, and acceleration of negative charge.
Charge Interaction in Polarizable Vacuum
Let’s speculate that the vacuum is not fully empty, instead having the property of vacuum polarization (
Figure 1). At time zero, a charge
assumed emerging at the origin. According to Weber’s electrodynamics, the emerging charge instantaneously exerts a force to all the positive-negative charge pairs in the vacuum. In the meantime, each charge pair also exerts a force to all other charge pairs (
Figure 3).
For example, the point charge
exerts a force
at a small volume of charge
at
. The force can be calculated with equation (1),
where is the acceleration of charge and is the charge density.
In the meantime, all other charges
(positive charges contained in the charge pairs of polarized vacuum) will exert a force
at the charge
at
. The force can be calculated with equation (1) and integration.
Where, is the position vector of , is the acceleration of , and the integration is with respect to . Because of spherical symmetry, .
And, all other charges
(negative charge in charge pairs of vacuum) will exert a force
at the charge
at
.
Where, is the position vector of and is the acceleration of .
Let’s neglect the mass of charge
(the mass of positive-negative charges in vacuum is likely much smaller compared to that of physical particles) and its acceleration force (mass times acceleration). Additionally, the polarization force within a positive–negative charge pair may also be neglected since it is likely small compared to the force exerted by the nearby volume of charges. Then the force summation of all the forces exerted at the charge
equals zero.
After some simplification,
Equation (10) holds for any at time zero. The unknown variable is . Since equation (10) is a nonlinear equation, it is difficult to solve it in a straightforward manner.
To get a solution of equation (10), let’s look into the point charge first. The point charge
may be seen as an integration of a delta function at the origin,
where
represents the whole vacuum space, and
Delta function has been routinely used in physics and engineering to model point properties. Delta function value is zero everywhere except at coordinate zero, and its integral over the entire coordinate is equal to one. Thus delta function can be seen as a “local” function, so does the gradient of delta function.
We can assume a solution that takes the form , where is a constant to be determined, and is the gradient of a delta function. For any given non-zero , , thus . Then the solution has a property that , for any given .
Let’s explore individual components of equation (10) with the assumed solution. Since
, we have
. Thus
Because
is anti-symmetric for
and
, and
for any given
, we have
Using equations (13, 14), equation (10) can be simplified to
Since
where
, equation (15) can be further simplified to
With further simplification, we have
If we insert our presumed expression for
into equation (17), we obtain
Here, we have used a spherical coordinate system. After integration along
and
,
Further integrating, we obtain
Re-arranging for
yields
,
Allowing us to arrive at a solution for the acceleration (solution uniqueness will be discussed in the next section),
Thus, the distribution of acceleration is:
Here, we have shown that our proposed solution satisfies equation (10) at time zero within the framework of the postulated polarizable vacuum. While the emerging charge is a local perturbation (delta function, equation 12), the acceleration caused by this charge is also a local phenomenon (gradient of delta function, equation 23) (
Figure 4). This “local” effect appears to be aligned with the principle of locality.
The above finding may be understood in this way: The emerging charge affects the nearby vacuum first, causing acceleration. The vacuum at a distance does not immediately react to the appearance of the emerging charge, even though the emerging charge already applied an action on it. The action from the emerging charge and the action from the neighboring accelerating vacuum cancel each other (as manifested by zero acceleration at a distance).