The cause of length contraction

The relativistic length contraction and time dilation factors in an inertial frame of reference Σ in which light propagates in symmetrical conditions in all directions and clocks have been synchronized using Einstein's method are of a very simple mathematical form and involve only two parameters: the speed v at which a body or clock travels in Σ and the speed c at which electric field disturbances, and perhaps any force field disturbances, travel in Σ.

This suggests that it should be possible to deduce length contraction and time dilation in Σ - and therefore the constancy of the two-way speed of light in any spatial inertial frame of reference - from a simple model of how field disturbances propagate in force fields.

In this post, I would like to make a start by attempting to deduce "electric length contraction" from a really simple model of electricity. In other words, I would like to show that all charged particles that are in a state of equilibrium in a measuring rod that is stationary in Σ will be in the same state of equilibrium in the same measuring rod moving at v through Σ if and only if the distances between them in the direction of movement are reduced by the relativistic factor. Is my sphere model of electricity up to the task?

Let me recall the basic idea behind the sphere model: the field around any charge that is stationary in Σ is represented as a series of concentric spheres with a particular sphere density λΣ:



When the charge is locally accelerated to a speed v in Σ and then continues to move at that speed, the information about the acceleration spreads through the spheres at the speed c, leading to the following situation after a time t has passed:

The basic sphere model law is that any electric field disturbance always traverses the same number of spheres in the same amount of time, as measured in Σ.

In this diagram, therefore, P1P2 = vt and P1P3 = ct. If the angle between P2P3 and v is α, the cosine rule can be applied to the triangle P1P2P3 to calculate P2P3, and that can in turn be used to determine the sphere density λα on P2P3 as a function of λΣ, v, c and α:


(Equation 1)

Let me now consider the Coulomb law for charges q1 and q2 which are stationary in Σ and separated by the distance r.


(Equation 2)

In the sphere model, what matters is not so much the distance r as such but, for a given sphere density λΣ, the number n1 of sphere surfaces around q1 cutting through r on the one hand, and the number n2 of sphere surfaces around q2 cutting through r on the other:


This is because what acts at q2 is the n1-th sphere surface around q1, and what acts at q1 is the n2-th sphere surface around q2.

For symmetry reasons, n2 = n1 , so we have:


or


Inserting this into Equation 2, the Coulomb law can be rewritten as follows:


(Equation 3)

Now suppose the two charges have been accelerated to a speed v in Σ and are still separated by the distance r as measured in Σ.

The number of sphere surfaces surrounding q1 and cutting through r is then increased to n1', and the number of sphere surfaces surrounding q2 and cutting through r is reduced to n2', as shown below:


Using Equation 1, we obtain for the sphere densities λα=0 on the leading side of q1 and λα=π on the trailing side of q2:



and



[PS: As indicated here, a more general analysis shows that the two densities that need to be considered are the density of q1 spheres over r (a) along the line connecting the two charges on the side facing towards q2 and (b) along the line connecting the two charges on the side facing away from q2 . The result in the case of two charges moving at the same velocity is the same, but in more general situations this is the approach that must be taken.]

Inserting n1' and n2' into (Equation 3) to replace n1 and n2, we thus obtain for the force between the two charges









(Equation 4)

It turns out that this modified Coulomb law for a pair of moving charges even holds for any pair of sphere densities λα and λα+π , in other words it holds for any angle α between the connecting line between the two charges and v. However, if π>α >0 , then F in the direction that is perpendicular to v is further modified by magnetic effects, and I have not yet incorporated such effects into my model.

Limiting myself therefore to the case of α=0, Equation 4 shows that the electric force between q1 and q2 moving at v through Σ and separated by r is the same as the electric force between two charges q1 and q2 that are stationary in Σ and separated by r(1 - v2/c2)-1/2.

Conversely then, if the stationary charges q1 and q2 are separated by r in Σ and are then both accelerated to v, if the force between the moving q1 and q2 is to be the same as it was between the stationary q1 and q2, then the distance between them in terms of Σ coordinates must be reduced to:



And this is the essence of "electric length contraction": all the charged particles that are in a state of equilibrium in a measuring rod that is stationary in Σ will be in the same state of equilibrium in the same measuring rod moving at v through Σ if and only if the distances between them in the direction of movement are reduced by the factor (1 - v2/c2)1/2.

What I have shown is that the "electric length contraction" of moving bodies by the relativistic factor as seen from an inertial frame of reference in which light propagates in symmetrical conditions in all directions and clocks are Einstein-adjusted is a natural consequence of my really simple sphere model of electricity.

I regard this as a major result, at least on a par with my earlier findings concerning "simultaneity" and "one-way speed" in special relativity.

Am I deluded? I will discuss that question in my next post.