Concluding remarks

The time has now come to draw my reflections to a close and to summarise what I have done. My starting point was the startling statement of special relativity that the one-way speed of light is constant for every observer. I have shown that this statement is only true in a very formal sense, based on a definition of time coordinates which violates the principles of causality and of simultaneity as normally understood.

This key finding left a big question wide open: why is the two-way speed of light constant for every observer? The answer I give can be summarised as follows:

In every point in space and time, there is one local coordinate system in which light travels in symmetric conditions in every direction. Movement of any body in relation to that coordinate system contracts the space occupied by that body because the forces that hold the body together change. This leads to the finding of a constant two-way speed of light.

I have demonstrated how this happens for electric forces between uniformly moving charges. Importantly, I have done so without using the concept of magnetism between moving electric charges, which is an unexplained and ultimately unphysical entity in classical physics.

I have extended this new approach to electrodynamics to accelerated charges. This has yielded results that only partially correspond to classical theory. I am unable to say how close my findings are to reality. I will have to leave verifying this, and if necessary further developing my approach to accelerated charges, to future generations of physicists. As for myself, I feel I have probed the questions which preoccupied me from my teens in sufficient depth for now. Others can take over from here.

The direction of the force on a moving charge caused by the acceleration of another charge

 It is now time to get on to the final bit: the direction of the force due to acceleration exerted by a charge q₁, moving at the velocity [u] and with the acceleration [a] at the retarded time, on a charge q₂ moving at the velocity v at the current time.

The retarded time is the current time minus the time it took for information on q₁, including [u] and [a], to reach q₂ at the current time.

I am considering this question for a frame of reference Σ in which electric conditions are isotropic for charges at rest. The one-way speed of light is thus c in this frame of reference. The following image presents the basic situation:

Here, q₁* is the position q₁ would be in if it had continued at the constant velocity [u] it had at the retarded time.

In a previous post, I have presented my result for the magnitude of the force on q₂ as measured in the rest frame of q₂. This turns out to be the same as the classical result for relatively small velocities u, as in electric currents, and any velocity v. It is also similar to classical results where the speeds of q₁ and q₂ are roughly < 0.1c. When both speeds are larger, there is a growing mismatch with the classical result.

In this post, I would like to determine the direction of the force on q₂ in Σ. Note that the direction of the force in the moving frame is not well defined because I have not introduced any coordinates for that frame. By contrast, the strength of the force on q₂ as measured in the rest frame of q₂, for example by a co-moving spring balance, does not rely on any time or space coordinates.

I would like to determine the direction of the force on q₂ without using the concept of the magnetic field or the results of special relativity. This is the way I have worked so far in this blog, because the magnetic field and special relativity both cry out for an explanation. My hope is to provide such an explanation. I have already presented my results for the case of charges moving at constant velocities, which is sufficient to provide a partial explanation for special relativity. It also explains magnetic effects on charges moving at constant velocities. This post will fit in the last missing piece, which is the direction of the force exerted by a charge q₁, moving at the velocity u and with the acceleration a at the retarded time, on a charge q₂ moving at the velocity v at the current time in Σ.

Let us take a deep breath after these preliminaries, and then proceed.

Let me first note down the classical result for the direction vector of the force on a moving charge q₂ caused by the acceleration of another charge q₁. The direction I am talking about is the direction of the force in the frame of reference moving at v as seen from the stationary frame of reference Σ.

Classically, we can calculate that direction by first determining the direction vector of the force in the frame of reference moving at v:

We can then multiply the x-component of that direction vector by (1-v²/c²)^0.5 to obtain the direction in Σ:

What do I obtain for the direction in my approach, which does not use magnetism or special relativity?

I think it makes sense to adapt the solution for the case of two charges moving at constant velocities (see Equation 9 in this article) to the case of q₁ being accelerated by [a]. For constant velocities, the direction of the force is given by

As is demonstrated here, this direction turns out to be the same as the classical direction.

Now consider the case of q₁ being accelerated by [a]. We have already seen that in this case we have to find the projection of [a] that is perpendicular to r₁₂*, and to project that onto a vector b_Ʇ that is perpendicular to n. Let the unit vector of b_Ʇ be

It then turned out that we had to use n_Ʇ and – n_Ʇ to calculate the magnitude, so I will use these vectors to calculate the direction, too. Similar to the case of constant velocities, I form

If we multiply this by the product of the two denominators, we get

The direction of this vector corresponds to about half the elements of the direction of the classical solution:

Is there something missing from my solution? I do not know. It is perhaps difficult to see, for example, why there should be a large z-component when neither n_Ʇ nor v contain such a component. Be that as it may, I cannot easily see what I should add to my solution, so I will have to leave it as it is.

The strength of the force an accelerated point charge exerts on another point charge – 2

I have made a bit more progress on this question, and this post will just serve to pull everything together that I know.

As I’ve said before, my intention is to derive the force an accelerated point charge q₁ exerts on another point charge q₂ without any reference to magnetism or special relativity. So far, I have looked into the magnitude of the force on q₂ as measured in the rest frame of q₂. Let me first repeat the basic image that describes the situation:

Here, q₁ is shown at the retarded time, in other words the time before the current time when information about the velocity and acceleration of q₁ was sent out to q₂ at the speed c to reach q₂ at the current time.

My formula for the magnitude of the force is:

with the frequency factor (originally developed here, here, here and here)

The frequency factor describes the frequency of the interaction compared to the frequency when both q₁ and q₂ are at rest. In the two equations above, if the acceleration at the retarded time is

  

and if we define

then

is the component of [a] that is vertical to r₁₂*, and

The question that still needs to be answered is the origin of the expressions

and

in the formula for the frequency factor, and of

in the initial denominator.

The following image may be helpful:

The vectors  and  and the length d are all explained in a previous post. We can define a vector

We can then establish a plane equation for a plane that is perpendicular to [r₁₂] and, at the time t = 0, at a distance d = d₁ from q₂, which is located at (0, 0, 0):

At t = 0, the plane can be visualised as follows:

The black dot lies in the plane, which moves at the speed  towards (0, 0, 0). We can now set

and set t = t₁ in the plane equation to calculate at what time t₁ the plane meets q₂, which moves at v along the x-axis.

We can then determine

where t₀ is the corresponding time for u=v=0.

We can set up a similar plane equation for the other solution of d = d₂ (see this previous post):

We can again set

and set t = t₂ in the plane equation to calculate at what time t₂ the plane meets q₂, which moves at v along the x-axis. We can then determine

where, again, t₀ is the corresponding time for u=v=0. From this, (2) follows. We can also begin to see the reason for (3) and (4), even if the explanation is not complete since I cannot at present give full explanations for the plane equations (6) and (8).

I have also yet to give a fuller explanation for the term sin²β in the denominator of (1). This term also appears in the denominator of my version of the equation describing the forces between charges moving at constant velocities, where it is introduced in section ‘IV. THE ANGLE FACTOR’. I think it may have the same origin in (1).

Equation (1) turns out to be very similar to the classical solution, but it is not identical. As can be seen in detail here, 35 terms of the expression under the square root are the same in both versions, but a few terms with exponents of c ≥ 4 in the denominator are not the same. In more detail, two terms with c⁴, two terms with c⁵, and five terms with c⁶ in the denominator are not the same. That is a total of 9 terms with a denominator of c⁴, c⁵, and c⁶. Provided all my calculations are correct.

I should perhaps say that the additions to my solution proposed in the earlier posts entitled ‘Progress’ and ‘Further progress’ are slightly uncertain. They have been made on the assumption that the classical solution is correct, but I do not know how certain that is.

I will now try to establish the direction in which the force acts according to my solution, and compare it with the classical result. I do not anticipate complete equivalence.