So, how did
I arrive at the sphere model equation describing the forces between
two uniformly moving point charges as measured by co-moving spring balances? I
started from Coulomb's law of electrostatics:
(1)
The law
describes the force between two stationary charges q1 and q2
separated by the distance r. We know from classical electromagnetism and
special relativity that the forces between two charges q1 and
q2 moving at constant velocities u and v, as
measured by co-moving spring balances, look rather different from this: they
depend on those velocities, the distance between the charges and the speed of
light in a rather complicated way. Classical electromagnetism describes the
force exerted by q1 on q2 in this more
general case by means of two fields: the electric field E associated
with q1 and the magnetic field B associated with q1.
These need to be inserted into the Lorentz force law
(2)
and a
relativistic force transformation needs to be applied to obtain the force on q2
as measured by a co-moving spring balance. I was convinced, however, that it
should be possible to explain the forces between moving charges as measured by
co-moving spring balances much more simply, in terms of a single 'electric'
field surrounding any charge in any state of motion. The first question I
needed to answer was what shape this single 'electric' field might take.
Step 1
I assumed
that in any point in space and time there is a uniformly moving frame of
reference S in which the electric field of any charge at rest - i.e. the electric properties
of the space surrounding such a charge - is isotropic (the same in every
direction). To me that meant that the field can be represented as a succession
of concentric and equally spaced sphere surfaces. In S, any electric
field disturbances resulting from the acceleration of a charge at rest would
thus propagate in isotropic conditions. It would therefore be possible to
synchronize clocks at rest in S using Einstein's clock adjustment
procedure, and as a result the speed c at which electric disturbances
propagate in S would be the same in every direction.
Note that
this set of assumptions is much weaker than those made in classical theory,
where it is somewhat implausibly assumed that the electric properties of the space surrounding
charges at rest are isotropic in any uniformly moving frame of reference.
What is more, in the sphere model there is no need to accept the constancy of
the two-way speed of light in every uniformly moving frame of reference
as a fundamental law of nature that cannot be explained any further, as is done
in classical theory.
With all
this in place, it was clear that accelerating a charge from rest in S to
a particular velocity u would distort the sphere surface configuration
surrounding it as follows:
Fig. 1
The sphere
fields of two charges q1 and q2 travelling
at constant velocities u and v in S and separated by the
distance r can consequently be represented as follows:
Fig. 2
My
hypothesis was now the following: the past accelerations of any two charges q1
and q2 moving at u and v in S bring
about changes in the sphere field configuration surrounding the charges, and
corresponding changes in the flow of information between them, which result in
changes in the forces between those charges, as measured by co-moving spring
balances, compared to the static case. More specifically, I hypothesised that
it should be possible to describe those changes in the sphere field
configuration, and corresponding changes in the flow of information, by a
series of sphere field parameters x1, x2, x3…
and a direction vector y so that (1) would become
(3)
Let me
pause here for a moment to reflect on the significance of this hypothesis: if
true, it would mean that the forces between uniformly moving charges as
measured by co-moving spring balances could be explained with reference to a
single field, the 'sphere field', and without the need to perform any
relativistic transformation. The sphere model would thus be able to explain the
mysterious appearance of 'magnetic forces' in classical theory. What is more,
given the distortions in the sphere fields of charges moving in S, it
would imply that locally Einstein clock adjustment would synchronize clocks
only in S and not in any other uniformly moving frame of reference. This
would instantly put paid to some rather outlandish ideas that are floating
around in the literature based on misinterpretations of special relativity,
such as the idea that we live in a 'block universe'. As if that wasn't enough,
the sphere model would explain how the electric forces between moving charges
are transformed in a manner that is consistent with length contraction. It
would thus provide a partial explanation of length contraction.
But how might
I be able to confirm my hypothesis? I decided to pursue a two-pronged approach.
Step 2
I assumed
that the classical results for the forces between uniformly moving charges as
measured by co-moving spring balances are empirically correct. This enabled me
to work out what I had to aim for in my search for the sphere model
parameters x1, x2,
x3… and the direction vector y. For simplicity, I
decided to define a cartesian coordinate system in which the charge q2
is located at the origin and moves in the direction of the x-axis, while
the charge q1 is located in the xy-plane, as follows:
Fig. 3
It is also
convenient to define an additional angle α as follows:
This is not
an independent parameter and the following relationship holds:
In terms of
this coordinate system, using the formulae for the electric and magnetic fields
of moving point charges, the Lorentz force law, and the transformations of
special relativity, classical theory tells us that the magnitude of the force
exerted by q1 on q2 as measured by a
co-moving spring balance is as follows:
(4)
And the
direction of that force as seen from S (i.e. the direction in which q2
is accelerated in S) is parallel to the vector
(5)
The task of
finding sphere model parameters that would produce this magnitude and this
direction seemed formidable. One of the difficulties I saw was that the
magnitude was not particularly factorised. Specifically, I saw little hope of
finding a single sphere model factor corresponding to the square root term in
(4). The first thing I did, therefore, was to play around a bit with (4). I
soon found that it could be expressed as follows:
(6)
This looked
much more promising. Note how there are now several terms in the equation which
are of the form a2 - b2. This factorises
nicely into (a + b)(a - b). My task now looked a
little bit more manageable but still quite daunting. How should I go about
finding sphere model parameters that would produce (5) and (6)?
Step 3
I started
by simply having a look to see which aspects of the sphere field configuration
in Figure 2 were different from the sphere field configuration in the static
case. The first that struck me was the q1 sphere surface
density along the line connecting the charges. This was clearly different from
the density that pertains in the static case (whatever precisely the magnitude
of that density might be).
I had soon
gathered a few pertinent results: a) the q1 sphere surface density
along the line connecting the charges is constant on the 'near side' of q1
(the side on which q2 is located) and on the 'far side' of q1
(the side on which q2 is not located); b) as can be seen in
Figure 4, information that travels from q1 to q2
from the location of q1 at the 'retarded time' (t0
in Figure 4), such that it arrives at q2 at the 'current
time' (t1 in Figure 4), travels through the q1
sphere field along the line connecting the charges at the current time, in
other words the sphere surface densities along that line in its current position are
potentially relevant bits of information transmitted from q1
to q2; c) if I define the 'density factors' as the factors by
which these densities are different from the density in the static case, and if
I multiply the near- and far-side density factors, then I obtain one of the
terms in (6), namely (1 - u2/c2). It looked
like I had found my first sphere model parameter x1!
Fig. 4
A second
parameter was soon to follow: I noticed that the angle at which the connecting
line cuts through neighbouring q1 sphere surfaces is
different from the 90-degree angle in the static case, and consequently the
path taken by electric information from one sphere surface to the next is
longer than the local perpendicular distance between the sphere surfaces. The
factor by which it is longer is the same on the near side and the far side. The
product of the two turned out to be another factor in (6), namely 1/(1 - u2sin2
α/c2).
But the
hard bit was yet to come. It was the remaining square root term in (6). It did
not seem likely that just taking a long hard look at the sphere field
configuration in Figure 2 would enable me to detect that factor in that
configuration. I suspected that the square root term had something to do with
the frequency at which electric information coming from q1
arrives at q2. That frequency could be expected to depend on
both u and v. I started to experiment with a number of sphere
model frequency factors, initially for special cases and in two dimensions
only. It was only when it occurred to me that I might have to look at
'near-side' and 'far-side' frequencies that I was able to make decisive
progress. Eventually I discovered that I had to form the geometric mean of
near-side and far-side information transmission rates, which depend on the q1
and q2 sphere surface configurations, to obtain the desired
term.
It thus
turned out that each of the factors I had identified - the density factor, the
angle factor and the frequency factor - could be regarded as the geometric mean
of near-side and far-side density, angle and frequency factors. Is that
plausible? In principle I think it is: it seems plausible that the information
that is transmitted from q1 to q2 concerns
a whole area around the mathematical point on which q1 is
centred. That information will have travelled along the line connecting the two
charges, so again it is plausible that it includes near-side and far-side
information on parameters along that line. The fact that what matters is the
geometric mean of near- and far-side factors, rather than some other kind of
combination of the two, is perhaps less immediately plausible. No doubt one day
somebody will find an explanation for it.
At this
point I felt like celebrating. I had succeeded in identifying just three simple
sphere field parameters by which (1) had to be modified to obtain the magnitude
of the force between two charges moving uniformly in S. But then I
remembered that I had not finished the job. I had yet to explain, in terms of
the sphere model, why the direction of the force as seen from S is
parallel to (5)!
Step 4
Again, the
task of obtaining the correct direction vector looked daunting, but this time I
succeeded more quickly than I had anticipated. I had a good idea where to look:
it would be plausible for the direction vector to depend on the direction from
which q1 information arrives
at q2. However, when I calculated the 'near-side
direction vector' n1< (see Figure 5), which
describes that direction in S, it wasn't parallel to (5). The near-side
direction vector is the vector pointing from the 'retarded' position of q1
at the centre of the q1 sphere surface on which q2
is located to the current position of q2. I then figured that
perhaps I had to include the 'far-side direction vector' n1>,
in other words the vector pointing from the retarded position of q1
to the far-side intersection of the line connecting q1 and q2
at the current time with the q1 sphere surface centred on the
retarded position. Still that didn't work.
Fig. 5
It was only
when I started to look at the situation from the point of view of q2
that I struck gold. I found that I had to consider the near-side and far-side
directions from which q1 information arrives at q2
from the point of view of q2. Those two directions are
given by n1< - v/c and n1>
- v/c. Simply adding up those two vectors still didn't do the
trick, however. I found that I had to modulate those two vectors by particular
factors. Just by looking at the x- and y-components, I was able
to calculate the factors that are required to obtain the correct result. They
turned out to be q2 sphere field parameters that I had
previously worked out for q1: the q2 angle
factors! What is more: the z-component worked with those same factors.
This could be no accident! I felt vindicated. My intuition had been right. All
aspects of the forces between uniformly moving charges as measured by co-moving
spring balances could be expressed simply in terms of a few basic sphere model
parameters. The end result was as follows:
(7)
Here d1,
e1 and f1 are the q1
density, angle and frequency factors, and e2<
and e2> are the near- and far-side q2
angle factors.
To my mind,
this is an amazing result. It means that the forces between uniformly moving
charges as measured by co-moving spring balances can be expressed simply in
terms of a few sphere model parameters, without reference to any 'magnetic
field' and without the need to perform any relativistic transformations.
Equation (7) can also be written purely in terms of u, v, n1<
and n1>, as follows:
(8)
However,
equation (7) shows more clearly how this force comes about, namely as a result
of changes in the sphere field configuration compared to the static case
described by Coulomb's law. And that concludes my account of how I arrived at
the sphere model equation describing the forces between two uniformly moving
point charges as measured by co-moving spring balances.
One last
thing. The need to systematically take into account near- and far-side sphere
field properties is a remarkable feature of the sphere model. I have tried to
explain it by assuming that information travelling from q1 to
q2 in the direction of n1<
includes full information on near- and far-side sphere field properties along
the line connecting the two charges at the current time. There is another
possibility. It is that in S the information travels well and truly in
the near- and far-side directions given by
n1< and n1>
and is exchanged instantly over the q1 sphere surface on
which q2 is located at the current time, t1,
as shown in Figure 6 below.
Fig.6
The idea
would be that, while energy in the form of sphere field disturbances can only
be transmitted at c in S, since it has to travel outwards from
one sphere surface to the next, information on interlinked near- and far-side
sphere field properties can be exchanged instantly on one and the same sphere
surface. The whole situation reminds me a bit of what I've read here and there
about the phenomenon of quantum entanglement. I am no expert on quantum
mechanics, so I don't know whether there is a connection. But it is intriguing
to think that, in addition to solving the mystery of the magnetic field in
classical electrodynamics, the sphere model might also shed some new light on
quantum entanglement, or vice versa.
