**T**he
Electrodynamics of Length Contraction And
Time Dilation

** Abstract.** Shows that length
contraction and time dilation are implicit in Maxwell's Equations, the Lorentz
force law, and Newtonian mechanics.

Fig. 1 depicts a negatively charged particle,
of rest mass m_{o}, orbiting a much more massive, positively charged
particle. The orbital speed is constant and equal to .95c. Owing to its much
greater mass, the central particle remains practically at rest at the origin of
inertial frame K’. At time t’ = 0 the satellite cuts across the positive
y’-axis.

Figure 1

Bohr-like "Atom" at Time t’ = 0

**
**Since |**v**’| is relativistic, the
satellite must experience a central force of magnitude

. (1)

Indeed (neglecting gravity),

. (2)

Viewed from inertial frame K, which moves in
the negative x’-direction at speed .95c, the satellite is momentarily at rest
on the y-axis at distance R = R’ from the origin and at time t = t’ = 0. And
relative to frame K the central body moves in the positive x-direction with
constant speed .95c. It has the magnetic (**B**) and electric (**E**)
fields of a charge moving with constant velocity. The Lorentz force law and
Newton’s second law can therefore be invoked to compute the motion of the
satellite for times t > 0. In its relativistic form Newton’s second law
states that

(3)

.

Or, in component form,

, (4a)

. (4b)

Solving Eq. 4b for a_{y} and
substituting in Eq. 4a produces

. (5a)

Similarly,

. (5b)

Knowing the satellite’s position, velocity, and acceleration at t = 0, and knowing the force acting on it, we can compute the position and velocity a short time later, and thence the force acting at that time. The process can be iterated, and the satellite’s motion (relative to K) can be numerically approximated. The "shape" of the "atom," as viewed from K, can then be obtained by subtracting .95ct from the computed satellite positions. The motion’s period can be determined by noting when the satellite again reaches its maximum, positive y-position.

Fig. 2 depicts the computed "shape."
It is a *foreshortened* circle
whose height is R = R’ but whose width is only (1 - .95^{2})^{1/2}R’.
Furthermore, whereas the orbital period in K’ is

, (6a)

The period in K is

. (6b)

Figure 2

The "Shape" of the "Atom" in Frame K

In this case, at least, the phenomena of length contraction and time dilation are consequences of Maxwell’s equations, the Lorentz force law, and Newton’s second law. To the extent the same results apply to actual, moving atoms, it is clear that the grid of a moving coordinate system is contracted in the direction of its motion, and clocks distributed throughout said grid run slowly. Add to this the idea that clocks in any inertial frame are synchronized by exploiting the constant speed of light in all directions (as experiment indicates is the case), and the Lorentz transformations result.

Of course the entire discussion could be repeated using an "atom" whose central body remains at rest at the origin of frame K. This "atom" would move in the negative x’-direction of K’. And the same length contraction and time dilation phenomena would result (from the perspective of K’) from applying Maxwell/Lorentz/Newton in K’.

One of the watershed events in physics was the
discovery by Michelson and Morley that light is measured to propagate with the
one, constant speed c relative to the earth's moving inertial frame of
reference. Fitzgerald suggested that length contraction might explain this
result. For a time it was thought that light must *really* propagate
in all directions with the one speed c, relative only to the frame in which the
luminiferous ether is hypothetically at rest. But the idea was that, owing to
the length contraction of inertial grids moving through the ether, light *appears* to
propagate with the one speed c relative to these frames also. Lorentz and
Einstein went further, adding time dilation, the relativity of simultaneity, and
the dependence of inertial mass on particle speed. (Evidently the electrodynamic
nature of length contraction and time dilation was not at first appreciated.)
Consequently it seemed that nature had concocted a complete conspiracy to thwart
mankind’s attempts to detect the ether’s rest frame. Poincare pointed out
that a complete conspiracy of nature is a *law* of
nature, and the whole idea of a luminiferous ether was eventually abandoned.

A caveat on the discussion in this article may
be in order, in view of the radiation reaction force of Abraham and Lorentz.
Given an "atom" with an orbiting, charged satellite (as depicted in
Fig. 1), d**a**’/dt points opposite to **v**’
at all times. Thus there is a radiation reaction force pointing opposite to **v**’,
and indeed such a system constantly emits radiant energy. The only way the
circular orbit can be maintained is if a tangential force counteracts the
radiation reaction force. This being the case, the radiation reaction force and
this counteraction sum to zero, and they have no mechanical effect on the
satellite’s motion. Without such a constant counteraction, the radiation
reaction force would act on the satellite, along with the central body’s
electric force, and the satellite would spiral into the "nucleus" as
radiation bleeds away into infinite space. Bohr understood this problem, but
said that at certain energies such radiation doesn’t occur, despite the
apparent contradiction with electromagnetic theory. His assertion of course
signaled the birth of quantum theory.