The 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 mo, 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.
Bohr-like "Atom" at Time t’ = 0
Since |v’| is relativistic, the satellite must experience a central force of magnitude
Indeed (neglecting gravity),
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
Or, in component form,
Solving Eq. 4b for ay and substituting in Eq. 4a produces
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 - .952)1/2R’. Furthermore, whereas the orbital period in K’ is
The period in K is
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), da’/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.