The Electrodynamics of Length Contraction And Time Dilation G.R.Dixon, 11/14/2004
The plotted results in this article were computed by the Visual Basic program provided in Appendix A. G.R.Dixon, 11/14/2004 Fig. 1 depicts a negatively charged particle, of rest mass m Figure 1
Bohr-like "Atom" at Time t’ = 0 Since | . (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 ( (3) . Or, in component form, , (4a) . (4b) Solving Eq. 4b for a . (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 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 , (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’. It is noteworthy that length contraction does not inevitably result when a system is accelerated out of one inertial frame into another. As pointed out in
a previous note, a system of 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 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 |