Beyond Length Contraction
1. A General "Snapshot" Algorithm.
Any extended body can be subdivided into incremental parts, and a complete specification of the motion of each part then specifies the motion of the body as a whole. An important subset consists of rigid body cases, where each part does not change its position with respect to the other parts.
Given the motion of each body part in inertial frame K, a "picture" of what will be observed at some particular instant (say at t=0) can be constructed. That is, the K observer(s) determine the space coordinates of the K-clocks that the parts coincide with when all of the K-clocks read zero. The Lorentz transformation then specifies what the K’ space coordinates and coincident K’ clocks read when this set of observations occurs.
Now in general the K’ clocks will not all read t’=0. If one wishes to know where the incremental parts are when all of the coincident K’ clocks read zero, then further processing is required.
If all of the body parts are permanently at rest in K, then they move with a single, constant velocity in K’ and the additional processing is quite simple. However, if the parts move with different and varying velocities relative to both K and K’, then the use of a computer is advisable.
The following "snapshot" algorithm is suggested for determining where K’ will find the parts when the coincident K’ clocks all read zero. (The algorithm is for a single incremental part with positive x-coordinate. It can be repeatedly applied to find the K’ locations for all the parts. All of the parts presumably move in the xy-plane. The origin clocks of K and K’ coincide when they mutually read t=t’=0.)
Start out with t=0.
1. Compute x, y, vx, vy, ax, ay at time t, using the formulas for the part’s motion.
2. Transform t to t’.
3. In the first iteration only, divide t’ into N increments, where N is suitably large. (Call each increment dt’)
4. Transform x, y, vx, vy, ax, ay into x’, y’, v’x, v’y, a’x, a’y.
5. Compute x’ and y’ a time dt’ later by assuming that a’ is constant over that short time period.
6. Add dt’ to t’.
7. Transform t’ to t.
8. Go to 1.
9. After N iterations, optionally save x’ and y’ for plotting purposes. This is approximately where that part will be observed to be when the coincident K’ clock reads zero. Repeat for all of the other parts.
10. Optionally plot y’ vs. x’, to see where all the parts will be observed to be at the single instant t’=0 in K’.
2. A Rigid Metal Rod at Rest in K.
Note: u is the speed of frame K’ relative to K.
Fig. 2_1 depicts a 2-meter rod, divided into 200 parts. The rod is centered on the origin of K and it lies on the x-axis. The time of the observations is t=0 (although the same observations would be found at any other value of t).
Rod Parts at t=0
The right end of the rod is at x=1 meter. According to the Lorentz transformation,
For u>0, g is greater than 1 and hence x’>x. The rod moves relative to K’. What happened to length contraction? The answer of course lies in the fact that the coincident K’ clock does not read zero (as the clock at the K’ origin does). It reads less than zero:
K’ wishes to know where the right end is some time later, when the coincident K’ clock reads t’=0.
Since the rod is at rest in K, a single calculation will provide the answer. But let us use the more general algorithm (the one that allows for any motion in K) to compute where the rod’s right end is in K’ when the coincident K’ clock reads t’=0. For u=0 Fig. 1_1 provides the answer (substituting x’ for x and y’ for y). Fig. 2_1 depicts the results for u=.95c. Here, then, is the celebrated length contraction of the rigid metal rod. (Note that only the right half of the rod is plotted. The left half is symmetric. The right half is divided into 100 segments, and each segment’s x-coordinate is reckoned at the segment’s midpoint.)
Rod Parts at t’=0, Rod at Rest in K
3. A Rigid Metal Rod that Spins in K.
Let Fig. 1_1 again depict the rod in K at time t=0. But in the present case let the rod be spinning counterclockwise about the origin at a constant angular ratew. If w=0 and u=.95c, then Fig. 2_1 again depicts the rod as observed using the grids and clocks of K’, at the single instant t’=0. Fig. 3_1 depicts things in K’, at time t’=0, when w=50000 and u=.95c. When the straight rod spins in K, it is both length-contracted and bent in K’. (Note that the y’ and x’ axes are at different scales; the actual bending is not as pronounced as that depicted. However, some bending is a consequence of any nonzero values of u and w.)
Rod Parts at t’=0, Rod Spinning in K
4. The Physics of Distortion.
Length contraction (and more generally the Lorentz transformation) can be shown to be a consequence of Maxwellian electrodynamics, as applied to a simplistic Bohr atom. Evidently the physics of the length contraction in Fig. 2_1, and of the length-contraction/bending in Fig. 3_1, lies in the atoms that comprise the iron bar. The length contraction depicted in Fig. 2_1, for example, is not the consequence of a compressive force couple acting in K’. Similarly for the bending depicted in Fig. 3_1. A stress-free bar in frame K implies a stress-free bar in frame K’. The effects are macroscopic manifestations of atomic distortions.
The general algorithm suggested herein (and the software that implements the algorithm) can be used, with simple modifications if necessary, for an unlimited variety of objects and motions in 3-dimensional K space. And of course one is not limited to the instants t=t’=0. Indeed one is not restricted to rigid objects. Plasmas, etc., can also be divided into parts (or individual particles) and the K’ observations, at any given K’ moment, can readily be computed and plotted from the known K motions.