2.4. Length Contraction and Time Dilation.
Fig. 2.4_1 depicts a 2-charge system. A positive charge is held fixed in inertial frame K. A negative charge travels at constant speed v around the positive charge.
Figure 2.4_1
A Simple, 2-Charge System
As previously pointed out, the negative charge will emit radiation at a constant rate. And a driving agent must exert a constant-magnitude tangential force in order to maintain the circular orbit. Let us assume that this radiative force is implicitly provided, and focus upon the radially inward inertial force,
. (2.4_1)
In the present case we shall assume that Finert is the electric (Coulombic) force that –q experiences in the electrostatic field of +q:
. (2.4_2)
Let us now view matters from frame K’, which moves in the positive x-direction of K at constant speed u=v. At the moment shown, +q moves in the negative x’-direction of K’ at constant speed u, and –q is momentarily at rest on the y’-axis of K’. We would like to compute the path of –q in K’ by using Newton’s 2nd law.
The force experienced by –q in frame K’ is generally specified by the Lorentz force law,
. (2.4_3)
Here E’ and B’ are the electric and magnetic fields of +q, at the instantaneous position of –q, and v’ is the velocity of –q relative to K’. Note that although B=0 in K, +q moves in K’ and hence B’ is generally nonzero.
The equation of motion for –q in K’ thus becomes
. (2.4_4)
(Note that the radiative force, driving –q around the circle, is not ascribable to the fields of +q.) Starting with initial conditions, the motion of –q is readily computed, producing the following results: (1) the motion of –q is quasi-cycloidal, but –q cuts "in front of" and "in back of" +q at distances of only (1-u2/c2)1/2R; (2) the time for one complete cycle in K’ is longer than the cycle time of t=2pR/v in K: t’=(1-u2/c2)-1/2t. Figs. 2.4_2 and 2.4_3 depict the computed "shape" in K’ for speeds of u=.01c and u=.95c respectively.
Figure 2.4_2
Shape in K’, u=.01c
Figure 2.4_3
Shape in K’, u=.95c
These two results, implicit in the dynamics of Maxwell-Lorentz-Newton, are generally referred to as length contraction and time dilation. They appear to be generally applicable to all moving systems, including atoms. For example, measuring rods that translate parallel to their lengths are shorter than they are when at rest. And moving clocks run more slowly than their resting counterparts.