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Changing Rest Frames

G.R.Dixon, 6/16/2006

If a particles motion (say x(t)) is given in inertial frame K, and if one does not require that the t epochs in K be evenly spaced, then the Lorentz transformation can be applied directly to find a table of values for plotting x(t) and higher order derivatives. The only requirement (if any) is that the single-valued spacing between successive values of t be small enough that no interval between successive values of t exceed an acceptable maximum.

For example, let a particle be at rest at the origin of K up until t=0. At t=0 the particle is given a constant acceleration of such magnitude that it comes to rest in K at xf. If u is the speed of K relative to K, then from the perspective of K:

, (1a)

and

. (1b)

tf, the moment the particle comes to rest in K, is

. (2)

From the perspective of K, xf and tf transform to

, (3a)

. (3b)

Fig. 1 and 2 plot x(t), v(t), a(t) and x(t), v(t), a(t) over the ranges 0<t<tf and 0<t<tf respectively, where u=.95c and xf=2000 meters. Note the time-varying a(t) a consequence of the acceleration transformation

. (4)

(Presumably ax and ax drop to zero when the particle comes to a rest in K an event both K and K agree upon, although the K and K values of x/x and t/t of said event will be a matter of disagreement)

Figure 1

 

 

Kinematics in K

Figure 2

 

 

Kinematics in K

The data point generating software can easily be modified to transform 3-dimensional motions in K. For reference, the ay transformation is

. (5)

(For a derivation of such transformations, see Special Relativity by A.P.French still in print.)