Changing Rest Frames
G.R.Dixon, 6/16/2006
If a particle’s 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.)