,
Copyright © 1999 David E. Rutherford
All Rights Reserved
Using the transformation matrix,
,
from "Additions: July 29, 1999" on this website, we can find the
measurements of lengths transverse to the direction of motion. We shall find
the components of a ruler at rest along the y-axis of an unprimed
frame of reference as measured by an observer at rest in a primed frame moving
with uniform four-velocity
relative
to the unprimed frame. The spacetime vector extending between the endpoints
of the ruler as measured by an observer at rest in the unprimed frame is
. The time
component, 
, is zero since the measurement of the length of the ruler is made
instantaneously. We will now use the transformation equations
where
to find the components of the spacetime vector between the endpoints of the ruler as measured in the primed frame.
Expanding Eqs. (2) and inserting the components of
and
, we get
or 
. This
represents a rotation of the
and
axes of
the primed frame relative to the
and
axes of
the unprimed frame by the angle
. Looking
from the unprimed frame in the direction of the positive x-axis, the
rotation of the primed frame relative to the unprimed frame is in the
counter-clockwise direction. The effect is reciprocal since an observer in
the primed frame sees a rotation of the unprimed frame in the counter-clockwise
direction when looking in the direction of relative motion of the unprimed
frame. The magnitudes,
and
, of the
spacetime vectors in the two frames, however, remain equal because
Copyright © 1999 David E. Rutherford
All Rights Reserved
E-mail: drutherford@softcom.net