Chronology | Current Month | Current Thread | Current Date |
[Year List] [Month List (current year)] | [Date Index] [Thread Index] | [Thread Prev] [Thread Next] | [Date Prev] [Date Next] |
The new problem. Let the traditional beam be replaced with aof
perfectly elastic ball bouncing up and down on the train, along the
y'-axis. On the train, its speed is w' and the time of a one-way
movement is t'. Special relativity assumes that distances
perpendicular to the axis of relative motion are equal in the two
frames so in the track frame the vertical, y-component of the path
the ball is h=w't'. Now the Pythagorean theorem gives
(w't')^2 + (vt)^2 = (st)^2 /1/
where s is the speed of the ball in the track frame (along a path
oblique to the axis of relative motion).
Eq. /1/ can be transformed into
t'/t = [(s/w')^2 - (v/w')^2]^(1/2) /2/
This is a general result giving time dilation for any speed
perpendicular to the axis of relative motion in the primed frame.
When this speed is c (the ball is replaced with a beam again and
w'=s=c), /2/ gives the familiar
t'/t = 1/gamma /3/