Proving the Relativistic Doppler Effect Using the K - Calculus
Suppose an inertial observer O' is moving along the positive x axis of an inertial observer O with uniform velocityAtthe origins of both inertial frames coincide. The world line of O' in the reference frame of O is shown below. Each observer carries a clock which are synchronized at the moment the origins coincide.
The clock carried by observer O ticks at timeseach tick results in a light signal being sent to O' which is received at timesand instantly reflected back to O.
Because the world line of O' in the inertial frame O is a straight line, there will be a linear relationship betweenandand because it passes through the origin, the relation is of the form so thatSimilarly, when light signal is received in O' and reflected,
The signals are received atso that
The signals take equal times to travel from O to O' and back, so the first signal is received by O' at a time in Os frame given and
Hence the speed of O' relative to O is
Henceand the frequencyof light received by a moving observer is reduced by the factor