## 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 velocity At the 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 times each tick results in a light signal being sent to O' which is received at times and 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 between and and because it passes through the origin, the relation is of the form  so that Similarly, when light signal is received in O' and reflected,

The signals are received at  so 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 so  Hence and the frequency of light received by a moving observer is reduced by the factor  