Proof of the Time Dilation Formula
To prove the time dilation formula consider the following observers.
The observer on the right is stationary in his reference frame, and the observer on the left sees the other moving to the right with speed
The observer on the right bounces a light signal between the ceiling and the floor. It takes a timeto return to the floor, so that
The observer on the left sees the moving framer of reference moving to the right, and will see the ray of light travel a distancein a timewhereHe will observe the ray of light to take a timeto return to the floor, so that
Square both sides to give
Move the last term to the left.
Divide byto give
Factorise withto give