Born Coordinates - The Sagnac Effect

The Sagnac Effect

Imagine that we have fastened a fiber-optic cable around the circumference of a ring which is rotating with steady angular velocity ω. We wish to compute the round trip travel time, as measured by a ring-riding observer, for a laser pulse sent clockwise and counterclockwise around the cable. For simplicity, we will ignore the fact that light travels through a fiber optic cable at somewhat less than the speed of light in a vacuum, and will pretend that the world line of our laser pulse is a null curve (but certainly not a null geodesic!).

In the Born line element, let us put . This gives

or

We obtain for the round trip travel time

Putting, we find so that the ring-riding observers can determine the angular velocity of the ring (as measured by a static observer) from the difference between clockwise and counterclockwise travel times. This is known as the Sagnac effect. It is evidently a global effect.