# Sagnac Effect and Special Relativity

## Main Question or Discussion Point

http://www.mathpages.com/rr/s2-07/2-07.htm

Today I met a man who was claiming that the Sagnac effect, particularly in the case of light moving in a circular pattern with a rotating observing "arm" as in the first figure in the link, is contrary to special relativity. He claimed that since the wave will hit the observing arm at a rate of c+v or c-v, that this contradicts the constancy of c.

Now, in my head, I am thinking that this speed of c+v is what we observe from our inertial system which holds the axis of the roation constant, but that an actual observation device placed on the arm would still observe the incoming light as having a speed of 3*10^8 m/s.

Is this to be explained by a slight time dilation, making the time difference in the frame of the observer located on the moving arm equal to the time difference observed by us in the reference frame holding the axis constant?

$$y = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$$

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Actually, I have been staying up all night working on this problem. I will post the solution when I am finished.

I've made a brief MAPLE worksheet demonstrating just one calculation to show that the Sagnac effect does not imply an inconstancy in the speed of light. Please see the attached PDF.

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A.T.
Now, in my head, I am thinking that this speed of c+v is what we observe from our inertial system
I think you have it backwards. Inertial frames always observe light traveling at c. In the rotating (non inertial) frame the speed of light can vary. That is what SR says and has been observed so far.

The webpage you quoted contains the time correction in there, but good job on going through the math yourself.

Demystifier
Inertial frames always observe light traveling at c. In the rotating (non inertial) frame the speed of light can vary. That is what SR says and has been observed so far.
Exactly. Although, the LOCAL velocity of light is allways equal to c, even for rotating and accelerating observers. That is, if you measure the velocity of light traveling NEAR you, you will allways get c. But in the Sagnac effect you don't measure the local velocity of light.

http://xxx.lanl.gov/abs/gr-qc/9904078 [Phys.Rev. A61 (2000) 032109]

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What is the structure that is described (propagating) in Special Relativity (Maxwell's equations?). Is there an induction experiment that emits light?

What light is used?

A laser

The Sagnac effect does not make any statement about SR nor does it depend upon GR for an explanation. You can get a time difference by setting up 4 mirrors each at the corner of a rectangle at an angle of 45 degrees - now put a source/receiver in one leg of the path so two beams travel in opposite directions around the path - now rotate the entire frame about an axis perpendicular to the plane of laser path(s) and there will be a difference in the time between the arrival of the photons because one photon has had to travel a longer distance to reach the moving receiver - but the same follows if you don't rotate the experiment, but simply move the transceiver linearly along one leg relative to the frame of the mirrors - the beam which is reflected by the mirrors in the clockwise direction will take a different amount of time than the counter-clockwise beam because one beam has to travel further to catch the receiver than the other