# On the general relativistic framework of the Sagnac effect

## Abstract

The Sagnac effect is usually considered as being a relativistic effect produced in an interferometer when the device is rotating. General relativistic explanations are known and already widely explained in many papers. Such general relativistic approaches are founded on Einstein’s equivalence principle (EEP), which states the equivalence between the gravitational “force” and the *pseudo-force* experienced by an observer in a non-inertial frame of reference, included a rotating observer. Typically, the authors consider the so-called Langevin-Landau-Lifschitz metric and the path of light is determined by null geodesics. This approach partially hides the physical meaning of the effect. It seems indeed that the light speed varies by \(c\pm \omega r\) in one or the other direction around the disk. In this paper, a slightly different general relativistic approach will be used. The different “gravitational field” acting on the beam splitter and on the two rays of light is analyzed. This different approach permits a better understanding of the physical meaning of the Sagnac effect.

## 1 Introduction

It can be useful to recall the context of the discovery of the Sagnac Effect. At the beginning of previous century, physicists were engaged in a very long debate concerning absolute space and its counterpart, the aether, the hypothetical medium of propagation of light. In the well known gedankenexperiment of the rotating bucket filled with water, Newton deduced the existence of an absolute rotation with respect to absolute space. In one of the most important work in the history of science (Principia), he expatiated on time, absolute and relative space and motion [1]. Mach criticized Newton’s reasoning in his book published in 1893 [2]. From his perspective, one must consider the rotation of water relative to all the matter in the Universe. It is well known that Mach’s ideas had a considerable influence on the development of Albert Einstein’s general theory of relativity (GTR), especially during the first years of the 20th century. Mach’s view led to a misconception about the GTR. A more complete analysis of the debate can be find in [3]. After the formulation of the special theory of relativity and before its generalization to the GTR, also the French physicist Georges Sagnac took part in the debate. In 1899, he indeed developed a theory of the existence of a motionless mechanical aether [4]. His aim was to explain all optics phenomena within this theoretical framework, with special attention to the Fresnel-Fizeau experiment for the drag of light in a moving medium [5, 6]. At the beginning of the 20th century, he conceived a rotating interferometer to test his ideas. Despite countless explanations, in more than a hundred years, there are still different interpretations of Sagnac experiment in the framework of the GTR. But this is not a rare thing in physics. In fact, it is not the only topic that, although it is well known in the scientific literature, still requires insights and explanations [7, 8]. In order to start, in next Sections, the Sagnac effect in the framework of Classical Mechanics will be briefly analyzed.

## 2 The Sagnac experiment within the framework of Classical Mechanics

*r*. Such light rays will arrive at the end point simultaneously. Instead, if the loop is rotating, the ray travelling in the same direction as the rotation of the loop must travel a distance greater than the ray travelling in the opposite direction. For this reason, the counter-rotating ray will arrive earlier than the co-rotating ray. The length of the path is \(L=2\pi r\) and, if there is not angular velocity of the loop, the duration of the path is

## 3 The Sagnac effect within the framework of the GTR

## 4 Coordinate velocity of light

*O*to a point infinitely near, the condition of null geodesics \(ds=0\) permits to obtain that temporal coordinate required for this as

*c*. Indeed,

## 5 Coriolis time delay

*m*is the total mass of the observer in the rotating system, see [32] for details. For non-relativistic velocities \(\left( v'\ll c\right) \) Eq. (28) reduces to [32]

## 6 Conclusions

In this paper some considerations about the Sagnac experiment have been made. It has been shown that, by considering the rotating metric and by imposing the cancellation of the line element, one has an unexceptionable explanation only from the mathematical point of view. In this way, it seems that the speed of light varies by \(c\pm \omega r\) in one or the other direction around the disk. Instead, as it happens for example in Rindler or Schwarzschild metric, the apparent variation of the speed of light is a consequence of time dilation. For this reason, it seems that the physics of the experiment is clearer by using the “gravitational” Coriolis time dilation.

## Notes

### Acknowledgements

The Authors thank an unknown Referee for useful comments. This work has been supported financially by the Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), project number 1/6025-63.

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