Notes on the Sagnac effect in General Relativity

TL;DR

This paper investigates the Sagnac effect within General Relativity, deriving general and special case formulas for the time difference in photon travel, and discusses implications for rotating frames and stationary space-times.

Contribution

It provides a general expression for the Sagnac time difference in arbitrary space-times and explores specific cases, including rotating frames and stationary space-times, linking to classical experiments.

Findings

Derived a general formula for the Sagnac effect in arbitrary space-times.

Identified special cases with closed-form expressions for the time difference.

Discussed the relationship between the Sagnac effect and Fizeau's experiment.

Abstract

The Sagnac effect can be described as the difference in travel time between two photons traveling along the same path in opposite directions. In this paper we explore the consequences of this characterisation in the context of General Relativity. We derive a general expression for this time difference in an arbitrary space-time for arbitrary paths. In general, this formula is not very useful since it involves solving a differential equation along the path. However, we also present special cases where a closed form expression for the time difference can be given. The main part of the paper deals with the discussion of the effect in a small neighbourhood of an arbitrarily moving observer in their arbitrarily rotating reference frame. We also discuss the special case of stationary space-times and point out the relationship between the Sagnac effect and Fizeau's "aether-drag" experiment.