Sagnac Effect of Goedel's Universe

TL;DR

This paper derives exact formulas for the Sagnac effect in G"odel's Universe, analyzing different experimental setups and relating the effect to invariant physical quantities, thus extending understanding of rotation effects in curved spacetime.

Contribution

It provides the first exact expressions for the Sagnac effect in G"odel's Universe and relates them to invariant physical quantities, bridging curved spacetime and rotating frame analyses.

Findings

Sagnac time delay formula derived for G"odel's Universe

Time delay vanishes at a specific detector rotation rate

Result closely resembles rotating Minkowski spacetime formula

Abstract

We present exact expressions for the Sagnac effect of Goedel's Universe. For this purpose we first derive a formula for the Sagnac time delay along a circular path in the presence of an arbitrary stationary metric in cylindrical coordinates. We then apply this result to Goedel's metric for two different experimental situations: First, the light source and the detector are at rest relative to the matter generating the gravitational field. In this case we find an expression that is formally equivalent to the familiar nonrelativistic Sagnac time delay. Second, the light source and the detector are rotating relative to the matter. Here we show that for a special rotation rate of the detector the Sagnac time delay vanishes. Finally we propose a formulation of the Sagnac time delay in terms of invariant physical quantities. We show that this result is very close to the analogous formula of the Sagnac time delay of a rotating coordinate system in Minkowski spacetime.