The manuscript discusses the Sagnac effect, that is, a clock effect in Special and General Relativity related to the impossibility to Einstein synchronise clocks along a spatially closed path. Compared to the existing extensive literature on the effect, the author chooses here a very general approach within General Relativity, thus including effects due to the curvature of spacetime. The results are then discussed in detail for the case that we can restrict to a temporally and spatially small section of spacetime. The different contributions arising from the rotation, acceleration, and gravitation are discussed separately. Finally, stationary spacetimes and the relation of the effect to the classical experiment of Fizeau, related to light moving in a medium, are considered.

The Sagnac effect can roughly be described as the difference in the travel time of two photons following a spatially closed path in opposite directions. In the present manuscript, the author models this situation in a four-dimensional Lorentzian spacetime given by Einstein’s equations. He introduces a generalised Fermi transport law along the worldline of the observer to model her rotating and accelerating lab. This Fermi frame also provides a foliation of spacetime by spaces of simultaneity, that is assumed to be valid globally. Then a general formula for the travel time, and thus the time difference, is derived. However, this general formula can only be evaluated if the spatial path and the metric are known.

To discuss the time difference more explicitly, the author derives in Fermi coordinates an expression for the components of the metric in terms of acceleration, rotation, and Riemann curvature in the vicinity of the observer worldline, up to cubic terms. By further assuming that the metric is time independent, this gives rise to three different characteristic terms contributing to the time difference. These are then discussed separately. Moreover, the author analysis these terms assuming a spatial path given by a general Lissajous curve. The paper closes with a discussion of two special cases: for a stationary spacetime, the results are rederived and found to be valid exactly without any additional assumptions. For a flat spacetime filled with an homogeneous and isotropic medium the result is found to coincide with the classical setup of Fizeau’s experiment.

The paper is very well written and clearly derives the Sagnac effect and it’s various contributions in a very general setup, including next to the classical rotation effects also the acceleration and the curvature. The results are illustrated and discussed using well justified approximations and curves given by Lissajous figures. I think the paper will become a standard reference in the discussion of the Sagnac effect.