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A Note on the Sagnac Effect in General Relativity as a Finslerian Effect

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Abstract

The geometry of the Sagnac effect in a stationary region of a spacetime is reviewed with the aim of emphasizing the role of asymmetry of a Finsler metric defined on a spacelike hypersurface associated to a stationary splitting and related to future-pointing null geodesics of the spacetime. We show also that an analogous asymmetry comes into play in the Sagnac effect for timelike geodesics.

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Notes

  1. We point out that in other references, as e.g. [7], the name Fermat metric has been attributed to the Riemannian metric h.

  2. Notice that if x is a reversible geodesic loop based at \(x(a)=x(b)\) and \(\bar{s}\in (a,b)\) then the two future-pointing lightlike curves defined by x are piecewise lightlike geodesics.

  3. Of course, if \(x=x(s)\) is a geodesic of \(F_\pm\) then \(\tilde{x}(s)=x(a+b-s)\) is a geodesic of \(F_{\mp }\).

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Acknowledgements

The authors would like to express their gratitude to an anonymous referee for her/his comments and, in particular, for an observation that has been taken into account in Remark 1 and for the suggestion to consider, in the last section, freely falling particles parameterized with proper time instead of any fixed affine parameter.

Funding

Both authors are partially supported by PRIN 2017JPCAPN Qualitative and quantitative aspects of nonlinear PDEs.

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  1. Department of Mechanics, Mathematics and Management, Politecnico di Bari, Via Orabona 4, 70125, Bari, Italy

    Erasmo Caponio & Antonio Masiello

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  1. Erasmo Caponio
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  2. Antonio Masiello
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The authors have contributed equally to the conception, writing, and development of this work.

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Correspondence to Erasmo Caponio.

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Caponio, E., Masiello, A. A Note on the Sagnac Effect in General Relativity as a Finslerian Effect. Found Phys 52, 5 (2022). https://doi.org/10.1007/s10701-021-00523-z

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