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A note on the description of plane gravitational waves in Fermi coordinates

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Abstract

We use the formalism of Fermi coordinates to describe the interaction of a plane gravitational wave in the proper detector frame. In doing so, we emphasize that in this frame the action of the gravitational wave can be explained in terms of a gravitoelectromagnetic analogy. In particular, up to linear displacements from the reference world-line, the effects of the wave on test masses can be described in terms of a Lorentz-like force equation. In this framework, we focus on the effects on time measurements provoked by the passage of the wave, and evaluate their order of magnitude. Eventually, we calculate the expression of the local spacetime metric in cylindrical coordinates adapted to the symmetries of the gravitational field and show its relevance in connection with the helicity–rotation coupling.

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Data Availability

Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

Notes

  1. Latin indices refer to space coordinates, while Greek indices to spacetime ones. We will use bold-face symbols like \(\textbf{W}\) to refer to vectors in the Fermi frame; the spacetime signature is \(+2\) in our convention.

  2. In \(e^{\mu }_{(\alpha )}\) tetrad indices like \((\alpha )\) are within parentheses, while \(\mu \) is a background spacetime index; however, for the sake of simplicity, we drop here and henceforth parentheses to refer to tetrad indices, which are the only ones used.

  3. Notice that this integration is path depending, since \(\varvec{\nabla }\times \textbf{A} = \varvec{B} \ne 0\).

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Acknowledgements

The author acknowledges the contribution of the local research project Modelli gravitazionali per lo studio dell’universo (2022)—Dipartimento di Matematica “G.Peano”, Università degli Studi di Torino; this work is done within the activity of the Gruppo Nazionale per la Fisica Matematica (GNFM).

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Authors and Affiliations

  1. Dipartimento di Matematica “G.Peano”, Università degli studi di Torino, Via Carlo Alberto 10, 10123, Torino, Italy

    Matteo Luca Ruggiero

  2. INFN - LNL, Viale dell’Università 2, 35020, Legnaro, PD, Italy

    Matteo Luca Ruggiero

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Correspondence to Matteo Luca Ruggiero.

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Ruggiero, M.L. A note on the description of plane gravitational waves in Fermi coordinates. Eur. Phys. J. Plus 138, 792 (2023). https://doi.org/10.1140/epjp/s13360-023-04424-0

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