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Gravitational waves in fourth order gravity

Abstract

In the post-Minkowskian limit approximation, we study gravitational wave solutions for general fourth-order theories of gravity. Specifically, we consider a Lagrangian with a generic function of curvature invariants \(f(R, R_{\alpha\beta}R^{\alpha\beta}, R_{\alpha\beta\gamma\delta }R^{\alpha\beta\gamma\delta})\). It is well known that when dealing with General Relativity such an approach provides massless spin-two waves as propagating degree of freedom of the gravitational field while this theory implies other additional propagating modes in the gravity spectra. We show that, in general, fourth order gravity, besides the standard massless graviton is characterized by two further massive modes with a finite-distance interaction. We find out the most general gravitational wave solutions in terms of Green functions in vacuum and in presence of matter sources. If an electromagnetic source is chosen, only the modes induced by \(R_{\alpha\beta}R^{\alpha\beta}\) are present, otherwise, for any \(f(R)\) gravity model, we have the complete analogy with tensor modes of General Relativity. Polarizations and helicity states are classified in the hypothesis of plane wave.

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Notes

  1. 1.

    Here we use the convention \(c=1\).

  2. 2.

    We are using the properties: \(2{R_{\alpha\beta}}^{;\alpha\beta}-\Box R=0\) and \({{R_{\mu}}^{\alpha\beta}}_{\nu;\alpha\beta}={{R_{\mu}}^{\alpha}}_{;\nu \alpha}-\Box R_{\mu\nu}\).

  3. 3.

    We set \(f_{X} =1\) i.e. \(G\rightarrow f_{X}(0)G\).

  4. 4.

    Any plane wave \(\psi\) transforming under a rotation of an angle \(\varphi\) about the direction of propagation into \(\tilde{\psi}\,=\,e^{\mathfrak{j} \xi\varphi}\psi\) has helicity \(\xi\).

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Acknowledgements

SC acknowledge INFN Sez. di Napoli (Iniziative Specifiche QGSKY, and TEONGRAV) for financial support.

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Capozziello, S., Stabile, A. Gravitational waves in fourth order gravity. Astrophys Space Sci 358, 27 (2015). https://doi.org/10.1007/s10509-015-2425-1

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Keywords

  • Gravitational waves
  • Modified theories of gravity
  • Weak field limit