Gravitational radiation in higher order non-local gravity

Abstract

In this paper we examine gravitational radiation in higher order non-local gravity described by the non-local gravitational Lagrangian density ${\mathcal{L}}_g=R+{\sum }_{h=1}^n{a}_{h}R{\square }^{-h}R$. This non-local theory of gravitation always exhibits the tensor transverse gravitational radiation for $k_1^2=0$, corresponding to the angular frequency ${\omega }_1$, composed of two standard $(+)$ and $(\times )$ polarization modes, massless and of helicity 2. Furthermore, it shows, under suitable constraint and $n\ge 2$, an additional massive transverse scalar gravitational radiation with helicity 0. It is composed of $n-1$ modes associated to $n-1$ angular frequencies ${\omega }_2,\dots ,{\omega }_n$, each of which of breathing polarization $(b)$ to lowest order in $\gamma$, a parameter that takes into account the difference in speed between the slightly massive wave and the massless one. Thanks to NP formalism, we find that the $E(2)$ class of non-local gravitational waves is $N_3$, according Petrov classification, where the presence or absence of all modes are observer independent. Also, the scalar radiation is forbidden for $n=1$ and $n=2$ cases, when some conditions are satisfied. Finally, in ${\square }^{-1}$ gravity where $n=1$, a possible degenerate case with a continuous infinity of transverse massive scalar breathing modes appears under a particular constraint, which reproduces in two-dimensional spacetime the Polyakov effective action.

31 pages, accepted for publication in International Journal of Geometric Methods in Modern Physics

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• The energy-momentum complex in non-local gravity
• Energy-Momentum Complex in Higher Order Curvature-Based Local Gravity
• The gravitational energy-momentum pseudo-tensor in $f(Q)$ non-metric gravity