In this paper, we examine gravitational radiation in higher order non-local gravity described by the non-local gravitational Lagrangian density ${{ℒ}}_g=R+{\sum }_{h=1}^n{a}_{h}R{\square }^{-h}R$. 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 is 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 to 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.

在本文中，我们研究了由非局域引力拉格朗日密度描述的高阶非局域引力中的引力辐射${{ℒ}}_G=R+{\sum }_{H=1}^n{\mathrm{一个}}_{H}R{\square }^{-H}R$. 这种非局域引力理论总是表现出张量横向引力辐射${ķ}_1^2=0$, 对应于角频率${\omega }_1$，由两个标准组成$(+)$和$(\times )$极化模式，无质量和螺旋性 2。此外，它表明，在适当的约束和$n\ge 2$，一个额外的大质量横向标量引力辐射，螺旋度为 0。它由$n-1$相关的模式$n-1$角频率${\omega }_2,\dots ,{\omega }_n$, 每个都是呼吸极化$(b)$降到最低的顺序$\gamma$，这个参数考虑了质量稍大的波和无质量波之间的速度差异。由于 NP 形式主义，我们发现$乙(2)$非局域引力波的类别是${ñ}_3$，根据彼得罗夫分类，其中所有模式的存在或不存在与观察者无关。此外，标量辐射是禁止的$n=1$和$n=2$在某些条件满足的情况下。最后，在${\square }^{-1}$重力在哪里$n=1$，在特定约束下出现了具有连续无穷大横向大规模标量呼吸模式的可能退化情况，该约束在二维时空中再现了波利亚科夫有效作用。