Exact solutions of non-local gravity in a class of almost universal spacetimes

Abstract

We study exact solutions of the infinite derivative gravity with null radiation which belong to the class of almost universal Weyl type III/N Kundt spacetimes. This class is defined by the property that all rank-2 tensors $B_{ab}$ constructed from the Riemann tensor and its covariant derivatives have traceless part of type N of the form $\mathcal{B}(\square )S_{ab}$ and the trace part constantly proportional to the metric. Here, $\mathcal{B}(\square )$ is an analytic operator and $S_{ab}$ is the traceless Ricci tensor. We show that the convoluted field equations reduce to a single non-local but linear equation, which contains only the Laplace operator $△$ on 2-dimensional spaces of constant curvature. Such a non-local linear equation is always exactly solvable by eigenfunction expansion or using the heat kernel method for the non-local form-factor $\exp (-{\ell }^2△)$ (with $\ell$ being the length scale of non-locality) as we demonstrate on several examples. We find the non-local analogues of the Aichelburg--Sexl and the Hotta--Tanaka solutions, which describe gravitational waves generated by null sources propagating in Minkowski, de Sitter, and anti-de Sitter spacetimes. They reduce to the solutions of the local theory far from the sources or in the local limit, $\ell \to 0$. In the limit $\ell \to \infty$, they become conformally flat. We also discuss possible hints suggesting that the non-local solutions are regular at the locations of the sources in contrast to the local solutions; all curvature components in the natural null frame are finite and specifically the Weyl components vanish.

22 pages, 3 figures, comments welcome!

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