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

TL;DR

This paper derives exact non-local gravity solutions within a specific class of spacetimes, revealing their properties and potential regularity at sources, extending classical solutions like Aichelburg--Sexl and Hotta--Tanaka.

Contribution

It provides explicit non-local solutions in infinite derivative gravity for a class of Kundt spacetimes, demonstrating their solvability and physical properties.

Findings

Exact non-local solutions for gravitational waves in various spacetimes.

Solutions reduce to local theory in the limit of zero non-locality.

Non-local solutions may be regular at source locations, unlike local counterparts.

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 $\triangle$ 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 $\exp(-\ell^2\triangle)$ (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, ${\ell\to0}$. 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.