Infinite-derivative linearized gravity in convolutional form

TL;DR

This paper reformulates ghost-free infinite-derivative linearized gravity into a non-local form using generalized functions and Fourier transforms, demonstrating regularity and absence of singularities in solutions.

Contribution

It introduces a non-local formulation of infinite-derivative gravity in the space of tempered distributions, analyzing the domain and regularity of solutions.

Findings

The non-local operator acts on a subset of the functional space.

The Riemann tensor remains regular, with no spacetime singularities.

Conditions for the existence of solutions in the distribution space are identified.

Abstract

This article aims to transform the infinite-order Lagrangian density for ghost-free infinite-derivative linearized gravity into non-local. To achieve it, we use the theory of generalized functions and the Fourier transform in the space of tempered distributions $\mathcal{S}^\prime$. We show that the non-local operator domain is not defined on the whole functional space but on a subset of it. Moreover, we prove that these functions and their derivatives are bounded in all $\mathbb{R}^3$ and, consequently, the Riemann tensor is regular and the scalar curvature invariants do not present any spacetime singularity. Finally, we explore what conditions we need to satisfy so that the solutions of the linearized equations of motion exist in $\mathcal{S}^\prime$.