Classical properties of non-local, ghost- and singularity-free gravity

In this paper we will show all the linearized curvature tensors in the infinite derivative ghost and singularity free theory of gravity in the static limit. We have found that in the region of non-locality, in the ultraviolet regime (at short distance from the source), the Ricci tensor and the Ricci scalar are not vanishing, meaning that we do not have a vacuum solution anymore due to the smearing of the source induced by the presence of non-local gravitational interactions. It also follows that, unlike in Einstein's gravity, the Riemann tensor is not traceless and it does not coincide with the Weyl tensor. Secondly, these curvatures are regularized at short distances such that they are singularity-free, in particular the same happens for the Kretschmann invariant. Unlike the others, the Weyl tensor vanishes at short distances, implying that the spacetime metric becomes conformally flat in the region of non-locality, in the ultraviolet. As a consequence, the non-local region can be approximated by a conformally flat manifold with non-negative constant curvatures. We briefly discuss the solution in the non-linear regime, and argue that 1/r metric potential cannot be the solution where non-locality is important in the ultraviolet regime.