Weak field limit of infinite derivative gravity

A form of infinite derivative gravity is free from ghostlike instabilities with improved small scale behavior. In this theory, we calculate the tree-level scattering amplitude and the corresponding weak field potential energy between two localized covariantly conserved spinning pointlike sources that also have velocities and orbital motion. We show that the spin-spin and spin-orbit interactions take the same form as in Einstein’s gravity at large separations, whereas at small separations, the results are different. We find that not only the usual Newtonian potential energy but also the spin-spin and spin-orbit interaction terms in the potential energy are nonsingular as one approaches $r\to 0$.