Non-singular rotating metric in ghost-free infinite derivative gravity

It is well-known that the vacuum solution of Einstein's theory of general relativity provides a rotating metric with a ring singularity, which is covered by the inner and outer horizons, and an ergo region. In this paper, we will discuss how ghost free, quadratic curvature, Infinite Derivative Gravity (IDG) may resolve the ring-type singularity in nature. It is well-known that a class of IDG actions admit linearized metric solutions which can avoid point-like singularity by a smearing process of the Delta-source distribution induced by non-locality, which makes the metric potential finite everywhere including at $r=0$, and even at the non-linear level may resolve the Schwarzschild singularity. The same action can also resolve the ring singularity in such a way that no horizons are formed in the linear regime, where in this case non-locality plays a crucial role in smearing out a delta-source distribution on a ring. We will also study the full non-linear regime. First, we will argue that the presence of non-local gravitational interaction will not allow the Kerr metric as an exact solution, as there are infinite order derivatives acting on the theta-Heaviside and the delta-Dirac distributions on a ring. Second, we will explicitly show that the Kerr-metric is not a pure vacuum solution when the Weyl squared term, with a non-constant form-factor, is taken into account in the action.