A regular metric does not ensure the regularity of spacetime

Abstract

In this paper we try to clarify that a regular metric can generate a singular spacetime. Our work focuses on a static and spherically symmetric spacetime in which regularity exists when all components of the Riemann tensor are finite. There is work in the literature that assumes that the regularity of the metric is a sufficient condition to guarantee it. We study three regular metrics and show that they have singular spacetime. We also show that these metrics can be interpreted as solutions for black holes whose matter source is described by nonlinear electrodynamics. We analyze the geodesic equations and the Kretschmann scalar to verify the existence of the curvature singularity. Moreover, we use a change of the line element $r \rightarrow \sqrt{r^2+a^2}$, which is a process of regularization of spacetime already known in the literature. We then recompute the geodesic equations and the Kretschmann scalar and show that all metrics now have regular spacetime. This process transforms them into black-bounce solutions, two of which are new. We have discussed the properties of the event horizon and the energy conditions for all models.