On the visibility of singularities in general relativity and modified gravity theories

We investigate the global causal structure of the end state of a spherically symmetric marginally bound Lemaitre-Tolman-Bondi (LTB) \cite{Lemaitre, Tolman, Bondi} collapsing cloud (which is well studied in general relativity) in the framework of modified gravity having the generalized Lagrangian $R+\alpha R^2$ in the action. Here $R$ is the Ricci scalar, and $\alpha \geq 0$ is a constant. By fixing the functional form of the metric components of the LTB spacetime, using up the available degree of freedom, we realize that the matching surface of the interior and the exterior metric are different for different values of $\alpha$. This change in the matching surface can alter the causal property of the first central singularity. We depict this by showing a numerical example. Additionally, for a globally naked singularity to have physical relevance, a congruence of null geodesics should escape from such singularity to be visible to an asymptotic observer for an infinite time. For this to happen, the first central singularity should be a nodal point. We here give a heuristic method to show that this singularity is a nodal point by considering the above class of theory of gravity, of which general relativity is a particular case.