Gravitational Collapse in Repulsive $R+\mu^4/R$ Gravity

In this paper we work out collapsing conditions for a spherical star in the weak field limit of the $R+\mu^4/R$ gravity and discuss the importance of the parameter $\mu$ to generate different criteria in the theory. Such criteria are proved to be resulting in a variety of different fates for the evolution of the outer shells of stars. Furthermore, we investigate the special case of violating the first junction condition and point out corresponding contradictions to the normal cases. These results show that the consistency of the $R+\mu^4/R$ theory of gravity with the common astrophysical predictions relies highly on the adoption of the parameter $\mu$ and satisfaction/violation of the first junction condition. For those anomalous results, further observational attempts are mandatory.