Bouncing compact objects. II. Effective theory of a pulsating Planck star

Abstract

This article presents an effective quantum extension of the seminal Oppenheimer-Snyder (OS) collapse in which the singularity resolution is modeled using the effective dynamics of the spatially closed loop quantum cosmology. Imposing the minimal junction conditions, namely the Israel-Darmois conditions, we glue this bouncing LQC geometry to the classical vacuum exterior Schwarzschild geometry across a time-like thin-shell. Consistency of the construction leads to several major deviations from the classical OS collapse model. Firstly, no trapped region can form and the bounce occurs always above, or at most at the Schwarzschild radius. Secondly, the bouncing star discussed here admits an IR cut-off, additionally to the UV cut-off and corresponds therefore to a pulsating compact object. Thirdly, the scale at which quantum gravity effects become non-negligible is encoded in the ratio between the UV cut-off of the quantum theory and the IR cut-off, which in turn, encodes the minimal energy density $\rho_{\text{min}}$ of the star prior to collapse. This energy density is no more fixed by the mass and maximal radius as in the classical OS model, but is now a free parameter of the model. In the end, while the present model cannot describe a black-to-white hole bounce as initially suggested by the Planck star model, it provides a concrete realization of a pulsating compact object based on LQC techniques. Consistency of the model shows that its regime of applicability is restricted to Planckian relics while macroscopic stellar objects are excluded. This first minimal construction should serve as a platform for further investigations in order to explore the physics of bouncing compact objects within the framework of loop quantum cosmology.