Relativistic Origin of the Cutoff Parameter in Exotic Compact Objects

Abstract

A Black Hole (BH) is a spacetime region with a horizon and where geodesics converge to a singularity. At such a point, the gravitational field equations fail. As an alternative to the problem of the singularity arises the existence of Exotic Compact Objects (ECOs) that prevent the problem of the singularity through a transition phase of matter once it has crossed the horizon. ECOs are characterized by a closeness parameter or cutoff, $\epsilon$, which measures the degree of compactness of the object. This parameter is established as the difference between the radius of the ECO's surface and the gravitational radius. Thus, different values of $\epsilon$ correspond to different types of ECOs. If $\epsilon$ is very big, the ECO behaves more like a star than a black hole. On the contrary, if $\epsilon$ tends to a very small value, the ECO behaves like a black hole. It is considered a conceptual model of the origin of the cutoff for ECOs, when a dust shell contracts gravitationally from an initial position to near the Schwarzschild radius. This allowed us to find that the cutoff makes two types of contributions: a classical one governed by General Relativity and one of a quantum nature, if the ECO is very close to the horizon, when estimating that the maximum entropy is contained within the material that composes the shell. Such entropy coincides with the Bekenstein--Hawking entropy. The established cutoff corresponds to a dynamic quantity dependent on coordinate time that is measured by a Fiducial Observer (FIDO). Without knowing the details about quantum gravity, parameter $\epsilon$ is calculated, which, in general, allows distinguishing the ECOs from BHs. Specifically, a black shell (ECO) is undistinguishable from a BH.