Beyond Buchdahl's limit: bilayered stars and thin-shell configurations

One of the theoretical motivations behind the belief that black holes as described by general relativity exist in nature is that it is hard to find matter configurations that mimic their properties, especially their compactness. One of the classic results that goes in this direction is the socalled Buchdahl limit: a bound for the maximum compactness that spherically symmetric isotropic fluid spheres in hydrostatic equilibrium can possibly achieve with an outward-decreasing energy density. However, physically realistic situations could violate both isotropy and the monotonicity of the density profile. Notably, Bondi already showed that if the density profile is allowed to be arbitrary (but remains non-negative), a less restrictive compactness bound emerges. Furthermore, if negative energy densities are permitted, configurations can approach the black hole compactness limit arbitrarily closely. In this work we introduce a set of simple bilayered and thin-shell toy models designed to illustrate the effect of relaxing separately the assumptions of Buchdahl's theorem. Within these models we highlight the existence of two special examples that we have called AdS stars and Einstein Static stars. We also discuss how these toy models may represent some of the main features of realistic systems, and how they could be extended to find more refined models.