Compactness bounds in general relativity

A foundational theorem due to Buchdahl states that, within general relativity (GR), the maximum compactness $\mathcal{C}\ensuremath{\equiv}GM/(R{c}^{2})$ of a static, spherically symmetric, perfect fluid object mass $M$ and radius $R$ is $\mathcal{C}=4/9$. As corollary, there exists gap between stars black holes (where $\mathcal{C}=1/2$). Here we generalize Buchdahl's result by introducing most equation state for elastic matter with constant longitudinal wave speeds apply it compute regular, self-gravitating objects in GR. We show that: (i) grows monotonically speed; (ii) can exceed bound reach hole $\mathcal{C}=1/2$ continuously; (iii) however, imposing subluminal propagation lowers $\mathcal{C}\ensuremath{\approx}0.462$, which conjecture be any static satisfying causality; (iv) also radial stability further decreases $\mathcal{C}\ensuremath{\approx}0.389$. Therefore, although anisotropies are often invoked as mechanism supporting horizonless ultracompact objects, argue that cannot reached physically reasonable GR true mimickers require either exotic or beyond-GR effects.