A compact theorem on the compactness of ultra-compact objects with monotonically decreasing matter fields

Abstract

Self-gravitating horizonless ultra-compact objects that possess light rings have attracted the attention of physicists and mathematicians in recent years. In the present compact paper we raise the following physically interesting question: Is there a lower bound on the global compactness parameters \(\mathcal{C}\equiv \text {max}_r\{2m(r)/r\}\) of spherically symmetric ultra-compact objects? Using the non-linearly coupled Einstein-matter field equations we explicitly prove that spatially regular ultra-compact objects with monotonically decreasing density functions (or monotonically decreasing radial pressure functions) are characterized by the lower bound \(\mathcal{C}\ge 1/3\) on their dimensionless compactness parameters.