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

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 ${\cal C}\equiv\text{max}_r\{2m(r)/r\}$ 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 ${\cal C}\geq1/3$ on their dimensionless compactness parameters.