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 $$\mathcal{C}\equiv \text {max}_r\{2m(r)/r\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo>≡</mml:mo> <mml:msub> <mml:mtext>max</mml:mtext> <mml:mi>r</mml:mi> </mml:msub> <mml:mrow> <mml:mo>{</mml:mo> <mml:mn>2</mml:mn> <mml:mi>m</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>r</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>/</mml:mo> <mml:mi>r</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> </mml:mrow> </mml:math> 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 $$\mathcal{C}\ge 1/3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> on their dimensionless compactness parameters.