A test of the conjectured critical black-hole formation -- null geodesic correspondence: The case of self-gravitating scalar fields

It has recently been conjectured [A. Ianniccari {\it et al.}, Phys. Rev. Lett. {\bf 133}, 081401 (2024)] that there exists a correspondence between the critical threshold of black-hole formation and the stability properties of null circular geodesics in the curved spacetime of the collapsing matter configuration. In the present compact paper we provide a non-trivial test of this intriguing conjecture. In particular, using analytical techniques we study the physical and mathematical properties of self-gravitating scalar field configurations that possess marginally-stable (degenerate) null circular geodesics. We reveal the interesting fact that the {\it analytically} calculated critical compactness parameter ${\mathcal{C}}^{\text{analytical}}\equiv {\text{max}}_r\{m(r)\mathrm{/}r\}=6\mathrm{/}25$, which signals the appearance of the first (marginally-stable) null circular geodesic in the curved spacetime of the self-gravitating scalar fields, agrees quite well (to within $\sim 10\mathrm{\%}$) with the exact compactness parameter ${\mathcal{C}}^{\text{numerical}}\equiv {\text{max}}_t\{{\text{max}}_r\{m(r)\mathrm{/}r\}\}\simeq 0.265$ which is computed {\it numerically} using fully non-linear numerical simulations of the gravitational collapse of scalar fields at the threshold of black-hole formation [here $m(r)$ is the gravitational mass contained within a sphere of radius $r$].