Analytic study of self-gravitating polytropic spheres with light rings

Ultra-compact objects describe horizonless solutions of the Einstein field equations which, like black-hole spacetimes, possess null circular geodesics (closed light rings). We study analytically the physical properties of spherically symmetric ultra-compact isotropic fluid spheres with a polytropic equation of state. It is shown that these spatially regular horizonless spacetimes are generally characterized by two light rings \(\{r^{\text {inner}}_{\gamma },r^{\text {outer}}_{\gamma }\}\) with the property \(\mathcal{C}(r^{\text {inner}}_{\gamma })\le \mathcal{C}(r^{\text {outer}}_{\gamma })\), where \(\mathcal{C}\equiv m(r)/r\) is the dimensionless compactness parameter of the self-gravitating matter configurations. In particular, we prove that, while black-hole spacetimes are characterized by the lower bound \(\mathcal{C}(r^{\text {inner}}_{\gamma })\ge 1/3\), horizonless ultra-compact objects may be characterized by the opposite dimensionless relation \(\mathcal{C}(r^{\text {inner}}_{\gamma })\le 1/4\). Our results provide a simple analytical explanation for the interesting numerical results that have recently presented by Novotný et al. (Phys Rev D 95:043009, 2017).