Polytropic spheres containing regions of trapped null geodesics

Abstract

We demonstrate that in the framework of standard general relativity, polytropic spheres with properly fixed polytropic index $n$ and relativistic parameter $\ensuremath{\sigma}$, giving a ratio of the central pressure ${p}_{\mathrm{c}}$ to the central energy density ${\ensuremath{\rho}}_{\mathrm{c}}$, can contain a region of trapped null geodesics. Such trapping polytropes can exist for $n>2.138$, and they are generally much more extended and massive than the observed neutron stars. We show that in the $n--\ensuremath{\sigma}$ parameter space, the region of allowed trapping increases with the polytropic index for intervals of physical interest, $2.138<n<4$. Space extension of the region of trapped null geodesics increases with both increasing $n$ and $\ensuremath{\sigma}>0.677$ from the allowed region. In order to relate the trapping phenomenon to astrophysically relevant situations, we restrict the validity of the polytropic configurations to their extension ${r}_{\mathrm{extr}}$ corresponding to the gravitational mass $M\ensuremath{\sim}2{M}_{\ensuremath{\bigodot}}$ of the most massive observed neutron stars. Then, for the central density ${\ensuremath{\rho}}_{\mathrm{c}}\ensuremath{\sim}1{0}^{15}\text{ }\text{ }\mathrm{g}\text{ }{\mathrm{cm}}^{\ensuremath{-}3}$, the trapped regions are outside ${r}_{\mathrm{extr}}$ for all values of $2.138<n<4$; for the central density ${\ensuremath{\rho}}_{\mathrm{c}}\ensuremath{\sim}5\ifmmode\times\else\texttimes\fi{}1{0}^{15}\text{ }\text{ }\mathrm{g}\text{ }{\mathrm{cm}}^{\ensuremath{-}3}$, the whole trapped regions are located inside ${r}_{\mathrm{extr}}$ for $2.138<n<3.1$; while for ${\ensuremath{\rho}}_{\mathrm{c}}\ensuremath{\sim}1{0}^{16}\text{ }\text{ }\mathrm{g}\text{ }{\mathrm{cm}}^{\ensuremath{-}3}$, the whole trapped regions are inside ${r}_{\mathrm{extr}}$ for all values of $2.138<n<4$, guaranteeing astrophysically plausible trapping for all considered polytropes. The region of trapped null geodesics is located close to the polytrope center and could have a relevant influence on the cooling of such polytropes or binding of gravitational waves in their interior.