Trapping of null geodesics in slowly rotating extremely compact Tolman VII spacetimes

TL;DR

This study investigates how slow rotation affects null geodesic trapping inside extremely compact Tolman VII objects, revealing enhanced trapping efficiency compared to non-rotating models, with implications for gravitational wave and neutrino physics.

Contribution

It extends previous models by analyzing the influence of rotation on null geodesic trapping in Tolman VII spacetimes, incorporating quadratic density profiles and comparing with Schwarzschild-based models.

Findings

Rotation enhances trapping for counter-rotating geodesics.

Trapping occurs in larger radii than in Schwarzschild models.

Trapping efficiency depends on spacetime parameters.

Abstract

Region of trapped null geodesics hidden inside of extremely compact objects is of astrophysical importance because of trapping of gravitational waves, or neutrinos. The trapping effect of null geodesics was extensively studied for spherically symmetric extremely compact objects. Recently, influence of rotation of the extremely compact objects on the trapping of null geodesics was treated in the simplest possible model of internal linearised \mbox{Hartle--Thorne} spacetime with uniform energy density distribution and uniform emissivity distribution of null geodesics. Here we extend the study of the rotation influence on the trapping effect in the case of linearized \mbox{Hartle--Thorne} spacetimes based on the Tolman VII spherically symmetric solutions, where we assume the emissivity of the null geodesics proportional to the energy density of the Tolman VII object having quadratic radial profile. We demonstrate enhancement (suppression) of the trapping effect in the case of counter-rotating (co-rotating) null geodesics due to the behavior of the effective potentials and escape cones of the null geodesics in the linearised Hartle--Thorne--Tolman VII spacetimes. In dependence on the parameters of these spacetimes we determine the "local" and "global" coefficients of efficiency of the trapping and compare the results to those related to the rotating spacetimes based on the internal Schwarzschild spacetimes. We demonstrate that in the Tolman VII spacetimes the trapping is more efficient, being allowed in objects with radii larger than those of the trapping internal Schwarzschild spacetimes, occurring even for $R>3.3M$.