Relativistic stars in $f(Q)$-gravity: Exact analytic solution for the power-law case $f(Q) = Q + b \: Q^\nu$

Abstract

We investigate static spherically symmetric spacetimes within the framework of symmetric teleparallel $f(Q)$ gravity in order to describe relativistic stars. We adopt a specific ansatz for the background geometry corresponding to a singularity-free space-time. We obtain an expression for the connection, which allows the derivation of solutions for any $f(Q)$ theory in this context. Our approach aims to address a recurring error appearing in the literature, where even when a connection compatible with spherical symmetry is adopted, the field equation for the connection is systematically omitted and not checked if it is satisfied. For the stellar configuration, we concentrate on the power-law model $f(Q)=Q+\alpha Q_{0}\left( \frac{Q}{Q_{0}}\right) ^{\nu }$. The de Sitter-Schwarzschild geometry naturally emerges as an attractor beyond a certain radius, we thus utilize it as the external solution beyond the boundary of the star. We perform a detailed investigation of the physical characteristics of the interior solution, explicitly determining the mass function, analyzing the resulting gravitational fluid properties and deriving the angular and radial speed of sound.