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

TL;DR

This work analyzes static spherically symmetric stars in symmetric teleparallel $f(Q)$ gravity, emphasizing that the connection equation must be satisfied alongside the metric equations. It constructs an exact interior solution for the power-law model $f(Q)=Q+\\alpha Q_{0}(Q/Q_{0})^{ u}$ using a singularity-free interior and a Schwarzschild-de Sitter exterior, ensuring a consistent matching. A symmetry-compatible second-family connection with $\\gamma_{2, ext{int}}(r)=r e^{-B/2}$ makes the interior solution valid for any $f(Q)$ and yields finite energy density and pressures, along with calculable radial and tangential sound speeds; the exterior solution corresponds to constant $Q$ and GR with an effective cosmological constant $\\Lambda_{ ext{eff}}$. The results demonstrate viable stellar configurations in $f(Q)$ gravity, with energy conditions satisfied and stability indicated by $\Gamma>4/3$, supporting the viability of nonmetricity-based gravity for relativistic stars and motivating a future Tolman-Oppenheimer-Volkoff formulation in this framework.

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 $f(Q)=Q+αQ_{0}\left( \frac{Q}{Q_{0}}\right) ^{ν}$. 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.

Figures (4)

• Figure 1: The radial (continuous lines) and tangential (dashed lines) sound speeds as functions of the star radius. The theory considered has $\nu =2$. The red lines correspond to $\alpha =0.1$ and the blue ones to $\alpha =0.01$. For the rest of the parameters we have $r_{1}=r_{2}=10$ km and $Q_{0}=0.01$ km$^{-2}$.
• Figure 2: The mass function in the interior of the star. The power index characterizing the theory is $\nu =2$. The red lines correspond to $\alpha =0.1$ and the blue ones to $\alpha =0.01$. For both curves $r_{1}=r_{2}=10$ km and $Q_{0}=0.01$ km$^{-2}$.
• Figure 3: Normalized energy density (dotted curves) and pressures in both directions (radial: solid curves, tangential: dashed curves) versus radial coordinate assuming $\alpha=0.01$ in blue color and $\alpha=0.1$ in red color. All three quantities are positive throughout the star, while at the same time $\rho > p_{r,t}$.
• Figure 4: Relativistic adiabatic index, $\Gamma$, in both directions (radial: solid, tangential: dashed) versus radial coordinate considering $\alpha=0.01$ in blue color and $\alpha=0.1$ in red color. The horizontal line in black represents the Newtonian value $4/3$.