Physical properties and the maximum compactness bound of a class of compact stars in $f(Q)$ gravity

TL;DR

This study investigates anisotropic compact stars within linear $f(Q)$ gravity, using the Karmarkar condition and the Vaidya-Tikekar metric to obtain a closed-form interior solution and matching it to Schwarzschild exterior. The authors show that, although the total mass increases with the linear gravity parameter $\alpha$, the maximum compactness bound $u= M/R$ is governed by the VT curvature parameter $K$ via $u \le \frac{2(K+2)}{5K+9}$, reducing to the Buchdahl limit when $K=0$. Varying $\alpha$ affects the internal density and pressures—making them larger for more negative $\alpha$—and shifts the mass-radius relation toward lighter, more compact configurations, enabling the tuning of radii to observed pulsars (e.g., XTE J1814-338) while keeping the bound below Buchdahl. The work highlights the role of non-metricity in stellar structure within the linear regime and suggests extensions to nonlinear $f(Q)$ models and additional physical effects. Overall, the paper provides a tractable framework to quantify how non-metricity modifies stellar interiors and observable M-R behavior in a controlled, embedding-based setting.

Abstract

Motivation: Motivated by the growing interest in understanding the role of non-metricity in describing dense stellar systems, in this paper, we study compact stellar configurations within the framework of linear $f(Q)$ gravity. Methodology: By adopting a linear modification of the form $f(Q) = αQ+β$, we analyze the internal structure and physical properties of an anisotropic relativistic star within the framework of $f(Q)$ gravity. We employ the Karmarkar's condition together with the Vaidya-Tikekar metric ansatz to obtain a closed-form interior solution of the star. The interior solution is then matched to the Schwarzschild exterior solution across the boundary of the star. By varying the model parameters, we analyze physical features of the resultant stellar configuration. Results: We note distinctive features in the density, pressure, anisotropy and total mass of the star under a such modification. By enforcing the condition that the central pressure remains finite, we obtain the maximum compactness bound which is shown to depend solely on the Vaidya-Tikekar curvature parameter $K$. We recover the Buchdahl bound for the curvature parameter $K=0$, which corresponds to the solution for an isotropic and homogeneous fluid sphere. Utilizing the energy density and radial pressure profiles, we numerically integrate the modified Tolman-Oppenheimer-Volkoff equations and obtain the mass-radius ($M-R$) relationships for different values of the model parameter $α$. We note that for higher values of $α$, the maximum mass and radius decrease, shifting the stable branch towards ultra-compact configurations. An interesting observation in our analysis is that a linearly modified $f(Q)$ gravity model can support comparatively low mass stars. Utilizing the observed mass of some known pulsars, we demonstrate how our model can be used to fine-tune the radius of the star.

Figures (8)

• Figure 1: Metric potentials $e^{\lambda(r)}$ and $e^{\nu(r)}$ plotted against radial coordinate $r$. The metric potentials are regular throughout the interior of the star.
• Figure 2: Plot of mass function $m(r)$ against radial distance $r$ for different values of $\alpha$. Mass $m(r)$ within a radial distance $r$ increases in linear $f(Q)$ gravity. Note that we have used the conversion $1~M_{\odot} = 1.475~$km to express the mass function $m(r)$ in terms of $M_\odot$.
• Figure 3: Radial ($p_{r}$) and transverse ($p_{t}$) pressure plotted against radial coordinate $r$ for different values of $\alpha$. The two pressures take higher values, particularly in the core region, in $f(Q)$ gravity.
• Figure 4: Energy density($\rho$) plotted against the radial distance $r$ for different values of $\alpha$. The energy density takes higher values in $f(Q)$ gravity.
• Figure 5: Anisotropic factor $\Delta$ plotted against radial distance $r$ for different values of $\alpha$. The anisotropy vanishes at the centre for all values of $\alpha$ and takes comparatively higher values in $f(Q)$ gravity.