Stability and Physical Properties of Compact Stars Beyond Einstein Gravity

Abstract: This manuscript discusses feasible features of anisotropic celestial sphere within the framework of $f(\mathbb{Q},\mathcal{L}<em>{m})$ gravity, where $\mathbb{Q}$ represents non-metricity scalar and $\mathcal{L}</em>{m}$ is the matter Lagrangian. The geometric configuration of static spherical symmetric structure is examined using a specific non-singular solution (Krori-Barua solution). A particular model of this theory is considered to derive explicit field equations. The Darmois matching conditions are used to evaluate unknown constants in the metric coefficients. To verify plausible existence of compact objects in this gravitational framework, we analyze their fundamental physical properties including fluid parameters, gradients, surface redshift, mass-radius relation, anisotropy measure, compactness factor, energy conditions and equations of state. The stability of the considered stellar objects is verified by adiabatic index and sound speed. Our results demonstrate that all required physical conditions are satisfied, confirming the existence of physically stable anisotropic celestial objects within this modified gravity.

• The paper introduces an extension of symmetric teleparallel gravity via f(Q, Lₘ) to examine anisotropic compact stars with non-minimal matter-geometry coupling.
• It derives static, spherically symmetric field equations using the Krori-Barua solution, ensuring regular density and pressure profiles that meet all energy conditions.
• The study demonstrates that enhanced central pressures and modified mass-radius relations yield stable stellar configurations within astrophysical bounds.

Stability and Physical Properties of Compact Stars in $f(\mathbb{Q},\mathcal{L}_m)$ Gravity

Introduction

The investigation addresses the structure and stability of compact stars within the framework of $f(\mathbb{Q},\mathcal{L}_m)$ gravity, a recently developed extension of symmetric teleparallel gravity. This approach generalizes coincident general relativity by allowing gravitational dynamics to depend both on the non-metricity scalar $\mathbb{Q}$ and the matter Lagrangian $\mathcal{L}_m$, introducing new matter-geometry couplings beyond those in $f(R)$ or $f(\mathbb{Q},\mathbb{T})$ models. The study's focus is on analyzing anisotropic compact stellar objects—systems where the radial and tangential pressures differ, which is a physically motivated scenario at supranuclear densities—by deriving explicit field equations, constructing interior solutions, and systematically evaluating stability and physical viability under these extended gravitational dynamics.

Theoretical Framework and Field Equations

The action underpinning $f(\mathbb{Q},\mathcal{L}_m)$ gravity is given by $$S = \frac{1}{2 \kappa} \int f(\mathbb{Q},\mathcal{L}_m) \sqrt{-g} \, d^4x$$, where $\mathbb{Q}$ is the non-metricity scalar reflecting symmetric teleparallel geometry, and $\mathcal{L}_m$ is the matter Lagrangian density. This formulation enables non-minimal coupling between spacetime geometry and the full matter Lagrangian, resulting in second-order field equations despite the additional degrees of freedom introduced.

A static, spherically symmetric metric is adopted, and an anisotropic fluid is modeled with $f(\mathbb{Q},\mathcal{L}_m)$0 (radial pressure), allowing for physically well-motivated matter-geometry interactions and manageable algebraic structure. The field equations are then derived for a specific linear model,

$f(\mathbb{Q},\mathcal{L}_m)$1

where $f(\mathbb{Q},\mathcal{L}_m)$2 governs the strength of matter-non-metricity coupling and $f(\mathbb{Q},\mathcal{L}_m)$3 acts as a background energy density. These parameters are constrained by requiring regularity, positive definite energy density and pressures, and physical viability of the resulting solutions.

Matching Conditions and Interior Solution

To construct complete stellar models, the interior Krori-Barua solution is employed:

$f(\mathbb{Q},\mathcal{L}_m)$4

with arbitrary constants $f(\mathbb{Q},\mathcal{L}_m)$5, $f(\mathbb{Q},\mathcal{L}_m)$6, $f(\mathbb{Q},\mathcal{L}_m)$7 determined by matching the interior solution to the exterior Schwarzschild metric at the stellar boundary through the Darmois conditions. This ensures that physical and mathematical continuity is enforced at the surface, where the pressure vanishes and the metric is continuous.

Physical Properties and Analysis

A comprehensive suite of physical properties is analytically evaluated and their parameter dependences are explored:

• Density and Pressure Profiles: Both quantities are maximal at the stellar center and decrease monotonically with radius. This behavior is essential for stable, physically realistic compact objects. Gradients of these profiles are negative-definite at the center, enforcing a dense core structure.
• Pressure Anisotropy: The difference $f(\mathbb{Q},\mathcal{L}_m)$8 is strictly positive throughout the star, indicating that anisotropy provides additional repulsive force counteracting gravity. This repulsive anisotropy is known to support higher-mass configurations and alter mass-radius relations compared to isotropic cases.
• Energy Conditions: All standard energy conditions (null, weak, strong, and dominant) are checked and found to be satisfied throughout the configuration, confirming that the matter content is non-exotic and physically acceptable under the new gravity model.
• Equation of State Parameters: The EoS parameters $f(\mathbb{Q},\mathcal{L}_m)$9 and $\mathbb{Q}$0 adhere to the physical range $\mathbb{Q}$1, as required for typical compact matter equations of state.

Compactness and Redshift Constraints

• Mass Function and Compactness: The enclosed mass increases monotonically with radius and vanishes at the center, signifying regularity. The compactness $\mathbb{Q}$2 is always below Buchdahl's limit ($\mathbb{Q}$3), and the surface redshift adheres to the Ivanov bound ($\mathbb{Q}$4 for anisotropic models).
• Mass-Radius Relation: The predicted mass-radius relation is consistent with observational data for known compact stars, including predictions for maximum masses up to $\mathbb{Q}$5 as found for Her X-1, demonstrating that the theoretical models remain within astrophysical limits.

Stability Assessment

Sound Speed and Adiabatic Index

• Causality and Subluminality: The squared sound speeds $\mathbb{Q}$6 and $\mathbb{Q}$7 are evaluated and remain within the interval $\mathbb{Q}$8, preserving causality and the absence of superluminal propagation.
• Adiabatic Index: The adiabatic indices $\mathbb{Q}$9 and $\mathcal{L}_m$0 both exceed the canonical threshold $\mathcal{L}_m$1, confirming dynamical stability against infinitesimal radial perturbations throughout the star.

Comparison with General Relativity and Modified Theories

Numerical comparison with standard GR and other modified gravity models is highlighted:

• Central Densities and Pressures: The $\mathcal{L}_m$2 models yield higher central density and central pressure compared to GR, and maintain greater stability ranges than those admissible in $\mathcal{L}_m$3 or $\mathcal{L}_m$4 theories.
• Physical Bounds: The enhanced coupling in $\mathcal{L}_m$5 enables physically viable and stable configurations even for objects that are unstable or unphysical under certain other modified gravity models.

Implications and Perspectives

The study clearly demonstrates that $\mathcal{L}_m$6 gravity provides a consistent and robust framework for modeling compact objects with anisotropic pressures, accommodating physical requirements not limited to but including regularity, stability, and compatibility with observed masses and radii. The additional degrees of freedom associated with non-metricity-matter coupling enable richer phenomenology—particularly in the maximum mass and mass-radius relations—while ensuring tractable and stable field equations.

On a practical level, this framework is a promising avenue for exploring gravitational theories that deviate from GR without invoking exotic matter or violating astrophysical constraints. Theoretically, these results support the viability of non-Riemannian geometries and matter-geometry couplings as cogent extensions for strong-field gravity and high-density regimes.

Future directions include incorporating more complex equations of state, relaxing the spherical symmetry assumption, and directly confronting high-precision observational data from gravitational wave detections and X-ray timing for compact stars. These investigations could further constrain the parameter space of $\mathcal{L}_m$7 and elucidate potential deviations from GR in astrophysical environments.

Conclusion

A detailed analysis of compact star interiors within $\mathcal{L}_m$8 gravity reveals that all requisite physical and stability criteria are met for anisotropic configurations. The field equations remain regular and physically interpretable, all energy conditions are satisfied, and stable stellar solutions exist for astrophysically relevant parameter ranges. These findings demonstrate the theoretical and phenomenological adequacy of $\mathcal{L}_m$9 models for modeling dense, compact objects, making them viable candidates for exploring gravitational phenomena beyond Einstein gravity.