White Dwarf Structure in $f(Q)$ Gravity

Abstract: In this work, we investigate the equilibrium structure of white dwarfs within the covariant formulation of symmetric teleparallel $f(Q)$ gravity, in which gravity is described by the nonmetricity scalar $Q$ instead of spacetime curvature. We consider static and spherically symmetric stellar configurations composed of cold, fully degenerate electron matter and adopt a quadratic form of the gravitational Lagrangian, $f(Q)=Q+αQ<sup>{2}$, where $α$ quantifies deviations from general relativity. The corresponding modified stellar structure equations are solved numerically in conjunction with the Chandrasekhar equation of state. We examine the impact of the parameter $α$ on the internal structure and global properties of white dwarfs, including the radial profiles of the metric potentials, pressure, density, nonmetricity scalar, and enclosed mass, as well as the mass--radius relation. While negative values of $α$ were explored, they lead to unstable or nonphysical configurations at high densities; therefore, the analysis is restricted to non-negative values of $α$. Our results show that nonmetricity corrections produce significant deviations from the general relativistic predictions in the high-density regime. In particular, increasing $α$ modifies the equilibrium configurations and leads to a reduction in the maximum mass relative to the Chandrasekhar limit, accompanied by corresponding changes in the stellar radius and interior profiles. For $α= 5\times10<sup>{18}\,\mathrm{cm<sup>2}$, we obtain a maximum mass $M_{\max}=1.3519\,M_{\odot}$ and radius $R=2228.85\,\mathrm{km}$, which are consistent with the observational constraints of the ultra-massive white dwarf ZTF J1901+1458. These findings suggest that white dwarfs can provide a complementary astrophysical probe for testing the viability of $f(Q)$ gravity in the strong-field regime.

• The paper solves covariant modified TOV equations with the quadratic model f(Q)=Q+αQ² and a Chandrasekhar equation of state, extending white-dwarf structure studies beyond the coincident gauge.
• Positive α values reduce maximum masses and increase radii relative to GR, from 1.4253 M☉ and 1,044 km at α=0 to 0.5452 M☉ and 8,106 km at α=10²³ cm², while low-density stars remain nearly unchanged.
• The model with α=5×10¹⁸ cm² predicts 1.3519 M☉ and 2,228.85 km, consistent with observations of ZTF J1901+1458, although magnetic fields, rotation, realistic microphysics, and independent gravity constraints remain unresolved.

Overview

This paper examines the equilibrium structure of white dwarfs within the covariant formulation of symmetric teleparallel $f(Q)$ gravity, in which the gravitational interaction is mediated by the nonmetricity scalar $Q$ rather than spacetime curvature. The author adopts a quadratic gravitational Lagrangian, $f(Q)=Q+\alpha Q^2$, where $\alpha$ parametrizes deviations from general relativity (GR), and solves the modified Tolman–Oppenheimer–Volkoff (TOV) equations numerically for cold, fully degenerate electron matter described by the Chandrasekhar equation of state. Unlike most prior work on compact objects in $f(Q)$ gravity, which employs the coincident gauge ($\Gamma^\alpha{}_{\beta\gamma}=0$), this analysis retains the affine connection as an independent geometric object, following the covariant formalism developed for neutron stars (Alwan et al., 2024).

Geometric and field-equation framework

The theory is built on the geometric trinity decomposition of the affine connection into Levi-Civita, contortion, and disformation parts. In symmetric teleparallel gravity both curvature and torsion vanish, so dynamics are governed by $Q = Q_{\alpha\beta\gamma}P^{\alpha\beta\gamma}$, with $P^\alpha{}_{\beta\gamma}$ the nonmetricity superpotential. Variation of the action $$S=\int[\frac{1}{2\kappa}f(Q)+\mathcal{L}_m]\sqrt{-g}\,d^4x$$ yields metric field equations of the form

$$f_Q\,\overset{\circ}{G}_{\beta\gamma}+\tfrac{1}{2}g_{\beta\gamma}(Qf_Q-f)+2f_{QQ}\,P^\alpha{}_{\beta\gamma}\,\overset{\circ}{\nabla}_\alpha Q=\kappa T_{\beta\gamma},$$

supplemented by the connection equation $Q$0. For a static, spherically symmetric line element with metric functions $Q$1 and $Q$2, these reduce to a coupled system for $Q$3, $Q$4, and the hydrostatic equilibrium condition $Q$5, which recovers the standard TOV system when $Q$6. Regularity at the center requires $Q$7, $Q$8, and the surface is defined by $Q$9.

Equation of state and numerical setup

Matter is modeled by the relativistic degenerate electron gas with mean molecular weight per electron $f(Q)=Q+\alpha Q^2$0, appropriate for helium, carbon, or oxygen compositions. The pressure and energy density are expressed through the relativity parameter $f(Q)=Q+\alpha Q^2$1, interpolating between polytropic indices $f(Q)=Q+\alpha Q^2$2 (non-relativistic) and $f(Q)=Q+\alpha Q^2$3 (ultra-relativistic). Sequences are integrated outward from specified central densities $f(Q)=Q+\alpha Q^2$4 until the pressure vanishes. The parameter range explored is $f(Q)=Q+\alpha Q^2$5 to $f(Q)=Q+\alpha Q^2$6; exploratory runs with negative $f(Q)=Q+\alpha Q^2$7 produced unstable or nonphysical configurations at high densities—a behavior consistent with neutron-star studies in the same framework—so the analysis is restricted to $f(Q)=Q+\alpha Q^2$8.

Internal structure results

All radial profiles remain smooth and singularity-free across the explored parameter space. Increasing $f(Q)=Q+\alpha Q^2$9 lowers the temporal metric function $\alpha$0 at fixed radius and suppresses the peak of $\alpha$1, indicating more extended configurations. The nonmetricity scalar $\alpha$2 is nearly zero at the center, reaches a negative minimum of order $\alpha$3 in the interior, and approaches zero at the surface; larger $\alpha$4 progressively flattens this profile. Central pressure and density both decrease with increasing $\alpha$5—for positive $\alpha$6, less pressure is required to support equilibrium against collapse—and the enclosed mass grows more slowly toward a reduced total mass.

Mass–radius relation and comparison with observation

The GR limit ($\alpha$7) reproduces the standard Chandrasekhar sequence with $\alpha$8 at $\alpha$9, serving as a consistency check. Representative maximum-mass configurations are:

$f(Q)$0 (cm²)	$f(Q)$1 (g cm⁻³)	$f(Q)$2 ($f(Q)$3)	$f(Q)$4 (km)
0	$f(Q)$5	1.4253	1044
$f(Q)$6	$f(Q)$7	1.4130	1327
$f(Q)$8	$f(Q)$9	1.3868	1731
$\Gamma^\alpha{}_{\beta\gamma}=0$0	$\Gamma^\alpha{}_{\beta\gamma}=0$1	1.3519	2229
$\Gamma^\alpha{}_{\beta\gamma}=0$2	$\Gamma^\alpha{}_{\beta\gamma}=0$3	1.2261	3378
$\Gamma^\alpha{}_{\beta\gamma}=0$4	$\Gamma^\alpha{}_{\beta\gamma}=0$5	0.5452	8106

Two systematic trends emerge: increasing $\Gamma^\alpha{}_{\beta\gamma}=0$6 reduces the maximum mass below the Chandrasekhar limit while shifting configurations toward larger radii, and the deviations are confined to the high-density regime—low-density solutions remain essentially indistinguishable from GR. Notably, for $\Gamma^\alpha{}_{\beta\gamma}=0$7 the model predicts $\Gamma^\alpha{}_{\beta\gamma}=0$8 and $\Gamma^\alpha{}_{\beta\gamma}=0$9, which fall within the observational constraints of the ultra-massive white dwarf ZTF J1901+1458 ($Q = Q_{\alpha\beta\gamma}P^{\alpha\beta\gamma}$0, $Q = Q_{\alpha\beta\gamma}P^{\alpha\beta\gamma}$1 km) (Caiazzo et al., 2021). This agreement suggests that quadratic $Q = Q_{\alpha\beta\gamma}P^{\alpha\beta\gamma}$2 corrections can accommodate observed compact white dwarf configurations, though it should be read as consistency rather than a unique determination of $Q = Q_{\alpha\beta\gamma}P^{\alpha\beta\gamma}$3, since ZTF J1901+1458 is also highly magnetized and rapidly rotating—effects not modeled here.

Stability

Dynamical stability is assessed via the relativistic adiabatic index $Q = Q_{\alpha\beta\gamma}P^{\alpha\beta\gamma}$4, requiring $Q = Q_{\alpha\beta\gamma}P^{\alpha\beta\gamma}$5 against radial perturbations. For all considered values of $Q = Q_{\alpha\beta\gamma}P^{\alpha\beta\gamma}$6, $Q = Q_{\alpha\beta\gamma}P^{\alpha\beta\gamma}$7 remains above the critical threshold throughout the stellar interior, although it decreases with increasing $Q = Q_{\alpha\beta\gamma}P^{\alpha\beta\gamma}$8. Combined with the turning-point behavior of the mass–central density relation, this supports the conclusion that positive-$Q = Q_{\alpha\beta\gamma}P^{\alpha\beta\gamma}$9 configurations are dynamically stable within the explored parameter range.

Limitations and open questions

The paper concedes several restrictions on its conclusions. Negative values of $P^\alpha{}_{\beta\gamma}$0 were excluded after producing unstable or nonphysical high-density configurations, so the viable sign of the coupling is asserted rather than derived from first principles. The equation of state neglects finite-temperature effects, Coulomb corrections, inverse beta decay, magnetic fields, and rotation—all relevant for ultra-massive white dwarfs such as ZTF J1901+1458, whose inferred parameters depend sensitively on its strong magnetic field. Stability is assessed only through the local adiabatic index criterion rather than full radial perturbation analysis, and no independent constraint on $P^\alpha{}_{\beta\gamma}$1 from other astrophysical or solar-system tests is imposed. Whether the value $P^\alpha{}_{\beta\gamma}$2 required to match ZTF J1901+1458 is compatible with cosmological and weak-field bounds remains an open question.

Conclusion

This work demonstrates that physically acceptable white dwarf equilibria can be constructed in covariant quadratic $P^\alpha{}_{\beta\gamma}$3 gravity using a non-coincident gauge. The quadratic nonmetricity correction systematically reduces the maximum mass below the Chandrasekhar limit and enlarges stellar radii, with effects concentrated near the relativistic density regime while leaving the weak-field limit close to GR. The compatibility of the $P^\alpha{}_{\beta\gamma}$4 model with the observed properties of ZTF J1901+1458 identifies massive white dwarfs as complementary probes of nonmetricity-based modifications of gravity in the strong-field regime, pending more realistic stellar modeling and cross-correlation with independent constraints on $P^\alpha{}_{\beta\gamma}$5.