White Dwarf Structure in $f(Q)$ Gravity

This study investigates the equilibrium structure of white dwarfs within the quadratic $f(Q)=Q+\alpha Q^2$ gravity framework, demonstrating that positive nonmetricity corrections reduce the maximum stellar mass below the Chandrasekhar limit while yielding configurations consistent with the observed properties of the ultra-massive white dwarf ZTF J1901+1458.

The Gist

Imagine the universe as a giant, cosmic dance floor. For nearly a century, our best understanding of how gravity works has been based on a set of rules written by Albert Einstein, known as General Relativity (GR). In Einstein's version, gravity is like the curvature of a trampoline: if you put a heavy bowling ball (a star) in the middle, the fabric bends, and smaller marbles (planets) roll toward it.

But recently, scientists have noticed some weird moves on the dance floor that Einstein's rules can't quite explain. The universe is expanding faster than it should, and galaxies spin in ways that suggest there's invisible "dark matter" holding them together. This has led physicists to ask: What if Einstein's rules are just a special case of a bigger, more complex dance?

This paper by Rajasmita Sahoo explores one of those "bigger dance rules," called $f(Q)$ gravity.

The New Dance Move: "Non-Metricity"

In Einstein's dance, gravity comes from the curvature of space. In this new theory ($f(Q)$ gravity), gravity comes from something called non-metricity.

Think of it this way:

• Einstein's View: Imagine a rubber sheet. Gravity is the sheet stretching and bending.
• $f(Q)$ View: Imagine a grid of rulers laid out across the universe. In Einstein's world, these rulers stay the same size no matter where they are. In the $f(Q)$ world, the rulers themselves can stretch or shrink depending on where they are. This "stretching of the rulers" is what creates the force we feel as gravity.

The author investigates a specific version of this theory where the stretching follows a simple rule: a little bit of the old Einstein rules, plus a "quadratic" (squared) bonus term. This bonus term is controlled by a knob called $\alpha$ (alpha).

• If you turn the knob to zero, you get Einstein's General Relativity.
• If you turn the knob up, you get this new, modified gravity.

The Test Subject: White Dwarfs

To see if this new dance works, the author didn't look at the whole universe or black holes. Instead, she looked at White Dwarfs.

What is a White Dwarf?
Imagine a star like our Sun that has run out of fuel and collapsed. It's incredibly dense—a teaspoon of white dwarf material would weigh as much as an elephant. It's held up not by heat, but by a quantum "push" from electrons (like a crowd of people refusing to sit in the same chair). This is called electron degeneracy pressure.

There is a famous limit to how heavy a white dwarf can get before it collapses. This is the Chandrasekhar Limit (about 1.4 times the mass of our Sun). If a white dwarf gets heavier than this, it usually explodes or collapses into a neutron star.

The Experiment: Turning the $\alpha$ Knob

The author used a supercomputer to simulate white dwarfs using the new $f(Q)$ rules. She asked: What happens to these stars if we turn up the $\alpha$ knob?

Here is what she found, translated into everyday terms:

1. The Stars Get "Fluffier"
In Einstein's world, a heavy white dwarf is very compact and small. In the new $f(Q)$ world, as you increase the $\alpha$ knob, the stars become larger and less dense.

• Analogy: Imagine a sponge. In Einstein's world, you squeeze the sponge tight. In the $f(Q)$ world, the sponge seems to have a hidden spring inside it that pushes back harder, keeping the sponge puffed up even when it's heavy.

2. The Weight Limit Drops
Because the new gravity rules push back harder, the stars can't get as heavy as they could in Einstein's universe before they become unstable.

• The Result: The maximum mass a white dwarf can hold drops from the classic 1.4 solar masses down to about 1.35 solar masses (or even lower if you turn the knob up high).

3. A Perfect Match for a Real Star
Here is the exciting part. Astronomers recently discovered a super-heavy white dwarf called ZTF J1901+1458. It's massive (1.35 solar masses) but surprisingly large (about 2,200 km wide).

• In Einstein's theory, a star this heavy should be much smaller.
• But when the author set her $\alpha$ knob to a specific value ($5 \times 10^{18}$), her new theory predicted a star with exactly that mass and size.
• The Metaphor: It's like trying to fit a square peg in a round hole (Einstein's theory struggling to explain the star's size). The author found a new shape for the peg (the $f(Q)$ theory) that fits the hole perfectly.

The Safety Check: Are They Stable?

Before celebrating, the author had to make sure these new stars wouldn't immediately collapse or explode. She checked the "stability index" (a measure of how well the star resists shaking).

• The Verdict: As long as the $\alpha$ knob is turned to a positive value, the stars are stable. They wobble but don't break.
• The Warning: If she tried to turn the knob to a negative value, the stars became unstable and nonsensical. Nature seems to prefer the "positive" version of this new gravity.

The Big Picture

This paper suggests that White Dwarfs are the perfect laboratories to test if Einstein's gravity is the whole story.

• In weak gravity (like our solar system), the new rules look almost identical to Einstein's.
• In extreme gravity (inside a white dwarf), the new rules make a big difference.

The author concludes that if we observe more of these "ultra-massive" white dwarfs, we might finally have proof that gravity works a little differently than Einstein thought, governed by the stretching of space-rulers (non-metricity) rather than just the bending of space.

In short: The universe might have a hidden "spring" inside its gravity. White dwarfs are the only places heavy enough to stretch that spring, and this paper shows that a specific type of spring could explain some of the strangest stars we've ever seen.

Technical Summary

1. Problem Statement

General Relativity (GR) has been highly successful but faces challenges in explaining cosmic acceleration and dark matter/energy without invoking new components. Modified Gravity Theories (MGTs) offer alternatives. Specifically, Symmetric Teleparallel Gravity (STG) describes gravity via nonmetricity ($Q$) rather than curvature or torsion. The $f(Q)$ gravity framework generalizes STG by replacing the linear nonmetricity scalar with an arbitrary function $f(Q)$.

While $f(Q)$ gravity has been studied cosmologically and in the context of neutron stars, its application to white dwarfs remains under-explored. White dwarfs are ideal laboratories for testing gravity in high-density regimes ($10^9$–$10^{10}$ g cm$^{-3}$) where electron degeneracy pressure balances gravitational collapse. The paper addresses the need to understand how nonmetricity corrections affect the equilibrium structure, mass-radius relations, and stability of white dwarfs, particularly in light of recent observations of ultra-massive white dwarfs (e.g., ZTF J1901+1458) that challenge standard GR predictions.

2. Methodology

The study employs a covariant formulation of $f(Q)$ gravity, avoiding the "coincident gauge" (where the affine connection vanishes) to maintain coordinate independence and a rigorous geometric treatment.

• Gravitational Model: The authors adopt a quadratic model for the gravitational Lagrangian:
  $$f(Q) = Q + \alpha Q^2$$
  where $\alpha$ is a coupling parameter quantifying deviations from GR ($\alpha = 0$ recovers GR).
• Stellar Configuration:
  • Spacetime: Static and spherically symmetric metric with line element $ds^2 = -e^{A(r)}dt^2 + e^{B(r)}dr^2 + r^2(d\theta^2 + \sin^2\theta d\phi^2)$.
  • Matter: Modeled as a perfect fluid with energy-momentum tensor $T_{\mu\nu} = \text{diag}(-\epsilon, p, p, p)$.
  • Equation of State (EoS): The Chandrasekhar equation of state for a cold, fully degenerate relativistic electron gas is used to relate pressure ($P$) and energy density ($\epsilon$).
• Field Equations:
  • The authors derive the modified Tolman-Oppenheimer-Volkoff (TOV) equations from the covariant field equations of $f(Q)$ gravity.
  • The system consists of coupled differential equations for the metric potentials ($A(r), B(r)$), pressure ($p$), and the nonmetricity scalar ($Q$).
• Numerical Approach:
  • The equations are integrated numerically from the stellar center ($r=0$) with boundary conditions $B(0)=0$, $A'(0)=0$, and central pressure $p_c$.
  • Integration proceeds until pressure vanishes ($p(R)=0$), defining the stellar radius $R$ and total mass $M$.
  • Simulations are run for $\alpha \geq 0$ (negative $\alpha$ was found to yield unstable configurations). The parameter range explored is $\alpha \in [0, 10^{23}]$ cm$^2$.

3. Key Contributions

• Covariant Derivation: The paper provides the first detailed derivation of modified TOV equations for white dwarfs specifically within the fully covariant $f(Q)$ framework, rather than the simplified coincident gauge.
• Quadratic Correction Analysis: It systematically analyzes the impact of the quadratic term $\alpha Q^2$ on stellar structure, demonstrating that nonmetricity acts as a repulsive or modifying force at high densities.
• Observational Connection: The study links theoretical modifications to specific observational data, notably the ultra-massive white dwarf ZTF J1901+1458.

4. Key Results

A. Metric Functions and Nonmetricity

• Metric Potentials ($A(r), B(r)$): Increasing $\alpha$ modifies the temporal and radial metric potentials. $A(r)$ grows slightly slower, and the peak of $B(r)$ decreases compared to GR, indicating a modification of the interior gravitational potential.
• Nonmetricity Scalar ($Q$): In GR, $Q \approx 0$. In $f(Q)$, $Q$ becomes non-zero, reaching a negative minimum in the interior. Increasing $\alpha$ suppresses the magnitude of this minimum, effectively altering the spacetime geometry.

B. Internal Structure Profiles

• Pressure and Density: For a given central density, increasing $\alpha$ leads to lower central pressure and lower central density compared to GR. The density profiles become flatter in the outer regions.
• Mass Distribution: The enclosed mass $m(r)$ grows more slowly with radius. Consequently, for a fixed central density, the resulting stars are less massive but larger in radius than their GR counterparts.

C. Mass-Radius (M-R) Relation

• Shift in Equilibrium: The M-R curves shift toward larger radii and lower maximum masses as $\alpha$ increases.
• Maximum Mass Reduction:
  • GR ($\alpha=0$): $M_{max} \approx 1.4253 M_\odot$ at $R \approx 1044$ km.
  • High $\alpha$ ($5 \times 10^{18}$ cm$^2$): $M_{max} \approx 1.3519 M_\odot$ at $R \approx 2229$ km.
• Observational Fit: The model with $\alpha = 5 \times 10^{18}$ cm$^2$ predicts a mass and radius that aligns remarkably well with the observed properties of ZTF J1901+1458 ($M \approx 1.35 M_\odot$, $R \approx 2140$ km). This suggests that nonmetricity corrections could explain the existence of such massive, extended white dwarfs without violating stability.

D. Stability Analysis

• Adiabatic Index ($\Gamma$): The relativistic adiabatic index was calculated to check for dynamical stability.
• Result: For all considered values of $\alpha$, $\Gamma$ remains greater than the critical value of $4/3$ throughout the stellar interior. This confirms that the modified configurations are dynamically stable against radial perturbations.

5. Significance and Conclusion

• Viability of $f(Q)$ Gravity: The study demonstrates that quadratic $f(Q)$ gravity admits physically consistent, stable white dwarf solutions.
• High-Density Probes: The deviations from GR are negligible in low-density regimes but become significant near the Chandrasekhar limit. This establishes white dwarfs as complementary probes to neutron stars for testing modified gravity in the strong-field regime.
• Parameter Constraints: The analysis restricts $\alpha$ to non-negative values, as negative values lead to unphysical or unstable configurations.
• Astrophysical Implications: The ability of the model to reproduce the properties of ZTF J1901+1458 suggests that nonmetricity-based modifications of gravity could be a viable explanation for ultra-massive white dwarfs, offering a new avenue to constrain the parameter space of $f(Q)$ theories using astrophysical observations.

In summary, the paper establishes that nonmetricity corrections in $f(Q)$ gravity significantly alter the equilibrium structure of white dwarfs, reducing their maximum mass and increasing their radius, while maintaining stability. This provides a theoretical framework that aligns with recent high-precision astronomical observations.