Two Parameter Deformation of Embedding Class-I Compact Stars in Linear $f(Q)$ Gravity

This paper proposes a two-parameter deformation framework combining gravitational decoupling with linear $f(Q)$ gravity to enlarge the mass window of compact stars, demonstrating that the interplay between the decoupling parameter $\epsilon$ and the coupling constant $\beta_1$ allows for physically viable, high-mass configurations compatible with recent observations without violating causal limits.

The Gist

The Big Picture: Building a Heavier Star Without Breaking the Rules

Imagine you are an architect trying to build a skyscraper (a neutron star) that is incredibly heavy. In our current understanding of physics (General Relativity), there is a strict limit to how heavy a building can get before it collapses into a black hole. However, recent observations have found "ghost" objects in the universe that are too heavy to be normal stars but too light to be black holes. They exist in a "mass gap."

The authors of this paper are trying to figure out how to build these heavy stars without breaking the laws of physics or making the material inside the star impossibly stiff (which would be unrealistic).

They propose a new blueprint using a modified version of gravity called linear $f(Q)$ gravity combined with a construction technique called gravitational decoupling.

The Two Tools in Their Toolbox

The paper introduces a "two-parameter" system. Think of this as having two different knobs on a control panel that you can turn to adjust the star.

1. The "Gravity Dial" ($\beta_1$)

In standard gravity (General Relativity), the strength of gravity is fixed. In this new theory, the authors introduce a knob called $\beta_1$.

• The Analogy: Imagine you are baking a cake. The recipe (the geometry of the star) stays exactly the same. However, you change the brand of flour you use. This new flour is slightly denser or lighter.
• What it does: Turning this knob doesn't change the shape of the star or how the walls are built. It simply rescales the "weight" of the ingredients. If you turn the dial down (make $\beta_1$ smaller), the star can hold more mass without collapsing, even though the shape of the star looks identical to the old one. It's like the star is "heavier" because the gravity holding it together is slightly different, not because the star itself changed shape.

2. The "Shape Shifter" ($\epsilon$)

This is the second knob, which comes from a technique called "gravitational decoupling."

• The Analogy: Imagine you have a balloon. The first knob just changed the air density inside. The second knob, however, actually stretches the rubber of the balloon. It changes the geometry, the pressure, and the internal structure.
• What it does: This knob physically deforms the star. It changes how the pressure is distributed inside, making the star "stiffer" and able to support more weight. It creates a new geometric shape that wasn't possible before.

Why This Combination is Special

The paper argues that previous models only had one of these tools, or used them in a way that didn't separate the effects.

• The Old Way (General Relativity): If you wanted a heavier star, you had to stretch the balloon (change the shape). But stretching the balloon also changed the internal pressure in a way that was hard to control. You couldn't tell if the extra weight was because you stretched the shape or because you changed the material.
• The New Way (This Paper): The authors show that by using both knobs together, they can do something unique:
  1. They can keep the "Shape Shifter" ($\epsilon$) fixed to ensure the star has a specific, realistic internal structure.
  2. Then, they can turn the "Gravity Dial" ($\beta_1$) to make the star even heavier without changing that structure.

It's like having a car where you can change the engine size (to go faster) without having to redesign the chassis. This allows them to build stars that are heavy enough to fit into that mysterious "mass gap" observed by astronomers, without violating the speed of light or other physical laws.

The Results: What Did They Find?

1. Solving the Mass Gap: By turning these two knobs, the authors found configurations that can support stars with masses around 2.6 to 2.8 times the mass of our Sun. This fits perfectly with the mysterious heavy objects detected by gravitational wave observatories (like GW190814) that were previously too heavy to be explained by standard models.
2. No "Magic" Stiffening: Usually, to make a star heavier, you have to assume the matter inside is incredibly stiff (like a super-hard diamond). The authors show that their method allows for heavy stars using more realistic, softer materials, because the "Gravity Dial" does the heavy lifting.
3. Physical Safety: They checked all the rules: the star doesn't collapse, the pressure doesn't go negative, and sound waves travel slower than light. The model is physically safe and stable.

The Bottom Line

The paper claims that by combining a specific type of modified gravity with a technique to deform the star's shape, they have created a "two-parameter" framework. This framework acts like a master control panel that allows physicists to tune the mass of a neutron star independently of its shape. This explains how we can have these incredibly heavy, mysterious stars in the universe without breaking the fundamental laws of physics.

Technical Summary

1. Problem Statement

The construction of realistic compact star models faces two primary challenges:

1. Geometric Rigidity: Embedding Class-I spacetimes (satisfying the Karmarkar condition) are highly restrictive. In General Relativity (GR), they often admit only idealized isotropic solutions or require specific, rigid metric ansätze (like the Vaidya-Tikekar metric).
2. Limitations of Linear $f(Q)$ Gravity: While $f(Q)$ gravity (based on non-metricity) is a promising alternative to GR, the linear form $f(Q) = \beta_1 Q + \beta_2$ is dynamically equivalent to GR at the geometric level. It only rescales the matter sector via the parameter $\beta_1$. Consequently, linear $f(Q)$ gravity alone cannot generate new geometric families of stellar solutions or alter classical compactness bounds; it merely shifts the mass scale.

Recent multi-messenger observations (e.g., GW190814, massive pulsars like PSR J0740+6620) have revealed objects in the "mass-gap" region ($2.5 - 5 M_\odot$) and high-mass pulsars ($>2 M_\odot$). Standard GR models often struggle to support these masses without violating causality or requiring unrealistically stiff equations of state (EOS). The authors seek a mechanism to enlarge the stellar mass window without arbitrarily stiffening the EOS, specifically within the context of linear $f(Q)$ gravity.

2. Methodology

The authors employ a hybrid approach combining Embedding Class-I geometry, Linear $f(Q)$ gravity, and Gravitational Decoupling (GD) via Minimal Geometric Deformation (MGD).

• Theoretical Framework:

  • Gravity: Linear $f(Q) = \beta_1 Q + \beta_2$. The field equations are derived in the coincident gauge (where the affine connection vanishes).
  • Geometry: A static, spherically symmetric spacetime satisfying the Karmarkar condition (reducing the 4D spacetime to an embedding in a 5D flat manifold).
  • Metric Ansatz: The Vaidya-Tikekar (VT) metric is used as the seed solution. This allows the $t=$ constant hypersurfaces to be embedded as spheroidal geometries rather than spherical ones, providing flexibility without assuming a specific EOS.

• Gravitational Decoupling (MGD):

  • The total energy-momentum tensor is split into a seed sector ($T_{\mu\nu}$) and an additional source sector ($\theta_{\mu\nu}$) coupled by a parameter $\epsilon$.
  • The metric potentials are deformed linearly:
    • Radial: $e^{-\lambda(r)} = W(r) + \epsilon \psi(r)$
    • Temporal: $\nu(r) = H(r) + \epsilon \eta(r)$
  • Constraint: To close the system, the authors impose the mimicking density condition ($\rho = \rho_\theta$), allowing the deformation function $\psi(r)$ to be solved analytically.

• The Two-Parameter Mechanism:

  • $\epsilon$ (Decoupling Parameter): Governs the geometric deformation. It modifies the metric potentials, alters the anisotropy profile, and effectively changes the stiffness of the Equation of State (EOS).
  • $\beta_1$ (Coupling Parameter): Governs the rescaling of the matter sector in linear $f(Q)$. It uniformly scales energy density and pressure without altering the underlying metric geometry.

3. Key Contributions

1. Structural Novelty: The paper establishes a genuine two-parameter framework $(\epsilon, \beta_1)$. Unlike GR-based decoupling (which relies solely on $\epsilon$) or pure linear $f(Q)$ (which relies solely on $\beta_1$), this model separates geometric deformation from matter rescaling.
2. Analytic Solution: The authors derive exact analytic solutions for the deformed metric potentials and physical variables (density, pressures, anisotropy) for an Embedding Class-I configuration in linear $f(Q)$ gravity.
3. Decoupling of Effects: The framework allows for a controlled comparison:
   • Fixing $\epsilon$ isolates the effect of $\beta_1$ (pure coupling-driven mass shift).
   • Fixing $\beta_1$ isolates the effect of $\epsilon$ (pure geometric deformation).
4. Compactness Bound Derivation: An analytic upper bound for compactness ($u = M/R$) is derived for the decoupled configuration, showing how it depends on $\epsilon$ and the curvature parameter $K$, but remains independent of $\beta_1$.

4. Results

• Physical Viability: The model satisfies all standard physical acceptability conditions:
  • Regularity at the center.
  • Monotonic decrease of density and pressures.
  • Satisfaction of Energy Conditions (NEC, WEC, SEC, DEC).
  • Causality ($v_s^2 \leq 1$) is maintained for a wide range of parameters.
• Mass-Radius Relationship:
  • Role of $\epsilon$: Increasing $\epsilon$ (stronger decoupling) stiffens the effective EOS, reduces anisotropy, and significantly increases the maximum mass.
  • Role of $\beta_1$: Decreasing $\beta_1$ (where $\beta_1 < 1$) rescales the matter sector, allowing the star to support higher masses for a fixed geometry and EOS stiffness compared to GR ($\beta_1=1$).
• Mass-Gap Accommodation:
  • In GR, the model struggles to reach masses $>2.5 M_\odot$ without violating causality.
  • In the proposed model, configurations with $\epsilon \approx 1$ and $\beta_1 \approx 0.8$ successfully support masses up to $\approx 2.8 M_\odot$, placing them within the neutron star-black hole mass-gap region.
• Comparison with Observations: The model successfully reproduces the mass-radius data for various pulsars (e.g., PSR J0740+6620, PSR J0348+0432, PSR J0952−0607) and mass-gap candidates (GW190814) by tuning $\epsilon$ and $\beta_1$. Notably, for very massive objects, GR fails to match observations even with strong decoupling, whereas linear $f(Q)$ succeeds due to the independent matter rescaling.

5. Significance

• Theoretical Insight: The work demonstrates that linear $f(Q)$ gravity is not merely a rescaling of GR but, when combined with gravitational decoupling, offers a structurally richer framework. It proves that geometric deformation and matter-sector rescaling are distinct mechanisms that can be independently manipulated to enhance stellar mass.
• Observational Relevance: The model provides a viable theoretical explanation for the existence of ultra-massive compact objects and mass-gap candidates without requiring exotic equations of state that violate causality. It suggests that the "missing" mass in standard models could be accounted for by the interplay of non-metricity coupling ($\beta_1$) and geometric deformation ($\epsilon$).
• Future Directions: The authors suggest that the distinct signatures of this two-parameter deformation (particularly in rotational observables like the moment of inertia) could serve as a discriminant between GR and $f(Q)$ gravity in future multi-messenger astronomy observations.

In summary, the paper successfully extends the Vaidya-Tikekar Embedding Class-I models from a single-parameter GR framework to a robust two-parameter system in linear $f(Q)$ gravity, offering a flexible and physically consistent solution to the challenge of modeling the most massive compact stars observed in the universe.