Analysis of Charged Compact Stars with Bardeen Black Hole in $f(\mathfrak{Q}, \mathcal{T})$ Gravity

This study investigates the physical properties and stability of charged compact stars in $f(\mathfrak{Q}, \mathcal{T})$ gravity by combining a Bardeen black hole exterior with a Finch-Skea interior metric, ultimately concluding that the proposed solutions are theoretically consistent and physically viable.

The Gist

Imagine the universe as a giant, cosmic construction site. For decades, scientists have been trying to understand the most extreme buildings on this site: Compact Stars. These are the "skyscrapers" of the cosmos—objects so dense that a teaspoon of their material would weigh a billion tons on Earth. Think of them as the ultimate heavyweights, like neutron stars or charged black holes, where gravity is so strong it tries to crush everything into a single point.

This paper is like a new architectural blueprint for these cosmic skyscrapers. The authors, M. Sharif and Iqra Ibrar, are asking a big question: Can we build a stable, charged compact star using a new set of physics rules, without it collapsing into a singularity (a point of infinite density where physics breaks down)?

Here is the breakdown of their project in simple terms:

1. The New Physics Rules: "f(Q, T) Gravity"

For a long time, we used Einstein's General Relativity (GR) as our rulebook. It works great for most things, but it struggles to explain why the universe is expanding faster and faster (like a balloon inflating on its own).

The authors are using a new, upgraded rulebook called f(Q, T) gravity.

• The Old Way (GR): Gravity is like a rubber sheet. Heavy objects bend the sheet, and other things roll toward them.
• The New Way (f(Q, T)): Imagine the rubber sheet isn't just bending; it's also stretching and changing its texture based on what's sitting on it. This theory adds a "smart material" element to space itself. It suggests that the geometry of space (how it's shaped) and the matter inside it (stars, gas, charge) talk to each other directly. It's like the building materials and the blueprint are having a conversation to decide how to hold up the structure.

2. The Exterior Design: The "Bardeen Black Hole"

Usually, when we model a star, we imagine the outside is empty space. But for this study, the authors wanted to see what happens if the star is surrounded by a Bardeen Black Hole.

• The Problem: Normal black holes have a "singularity" in the center—a place where the math explodes and reality breaks. It's like a hole in the blueprint where the building shouldn't exist.
• The Solution: The Bardeen model is a "regular" black hole. Think of it as a black hole with a soft, fuzzy core instead of a sharp, infinite point. It's a black hole that doesn't break the laws of physics at its center. The authors used this "safe" black hole as the outer shell for their star.

3. The Interior Design: The "Finch-Skea" Structure

Inside the star, the authors needed a way to describe how the pressure and density change from the center to the edge. They used a mathematical recipe called the Finch-Skea metric.

• The Analogy: Imagine a giant, charged jellybean. The center is super hard and dense, and as you move toward the candy shell, it gets softer. The Finch-Skea model is the specific recipe that tells the jellybean exactly how to be hard in the middle and soft on the outside without cracking. It ensures the star doesn't have a "crack" or a "hole" in its center.

4. The Stress Test: Is the Building Stable?

Once they built their theoretical star, they put it through a series of stress tests to see if it would stand up or collapse. They checked:

• The "Squeeze" Test (Energy Conditions): They checked if the star is made of "normal" stuff or "exotic" magic dust. The results showed it's made of normal, stable matter.
• The "Tug-of-War" (Equilibrium): Inside the star, four forces are fighting:
  1. Gravity: Trying to crush the star inward.
  2. Pressure: Pushing outward to stop the crush.
  3. Electric Charge: Since the star is charged, the electric repulsion pushes outward (like two magnets pushing apart).
  4. Anisotropy: A fancy word for "directional pressure." Imagine the star is slightly squashed or stretched in different directions, creating a balancing force.
  • The Result: The authors found that these four forces perfectly balance each other out. It's a cosmic tug-of-war where the rope isn't moving; the star is in perfect equilibrium.
• The "Speed Limit" Test (Causality): They checked if sound waves inside the star travel faster than light. If they did, the star would break physics. The test passed: sound travels slower than light, so the star is safe.
• The "Cracking" Test: They checked if the star would shatter like a dry cookie. The math showed the star is flexible and stable, not brittle.

5. The Conclusion: A Successful Blueprint

The paper concludes that yes, you can build a stable, charged compact star using these new rules (f(Q, T) gravity) and this specific design (Bardeen exterior + Finch-Skea interior).

Why does this matter?
It's like finding a new way to build skyscrapers that are stronger and more efficient than before. It suggests that our universe might be governed by these "smart" gravity rules, and that the densest objects in the cosmos might be stable, regular structures rather than chaotic, physics-breaking singularities.

In a nutshell: The authors designed a theoretical "super-star" using a new gravity theory. They proved that if you give this star a charge and surround it with a "soft-core" black hole, it stays together, doesn't collapse, and follows all the laws of physics. It's a successful blueprint for the universe's most extreme buildings.

Technical Summary

1. Problem Statement

The study addresses the theoretical modeling of charged compact stars (such as neutron stars or strange stars) within the framework of $f(Q, T)$ gravity, a modified theory of gravity where $Q$ is the non-metricity scalar and $T$ is the trace of the energy-momentum tensor.

Key challenges and objectives include:

• Singularity Avoidance: Standard General Relativity (GR) often predicts singularities in black holes and stellar cores. The authors aim to model stellar interiors that avoid these singularities by coupling them with the Bardeen black hole metric, a regular (singularity-free) solution derived from non-linear electrodynamics.
• Anisotropy and Charge: Realistic compact stars often exhibit anisotropic pressure (radial pressure $\neq$ tangential pressure) and possess significant electric charge. The study seeks to understand how these factors influence stability and structure within the $f(Q, T)$ framework.
• Theoretical Gap: While $f(Q, T)$ gravity has been explored cosmologically, its application to charged, anisotropic compact stars using the Finch-Skea metric and Bardeen exterior has not been previously investigated.

2. Methodology

A. Theoretical Framework

• Gravity Model: The authors utilize a linear $f(Q, T)$ model defined as:
  $$f(Q, T) = \zeta Q + \eta T$$
  where $\zeta$ and $\eta$ are arbitrary non-zero constants. This linear choice simplifies the field equations while retaining the coupling between geometry (non-metricity) and matter.
• Spacetime Geometry:
  • Interior: The stellar interior is modeled using the Finch-Skea metric potential, chosen for its regular behavior at the core and ability to satisfy physical acceptability conditions.
    $$ds^2 = -e^{\alpha(r)}dt^2 + e^{\beta(r)}dr^2 + r^2(d\theta^2 + \sin^2\theta d\phi^2)$$
    where $\alpha(r) = \ln[\chi(1 + \phi r^2)^2]$ and $\beta(r) = \ln[1 + 16\chi\phi^2\gamma r^2]$.
  • Exterior: The exterior spacetime is matched to the Bardeen black hole metric, which includes a non-linear electromagnetic field:
    $$A(r) = 1 - \frac{2Mr^2}{(q^2 + r^2)^{3/2}}$$
    where $M$ is mass and $q$ is electric charge.

B. Field Equations and Solving

• The Einstein-Maxwell field equations are derived for the $f(Q, T)$ framework, incorporating the stress-energy tensor for an anisotropic fluid and the electromagnetic field tensor.
• The authors solve the resulting system of differential equations to obtain analytical expressions for energy density ($\rho$), radial pressure ($p_r$), and tangential pressure ($p_t$).
• Matching Conditions: To determine the unknown constants ($\chi, \phi, \gamma$) in the Finch-Skea metric, the interior solution is matched to the exterior Bardeen metric at the stellar surface ($r=R$). This requires the continuity of the metric components ($g_{tt}, g_{rr}$) and their first derivatives.

C. Physical Analysis

The viability of the model is assessed through graphical analysis (varying the parameter $\zeta$) of several physical criteria:

1. Fluid Dynamics: Behavior of $\rho, p_r, p_t$ and their gradients.
2. Anisotropy: Analysis of the anisotropic factor $\Delta = p_t - p_r$.
3. Energy Conditions: Verification of Null (NEC), Weak (WEC), Strong (SEC), and Dominant (DEC) energy conditions.
4. Equation of State (EoS): Calculation of radial ($\omega_r$) and tangential ($\omega_t$) EoS parameters.
5. Equilibrium: Verification of the Tolman-Oppenheimer-Volkoff (TOV) equation, balancing gravitational, hydrostatic, anisotropic, and electric forces.
6. Stability Criteria:
   • Causality: Sound speeds ($v_r^2, v_t^2$) must be $\leq 1$.
   • Cracking Condition: Herrera's stability condition $|v_t^2 - v_r^2| \leq 1$.
   • Adiabatic Index: $\Gamma > 4/3$ to prevent gravitational collapse.
   • Compactness & Redshift: Ensuring $2M/R \leq 8/9$ and surface redshift $Z \leq 5.211$.

3. Key Contributions

1. Novel Model Construction: This is the first study to combine the Finch-Skea metric (interior) with the Bardeen black hole (exterior) within the $f(Q, T)$ gravity framework for charged compact stars.
2. Singularity-Free Core: By utilizing the Bardeen geometry and Finch-Skea potential, the model successfully describes a stellar configuration with a regular, non-singular core, addressing a major limitation of standard GR models.
3. Analytical Solutions: The authors derive exact analytical solutions for the physical parameters under the linear $f(Q, T)$ model, providing a tractable tool for further astrophysical research.
4. Comprehensive Stability Verification: The study provides a rigorous, multi-faceted stability analysis (forces, causality, cracking, adiabatic index) specifically tailored to the $f(Q, T)$ context, demonstrating that the model is physically viable.

4. Results

• Fluid Parameters: Energy density ($\rho$) and pressures ($p_r, p_t$) are maximum at the core and decrease monotonically towards the surface. $p_r$ vanishes at the boundary ($r=R$), satisfying the boundary condition for a finite star.
• Anisotropy: The anisotropic factor $\Delta$ is positive throughout the star, indicating an outward-directed repulsive force that helps counteract gravitational collapse and supports the star's structure.
• Energy Conditions: All energy conditions (NEC, WEC, SEC, DEC) are satisfied for the entire stellar interior, confirming the presence of ordinary (non-exotic) matter.
• Equilibrium: The TOV equation analysis shows that the sum of gravitational ($F_g$), hydrostatic ($F_h$), electric ($F_e$), and anisotropic ($F_a$) forces equals zero, confirming the system is in static equilibrium.
• Stability:
  • Causality: Radial and tangential sound speeds remain within the physical limit ($0 \leq v^2 \leq 1$).
  • Cracking: The difference in sound speeds satisfies the stability condition $|v_t^2 - v_r^2| \leq 1$.
  • Adiabatic Index: Both radial ($\Gamma_r$) and tangential ($\Gamma_t$) adiabatic indices exceed the critical value of $4/3$, ensuring stability against collapse.
  • Compactness & Redshift: The mass-to-radius ratio and surface redshift remain within the theoretical limits for stable compact stars.

5. Significance

• Validation of $f(Q, T)$ Gravity: The study demonstrates that $f(Q, T)$ gravity is a robust alternative to General Relativity for modeling dense stellar objects, offering a consistent framework that unifies early-universe inflation and late-time acceleration without needing dark energy.
• Comparison with Other Theories: The authors note that while similar conformal models in $f(Q)$ gravity suffer from central singularities, their $f(Q, T)$ model with the Bardeen/Finch-Skea combination remains stable and singularity-free. Furthermore, the second-order nature of $f(Q, T)$ equations simplifies calculations compared to fourth-order theories like $f(R)$.
• Astrophysical Relevance: The results provide a theoretical basis for understanding the internal structure of highly dense, charged compact stars, potentially aiding in the interpretation of observational data from pulsars and gravitational wave events.

In conclusion, the paper establishes that charged compact stars modeled with the Finch-Skea metric and Bardeen exterior in $f(Q, T)$ gravity are theoretically consistent, physically valid, and stable, offering a promising avenue for future research in modified gravity and compact object astrophysics.