Structural Properties of Magnetized Neutron Stars under f (R, T ) Gravity Framework

This study numerically investigates the structural properties of magnetized neutron stars within the f(R,T) gravity framework, revealing that negative modified gravity parameters increase maximum mass while strong magnetic fields cause only slight reductions, with results consistent with observational data from GW170817, PSR, and NICER.

The Gist

Imagine the universe as a giant cosmic kitchen. In this kitchen, Neutron Stars are the most extreme ingredients you can find. They are the leftover "crumbs" from massive stars that exploded, but they are so incredibly dense that a single teaspoon of neutron star material would weigh as much as a mountain on Earth.

For a long time, scientists have tried to understand how these cosmic mountains hold themselves together. They use a set of rules called General Relativity (Einstein's theory of gravity) to predict how big and heavy these stars can get. But, just like a recipe might need a tweak to make a better cake, some scientists think Einstein's rules might need a little adjustment in these extreme environments.

This paper is about testing a new "recipe" for gravity while also adding a very spicy ingredient: Super Strong Magnetic Fields.

Here is the breakdown of their experiment in simple terms:

1. The New Gravity Recipe: $f(R, T)$

Think of General Relativity as a standard recipe for a cake. It works great for most things, but when you get to the very center of a neutron star, the pressure is so high that the recipe might need a secret ingredient.

The authors are testing a modified recipe called $f(R, T)$ gravity.

• The Old Way: Gravity depends only on how much stuff (matter) is there.
• The New Way: In this model, gravity also depends on how that matter is moving and pressing against itself. It's like saying the taste of the cake depends not just on the flour, but on how hard you kneaded the dough.
• The Result: They found that if you tweak this "kneading" factor (represented by a number called $\lambda$), the neutron stars can get bigger and heavier than Einstein's original rules predicted. Specifically, if the tweak is "negative," the star becomes a bit "fluffier" and can support more mass without collapsing.

2. The Spicy Ingredient: Magnetic Fields

Neutron stars are already magnets, but some (called Magnetars) have magnetic fields so strong they could wipe a credit card from halfway across the galaxy. The authors wanted to see what happens if you crank that magnetic field up to the maximum possible limit ($10^{18}$ Gauss).

• The Expectation: You might think such a powerful magnetic force would blow the star apart or make it squish into a weird shape.
• The Reality: Surprisingly, the magnetic field acts like a tiny bit of extra weight. It adds a little bit of mass (about 2% of the Sun's mass) because energy itself has weight. However, it doesn't change the star's shape much. The star stays round (spherical), and the magnetic field doesn't break the rules of the "gravity recipe." It's like adding a pinch of salt to a cake; it changes the flavor slightly, but the cake still rises the same way.

3. The Experiment: Mixing the Ingredients

The team used a supercomputer to solve complex math equations (called the TOV equations) that describe how a star balances its own gravity against its internal pressure. They tested three different "flavors" of neutron star matter (called APR, FPS, and SLy) to make sure their results weren't just a fluke of one specific model.

They ran the simulation in two scenarios:

1. Normal Gravity (Einstein's rules).
2. Modified Gravity (The new $f(R, T)$ rules).

They did this both without magnetic fields and with the super-strong magnetic fields.

4. The Findings: What Did They Discover?

• Heavier Stars: When they used the modified gravity rules (with a negative $\lambda$), the neutron stars could become much heavier—up to 2.7 times the mass of our Sun. This is a big deal because we have observed neutron stars that are about 2 solar masses, and standard Einstein gravity struggles to explain how they stay stable without collapsing into black holes.
• The Magnetic Effect: The strong magnetic field made the stars slightly smaller and slightly less massive, but the change was very small. It confirmed that even with these crazy magnetic fields, the stars remain stable spheres.
• Real-World Check: They compared their computer models with real data from space telescopes (like NICER) and gravitational wave detectors (like LIGO from the event GW170817). Their new "recipe" fits the real-world data perfectly!

The Big Picture Analogy

Imagine you are trying to stack a tower of Jenga blocks.

• General Relativity is the rule that says, "You can only stack this many blocks before the tower falls."
• Neutron Stars are those towers.
• Magnetic Fields are like blowing a gentle breeze on the tower. The authors found that the breeze doesn't knock the tower over; it just shifts the weight slightly.
• Modified Gravity ($f(R, T)$) is like discovering that the blocks are actually made of a slightly stretchy rubber. Because they are stretchy, you can stack more blocks than you thought possible before the tower falls.

Conclusion

This paper tells us that the universe might be a bit more flexible than we thought. By tweaking the rules of gravity to account for how matter interacts with space, we can explain how the heaviest neutron stars exist without turning into black holes. The super-strong magnetic fields are impressive, but they are just a side note in this grand cosmic dance; the real game-changer is the new way we are looking at gravity itself.

This helps astronomers understand the "cosmic kitchen" better and suggests that future observations of colliding stars will help us fine-tune this new gravity recipe even more.

Technical Summary

1. Problem Statement

Neutron stars (NSs) serve as extreme laboratories for testing the interplay between dense matter physics, strong magnetic fields, and gravitational theories. While General Relativity (GR) has been successful, anomalies in cosmology (e.g., dark matter, galaxy rotation curves) suggest that gravity may deviate from GR at high densities or large scales.

• The Gap: Previous studies on NSs in modified gravity (specifically $f(R, T)$ gravity) have largely ignored strong magnetic fields. Conversely, studies on magnetized NSs within GR have not incorporated modified gravity effects.
• The Objective: To systematically investigate how a matter-geometry coupling term in $f(R, T)$ gravity, combined with ultra-strong central magnetic fields ($B_c \sim 10^{18}$ Gauss), influences the structural properties (mass-radius relations, internal pressure, and density profiles) of neutron stars using realistic Equations of State (EoS).

2. Methodology

A. Theoretical Framework

The study employs the $f(R, T)$ modified gravity theory, specifically the linear model:
$$f(R, T) = R + 2\lambda\kappa T$$
Where:

• $R$ is the Ricci scalar.
• $T$ is the trace of the energy-momentum tensor.
• $\lambda$ is the coupling parameter quantifying the strength of the matter-geometry interaction.
• $\kappa = 8\pi G/c^4$.

This model introduces a direct coupling between geometry and matter, altering the conservation laws and equilibrium conditions without introducing extra scalar degrees of freedom (keeping field equations second-order).

B. Equations of State (EoS)

Three realistic nuclear EoS models were used to close the system of equations:

1. APR (Akmal-Pandharipande-Ravenhall): Models isospin asymmetry and baryon density interactions, capturing high-to-low density phase transitions.
2. FPS (Friedman–Pandharipande–Skyrme): Describes the inner crust and liquid core, with a crust-core transition at $\rho \approx 1.6 \times 10^{14}$ g/cm$^3$.
3. SLy (Skyrme–Lyon): Uses a unified nuclear Hamiltonian for the core and crust, noted for being stiffer at the crust-core interface compared to FPS.

C. Magnetic Field Model

The study incorporates a strong central magnetic field ($B_c$) up to $10^{18}$ Gauss.

• Isotropic Approximation: Despite ultra-strong fields potentially causing pressure anisotropy ($P_{\parallel} \neq P_{\perp}$), the authors retain an isotropic pressure approximation. This is justified because the magnetization contribution ($M_B/P$) remains small ($\lesssim 0.05$) even at $10^{18}$ Gauss.
• Field Profile: The magnetic field $B(r)$ is modeled using an empirical polynomial profile that decreases from the center to the surface.
• Energy-Momentum Tensor: The magnetic energy density ($B^2/8\pi$) is added to the total energy density, and a Lorentz force term $L(r)$ is included in the pressure gradient equation.

D. Modified TOV Equations

The authors derived and numerically solved the modified Tolman-Oppenheimer-Volkoff (TOV) equations for $f(R, T)$ gravity with magnetic fields. The system consists of coupled first-order differential equations for mass $m(r)$, pressure $P(r)$, and metric potential $\phi(r)$:

1. Mass Equation: Modified to include magnetic energy density and the $f(R, T)$ trace term $h(T)$.
2. Hydrostatic Equilibrium: The pressure gradient equation includes the Lorentz force and modified gravitational terms dependent on $\lambda$.
3. Numerical Solution: Solved using a fourth-order Runge-Kutta method with adaptive step control, integrating from the center ($r=0$) to the surface ($P=0$).

3. Key Contributions

• Unified Framework: This is one of the first studies to simultaneously solve for the structural properties of magnetized neutron stars within the $f(R, T)$ gravity framework using multiple realistic EoSs.
• Parameter Space Exploration: The study systematically varies the coupling parameter $\lambda$ (specifically negative values: $0, -1/8\pi, -2/8\pi, -3/8\pi$) and the central magnetic field ($B_c = 0$ vs. $10^{18}$ Gauss).
• Validation: The results are rigorously compared against multimessenger observational constraints, including NICER data, massive pulsar masses (PSR J0348+0432, PSR J0740+6620), and the GW170817 tidal deformability limits.

4. Key Results

A. Mass-Radius ($M-R$) Relations

• Effect of $\lambda$: Negative values of the coupling parameter $\lambda$ lead to a stiffening of the equation of state. As $\lambda$ becomes more negative, the maximum gravitational mass ($M_{max}$) increases, and the stellar radius expands.
  • Example: For the APR EoS, $M_{max}$ increases from $\approx 2.20 M_\odot$ (GR, $\lambda=0$) to $\approx 2.74 M_\odot$ ($\lambda = -3/8\pi$).
• Effect of Magnetic Field ($B_c$): A central magnetic field of $10^{18}$ Gauss causes a slight decrease in the maximum mass and a contraction in the radius (by a few percent).
  • Mechanism: The magnetic energy density adds to the gravitational mass but does not provide sufficient isotropic pressure support to counteract the increased gravity, leading to higher compactness.
• Combined Effect: The negative $\lambda$ effect (increasing mass/radius) dominates over the magnetic field effect (slightly decreasing mass/radius).

B. Internal Structure Profiles

• Mass Distribution: Magnetic fields cause the mass profile to shift upward, meaning a given mass is contained within a slightly smaller radius compared to non-magnetized stars. The magnetic contribution to the total mass ($\Delta M$) is estimated at $\approx 0.02 M_\odot$.
• Pressure Profiles:
  • Magnetic fields significantly impact pressure near the core but diminish toward the surface.
  • More negative $\lambda$ values result in lower central pressures and flatter pressure gradients, indicating a "fluffier" star with weaker effective gravitational binding.

C. Observational Consistency

The predicted $M-R$ curves for various $\lambda$ values and EoSs remain consistent with:

• NICER Observations: Posterior distributions for PSR J0030+0451 and PSR J0740+6620.
• Massive Pulsars: The model supports masses up to $\sim 2.14 M_\odot$ (PSR J0740+6620) and even higher for negative $\lambda$.
• GW170817: The radii for $1.4 M_\odot$ stars fall within the tidal deformability constraints ($R_{1.4} \sim 11-13$ km).

5. Significance and Conclusion

• Viability of $f(R, T)$ Gravity: The study demonstrates that moderate matter-geometry coupling (negative $\lambda$) can explain the existence of very massive neutron stars without violating current observational constraints.
• Role of Magnetic Fields: While strong magnetic fields ($10^{18}$ G) do not drastically alter the global structure or break spherical symmetry in this regime, they act as a secondary correction that slightly increases compactness.
• Future Directions: The authors suggest extending this work to include:
  • Tidal Love numbers and deformabilities for gravitational wave analysis.
  • Anisotropic pressure models (poloidal-toroidal fields) and rotation.
  • Higher-order curvature terms (e.g., $R^2$) and thermal evolution.

Conclusion: The paper establishes that $f(R, T)$ gravity provides a robust framework for modeling magnetized neutron stars, where the matter-geometry coupling can significantly enhance the maximum mass capacity of these stars, offering a potential alternative to standard GR explanations for the most massive observed pulsars.