⚛️ general relativity

Mass-radius relation, moment of inertia, and tidal love numbers of anisotropic neutron stars in f (R,T) gravity

This study investigates the mass-radius relation, moment of inertia, and tidal Love numbers of anisotropic neutron stars within the $f(R,T)=R+2\beta T$ gravity framework using the Horvat anisotropy model, demonstrating that while both anisotropy and the gravity parameter influence physical properties, the former has a dominant effect, ultimately identifying specific configurations that satisfy observational constraints from GW170817 and GW190814.

Original authors: Yusmantoro Yusmantoro, Freddy Permana Zen, Muhammad Lawrence Pattersons

Published 2026-02-02

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CC BY 4.0

Original authors: Yusmantoro Yusmantoro, Freddy Permana Zen, Muhammad Lawrence Pattersons

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant cosmic construction site. In the middle of this site, there are some of the most extreme buildings imaginable: Neutron Stars. These are the collapsed cores of dead stars, so dense that a single teaspoon of their material would weigh as much as a mountain.

For a long time, scientists have tried to understand how these "cosmic skyscrapers" are built. They use a set of blueprints called General Relativity (Einstein's theory of gravity) to predict how heavy a star can get before it collapses, how big it is, and how it reacts when squeezed by a neighbor.

But, just like architects sometimes wonder if there are better building materials or stronger glue, physicists have been asking: What if gravity works slightly differently than Einstein thought?

This paper is a team of researchers from Indonesia who decided to test a new "blueprint" for gravity called $f(R, T)$ gravity. Think of this as a new rulebook where gravity isn't just about the shape of space (geometry), but also has a direct conversation with the matter inside the star. They also added a twist: they assumed the pressure inside these stars isn't the same in all directions (like a balloon that is squeezed harder from the sides than from the top). They call this anisotropy.

Here is what they found, explained simply:

1. The "Squeezing" Game (Mass and Size)

The researchers asked: If we change the rules of gravity and the internal pressure, how big and heavy can these stars get?

The Analogy: Imagine a sponge. If you squeeze it from the sides (anisotropy), it might hold more water (mass) before it crumbles.
The Finding: They found that the "squeezing" direction (the anisotropy parameter) matters a lot more than the new gravity rule. By adjusting these settings, they could build stars that are incredibly heavy—up to 2.67 times the mass of our Sun.
Why it matters: In 2019, scientists detected a gravitational wave (a ripple in space-time) from a collision involving a mysterious object that was too heavy to be a normal neutron star but too light to be a typical black hole. This paper suggests that if you use their new gravity rules and the right internal pressure, a neutron star can actually reach that heavy weight. This supports the idea that the mysterious object was indeed a super-heavy neutron star, not a black hole.

2. The "Spinning Top" Test (Moment of Inertia)

Next, they looked at how hard it is to spin these stars. This is called the Moment of Inertia.

The Analogy: Think of a figure skater. If they pull their arms in, they spin faster. If they have a heavy, wide body, it's harder to get them spinning.
The Finding: They calculated how these stars would spin and compared it to real observations of pulsars (spinning neutron stars). Their new models fit the real-world data perfectly. It's like checking if their new blueprint produces a spinning top that behaves exactly like the ones we see in the sky.

3. The "Jelly" Test (Tidal Love Numbers)

This is the most complex part. When two neutron stars dance around each other before crashing, their gravity pulls on each other, stretching them like taffy or jelly. This stretching ability is called Tidal Deformability.

The Analogy: Imagine two people holding hands and spinning. If they are made of hard rock, they don't change shape. If they are made of soft jelly, they stretch out.
The Finding:
- Scenario A (The "Rock" Star): Using one type of internal material (QHD EoS), the stars were so stiff they barely stretched at all. Their "stretchiness" was almost zero. This is too small to match the famous 2017 collision (GW170817), but it perfectly explains the mysterious heavy object from 2019 (GW190814). Why? Because if the object is so stiff, the detectors couldn't feel it stretching, which is why we didn't see that data.
- Scenario B (The "Jelly" Star): Using a different type of material (BPS+β EoS), the stars were stretchier. These models matched the 2017 collision data perfectly.

The Big Picture Conclusion

The researchers didn't just build one model; they built two different types of neutron stars using their new gravity rules:

The "Rock" Star: Very heavy, very stiff, barely stretches. This looks like the mysterious heavy object from the 2019 event.
The "Jelly" Star: Slightly lighter, stretchy, and matches the 2017 event.

The Takeaway:
The paper argues that by using this new gravity rulebook ( $f(R, T)$ ) and allowing for uneven internal pressure, we can explain both of these mysterious cosmic events. It suggests that the heavy object in 2019 wasn't a black hole after all, but a neutron star that was just too stiff to show any stretching.

In short, they used a new set of physics rules to show that neutron stars are more versatile than we thought, capable of being both the "rock" and the "jelly" needed to explain the universe's biggest collisions.

Technical Summary: Mass-Radius Relation, Moment of Inertia, and Tidal Love Numbers of Anisotropic Neutron Stars in $f(R, T)$ Gravity

Problem Statement
Neutron stars (NSs) serve as critical laboratories for testing theories of gravity and the equation of state (EoS) of dense nuclear matter. While General Relativity (GR) successfully describes many stellar properties, it faces challenges in cosmology, such as explaining dark matter and dark energy, prompting the investigation of Modified Gravity (MG). Furthermore, recent gravitational wave (GW) observations, specifically GW170817 (binary NS merger) and GW190814 (merger involving a compact object of $2.50-2.67 M_\odot$ ), have introduced new constraints on NS mass, radius, and tidal deformability. The nature of the secondary object in GW190814 remains debated: it could be the lightest black hole or the most massive NS ever observed. Additionally, the internal pressure of NSs is likely anisotropic (radial pressure $\neq$ tangential pressure), a factor often neglected in standard isotropic models. This paper investigates whether anisotropic NSs within the framework of $f(R, T)$ gravity can satisfy observational constraints from GW170817 and GW190814, thereby offering a theoretical candidate for the mysterious GW190814 secondary object.

Methodology
The authors employ the simplest linear form of $f(R, T)$ gravity, defined as $f(R, T) = R + 2\beta T$ , where $R$ is the Ricci scalar, $T$ is the trace of the energy-momentum tensor, and $\beta$ is a free coupling parameter. The study incorporates anisotropy using the Horvat model, where the anisotropy factor is defined as $\sigma = p_t - p_r = 2\alpha m_p r / r$ , with $\alpha$ representing the anisotropy strength.

The investigation proceeds through the following steps:

Field Equations: The authors derive the modified Tolman-Oppenheimer-Volkoff (TOV) equations for static, spherically symmetric anisotropic stars in $f(R, T)$ gravity.
Equations of State (EoS): Two distinct EoS models are utilized to model the stellar interior:
- Quantum Hadrodynamics (QHD): A model where baryons interact via meson exchange.
- BPS + $\beta$ : A composite model combining the BPS crust (low density) and a $\beta$ -stable core derived from the Hartree-Fock Quark-Meson Coupling model.
Parameter Space: The study explores three values of the gravity parameter ( $\beta = 0, -0.01, -0.02$ ) and four values of the anisotropy parameter ( $\alpha = -0.12, -0.06, 0.06, 0.12$ ).
Physical Quantities: The authors calculate the mass-radius ( $M-R$ ) relation, the moment of inertia ( $I$ ) using the slow-rotation approximation, and the Tidal Love Numbers (TLNs). Both polar (electric) and axial (magnetic) TLNs are derived via metric perturbations (Regge-Wheeler formalism).
Constraints: Results are validated against observational data:
- GW170817: Constraints on the mass-radius plane, allowed radii for $1.4 M_\odot$ and $2.0 M_\odot$ stars, and tidal deformability limits ( $580 < \Lambda_{1.4} < 800$ ).
- GW190814: The mass range of the secondary object ( $2.50 - 2.67 M_\odot$ ).
- Radio Observations: Moment of inertia constraints from heavy pulsar observations (Landry et al.).

Key Contributions and Results

Mass-Radius Relation:
- The study finds that both $\alpha$ and $\beta$ influence the maximum mass, but the anisotropy parameter $\alpha$ has a significantly larger impact. Negative $\alpha$ (where $p_t < p_r$ ) introduces an additional repulsive force, allowing stars to support higher masses and increasing compactness.
- Negative $\beta$ increases the maximum mass but reduces compactness (stars have larger radii for the same mass) compared to GR.
- GW190814 Candidate: Only NSs modeled with the QHD EoS and negative anisotropy ( $\alpha = -0.12$ ) can achieve the high mass range ( $2.50 - 2.67 M_\odot$ ) observed in GW190814. Specifically, the combination $\alpha = -0.12$ and $\beta = -0.02$ reaches the upper bound of $2.67 M_\odot$ without requiring rotation, supporting the hypothesis that the GW190814 secondary object is a static NS.
- GW170817 Candidate: NSs modeled with the BPS + $\beta$ EoS successfully satisfy the GW170817 mass-radius constraints for various $\alpha$ and $\beta$ values.
Moment of Inertia:
- The calculated moments of inertia for all considered parameters are consistent with the constraints derived from radio observations of heavy pulsars.
- Negative $\alpha$ significantly increases the moment of inertia, while negative $\beta$ has a minor increasing effect. The BPS + $\beta$ models generally yield higher moments of inertia than QHD models.
Tidal Love Numbers (TLNs) and Deformability:
- Polar vs. Axial: Polar (electric) TLNs are significantly larger than axial (magnetic) TLNs. The axial TLNs are extremely small, consistent with the assumption in GW data analysis that they do not significantly affect the GW phase.
- EoS Stiffness: The QHD EoS is found to be much "stiffer" than the BPS + $\beta$ EoS. Consequently, QHD stars exhibit very low polar TLNs and tidal deformability ( $\Lambda$ ), while BPS + $\beta$ stars show higher values.
- GW170817 Compatibility: The tidal deformability of NSs with the BPS + $\beta$ EoS (specifically $\alpha = -0.06$ ) falls within the GW170817 constraint range ( $580 < \Lambda_{1.4} < 800$ ).
- GW190814 Interpretation: The tidal deformability of NSs with the QHD EoS is extremely low ( $\Lambda < 200$ ), failing the GW170817 constraint. However, the authors interpret this small value as a plausible explanation for the lack of tidal deformability data in GW190814. If the secondary object were a QHD NS, its extreme rigidity would render its tidal signature undetectable, explaining why no $\Lambda$ constraint was established for it.

Significance and Claims
The paper claims that its theoretical investigation provides a viable framework for interpreting recent GW events within the context of modified gravity and anisotropic matter.

GW190814 Resolution: The study supports the identification of the GW190814 secondary object as a neutron star rather than a black hole. This is achieved by demonstrating that anisotropic NSs with the QHD EoS in $f(R, T)$ gravity can reach masses up to $2.67 M_\odot$ without invoking rapid rotation, a condition often required in GR to reach such masses.
Undetected Tidal Deformability: The authors argue that the non-detection of tidal deformability in GW190814 is not evidence against the NS hypothesis but rather a consequence of the object's specific internal structure (QHD EoS) and the modified gravity coupling, which result in extremely low deformability.
Role of Anisotropy and MG: The work highlights that the anisotropy parameter $\alpha$ is the dominant factor in determining the maximum mass and compactness, surpassing the influence of the $f(R, T)$ coupling parameter $\beta$ . It further demonstrates that the $2\beta T$ term in $f(R, T)$ gravity can reduce polar TLNs below GR values, contrasting with other MG models (like $f(R)$ ) where scalar fields often increase deformability.

In conclusion, the authors assert that anisotropic neutron stars in $f(R, T)$ gravity, modeled with specific EoS, can simultaneously satisfy the mass constraints of GW190814 and the tidal/radius constraints of GW170817, offering a unified theoretical candidate for the compact objects observed in these events.

1. The "Squeezing" Game (Mass and Size)

2. The "Spinning Top" Test (Moment of Inertia)

3. The "Jelly" Test (Tidal Love Numbers)

The Big Picture Conclusion

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