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Slowly rotating white dwarfs in f(R,Lm, T) = R + α LmT gravity: moment of inertia, rotational mass shift, and quadrupole

This paper presents the first comprehensive framework for slowly rotating white dwarf models in f(R,Lm, T) = R + αLmT gravity, utilizing the Hartle-Thorne formalism to derive structure equations and numerically demonstrate that positive coupling constants increase the maximum mass beyond the Chandrasekhar limit while confirming the stability and calculating key rotational properties like moment of inertia and quadrupole moment.

Original authors: Kazi Abu Rousan

Published 2026-07-07

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CC BY 4.0

Original authors: Kazi Abu Rousan

Original paper licensed under CC BY 4.0 (https://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 a white dwarf star as a cosmic dead end, the dense, glowing ember left behind after a star like our Sun runs out of fuel. For decades, physicists believed there was a strict "weight limit" for these stars, known as the Chandrasekhar limit (about 1.4 times the mass of our Sun). If a white dwarf gets heavier than this, it was thought to collapse or explode.

However, astronomers have spotted some "super-heavy" white dwarfs that seem to break this rule. This paper asks: Could a slight tweak to the laws of gravity explain how these stars get so heavy without exploding?

Here is a breakdown of the research using simple analogies:

1. The New Gravity Recipe

The author, Kazi Abu Rousan, isn't changing the whole recipe of gravity (Einstein's General Relativity). Instead, he adds a tiny, specific "spice" to the mix. He uses a modified gravity theory called $f(R, L_m, T)$ gravity.

The Analogy: Think of General Relativity as a standard cake recipe. This new theory adds a specific ingredient (a coupling constant called $\alpha$ ) that only kicks in when you have a lot of "stuff" (matter) packed tightly together.
The Result: When this "spice" is added, the rules for how much weight a white dwarf can hold change. The paper finds that with the right amount of this spice, white dwarfs can support up to 1.51 times the Sun's mass, explaining those mysterious super-heavy stars.

2. Spinning the Star (Rotation)

Most previous studies looked at these stars as if they were sitting perfectly still. But stars spin. The author wanted to see what happens when you spin these "spiced-up" white dwarfs.

The Analogy: Imagine a figure skater. When they spin, they bulge out at the waist and flatten at the poles. This paper calculates exactly how much the star bulges and how its internal structure shifts when it spins.
The Discovery: The spinning adds a tiny bit of extra support (like centrifugal force), but the paper concludes that spinning alone isn't enough to explain the super-heavy stars. The real hero is the new gravity "spice" ( $\alpha$ ). The spin just adds a small, predictable tweak to the mass and shape.

3. The "Inertia" and "Mass Shift"

The paper calculates two very specific things about the spinning stars:

Moment of Inertia: How hard it is to stop the star from spinning. The author found that as the gravity "spice" gets stronger, the star becomes slightly easier to spin up (it has a different inertia).
Rotational Mass Shift: How much heavier the star gets just because it's spinning. The paper shows this effect is tiny for white dwarfs. Even if they spin very fast, the extra weight they gain from spinning is negligible compared to the extra weight allowed by the new gravity theory.

4. The "Quadrupole" (The Shape Problem)

The author also tried to measure the star's "squishiness" or shape (called the quadrupole moment).

The Challenge: White dwarfs are not very dense compared to black holes or neutron stars. Because they are so "fluffy" (in cosmic terms), trying to measure their shape by looking at the space far away from them is like trying to hear a whisper in a hurricane. The signal is too weak and gets lost in the noise.
The Solution: Instead of listening from the outside, the author used a clever mathematical shortcut (Radau's equation) to calculate the shape from the inside out. This gave a clear, reliable number for the star's shape, which previous methods missed.

5. The "Regular" Limit (Why they don't explode)

In standard physics, a star becomes unstable and collapses when it hits a specific maximum mass.

The Twist: In this new gravity theory, for certain values of the "spice" ( $\alpha > 0$ ), the star doesn't hit a "collapse wall." Instead, the math simply stops working at a certain high density because the equations get "regular" (smooth) but hit a boundary.
The Takeaway: The paper confirms that all the stars they modeled are stable. They won't collapse just because they are heavy; they are held up by the modified gravity rules.

Summary

This paper builds the first detailed models of spinning white dwarfs in a new theory of gravity.

Gravity is tweaked: A new parameter ( $\alpha$ ) allows white dwarfs to be heavier than the traditional limit.
Spin is minor: Spinning the star adds a tiny bit of mass, but it's not the main reason they are super-heavy.
Shape is calculated: The author found a new way to calculate the star's shape that works better than old methods for these specific stars.
Stability confirmed: These heavy, spinning stars are mathematically stable and won't collapse, provided the gravity "spice" is set correctly.

The paper essentially says: "If you change gravity just a little bit, you can explain why some white dwarfs are heavier than we thought possible, and here is exactly how they would look and behave if they were spinning."

Technical Summary: Slowly Rotating White Dwarfs in $f(R, L_m, T) = R + \alpha L_m T$ Gravity

Problem Statement
White dwarfs (WDs) serve as progenitors for Type Ia supernovae, yet observations of over-luminous events suggest progenitors may exceed the standard Chandrasekhar mass limit ( $M_{Ch} \approx 1.44 M_\odot$ ). While modifications to the gravitational sector have been proposed to explain super-Chandrasekhar masses, previous studies in $f(R, L_m, T)$ gravity were restricted to static configurations. This paper addresses the gap in understanding how slow rotation affects the structure, stability, and observable properties of white dwarfs within the specific model $f(R, L_m, T) = R + \alpha L_m T$ , where $\alpha$ is a matter-geometry coupling constant and the matter Lagrangian is chosen as $L_m = p$ .

Methodology
The authors employ the Hartle-Thorne formalism, expanding the metric and field equations to second order in the angular velocity $\Omega$ .

Field Equations: The theory is defined by an action where the gravitational Lagrangian depends on the Ricci scalar $R$ , the matter Lagrangian $L_m$ , and the trace of the energy-momentum tensor $T$ . For the specific model $f = R + \alpha L_m T$ , the field equations reduce to $G_{\mu\nu} = S_{\mu\nu}$ , where the effective source $S_{\mu\nu}$ is algebraically dependent on matter variables ( $\rho, p$ ) and the coupling $\alpha$ . Crucially, because $f_R = 1$ , the theory propagates no extra scalar degrees of freedom, and the exterior vacuum solution remains exactly General Relativity (GR).
Structure Equations: The authors derive the full system of differential equations:
- Zeroth Order ( $O(\Omega^0)$ ): A modified Tolman-Oppenheimer-Volkoff (TOV) system describing the static background.
- First Order ( $O(\Omega)$ ): A frame-dragging equation for the angular velocity of local inertial frames.
- Second Order ( $O(\Omega^2)$ ): Equations for monopole ( $\ell=0$ ) and quadrupole ( $\ell=2$ ) deformations, including mass shifts and shape functions.
Verification: All derived closure relations are verified via an independent symbolic residual test using exact rational arithmetic, confirming that the field equations and conservation laws ( $\nabla_\mu S^\mu_\nu = 0$ ) are satisfied to literal zero.
Numerical Integration: The system is integrated using a realistic zero-temperature equation of state (EoS) for fully ionized $^{12}\text{C}$ , incorporating a relativistic degenerate electron gas and a body-centered cubic (bcc) lattice Coulomb correction. Calculations are performed over a grid of central densities and seven values of the coupling parameter $\alpha/u_1 \in \{-1.6, \dots, +1.6\}$ .
Stability Analysis: Radial stability is assessed by solving the Sturm-Liouville eigenvalue problem for the fundamental radial pulsation mode ( $\omega_0^2$ ) of the effective barotropic fluid, confirming that the turning-point criterion ( $dM/d\rho_c = 0$ ) remains rigorous in this theory.

Key Results

Mass-Radius Relations: The maximum mass of white dwarfs increases with positive $\alpha$ . For $\alpha = -1.6 u_1$ , $M_{max} \approx 1.34 M_\odot$ ; for $\alpha \approx +0.8 u_1$ , $M_{max} \approx 1.51 M_\odot$ .
Nature of the Mass Maximum: For $\alpha \leq 0$ , the mass sequence terminates at a genuine turning point ( $dM/d\rho_c = 0$ ). However, for all $\alpha > 0$ , the sequence terminates at an interior regularity bound where the denominator of the modified hydrostatic equation vanishes while the mass is still rising. Consequently, the reported "maximum" masses for positive $\alpha$ are limits of regularity rather than dynamical instability points.
Moment of Inertia: The dimensionless moment of inertia ( $\bar{I} = I/M^3$ ) varies smoothly with mass and coupling. For a representative star at $\rho_c = 2 \times 10^9 \text{ g cm}^{-3}$ , $\bar{I} \approx 1.20 \times 10^5$ .
Rotational Mass Shifts: Two constants are reported:
- $\delta M_N$ (fixed baryon number): Represents the rotational kinetic energy ( $\delta M_N = \frac{1}{2}\bar{I}$ ). This value is small ( $\sim 6 \times 10^4$ ), indicating rotation alone cannot significantly lift a white dwarf toward the Chandrasekhar limit.
- $\delta M_c$ (fixed central density): Includes the rest mass of additional baryons supported at fixed density, scaling as $C^{-3}$ and reaching $\sim 1.5 \times 10^8$ .
Quadrupole Moment: Asymptotic mode matching fails to recover the physical quadrupole moment at white dwarf compactness ( $C \sim 10^{-4}$ ) because the decaying physical mode is suppressed by a factor of $\sim C^4 \sim 10^{-12}$ relative to the growing mode, leading to numerical noise dominance. Instead, the authors compute the dimensionless quadrupole $\bar{Q}$ via a well-conditioned volume integral of the Radau equation, yielding $\bar{Q} \approx 2.0 \times 10^3$ for the representative star, with the relation $\bar{Q}C \approx 2$ holding along the sequence.
Stability: All reported configurations lie on the stable branch ( $\omega_0^2 > 0$ ). For positive $\alpha$ , the models remain stable up to the regularity bound.

Significance and Claims
The paper claims to provide the first slowly rotating white dwarf models in $f(R, L_m, T)$ gravity that include the moment of inertia, rotational mass shift, and quadrupole moment.

Theoretical Rigor: By demonstrating that $f_R=1$ reduces the dynamics to GR sourced by a conserved effective fluid, the authors establish that the standard radial-pulsation stability criterion applies exactly, avoiding the ambiguities often associated with non-minimal coupling theories.
Observational Consistency: The models reproduce the mass-radius sequences of observed white dwarfs (including the ultra-massive ZTF J1901) and remain consistent with the static constraints on $\alpha$ derived from Gaia data.
Methodological Contribution: The work highlights a numerical limitation in extracting quadrupole moments for low-compactness stars via exterior matching and proposes the Radau-equation volume integral as a robust alternative.
Limitations: The authors explicitly state that the super-Chandrasekhar masses in this theory are supported by the coupling $\alpha$ , not by rotation, as the rotational mass shift is negligible ( $\delta M/M \sim 10^{-4}$ ). The study is limited to zero-temperature EoS and neglects magnetic fields, which are identified as necessary for future work to fully separate matter-curvature coupling from magnetohydrostatic support.