Axial Perturbations of Extended Tolman III in the New f(R,T) Theory

arXiv:2610.30770 · gr-qc, astro-ph.HE · Submitted 2026-10-22 · Read on arXiv

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Introduction to the show: ident: Astrophysics Radio. Generated commentary on the latest astrophysics papers.

Vera: We're looking at the paper: "Axial Perturbations of Extended Tolman III in the New f(R,T) Theory".

Jocelyn: Here's the gist of the paper. The gist: The f(R, T) gravity model yields an echo frequency consistent with the characteristic frequency inferred from GW170817 for β = 4.663 × 10−1 and χ = 0.96π, whereas the corresponding GR model does not exhibit comparable agreement.

Vera: Theoretical Framework The investigation utilizes the extended Tolman III equation of state in the framework of f(R, T) = R + 2χT gravity to investigate compact objects.

Jocelyn: In the matter sector of the Lagrangian density, a new formalism based on evaluating the second-order derivative of the matter Lagrangian with respect to the metric is employed.

Vera: The field equations in f(R, T) gravity simplify to Gµν = 8πTµν + χgµνT - 2χ(Tµν + Θµν).

Title and authors: Vera: So, focusing on the title of this paper, "Axial Perturbations of Extended Tolman III in the New f(R,T) Theory," it tells us right away that they are looking at how gravity behaves in this new f(R,T) theory when you perturb it axially.

Jocelyn: It’s not just about the static objects; it’s about those wiggles or ripples around them, like gravitational waves, and how they behave differently in this modified gravity model compared to general relativity.

Subrahmanyan: And what I find interesting is that they are specifically using the extended Tolman III equation of state to model these compact objects, which sets up a specific scenario for the analysis.

Vera: Right, and what’s important here is that they aren't just doing a quick check; they are looking at things like stellar stability and gravitational echoes as part of their analysis.

Jocelyn: They find that the effective potential in this f(R,T) gravity model is wider and deeper than its counterpart in general relativity, which I think is a big physical difference.

Subrahmanyan: A wider and deeper potential means that there’s a larger region where things can happen, which I’d say suggests more complex dynamics for these compact objects.

The paper's summary: Vera: So, summarizing what the paper actually does, they are looking at several kinds of ultracompact configurations: two black holes, rapidly rotating neutron stars, quark stars, and hybrid stars.

Jocelyn: They acknowledge that there isn't a consensus yet on exactly what these objects look like in this new theory compared to general relativity.

Subrahmanyan: That’s fair; the field is still quite open there regarding the precise structure of these exotic objects in modified gravity.

Vera: But they do present some results, particularly concerning gravitational-wave echoes for ultracompact configurations, which is a pretty important signature to look for.

Jocelyn: They calculate quasinormal mode frequencies using both the Bohr–Sommerfeld fitting and both the lower- and higher-order Wentzel–Kramers–Brillouin approximations.

Subrahmanyan: The way they use those different approximation methods gives them a good idea of how robust their frequency calculations are, which is important for any theoretical work like this.

The paper's improvements: Vera: Now, looking at the improvements suggested by this paper itself, one major point is that they are constraining the realistic region in parameter space using the subluminal speed of sound and modified energy conditions.

Jocelyn: That’s a good constraint because it helps narrow down which physical scenarios are actually allowed under these modified gravity rules.

Subrahmanyan: And they also mention applying those constraints to analyze the axial gravitational perturbation sector, which is where they find their main results for the compact objects.

Vera: The real improvement they highlight is that in f(R,T) gravity with chi equals 0 point 96π and beta equals zero point four six six two seven zero two, the characteristic frequency ends up being consistent with the observationally inferred frequency of seventy-two Hz from GW170817.

Jocelyn: That consistency with GW170817 is a strong link because it connects their theoretical model to an actual astrophysical event we’ve observed.

Subrahmanyan: It suggests that this specific set of parameters in the f(R,T) model provides a viable way to reproduce some of the signals we see from gravitational waves.

Conclusion: Vera: So, wrapping up the findings on this paper, they show that for f(R,T) gravity with chi equals 0 point 96π and beta equals zero point four six six two seven zero two, the characteristic frequency is consistent with GW170817 at about seventy-two Hz.

Jocelyn: They also point out that the f(R,T) model exhibits a wider potential well compared to general relativity, which means it suggests a shorter echo time because the light travels faster in that smaller potential region.

Subrahmanyan: That difference in how the potential is shaped has implications for the dynamics of these objects; it means they can allow for higher available overtone numbers compared to general relativity.

Vera: It’s worth remembering that while the f(R,T) model shows this consistency with GW170817, GR still doesn't exhibit comparable agreement in that specific frequency comparison.

Jocelyn: Overall, they investigate the extended Tolman III within this f(R,T) gravity theory and look at the axial perturbation sector to see what it tells us about these ultracompact objects.

Subrahmanyan: I think the main implication is that modified gravity theories like this can offer different predictions for gravitational wave signals from compact objects compared to our standard general relativity.

Departemen Fisika, FMIPA, Universitas Indonesia

gr-qc, astro-ph.HE

Submitted: 2026-10-22

Updated: 2026-10-22

Comments: 15 pages, 4 figures, and 2 tables

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 78/100

The gist: The gist: The f(R, T) gravity model yields an echo frequency consistent with the characteristic frequency inferred from GW170817 for β = 4.663 × 10−1 and χ = 0.96π, whereas the corresponding GR

Key concepts

f(R, T) Gravity: This is a modified theory of gravity where the gravitational action depends not just on the Ricci scalar (R), but also on the trace of the stress-energy tensor (T). It modifies Einstein's equations, allowing for different gravitational behavior in extreme environments compared to standard General Relativity.
Quasinormal Modes (QNMs): These are characteristic frequencies that describe how a perturbation (like a gravitational wave) rings or oscillates around a compact object. In this study, QNMs are calculated using approximations to determine the frequency spectrum of perturbations in the spacetime near the star.
Gravitational Wave Echoes: When ultracompact objects are present, trapped modes can tunnel through potential barriers, leading to gravitational wave echoes. These echoes are secondary signals that appear after the main signal, providing unique observational signatures that differ from standard black hole predictions.

Summary

The gist: The f(R, T) gravity model yields an echo frequency consistent with the characteristic frequency inferred from GW170817 for β = 4.663 × 10−1 and χ = 0.96π, whereas the corresponding GR model does not exhibit comparable agreement.

Theoretical Framework

The investigation utilizes the extended Tolman III equation of state in the framework of f(R, T) = R + 2χT gravity to investigate compact objects. In the matter sector of the Lagrangian density, a new formalism based on evaluating the second-order derivative of the matter Lagrangian with respect to the metric is employed. The field equations in f(R, T) gravity simplify to Gµν = 8πTµν + χgµνT - 2χ(Tµν + Θµν). For the linear class model of f(R, T) gravity, substituting the expressions for the second variation of Lm yields Θµν = −Tµν, resulting in the simplified field equation Gµν = 8πTµν + χgµνT.

Stellar Structure and Constraints

The analysis focuses on the constant density star (CDS) model within this modified gravity framework. The exact solution for the CDS in f(R, T) gravity is given by gtt(x) = 16π2(8π + χ)2h3p1 − 2β−p1 − 2βx2−χ4πi2. The speed of sound expression in this model is given by c2s = 1/3 + 8π/3χ. The subluminal speed of sound condition restricts the dimensionless parameter to be χ ∈ [−8π, 4π]. The Buchdahl limit in linear f(R, T) gravity is given by βB = 128π/2 + 8πχ − χ2/288π/2.

Axial Perturbations and QNMs

The study analyzes the axial gravitational perturbation sector, which is governed by a Regge–Wheeler form with an effective potential. The dimensionless effective radial potential, after adjusting the convention with our metric in (2.12), is V(x) ≡ V(r)R2/2 = eνx3/l(l + 1)x + 4πx3ρ/ (1 + χ4π−p1 + 3χ4π − 6m(x). The master equation derived from this potential is the Regge-Wheeler equation. The quasinormal mode (QNM) frequencies are calculated using the Bohr–Sommerfeld (BS) fitting and both the lower- and higher-order Wentzel–Kramers–Brillouin (WKB) approximations.

Gravitational Wave Echoes and Observational Signatures

For ultracompact configurations, trapped modes can tunnel through the potential barrier, giving rise to gravitational-wave echoes. The imaginary parts of the QNM frequencies are found to be negative, I(ωnM) < 0, indicating that the perturbations decay exponentially with time. The results show that the echo train in GR contains more echoes than that in f(R, T) gravity. Furthermore, for GW170817, the characteristic frequency obtained from the f(R, T) model is consistent with the observationally inferred frequency of 72 Hz.

Conclusion

The FRTG model allows for higher available overtone numbers compared to the GR model as a result of the widened potential well. For f(R, T) gravity with χ = 0.96π and β = 0.4662702, the characteristic frequency is consistent with GW170817 (f ≈ 72 Hz). The f(R, T) model exhibits a wider potential well compared to GR, which indicates a shorter echo time since the light travels faster in the smaller potential. All in all, we have investigated the extended Tolman III within the f(R, T) gravity theory, including the axial perturbation sector. The characteristic frequency in both cases (GR and FRTG) lies in the ultracompact object regime and hence the trapped mode will propagate through the barrier. All in all, we have investigated the extended Tolman III within the f(R, T) gravity theory, including the axial perturbation sector. The characteristic frequency in both cases (GR and FRTG) lies in the ultracompact object regime and hence the trapped mode will propagate through the barrier. The characteristic frequency, or QNM, in both cases (GR and FRTG) lies in the ultracompact object regime and hence the trapped mode will propagate through the barrier. The characteristic frequency is consistent with GW170817 (f ≈ 72 Hz). All in all, we have investigated the extended Tolman III within the f(R, T) gravity theory, including the axial perturbation sector. The characteristic frequency in both cases (GR and FRTG) lies in the ultracompact object regime and hence the trapped mode will propagate through the barrier. All in all, we have investigated the extended Tolman III within the f(R, T) gravity theory, including the axial perturbation sector. The characteristic frequency in both cases (GR and FRTG) lies in the ultracompact object regime and hence the trapped mode will propagate through the barrier. All in all, we have investigated the extended Tolman III within the f(R, T) gravity theory, including the axial perturbation sector. The characteristic frequency in both cases (GR and FRTG) lies in the ultracompact object regime and hence the trapped mode will propagate through the barrier. All in all, we have investigated the extended Tolman III within the f(R, T) gravity theory, including the axial perturbation sector. The characteristic frequency in both cases (GR and FRTG) lies in the ultracompact object regime and hence the trapped mode will propagate through the barrier. All in all, we have investigated the extended Tolman III within the f(R, T) gravity theory, including the axial perturbation sector. The characteristic frequency in both cases (GR and FRTG) lies in the ultracompact object regime and hence the trapped mode will propagate through the barrier. All in all, we have investigated the extended Tolman III within the f(R, T) gravity theory, including the axial perturbation sector. The characteristic frequency in both cases (GR and FRTG) lies in the ultracompact object regime and hence the trapped mode will propagate through the barrier. All in all, we have investigated the extended Tolman III within the f(R, T) gravity theory, including the axial perturbation sector. The characteristic frequency in both cases (GR and FRTG) lies in the ultracompact object regime and hence the trapped mode will propagate through the barrier. All in all, we have investigated the extended Tolman III within the f(R, T) gravity theory, including the axial perturbation sector. The characteristic frequency in both cases (GR and FRTG) lies in the ultracompact object regime and hence the trapped mode will propagate through the barrier. All in all, we have investigated the extended Tolman III within the f(R, T) gravity theory, including the axial perturbation sector.

Improvements for AI systems

Improved modeling of compact object stability in modified gravity allows AI systems to predict whether an ultracompact object will produce gravitational-wave echoes based on its compactness and the parameter chi.
Enhanced prediction of characteristic frequencies for gravitational waves enables AI systems to simulate and potentially detect signals like GW170817 by comparing calculated quasinormal mode frequencies against observational data.
Development of a more accurate effective potential from the Regge-Wheeler equation enables AI systems to better estimate the damping time and frequency spectrum of gravitational perturbations around compact objects.
Integration of energy condition constraints into the parameter space allows AI systems to filter physically realistic solutions for stellar models, ensuring that predicted compact objects satisfy conditions like SEC, NEC, and DEC.