Slowly rotating Black Holes in DHOST Theories

arXiv:2512.17614 · gr-qc, astro-ph.HE, hep-th · Submitted 2025-12-19 · Read on arXiv

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

Vera: I'm Vera, and with me are Jocelyn and Subrahmanyan, guest researcher.

Jocelyn: Today's paper: "Slowly rotating Black Holes in DHOST Theories".

Vera: This research investigates slowly rotating black hole solutions within Degenerate Higher Order Scalar Tensor (DHOST) theories,

Jocelyn: First, who's behind it and why it matters.

Title and authors: Vera: Well team, we're starting with the paper "Slowly rotating Black Holes in DHOST Theories," and it looks like Candana and Langloisc are tackling a really important area in modified gravity. It’s about finding analytic solutions for rotating black holes within these DHOST theories, which is a big deal because these theories modify how gravity works compared to General Relativity.

Jocelyn: I agree, Vera, it seems like the title itself points to the core of what they are doing: looking at how rotation affects black holes when you introduce those higher-order scalar tensor terms. It’s interesting that they’re focusing on slowly rotating solutions first; that usually makes the math manageable before tackling full general relativity.

Subrahmanyan: From a theoretical standpoint, this work is significant because it provides analytical general solutions for these rotating black holes in DHOST theories, and they are often cited as the first known fully analytic examples of such objects that possess scalar hair. That connection between rotation and scalar fields is something we need to explore deeply in cosmology.

Vera: Exactly, Subrahmanyan, and I'm really curious about how they achieved these analytical solutions for the frame-dragging function omega. It sounds like they managed to make a difficult equation integrable for any DHOST theory, which is quite an achievement.

Jocelyn: And that integrability seems key because it lets them get an explicit form for omega(r), which is what describes the frame dragging effect as you move away from the black hole. I wonder if that explicit form holds up when you look at more complex theories than just quadratic ones.

Subrahmanyan: The paper does explore that angular dependence in omega, and they show through regularity conditions at both the horizon and infinity that this dependence is forbidden, which mirrors the results we see in General Relativity. That confirms a lot about the structure of these solutions.

Vera: So, it sounds like they proved that for these DHOST theories, you can expect a certain kind of symmetry to hold regarding how rotation affects the spacetime geometry around a black hole. What’s next in their analysis?

Jocelyn: They then move on to examining the influence of rotation on key orbital parameters, specifically the Innermost Stable Circular Orbit, or ISCO, and circular light trajectories. These are directly relevant for interpreting data coming from gravitational wave detectors and direct imaging instruments like Event Horizon Telescope observations.

Subrahmanyan: That’s where the connection to astrophysics gets really tangible; understanding how rotation shifts these critical orbits tells us exactly how much modified gravity theory would alter the expected signal from a merging binary black hole system.

Title and authors: Vera: It sounds like they’ve done more than just find solutions, though; they're showing that for specific subclasses of theories, like shift-symmetric ones, the first-order rotational term is identical to what we see in Kerr solutions. That’s a strong result when you're dealing with modified gravity.

Jocelyn: That’s impressive because it suggests that even if the static solution looks different from Schwarzschild, the immediate effect of rotation on those orbits can still align with known results for Kerr black holes under certain conditions. It simplifies things considerably for us when we try to fit data.

Subrahmanyan: Indeed, and if you look at quadratic shift-symmetric DHOST theories specifically, the function Q simplifies in a way that makes omega(r) identical to the Kerr expression, which is omega(r) = 2J/r cubed. That’s a very neat simplification for those specific cases.

Vera: That's quite a finding for quadratic theories, Subrahmanyan. It shows that even in these more complex DHOST frameworks, there are still regimes where the physics behaves very much like what we expect from standard General Relativity when it comes to rotation parameters.

Jocelyn: But I wonder if this Kerr-like result is just an approximation valid in the slow-rotation limit, or if it’s a more general feature of those quadratic models that we haven't fully explored yet. That distinction is important for how much we can trust these predictions for observational work.

Subrahmanyan: The paper does touch on the relationship between different DHOST theories using disformal transformations, showing that the frame dragging functions are related by omega(r) = (R(r)), where R(r) = r p C(r). This structural consistency across different theories is a very powerful piece of information for connecting these modified gravity models.

Vera: That structural consistency across different actions is what I find compelling, Subrahmanyan. It suggests that the underlying geometry, despite the modifications to gravity, maintains a certain kind of relationship when you transform between these related theories. It really helps us organize the landscape of possible DHOST physics.

Jocelyn: And this is where I think the real practical application lies for researchers trying to interpret data; if we have multiple theoretical frameworks that are linked by these transformations, it gives us a systematic way to test which framework matches our observations.

Subrahmanyan: Precisely, and looking at the broader cosmic picture, finding these analytical solutions helps us map out the parameter space of modified gravity theories that could realistically manifest in astrophysical objects we observe. It narrows down the possibilities quite a bit.

Title and authors: Vera: So, to wrap up this part of our discussion on "Slowly rotating Black Holes in DHOST Theories," we’ve seen how they establish integrability for omega, showed Kerr-like behavior for certain quadratic theories, and demonstrated consistency across different theories via disformal maps. This work lays a solid foundation for analyzing rotating black holes in these new gravity frameworks.

Jocelyn: And the implication for us is that when we analyze gravitational wave signals or image data, we can use this formalism to predict how scalar hair might affect orbital dynamics, even if those effects are small at first order. It gives us a predictive tool beyond just fitting to standard GR templates.

Subrahmanyan: I think the main implication is the confirmation that analytical solutions with scalar hair are not just theoretical curiosities but can be rigorously derived within this DHOST framework, providing concrete targets for observational tests.

Vera: It’s certainly a step forward in our understanding of how gravity behaves when we move beyond Einstein's theory, especially since they explicitly ruled out angular dependence for omega based on horizon and infinity conditions. We’re definitely getting clearer boundaries here.

Jocelyn: That boundary setting is crucial because it means we don't have to worry about the frame dragging suddenly becoming direction-dependent in a way that would be physically impossible near a black hole. It keeps the physics constrained, which is what we need when interpreting signals.

Subrahmanyan: And as an engineer, I see this formalism as a validation of the mathematical machinery itself; showing that these complex differential equations have hidden symmetries that lead to clean solutions is valuable for developing better numerical techniques for other modified gravity problems down the line.

Vera: So, we’ve seen how they move from a general setup to specific Kerr-like results under certain constraints, which is a very structured path through the theory. It’s all about building up certainty step by step.

Jocelyn: And that structure is exactly what helps us when we try to connect these abstract solutions back to the messy reality of observing pulsars and galaxy clustering; it provides a template for deviation.

Subrahmanyan: I think the overall impact of "Slowly rotating Black Holes in DHOST Theories" is setting a clearer theoretical benchmark for how modified gravity theories should behave when they interact with astrophysical compact objects.

Vera: It’s certainly a solid piece of work, and it opens up new avenues for us to look at rotating black holes that have scalar hair in these theories. We’ll be looking forward to seeing how this formalism helps us tackle more complex astrophysical scenarios next.

The paper's summary: Vera: So, to summarize what we just heard about "Slowly rotating Black Holes in DHOST Theories," the paper essentially shows how to find exact mathematical solutions for black holes that are slowly spinning within these modified gravity frameworks, and it’s significant because these solutions show how scalar fields—the 'hair'—actually interact with rotation.

Jocelyn: I mean, it’s interesting because they managed to get a concrete analytical form for the frame-dragging function omega, which is usually a really messy calculation in these modified theories, and they proved that this function is integrable for any DHOST theory. That makes the whole solution much more accessible than before.

Subrahmanyan: Exactly, Jocelyn, that integrability is the core mathematical achievement because it unlocks an explicit formula for omega(r), which describes how spacetime is dragged around the black hole's rotation in these scalar tensor theories, and that’s a big step since we often struggle to get such clean results.

Vera: And they didn't just stop there; they used disformal transformations to show that solutions from different DHOST theories can be related through specific scaling functions, which is a really cool way to map out the entire landscape of these theories. It suggests there’s a deep structural connection between them that we can exploit.

Jocelyn: That structural mapping is what really gets my attention because it means we might be able to use observational data from pulsars or black holes to tell which specific DHOST model is actually producing the signals we see, rather than just testing one theory in isolation.

Subrahmanyan: And that’s where the real cosmic impact comes in; if these analytical solutions hold up under scrutiny, it gives us a predictable way to test theories of modified gravity against actual astrophysical observations of compact objects. It moves these concepts out of purely mathematical papers and into something we can actually measure with instruments like LIGO or future X-ray observatories.

Vera: I agree, Subrahmanyan; this isn't just abstract math for math’s sake, it provides a concrete theoretical benchmark for what rotating black holes look like when you introduce these scalar fields in DHOST theories. It sets a clear target for observational constraints on modified gravity.

Jocelyn: And thinking about the implications for gravitational waves, if we can predict how rotation and scalar hair change the ISCO or light paths with this kind of precision, it gives us much better templates to compare against real merger signals from black hole binaries.

Subrahmanyan: Precisely; that enhanced predictive power is what makes these results relevant for interpreting data from gravitational wave detectors, allowing us to separate the effects of modified gravity from standard General Relativity more effectively than we currently can.

Vera: It’s exciting because it bridges the gap between complex theoretical modifications and tangible astronomical predictions, giving us a roadmap for how to look for scalar hair in astrophysical systems.

Jocelyn: So what I’m curious about next is how these predicted deviations in orbital parameters translate into actual observable shifts in the data we collect from pulsar timing or sky surveys.

Subrahmanyan: That’s where the work needs to be applied; moving from a general analytic solution to calculating those specific orbital shifts, like the effective potential V eff(r), is the next crucial step for testing these models.

The paper's improvements: Tom: So, we’ve just been talking about how researchers found analytical solutions for slowly rotating black holes in DHOST theories, and now we're looking at what they suggest to do next with this research. The authors point out several directions where they think future work should focus to push these findings further.

Vera: They suggest a few major areas for improvement, starting with developing more robust methods for handling the angular dependence of the frame-dragging function omega, because even though they proved it must vanish at infinity and the horizon, getting that precise mathematical behavior is still challenging.

Jocelyn: I’m interested in their suggestion to use techniques like spectral decomposition to automatically analyze those angular modes, since that sounds like it would help us handle complex perturbations in these spacetimes much more efficiently than solving individual differential equations for each mode.

Subrahmanyan: From a theoretical standpoint, the authors also hint at exploring the full range of disformal transformations more deeply, perhaps looking at how the relationship between different DHOST theories behaves under even more complex metric changes beyond just quadratic ones. They want to see if those consistency relations hold true across a wider class of modified gravity actions.

Vera: It sounds like they are pushing for a broader test of the theory’s structural consistency, which is important because it helps us understand the underlying principles that link these different gravitational theories together.

Jocelyn: I think their suggestion to use conserved quantities derived from hidden symmetries to build PINNs could be really useful for simulating these fields; if an AI system can be structured around those conservation laws, it should lead to much more stable and accurate predictions for complex gravitational fields.

Subrahmanyan: That aligns with my thoughts on using AI, as you mentioned earlier, because if the AI is built on those underlying symmetries, it could potentially serve as a powerful Theory Explorer capable of inferring the action parameters from observational data without needing exhaustive brute-force fitting.

Vera: So they’re looking at improving the mathematical tools to make these solutions more robust and easier to use for both theoretical prediction and potential future AI applications in astrophysics.

Jocelyn: And this focus on better computational methods is crucial because as we get better data from pulsar surveys, we need predictive tools that can handle the non-linearity of these modified gravity effects without getting bogged down in numerical instability.

Subrahmanyan: They are also emphasizing the need to connect these analytic results directly to observable quantities like the ISCO and light trajectories, not just keep them as mathematical curiosities; they want a clearer path showing how this theoretical framework influences actual astrophysical measurements.

Vera: I’m glad they are focusing on that connection because ultimately, our job as observational astronomers is to see these effects in the data, so having a clear link between the math and what we actually measure is essential.

Jocelyn: So, the paper isn't just giving us an answer for black holes; it's providing a framework that tells us how to better analyze and predict deviations from General Relativity in any new modified gravity theory.

Conclusion: Tom: So we've reached the conclusion of our discussion on "Slowly rotating Black Holes in DHOST Theories," where Vera and Jocelyn summarize what this paper means for modified gravity research and then let Subrahmanyan weigh in on the bigger picture before we wrap up.

Vera: To recap, this paper shows that even within these higher-order scalar tensor theories, you can derive explicit analytical solutions for slowly rotating black holes, proving that the frame-dragging function is integrable and showing how different theories are structurally related through disformal transformations.

Jocelyn: It really boils down to finding a concrete mathematical pathway to understand how rotation behaves when gravity isn't exactly General Relativity. It gives us a solid starting point for testing new theories against what we actually observe in the sky or from gravitational waves.

Subrahmanyan: I think the main implication here is that we have a rigorous way to map out which modified gravity models are physically viable candidates for describing astrophysical objects like black holes, which is really important for guiding future observational tests.

Vera: That’s right, Subrahmanyan; it moves us from just guessing what might work to having a structured way of testing those structural relationships we found in the paper. It opens up new avenues for looking at rotating black holes that have scalar hair in these theories.

Jocelyn: I'm excited to see how this mathematical framework helps us refine our analysis of gravitational wave signals, because if we can predict the ISCO shifts with this much accuracy, it makes interpreting those signals from binary mergers much more precise.

Subrahmanyan: Exactly, Jocelyn; that predictive power is what separates theoretical work from observational science when you’re trying to constrain fundamental physics. It provides concrete targets for future observations.

Vera: So we’ve covered how this paper establishes a clear link between modified gravity theories and the geometry of rotating black holes, providing us with a powerful tool for both theory and observation.

Jocelyn: It was a really insightful look at how these complex equations simplify under certain conditions, which is exactly what we need when trying to process the huge amounts of data coming from surveys like Euclid.

Subrahmanyan: And that’s the path forward; using this kind of analytical rigor to guide our simulations and our observational searches for signs of modified gravity in the cosmos.

Hugo Candana, Karim Nouib, David Langloisc

LUX, Observatoire de Paris, University of Paris-Saclay, CNRS/IN2P3, IJCLab, Université Paris Cité

gr-qc, astro-ph.HE, hep-th

Submitted: 2025-12-19

Updated: 2026-09-29

Comments: 22 pages, 2 figures

DOI: 10.1103/8d8w-hnjg

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 76/100

The gist: This research investigates slowly rotating black hole solutions within Degenerate Higher Order Scalar Tensor (DHOST) theories, exploring how these modified gravity frameworks deviate from General

Key concepts

DHOST Theories
These are modified gravity theories that go beyond General Relativity by including higher-order scalar tensor terms. The paper investigates how black holes behave within these complex frameworks, exploring how the added complexity changes the spacetime geometry compared to standard GR.
Hartle–Thorne Ansatz
This is a mathematical model used to describe slowly rotating spacetimes around a black hole. It simplifies the complex equations of motion by assuming that the rotation is very small and depends only on the radial distance, allowing researchers to find manageable solutions.
Frame-Dragging Function ($\omega$)
This function describes how spacetime itself is dragged around by a rotating mass. The study shows that in DHOST theories, this function can be explicitly calculated. Crucially, it proves that for these specific theories, the frame-dragging effect does not depend on the angular position ($\theta$), similar to how it behaves in standard black holes.

Terminology

Summary

This research investigates slowly rotating black hole solutions within Degenerate Higher Order Scalar Tensor (DHOST) theories, exploring how these modified gravity frameworks deviate from General Relativity and how rotation affects astrophysical phenomena. The study is significant because it provides analytical general solutions for rotating black holes in DHOST theories, which are often the first known fully analytic examples of such objects with scalar hair. Furthermore, the work examines the influence of rotation on key orbital parameters like the Innermost Stable Circular Orbit (ISCO) and circular light trajectories, providing insights relevant to data from gravitational wave detectors and direct imaging.

General Framework and Methodology

The study begins by starting from a static, spherically symmetric metric solution of a DHOST theory and employing the Hartle–Thorne ansatz to model a slowly rotating spacetime. The core mathematical achievement is showing that the differential equation governing the frame-dragging function, denoted as ω (which is supposed to depend on the radial coordinate only), is integrable for any DHOST theory, allowing for an explicit form of this function. The authors also consider angular dependence in ω and demonstrate that regularity conditions at both the horizon and at infinity forbid such dependence, mirroring results from General Relativity.

Derivation of the Frame-Dragging Function

The derivation relies on substituting the Hartle–Thorne ansatz into the equations of motion for the metric, expanding these equations up to linear order in J (the angular momentum), and then focusing on a specific linear combination of equations, namely Etφ + ωEφφ. This combination reduces to a simple form:

)&Etφ + ωEφφ = 1/2r squared p hf sin 2(θ) d/dr Q(r)ω'(r). (Equation 7)

The integrability of the equation for ω is directly linked to a conservation law derived from a shift-symmetry under the transformation φ → φ+ω0t. This symmetry implies that the Lagrangian density L does not depend directly on ω but on its first and second derivatives, leading to the conservation equation:

)&δS/δω = −dJ/dr, where J = ∂L/∂ω'−d/dr(∂L/∂ω''). (Equation 16)

This conserved quantity is explicitly calculated as:

)&J = sin 3(θ)Q(r)ω'(r) + O(ω 3). (Equation 17)

Transformation and Equivalence of Theories

The paper investigates the relationship between different DHOST theories using disformal transformations, defined by gµν −→ g˜µν = Cgµν + D∂µϕ∂νϕ. By applying this transformation to a slowly rotating metric, the authors show that the new metric g˜µν can be written in a form similar to the original Hartle–Thorne ansatz (Equation 21). Crucially, they demonstrate that for quadratic theories, the frame dragging functions are related by:

)&ω(r) = ˜ω (R(r)), with R(r) = r p C(r). (Equation 27)

This relationship holds because the consistency relation R'Q/Q˜ = 1 is satisfied, and in the case of quadratic theories, this implies that the integration constants k and ˜k are identical.

Application to Specific Theories and Geodesics

The results are applied to specific subclasses of theories:

  1. In shift-symmetric theories, the first-order rotational term is always identical to that of Kerr, even if the static solution differs from Schwarzschild.

  2. For quadratic shift-symmetric DHOST theories (related to quadratic Horndeski theories), the function Q simplifies such that Q is proportional to r 4, and consequently, ω(r) is identical to the Kerr expression: ω(r) = 2J/r cubed (Equation 31).

  3. For specific black holes with primary hair in quadratic Beyond Horndeski theories, the study of time-like and light-like circular geodesics shows how they deviate from Kerr predictions in the slow-rotation limit, quantified by an effective potential Veff(r) (Equation 40).

Angular Dependence Analysis

The possibility of angular dependence in ω is explored by computing the partial differential equation governing a θ-dependent ω. This leads to an ordinary differential equation for each Legendre polynomial mode l:

)&d/dr Q dωl/dr + 2 − l(l + 1) Qˆ l = 0. (Equation 47)

The analysis of the asymptotic behavior at spatial infinity and near the horizon reveals that all modes l ≥ 2 must vanish, confirming that ω cannot depend on θ, consistent with the no-hair theorem in General Relativity.

Improvements for AI systems

Here are the specific improvements that can be made to AI systems based on the insights derived from this scientific paper, along with what those improved systems could accomplish:

)1. Improved Physical Modeling and Simulation (for Physics-Informed Neural Networks - PINNs):

The paper demonstrates a method for deriving an explicit, integrable differential equation for the frame-dragging function ω in DHOST theories by leveraging a hidden symmetry and a specific conservation law (equation 10).

  1. AI System Capability: These systems could be used to develop PINNs that are not just solving PDEs, but are inherently structured around conserved quantities derived from hidden symmetries. This allows the AI to predict the evolution of complex, non-linear gravitational fields (like those in rotating black holes) with significantly higher accuracy and stability than standard solvers, especially when dealing with modified gravity theories where traditional numerical methods struggle due to complexity.

)2. Enhanced Model Generalization and Theory Discovery:

The paper shows that a general result for slowly rotating black holes in DHOST theories can be derived, which is then shown to hold under disformal transformations (Section 3.1). Furthermore, it establishes a consistent relationship between the solutions of different theories (e.g., DHOST vs. Horndeski via disformal maps) through consistency relations like the one involving Q and Q̃ (Equation 83).

  1. AI System Capability: An AI system could be designed as a Theory Explorer. Given a set of observed gravitational wave data or astrophysical compact object metrics, the system could use this structural relationship to rapidly map which modified gravity theory (DHOST subclass) is most likely responsible. It would be able to infer the underlying action parameters by checking if the observed metric solutions satisfy the consistency relations derived from disformal invariance, effectively bypassing lengthy brute-force parameter fitting.

)3. Robust Analysis of Singularities and Boundary Conditions:

The paper rigorously proves that for any DHOST theory, the frame-dragging function ω cannot possess angular dependence due to regularity conditions at both the horizon and spatial infinity (Section 4). It further analyzes the near-horizon behavior of modes (Equation 70) to show that solutions must be regular and differentiable.

  1. AI System Capability: This capability allows for a Singularity Auditor. For any new theoretical model or simulation output, this AI could automatically perform a structural check to ensure the resulting metric does not exhibit unphysical singularities or non-differentiable behavior at crucial boundaries (like the event horizon). This is invaluable in experimental data analysis where precision near horizons is paramount, ensuring that only physically viable solutions are retained.

)4. High-Fidelity Prediction of Astrophysical Observables:

The paper applies the formalism to calculate the Innermost Stable Circular Orbit (ISCO) and circular light trajectories (Equations 39, 40) for rotating black holes with primary hair. It explicitly shows how these quantities deviate from Kerr predictions based on rotation parameters and scalar field coupling constants.

  1. AI System Capability: This system could serve as a Gravitational Wave Emulator. When analyzing gravitational wave signals from merging compact objects (LIGO/Virgo/KAGRA), the AI could use this formalism to predict the precise shifts in orbital dynamics caused by scalar hair or modified gravity effects, allowing for much more accurate parameter estimation of the underlying physics than is possible with standard GR templates.

)5. Automated Mode Decomposition and Spectral Analysis:

The paper decomposes an angularly dependent function ω(r, θ) using Legendre polynomials (Equation 46), leading to an ODE for each mode that must be solved (Equation 47). It then analyzes the asymptotic behavior of the solutions at infinity and the horizon based on the index parameters (n±) derived from coefficients like α0H and α2H.

  1. AI System Capability: This system could be a Spectral Analyzer. For complex, non-axisymmetric spacetime perturbations, this AI could automatically decompose them into spherical harmonic modes, solve the resulting ODEs efficiently, and immediately diagnose whether the solution is regular or pathological at critical points (like the horizon). This speeds up the analysis of perturbation theory in modified gravity by automating the challenging task of mode matching.

Abstract

We study slowly rotating black hole solutions within Degenerate Higher Order Scalar Tensor (DHOST) theories. Starting from a static, spherically symmetric metric solution of a DHOST theory, we employ the Hartle-Thorne ansatz to model a slowly rotating spacetime. We show that the differential equation governing the frame-dragging function ω (which is supposed to depend on the radial coordinate only) is integrable for any DHOST theory allowing us to obtain its explicit form. We also consider angular dependence in ω and show that regularity at the horizon and at infinity forbids it, as in General Relativity. As an illustration of the formalism introduced here, we study the slowly-rotating version of black hole solutions with primary hair obtained recently, examining the influence of the rotation on the Innermost Stable Circular Orbit (ISCO) and on the circular light trajectories in the equatorial plane.

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