⚛️ general relativity

Slowly rotating Black Holes in DHOST Theories

This paper establishes that slowly rotating black hole solutions in Degenerate Higher Order Scalar Tensor (DHOST) theories are governed by an integrable frame-dragging equation that forbids angular dependence, and applies this formalism to analyze how rotation affects the Innermost Stable Circular Orbit and light trajectories in recently discovered hairy black hole solutions.

Original authors: Hugo Candan, Karim Noui, David Langlois

Published 2026-09-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Hugo Candan, Karim Noui, David Langlois

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

Deep in the fabric of our universe, gravity is the invisible hand that shapes everything from the fall of an apple to the orbit of distant stars. For over a century, our best map of this force has been Albert Einstein's theory of General Relativity, which describes gravity not as a pull, but as a curvature of space and time caused by mass. While this theory has passed every test thrown at it, from the bending of starlight to the ripples of gravitational waves, scientists remain curious. They wonder if gravity behaves differently in the most extreme environments, such as the swirling maws of black holes, or if there are hidden layers to the cosmic story that Einstein's equations do not tell. To explore this, researchers have developed new mathematical frameworks that add a subtle, invisible field to the mix, a scalar field that acts like an extra messenger alongside the familiar fabric of space-time. These new theories, known as Degenerate Higher Order Scalar-Tensor theories, or DHOST, offer a playground to test if the rules of gravity change when the stakes are highest.

The focus of this new work is a specific type of cosmic object: a black hole that is slowly spinning. In our universe, almost nothing is perfectly still; stars, planets, and black holes all rotate. When a black hole spins, it drags the very space around it, a phenomenon called frame-dragging, much like a spoon stirring a thick liquid pulls the liquid along with it. For decades, physicists have struggled to write down the exact mathematical description of a spinning black hole in these new gravity theories. The equations are notoriously difficult, often requiring powerful computers to find numerical approximations rather than clean, exact formulas. However, a team of researchers has now found a way to solve these equations exactly for the case of slow rotation, revealing a surprising simplicity hidden within the complexity.

The researchers started with a known, non-spinning black hole solution from a DHOST theory and asked what would happen if they gently set it spinning. Instead of tackling the full, chaotic problem of a rapidly rotating object, they used a method that treats the spin as a tiny addition to the static state. This approach allowed them to isolate the specific part of the equations that describes how the black hole drags space around it. What they discovered was remarkable: the equation governing this dragging effect could be solved directly for any theory within this broad class. They found that the dragging force follows a specific, integrable path, meaning they could write down a precise formula for it based solely on the properties of the static black hole they started with. This was not just a lucky guess for one specific theory; it worked for the entire family of these modified gravity models, suggesting a deep, underlying order in how these theories respond to rotation.

One of the most striking findings concerns the shape of this dragging effect. In the standard theory of General Relativity, the way a black hole drags space depends only on how far you are from the center, not on your latitude or angle. This is a consequence of the "no-hair" theorem, which states that black holes are simple objects defined only by their mass and spin. In the new theories, where black holes can have extra "hair"—meaning they carry additional scalar charges that change their structure—one might expect the dragging effect to become lopsided, varying as you move from the pole to the equator. The researchers tested this possibility by allowing the dragging function to depend on the angle. They found that for the solution to remain physically sensible—smooth and regular at the edge of the black hole and far away in space—this angular dependence is strictly forbidden. Just as in Einstein's original theory, the dragging effect must depend only on distance, not direction. This result holds true even for the most exotic black holes in these new theories, reinforcing the idea that the geometry of a spinning black hole is far more rigid than one might expect.

To see how these theoretical discoveries play out in the real world, the team applied their formulas to a specific, recently discovered type of black hole that possesses this extra scalar hair. They examined how the slow spin of such a black hole would affect the paths of objects orbiting it. Specifically, they looked at the innermost stable circular orbit, the closest distance a massive particle can orbit without inevitably falling in, and the path of light circling the black hole. They found that while the presence of the extra hair changes the size of these orbits compared to a standard black hole, the introduction of slow rotation shifts them in a predictable way. For light and matter orbiting against the spin, the stable path moves further out; for those orbiting with the spin, it moves closer in. These shifts are small but measurable, offering a potential signature that future telescopes and gravitational wave detectors could use to distinguish between Einstein's gravity and these more complex alternatives.

The work does more than just provide a new formula; it demonstrates that even in the most complicated modified gravity theories, there are pockets of mathematical elegance that allow for exact solutions. By showing that the equations for a slowly spinning black hole can be integrated directly, the researchers have opened a door to understanding how rotation interacts with the extra fields that might exist in our universe. Their findings suggest that while these new theories allow for black holes with more complex internal structures, the way they twist space-time remains surprisingly constrained. As we gather more data from the edges of black holes with instruments like the Event Horizon Telescope and future gravitational wave observatories, these precise theoretical predictions will serve as a crucial guide, helping us determine if the universe follows Einstein's script or if there is a new, hidden chapter waiting to be read.

Technical Summary: Slowly Rotating Black Holes in DHOST Theories

Problem Statement
While Degenerate Higher Order Scalar-Tensor (DHOST) theories have yielded numerous static, spherically symmetric black hole solutions (including those with "primary hair"), analytical rotating solutions remain scarce. Existing rotating solutions are either "stealth" (where the metric is identical to the Kerr metric), obtained via disformal transformations of stealth solutions, or derived numerically. A general analytical framework for slowly rotating black holes in arbitrary DHOST theories, starting from a static background, has been lacking. This paper addresses the challenge of solving the equations of motion for rotating spacetimes in DHOST theories, specifically investigating the integrability of the frame-dragging function and the constraints on its angular dependence.

Methodology
The authors employ the Hartle–Thorne ansatz to model a slowly rotating spacetime as a first-order perturbation in angular momentum $J$ around a static, spherically symmetric background metric $g^{(0)}_{\mu\nu}$ .

Metric Ansatz: The metric is expanded as $g_{\mu\nu} = g^{(0)}_{\mu\nu} + J g^{(1)}_{\mu\nu} + O(J^2)$ , where the perturbation introduces a frame-dragging term proportional to $\omega(r)$ .
Equations of Motion: The authors substitute this ansatz into the DHOST field equations. They demonstrate that the diagonal equations ( $E_{tt}, E_{rr}$ ) and the mixed radial-time equation ( $E_{tr}$ ) remain satisfied by the static solution. The non-trivial dynamics arise from the off-diagonal equation $E_{t\phi}$ .
Integrability Argument: Instead of solving the second-order differential equation for $\omega(r)$ directly, the authors consider the linear combination $E_{t\phi} + \omega E_{\phi\phi}$ . They show this combination reduces to a total derivative form, $\frac{d}{dr}[Q(r)\omega'(r)] = 0$ , where $Q(r)$ is a function determined by the static background and the specific DHOST Lagrangian functions.
Symmetry Analysis: The integrability is linked to a "shift-symmetry" in the space of solutions. By modifying the ansatz to include quadratic terms in $\omega$ (which do not affect the linear order), the authors show that the transformation $\phi \to \phi + \omega_0 t$ corresponds to a shift $\omega \to \omega + \omega_0$ . This continuous symmetry implies a conserved quantity (Noether current), leading directly to the integrable form of the equation.
Angular Dependence: The authors extend the analysis to allow $\omega$ to depend on the polar angle $\theta$ . They decompose $\omega(r, \theta)$ into Legendre polynomial modes and analyze the regularity of higher-order modes ( $l \geq 2$ ) at the event horizon and spatial infinity.

Key Contributions and Results

General Integral Solution: For any DHOST theory (quadratic and cubic), the frame-dragging function $\omega(r)$ is found to be integrable. The general solution is given by:
$\omega(r) = k \int \frac{dr}{Q(r)}$
where $k$ is an integration constant proportional to the angular momentum $J$ , and $Q(r)$ is explicitly derived in terms of the static metric functions ( $h, f$ ), the scalar field profile, and the DHOST coupling functions ( $F_i, A_i, B_i$ ).
Shift-Symmetric Theories: In theories with shift symmetry ( $\phi \to \phi + \text{const}$ ), the authors derive specific behaviors:
- For quadratic shift-symmetric Horndeski theories, $Q(r) \propto r^4$ , leading to $\omega(r) = 2J/r^3$ , identical to the Kerr solution, even if the static metric differs from Schwarzschild.
- For quadratic Beyond-Horndeski theories related to Horndeski via disformal transformations, the frame-dragging function transforms as $\omega(r) = \tilde{\omega}(R(r))$ , where $R(r)$ depends on the disformal factor. Asymptotic flatness requires the disformal factor to approach unity at infinity, preserving the Kerr-like asymptotic behavior.
Application to Primary Hair Solutions: The formalism is applied to recently discovered black hole solutions with primary hair (scalar charge independent of mass). The authors calculate the influence of slow rotation on:
- Innermost Stable Circular Orbit (ISCO): The ISCO radius shifts depending on the rotation parameter $a=J/M$ and the hair parameter $\xi$ . For retrograde orbits, the ISCO radius increases with rotation; for prograde, it decreases. The sensitivity to rotation is highest when the hair parameter is small.
- Circular Light Trajectories: Similar shifts are observed for photon orbits, with the radius varying with both the hair parameter and rotation.
Constraints on Angular Dependence: The paper demonstrates that for specific subclasses of DHOST theories (General Relativity, quadratic shift-symmetric theories, and 4D Einstein-Gauss-Bonnet), regularity conditions at the horizon and at spatial infinity forbid angular dependence ( $\theta$ -dependence) of $\omega$ .
- In General Relativity, modes with $l \geq 2$ are either divergent at infinity or non-differentiable at the horizon.
- In the studied DHOST classes, similar analysis shows that requiring the metric to be continuous and differentiable eliminates all higher-order modes ( $l \geq 2$ ), leaving only the $l=1$ mode (the standard frame-dragging term).
- Note on General Case: The authors explicitly state that for the general case (specifically in 4D Einstein-Gauss-Bonnet), while they can eliminate divergent branches at infinity and non-differentiable branches at the horizon, they do not provide a proof that these eliminated branches are distinct. Consequently, a regular branch of solution with angular dependence cannot be definitively ruled out for the general case, though it is expected to be absent based on the continuity with GR.

Significance
The paper establishes that the equations governing slowly rotating black holes in DHOST theories possess a remarkable integrability property, allowing for the derivation of explicit analytical expressions for the frame-dragging function based solely on static solutions. This result generalizes previous findings limited to specific subclasses or numerical approaches.

The authors emphasize that this integrability is likely not exclusive to scalar-tensor theories and could be extended to other modified gravity frameworks. Furthermore, the analysis of regularity conditions reinforces the "no-hair" nature of the rotational sector in specific DHOST subclasses, even in the presence of scalar hair in the static sector, while highlighting the subtleties in proving this for the most general cases. These analytical tools provide a foundation for testing DHOST theories against observational data from gravitational wave detectors (LIGO-Virgo-KAGRA) and the Event Horizon Telescope, particularly regarding deviations in ISCO and photon ring properties.

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