⚛️ general relativity

Radial Perturbations of Black Holes in DHOST Theories

This paper demonstrates that static black holes with primary hair in a subfamily of DHOST theories are radially stable by recasting the monopole mode equation into a flat wave equation within the unitary gauge, which extends beyond the event horizon and applies universally to disformally related solutions, while also revealing that stealth black holes exhibit either a lack of propagating modes or stable radial degrees of freedom depending on their kinetic term behavior.

Original authors: Christos Charmousis, Simon Iteanu, David Langlois, Karim Noui

Published 2026-06-25

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CC0 1.0

Original authors: Christos Charmousis, Simon Iteanu, David Langlois, Karim Noui

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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 gravity as a giant, invisible trampoline. In our standard understanding of the universe (General Relativity), if you place a heavy bowling ball (a black hole) on this trampoline, it creates a deep, smooth dip. If you wiggle the trampoline, the waves travel in predictable ways.

But what if the trampoline itself is made of a slightly different, more complex material? This is the world of DHOST theories. These are new, experimental ideas about how gravity might work, suggesting that space-time isn't just a simple fabric but has an extra "ingredient" called a scalar field mixed in.

This paper is like a stress test for black holes made in these new, complex materials. The authors ask: "If we poke these black holes, do they wobble and settle down (stable), or do they explode and fall apart (unstable)?"

Here is the breakdown of their findings using simple analogies:

1. The "Hair" on the Black Hole

In standard physics, black holes are famously "bald." They are defined only by their mass, spin, and electric charge. But in these new theories, black holes can have "primary hair."

The Analogy: Imagine a standard black hole is a smooth, featureless stone. A "hairy" black hole is like a stone covered in a specific type of moss that grows independently of the stone's weight. This "moss" (the scalar field) is a new property the black hole possesses, changing how it interacts with the universe.

2. The "Flat" Map vs. The "Tortoise" Map

To study how these black holes wiggle, the authors had to change the way they measured space and time.

The Old Way (General Relativity): Imagine trying to map a cave that gets infinitely long as you go deeper. You use a "tortoise coordinate" (like a very slow, winding path) where the entrance to the cave (the event horizon) stretches out forever on your map. You can never actually reach the center on the map.
The New Way (This Paper): The authors found a special "unitary gauge" (a specific way of looking at the problem) where the scalar field acts like a clock. In this view, the map of the black hole becomes flat and finite.
The Result: Instead of a path that stretches to infinity, the map is a short, closed loop. You can walk from the outside of the black hole, cross the horizon, and reach the center without your map breaking. This allowed them to see the "inside" of the black hole clearly, which is usually impossible.

3. The "Musical Instrument" Test

To check for stability, the authors turned the black hole into a musical instrument.

The Setup: They treated the black hole like a drum or a guitar string. When you pluck a string, it vibrates at specific frequencies. If the string is tight and healthy, it vibrates and eventually stops (stable). If the string is loose or broken, it might snap or vibrate uncontrollably (unstable).
The Schrödinger Equation: They used a famous physics equation (usually used for tiny particles like electrons) to describe these vibrations. They calculated the "potential energy" (the tension of the string) for the black hole.
The Finding: For most of the "hairy" black holes they studied, the "tension" was positive. This means the black hole is like a well-tuned guitar string: if you poke it, it wiggles a bit and then settles back down. It is stable.

4. The "Stealth" Black Holes

The paper also looked at "stealth" black holes. These are tricky cases where the black hole looks exactly like a normal one from the outside, but has a hidden, complex internal structure.

Case A (Constant Hidden Energy): If the hidden energy is constant, the black hole is so quiet that it doesn't even have a "voice." There is no vibration to study at all. It's like a drum with no skin.
Case B (Changing Hidden Energy): If the hidden energy changes, the black hole does have a voice. However, the authors found that for most settings, this voice is stable. But, if you tune the "knobs" of the theory (the coupling constants) to very specific, special values, the black hole becomes unstable and could theoretically fall apart.

5. The "Mirror" Effect

One of the coolest discoveries is about disformal transformations. This is a mathematical trick where you can stretch and squeeze the fabric of space-time to turn one theory into another.

The Analogy: Imagine you have a photo of a black hole. You can stretch the photo horizontally or vertically (a disformal transformation) to make it look like a different black hole in a different theory.
The Result: The authors found that the "song" (the vibration equation) the black hole sings is exactly the same before and after you stretch the photo. Whether you look at the original hairy black hole or its "stretched" cousin, the rules for stability don't change. This suggests a deep, unbreakable link between these different types of gravity theories.

Summary

The paper concludes that black holes with this new "hair" are generally stable. They don't spontaneously explode when poked. The authors achieved this by inventing a new way to map the inside of a black hole (making the horizon crossable on the map) and treating the black hole like a musical instrument to check its vibrations. They found that as long as the "tuning knobs" of the theory aren't set to very specific, weird numbers, these exotic black holes are safe and sound.

Technical Summary: Radial Perturbations of Black Holes in DHOST Theories

Problem Statement
The paper addresses the stability of static black hole solutions with "primary hair" within the framework of Degenerate Higher-Order Scalar-Tensor (DHOST) theories. While axial perturbations of these solutions have been previously studied, radial (monopole) perturbations present a unique challenge. In General Relativity (GR), Birkhoff's theorem implies the absence of propagating radial modes; however, in DHOST theories, the scalar degree of freedom couples with metric perturbations to create a single propagating radial degree of freedom. The central problem is to derive the equation of motion for this mode, determine its stability, and analyze the behavior of the perturbation across the event horizon and into the black hole interior.

Methodology
The authors employ a systematic perturbative approach around static, spherically symmetric backgrounds with a scalar field exhibiting linear time dependence ( $\bar{\phi} = qt + \psi(r)$ ). The methodology proceeds through the following steps:

Gauge Fixing and Master Variable Construction: Working in the unitary gauge where the scalar perturbation vanishes ( $\delta\phi = 0$ ), the authors reduce the system of linearized metric equations to a single second-order differential equation for a gauge-invariant master variable, $\sigma$ . This involves eliminating auxiliary metric perturbation components ( $H_0, H_1, H_2$ ) to isolate the dynamics of the monopole mode.
Coordinate Transformation: To simplify the master equation, the authors introduce a new time coordinate $t^*$ $t^{*}$ and a new radial coordinate $r^*$ $r^{*}$ .
- The new time coordinate $t^*$ is shown to coincide with the scalar field itself (up to a rescaling), allowing the coordinate system to remain regular across the event horizon and extend into the black hole interior.
- The new radial coordinate $r^*$ is defined via an integral involving the background metric functions, effectively compactifying the radial domain into a finite interval $[0, r^*_{max}]$ .
Schrödinger-like Reformulation: The master equation is transformed into a flat radial wave equation of the form:
$\left[ \frac{\partial^2}{\partial t^{*2}} - \frac{\partial^2}{\partial r^{*2}} + V_{\text{eff}}(r^*) \right] \Psi = 0$
where $\Psi$ is a renormalized version of $\sigma$ . This allows the stability analysis to be framed as a spectral problem for a Schrödinger operator $\mathcal{L} = -\partial_{r^*}^2 + V_{\text{eff}}$ .
Stability Analysis: Stability is established by proving that the operator $\mathcal{L}$ can be extended to a positive self-adjoint operator. This requires analyzing the asymptotic behavior of solutions at the boundaries of the compact interval ( $r^* \to 0$ and $r^* \to r^*_{max}$ ) to define appropriate boundary conditions (specifically, the Friedrichs extension) that ensure the operator's positivity.

Key Contributions and Results

Radial Stability of Primary Hair Black Holes: For a subfamily of shift-symmetric DHOST theories (specifically the Beyond Horndeski subclass) admitting black holes with primary hair, the authors demonstrate that the radial mode is stable. They show that the associated Schrödinger operator admits a positive self-adjoint extension. This is achieved by imposing regularity conditions at the origin and specific boundary conditions at spatial infinity, ensuring that the time evolution of perturbations remains bounded.
Coordinate Regularity and Horizon Penetration: A significant finding is that the coordinate choice leading to the flat wave equation corresponds to the unitary gauge. Unlike the standard tortoise coordinate in GR, which diverges at the horizon, the new radial coordinate $r^*$ remains finite and well-defined across the event horizon, extending to the central singularity. This allows for a unified description of perturbations both outside and inside the black hole.
Invariance under Disformal Transformations: The authors prove that the Schrödinger-like equation governing radial perturbations is invariant under disformal transformations of the metric. Consequently, the stability analysis derived for the original "homogeneous" solutions applies equally to "disformed" black hole solutions (where $A(r) \neq B(r)$ ) generated by mapping the theory to the Horndeski subclass.
Stealth Solutions:
- Constant Kinetic Term ( $X$ ): For stealth Schwarzschild solutions with a constant kinetic term, the analysis confirms the absence of a propagating radial degree of freedom at the linear level.
- Non-Constant Kinetic Term: For stealth solutions with a non-constant $X$ , a stable radial degree of freedom is identified, except for specific values of the coupling constants where the determinant of the system vanishes.
Potential Behavior: The effective potential $V_{\text{eff}}$ is derived explicitly. While not strictly positive for all parameter values, the authors utilize a factorization method ( $V_{\text{eff}} = S^2 - dS/dr^*$ ) to demonstrate that the operator remains positive under the chosen boundary conditions. Numerical computations of the fundamental mode frequency confirm $\omega^2 > 0$ for various parameter sets.

Significance
The paper provides a rigorous proof of the linear stability of radial perturbations for a class of black holes with primary hair in DHOST theories. By recasting the complex coupled scalar-metric perturbations into a simple Schrödinger-like equation, the authors establish a robust mathematical framework for analyzing stability in modified gravity. The extension of the radial coordinate into the black hole interior offers a novel perspective on the causal structure of perturbations in these theories, distinct from the GR paradigm. Furthermore, the demonstration of invariance under disformal transformations suggests that stability results are robust across a wide class of related theories, including those mapped to the Horndeski subclass. This work complements previous studies on axial perturbations, providing a more complete picture of the stability of hairy black holes in the strong-field regime.