⚛️ general relativity

Nonlinear Stability of Kerr-Sen Black Holes in Merging Binaries

This paper employs numerical relativity simulations of merging Kerr-Sen black holes to demonstrate the long-term stability of their associated dilaton and axion fields and confirms that black holes immersed in a scalar background, including initially unscalarized Kerr-Newman black holes, acquire and retain scalar hair.

Original authors: Andrew Carroll, Eric W. Hirschmann, Hyun Lim, David Neilsen, David F. Van Komen, Sebastian Vander Ploeg Fallon

Published 2026-06-24

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CC BY 4.0

Original authors: Andrew Carroll, Eric W. Hirschmann, Hyun Lim, David Neilsen, David F. Van Komen, Sebastian Vander Ploeg Fallon

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

The Big Picture: Testing the "Cosmic Rules"

Imagine the universe follows a specific rulebook called General Relativity (our current best theory of gravity). For a long time, this rulebook has passed every test we've thrown at it. But scientists suspect there might be hidden chapters in the book—extensions to the theory that only show up in extreme situations, like inside black holes.

This paper investigates one of those potential "hidden chapters." It looks at a specific theory called Einstein-Maxwell-Dilaton-Axion (EMDA). Think of EMDA as General Relativity with two extra ingredients added to the mix:

The Dilaton: A field that acts like a "volume knob" for the strength of forces.
The Axion: A field that acts like a "twist" or a hidden magnetic-like property.

The authors wanted to know: If we smash two black holes together in this modified universe, do these extra ingredients survive the crash, or do they get destroyed?

The Experiment: A Cosmic Head-On Collision

To find out, the researchers didn't use a real telescope. Instead, they built a super-complex computer simulation—a virtual laboratory where they could break the laws of physics to see what happens.

They set up a "head-on collision" between two black holes. They tested two main scenarios:

Scenario 1: The "Hairy" Black Holes

First, they started with black holes that already had these extra fields attached to them. In physics, we often say black holes are "bald" (they only have mass, spin, and charge). But in this theory, they can have "hair" (the dilaton and axion fields).

The Analogy: Imagine two snowballs covered in glitter (the extra fields) crashing into each other.
The Result: When the snowballs merged into one giant snowball, the glitter didn't fly off into the air. It stayed stuck to the new, bigger snowball.
The Finding: The simulation showed that as long as the black hole has an electric charge, the dilaton field (the "volume knob") survives the merger. If the black hole is also spinning, the axion field (the "twist") also survives. The new black hole settles down, still wearing its "hair."

Scenario 2: The "Bald" Black Holes

Next, they tried something trickier. They started with "bald" black holes (standard black holes with no extra fields) but placed them in a sea of these extra fields (like putting a plain snowball in a room full of glitter dust).

The Analogy: Imagine a plain snowball sitting in a room where glitter is floating around. Does the snowball stay plain, or does it start picking up glitter?
The Result: The plain black holes quickly started "growing" hair. They absorbed the surrounding fields and transformed into the "hairy" version of themselves.
The Finding: Even if you start with a "bald" black hole, the universe of this theory forces it to grow these extra fields. It cannot stay bald.

Why This Matters (According to the Paper)

The main goal of this paper was to test stability.

In the world of physics, if a theory predicts that black holes would instantly explode or lose their defining features when they merge, that theory is probably wrong. It's like a house of cards that collapses if you blow on it.

The authors found that these "hairy" black holes are stable.

They don't explode.
They don't lose their extra fields during the violent crash.
They settle down into a calm, new state that still has all the same "hair."

This suggests that this specific alternative theory of gravity is a valid candidate for describing our universe. It behaves consistently even in the most violent, nonlinear events (like black hole mergers).

The Bottom Line

The paper concludes that if our universe follows the rules of this specific "EMDA" theory, black holes are robust. They can carry extra "hair" (dilaton and axion fields) through a merger without losing it. Furthermore, even if a black hole starts without this hair, the environment will force it to grow it.

The authors note that while their results look good, we still need to watch real black holes merging (using gravitational wave detectors like LIGO) to see if nature actually behaves this way. For now, the computer says: "Yes, these hairy black holes are stable."

Technical Summary: Nonlinear Stability of Kerr–Sen Black Holes in Merging Binaries

Problem Statement
While General Relativity (GR) has passed all current experimental tests, alternative theories of gravity remain viable candidates for describing strong-field regimes. A critical challenge in testing these theories is the lack of robust nonlinear simulations, particularly for theories that introduce additional propagating degrees of freedom (such as scalar fields) coupled to gravity. Specifically, the Einstein-Maxwell-Dilaton-Axion (EMDA) theory, which arises as a low-energy limit of heterotic string theory, predicts the existence of Kerr–Sen black holes. These solutions possess nontrivial dilaton and axion fields in addition to mass, spin, and charge.

The central problem addressed in this work is the nonlinear stability of these Kerr–Sen black holes during highly dynamical processes, specifically binary black hole mergers. It remains unclear whether the scalar hair (dilaton and axion fields) associated with these black holes persists through the merger, disperses via radiation, or settles into a stable remnant configuration. Furthermore, it is unknown whether black holes initially devoid of scalar hair (e.g., Kerr–Newman solutions) can dynamically acquire such hair when immersed in a scalar background or during the merger process.

Methodology
The authors perform fully nonlinear numerical relativity simulations of head-on binary black hole collisions within the EMDA framework.

Theoretical Framework: The study utilizes the EMDA action, which couples the Einstein-Maxwell system to a dilaton ( $\phi$ ) and an axion ( $\kappa$ ). The authors focus on the specific coupling values $(\alpha_0, \alpha_1) = (1, 1)$ , corresponding to the heterotic string theory limit, which admits the Kerr–Sen family of solutions.
Numerical Infrastructure: Simulations are conducted using Dendro-GR, a numerical relativity framework employing an octree-based adaptive mesh refinement (AMR) scheme with wavelet adaptive multiresolution (WAMR). This allows for dynamic refinement near black holes and regions of high field gradients.
Formulation: The system is evolved as a Cauchy problem using the BSSN formulation for the metric, supplemented by evolution equations for the electromagnetic, dilaton, and axion fields. Puncture gauge conditions are employed.
Initial Data: The authors construct approximate initial data for binary black holes using a modified elliptic solver based on the two-charged-punctures method. Two distinct classes of initial configurations are tested:
1. Initially Scalarized: Binary systems where both black holes possess nontrivial dilaton and axion fields (Kerr–Sen).
2. Initially Unscalarized: Binary systems where black holes are initially Kerr–Newman (no scalar hair), potentially immersed in a perturbative scalar background.
Diagnostics: The study monitors Hamiltonian and momentum constraints, electromagnetic divergence constraints, and quasilocal quantities (mass, spin, charge) on apparent horizons. Radiation is extracted via the Newman-Penrose scalar $\Psi_4$ (gravitational) and electromagnetic scalars $\Phi_1, \Phi_2$ , as well as scalar field radiation.

Key Contributions and Results

Persistence of Scalar Hair:
The simulations demonstrate that for nontrivial electric charge, the dilaton field persists through the merger and settles onto the remnant black hole. In the presence of both charge and spin, the axion field also persists. The authors find that these fields do not radiate away significantly; the scalar radiation energy is at least three orders of magnitude smaller than gravitational or electromagnetic radiation.
Stability of the Remnant:
The final remnant of the merger settles into a stationary configuration identified as a larger-mass Kerr–Sen black hole. Quasilocal quantities (horizon mass, dimensionless spin, and electric charge) stabilize to constant values, confirming the nonlinear stability of the Kerr–Sen solution within the EMDA theory under the tested conditions.
Dynamic Scalarization:
The study investigates whether initially unscalarized black holes (Kerr–Newman) can acquire scalar hair. The results show that even without pre-existing scalar fields, the system dynamically scalarizes. Kerr–Newman black holes rapidly develop both dilaton and axion hair during the evolution, eventually settling into a Kerr–Sen configuration. This indicates that scalarization is a robust dynamical feature of the theory and does not require pre-existing hair.
Radiation Characteristics:
In head-on collisions, gravitational radiation is the dominant form of energy release, followed by electromagnetic radiation. Scalar radiation (both dilaton and axion) is present but negligible in terms of total energy budget compared to the tensor and vector modes.

Significance and Claims
The paper claims to provide the first fully nonlinear simulations of binary black hole mergers in the full EMDA system. Its primary significance lies in establishing the nonlinear stability of Kerr–Sen black holes. The authors conclude that the additional degrees of freedom (dilaton and axion) behave as stable "hair" that survives the highly nonlinear merger process.

The work suggests that:

Kerr–Sen black holes are stable endpoints for binary coalescence in EMDA theory.
Scalar hair is not merely a static feature but a dynamically generated and retained property of black holes in this theory.
The persistence of these fields implies that future gravitational wave observations could potentially distinguish between GR and EMDA if the scalar radiation or the ringdown of the remnant deviates sufficiently from GR predictions, although the current simulations show scalar radiation is weak.

The authors note that while their results are specific to the heterotic string limit $(\alpha_0, \alpha_1) = (1, 1)$ , the broader parameter space of EMDA theories remains to be explored. They identify the analysis of inspiraling waveforms and the exploration of different coupling constants as necessary future directions to fully assess the observational phenomenology of these theories.