Two asymptotically flat spinning black holes balanced by their self-interacting, synchronised scalar hair

This paper investigates how quartic scalar self-interactions influence asymptotically flat, balanced configurations of two spinning black holes with synchronised scalar hair, revealing that repulsive interactions drive topological changes in ergoregions and broaden analytical models but cannot increase horizon mass, whereas attractive interactions are required to achieve larger mass fractions.

The Gist

Imagine the universe as a giant, cosmic dance floor. Usually, when two heavy dancers (black holes) try to stand still next to each other, they inevitably crash into one another because their gravity pulls them together. In the empty vacuum of space, the only way to keep them apart is to jam a rigid, unbreakable pole (a "strut") between them to push them apart. But in this paper, the authors propose a much more elegant solution: they use a "cloud" of invisible, wavy energy to hold the dancers in place.

Here is a simple breakdown of what the paper explores, using everyday analogies:

The Cast of Characters

1. The Dancers (Black Holes): These are the spinning black holes. In this story, they are "hairy," meaning they aren't just bare, empty spheres; they are covered in a fuzzy coat of scalar fields (a type of energy wave).
2. The Cloud (Scalar Hair): Think of this as a mist or a cloud of energy surrounding the black holes. In the authors' previous work, they found that if you have two spinning black holes, this cloud can arrange itself in a specific way to push the black holes apart, balancing their mutual gravity without needing a physical pole.
3. The "Self-Interaction" (The Magic Ingredient): This is the main focus of the paper. Imagine the particles in that energy cloud can talk to each other.
   • Repulsive Interaction: They push each other away (like magnets with the same pole facing each other).
   • Attractive Interaction: They pull toward each other (like magnets with opposite poles).
     The paper specifically studies what happens when you turn up the "repulsive" setting.

The Three Scenarios Studied

The authors looked at three different "dance formations" to see how this repulsive magic ingredient changes things:

1. The Floating Clouds (Two Spinning Boson Stars)
Before putting black holes in the mix, they looked at just the energy clouds themselves (called Boson Stars).

• The Discovery: When the repulsion is weak, the cloud looks like a single, giant donut (a torus) wrapping around the center. But as the repulsion gets stronger, the cloud gets pushed apart, splitting into two separate donuts.
• The Analogy: Imagine a single, thick ring of dough. If you add a lot of baking powder (repulsion) that makes the dough expand, the ring might eventually snap and become two smaller, separate rings. This change in shape is called a "topological transition."

2. The Single Dancer with a Coat (One Black Hole)
Next, they looked at a single black hole covered in this energy coat.

• The Discovery: Adding the repulsive force makes the coat much "fluffier" and larger. The black hole can hold a lot more of this energy.
• The Catch: However, the black hole itself doesn't get heavier. It's like putting a massive, fluffy winter coat on a person. The person (the black hole's core) stays the same weight, but the total package (person + coat) becomes much heavier. The paper calls this "hairier but not heavier."
• Bonus: The repulsion also makes it easier for scientists to use simple math models to predict how these systems behave, even when the "coat" is very thick.

3. The Two Dancers in Sync (Two Black Holes)
Finally, they looked at the main event: two black holes balanced by the cloud.

• The Discovery: The repulsive force changes the "dance steps" (the mathematical structure) of how these pairs can exist.
  • Just like the single black hole, the repulsion makes the energy cloud bigger, but it does not make the black holes themselves heavier. In fact, the black holes end up carrying a smaller percentage of the total mass because the cloud is so huge.
  • If you were to use an attractive force instead (pulling the cloud together), the black holes could actually become heavier, but that's a different story.
• The Balance: The repulsive force is so effective that it completely removes the need for that "unbreakable pole" (the conical singularity) that usually holds two black holes apart in empty space. The cloud does all the work.

The Big Picture

The paper essentially asks: "If we make the energy cloud around these black holes push itself apart, what happens?"

The answer is that the cloud becomes more spread out and changes shape (splitting from one donut to two). This allows the black holes to sit in a stable, balanced state without crashing into each other or needing a physical strut. However, this extra "push" doesn't make the black holes themselves more massive; it just makes the fuzzy energy surrounding them much more prominent.

The authors conclude that while these repulsive forces create beautiful, balanced cosmic structures, they cannot make the black holes themselves "heavier" in terms of their event horizon mass. To get heavier black holes, you would need the opposite effect (attraction), which is a different scenario entirely.

Technical Summary

Technical Summary: Two Asymptotically Flat Spinning Black Holes Balanced by Their Self-Interacting, Synchronised Scalar Hair

Problem Statement
The paper addresses the construction of asymptotically flat, stationary configurations of two spinning black holes (2sBHs) in equilibrium. In vacuum General Relativity (GR), such configurations are known to require a conical singularity (a "strut") on the symmetry axis to balance the mutual gravitational attraction, rendering them physically unrealistic. Previous work [1] demonstrated that placing horizons at the equilibrium points of two spinning boson stars (2sBSs) with synchronised scalar hair allows for regular, balanced 2sBHs without conical singularities. However, the effects of scalar self-interactions on these configurations remained unexplored. Motivated by astrophysical models (e.g., axion-like dark matter) and theoretical considerations regarding the mass limits of boson stars, this study investigates how quartic repulsive self-interactions ($\lambda |\Psi|^4$) modify the existence, topology, and physical properties of 2sBSs, single spinning black holes with quadrupolar scalar hair (1sBHs), and the balanced 2sBHs.

Methodology
The authors work within the Einstein-Klein-Gordon framework with a complex scalar field $\Psi$ possessing a potential $U(|\Psi|^2) = \mu^2|\Psi|^2 + \lambda|\Psi|^4$, where $\lambda > 0$ represents a repulsive self-interaction. The study employs a generalized Bach-Weyl framework extended to spinning, hairy spacetimes.

1. 2sBSs (Horizonless): The authors solve the coupled Einstein-Klein-Gordon equations using a metric ansatz with two commuting Killing vectors ($\partial_t, \partial_\phi$) and a scalar field ansatz $\Psi = \psi(r, \theta)e^{i(m\phi-\omega t)}$ with $m=1$. Boundary conditions enforce regularity at the origin, asymptotic flatness, and $Z_2$ symmetry across the equatorial plane. Numerical solutions are obtained using a finite difference solver with a Newton-Raphson method on a compactified radial grid.
2. 1sBHs (Single Hairy Black Hole): By placing a horizon at the center of a 2sBS, the authors construct 1sBHs. A coordinate transformation $u = \sqrt{r^2 - r_H^2}$ is used to map the horizon to the origin. The synchronisation condition $\Omega_H = \omega/m$ is imposed at the horizon.
3. 2sBHs (Double Hairy Black Holes): To construct the equilibrium 2sBHs, the authors generalize the vacuum Bach-Weyl metric to include scalar hair and rotation. They introduce a coordinate transformation $(\rho, z) \to (r, \theta)$ to improve numerical accuracy near the horizons and infinity. A specific numerical strategy is developed to find equilibrium solutions (where the conical excess/deficit $\delta = 0$):
   • For fixed parameters $(\lambda, \omega)$, the conical deficit $\delta$ is calculated as a function of the rod parameters $(a, b)$.
   • Equilibrium points ($\delta \approx 0$) are identified by fitting curves in the $(a, b)$ parameter space.
   • Solution branches are constructed by scanning along these fitted curves.

Key Contributions and Results

• Two Spinning Boson Stars (2sBSs):

  • Mass and Distance: Repulsive self-interactions increase the maximum ADM mass of 2sBSs. The proper distance $D$ between the two components at maximum mass remains approximately constant ($D \approx 10$) regardless of $\lambda$, but for a fixed $D$, the mass increases with $\lambda$.
  • Ergoregion Topology: In the strong-gravity regime, increasing $\lambda$ induces a topological transition in the ergoregion. The ergosurface changes from a single torus (at $\lambda=0$) to a double torus. This is attributed to the repulsive force suppressing scalar matter accumulation between the two components, disconnecting the $g_{tt}=0$ surface along the symmetry axis.

• Single Hairy Kerr Black Holes (1sBHs):

  • Existence Domain: The domain of existence for 1sBHs is bounded by the 2sBSs line, the Kerr existence line (scalar clouds), and the extremal 1sBH line. As $\lambda$ increases, the maximum mass of the 2sBSs and extremal 1sBHs increases, while the range of horizon angular velocities decreases.
  • "Hairier but Not Heavier": While self-interactions significantly increase the total ADM mass ($M$) and angular momentum ($J$) of the system, the horizon mass ($M_H$) and horizon angular momentum ($J_H$) remain bounded by the "Hod point" (the intersection with extremal Kerr). Thus, increasing $\lambda$ makes the black holes "hairier" (more scalar mass) but does not make the horizons "heavier."
  • Effective Model Accuracy: The authors test an analytical effective model [2,3] against numerical results. They find that while the model is accurate for small scalar fractions ($p < 0.1$), increasing the self-interaction coupling $\lambda$ actually improves the model's accuracy for larger $p$. This is because repulsive interactions dilute the scalar matter distribution, making the system behave more like a test particle in a background, which the effective model approximates well.

• Double Hairy Kerr Black Holes (2sBHs):

  • Bifurcation Structure: The solution sequences for 2sBHs bifurcate from the 2sBSs. The authors classify sequences based on whether horizons emerge from stable or unstable equilibrium points of the scalar environment.
  • Topological Changes in Sequences: As $\lambda$ increases, the mass gap between the two branches of type (i) sequences (one horizon from a stable point, one from an unstable point) closes. Eventually, these transform into a single sequence of type (ii) (both from unstable) and a sequence of type (iii) (both from stable).
  • Horizon Mass Fraction: Similar to the 1sBH case, repulsive self-interactions ($\lambda > 0$) reduce the fraction of total mass contained in the horizons ($M_H/M$), making the horizons lighter. Conversely, the authors note that attractive self-interactions ($\lambda < 0$) would allow for heavier horizons, though such potentials are unbounded from below.
  • Equilibrium: The scalar environment of the 2sBSs provides the necessary repulsive force to balance the gravitational attraction of the two black holes, eliminating the need for a conical singularity.

Significance and Claims
The paper claims to provide the first systematic investigation of quartic self-interactions on balanced multi-black-hole configurations with synchronised scalar hair. The primary significance lies in demonstrating that:

1. Self-interactions fundamentally alter the topology of ergoregions in horizonless boson stars.
2. The "hairier but not heavier" phenomenon, previously observed for fundamental scalar hair, extends to quadrupolar scalar hair and multi-black-hole systems.
3. Repulsive self-interactions can enhance the validity of analytical effective models for hairy black holes by diluting the scalar matter distribution.
4. A robust numerical strategy can be developed to construct equilibrium 2sBHs within the generalized Bach-Weyl framework, revealing complex bifurcation structures in the solution space that depend on the strength of the self-interaction.

The authors conclude that while repulsive interactions cannot increase the horizon mass, they are crucial for stabilizing these configurations and modifying their global properties, offering a richer landscape for potential astrophysical mimickers and theoretical exploration.