Spin-up and mass-gain in hyperbolic encounters of spinning black holes

Through numerical relativity simulations of equal-mass black hole scattering, this study demonstrates that spin-up and mass-gain are maximized near the merger threshold with high momenta and anti-aligned spins, revealing that sufficiently high initial spin can paradoxically lead to a spin-down despite an increase in total angular momentum due to significant mass accretion.

The Gist

Imagine two massive, spinning tops (black holes) zooming through space on a collision course, but instead of crashing into each other and sticking together, they are moving so fast that they might just swing past one another and fly apart. This is called a hyperbolic encounter.

This paper is like a high-speed camera study of what happens to these spinning tops when they get close enough to "kiss" but not quite merge. The researchers used powerful supercomputers to simulate these cosmic dance-offs and discovered some surprising things about how the black holes change their spin and size.

Here is the breakdown in simple terms:

1. The Setup: A Cosmic Billiard Game

Think of the black holes as two heavy billiard balls on a table, but instead of a stick, they are pulled by gravity.

• The Spin: Some balls are spinning clockwise, some counter-clockwise, and some aren't spinning at all.
• The Aim: The researchers changed the angle at which the balls approached each other. If they aimed perfectly head-on, they would crash and merge. If they aimed wide, they would just bounce off.
• The "Threshold": There is a very specific, razor-thin angle where the balls are just about to crash but instead get pulled into a wild, spiraling dance before flying apart. This is the "edge of chaos."

2. The Big Discovery: The "Spin-Up" and "Spin-Down"

When these black holes swing past each other, they don't just bounce; they interact with the ripples in space-time (gravitational waves) they create. It's like two people running past each other and grabbing a handful of the air between them.

• The Spin-Up: If a black hole is spinning the "wrong" way (counter-rotating) or not spinning at all, the encounter acts like a cosmic slingshot. It steals some of the orbital energy and dumps it into the black hole's own spin, making it spin faster.

  • Analogy: Imagine a figure skater spinning slowly. As they pass a spinning fan, the air currents grab their arms and pull them, making them spin much faster.
  • Result: The fastest spin-up happened when the black holes were spinning in the opposite direction of their orbit and the encounter was very close to the "crash" threshold. They could spin up by about 30%.

• The Spin-Down (The Twist): Here is the weird part. If a black hole was already spinning very fast in the "right" direction, it actually slowed down its spin rate after the encounter.

  • Wait, didn't it gain energy? Yes! But here is the catch: It also got heavier.
  • Analogy: Imagine a spinning top. If you suddenly glue a heavy weight to its side, it will spin slower even if you didn't touch the motor. The black hole gained so much mass (energy) that even though it absorbed more angular momentum, its "spin speed" (spin divided by mass squared) actually dropped.

3. The "Tidal Heating": Getting Fatter

Just like the spin, the black holes also gained mass. This is called "tidal heating."

• As the black holes swing past each other, they emit gravitational waves (ripples).
• Usually, these waves fly away into the universe. But in these close encounters, some of those waves get "sucked back in" by the black holes, like a vacuum cleaner sucking up dust.
• The Result: The black holes eat their own waste. They gain mass. In the most extreme cases, a black hole could grow by 15% in a single flyby.

4. Why Does This Matter?

You might ask, "Who cares about black holes swinging past each other?"

• The Universe is Full of Clusters: In dense star clusters (like crowded dance floors in space), black holes are constantly bumping into each other. They don't always merge; they often just swing by.
• The "Remnant" Effect: If a black hole survives a close call, it leaves the encounter spinning faster (or slower) and heavier than before. If it gets hit again later, it starts with a different "personality."
• Detecting the Invisible: Future telescopes (like the Einstein Telescope or Cosmic Explorer) will be able to hear these "flyby" sounds. By understanding how the spin and mass change, scientists can tell if they are listening to a merger or a near-miss.

Summary in One Sentence

When two black holes swing past each other at high speed, they can steal energy from their own orbit to spin faster and get heavier, but if they were already spinning too fast, the extra weight they gain actually makes them spin slower—a cosmic balancing act that changes their identity forever.

Technical Summary

1. Problem Statement

While the majority of gravitational wave (GW) research focuses on quasi-circular binary black hole (BH) coalescences, hyperbolic encounters (scattering events) are increasingly recognized as a significant astrophysical channel, particularly in dense stellar clusters. In these events, BHs are not initially gravitationally bound but may become so through GW emission (dynamical capture) or simply scatter.

Previous studies have established that scattering BHs can gain spin ("spin-up") and mass ("mass-gain") by re-absorbing orbital angular momentum and energy radiated as GWs. However, existing literature largely focused on:

• Non-spinning initial BHs.
• Ultra-relativistic regimes.
• Limited parameter spaces regarding initial spin magnitude and alignment.

This work addresses the gap by investigating the spin evolution and mass gain of equal-mass, initially spinning BHs across a wide range of initial spins ($\chi_i \in [-0.7, 0.7]$) and moderate initial momenta. The authors specifically aim to understand how the initial spin alignment (parallel vs. anti-parallel to orbital angular momentum) and incident angle influence the final state of the BHs, particularly near the threshold between scattering and merging.

2. Methodology

The authors performed a comprehensive suite of numerical relativity (NR) simulations using the Einstein Toolkit and the Canuda code.

• System Setup:

  • Binary Configuration: Equal-mass BHs ($m_1 = m_2 = 0.5M$, total mass $M=1$).
  • Initial Separation: $d = 100M$ along the x-axis.
  • Initial Spins: Aligned or anti-aligned with the orbital angular momentum (z-axis). The dimensionless spin $\chi_i$ ranges from $-0.7$ to $0.7$.
  • Initial Momenta: Two primary sets ($|P_i|/M = 0.245$ and $0.490$) and an extended set for $\chi_i=0.7$ ranging from $0.06125$ to $0.6125$.
  • Incident Angle ($\theta$): Varied to identify the threshold angle ($\theta_{th}$) separating scattering from merger (or zoom-whirl) outcomes.

• Numerical Framework:

  • Evolution: 3+1 formulation of Einstein's equations using the BSSN formalism with moving puncture gauge.
  • Resolution: Adaptive Mesh Refinement (AMR) with up to 7 levels; innermost resolution $dx \approx M/64$.
  • Convergence: Fourth-order finite differencing for spatial derivatives and Runge-Kutta time integration. Convergence tests were performed on representative cases (zoom-whirl, far-from-threshold scattering, near-threshold scattering) to quantify numerical errors.

• Observables:

  • Spin ($\chi$) and Angular Momentum ($S$): Extracted from the apparent horizon area ($A_H$) and equatorial circumference ($C_e$) using the Christodoulou mass formula.
  • Mass ($m$): Calculated via the Christodoulou formula, distinguishing between total mass and irreducible mass ($m_{irr}$).
  • GW Radiation: Computed via the Weyl scalar $\Psi_4$ projected onto spin-weighted spherical harmonics.
  • Orbital Angular Momentum ($J$): Computed via global conservation laws, accounting for radiated GW angular momentum ($J_{GW}$).

3. Key Contributions

1. Systematic Parameter Scan: The first extensive NR study of scattering BHs with non-zero initial spins across a broad range ($\chi \in [-0.7, 0.7]$) and moderate momenta, moving beyond the non-spinning or ultra-relativistic limits.
2. Discovery of Spin-Down: Identification of a counter-intuitive phenomenon where BHs with high initial aligned spins ($\chi_i = 0.7$) experience a spin-down (decrease in dimensionless spin $\chi$) despite an increase in total angular momentum. This is attributed to a significant increase in BH mass.
3. Threshold Angle Characterization: Detailed mapping of the threshold angle $\theta_{th}$ as a function of initial spin and momentum, showing a linear decrease with increasing spin and momentum.
4. Quantification of Mass Gain: A systematic analysis of "tidal heating" (mass gain) in scattering BHs, linking it to the second law of black hole thermodynamics and the interplay between irreducible mass and spin.

4. Key Results

A. Morphology and Thresholds

• Threshold Angle ($\theta_{th}$): The angle separating scattering from merger decreases linearly as initial spin increases (for aligned spins) and as initial momentum increases.
• Morphologies: Three distinct outcomes were observed based on $\theta$:
  1. Merger: Small angles.
  2. Zoom-Whirl: Intermediate angles (BHs perform multiple orbits before merging).
  3. Scattering: Large angles (BHs escape to infinity).

B. Spin Evolution (Spin-up vs. Spin-down)

• General Trend: For most initial spins (especially negative/anti-aligned), BHs spin up ($\chi_f > \chi_i$). The spin-up is maximized near the threshold angle $\theta_{th}$.
• Dependence on Initial Spin:
  • Anti-aligned ($\chi_i < 0$): Significant spin-up occurs as the magnitude of the negative spin decreases (becoming less negative).
  • Aligned ($\chi_i > 0$): Spin-up is generally observed for small positive spins.
  • High Aligned ($\chi_i = 0.7$): A spin-down is observed for certain incident angles and momenta.
• The Spin-Down Mechanism: For $\chi_i = 0.7$, the BH angular momentum ($S$) increases (or stays constant), but the dimensionless spin $\chi = S/m^2$ decreases. This is because the mass gain ($m$) is substantial enough to outweigh the angular momentum gain.
• Linear Relationship: At the threshold angle, the change in spin ($\Delta \chi$) decreases linearly with increasing initial spin $\chi_i$.
• Maximum Spin-up: The maximum observed spin-up is $\Delta \chi \approx 0.3$, occurring for $\chi_i = -0.7$ and $|P_i| = 0.490M$.

C. Mass Gain (Tidal Heating)

• Irreducible Mass: Always increases (consistent with the second law of black hole thermodynamics).
• Total Mass: Always increases due to the re-absorption of GW energy.
• Magnitude:
  • Mass gain is largest near the threshold angle, for large initial momenta, and for anti-aligned spins.
  • Maximum mass increase observed: ~15% (for $\chi_i = 0.7$ and high momentum).
  • Maximum irreducible mass increase: ~11% (for $\chi_i = -0.7$).
• Spin-Mass Interplay: In systems with negative initial spins, the spin magnitude decreases (spin-up), which reduces the rotational contribution to the total mass. Consequently, the relative increase in total mass is smaller than the increase in irreducible mass.

D. Spin-Up Efficiency

• Defined as the fraction of initial orbital angular momentum transferred to the BHs: $2(S_f - S_i) / J_i$.
• Maximum efficiency is < 5%.
• Efficiency increases with more negative initial spins and higher initial momenta.

5. Significance and Implications

• Astrophysical Relevance: These results are crucial for modeling BH populations in dense clusters (globular clusters, nuclear star clusters) where hyperbolic encounters are frequent. Repeated scattering events could significantly alter the spin distribution of BHs, potentially explaining the spin distribution of primordial BHs or remnants of previous mergers.
• Gravitational Wave Astronomy: Understanding the final spin and mass of scattered or captured BHs is vital for interpreting GW signals from eccentric mergers or dynamical captures, which are expected targets for third-generation detectors (Cosmic Explorer, Einstein Telescope) and space-based detectors (LISA).
• Theoretical Insight: The observation of "spin-down" due to mass gain highlights the complex thermodynamic relationship between a BH's spin, mass, and irreducible mass in dynamic, strong-field regimes. It demonstrates that an increase in angular momentum does not guarantee an increase in dimensionless spin if the mass gain is dominant.
• Future Directions: The authors suggest extending these studies to unequal-mass binaries, precessing spins, and exploring the ultra-relativistic limit where spin dependence may saturate.

In summary, this paper provides a rigorous numerical characterization of how spinning black holes evolve during hyperbolic encounters, revealing that initial spin alignment and incident angle are critical determinants of whether a BH spins up or down, and quantifying the significant mass gain (up to 15%) possible in these high-energy scattering events.