Radiated Angular Momentum from Spinning Black Hole Scattering Trajectories

Using worldline quantum field theory and modern diagrammatic techniques, this paper derives spinning black hole scattering trajectories up to quadratic order in spins and establishes a new framework for computing radiated angular momentum up to 2PM order, thereby advancing high-precision analytical gravitational wave physics.

The Gist

The Big Picture: A Cosmic Dance of Black Holes

Imagine two massive, spinning figure skaters (our black holes) gliding across a frozen lake (space-time). They don't collide; instead, they glide past each other, close enough that their gravity pulls on one another, causing them to swerve and spin faster or slower. This is called a scattering event.

For decades, physicists have been trying to predict exactly how these skaters move and what happens to the energy they lose. When they swerve, they create ripples in the ice—gravitational waves. These waves carry away energy and, crucially, angular momentum (the "spin" or rotational energy of the system).

This paper is a major step forward in calculating exactly how much "spin" is lost to the universe during this dance, specifically when the black holes are spinning themselves.

The Problem: It's Too Complicated to Solve by Hand

Calculating the path of these black holes is like trying to predict the exact path of a leaf in a hurricane while the leaf is also spinning.

• The Old Way: Physicists usually calculate the final result (how much energy was lost) by looking at the "aftermath" (the gravitational waves far away). It's like trying to figure out how hard a car braked by only looking at the skid marks left on the road, without ever seeing the car itself.
• The Difficulty: When the black holes are spinning, the math gets incredibly messy. The "skid marks" (waves) are hard to calculate directly, and the "path" (trajectory) is even harder to solve because the spin interacts with the gravity in complex ways.

The New Tool: "Worldline Quantum Field Theory" (WQFT)

The authors of this paper used a new, powerful toolkit called Worldline Quantum Field Theory.

The Analogy: The LEGO Construction
Imagine you want to know exactly how a house will shake during an earthquake.

• Traditional Method: You build the house, shake it, and measure the cracks.
• The WQFT Method: Instead of building the whole house, you break the problem down into tiny, individual LEGO bricks (diagrams). You calculate how each brick interacts with its neighbors using rules from quantum physics (usually used for subatomic particles), and then you snap them all together to reconstruct the path of the black holes.

This method allows them to draw "Feynman diagrams" (blueprints of interactions) to solve the equations of motion directly, rather than guessing the answer and checking if it fits.

What They Actually Did

1. Mapping the Path (The Trajectory):
   They calculated the exact path the black holes take as they swing past each other.

   • Leading Order (The First Swerve): They figured out the path up to the point where the black holes' spins interact with each other twice (quadratic order). Think of this as mapping the first big curve in the dance.
   • Sub-Leading Order (The Second Swerve): They went deeper, calculating the next level of detail (2PM order). This is like mapping the tiny wobbles and corrections that happen after the main curve.

2. The "Mushroom" Diagrams:
   In their calculations, they had to deal with new types of mathematical shapes they call "mushroom diagrams." These represent complex loops of interaction that happen during the scattering, not just at the beginning or end. It's like realizing that while the skaters are turning, they are also creating tiny whirlpools in the ice that push back on them.

3. The Result: Radiated Angular Momentum:
   Once they had the full path, they could finally calculate how much "spin" was radiated away into the universe as gravitational waves.

   • The Finding: They confirmed that when spinning black holes scatter, they lose a specific amount of rotational energy. Their calculation matches previous results for simpler cases but now includes the complex effects of the black holes spinning.

Why This Matters

1. A New Shortcut:
Previously, to find out how much spin is lost, you had to calculate the entire gravitational wave signal first (which is like calculating the sound of the whole concert). This paper shows you can calculate the path of the dancers first, and the "lost spin" pops out naturally. It's a more direct, efficient route to the answer.

2. Future Proofing:
As our telescopes (like LIGO and the future Einstein Telescope) get better, they will hear fainter, more complex signals from black hole collisions. To understand these signals, we need "maps" that are incredibly precise. This paper provides a framework to calculate these maps to higher and higher levels of precision.

3. The "Spin" Factor:
Most previous calculations ignored the fact that black holes spin. This paper proves that we can handle the spin mathematically. Since real black holes do spin, this is essential for matching theory with the real universe.

The Takeaway

Think of this paper as the physicists finally solving the "perfect dance steps" for two spinning black holes. By using a clever new way of breaking the problem into tiny pieces (diagrams), they mapped out the dance floor with high precision. This allows us to predict exactly how much "spin" the universe steals from these cosmic dancers, helping us decode the gravitational waves we detect on Earth.

It's a bridge between the messy, complex reality of spinning black holes and the clean, precise math needed to understand them.

Technical Summary

1. Problem Statement

The paper addresses the challenge of computing radiated angular momentum ($J_{\text{rad}}$) in the gravitational scattering of two spinning black holes. While significant progress has been made in calculating conservative dynamics (scattering angles, momentum impulses) and radiated energy-momentum up to high Post-Minkowskian (PM) orders (up to 5PM), the calculation of radiated angular momentum remains intricate.

Current methods for obtaining $J_{\text{rad}}$ typically rely on:

• Extracting it from the far-field gravitational waveform.
• Using linear response relations or eikonal approximations.
• Integrating equations of motion directly (limited to 2PM).

A critical gap exists: explicit, time-dependent scattering trajectories for spinning bodies have not been fully derived using modern Quantum Field Theory (QFT) techniques beyond the leading order. Furthermore, existing QFT methods usually focus on asymptotic observables (momentum kicks) rather than the full trajectory required to compute angular momentum directly without intermediate waveform reconstruction.

2. Methodology

The authors employ the Worldline Quantum Field Theory (WQFT) approach. This framework treats classical black holes as point particles moving along worldlines, coupled to a bulk gravitational field.

• Action and Spin Formalism: They utilize a worldline action where spins are modeled via bosonic oscillators ($\alpha^\mu, \bar{\alpha}^\mu$) rather than supersymmetric formalisms. This allows for a description valid up to quadratic order in spins without dealing with anti-commuting variables. The spin tensor is defined as $S^{\mu\nu} = -2m i \bar{\alpha}^{[\mu} \alpha^{\nu]}$.
• Perturbative Expansion: The equations of motion are solved perturbatively in Newton's constant $G$ (PM expansion).
  • 1PM (Leading Order): Solutions are obtained in the time domain up to quadratic order in spins.
  • 2PM (Sub-leading Order): Solutions are derived in the frequency (momentum) domain up to linear order in spins.
• Diagrammatic Techniques: The trajectories are generated by summing tree-level Feynman diagrams. Unlike standard impulse calculations where the outgoing energy is set to zero, the authors keep the energy $\omega$ on the outgoing worldline line finite to capture the full time-dependent trajectory.
• Integration Techniques: The calculation utilizes modern collider physics tools:
  • Integration-by-Parts (IBP) reductions.
  • Differential Equations (DEs) for master integrals.
  • The Method of Regions to handle boundary conditions and divergences.
• Novel Integral Families: The authors encounter a new family of one-loop Feynman integrals (involving two scales: momentum transfer $|q|$ and energy $q \cdot v$) that differ from those in standard asymptotic PM calculations. These integrals are a subset of those appearing in the $O(G^3)$ gravitational waveform.

3. Key Contributions

A. Derivation of Spinning Trajectories

The paper provides the first explicit derivation of scattering trajectories for spinning black holes using WQFT:

• 1PM Trajectory: Explicit time-domain solutions for the position $z^\mu(\tau)$ and spin vector $a^\mu(\tau)$ up to quadratic order in spins.
• 2PM Trajectory: Momentum-space solutions ($z^\mu(q)$) up to linear order in spins. These are presented in terms of gauge-invariant mass sectors ($m_1 m_2$ and $m_2^2$).

B. New Mechanism for Radiated Angular Momentum

The authors establish a direct computational pathway to $J_{\text{rad}}$ using the derived trajectories, bypassing the need to first compute the full gravitational waveform.

• They define the total angular momentum $J^\mu$ in the Center-of-Mass (COM) frame, accounting for the boost required to align the individual spin vectors with the COM frame.
• The change in angular momentum $\Delta J^\mu$ is computed by evaluating the asymptotic limits ($\tau \to \pm \infty$) of the orbital and spin contributions derived from the trajectories.

C. Resolution of Divergences and Checks

• Logarithmic Divergences: The authors address the intrinsic logarithmic divergence arising from fixing boundary conditions at $\tau = -\infty$. They demonstrate that subtracting the result at a finite reference time yields finite, physical trajectories.
• On-Shell Checks: They verify their results by confirming the on-shell condition ($g_{\mu\nu}\dot{x}^\mu \dot{x}^\nu = 1$) and the conservation of the spin-supplementary condition (SSC) and spin magnitude.
• Consistency: The 1PM and 2PM trajectories successfully reproduce known momentum impulses ($\Delta p^\mu$) and spin kicks in the asymptotic limit.

4. Results

Radiated Angular Momentum at 2PM

The primary result is the calculation of the radiated angular momentum at order $G^2$ (2PM) up to linear order in spins.

• Non-Spinning Case: The result reproduces the known literature value, showing that radiation starts at $G^2$ (while momentum radiation starts at $G^3$). The result depends on the boost factor $\gamma$ and the impact parameter $b$.
• Spinning Case (Linear Spin): The radiated angular momentum $\Delta J^\mu_{S1}$ is found to have components along the impact parameter direction ($\hat{b}$) and the orbital angular momentum direction ($\hat{l}$).
  • For aligned spins ($a_i \parallel \hat{l}$), the result simplifies to a term proportional to $(s_1 + s_2)\hat{l}^\mu$.
  • The expression involves terms with $\text{arccosh}(\gamma)$, which typically appear only at higher loops in other formalisms, highlighting the efficiency of the WQFT approach.
• Quadratic Spin: While the full quadratic spin trajectory is derived, the explicit 2PM radiated angular momentum for quadratic spins is not fully expanded in the text but is noted to be available in the supplementary repository.

The final formula for the aligned spin case (Eq. 69) is:
$$\Delta J^{(2)\mu}_{S1} = \frac{8\gamma m_1 m_2 (s_1 + s_2)}{3(\gamma v)^2 |b|^2} \left[ (8 - 5\gamma^2)\gamma v + \gamma(6\gamma^2 - 9)\text{arccosh}(\gamma) \right] \hat{l}^\mu$$

5. Significance and Impact

1. New Computational Paradigm: The paper demonstrates that calculating full scattering trajectories is a viable and efficient alternative to waveform reconstruction for determining radiated angular momentum. This is crucial for higher-order PM calculations where waveform complexity grows exponentially.
2. High-Precision Frontier: By establishing a framework for 2PM spinning trajectories, the authors pave the way for extending radiative calculations to 3PM and beyond, which is essential for the next generation of gravitational wave detectors (e.g., Einstein Telescope, LISA).
3. Technical Advancement: The identification and solution of a new family of multi-scale one-loop integrals contributes to the mathematical toolkit of classical gravity, linking trajectory calculations directly to the integrals appearing in the $O(G^3)$ waveform.
4. Effective One-Body (EOB) Models: The results provide high-precision inputs for calibrating EOB models, which are used to generate waveform templates for data analysis in gravitational wave astronomy.

In summary, this work bridges the gap between perturbative QFT techniques and the classical dynamics of spinning black holes, providing a robust, diagrammatic method to compute radiated angular momentum that complements and extends existing asymptotic methods.