⚛️ general relativity

Radiative losses from unbound orbits at quadratic order in spin from MPM formalism

This paper utilizes the MPM formalism to compute all radiative losses for spinning two-body systems in hyperbolic-like orbits up to 2PN accuracy including spin-squared corrections, generalizing previous linear-spin results and demonstrating that deviations from radial fall inevitably emerge as post-Newtonian accuracy increases.

Original authors: Donato Bini, Giorgio Di Russo

Published 2026-09-18

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Donato Bini, Giorgio Di Russo

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

Gravity is the invisible architect of the cosmos, shaping the paths of stars, planets, and black holes. For decades, scientists have relied on a set of rules called the post-Newtonian approximation to predict how these massive objects move when they are far apart and moving slowly. This framework works beautifully for many situations, but it begins to fray when objects spin rapidly or when they are locked in a violent, high-speed encounter. To understand the universe fully, particularly the ripples in spacetime known as gravitational waves, researchers must refine these rules to account for the complex ways spinning bodies influence one another. The challenge is immense: calculating the energy and momentum lost as these objects fly past each other requires tracking not just their mass and speed, but also how their internal rotation twists the fabric of space around them.

A team of researchers has taken a significant step forward in this complex calculation by focusing on two spinning objects, such as black holes or neutron stars, that are on a hyperbolic trajectory. Imagine two massive bodies approaching each other from deep space, swinging around one another in a wide arc, and then flying apart again, never to meet. This is a scattering event, distinct from the slow spiral of two objects merging. The scientists wanted to know exactly how much energy, angular momentum, and linear momentum these spinning bodies lose to gravitational radiation during such a flyby. They focused specifically on a simplified scenario where the spins of both objects are aligned with the axis of their orbit, a configuration that makes the math more tractable while still revealing deep physical truths.

Using a sophisticated mathematical tool known as the Multipolar-Post-Minkowskian formalism, the team computed the radiative losses for these encounters up to a high level of precision. This method allows them to break down the complex gravitational field into a series of terms, calculating the effects of the objects' spins on the radiation they emit. Their work goes beyond previous studies that only considered the effects of spin in a straight line. Here, they calculated the corrections that arise when the spin interacts with itself, a quadratic effect that had not been fully mapped out for these types of scattering orbits before. They found that the spin of the objects significantly alters the amount of energy and momentum carried away by gravitational waves, with the effects depending on the square of the spin magnitude.

The researchers also turned their attention to a different, more extreme scenario: a radial infall, where one object drops straight toward another without any sideways motion. In a world without spin, such an object would fall perfectly straight down. However, the team discovered that once the internal rotation of the bodies is taken into account, even a tiny bit, the motion can no longer remain perfectly radial as the precision of the calculation increases. The spin introduces a subtle push that forces the object off its straight path. This finding is crucial because it shows that the idealized picture of a straight drop is an illusion that breaks down as soon as one looks closely enough at the physics of spinning bodies.

To ensure their results were solid, the team cross-checked their findings against existing theories and known limits. They confirmed that their new calculations for the leading effects matched previous work, giving them confidence in the more complex, higher-order results they had derived. These new results provide a detailed map of how spinning objects lose energy and momentum during high-speed encounters. This map is essential for the next generation of gravitational wave detectors, which will need to compare their observations against highly accurate theoretical templates to identify the properties of the cosmic collisions they detect. By clarifying how spin shapes the radiation emitted during these flybys and falls, the study helps refine the tools scientists use to listen to the universe.

Technical Summary: Radiative losses from unbound orbits at quadratic order in spin from MPM formalism

Problem Statement
This study investigates the radiative losses (energy, angular momentum, and linear momentum) emitted by a spinning two-body system undergoing hyperbolic-like (unbound) scattering motions. The analysis focuses specifically on the "aligned spin" case, where the spins of the bodies are perpendicular to the orbital plane and aligned with the system's orbital angular momentum. While extensive theoretical results exist for non-spinning hyperbolic encounters and for spinning systems at linear order in spin, there is a lack of analytical results for radiative losses at quadratic order in spin ( $O(S^2)$ ) within the Post-Newtonian (PN) framework. Additionally, the paper addresses the specific case of radial infall in the presence of aligned spins, a scenario where linear spin corrections vanish, necessitating a study of quadratic effects.

Methodology
The authors employ the Multipolar-Post-Minkowskian (MPM) formalism to compute the radiative losses. The methodology involves the following steps:

Dynamics: The equations of motion for two spinning bodies are derived in center-of-mass harmonic coordinates, truncated at the 2PN absolute accuracy level. This includes Newtonian, 1PN, and 2PN conservative terms, as well as spin-orbit (SO) and spin-spin (SS) contributions. The study explicitly excludes hereditary (tail) effects and radiation-reaction forces in the conservative dynamics, focusing on the conservative trajectory to compute the fluxes.
Orbital Representation: For hyperbolic-like orbits, the authors utilize a quasi-Keplerian representation. They derive the orbital parameters (semi-major axis, eccentricity, etc.) as expansions in the spin variables, including terms up to $O(S^2)$ .
Multipole Moments: The source multipole moments (mass and current multipoles) are constructed up to spin-squared order. These are expressed in terms of the spin variables $S$ (sum of spins) and $S^*$ (a mass-weighted combination).
Flux Calculation: Using the MPM formalism, the radiative multipoles are related to the source multipoles. The energy, angular momentum, and linear momentum fluxes are calculated using standard formulas involving time derivatives of these multipoles. The integration is performed along the spin-corrected hyperbolic trajectories.
Radial Infall: A specific limit is analyzed where the motion is purely radial. In this configuration, the authors show that linear-in-spin corrections to the orbit cancel out, making the $O(S^2)$ terms the leading spin-dependent corrections. However, they demonstrate that as PN accuracy increases beyond the current level, deviations from purely radial motion necessarily appear.

Key Contributions and Results

Quadratic Spin Corrections to Hyperbolic Scattering: The primary contribution is the derivation of exact expressions for radiative losses at $O(S^2)$ $O (S^{2})$ for hyperbolic-like orbits. This includes:
- Energy Loss: The authors provide the exact integrated energy loss $\Delta E_{rad}$ and its expansion in the large impact parameter ( $b$ ) limit. The results are separated into terms proportional to $(S^2 + S^{*2})$ and $S S^*$ .
- Angular Momentum Loss: Exact expressions for the radiated angular momentum $\Delta J_{rad}$ are derived, including the $O(S^2)$ contributions in the large $b$ expansion.
- Linear Momentum Loss: The paper computes the radiated linear momentum kick, specifically the $y$ -component (perpendicular to the scattering plane), finding non-vanishing $O(S^2)$ contributions.
Scattering Angle: The study derives the scattering angle corrections at $O(S)$ and $O(S^2)$ , confirming consistency with existing literature at linear order and providing new results at quadratic order.
Radial Infall Analysis: The authors demonstrate that for radial infall with aligned spins, the linear-in-spin corrections to the orbit cancel out, leaving $O(S^2)$ as the leading spin-dependent correction. Consequently, the motion remains purely radial only at the current PN accuracy level; as PN accuracy increases, deviations from radial fall appear necessarily. They compute the $O(S^2)$ corrections to the energy flux, angular momentum flux (which becomes non-zero along the z-axis), and linear momentum flux for this specific trajectory.
Validation: The leading Post-Minkowskian (PM) order results are cross-checked against existing literature derived from Effective Field Theory (EFT) and scattering amplitudes. The $O(S^2)$ terms in the large impact parameter expansion are shown to agree with the leading PM results found in Ref. [36], validating the new higher-order PN results.

Significance and Claims
The paper claims that its results represent the first computation of all radiative losses (energy, angular, and linear momentum) for spinning unbound orbits at quadratic order in spin using the MPM formalism. The authors emphasize that while leading PM order results are consistent with known literature, the higher-order PM results (within the 2PN accuracy) presented here are novel.

The study serves as a benchmark for future theoretical developments. The authors note that their results provide "valuable materials to support future accomplishments," particularly for cross-checking similar expressions obtained via different formalisms, such as scattering amplitudes. Furthermore, the analysis of the radial infall case highlights a fundamental limitation of the PN approximation in the presence of spin: as PN accuracy increases, the assumption of purely radial motion becomes inconsistent, necessitating the inclusion of precession and non-radial deviations.

The work is modestly framed as a step toward more complete models; the authors explicitly state that hereditary effects and NLO spin-orbit corrections (which would introduce radiation-reaction and precession) are topics for future studies. The current results are limited to the 2PN accuracy and the aligned spin configuration.

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