⚛️ high-energy theory

All Order Gravitational Wave Tails for Subleading Post-Newtonian Expansion

This paper extends the universality of the infinite tower of gravitational wave tail terms from the Newtonian limit to the first subleading post-Newtonian order for nonrelativistic gravitational scattering, demonstrating that these corrections are fully determined by asymptotic trajectories and confirming a previous relativistic conjecture.

Original authors: Debanjan Karan, Babli Khatun

Published 2026-10-16

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Debanjan Karan, Babli Khatun

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

When massive objects like stars or black holes move through space, they do not merely travel; they ripple the very fabric of the universe. This fabric, known as spacetime, stretches and squeezes in response to the motion of matter, sending out waves of gravity that travel at the speed of light. For decades, scientists have sought to predict exactly what these waves look like when two massive bodies scatter off one another, a process that happens when they pass close enough to feel each other's pull but not close enough to collide. The challenge is that the interaction is incredibly complex. The path a body takes depends on its speed, its mass, and the subtle, lingering influence of the gravitational force itself, which never truly switches off, even as the objects move far apart.

In recent years, a specific pattern has emerged in the mathematical description of these waves. When the waves are observed long after the scattering event, they do not simply fade away into silence. Instead, they leave behind a faint, lingering echo known as a "tail." This tail is not a reflection off a physical object, but a consequence of the long-range nature of gravity itself. The force of gravity travels at the speed of light, but because it is so weak and extends infinitely, parts of the wave interact with the curvature of spacetime created by the moving objects, causing a delayed signal to arrive later than the main burst. Scientists have recently discovered that the strength of these tails follows a universal rule: it depends only on the mass and speed of the objects as they enter and leave the interaction, not on the messy details of how they swerved or accelerated during the encounter. This universality suggests that the universe has a simple, underlying order that persists even in chaotic collisions.

A team of researchers at the International Centre for Theoretical Sciences in Bangalore has now pushed this understanding further. They asked whether this simple rule holds true when the objects are moving slightly faster, or when the calculation is refined to include the next level of detail in the physics of motion. In the language of physics, they moved from the "Newtonian" limit, where speeds are very slow, to a "subleading" order, which accounts for the first corrections that appear as objects gain a bit more speed. Their goal was to see if the universal nature of the gravitational tail survives this refinement, or if the extra complexity of the interaction introduces new, unpredictable variables.

The researchers focused on a scenario where any number of massive objects approach each other, interact through gravity, and then fly apart. They did not assume the objects were moving at speeds close to light, but rather that they were moving slowly enough that their speeds could be treated as a small fraction of the speed of light. By carefully expanding the equations that describe the motion of these objects and the gravitational waves they emit, they derived a complete formula for the tail of the wave at this new level of precision. They found that the tail is still determined entirely by the starting and ending states of the objects. The correction to the previous, simpler formula arises from a specific effect related to the time it takes for the gravitational influence to travel from one object to another, a delay that becomes significant when the objects are moving.

Crucially, the team showed that even with this added complexity, the details of the interaction during the collision do not matter. The strength of the tail depends only on the masses of the objects and their velocities as they arrive and depart. The researchers identified that the new terms in the formula are linked to the distribution of mass in a more complex shape than a simple sphere, and to how the mass is distributed relative to the rotation of the system. Despite this added geometric complexity, the result remains universal. The coefficients that determine the size of the tail are fixed by the asymptotic data—the final, free-moving state of the particles—just as they were in the simpler, slower case.

To verify their findings, the researchers applied their new formula to a specific, common case: the scattering of two particles. In this scenario, the infinite series of terms that make up the tail can be summed up into a single, closed mathematical expression. This allowed them to compare their result directly with a bold, all-encompassing hypothesis proposed in earlier work, which suggested that a single formula could describe these tails for objects moving at any speed, from very slow to nearly the speed of light. The researchers found that their new, more precise calculation for slow-moving objects matched the prediction of this all-speed hypothesis perfectly. This agreement serves as a strong confirmation that the hypothesis is on the right track, even though it has not yet been proven for all possible speeds.

The work also clarifies what is and is not included in this picture. The researchers noted that their calculation accounts for the matter of the objects themselves but does not include a specific type of memory effect caused by the energy carried away by the gravitational waves themselves. This "null memory" is a separate phenomenon that would need to be added on top of their results if it were significant at this level of precision. However, for the specific order of speed they studied, the contribution from the matter trajectories is the dominant factor. They also confirmed that the intrinsic spin, or the internal rotation, of the particles does not affect the tail at this level of calculation, as its influence appears only at much higher orders of speed.

By extending the universal description of gravitational tails to the first subleading order, this research strengthens the bridge between the simple, slow-motion world of Newton and the complex, relativistic world of Einstein. It demonstrates that the universe's tendency to simplify the description of gravitational waves into a few key parameters is robust. Even as the objects move faster and the equations become more intricate, the fundamental rule remains: the echo of a gravitational scattering event is written in the final state of the participants, independent of the chaotic dance they performed in between. This finding provides a deeper layer of confidence in the theoretical tools used to interpret the signals detected by observatories on Earth, ensuring that the stories we tell about colliding black holes and neutron stars are built on a foundation that holds up under closer scrutiny.

Technical Summary: All Order Gravitational Wave Tails for Subleading Post-Newtonian Expansion

Problem Statement
The classical soft graviton theorem establishes that the low-frequency gravitational waveform emitted during scattering processes is universal, determined solely by the asymptotic masses and momenta of the scattering objects, independent of the interaction details. Recent work [1] demonstrated that in the Newtonian limit (leading order in the velocity expansion), the entire infinite tower of logarithmic tail terms—arising from the long-range nature of gravity—can be expressed in closed form. These terms take the form $u^{-r}(\ln|u|)^{r-1}$ in the time domain (or $\omega^{r-1}(\ln \omega)^r$ in the frequency domain).

However, a gap remained regarding whether this universality and the closed-form summation extend beyond the leading Newtonian order. Specifically, it was unclear if the leading-logarithmic contributions at the first subleading order in the Post-Newtonian (PN) expansion (order $\alpha^3$ , where $\alpha$ parametrizes the small velocity) would require new dynamical information about the scattering encounter or if they remained fixed entirely by asymptotic data. Furthermore, while a relativistic conjecture for all-order tails existed [1], it had not been explicitly verified at this subleading PN order.

Methodology
The authors analyze the scattering of an arbitrary number of non-relativistic massive objects interacting solely via long-range gravity. The calculation is performed within a fixed Newton's constant $G$ framework, retaining the full tower of tail terms to determine the leading-logarithmic waveform to all orders in $G$ at the specific velocity order considered.

Expansion Scheme: The asymptotic 3-velocities are parametrized as $\alpha \vec{v}_a$ , where $\alpha$ is a small expansion parameter. The leading Newtonian results correspond to $O(\alpha^2)$ , while this paper targets the $O(\alpha^3)$ correction.
Source Derivation: The waveform is derived from the retarded solution to the linearized Einstein equations. The correction at $O(\alpha^3)$ arises primarily from the first-order expansion of the retarded-time argument ( $u + \vec{n} \cdot \vec{x}$ ) in the source integral.
Multipole Decomposition: The stress-current moment $S^{kij}$ is decomposed into a fully symmetric part (related to the mass octupole moment, $I^{kij}$ ) and a part related to the spatial moment of the angular-momentum density ( $J^{ki|j}$ ). Using energy-momentum conservation, the waveform correction is expressed in terms of these moments.
Trajectory Ansatz: The asymptotic trajectories of the particles are expanded in powers of the rescaled retarded time $w = \alpha^3 u$ . The coefficients of the logarithmic terms in the trajectory ( $\vec{c}_a, \vec{d}_{r,a}$ ) are determined by solving the Newtonian force equation iteratively.
Verification: The derived PN waveform is compared against the all-orders relativistic conjecture proposed in [1]. The conjecture's expression is expanded in the small-velocity limit to isolate the $O(\alpha^3)$ terms, and the matter and gravitational contributions are matched against the direct PN calculation.

Key Contributions and Results

Universality at Subleading Order: The paper derives the leading-logarithmic tail terms at the first subleading PN order ( $O(\alpha^3)$ ) and finds they are universal. Their coefficients are determined entirely by the asymptotic Newtonian trajectories (masses and velocities) of the scattering objects. No detailed knowledge of the interaction during the collision is required.
New Physical Dependencies: Unlike the leading Newtonian order ( $O(\alpha^2)$ ), which depends only on the mass quadrupole, the $O(\alpha^3)$ correction depends on the mass octupole moment and the angular-momentum density.
Closed-Form Summation for 2→2 Scattering: For the specific case of two incoming and two outgoing particles, the infinite series of logarithmic tail terms is summed explicitly into a closed-form geometric series. The resulting late-time waveform is:
$e^{(1)}_{ij}(w) = \frac{2G\alpha^3}{R} \sum_{a=1}^2 m_a (\vec{n} \cdot \vec{v}_a) v^i_a v^j_a + \frac{2G m_1 m_2 \alpha^3}{R(m_1+m_2)} \sum_{r=1}^\infty \frac{(-1)^{r-1} \gamma^r (\ln w)^{r-1}}{w^r} \left[ \dots \right]$
where $\gamma = G(m_1+m_2)/v^3$ .
Mass Difference Dependence: In the center-of-mass frame, the waveform at this order is found to be proportional to the mass difference $\delta m = m_2 - m_1$ . Consequently, the waveform vanishes for equal-mass scattering at this specific order due to reflection symmetry.
Validation of Relativistic Conjecture: The direct PN derivation matches the expansion of the all-orders relativistic conjecture from [1]. This agreement tests the conjecture at an additional velocity order, confirming that the conjecture correctly captures the subleading logarithmic structure, though it does not constitute a proof at general velocities.

Significance
The paper establishes that the universality of gravitational wave tails, previously known only in the Newtonian limit, persists at the first subleading Post-Newtonian order. It demonstrates that even when probing properties beyond the mass quadrupole (specifically the octupole and angular momentum density), the leading-logarithmic waveform remains fixed by asymptotic data alone. This result strengthens the connection between classical scattering waveforms and soft theorems, providing a rigorous test of the all-orders relativistic conjecture proposed in earlier work. The ability to sum the tail series in closed form for 2→2 scattering offers a precise analytical tool for comparing with future high-precision gravitational wave observations or numerical relativity simulations in the appropriate limits.

Limitations and Scope
The authors note that the calculation is restricted to the radiative order ( $1/R$ ) and assumes non-relativistic asymptotic states. The intrinsic spin of the particles is neglected, as its contribution enters at a higher PN order. Additionally, the "null memory" contribution (radiative memory) is not included; the results are complete only if this contribution is negligible at $O(\alpha^3)$ or added separately. The error terms arising from approximations in the source and trajectory are analyzed in the appendices and shown to be subleading to the order considered.

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