- The paper presents a systematic analytic computation of gravitational waveforms from radial infall, capturing both instantaneous and hereditary contributions up to 2.5PN order.
- It employs the Multipolar Post-Minkowskian formalism with a time-domain approach to derive explicit multipole moments and radiation-reaction effects, validated against numerical data.
- The study benchmarks energy and recoil fluxes while outlining extensions to 4.5PN corrections and hybrid analytic-numerical models for gravitational wave analysis.
Gravitational Waveforms from Radial Infall: Computation up to 2.5 Post-Newtonian Order
Introduction
The paper "Radial fall: the gravitational waveform up to the second-and-half Post-Newtonian order" (2604.01699) presents a comprehensive analytical treatment of gravitational wave (GW) emission from a two-body system undergoing head-on (radial) collision, computed within the Multipolar Post-Minkowskian (MPM) formalism up to 2.5 Post-Newtonian (PN) order. The focus is on the regime where the PN expansion remains valid, i.e., while the infalling bodies remain outside the strong field region near the black hole (BH) horizon. This scenario is critical for modeling GW signals from sources such as nearly-radially plunging binaries, and for benchmarking full numerical relativity approaches.
The authors exploit the structure of the MPM formalism, which enables a systematic decomposition of the GW metric perturbations at future null infinity into a sum of radiative multipole moments. At each PN order, both instantaneous and hereditary contributions (notably tails and non-linear memory effects) are accounted for.
The computation is anchored in a time-domain approach, with retarded time defined using a Bondi-type parameterization. All source and gauge multipole moments are explicitly expressed in terms of the two-body system's positions and velocities in the center-of-mass (CM) frame, adopting modified harmonic coordinates for the required PN accuracy. The main result is the TT, asymptotic waveform, specified up to 2.5PN—including corrections from radiation-reaction effects entering at this order.
The motion is initialized as a release from rest at infinity, so that during the early infall phase, the weak-field and slow-motion approximations underlying the PN expansion hold. The analysis is terminated near the horizon, before the breakdown of the PN scheme.
The GW waveform is given as a multipole expansion, including explicit forms for quadrupole (U2), octupole (U3), and higher moment contributions. All hereditary (e.g., tail and memory) effects relevant up to 2.5PN are included.
A salient feature at 2.5PN is the emergence of radiation-reaction, incorporated via the Blanchet-Damour-Deruelle prescription for the corresponding acceleration. Analytical expressions for the radiation-reaction modification to the radial trajectory are provided, with the influence of radiation damping quantified up to high PN order. The energy, angular momentum, and linear momentum fluxes are derived in both time and frequency domains:
- Energy Flux: The main analytic expression for the GW energy spectrum dE/dω matches established numerical and perturbative results in the weak-field regime and peaks at Mω in the range $0.26$–$0.30$, depending on the symmetric mass ratio ν. This reproduces, within analytic control, the location of the spectral maximum observed in numerical studies, e.g., [Mitsou:2010jv, Davis:1971gg].
- Angular Momentum Flux: For pure radial infall, all angular momentum emissions identically vanish, as a direct consequence of the system's axisymmetry.
- Linear Momentum Flux: While zero for the equal-mass case, a non-vanishing recoil (or “kick”) is computed for asymmetric mass ratios, beginning at higher PN order (η4).
The hereditary (nonlocal-in-time) contributions—in particular the non-linear memory entering at 2.5PN—are explicitly isolated and their analytic dependence provided.
Inertial Force Effects at 4.5PN and Center-of-Mass Corrections
The work discusses the onset, at 3.5PN and beyond, of non-inertial effects in the CM frame, necessary for physical self-consistency once radiative losses of linear momentum are included. Employing the formalism of [Blanchet:2026suq], the authors analyze the structure of the induced inertial forces at 4.5PN order, explicitly anticipating the structure of CM acceleration terms required for future 4PN-to-4.5PN accurate waveform computations.
Analytic/Numerical Comparison and Physical Interpretation
The analytic results agree well with existing numerical results in the weak-field domain. Specifically, the position of the peak in the ω-spectrum of the energy flux coincides with that observed in discrete numerical relativity simulations and perturbation-theory calculations based on the Zerilli or Teukolsky equations. Quantitative agreement is shown up to the regime where the PN expansion remains reliable.
A remarkable analytic result is the vanishing of angular momentum emission for head-on infall, as enforced by the symmetry of the dynamics, which provides an exact consistency check independent of PN order.
The present approach, while limited to the weak-field regime due to the breakdown of PN theory near horizon-crossing, enables a controlled evaluation of all waveform multipoles, energy fluxes, and hybridizes naturally with complementary approaches for the strong-field merger phase.
Implications and Perspectives
From a practical perspective, this work delivers PN-accurate templates for the GW emission from head-on or highly eccentric compact object mergers, applicable in the modeling of extreme-mass-ratio inspirals and in GW data analysis for events with low residual angular momentum. The analytic control up to 2.5PN allows for rigorous benchmarking of more general numerical relativity or self-force computations.
Theoretically, the proper PN characterization of hereditary effects (tails, memory) and inertial force corrections at high PN order paves the way for more refined analytic/numerical hybrid approaches and for cross-validation with amplitude-based methods in the high-energy regime [Brandhuber:2023hhy, Herderschee:2023fxh, Georgoudis:2023lgf]. The explicit identification of when the PN approximation breaks down underlines the need for systematic matching to strong-field results, which remains an open problem in analytic relativity.
Extensions to non-radial infall (finite impact parameter) and the inclusion of self-force effects are naturally implied follow-ups, as is the synthesis with effective one-body and amplitude-based frameworks for broader modeling of binary coalescence and gravitational recoil.
Conclusion
This paper delivers a systematic analytical computation of gravitational waveforms and energy/momentum emissions for a two-body system in head-on plunge, valid up to 2.5PN in the framework of the MPM formalism. The explicit inclusion of all hereditary and radiation-reaction effects, as well as readiness for higher-order (4.5PN) inertial force corrections, provides a robust foundation for further analytic and hybrid approaches. These results constitute a necessary reference for both comparison with numerical relativity and the analytic modeling of GW sources exhibiting near-radial encounters.
