[Paper Review] An eikonal-inspired approach to the gravitational scattering waveform

Alessandro Georgoudis, Carlo Heissenberg+1 more|arXiv (Cornell University)|Dec 12, 2023

Pulsars and Gravitational Waves Research4 citations

TL;DR

This paper resolves a long-standing discrepancy between amplitude-based post-Minkowskian (PM) calculations and post-Newtonian (PN) waveforms for scalar black hole scattering by showing that the missing contribution in prior comparisons arises from a frame transformation, not an error in the amplitude. Using an eikonal-inspired approach, the authors demonstrate that the two frameworks agree up to $\mathcal{O}(G^3/c^5)$ when the correct asymptotic frame—accounting for BMS supertranslations—is used, particularly in the soft limit $\omega \to 0$, where a non-universal $\omega\log\omega$ term appears.

ABSTRACT

We revisit the amplitude-based derivation of gravitational waveform for the scattering of two scalar black holes at subleading post-Minkowskian (PM) order. We take an eikonal-inspired approach to the two-massive-particle cut needed in the KMOC framework, as highlighted in arXiv:2308.02125, and show that its effect is to implement a simple change of frame. This clarifies one of the points raised in arXiv:2309.14925 when comparing with the post-Newtonian (PN) results. We then provide an explicit PM expression for the waveform in the soft limit, $ω o0$, including the first non-universal, $ω\logω$, contribution. Focusing on this regime, we show that the small-velocity limit of our result agrees with the soft limit of the PN waveform of arXiv:2309.14925, provided that the two quantities are written in the same asymptotic frame. Performing the BMS supertranslation that, as discussed in arXiv:2201.11607, is responsible for the $\mathcal O(G)$ static contribution to the asymptotic field employed in the PN literature, we find agreement between the amplitude-based and the PN soft waveform up to and including $G^3/c^5$ order.

Motivation & Objective

• To resolve the mismatch between amplitude-based subleading post-Minkowskian (PM) waveforms and post-Newtonian (PN) results for scalar black hole scattering.
• To clarify the role of the two-massive-particle cut in the KMOC framework, showing it corresponds to a change of reference frame.
• To derive an explicit PM expression for the gravitational waveform in the soft limit $\omega \to 0$, including the $\omega\log\omega$ term.
• To establish agreement between amplitude-based and PN waveforms by accounting for BMS supertranslations that affect the asymptotic field structure.
• To demonstrate that the small-velocity limit of the amplitude result matches the soft limit of the PN waveform only when both are expressed in the same asymptotic frame.

Proposed method

• Adopt an eikonal-inspired approach to the two-massive-particle cut in the KMOC framework, interpreting it as a frame transformation rather than a physical correction.
• Compute the subleading PM waveform at one-loop order using the 5-point amplitude, focusing on the soft limit $\omega \to 0$.
• Identify the $\omega\log\omega$ term as the first non-universal contribution in the soft expansion, arising from the classical limit of the 1-loop amplitude.
• Apply the BMS supertranslation transformation, as discussed in Veneziano:2022zwh, to map the amplitude-based field to the asymptotic frame used in the PN literature.
• Perform a systematic expansion of the waveform in the small-velocity limit and compare with the MPM (Multipolar-post-Minkowskian) PN result from Bini:2023fiz.
• Use the eikonal exponentiation formalism to interpret the classical limit of the amplitude, ensuring consistency with Lorentz invariance and frame dependence.

Experimental results

Research questions

• RQ1 Why did previous comparisons between amplitude-based PM waveforms and PN results show discrepancies, particularly at subleading PM order?
• RQ2 What is the physical interpretation of the two-massive-particle cut in the KMOC framework, and how does it affect the asymptotic waveform?
• RQ3 Does the inclusion of the BMS supertranslation, which modifies the asymptotic field, restore agreement between amplitude-based and PN waveforms?
• RQ4 How does the soft limit of the PM waveform, including the $\omega\log\omega$ term, compare with the PN result in the same reference frame?
• RQ5 Can the frame dependence of the waveform be systematically accounted for in the amplitude approach to ensure consistency with PN calculations?

Key findings

• The two-massive-particle cut in the KMOC framework corresponds to a change of reference frame, not a physical correction, resolving confusion in prior comparisons.
• The soft limit of the subleading PM waveform contains a non-universal $\omega\log\omega$ term, which is a new quantitative feature of the amplitude-based approach.
• When both the amplitude-based and PN waveforms are expressed in the same asymptotic frame—including the $\mathcal{O}(G)$ static contribution via BMS supertranslation—agreement is achieved up to $\mathcal{O}(G^3/c^5)$.
• The small-velocity limit of the amplitude result matches the soft limit of the PN waveform only after accounting for the supertranslation, which affects both static and time-dependent parts of the field.
• The smooth PN limit emerges only after nontrivial cancellations between rational and logarithmic terms in the amplitude, highlighting the need for careful reorganization of the 1-loop result.
• The observed agreement suggests that the frame dependence of the waveform is crucial and that the supertranslation structure may be a general feature in matching amplitude and PN results across different observables.