Scattering Amplitudes and Conservative Binary Dynamics at ${\cal O}(G^4)$ (2101.07254v2)

Abstract: Using scattering amplitudes, we obtain the potential contributions to conservative binary dynamics in general relativity at fourth post-Minkowskian order, ${\cal O}(G^4)$. As in previous lower-order calculations, we harness powerful tools from the modern scattering amplitudes program including generalized unitarity, the double copy, and advanced multiloop integration methods, in combination with effective field theory. The classical amplitude involves polylogarithms with up to transcendental weight two and elliptic integrals. We derive the radial action directly from the amplitude, and determine the corresponding Hamiltonian in isotropic gauge. Our results are in agreement with known overlapping terms up to sixth post-Newtonian order, and with the probe limit. We also determine the post-Minkowskian energy loss from radiation emission at ${\cal O}(G^3)$ via its relation to the tail effect.

• The paper demonstrates that scattering amplitude methods can compute conservative binary dynamics at the fourth post-Minkowskian order.
• It employs advanced techniques such as generalized unitarity, double copy, and effective field theory to tackle complex multiloop integrations.
• The results validate the approach by matching known physics up to the sixth post-Newtonian order and indicate promising directions for gravitational-wave research.

Scattering Amplitudes and Conservative Binary Dynamics at $\mathcal{O}(G^4)$

The paper of gravitational-wave science has advanced significantly, necessitating the development of sophisticated theoretical frameworks to interpret the data obtained from gravitational-wave detectors. The utilization of scattering amplitudes in the context of general relativity has proven to be a particularly effective approach. The paper "Scattering Amplitudes and Conservative Binary Dynamics at $\mathcal{O}(G^4)$" addresses the application of scattering amplitude techniques for calculating conservative binary dynamics up to the fourth post-Minkowskian (PM) order.

The authors of the paper extend methodologies based on modern scattering amplitude techniques, such as generalized unitarity and the double copy, combined with effective field theory (EFT), to derive the contributions of the potential binary dynamics at the fourth post-Minkowskian order, $\mathcal{O}(G^4)$, in general relativity. This work follows previous efforts at lower orders and offers insights into how classical binary dynamics can be derived from the quantum framework of scattering amplitudes.

Technical Highlights

This paper harnesses advanced techniques in theoretical physics, such as:

1. Generalized Unitarity and Double Copy: These methods allow one to connect scattering amplitudes in gauge theories to corresponding processes in gravity, facilitating the calculation of gravitational corrections using well-developed gauge theory techniques.
2. Post-Minkowskian (PM) Expansion: This non-perturbative approach doesn't rely on the small velocity assumption inherent in post-Newtonian expansions, thus allowing the paper of relativistic effects in gravitational dynamics over longer ranges.
3. Advanced Multiloop Integration Methods: The paper details the complex multiloop integrals involved in computing the amplitudes necessary for extracting the binary dynamics, integrating these using new techniques in differential equations and asymptotic expansions.
4. Amplitudes-Based Classical Dynamics: By expressing classical gravitation interactions as quantum scattering amplitude processes, the work presents a way to evaluate dynamics traditionally tackled with astrophysical and classical mechanics methods.

Key Results and Implications

• Agreement with Known Physics: The calculated dynamics at $\mathcal{O}(G^4)$ align with known results up to the sixth post-Newtonian order in their overlapping domain and are consistent with the probe limit, confirming the validity of their approach.
• Potential Contributions and Energy Loss: The work extends to consider the energy loss from radiation emission at $\mathcal{O}(G^3)$, linked to the tail effect in gravitational scattering, which relates to the integration of radiation backscatter into the calculations.
• Elliptic Integrals and Complexity: At $\mathcal{O}(G^4)$, the inclusion of elliptic integrals reflects increased complexity and the rich structure of gravitational interactions at higher orders, suggesting complex dynamics that necessitate advanced mathematical tools.

Future Directions

The implications of these results are significant for the future of gravitational wave astronomy and theoretical gravitational physics, predicting new aspects of gravitational interactions observable in strong field regimes. Future developments could include:

• Full integration of radiation effects with potential dynamics for a complete understanding of the conservative contributions at high PM orders.
• Application of these techniques to spinning bodies or with tidal interactions which require extensions of the current formalism.
• Further development of the amplitude-action relations and integration techniques to facilitate efficient calculations at even higher precision and for more complex systems.

In conclusion, this paper marks an important milestone in leveraging quantum field theory techniques to gain insights into classical gravity, enriching both our theoretical understanding and the practical tools available for modeling gravitational-wave sources.