Classical black hole scattering from a worldline quantum field theory (2010.02865v3)

Abstract: A precise link is derived between scalar-graviton S-matrix elements and expectation values of operators in a worldline quantum field theory (WQFT), both used to describe classical scattering of a pair of black holes. The link is formally provided by a worldline path integral representation of the graviton-dressed scalar propagator, which may be inserted into a traditional definition of the S-matrix in terms of time-ordered correlators. To calculate expectation values in the WQFT a new set of Feynman rules is introduced which treats the gravitational field $h_{\mu\nu}(x)$ and position $x_i^\mu(\tau_i)$ of each black hole on equal footing. Using these both the next-order classical gravitational radiation $\langle h^{\mu\nu}(k)\rangle$ (previously unknown) and deflection $\Delta p_i^\mu$ from a binary black hole scattering event are obtained. The latter can also be obtained from the eikonal phase of a $2\to2$ scalar S-matrix, which we show to correspond to the free energy of the WQFT.

Overview of Classical Black Hole Scattering from a Worldline Quantum Field Theory

This paper, authored by Gustav Mogull, Jan Plefka, and Jan Steinhoff, explores the classical scattering of black holes by utilizing concepts from worldline quantum field theory (WQFT). The paper proposes a method to link scalar-graviton S-matrix elements to expectation values of operators in a WQFT, highlighting a systematic approach to analyzing classical scattering events in gravity. The authors present a path integral formulation that brings together the graviton-dressed scalar propagator with a traditional definition of the S-matrix. This methodology is augmented by a novel set of Feynman rules treating both gravitational fields and black hole trajectories consistently, allowing insights into both two-body and three-body interactions.

Formalism and Methodology

The authors begin by setting out a worldline representation that permits a transformation between quantum field theory and expectations of classical mechanics. The approach employs path integrals to represent the dynamics of massive scalars combined with gravitation, which can be inserted into quantum field theoretical correlators. A key part of this representation is the handling of the graviton-dressed propagator, which encapsulates both the scalar’s quantum excitations and gravitational interactions.

A significant contribution of this paper is the derivation of momentum space representations for these propagators, helping to bridge the divide from classical mechanics to quantum formulations. The authors perform LSZ reduction, a method to relate field theory correlators to S-matrix elements by amputating the external legs, thereby enabling analysis over infinite proper-time domains. This approach is crucial for understanding scattering processes without a dependence on virtual loops, simplifying calculations in the classical limit.

Numerical Results and Diagrams

The paper focuses on two primary outcomes: the gravitational radiation emitted during scattering events and the impulse imparted to black holes during these events. For the radiation, expressions are derived for the two-body and three-body systems at leading orders in the Post-Minkowskian (PM) expansion. Here, the authors elucidate the connections between scattering amplitudes and the emitted gravitational frequencies, ensuring compatibility with known scalar-graviton amplitude calculations.

Regarding the impulse on black holes, the paper introduces a method for calculating the deflection of a single black hole in a binary scattering event. The methods presented here provide clarity in computing these values without the necessity of deriving an effective potential, thus advancing computational efficiency. These impulse results are framed within the understanding of classical eikonal phases, serving as generators for observable scattering behaviors.

Implications and Future Prospects

The insights brought forward by Mogull, Plefka, and Steinhoff have practical and theoretical implications in the field of classical gravitation and quantum field interactions. This research sets the groundwork for precision computations of classical gravitational phenomena and could serve as a basis for analyzing more complex systems involving spin or tidal forces. While focused on non-spinning bodies currently, the methodology suggests possible extensions into the field of spinning black holes and multiple-body interactions.

Future directions could further explore 4PM or higher-order integrations within similar frameworks, potentially leveraging the synergy between scattering amplitudes and the classical observables covered in this paper. Additionally, introducing classical spin vectors and accounting for spin effects could enrich the models, making them more applicable to real-world gravitational wave observations.

Conclusion

Overall, this paper offers a significant step forward in understanding classical gravitational scattering from the vantage point of quantum field theory. By employing WQFT, the authors provide a refined mechanism to evaluate interactions that traditional methods would struggle to compute efficiently. The synthesis of complex mathematical frameworks with tangible gravitational phenomena establishes this work as a foundation for further exploration into classical and quantum gravitational realms.