Gravitational scattering, post-Minkowskian approximation and Effective One-Body theory

Abstract: A novel approach to the Effective One-Body description of gravitationally interacting two-body systems is introduced. This approach is based on the post-Minkowskian approximation scheme (perturbation theory in G, without assuming small velocities), and employs a new dictionary focussing on the functional dependence of the scattering angle on the total energy and the total angular momentum of the system. Using this approach, we prove to all orders in v/c two results that were previously known to hold only to a limited post-Newtonian accuracy: (i) the relativistic gravitational dynamics of a two-body system is equivalent, at first post-Minkowskian order, to the relativistic dynamics of an effective test particle moving in a Schwarzschild metric; and (ii) this equivalence requires the existence of an exactly quadratic map between the real (relativistic) two-body energy and the (relativistic) energy of the effective particle. The same energy map is also shown to apply to the effective one-body description of two masses interacting via tensor-scalar gravity.

• The paper presents a post-Minkowskian method to calculate the scattering angle in gravitational two-body systems with full relativistic accuracy.
• It reveals an exact quadratic relationship between the real energy and the effective one-body energy using a Schwarzschild-type metric.
• The approach extends to tensor-scalar gravity, enhancing waveform modeling in gravitational wave research.

Gravitational Scattering, Post-Minkowskian Approximation, and Effective One-Body Theory

In the field of gravitational dynamics, the work titled "Gravitational scattering, post-Minkowskian approximation and Effective One-Body theory" by Thibault Damour introduces a post-Minkowskian approach to the Effective One-Body (EOB) description of gravitationally interacting two-body systems. Unlike the traditional post-Newtonian (PN) expansions that are common in this area of study, this paper focuses on full relativistic computations by leveraging the post-Minkowskian (PM) scheme, which organizes corrections in powers of $G$ without assuming small velocities.

Key Contributions

1. Scattering Function Analysis:

The paper develops a methodology to compute the scattering angle of two-body systems through a PM approximation. The scattering angle $\chi$, expressed as a function of the center-of-mass energy and angular momentum, is calculated to all orders in $v/c$ for first-order PM approximations. The study demonstrates that this angle is consistent over the given parametrization, yielding a number of equivalencies to known Newtonian results.

2. Exact Energy Mapping:

One of the principal results is demonstrating an exact, quadratic relationship between the real two-body energy of the system and the effective one-body energy within linear approximations in $G$. This findings expand upon previous work which could only claim this result to low orders in the PN expansion.

3. Identification with Effective Metrics:

Through careful comparison of real and effective scattering functions, the work identifies that the effective dynamics can be encapsulated via motion in a Schwarzschild-type metric with specific mass assignment. This includes the emergent realization that the effective dynamics mandates an exactly quadratic energy map, applicable at various levels of approximation.

4. Application to Tensor-Scalar Gravity:

The methodology extends to alternative theories of gravity, specifically tensor-scalar gravitational models. This aids in illustrating how different interaction characteristics reshape the effective metric approach and showcases the malleability and broad applicability of this theoretical framework.

Implications and Future Directions

The research presented in Damour's paper provides a method for improving our understanding of gravitational interactions beyond the Newtonian regime, allowing explorations into relativistic binary systems that may include high-velocity interactions. This approach can play a significant role in refining waveform models for gravitational wave observatories like LIGO and Virgo, which benefit from improved accuracy in predicting the final stages of binary coalescence.

Moreover, the potential extension of this work to higher orders in $G$ and inclusion of bodies with spin might offer further insights into complex dynamical phenomena. The adaptability to alternate gravitational theories could open new avenues for understanding fundamental physics, particularly in regimes where standard General Relativity may present limitations or encounter anomalies.

In conclusion, this work establishes a robust formalism for analyzing gravitational dynamics in two-body systems using the EOB framework with a PM approach. The findings consolidate the credibility of using relativistic methodologies to further our grasp over astrophysical events observed through gravitational waves, potentially proving instrumental in refining theoretical predictions in gravitational physics.