“High Order Calculation of Singular Field for Self-Force Calculations”
arXiv:2404.10082v1 Announce Type: new
Abstract: Most self-force calculations rely, in one way or another, on representations of a particle’s Detweiler- Whiting singular field. We present a simple method of calculating the singular field to high order in a local expansion in powers of distance from the particle. As a demonstration, we compute the singular field to 14th order in distance, 10 orders beyond the previous state of the art, in the simple case of a scalar charge in circular orbit around a Schwarzschild black hole. We provide the result in both a 4-dimensional form and a decomposed form suitable for use in an m-mode puncture scheme. Our method should have applications in overcoming bottlenecks in current self-force calculations at both first and second order in perturbation theory.
Self-Force Calculations: Future Roadmap
Most self-force calculations rely on representations of a particle’s Detweiler-Whiting singular field. A simple method of calculating the singular field to high order in a local expansion in powers of distance from the particle has been presented. This method demonstrates the computation of the singular field to 14th order in distance, which is 10 orders beyond the previous state of the art. The case studied is a scalar charge in circular orbit around a Schwarzschild black hole.
The results of the calculation are provided in both a 4-dimensional form and a decomposed form suitable for use in an m-mode puncture scheme. The method presented in this study is expected to have applications in overcoming bottlenecks in current self-force calculations at both first and second order in perturbation theory.
Roadmap for Future Readers
- Study current self-force calculations and their reliance on representations of the Detweiler-Whiting singular field.
- Understand the limitations and challenges faced in existing methods.
- Learn the simple method presented in this study for calculating the singular field to high order in a local expansion.
- Implement the method to calculate the singular field to a desired order, potentially going beyond the current state of the art (14th order).
- Apply the results obtained in both the 4-dimensional form and the decomposed form to relevant problems and scenarios in self-force calculations.
- Explore the potential applications of the method in overcoming current bottlenecks in self-force calculations, particularly at first and second order in perturbation theory.
- Keep an eye on further advancements in self-force calculations and related areas of research.
- Collaborate with other researchers to improve and refine the method, and contribute to the advancement of self-force calculations.
Potential Challenges and Opportunities
Challenges:
- Understanding the mathematical foundations and techniques involved in self-force calculations and the Detweiler-Whiting singular field representation.
- Implementing the method to calculate the singular field to higher orders may require advanced computational resources.
- Applying the results to complex systems or scenarios may involve additional complexities and may require interdisciplinary approaches.
- Further research and collaboration may be needed to address any limitations or issues that arise during the implementation and application of the method.
Opportunities:
- Advancing the state of the art in self-force calculations by going beyond the previous limitations and achieving higher-order accuracy.
- Improving the efficiency and accuracy of self-force calculations through the use of the presented method.
- Exploring new avenues of research and applications in self-force calculations.
- Contributing to the broader understanding of gravitational physics and its implications in fundamental physics and astrophysics.