arXiv:2501.13176v1 Announce Type: new
Abstract: We consider the self-force acting on a pointlike (electromagnetic or conformal-scalar) charge held fixed on a spacetime with a spherically-symmetric mass distribution of constant density (the Schwarzschild star). We calculate the self-force with two complementary regularization methods, direct and difference regularization, and we find agreement. The Schwarzschild star is shown to be conformal to a three-sphere geometry; we use this conformal symmetry to obtain closed-form expressions for mode solutions. The new results for the self-force come in three forms: series expansions for the self-force in the far field; an approximation that captures the divergence in the self-force near the star’s boundary; and as numerical data presented in a selection of plots. We conclude with a discussion of the logarithmic divergence in the self-force in the approach to the star’s surface.
Future Roadmap: Challenges and Opportunities
Introduction
The article investigates the self-force acting on a pointlike charge fixed on a spacetime with a spherically-symmetric mass distribution of constant density, known as the Schwarzschild star. The self-force is calculated using two regularization methods, direct and difference regularization, with agreement found between them. The article also discusses the conformal symmetry of the Schwarzschild star and its implications for obtaining closed-form expressions for mode solutions. The self-force is presented in three different forms: series expansions in the far field, an approximation for the divergence near the star’s boundary, and numerical data presented in plots.
Roadmap
To fully understand the conclusions of the article and explore potential future directions, readers should consider the following roadmap:
- Understanding the Self-Force: Dive deeper into the concept of the self-force and its significance in the context of a pointlike charge held fixed on a Schwarzschild star. Gain a clear understanding of the self-force calculations performed using direct and difference regularization methods.
- Exploring Conformal Symmetry: Explore the conformal symmetry of the Schwarzschild star and its implications for obtaining closed-form expressions for mode solutions. Understand how this symmetry contributes to the understanding of the self-force acting on the charge.
- Series Expansions in the Far Field: Examine the series expansions for the self-force in the far field. Analyze the implications of these expansions and their usefulness in practical applications. Consider potential challenges in extending these series expansions to more complex mass distributions.
- Approximation for Divergence Near the Star’s Boundary: Study the approximation presented to capture the divergence in the self-force near the star’s boundary. Evaluate the accuracy of the approximation and potential limitations in real-world scenarios.
- Numerical Data and Plots: Analyze the numerical data presented in a selection of plots. Identify patterns, trends, and potential correlations between the self-force and various parameters. Consider the limitations and challenges in extrapolating these numerical results to other scenarios.
- Discussion of Logarithmic Divergence: Engage in the discussion regarding the logarithmic divergence in the self-force as the charge approaches the star’s surface. Understand the implications of this divergence and potential future research directions to mitigate or utilize its effects.
Conclusion
The article provides an in-depth analysis of the self-force acting on a pointlike charge held fixed on a Schwarzschild star. It offers various insights into the calculations, conformal symmetry, series expansions, approximations, and numerical data related to the self-force. Readers can further explore the topics outlined in the roadmap to gain a deeper understanding of the conclusions and potentially uncover future challenges and opportunities in this area of study.