[论文解读] The Self-Force Problem: Local Behaviour of the Detweiler-Whiting Singular Field

Anna Heffernan|arXiv (Cornell University)|Mar 24, 2014

Pulsars and Gravitational Waves Research参考文献 1被引用 3

一句话总结

本论文通过在弯曲时空中的高阶展开德特维尔-惠廷奇奇异场，提升了引力自力计算的精度。通过计算超过14个新的正则化参数并构建高阶有效源，该工作使模式求和与二阶自力方案的精度得以提高，推动了极端质量比旋进的建模，并有助于检验宇宙审查猜想。

ABSTRACT

The growing reality of gravitational wave astronomy is giving age-old problems a new lease of life; one such problem is that of the self-force. A charged or massive particle moving in a curved background space-time produces a field that affects its motion, pushing it off its expected geodesic. This self-field gives rise to a so-called self-force acting on the particle. In modelling this motion, the self-force approach uses a perturbative expansion in the mass ratio. One of the most interesting sources of gravitational waves are extreme mass ratio inspirals - systems perfectly suited to self-force modelling. One of the key problems within the self-force model is the divergence of the field at the particle. To resolve this, the field is split into a singular component and a smooth regular field. This regular-singular split, introduced by Detweiler and Whiting, is used in most modern self-force calculations. In this thesis, we derive high-order expansions of the Detweiler-Whiting singular field, and use these to push the boundaries on current precision limits of self-force calculations. Within the mode-sum scheme, we give over 14 previously unknown regularisation parameters, almost doubling the current regularisation parameter database. We also produce smooth effective sources to high order, and propose an application of the higher terms to improve accuracy in the m-mode scheme. Finally, we investigate the status of the cosmic censorship conjecture and the role that the self-force plays. To this end, we give regularisation parameters for non-geodesic motion. We also show the necessity of our results in the exciting area of second order self-force calculations, which benefit significantly from high-order coordinate expansions of the singular field. We calculate several parameters that these schemes require, and highlight the further advancements possible from the results of this thesis.

研究动机与目标

• 通过改进德特维尔-惠廷奇正则-奇异场分解，解决弯曲时空点粒子处场的发散问题。
• 通过奇异场的高阶坐标展开，将自力计算的精度推向极限。
• 为模式求和方案和二阶自力计算提供新的正则化参数。
• 通过非测地线运动研究自力在检验宇宙审查猜想中的作用。

提出的方法

• 在粒子世界线周围的黎曼法坐标中，推导德特维尔-惠廷奇奇异场的高阶泰勒展开。
• 利用推导出的展开式，应用模式求和正则化方案计算正则化参数。
• 构建高阶光滑有效源，以提高数值自力实现的精度。
• 使用辛格的世界函数和双张量微积分，系统地计算场导数及其重合极限。
• 应用里奇恒等式和辛格法则，计算高阶双张量分量及其重合极限。
• 通过与已知恒等式的一致性验证结果，并推导非测地线运动的新参数。

实验结果

研究问题

• RQ1 德特维尔-惠廷奇奇异场在黎曼法坐标中的高阶展开项是什么？
• RQ2 从高阶展开中可推导出多少个新的正则化参数？它们如何改善模式求和方案？
• RQ3 奇异场的高阶项在实现二阶自力计算中起到什么作用？
• RQ4 非测地线运动的正则化参数如何影响宇宙审查猜想？
• RQ5 高阶有效源在多大程度上提升了数值自力计算的精度？

主要发现

• 论文计算了超过14个此前未知的正则化参数，几乎使现有模式求和方案数据库翻倍。
• 已推导出奇异场的高阶展开，包含至δz^4阶项，显著提升了自力计算的精度。
• 推导出的有效源在高阶下保持光滑且精确，可直接用于增强m-模方案。
• 该工作为当前正在积极发展的二阶自力计算提供了关键参数。
• 已计算出非测地线运动的正则化参数，首次使自力在宇宙审查猜想中的作用得以系统研究。
• 系统推导并验证了高阶双张量的重合极限恒等式，包括σ_abc和σ_abcd。