[论文解读] Hyperboloidal discontinuous time-symmetric numerical algorithm with higher order jumps for gravitational self-force computations in the time domain

Lidia J. Gomes Da Silva, Rodrigo Panosso Macedo+4 more|arXiv (Cornell University)|Jun 22, 2023

Radio Astronomy Observations and Technology被引用 4

一句话总结

本文提出了一种基于双曲面切片的高阶间断时间对称数值算法，用于时域引力自力计算，通过引入间断拉格朗日与埃尔米特插值，即使在粒子处存在奇异性，也能实现高阶精度。该方法可实现精确的通量辐射测量，并准确计算粒子极限处的场与导数，证明其适用于极端质量比旋进（EMRI）中的全自洽轨道演化。

ABSTRACT

Within the next decade the Laser Interferometer Space Antenna (LISA) is due to be launched, providing the opportunity to extract physics from stellar objects and systems, such as extit{Extreme Mass Ratio Inspirals}, (EMRIs) otherwise undetectable to ground based interferometers and Pulsar Timing Arrays (PTA). Unlike previous sources detected by the currently available observational methods, these sources can extit{only} be simulated using an accurate computation of the gravitational self-force. Whereas the field has seen outstanding progress in the frequency domain, metric reconstruction and self-force calculations are still an open challenge in the time domain. Such computations would not only further corroborate frequency domain calculations and models, but also allow for full self-consistent evolution of the orbit under the effect of the self-force. Given we have extit{a priori} information about the local structure of the discontinuity at the particle, we will show how to construct discontinuous spatial and temporal discretisations by operating on discontinuous Lagrange and Hermite interpolation formulae and hence recover higher order accuracy. In this work we demonstrate how this technique in conjunction with well-suited gauge choice (hyperboloidal slicing) and numerical (discontinuous collocation with time symmetric) methods can provide a relatively simple method of lines numerical algorithm to the problem. This is the first of a series of papers studying the behaviour of a point-particle prescribing circular geodesic motion in Schwarzschild in the extit{time domain}. In this work we describe the numerical machinery necessary for these computations and show not only our work is capable of highly accurate flux radiation measurements but it also shows suitability for evaluation of the necessary field and it's derivatives at the particle limit.

研究动机与目标

• 开发一种稳健的时域数值框架，用于极端质量比旋进（EMRIs）中的引力自力计算，这对LISA任务科学至关重要。
• 解决在时域中准确计算自力的挑战，尽管频域研究已取得进展，但度规重构与粒子处场量评估仍是开放问题。
• 通过将粒子处不连续结构的先验知识融入空间与时间离散化，克服点粒子奇异性。
• 通过采用双曲面切片与时间对称间断配点法的时行方法，展示通量辐射与场导数测量的高阶精度与稳定性。

提出的方法

• 该方法采用双曲面切片，自然吸收未来类光无穷远处的出射辐射，提升时域模拟的长期稳定性。
• 使用间断拉格朗日与埃尔米特插值公式，构建能捕捉粒子处不连续局部结构的空间与时间离散化。
• 通过将场奇异行为导出的跳跃条件融入插值框架，恢复高阶精度。
• 在时行法框架中应用时间对称间断配点法，提升稳定性和收敛性。
• 该算法实现于史瓦西时空中作圆周测地线运动的点粒子，作为EMRI波形的基础模型。
• 通过将时域结果与文献[39]的频域数据对比，对RWZ主函数及其导数进行数值验证，误差估计基于公式(76)。

实验结果

研究问题

• RQ1 在存在点粒子奇异性的情况下，时域数值算法能否实现引力自力计算的高阶精度？
• RQ2 如何构建间断的空间与时间离散化，以在场在粒子位置出现跳跃时仍保持精度？
• RQ3 双曲面切片在多大程度上提升了长期时域模拟中自力效应的稳定性和精度？
• RQ4 该方法能否准确计算粒子处的通量与场导数，从而实现自洽的轨道演化？
• RQ5 与频域结果相比，该算法在RWZ主函数的收敛性与数值误差方面表现如何？

主要发现

• 该算法在RWZ主函数的内部与外部解中均实现高阶收敛，部分模态的数值误差低至$1.7 \times 10^{-9}$。
• 通量辐射测量高度精确，前$l=5$个模态的误差范围为$[1.7 \times 10^{-9}, 7.9 \times 10^{-8}]$。
• 该方法成功计算了粒子位置处RWZ函数的一阶时间与径向导数，多数模态的误差低于$10^{-8}$。
• 在$ au=10,000$时，$(l,m)=(2,2)$模态的相图中观察到辛结构保持，表明长期稳定性。
• 数值收敛性研究证实了时间离散化步长的最优选择，误差标度与理论预期一致。
• 该方法与粒子处自力评估兼容，这是EMRI模型中实现自洽轨道演化的关键要求。