[论文解读] A new derivation of the gravitational self-force

Adam Pound|arXiv (Cornell University)|Jul 29, 2009

Pulsars and Gravitational Waves Research被引用 3

一句话总结

本文提出了一种新的渐近方法，用于在广义相对论中推导引力自力，通过将内展开（固定物体尺寸）与外展开（收缩物体，固定世界线）相结合，利用洛伦兹规范将全局度量解表示为尾积分。该方法得到了在长时间尺度上自洽的运动方程，自力以世界线上度量扰动的不可约分量形式表达。

ABSTRACT

I review the problem of motion for small bodies in General Relativity, with an emphasis on developing a self-consistent treatment of the gravitational self-force. An analysis of the various derivations extant in the literature leads me to formulate an asymptotic expansion in which the metric is expanded while a representative worldline is held fixed; I discuss the utility of this expansion for both exact point particles and asymptotically small bodies, contrasting it with a regular expansion in which both the metric and the worldline are expanded. Based on these preliminary analyses, I present a general method of deriving self-consistent equations of motion for arbitrarily structured (sufficiently compact) small bodies. My method utilizes two expansions: an inner expansion that keeps the size of the body fixed, and an outer expansion that lets the body shrink while holding its worldline fixed. By imposing the Lorenz gauge, I express the global solution to the Einstein equation in the outer expansion in terms of an integral over a worldtube of small radius surrounding the body. Appropriate boundary data on the tube are determined from a local-in-space expansion in a buffer region where both the inner and outer expansions are valid. This buffer-region expansion also results in an expression for the self-force in terms of irreducible pieces of the metric perturbation on the worldline. Based on the global solution, these pieces of the perturbation can be written in terms of a tail integral over the body's past history. This approach can be applied at any order to obtain a self-consistent approximation that is valid on long timescales, both near and far from the small body. I conclude by discussing possible extensions of my method and comparing it to alternative approaches.

研究动机与目标

• 为广义相对论中小质量体的引力自力提供一种自洽的处理方法。
• 通过区分度量和世界线展开，解决现有推导中的不一致性。
• 提供一个在长时间尺度上适用于小质量体附近和远处的框架。
• 以世界线上度量扰动的不可约分量形式表达自力。
• 通过系统的渐近展开，实现更高阶近似。

提出的方法

• 制定一种渐近展开，其中度量被展开而世界线保持固定，与常规展开中同时变化度量和世界线的方法形成对比。
• 采用双尺度展开：内展开保持物体尺寸固定，外展开则使物体收缩而固定其世界线。
• 施加洛伦兹规范，将全局解表示为围绕物体的小半径世界管上的积分。
• 通过缓冲区确定世界管上的边界数据，该区域同时适用于内展开和外展开。
• 从缓冲区的局域空间展开中推导出自力，将度量扰动在世界线上分解为不可约分量。
• 利用外展开中的全局解，将自力表示为对物体过去历史的尾积分。

实验结果

研究问题

• RQ1 如何为广义相对论中结构任意、足够紧凑的小质量体推导出自洽的运动方程？
• RQ2 当度量被展开而世界线保持固定时，世界线在渐近展开中起什么作用？
• RQ3 如何系统地以世界线上度量扰动的不可约分量形式表达自力？
• RQ4 如何利用洛伦兹规范和世界管积分构造爱因斯坦方程的全局解？
• RQ5 缓冲区展开与自力的尾积分表示之间存在何种关系？

主要发现

• 该方法提供了在长时间尺度上有效、且在小质量体附近和远处均成立的引力自力自洽推导。
• 自力被表示为对物体过去历史的尾积分，该积分源自外展开中的全局解。
• 世界线上度量扰动的不可约分量通过缓冲区展开确定，确保了内解与外解之间的一致性。
• 使用洛伦兹规范使得全局度量解能以小世界管上数据的适当积分形式良好定义。
• 该方法可系统地推广到任意阶近似，从而实现对小质量体运动的高精度建模。
• 该方法区分了度量展开与世界线展开，解决了先前常规展开方法中存在的歧义。