构建克尔度规的正则调和坐标至第四后闵可夫斯基阶
Constructing the canonical harmonic coordinates of Kerr metric to the fourth post-Minkowskian order
AI总结：
本文在多极后闵可夫斯基形式体系中构建出无规范矩的克尔度规正则调和坐标至第四后闵可夫斯基阶，其宇称特性区别于其他调和坐标，且该正则坐标为MPM框架中克尔度规最具规范纯净性的表示。
AI中文摘要：
本文在多极后闵可夫斯基（MPM）形式体系中构建克尔度规的正则调和坐标，精度达到第四后闵可夫斯基（4PM）阶。基于已知的克尔度规的Geroch–Hansen矩和Gürsel定理，我们推导了无任何规范矩的精确正则MPM矩$\boldsymbol{\rm M}_L$、$\boldsymbol{\rm S}_L$。利用这些矩，我们迭代计算哥特式度规扰动$\boldsymbol{h}^{\boldsymbol{\rm \nu}}_{\boldsymbol{\rm can}}$至4PM阶，并计算4PM正则度规$\boldsymbol{g}_{\boldsymbol{\rm can}}^{\boldsymbol{\rm \nu}}$。所得度规的空间和时间分量是自旋参数$\boldsymbol{a}$的偶函数，而混合分量是奇函数。这种宇称特性使正则坐标区别于其他调和坐标。为对比这种最小规范构造，我们还在Jiang–Lin坐标中提取了克尔度规的1PM源矩。我们发现，Jiang–Lin表示从1PM阶开始就具有非零规范矩，而在正则表示中，所有阶的规范矩均为零。这种对比凸显了正则坐标是MPM框架中克尔度规最具规范纯净性的表示。我们还计算了史瓦西情形的完整正则度规至所有PM阶。最近Damgaard等人利用动量空间递归的独立构造得到了与4PM等价的结果，以$\boldsymbol{a}$的幂级数形式表示，为我们的闭式4PM正则度规提供了交叉验证。将正则克尔坐标与先前已知的调和克尔坐标关联的坐标变换仍是一个未解决的问题。
英文摘要：
In this paper we construct the canonical harmonic coordinates of the Kerr metric within the multipolar post-Minkowskian (MPM) formalism to the fourth post-Minkowskian (4PM) order. Based on the well known Geroch--Hansen moments of Kerr metric and Gürsel's theorem, we derive the exact canonical MPM moments $\mathrm{M}_L,\mathrm{S}_L$, which are free of any gauge moments. With these moments, we iteratively compute the gothic metric perturbation $h^{μν}_{\mathrm{can}}$ up to 4PM order and compute the 4PM canonical metric $g_{μν}^{\mathrm{can}}$. The resulting spatial and time components of these metrics are even functions of the spin parameter $a$ while the mixed components are odd. This parity property distinguishes the canonical coordinates from other harmonic coordinates. To contrast this minimal-gauge construction, we also extract the 1PM source moments of the Kerr metric in the Jiang--Lin coordinates. We find that the Jiang--Lin representation possesses non-vanishing gauge moments starting from the 1PM order, whereas in the canonical representation gauge moments vanish to all orders. This comparison highlights the canonical coordinates as the most gauge-pure representation of the Kerr metric in the MPM framework. The complete canonical metric for the Schwarzschild case is also computed to all PM orders. A recent independent construction by Damgaard et al. using momentum-space recursion yields 4PM equivalent results expressed as a power series in $a$, providing a cross-validation of our closed-form 4PM canonical metric. The coordinate transformation linking the canonical Kerr coordinates to previously known harmonic Kerr coordinates remains an open problem.
