We recently introduced in [ 9 ] a boundary-to-bound dictionary between gravitational scattering data and observables for bound states of non-spinning bodies. In this paper, we elaborate further on this holographic map. We start by deriving the following — remarkably simple — formula relating the periastron advance to the scattering angle: ΔΦ J E = χ J E + χ − J E $$ \Delta \Phi \left(J,\mathcal{E}\right)=\upchi \left(J,\mathcal{E}\right)+\upchi \left(-J,\mathcal{E}\right) $$ , via analytic continuation in angular momentum and binding energy. Using explicit expressions from [ 9 ], we confirm its validity to all orders in the Post-Minkowskian (PM) expansion. Furthermore, we reconstruct the radial action for the bound state directly from the knowledge of the scattering angle. The radial action enables us to write compact expressions for dynamical invariants in terms of the deflection angle to all PM orders, which can also be written as a function of the PM-expanded amplitude. As an example, we reproduce our result in [ 9 ] for the periastron advance, and compute the radial and azimuthal frequencies and redshift variable to two-loops. Agreement is found in the overlap between PM and Post-Newtonian (PN) schemes. Last but not least, we initiate the study of our dictionary including spin. We demonstrate that the same relation between deflection angle and periastron advance applies for aligned-spin contributions, with J the (canonical) total angular momentum. Explicit checks are performed to display perfect agreement using state-of-the-art PN results in the literature. Using the map between test- and two-body dynamics, we also compute the periastron advance up to quadratic order in spin, to one-loop and to all orders in velocity. We conclude with a discussion on the generalized ‘impetus formula’ for spinning bodies and black holes as ‘elementary particles’. Our findings here and in [ 9 ] imply that the deflection angle already encodes vast amount of physical information for bound orbits, encouraging independent derivations using numerical and/or self-force methodologies.

中文翻译：

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从边界数据到绑定状态。第二部分。动态不变量的散射角（带扭曲）

我们最近在 [9] 中引入了引力散射数据和非旋转体束缚态的可观测值之间的边界到边界字典。在本文中，我们进一步阐述了这张全息图。我们首先推导出以下 - 非常简单 - 将近星体前进与散射角相关联的公式：ΔΦ JE = χ JE + χ − JE $$ \Delta \Phi \left(J,\mathcal{E}\right)=\ upchi \left(J,\mathcal{E}\right)+\upchi \left(-J,\mathcal{E}\right) $$ ，通过角动量和结合能的解析延拓。使用 [9] 中的显式表达式，我们确认其对 Post-Minkowskian (PM) 扩展中的所有订单的有效性。此外，我们直接根据散射角的知识重建束缚态的径向作用。径向作用使我们能够根据所有 PM 阶次的偏转角编写动态不变量的紧凑表达式，这也可以写成 PM 扩展幅度的函数。作为一个例子，我们在 [9] 中重现了我们的结果，用于近天体前进，并将径向和方位角频率以及红移变量计算为两个循环。在 PM 和后牛顿 (PN) 方案之间的重叠中发现了一致。最后但并非最不重要的一点是，我们开始研究我们的字典，包括自旋。我们证明了偏转角和近天体推进之间的相同关系适用于对齐自旋贡献，J 是（规范的）总角动量。使用文献中最先进的 PN 结果执行显式检查以显示完美的一致性。使用测试和两体动力学之间的映射，我们还计算了近天体推进到自旋的二次阶、单环和速度的所有阶。我们最后讨论了作为“基本粒子”的自旋体和黑洞的广义“动力公式”。我们在此处和 [9] 中的发现意味着偏转角已经为绑定轨道编码了大量物理信息，鼓励使用数值和/或自力方法进行独立推导。