The recent detection of gravitational waves emanating from inspiralling black hole binaries has triggered a renewed interest in the dynamics of relativistic two-body systems. The conservative part of the latter are given by Hamiltonian systems obtained from so-called post-Newtonian expansions of the general relativistic description of black hole binaries. In this paper we study the general question of whether there exist relativistic binaries that display Kepler-like dynamics with elliptical orbits. We show that an orbital equivalence to the Kepler problem indeed exists for relativistic systems with a Hamiltonian of a Kepler-like form. This form is realised by extremal black holes with electric charge and scalar hair to at least first order in the post-Newtonian expansion for arbitrary mass ratios and to all orders in the post-Newtonian expansion in the test-mass limit of the binary. Moreover, to fifth post-Newtonian order, we show that Hamiltonians of the Kepler-like form can be related explicitly through a canonical transformation and time reparametrisation to the Kepler problem, and that all Hamiltonians conserving a Laplace – Runge – Lenz-like vector are related in this way to Kepler.

最近探测到从吸气黑洞双星发出的引力波引发了人们对相对论双体系统动力学的新兴趣。后者的保守部分由哈密顿系统给出，该系统是从黑洞双星的广义相对论描述的所谓后牛顿展开中获得的。在本文中，我们研究了是否存在具有椭圆轨道的类开普勒动力学的相对论双星这一普遍问题。我们证明，对于具有类开普勒形式的哈密顿量的相对论系统，确实存在开普勒问题的轨道等效性。这种形式是通过带有电荷和标量毛的极值黑洞实现的，它至少达到任意质量比的后牛顿展开式中的一阶，并且达到双星测试质量极限的后牛顿展开式中的所有阶。此外，对于后牛顿第五阶，我们表明类开普勒形式的哈密顿量可以通过规范变换和时间重参数化与开普勒问题明确相关，并且所有守恒拉普拉斯-龙格-楞次类向量的哈密顿量都是以这种方式与开普勒相关。