Approximate and Exact GR-Solutions for the Two-Body Problem  

作　　者：Jan Helm Jan Helm(Technical University of Berlin, Berlin, Germany)

机构地区：[1]Technical University of Berlin, Berlin, Germany

出　　处：《Journal of High Energy Physics, Gravitation and Cosmology》2022年第3期690-723,共34页高能物理（英文）

摘　　要：A binary gravitational rotator, also called the two-body problem, is a pair of masses m₁, m₂ moving around their center-of-mass (com) in their own gravitational field. In Newtonian gravitation, the two-body problem can be described by a single reduced mass (gravitational rotator) m_r = m₁m₂/(m₁+m₂) orbiting around the total mass m = m₁+m₂ situated in com in the distance r, which is the distance between the two original masses. In this paper, we discuss the rotator in Newtonian, Schwarzschild and Kerr spacetime context. We formulate the corresponding Kerr orbit equations, and adapt the Kerr rotational parameter to the Newtonian correction of the rotator potential. We present a vacuum solution of Einstein equations (Manko-Ruiz), which is a generalized Kerr spacetime with five parameters g_μν (m₁, m₂, R, a₁, a₂), and adapt it to the Newtonian correction for observer orbits. We show that the Manko-Ruiz metric is the exact solution of the GR-two-body problem (i.e. GR-rotator) and express the orbit energy and angular momentum in terms of the 5 parameters. We calculate and discuss Manko-Ruiz rotator orbits in their own field, and present numerical results for two examples. Finally, we carry out numerical calculations of observer orbits in the rotator field for all involved models and compare them.A binary gravitational rotator, also called the two-body problem, is a pair of masses m₁, m₂ moving around their center-of-mass (com) in their own gravitational field. In Newtonian gravitation, the two-body problem can be described by a single reduced mass (gravitational rotator) m_r = m₁m₂/(m₁+m₂) orbiting around the total mass m = m₁+m₂ situated in com in the distance r, which is the distance between the two original masses. In this paper, we discuss the rotator in Newtonian, Schwarzschild and Kerr spacetime context. We formulate the corresponding Kerr orbit equations, and adapt the Kerr rotational parameter to the Newtonian correction of the rotator potential. We present a vacuum solution of Einstein equations (Manko-Ruiz), which is a generalized Kerr spacetime with five parameters g_μν (m₁, m₂, R, a₁, a₂), and adapt it to the Newtonian correction for observer orbits. We show that the Manko-Ruiz metric is the exact solution of the GR-two-body problem (i.e. GR-rotator) and express the orbit energy and angular momentum in terms of the 5 parameters. We calculate and discuss Manko-Ruiz rotator orbits in their own field, and present numerical results for two examples. Finally, we carry out numerical calculations of observer orbits in the rotator field for all involved models and compare them.

关 键 词：General Relativity Two-Body Problem Gravitational Rotator Kerr Metric Generalized Kerr Metric 

分 类 号：TE3[石油与天然气工程—油气田开发工程]