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) mr = 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 gμν (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.
References
[1]
Helm, J. (2022) Binary GR-Gravitational Rotator. https://www.researchgate.net/publication/358270847_Binary_GR-gravitational_rotator
[2]
Helm, J. (2022) General Relativity. https://www.researchgate.net/publication/358270510_General_Relativity
[3]
Chandrasekhar, S. (1992) The Mathematical Theory of Black Holes. Oxford University Press, Oxford.
[4]
Visser, M. (2008) The Kerr Spacetime: A Brief Introduction. arXiv:0706.0622.
[5]
Fliessbach, T. (1993) Allgemeine Relativitätstheorie. Bibliographisches Institut, Gotha.
[6]
Soff, G. (1993) Allgemeine Relativitätstheorie. Goethe University Frankfurt, Frankfurt.
[7]
Blau, M. (2000) Lecture Notes on General Relativity. Bern University, Bern.
[8]
Ciufolini, I. and Wheeler, J.A. (1996) Gravitation and Inertia. Princeton University Press, Princeton. https://doi.org/10.1515/9780691190198
[9]
Goldstein, H. (1980) Classical Mechanics. Addison Wesley, Boston.
[10]
Helm, J. (2019) Approximate and Exact GR-Solutions for the Two-Body Problem. https://www.researchgate.net/publication/330737887_An_exact_General-Relativistic_solution_for_the_relativistic_rotator
[11]
Damour, T. (2013) The General Relativistic Two Body Problem. arXiv:1312.3505.
[12]
Baumgarte, T.W., Cook, G.B., Scheel, M.A., et al. (1997) General Relativistic Models of Binary Neutron Stars in Quasi-Equilibrium, Physical Review D, 57, 7299-711.
[13]
Cabrera-Munguia, I. (2018) Unequal Binary Configurations of Interacting Kerr Black Holes. Physics Letters B, 786, 466-471. https://doi.org/10.1016/j.physletb.2018.10.037
[14]
Stefani, H., Kramer, D., MacCallum, M., et al. (2003) Exact Solutions of Einstein’s Field Equations. Cambridge University Press, Cambridge.
[15]
Manko, V. and Ruiz, E. (2019) Metric for Two Arbitrary Kerr Sources. Physics Letters B, 794, 36-40. https://doi.org/10.1016/j.physletb.2019.05.027
[16]
Manko, V. and Ruiz, E. (2011) On a Simple Representation of the Kinnersley-Chitre Metric. Progress of Theoretical Physics, 125, 1241-1254. https://doi.org/10.1143/PTP.125.1241
