Harmonic Metric For Charged Rotating Black Hole And Its Post-Newtonian Dynamics

Einstein’s field equations, describing the relationship between the curved spacetime and mass, are the fundamental equations of general relativity. However, due to symmetric and Bianchi identities, Einstein’s field equations can not completely determine a unique solution, unless they are imposed on coordinate condition. Therefore, once chosen the specific coordinate condition, the reference coordinate system in which we investigate the problem of physics has been chosen.The paper investigates mainly the exact solutions of the Einstein’s field equations about the charged rotating black hole in the harmonic coordinate condition. The harmonic coordinate condition is one of the most important coordinate conditions in general relativity. Doctor Vladimir A. Fock, an academician of the Soviet academy of sciences, en believe that only in this condition, the solutions of the Einstein’s field equations have a very clearly physical significance. Peiyuan Zhou, an academician of the Chinese academy of sciences, also holds a similar viewpoint. Although this viewpoint is not accepted by the mainstream relativity community, Einstein’s field equations can deduce to Poisson equations in the harmonic coordinate condition. Therefore, the harmonic coordinate condition is often used to derive the post-Newtonian dynamics and gravitational radiation in an isolated system. We start with the solution of the charged rotating black hole in the Boyer-Lindquist coordinate system to construct the harmonic coordinate system of the charged rotating black hole. We obtain a series of harmonic coordinates for the charged rotating black hole by solving the harmonic coordinate condition. Then utilize the post-Newtonian approximation theory, we determine the physical harmonic coordinate system for the charged rotating black hole according with the relation between the leading terms of space-time component of metric tensor with the one of space-time component of energy-momentum tensor, and obtain the exact and unique metric for the charged rotating black hole.Basing on harmonic metric for the charged rotating block hole, we first investigate the charged particle’s dynamics in the far-field of the charged rotating black hole and calculate the orbital precession in the equatorial plane. The charged particle moves in the gravitational field of the charged rotating black hole, not only suffer the action of gravitation, but also the electromagnetic force. The first post-Newtonian motion of equation for the charged particle can be obtained by the geodesic equation imposed with Lorentz force. We employ the approach of the rate of change of the Runge-Lenz vector to calculate the charged particle’s orbital precession in the equatorial plane of the charged rotating black hole, and obtain a new effect which couples the black hole’s mass, angular momentum and charge. The orbital precess caused by the new effect may be large than the effect of the black hole’s rotation.Subsequently, we investigate the photon’s dynamics and gravitational deflection in the far-field of the charged rotating black hole. Photon moves along the geodesic of the gravitational field. Photon’s second post-Newtonian acceleration can be obtained by the harmonic metric of the charged rotating black hole. In the Newtonian approximation, the acceleration of photon is zero, so the trajectory of the photon is a straight line. In the first post-Newtonian approximation, photon will deviate from the straight line, and the deviation is called first post-Newtonian correction. We decompose the correction into two components:one component, paralleling the orientation of the initial velocity of the photon, the velocity correction along this orientation can be obtained by utilized that the proper time interval of the photon is zero. The other perpendicular to the orientation of the initial velocity of the photon, the acceleration correction along this orientation can be determined by using the photon’s second post-Newtonian acceleration. The acceleration correction can be integrated along the straight line to obtain the velocity correction along the perpendicular to straight line, and integrated again to obtain the trajectory correction along this orientation. We utilize the iterative method to obtain the photon’s second post-Newtonian velocity correction. The deflection angle can be express by the initial velocity multiplication cross with the photon’s second post-Newtonian velocity. We obtain the formula of calculating the deflection angle of light with arbitrary orientation in the far-field of the charged rotating black hole.At last, we investigate the spinning particle’s dynamics in the far-field of the charged rotating black hole. A spinning particle moves along an orbit in the gravitational field described by Mathisson-Papapetrou equations. However, Mathisson-Papapetrou equations are not a closed system, and the spin supplementary condition needs to be imposed to fix the representative worldline of the spinning particle. We derive the spinning particle’s motion of equations and spin evolution equations under the three different spin supplementary conditions in the background spacetime of the charged rotating black hole, and demonstrate that Mathisson-Papapetrou equations with three different spin supplementary conditions are describing the same physical motion of the spinning particle.