The Second Post-Newtonian Motion in Schwarzschild Spacetime
- PDF / 505,618 Bytes
- 6 Pages / 612 x 792 pts (letter) Page_size
- 54 Downloads / 146 Views
The Second Post-Newtonian Motion in Schwarzschild Spacetime Bo Yang1 and Wenbin Lin1, 2* 1
School of Physical Science and Technology, Southwest Jiaotong University, Chengdu, 610031, China 2 School of Mathematics and Physics, University of South China, Hengyang, 421001, China Received June 17, 2020; revised July 7, 2020; accepted July 18, 2020
Abstract—We present the second-order post-Newtonian solution for the quasi-Keplerian motion in the Schwarzschild space-time under the Wagoner-Will-Epstein-Haugan representation. Detailed derivations are provided for readers’ convenience. DOI: 10.1134/S0202289320040118
1. INTRODUCTION The post-Newtonian (PN) approximations have been extensively used in general relativity to solve the motion of the bodies in the strong gravitational fields, and a number of analytical solutions have been achieved. These solutions include not only the small-deflection motion of photons [1–3], but also the quasi-Keplerian motion of test particles as well as binary systems [4–20]. The solutions of the quasi-Keplerian motion are mainly classified into two representations: one is the Wagoner–Will–Epstein– Haugan representation [5, 6, 8], and the other is the Brumberg–Damour–Deruelle representation [4, 7]. Notice that the Brumberg formulation is slightly different from the Damour–Deruelle formulation, and this is due to using different definitions of the true anomaly for the quasi-Keplerian motion. Klioner and Kopeikin compare various parameterizations of the two-body problem in the 1PN approximation and prove that they are all equivalent, and also give exact transformation formulas between the parameters of the orbit in different parametrizations [11]. The solutions for the mass effects in the Wagoner–Will– Epstein–Haugan representation are obtained via an iterative method based on the Keplerian motion in the Newton theory, and are only calculated to the 1PN order. In contrast, the solutions under the Brumberg–Damour–Deruelle representation are obtained via making use of the conservation of orbital energy and angular momentum and fitting the orbit with the positions of perihelion and aphelion, and have been calculated to the 3PN order. In this work, following the approaches given by Refs. [5, 6, 8, 10], we derive the 2PN solution for the quasi-Keplerian motion of a test particle
in the Schwarzschild space-time in the Wagoner– Will–Epstein–Haugan representation. Although the achieved solution is effectively equivalent to the 2PN solution in the Brumberg–Damour–Deruelle representation, it adds a different formulation and the corresponding derivations for the 2PN quasiKeplerian motion to the body of literature. 2. THE 2PN METRIC, LAGRANGIAN, AND DYNAMICS IN SCHWARZSCHILD SPACETIME The second-order PN approximations for the Schwarzschild metric in the harmonic coordinates can be written as [21] 2m 2m2 2m3 − 2 + 3 , r r r g0i = 0, 2m m2 m 2 xi xj + 2 δij + 2 2 , gij = 1 + r r r r g00 = −1 +
(2) (3)
where m denotes the body’s mass. r ≡ |x| denotes the distance from the field position x ≡ (x1 , x2 , x3 ) to the
Data Loading...
