The Post-Newtonian Approximation for Relativistic Compact Binaries

Abstract

We discuss various aspects of the post-Newtonian approximation in general relativity. After presenting the foundation based on the Newtonian limit, we show a method to derive post-Newtonian equations of motion for relativistic compact binaries based on a surface integral approach and the strong field point particle limit. As an application we derive third post-Newtonian equations of motion for relativistic compact binaries which respect the Lorentz invariance in the post-Newtonian perturbative sense, admit a conserved energy, and are free from any ambiguity.

Figure 1

Body zone coordinates and near zone coordinates. In the near zone coordinates (τ, xⁱ), both the body zone and the star shrink as ϵ (thin dotted arrow) and ϵ² (thick dotted arrow) respectively. Both the thin and thick arrows point inside. In the body zone coordinates , the star does not shrink while the body zone boundary goes to infinity as ϵ⁻¹ (thin dotted line pointing outside).

Figure 2

Gravitational energy momentum flux through the body zone boundary. The meshed two circles represent stars 1 and 2. Each star is surrounded by the body zone represented here by a striped area. The arrows around star 1 represent the gravitational energy momentum flux flowing through the body zone boundary.

Figure 3

The vectors used in the surface integral over the boundary of the body zone 1.

Figure 4

Flowchart of the post-Newtonian iteration.