Post-Newtonian gravitational radiation and equations of motion via direct integration of the relaxed Einstein equations: Foundations

We present a self-contained framework called direct integration of the relaxed Einstein equations for calculating equations of motion and gravitational radiation emission for isolated gravitating systems based on the post-Newtonian approximation. We cast the Einstein equations into their “relaxed” form of a flat-spacetime wave equation together with a harmonic gauge condition, and solve the equations formally as a retarded integral over the past null cone of the field point (chosen to be within the near zone when calculating equations of motion and in the far zone when calculating gravitational radiation). The “inner” part of this integral (within a sphere of radius $\mathcal{R}\sim$ one gravitational wavelength) is approximated in a slow-motion expansion using standard techniques; the “outer” part, extending over the radiation zone, is evaluated using a null integration variable. We show generally and explicitly that all contributions to the inner integrals that depend on $\mathcal{R}$ cancel corresponding terms from the outer integrals, and that the outer integrals converge at infinity, subject only to reasonable assumptions about the past behavior of the source. The method cures defects that plagued previous “brute-force” slow-motion approaches to motion and gravitational radiation for isolated systems. We detail the procedure for iterating the solutions in a weak-field, slow-motion approximation, and derive expressions for the near-zone field through 3.5 post-Newtonian order in terms of Poisson-like potentials.