Fourth-post-Newtonian-exact approximation to General Relativity

TL;DR

This paper introduces a fourth post-Newtonian approximation to General Relativity formulated with elliptic equations, improving accuracy for stationary and dynamic systems, including black holes, and extends previous waveless approaches.

Contribution

It presents a fully constrained, elliptic scheme for a 4PN approximation that encompasses existing methods and introduces a 2PN waveless variant for better modeling of gravitational systems.

Findings

Achieves agreement with GR up to 5PN for stationary black holes.

Develops a 2PN waveless approximation with restrictive conditions.

Extends the conformal-flat-condition and CFC+ approaches.

Abstract

An approximation to General Relativity is presented that agrees with the Einstein field equations up to and including the fourth post-Newtonian (PN) order. This approximation is formulated in a fully constrained scheme: all involved equations are explicitly elliptic except the wave equation that describes the two independent degrees of freedom of the gravitational field. The formalism covers naturally the conformal-flat-condition (CFC) approach by Isenberg, Wilson, and Mathews and the improved second PN-order exact approach CFC+. For stationary configurations, like Kerr black holes, agreement with General Relativity is achieved even through 5PN order. In addition, a particularly interesting 2PN-exact waveless approximation is analyzed in detail, which results from imposing more restrictive conditions. The proposed scheme can be considered as a further development on the waveless approach suggested by Schaefer and Gopakumar [Phys. Rev. D {\bf 69}, 021501 (2004)].