Fourth-post-Newtonian exact approximation to general relativity

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 approach by Isenberg, Wilson, and Mathews and the improved second PN-order exact approach conformal-flat condition plus. 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 Schäfer and Gopakumar [26].