Nonlinear radiation gauge for near Kerr spacetimes

In this paper, we introduce and explore the properties of a new gauge choice for the vacuum Einstein equation inspired by the ingoing and outgoing radiation gauges (IRG, ORG) for the linearized vacuum Einstein equation introduced by Chrzanowski in his work on metric reconstruction [DOI:https://doi.org/10.1103/PhysRevD.11.2042] on the Kerr background. It has been shown by Price, Shankar and Whiting [arXiv:gr-qc/0611070] that the IRG/ORG are consistent gauges for the linearized vacuum Einstein equation on Petrov type II backgrounds. In [arXiv:1903.03859], the ORG was used in proving linearized stability for the Kerr spacetime, and the new non-linear radiation gauge introduced here is a direct generalization of that gauge condition, and is intended to be used to study the stability of Kerr black holes under the evolution generated by the vacuum Einstein equation.