Nonlinear radiation gauge for near Kerr spacetimes

Abstract

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 [9] on the Kerr background. It has been shown by Price, Shankar and Whiting [29] that IRG/ORG are consistent gauges for the linearized vacuum Einstein equation on Petrov type II backgrounds. In [1] 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.

Sharp decay for Teukolsky equation in Kerr spacetimes

“…• A scalar which has proper spin weight and zero boost weight in the sense of Geroch, Held and Penrose [37] is called a spin-weighted scalar. 4 Unless otherwise stated, we shall always denote s the spin weight, and we call a spin-weighted scalar with spin weight s as a spin s scalar.…”

“…In the end, we draw attention to the progress on black hole stability problem in recent years. Linear stability of a Schwarzschild or a subextremal Reissner-Nordström spacetime has been shown by [28,52,7,53,50,51,38], and linear stability of a slowly rotating Kerr spacetime is proven in [3,44,4]. For nonlinear stability results, we refer to [61,29] for Schwarzschild, [48] for slowly rotating Kerr-de Sitter, and [60,39,62] for slowly rotating Kerr.…”

Wave equations estimates and the nonlinear stability of slowly rotating Kerr black holes

“…We further restrict the discussion to asymptotically flat initial data sets, i.e. we assume that outside a sufficiently large compact set K, Σ (0) \ K is diffeomorphic to the complement of the unit ball in R 3 and admits a system of coordinates in which g (0) is asymptotically euclidean and k (0) vanishes at appropriate order.…”

“…Proposition 2.1.27. For a tensor Ψ ∈ O 1 (M), we have the curvature formula 3 ( Ḋµ Ḋν − Ḋν Ḋµ )Ψ a = Ṙabµν Ψ b (2.1.12)…”

“…This loss is due to the interpolation between the bootstrap estimates for k large and those for k small , see for example Lemma 5.1 in [50]. The loss is more than compensated by the fact that the resulting gain 3 4 δ dec is doubled in nonlinear estimates. The remark also applies to Chapter 12.…”