Late-time tails and mode coupling of linear waves on Kerr spacetimes

Abstract

We provide a rigorous derivation of the precise late-time asymptotics for solutions to the scalar wave equation on subextremal Kerr backgrounds, including the asymptotics for projections to angular frequencies ≥ 1 and ≥ 2. The -dependent asymptotics on Kerr spacetimes differ significantly from the non-rotating Schwarzschild setting ("Price's law"). The main differences with Schwarzschild are slower decay rates for higher angular frequencies and oscillations along the null generators of the event horizon. We introduce a physical space-based method that resolves the following two main difficulties for establishing -dependent asymptotics in the Kerr setting: 1) the coupling of angular modes and 2) a loss of ellipticity in the ergoregion. Our mechanism identifies and exploits the existence of conserved charges along null infinity via a time invertibility theory, which in turn relies on new elliptic estimates in the full black hole exterior. This framework is suitable for resolving the conflicting numerology in Kerr late-time asymptotics that appears in the numerics literature.

Sharp decay for Teukolsky equation in Kerr spacetimes

“…Meanwhile, if we introduce a coordinate φH + = φ− a 2Mr+ τ mod 2π such that it is invariant under the null Killing generator K = ∂ τ + a 2Mr+ ∂ φ along H + , then the asymptotics of the spin ±s components on H + exhibit the so-called horizon oscillation [13] in the sense that the asymptotic profiles for each azimuthal m-mode contain an oscillatory factor e iam 2M r + τ . This is predicted in [13] and first rigorously proven for ℓ = 1 mode of the scalar field on Kerr in [11].…”

“…Donninger-Schlag-Soffer [34] then obtained in a compact region outside a Schwarzschild black hole t −2ℓ−2 decay (and t −2ℓ−3 decay for static initial data) for an ℓ mode. The globally sharp v −1 τ −2 pointwise decay is first proven by Angelopoulos-Aretakis-Gajic [10,9] and the precise late-time asymptotic profile is calculated therein; Hintz [47] computed the v −1 τ −2 leading order term on both Schwarzschild and subextreme Kerr spacetimes and further obtained v −1 τ −2ℓ−2 sharp asymptotics for ≥ ℓ modes in a compact region on Schwarzschild; Luk-Oh [65] derived sharp decay for the scalar field on a Reissner-Nordström background and used it to obtain linear instability of the Reissner-Nordström Cauchy horizon (see also their works [66,67] on a generalization to a nonlinear setting); Angelopoulos-Aretakis-Gajic based on their own earlier works and re-derived in [12] v −1 τ −2ℓ−2 late time asymptotics for ≥ ℓ 0 modes in a finite radius region on Schwarzschild, and they further computed in [11] the asymptotic profiles of the ℓ = 0, ℓ = 1, and ℓ ≥ 2 modes in a subextreme Kerr spacetime; we [72] independently computed the global v −1 τ −2ℓ−2 late time asymptotics for ≥ ℓ modes in a Schwarzschild spacetime. Additionally, Kehrberger [54,55,56] considered the precise structure of gravitational radiation near infinity for the scalar field on Schwarzschild.…”

Instability of the Kerr Cauchy horizon under linearised gravitational perturbations

“…If we do not assume the initial data to be compactly supported, but still to be smooth with respect to the conformal compactification at future null infinity, we expect the generic decay rates to be slower by a power of v ´1 `, see also [2]. There is evidence that the assumption of smoothness at future null infinity is not satisfied in many physically interesting situations (see [9], [38]) and that this impacts on the late time tails [39].…”

“…(The wave itself remains bounded [31], [24].) It was later shown by Hintz [32] and Angelopoulos-Aretakis-Gajic [2] that the assumed bounds on the event horizon are generically satisfied.…”

“…See[40],[16],[41] for recent results on the black hole stability problem 2. See also[45] for the linearised case.…”