Price's law and precise late-time asymptotics for subextremal Reissner-Nordström black holes

Abstract

In this paper, we prove precise late-time asymptotics for solutions to the wave equation supported on angular frequencies greater or equal to on the domain of outer communications of subextremal Reissner-Nordström spacetimes up to and including the event horizon. Our asymptotics yield, in particular, sharp upper and lower decay rates which are consistent with Price's law on such backgrounds. We present a theory for inverting the time operator and derive an explicit representation of the leadingorder asymptotic coefficient in terms of the Newman-Penrose charges at null infinity associated with the time integrals. Our method is based on purely physical space techniques. For each angular frequency we establish a sharp hierarchy of r-weighted radially commuted estimates with length 2 + 5. We complement this hierarchy with a novel hierarchy of weighted elliptic estimates of length + 1.

Price's law for spin fields on a Schwarzschild background

“…For the scalar field (s = 0), it is manifest that if, additionally, the time derivative of the initial data vanishes, then the decay rate is faster by an extra τ −1 globally. Note that this has been verified in [38,11] as well. For s = 1, 2, this is in fact compatible with the expectation in [45,63] that solutions to the Regge-Wheeler equation with smooth, compactly supported in a compact region in (2M, +∞), static (in the sense that its time derivative on the initial hypersurface t = τ 0 vanishes) initial data will develop power tails τ −2ℓ0−4 in a finite radius region.…”

“…Very recently, much important progress were made in proving the Price's law for the spin-s fields on Schwarzschild, Reissner-Nördstrom, and Kerr backgrounds. For the scalar field, Hintz [38] computed the v −1 τ −2 leading order term on both Schwarzschild and subextremal Kerr spacetimes and obtained t −2ℓ0−3 sharp asymptotics for ℓ ≥ ℓ 0 modes in a compact region on Schwarzschild; Angelopoulos-Aretakis-Gajic derived in [10] the asymptotic profiles of the ℓ = 0, ℓ = 1, and ℓ ≥ 2 modes in a subextremal Kerr spacetime and computed in [11] the v −1 τ −2ℓ0−2 asymptotics for ℓ ≥ ℓ 0 modes on a subextremal Reissner-Nordström background. For non-zero spin fields, in an earlier work [50] of the first author of our current work, v −2−s τ − 3 2 +s decay in non-static Kerr and v −2−s τ −3+s+ǫ decay in Schwarzschild towards a stationary/static Coulomb solution are proven, and it also proves the almost Price's law v −2−s τ −2−ℓ0+s+ε for any ℓ ≥ ℓ 0 modes for the Maxwell field in the region ρ ≥ τ on a Schwarzschild background; the authors of this current work obtained in [53] the energy and Morawetz estimates and calculated the asymptotic profiles with decay v − 3 2 −s τ − 5 2 +s for the spin s = ± 1 2 components of the massless Dirac field on Schwarzschild.…”

The non-linear stability of the Schwarzschild family of black holes

“…In fact, in the context of the linear theory, vanishing of the coefficient of each polynomial term in the asymptotic expansion along H + or I + imposes an additional constraint on initial data. See for instance [AAG18b,Hin20,AAG21a,AAG21b]. Thus, in view of the strong assumption on scattering data, the solutions constructed in [DHR13] are expected to be infinite codimension in the moduli space of initial data.…”

Sharp decay for Teukolsky equation in Kerr spacetimes

“…To deduce further energy decay, it is convenient to decompose the field into spin-weighted spherical harmonic modes and employ different techniques to obtain almost sharp decay for the modes. See [10,9,12] for s = 0 and [72] for general s in Schwarzschild spacetimes.…”

“…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.…”