Price's law for spin fields on a Schwarzschild background

Abstract

In this work, we give a proof of the globally sharp asymptotic profiles for the spin-s fields on a Schwarzschild background, including the scalar field (s = 0), the Maxwell field (s = ±1) and the linearized gravity (s = ±2). This confirms the conjectured Price's law in the physics literature which predicts the sharp estimates of the spin s = ±s components towards the future null infinity as well as in a compact region. The asymptotics are derived via a unified, detailed analysis of the Teukolsky master equation which is satisfied by all these components. Contents 1. Introduction 1.1. Price's law 1.2. Main results 1.3. Outline of the proof 1.4. Other relevant works Overview of the paper 2. Preliminaries 2.1. Coordinates and foliation of the spacetime 2.2. General conventions 2.3. Basic definitions 2.4. Teukolsky master equation 2.5. Spin-weighted spherical harmonic decomposition 2.6. The full Maxwell system of equations 2.7. Basic estimates 3. Almost Price's law 3.1. Energy and Morawetz estimates 3.2. Extended wave systems 3.3. Initial energies 3.4. Weak decay estimates 3.5. Teukolsky-Starobinsky identities 3.6. Newman-Penrose constants 3.7. Almost Price's law for a fixed mode in the exterior region {ρ ≥ τ } 3.8. Equations for a fixed mode 3.9. Almost Price's law for a fixed ℓ mode in the interior region {ρ ≤ τ } 4. Price's law under nonvanishing Newman-Penrose constant condition 4.1. Price's law in a region {v − u ≥ v α } 4.2. Price's law in the region {v − u ≤ v α } 4.3. Global Price's law estimates 5. Price's law under vanishing Newman-Penrose constant condition 5.1. Time integral

Sharp decay for Teukolsky equation in Kerr spacetimes

“…• Our result confirms both the heuristic Price's law [81,82,49,40] in the region r ≥ r + of a Kerr spacetime and the claim of Barack-Ori [13] that the spin +s (s = 1, 2) component enjoys faster decay than the Price's law on H + if am = 0, and generalizes the statements in [72] from Schwarzschild to subextreme Kerr backgrounds. 2 Note that it is shown in [71] that Barack-Ori's claim can not be generalized to s = 1 2 case which corresponds to the massless Dirac field.…”

“…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.…”

“…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.…”

“…The first author of this current work derived in [68] v −2−s τ − 3 2 +s decay in non-static Kerr and v −2−s τ −3+s+ǫ almost sharp decay for all spin s components of the Maxwell field in Schwarzschild towards a stationary/static Coulomb solution, and it also proved the almost sharp v −2−s τ −2−ℓ+s+ε decay for any ≥ ℓ modes for the Maxwell field in the region ρ τ on a Schwarzschild background. If restricted to a Schwarzschild background, we [72] computed v −1−s−s τ −2−s+s late time asymptotic profiles for the spin ±s components globally in the DOC, and, for ≥ ℓ modes of the spin s components, computed v −1−s−s τ −2−ℓ0+s asymptotics in region ρ ≥ τ , r ℓ−s τ −3−2ℓ0 asymptotics in region ρ ≤ τ , and achieved τ −4−2ℓ0 asymptotics for the ≥ ℓ 0 modes for the spin +s (s = 1, 2) components on H + ; hence, we have confirmed in [72] both the Price's law (for s = 1, 2) and Barack-Ori's claim (for s = 1, 2) for the spin s component on a Schwarzschild background. Let us also mention that we [71] generalized the Price's law to the massless Dirac field on Schwarzschild by calculating v − 3 2 −s τ − 5 2 +s asymptotic profiles for its spin s = ± 1 2 components.…”

“…Our proof can be divided into three steps, each of which is discussed in the following three subsubsections respectively. The first two steps are based on a generalization of the approach developed in our earlier work [72] on Schwarzschild to Kerr spacetimes, and the main ingredient of the third step is a novel global conservation law that can be applied to other problems, cf. Section 1.3.…”