Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Price's law for spin fields on a Schwarzschild background

Siyuan Ma, Lin Zhang

TL;DR

The paper delivers a rigorous, globally sharp Price’s law for massless spin-s fields on Schwarzschild, covering scalar, Maxwell, and linearized gravity cases (s=0,1,2). It unifies the analysis through the Teukolsky master equation, leveraging extended wave systems, $r^p$-weighted estimates, and Newman–Penrose constants to obtain both exterior and interior decay with horizon refinements for +1 and +2. Central technical tools include Teukolsky–Starobinsky identities and a detailed mode-by-mode (m,ℓ) decomposition, enabling explicit leading-order asymptotics and global decay. The results illuminate the role of NP constants and time integrals in determining tail behavior, and lay groundwork for extensions to charged or rotating backgrounds and related stability problems. The methods open avenues for nonlinear stability analyses and for approaching Strong Cosmic Censorship questions in charged or rotating spacetimes.

Abstract

In this work, we derive the globally precise late-time asymptotics for the spin-$\mathfrak{s}$ fields on a Schwarzschild background, including the scalar field $(\mathfrak{s}=0)$, the Maxwell field $(\mathfrak{s}=\pm 1)$ and the linearized gravity $(\mathfrak{s}=\pm 2)$. The conjectured Price's law in the physics literature which predicts the sharp rates of decay of the spin $s=\pm \mathfrak{s}$ components towards the future null infinity as well as in a compact region is shown. Further, we confirm the heuristic claim by Barack and Ori that the spin $+1, +2$ components have an extra power of decay at the event horizon than the conjectured Price's law. The asymptotics are derived via a unified, detailed analysis of the Teukolsky master equation that is satisfied by all these components.

Price's law for spin fields on a Schwarzschild background

TL;DR

The paper delivers a rigorous, globally sharp Price’s law for massless spin-s fields on Schwarzschild, covering scalar, Maxwell, and linearized gravity cases (s=0,1,2). It unifies the analysis through the Teukolsky master equation, leveraging extended wave systems, -weighted estimates, and Newman–Penrose constants to obtain both exterior and interior decay with horizon refinements for +1 and +2. Central technical tools include Teukolsky–Starobinsky identities and a detailed mode-by-mode (m,ℓ) decomposition, enabling explicit leading-order asymptotics and global decay. The results illuminate the role of NP constants and time integrals in determining tail behavior, and lay groundwork for extensions to charged or rotating backgrounds and related stability problems. The methods open avenues for nonlinear stability analyses and for approaching Strong Cosmic Censorship questions in charged or rotating spacetimes.

Abstract

In this work, we derive the globally precise late-time asymptotics for the spin- fields on a Schwarzschild background, including the scalar field , the Maxwell field and the linearized gravity . The conjectured Price's law in the physics literature which predicts the sharp rates of decay of the spin components towards the future null infinity as well as in a compact region is shown. Further, we confirm the heuristic claim by Barack and Ori that the spin components have an extra power of decay at the event horizon than the conjectured Price's law. The asymptotics are derived via a unified, detailed analysis of the Teukolsky master equation that is satisfied by all these components.

Paper Structure

This paper contains 40 sections, 46 theorems, 290 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Heuristic Price's law
  3. Main results
  4. Outline of the proof
  5. Overview of the approach
  6. Remarks on other most relevant works
  7. Future applications
  8. Other relevant works
  9. Preliminaries
  10. Coordinates and foliation of the spacetime
  11. General conventions
  12. Operators and norms
  13. Spin-weighted spherical harmonic decomposition
  14. Basic analytic estimates
  15. An $r^p$ lemma for wave equations
  16. ...and 25 more sections

Key Result

Theorem 1.1

Let $j\in \mathbb{N}$, let $\mathfrak{s}=0,1,2$, let $\ell_0\geq \mathfrak{s}$, and let $\tau_0\geq 1$. Let $Y_{m,\ell}^{s}(\cos\theta)e^{im{\phi}}$ be the spin-weighted spherical harmonics defined as in Section sect:decompIntoModes. Let function $h_{\mathfrak{s},\ell_0}$ be defined as in Definition

Figures (4)

  • Figure 1: Hyperboloidal foliation and related definitions.
  • Figure 2: Initial hypersurface $\Sigma_{\tau_0}$ on which initial data will be imposed.
  • Figure 3: Useful hypersurfaces
  • Figure 4: For any point $(u,v)$ in $\{r\geq R\}\cap\{v-u\geq v^\alpha\}$, i.e. $v\geq v_{\gamma_\alpha}(u)$, one integrates along $u=const$ from $(u, v_{\gamma_\alpha}(u))\in \gamma_\alpha$.

Theorems & Definitions (121)

  • Theorem 1.1: Rough version of the globally precise late-time asymptotics of the spin $s$ components
  • Remark 1.2
  • Remark 1.3: Asymptotics for the middle component of the Maxwell field
  • Remark 1.4: Discussions on other spin fields
  • Remark 1.5: Generalization to wave equations with a potential
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • ...and 111 more