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Sharp decay estimates for massless Dirac fields on a Schwarzschild background

Siyuan Ma, Lin Zhang

TL;DR

The paper establishes sharp, Price’s-law-type decay for massless Dirac fields on a Schwarzschild background by proving uniform energy boundedness and integrated local energy decay through a coupled symmetric hyperbolic wave system. It then derives almost sharp decay estimates via an r^p hierarchy and decomposes the field into angular modes, with NP constants determining leading-order asymptotics. A dichotomy arises between nonvanishing and vanishing first Newman–Penrose constants: nonzero constants fix the leading decay and asymptotics, while vanishing constants yield improved decay rates and require a refined time-integral construction to capture sharper behavior. The results provide a comprehensive, mode-by-mode account of Dirac field decay outside Schwarzschild, including precise asymptotics and conditions under which Price’s law holds, with implications for black hole stability and related conjectures.

Abstract

We consider the explicit asymptotic profile of massless Dirac fields on a Schwarzschild background. First, we prove for the spin $s=\pm \frac{1}{2}$ components of the Dirac field a uniform bound of a positive definite energy and an integrated local energy decay estimate from a symmetric hyperbolic wave system. Based on these estimates, we further show that these components have globally pointwise decay $fv^{-3/2-s}τ^{-5/2+s}$ as both an upper and a lower bound outside the black hole, with function $f$ finite and explicitly expressed in terms of the initial data and the coordinates. This establishes the validity of the conjectured Price's law for massless Dirac fields outside a Schwarzschild black hole.

Sharp decay estimates for massless Dirac fields on a Schwarzschild background

TL;DR

The paper establishes sharp, Price’s-law-type decay for massless Dirac fields on a Schwarzschild background by proving uniform energy boundedness and integrated local energy decay through a coupled symmetric hyperbolic wave system. It then derives almost sharp decay estimates via an r^p hierarchy and decomposes the field into angular modes, with NP constants determining leading-order asymptotics. A dichotomy arises between nonvanishing and vanishing first Newman–Penrose constants: nonzero constants fix the leading decay and asymptotics, while vanishing constants yield improved decay rates and require a refined time-integral construction to capture sharper behavior. The results provide a comprehensive, mode-by-mode account of Dirac field decay outside Schwarzschild, including precise asymptotics and conditions under which Price’s law holds, with implications for black hole stability and related conjectures.

Abstract

We consider the explicit asymptotic profile of massless Dirac fields on a Schwarzschild background. First, we prove for the spin components of the Dirac field a uniform bound of a positive definite energy and an integrated local energy decay estimate from a symmetric hyperbolic wave system. Based on these estimates, we further show that these components have globally pointwise decay as both an upper and a lower bound outside the black hole, with function finite and explicitly expressed in terms of the initial data and the coordinates. This establishes the validity of the conjectured Price's law for massless Dirac fields outside a Schwarzschild black hole.

Paper Structure

This paper contains 46 sections, 47 theorems, 430 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Outline of the proof
  3. Basic energy and Morawetz estimates
  4. Almost sharp energy decay estimates
  5. Almost sharp pointwise decay estimates
  6. Asymptotics in the case of nonvanishing first Newman--Penrose constant
  7. Asymptotics in the case of vanishing first Newman--Penrose constant
  8. Related works
  9. Preliminaries
  10. General conventions
  11. Further definitions
  12. General facts
  13. Decomposition into modes for spin-weight $s$ scalars
  14. Hardy and Sobolev estimates
  15. $r^p$ estimate for a general spin-weighted wave equation on Schwarzschild
  16. ...and 31 more sections

Key Result

Theorem 1.1

(Price's law in nonvanishing first Newman--Penrose constant case) Let $j\in \mathbb{N}$. Assume Then there exists an $\epsilon>0$ such that in $\Omega_{\tau_0,\infty}$, where

Figures (6)

  • Figure 1: Hyperboloidal foliation and some related definitions.
  • Figure 2: Initial hypersurface $\Sigma_{\tau_0}$.
  • Figure 3: For $v$ large enough, there are some useful curves in spacetime, where $\gamma_\alpha=\{v-u=v^\alpha\}$ for an $\alpha\in (0,1)$.
  • Figure 4: For any point $(u,v)$ in $\{r\geq R\}\cap\{v-u\geq v^\alpha\}$, i.e. $u_{\Sigma_{\tau_0}}(v)\leq u\leq u_{\gamma_\alpha}(v)$, integrate along $v=const$ from $(u_{\Sigma_{\tau_0}}(v),v)\in \Sigma_{\tau_0}$.
  • Figure 5: For any point $(u,v)$ in $\{r\geq R\}\cap\{v-u\geq v^\alpha\}$, i.e. $v\geq v_{\gamma_\alpha}(u)$, integrate along $u=const$ from $(u, v_{\gamma_\alpha}(u))\in \gamma_\alpha$.
  • ...and 1 more figures

Theorems & Definitions (120)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 110 more